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Maxima is a computer algebra system, implemented in Lisp.
Maxima is derived from the Macsyma system, developed at MIT in the years 1968 through 1982 as part of Project MAC. MIT turned over a copy of the Macsyma source code to the Department of Energy in 1982; that version is now known as DOE Macsyma. A copy of DOE Macsyma was maintained by Professor William F. Schelter of the University of Texas from 1982 until his death in 2001. In 1998, Schelter obtained permission from the Department of Energy to release the DOE Macsyma source code under the GNU Public License, and in 2000 he initiated the Maxima project at SourceForge to maintain and develop DOE Macsyma, now called Maxima.
Maxima infrastructure
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Start Maxima with the command "maxima". Maxima will display version information and a prompt. End each Maxima command with a semicolon. End the session with the command "quit();". Here's a sample session:
[wfs@chromium]$ maxima
Maxima 5.9.1 http://maxima.sourceforge.net
Using Lisp CMU Common Lisp 19a
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
This is a development version of Maxima. The function bug_report()
provides bug reporting information.
(%i1) factor(10!);
8 4 2
(%o1) 2 3 5 7
(%i2) expand ((x + y)^6);
6 5 2 4 3 3 4 2 5 6
(%o2) y + 6 x y + 15 x y + 20 x y + 15 x y + 6 x y + x
(%i3) factor (x^6 - 1);
2 2
(%o3) (x - 1) (x + 1) (x - x + 1) (x + x + 1)
(%i4) quit();
[wfs@chromium]$
Maxima can search the info pages. Use the describe command to show
information about the command or all the commands and variables containing
a string.
The question mark ? (exact search) and double question mark ??
(inexact search) are abbreviations for describe:
(%i1) ?? integ
0: Functions and Variables for Elliptic Integrals
1: Functions and Variables for Integration
2: Introduction to Elliptic Functions and Integrals
3: Introduction to Integration
4: askinteger (Functions and Variables for Simplification)
5: integerp (Functions and Variables for Miscellaneous Options)
6: integer_partitions (Functions and Variables for Sets)
7: integrate (Functions and Variables for Integration)
8: integrate_use_rootsof (Functions and Variables for Integration)
9: integration_constant_counter (Functions and Variables for
Integration)
10: nonnegintegerp (Functions and Variables for linearalgebra)
Enter space-separated numbers, `all' or `none': 5 4
-- Function: integerp (<expr>)
Returns `true' if <expr> is a literal numeric integer, otherwise
`false'.
`integerp' returns false if its argument is a symbol, even if the
argument is declared integer.
Examples:
(%i1) integerp (0);
(%o1) true
(%i2) integerp (1);
(%o2) true
(%i3) integerp (-17);
(%o3) true
(%i4) integerp (0.0);
(%o4) false
(%i5) integerp (1.0);
(%o5) false
(%i6) integerp (%pi);
(%o6) false
(%i7) integerp (n);
(%o7) false
(%i8) declare (n, integer);
(%o8) done
(%i9) integerp (n);
(%o9) false
-- Function: askinteger (<expr>, integer)
-- Function: askinteger (<expr>)
-- Function: askinteger (<expr>, even)
-- Function: askinteger (<expr>, odd)
`askinteger (<expr>, integer)' attempts to determine from the
`assume' database whether <expr> is an integer. `askinteger'
prompts the user if it cannot tell otherwise, and attempt to
install the information in the database if possible. `askinteger
(<expr>)' is equivalent to `askinteger (<expr>, integer)'.
`askinteger (<expr>, even)' and `askinteger (<expr>, odd)'
likewise attempt to determine if <expr> is an even integer or odd
integer, respectively.
(%o1) true
To use a result in later calculations, you can assign it to a variable or
refer to it by its automatically supplied label. In addition, %
refers to the most recent calculated result:
(%i1) u: expand ((x + y)^6);
6 5 2 4 3 3 4 2 5 6
(%o1) y + 6 x y + 15 x y + 20 x y + 15 x y + 6 x y + x
(%i2) diff (u, x);
5 4 2 3 3 2 4 5
(%o2) 6 y + 30 x y + 60 x y + 60 x y + 30 x y + 6 x
(%i3) factor (%o2);
5
(%o3) 6 (y + x)
Maxima knows about complex numbers and numerical constants:
(%i1) cos(%pi); (%o1) - 1 (%i2) exp(%i*%pi); (%o2) - 1
Maxima can do differential and integral calculus:
(%i1) u: expand ((x + y)^6);
6 5 2 4 3 3 4 2 5 6
(%o1) y + 6 x y + 15 x y + 20 x y + 15 x y + 6 x y + x
(%i2) diff (%, x);
5 4 2 3 3 2 4 5
(%o2) 6 y + 30 x y + 60 x y + 60 x y + 30 x y + 6 x
(%i3) integrate (1/(1 + x^3), x);
2 x - 1
2 atan(-------)
log(x - x + 1) sqrt(3) log(x + 1)
(%o3) - --------------- + ------------- + ----------
6 sqrt(3) 3
Maxima can solve linear systems and cubic equations:
(%i1) linsolve ([3*x + 4*y = 7, 2*x + a*y = 13], [x, y]);
7 a - 52 25
(%o1) [x = --------, y = -------]
3 a - 8 3 a - 8
(%i2) solve (x^3 - 3*x^2 + 5*x = 15, x);
(%o2) [x = - sqrt(5) %i, x = sqrt(5) %i, x = 3]
Maxima can solve nonlinear sets of equations. Note that if you don't want a result printed, you can finish your command with $ instead of ;.
(%i1) eq_1: x^2 + 3*x*y + y^2 = 0$
(%i2) eq_2: 3*x + y = 1$
(%i3) solve ([eq_1, eq_2]);
3 sqrt(5) + 7 sqrt(5) + 3
(%o3) [[y = - -------------, x = -----------],
2 2
3 sqrt(5) - 7 sqrt(5) - 3
[y = -------------, x = - -----------]]
2 2
Maxima can generate plots of one or more functions:
(%i1) plot2d (sin(x)/x, [x, -20, 20])$
(%i2) plot2d ([atan(x), erf(x), tanh(x)], [x, -5, 5], [y, -1.5, 2])$
(%i3) plot3d (sin(sqrt(x^2 + y^2))/sqrt(x^2 + y^2),
[x, -12, 12], [y, -12, 12])$
Categories: Help
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| 2.1 Functions and Variables for Bug Detection and Reporting |
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Run the Maxima test suite. Tests producing the desired answer are considered "passes," as are tests that do not produce the desired answer, but are marked as known bugs.
run_testsuite takes the following optional keyword arguments
display_allDisplay all tests. Normally, the tests are not displayed, unless the test
fails. (Defaults to false).
display_known_bugsDisplays tests that are marked as known bugs. (Default is false).
testsThis is a single test or a list of tests that should be run. Each test can be specified by
either a string or a symbol. By default, all tests are run. The complete set
of tests is specified by testsuite_files.
timeDisplay time information. If true, the time taken for each
test file is displayed. If all, the time for each individual
test is shown if display_all is true. The default is
false, so no timing information is shown.
share_testsLoad additional tests for the share directory. If true,
these additional tests are run as a part of the testsuite. If
false, no tests from the share directory are run. If
only, only the tests from the share directory are run.
Of course, the actual set of test that are run can be controlled by
the tests option. The default is false.
For example run_testsuite(display_known_bugs = true, tests=[rtest5])
runs just test rtest5 and displays the test that are marked as
known bugs.
run_testsuite(display_all = true, tests=["rtest1", rtest1a]) will
run tests rtest1 and rtest2, and displays each test.
run_testsuite changes the Maxima environment.
Typically a test script executes kill to establish a known environment
(namely one without user-defined functions and variables)
and then defines functions and variables appropriate to the test.
run_testsuite returns done.
Categories: Debugging
testsuite_files is the set of tests to be run by
run_testsuite. It is a list of names of the files containing
the tests to run. If some of the tests in a file are known to fail,
then instead of listing the name of the file, a list containing the
file name and the test numbers that fail is used.
For example, this is a part of the default set of tests:
["rtest13s", ["rtest14", 57, 63]]
This specifies the testsuite consists of the files "rtest13s" and "rtest14", but "rtest14" contains two tests that are known to fail: 57 and 63.
Categories: Debugging · Global variables
share_testsuite_files is the set of tests from the share
directory that is run as a part of the test suite by
run_testsuite..
Categories: Debugging · Global variables
Prints out Maxima and Lisp version numbers, and gives a link
to the Maxima project bug report web page.
The version information is the same as reported by build_info.
When a bug is reported, it is helpful to copy the Maxima and Lisp version information into the bug report.
bug_report returns an empty string "".
Categories: Debugging
Returns a summary of the parameters of the Maxima build,
as a Maxima structure (defined by defstruct).
The fields of the structure are:
version, timestamp, host, lisp_name, and lisp_version.
When the pretty-printer is enabled (via display2d),
the structure is displayed as a short table.
See also bug_report.
Examples:
(%i1) build_info (); (%o1) Maxima version: "5.36.1" Maxima build date: "2015-06-02 11:26:48" Host type: "x86_64-unknown-linux-gnu" Lisp implementation type: "GNU Common Lisp (GCL)" Lisp implementation version: "GCL 2.6.12" (%i2) x : build_info ()$ (%i3) x@version; (%o3) 5.36.1 (%i4) x@timestamp; (%o4) 2015-06-02 11:26:48 (%i5) x@host; (%o5) x86_64-unknown-linux-gnu (%i6) x@lisp_name; (%o6) GNU Common Lisp (GCL) (%i7) x@lisp_version; (%o7) GCL 2.6.12 (%i8) x; (%o8) Maxima version: "5.36.1" Maxima build date: "2015-06-02 11:26:48" Host type: "x86_64-unknown-linux-gnu" Lisp implementation type: "GNU Common Lisp (GCL)" Lisp implementation version: "GCL 2.6.12"
The Maxima version string can (here 5.36.1) can look very different:
(%i1) build_info(); (%o1) Maxima version: "branch_5_37_base_331_g8322940_dirty" Maxima build date: "2016-01-01 15:37:35" Host type: "x86_64-unknown-linux-gnu" Lisp implementation type: "CLISP" Lisp implementation version: "2.49 (2010-07-07) (built 3605577779) (memory 3660647857)"
In that case, Maxima was not build from a released sourcecode, but directly from the GIT-checkout of the sourcecode. In the example, the checkout is 331 commits after the latest GIT tag (usually a Maxima (major) release (5.37 in our example)) and the abbreviated commit hash of the last commit was "8322940".
Categories: Debugging
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| 3.1 Documentation | ||
| 3.2 Functions and Variables for Help |
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The Maxima on-line user's manual can be viewed in different forms. From the
Maxima interactive prompt, the user's manual is viewed as plain text by the
? command (i.e., the describe function). The user's manual is
viewed as info hypertext by the info viewer program and as a
web page by any ordinary web browser.
example displays examples for many Maxima functions. For example,
(%i1) example (integrate);
yields
(%i2) test(f):=block([u],u:integrate(f,x),ratsimp(f-diff(u,x)))
(%o2) test(f) := block([u], u : integrate(f, x),
ratsimp(f - diff(u, x)))
(%i3) test(sin(x))
(%o3) 0
(%i4) test(1/(x+1))
(%o4) 0
(%i5) test(1/(x^2+1))
(%o5) 0
and additional output.
Categories: Console interaction
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Searches for Maxima names which have string appearing anywhere within
them. Thus, apropos (exp) returns a list of all the flags and functions
which have exp as part of their names, such as expand, exp,
and exponentialize. Thus if you can only remember part of the name of
something you can use this command to find the rest of the name. Similarly, you
could say apropos (tr_) to find a list of many of the switches relating
to the translator, most of which begin with tr_.
apropos("") returns a list with all Maxima names.
apropos returns the empty list [], if no name is found.
Example:
Show all Maxima symbols which have "gamma" in the name:
(%i1) apropos("gamma");
(%o1) [%gamma, gamma, gammalim, gamma_expand, gamma_greek,
gamma_incomplete, gamma_incomplete_generalized,
gamma_incomplete_regularized, Gamma, log_gamma, makegamma,
prefer_gamma_incomplete, gamma-incomplete,
gamma_incomplete_generalized_regularized]
Categories: Help
Evaluates Maxima expressions in filename and displays the results.
demo pauses after evaluating each expression and continues after the
user enters a carriage return. (If running in Xmaxima, demo may need
to see a semicolon ; followed by a carriage return.)
demo searches the list of directories file_search_demo to find
filename. If the file has the suffix dem, the suffix may be
omitted. See also file_search.
demo evaluates its argument.
demo returns the name of the demonstration file.
Example:
(%i1) demo ("disol");
batching /home/wfs/maxima/share/simplification/disol.dem
At the _ prompt, type ';' followed by enter to get next demo
(%i2) load(disol)
_
(%i3) exp1 : a (e (g + f) + b (d + c))
(%o3) a (e (g + f) + b (d + c))
_
(%i4) disolate(exp1, a, b, e)
(%t4) d + c
(%t5) g + f
(%o5) a (%t5 e + %t4 b)
_
Categories: Help · Console interaction · File input
describe(string) is equivalent to
describe(string, exact).
describe(string, exact) finds an item with title equal
(case-insensitive) to string, if there is any such item.
describe(string, inexact) finds all documented items which contain
string in their titles. If there is more than one such item, Maxima asks
the user to select an item or items to display.
At the interactive prompt, ? foo (with a space between ? and
foo) is equivalent to describe("foo", exact), and ?? foo
is equivalent to describe("foo", inexact).
describe("", inexact) yields a list of all topics documented in the
on-line manual.
describe quotes its argument. describe returns true if
some documentation is found, otherwise false.
See also Documentation.
Example:
(%i1) ?? integ
0: Functions and Variables for Elliptic Integrals
1: Functions and Variables for Integration
2: Introduction to Elliptic Functions and Integrals
3: Introduction to Integration
4: askinteger (Functions and Variables for Simplification)
5: integerp (Functions and Variables for Miscellaneous Options)
6: integer_partitions (Functions and Variables for Sets)
7: integrate (Functions and Variables for Integration)
8: integrate_use_rootsof (Functions and Variables for
Integration)
9: integration_constant_counter (Functions and Variables for
Integration)
10: nonnegintegerp (Functions and Variables for linearalgebra)
Enter space-separated numbers, `all' or `none': 7 8
-- Function: integrate (<expr>, <x>)
-- Function: integrate (<expr>, <x>, <a>, <b>)
Attempts to symbolically compute the integral of <expr> with
respect to <x>. `integrate (<expr>, <x>)' is an indefinite
integral, while `integrate (<expr>, <x>, <a>, <b>)' is a
definite integral, [...]
-- Option variable: integrate_use_rootsof
Default value: `false'
When `integrate_use_rootsof' is `true' and the denominator of
a rational function cannot be factored, `integrate' returns
the integral in a form which is a sum over the roots (not yet
known) of the denominator.
[...]
In this example, items 7 and 8 were selected (output is shortened as indicated
by [...]). All or none of the items could have been selected by entering
all or none, which can be abbreviated a or n,
respectively.
Categories: Help · Console interaction
example (topic) displays some examples of topic, which is a
symbol or a string. To get examples for operators like if, do,
or lambda the argument must be a string, e.g. example ("do").
example is not case sensitive. Most topics are function names.
example () returns the list of all recognized topics.
The name of the file containing the examples is given by the global option
variable manual_demo, which defaults to "manual.demo".
example quotes its argument. example returns done unless
no examples are found or there is no argument, in which case example
returns the list of all recognized topics.
Examples:
(%i1) example(append);
(%i2) append([y+x,0,-3.2],[2.5e+20,x])
(%o2) [y + x, 0, - 3.2, 2.5e+20, x]
(%o2) done
(%i3) example("lambda");
(%i4) lambda([x,y,z],x^2+y^2+z^2)
2 2 2
(%o4) lambda([x, y, z], x + y + z )
(%i5) %(1,2,a)
2
(%o5) a + 5
(%i6) 1+2+a
(%o6) a + 3
(%o6) done
Categories: Help · Console interaction
Default value: "manual.demo"
manual_demo specifies the name of the file containing the examples for
the function example. See example.
Categories: Help · Global variables
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| 4.1 Introduction to Command Line | ||
| 4.2 Functions and Variables for Command Line | ||
| 4.3 Functions and Variables for Display |
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__ is the input expression currently being evaluated. That is, while an
input expression expr is being evaluated, __ is expr.
__ is assigned the input expression before the input is simplified or
evaluated. However, the value of __ is simplified (but not evaluated)
when it is displayed.
__ is recognized by batch and load. In a file processed
by batch, __ has the same meaning as at the interactive prompt.
In a file processed by load, __ is bound to the input expression
most recently entered at the interactive prompt or in a batch file; __
is not bound to the input expressions in the file being processed. In
particular, when load (filename) is called from the interactive
prompt, __ is bound to load (filename) while the file is
being processed.
Examples:
(%i1) print ("I was called as", __);
I was called as print(I was called as, __)
(%o1) print(I was called as, __)
(%i2) foo (__);
(%o2) foo(foo(__))
(%i3) g (x) := (print ("Current input expression =", __), 0);
(%o3) g(x) := (print("Current input expression =", __), 0)
(%i4) [aa : 1, bb : 2, cc : 3];
(%o4) [1, 2, 3]
(%i5) (aa + bb + cc)/(dd + ee + g(x));
cc + bb + aa
Current input expression = --------------
g(x) + ee + dd
6
(%o5) -------
ee + dd
Categories: Global variables
_ is the most recent input expression (e.g., %i1, %i2,
%i3, …).
_ is assigned the input expression before the input is simplified or
evaluated. However, the value of _ is simplified (but not evaluated)
when it is displayed.
_ is recognized by batch and load. In a file processed
by batch, _ has the same meaning as at the interactive prompt.
In a file processed by load, _ is bound to the input expression
most recently evaluated at the interactive prompt or in a batch file; _
is not bound to the input expressions in the file being processed.
Examples:
(%i1) 13 + 29; (%o1) 42 (%i2) :lisp $_ ((MPLUS) 13 29) (%i2) _; (%o2) 42 (%i3) sin (%pi/2); (%o3) 1 (%i4) :lisp $_ ((%SIN) ((MQUOTIENT) $%PI 2)) (%i4) _; (%o4) 1 (%i5) a: 13$ (%i6) b: 29$ (%i7) a + b; (%o7) 42 (%i8) :lisp $_ ((MPLUS) $A $B) (%i8) _; (%o8) b + a (%i9) a + b; (%o9) 42 (%i10) ev (_); (%o10) 42
Categories: Console interaction · Global variables
% is the output expression (e.g., %o1, %o2, %o3,
…) most recently computed by Maxima, whether or not it was displayed.
% is recognized by batch and load. In a file processed
by batch, % has the same meaning as at the interactive prompt.
In a file processed by load, % is bound to the output expression
most recently computed at the interactive prompt or in a batch file; %
is not bound to output expressions in the file being processed.
Categories: Console interaction · Global variables
In compound statements, namely block, lambda, or
(s_1, ..., s_n), %% is the value of the previous
statement.
At the first statement in a compound statement, or outside of a compound
statement, %% is undefined.
%% is recognized by batch and load, and it has the
same meaning as at the interactive prompt.
See also %.
Examples:
The following two examples yield the same result.
(%i1) block (integrate (x^5, x), ev (%%, x=2) - ev (%%, x=1));
21
(%o1) --
2
(%i2) block ([prev], prev: integrate (x^5, x),
ev (prev, x=2) - ev (prev, x=1));
21
(%o2) --
2
A compound statement may comprise other compound statements. Whether a
statement be simple or compound, %% is the value of the previous
statement.
(%i3) block (block (a^n, %%*42), %%/6);
n
(%o3) 7 a
Within a compound statement, the value of %% may be inspected at a break
prompt, which is opened by executing the break function. For example,
entering %%; in the following example yields 42.
(%i4) block (a: 42, break ())$ Entering a Maxima break point. Type 'exit;' to resume. _%%; 42 _
Categories: Global variables
The value of the i'th previous output expression. That is, if the next
expression to be computed is the n'th output, %th (m) is the
(n - m)'th output.
%th is recognized by batch and load. In a file processed
by batch, %th has the same meaning as at the interactive prompt.
In a file processed by load, %th refers to output expressions most
recently computed at the interactive prompt or in a batch file; %th does
not refer to output expressions in the file being processed.
Example:
%th is useful in batch files or for referring to a group of
output expressions. This example sets s to the sum of the last five
output expressions.
(%i1) 1;2;3;4;5; (%o1) 1 (%o2) 2 (%o3) 3 (%o4) 4 (%o5) 5 (%i6) block (s: 0, for i:1 thru 5 do s: s + %th(i), s); (%o6) 15
Categories: Console interaction
As prefix to a function or variable name, ? signifies that the name is a
Lisp name, not a Maxima name. For example, ?round signifies the Lisp
function ROUND. See Lisp and Maxima for more on this point.
The notation ? word (a question mark followed a word, separated by
whitespace) is equivalent to describe("word"). The question mark must
occur at the beginning of an input line; otherwise it is not recognized as a
request for documentation. See also describe.
Categories: Help · Console interaction
The notation ?? word (?? followed a word, separated by whitespace)
is equivalent to describe("word", inexact). The question mark must occur
at the beginning of an input line; otherwise it is not recognized as a request
for documentation. See also describe.
Categories: Help · Console interaction
The dollar sign $ terminates an input expression,
and the most recent output % and an output label, e.g. %o1,
are assigned the result, but the result is not displayed.
See also ;.
Example:
(%i1) 1 + 2 + 3 $ (%i2) %; (%o2) 6 (%i3) %o1; (%o3) 6
The semicolon ; terminates an input expression,
and the resulting output is displayed.
See also $.
Example:
(%i1) 1 + 2 + 3; (%o1) 6
Default value: %i
inchar is the prefix of the labels of expressions entered by the user.
Maxima automatically constructs a label for each input expression by
concatenating inchar and linenum.
inchar may be assigned any string or symbol, not necessarily a single
character. Because Maxima internally takes into account only the first char of
the prefix, the prefixes inchar, outchar, and
linechar should have a different first char. Otherwise some commands
like kill(inlabels) do not work as expected.
See also labels.
Example:
(%i1) inchar: "input";
(%o1) input
(input2) expand((a+b)^3);
3 2 2 3
(%o2) b + 3 a b + 3 a b + a
Categories: Display flags and variables
Default value: []
infolists is a list of the names of all of the information
lists in Maxima. These are:
labelsAll bound %i, %o, and %t labels.
valuesAll bound atoms which are user variables, not Maxima options or switches,
created by : or :: or functional binding.
functionsarraysAll declared and undeclared arrays, created by :, ::, or
:=.
macrosAll user-defined macro functions, created by ::=.
myoptionsAll options ever reset by the user (whether or not they are later reset to their default values).
rulesAll user-defined pattern matching and simplification rules, created
by tellsimp, tellsimpafter, defmatch, or
defrule.
aliasesAll atoms which have a user-defined alias, created by the alias,
ordergreat, orderless functions or by declaring the atom as a
noun with declare.
dependenciesAll atoms which have functional dependencies, created by the
depends, dependencies, or gradef functions.
gradefsAll functions which have user-defined derivatives, created by the
gradef function.
propsAll atoms which have any property other than those mentioned above, such as
properties established by atvalue or matchdeclare, etc.,
as well as properties established in the declare function.
let_rule_packagesAll user-defined let rule packages
plus the special package default_let_rule_package.
(default_let_rule_package is the name of the rule package used when
one is not explicitly set by the user.)
Categories: Declarations and inferences · Global variables
Removes all bindings (value, function, array, or rule) from the arguments
a_1, …, a_n. An argument a_k may be a symbol or a
single array element. When a_k is a single array element, kill
unbinds that element without affecting any other elements of the array.
Several special arguments are recognized. Different kinds of arguments
may be combined, e.g., kill (inlabels, functions, allbut (foo, bar)).
kill (labels) unbinds all input, output, and intermediate expression
labels created so far. kill (inlabels) unbinds only input labels which
begin with the current value of inchar. Likewise,
kill (outlabels) unbinds only output labels which begin with the current
value of outchar, and kill (linelabels) unbinds only
intermediate expression labels which begin with the current value of
linechar.
kill (n), where n is an integer,
unbinds the n most recent input and output labels.
kill ([m, n]) unbinds input and output labels m through
n.
kill (infolist), where infolist is any item in
infolists (such as values, functions, or
arrays) unbinds all items in infolist.
See also infolists.
kill (all) unbinds all items on all infolists. kill (all) does
not reset global variables to their default values; see reset on this
point.
kill (allbut (a_1, ..., a_n)) unbinds all items on all
infolists except for a_1, …, a_n.
kill (allbut (infolist)) unbinds all items except for the ones on
infolist, where infolist is values,
functions, arrays, etc.
The memory taken up by a bound property is not released until all symbols are unbound from it. In particular, to release the memory taken up by the value of a symbol, one unbinds the output label which shows the bound value, as well as unbinding the symbol itself.
kill quotes its arguments. The quote-quote operator ''
defeats quotation.
kill (symbol) unbinds all properties of symbol. In contrast,
the functions remvalue, remfunction,
remarray, and remrule unbind a specific property.
kill always returns done, even if an argument has no binding.
Categories: Evaluation · Console interaction · Session management
with symbol. Typically symbol is the value of
inchar, outchar, or linechar.
If no labels begin with symbol, labels returns an empty list.
By default, Maxima displays the result of each user input expression, giving the
result an output label. The output display is suppressed by terminating the
input with $ (dollar sign) instead of ; (semicolon). An output
label is constructed and bound to the result, but not displayed, and the label
may be referenced in the same way as displayed output labels. See also
%, %%, and %th.
Intermediate expression labels can be generated by some functions. The option
variable programmode controls whether solve and some other
functions generate intermediate expression labels instead of returning a list of
expressions. Some other functions, such as ldisplay, always generate
intermediate expression labels.
See also inchar, outchar, linechar, and
infolists.
Categories: Display functions · Console interaction
The variable labels is the list of input, output, and intermediate
expression labels, including all previous labels if inchar,
outchar, or linechar were redefined.
Categories: Display flags and variables · Console interaction
Default value: %t
linechar is the prefix of the labels of intermediate expressions
generated by Maxima. Maxima constructs a label for each intermediate expression
(if displayed) by concatenating linechar and linenum.
linechar may be assigned any string or symbol, not necessarily a single
character. Because Maxima internally takes into account only the first char of
the prefix, the prefixes inchar, outchar, and
linechar should have a different first char. Otherwise some commands
like kill(inlabels) do not work as expected.
Intermediate expressions might or might not be displayed.
See programmode and labels.
Categories: Display flags and variables
The line number of the current pair of input and output expressions.
Categories: Display flags and variables · Console interaction
Default value: []
myoptions is the list of all options ever reset by the user,
whether or not they get reset to their default value.
Categories: Global variables · Session management · Console interaction
Default value: false
When nolabels is true, input and output result labels (%i
and %o, respectively) are displayed, but the labels are not bound to
results, and the labels are not appended to the labels list. Since
labels are not bound to results, garbage collection can recover the memory taken
up by the results.
Otherwise input and output result labels are bound to results, and the labels
are appended to the labels list.
Intermediate expression labels (%t) are not affected by nolabels;
whether nolabels is true or false, intermediate expression
labels are bound and appended to the labels list.
See also batch, load, and labels.
Categories: Global flags · Session management
Default value: false
When optionset is true, Maxima prints out a message whenever a
Maxima option is reset. This is useful if the user is doubtful of the spelling
of some option and wants to make sure that the variable he assigned a value to
was truly an option variable.
Example:
(%i1) optionset:true; assignment: assigning to option optionset (%o1) true (%i2) gamma_expand:true; assignment: assigning to option gamma_expand (%o2) true
Categories: Global flags · Session management · Console interaction
Default value: %o
outchar is the prefix of the labels of expressions computed by Maxima.
Maxima automatically constructs a label for each computed expression by
concatenating outchar and linenum.
outchar may be assigned any string or symbol, not necessarily a single
character. Because Maxima internally takes into account only the first char of
the prefix, the prefixes inchar, outchar and
linechar should have a different first char. Otherwise some commands
like kill(inlabels) do not work as expected.
See also labels.
Example:
(%i1) outchar: "output";
(output1) output
(%i2) expand((a+b)^3);
3 2 2 3
(output2) b + 3 a b + 3 a b + a
Categories: Display flags and variables
Displays input, output, and intermediate expressions, without recomputing them.
playback only displays the expressions bound to labels; any other output
(such as text printed by print or describe, or error messages)
is not displayed. See also labels.
playback quotes its arguments. The quote-quote operator ''
defeats quotation. playback always returns done.
playback () (with no arguments) displays all input, output, and
intermediate expressions generated so far. An output expression is displayed
even if it was suppressed by the $ terminator when it was originally
computed.
playback (n) displays the most recent n expressions.
Each input, output, and intermediate expression counts as one.
playback ([m, n]) displays input, output, and intermediate
expressions with numbers from m through n, inclusive.
playback ([m]) is equivalent to
playback ([m, m]); this usually prints one pair of input and
output expressions.
playback (input) displays all input expressions generated so far.
playback (slow) pauses between expressions and waits for the user to
press enter. This behavior is similar to demo.
playback (slow) is useful in conjunction with save or
stringout when creating a secondary-storage file in order to pick out
useful expressions.
playback (time) displays the computation time for each expression.
playback (grind) displays input expressions in the same format as the
grind function. Output expressions are not affected by the grind
option. See grind.
Arguments may be combined, e.g., playback ([5, 10], grind, time, slow).
Categories: Display functions · Console interaction
Default value: _
prompt is the prompt symbol of the demo function,
playback (slow) mode, and the Maxima break loop (as invoked by
break).
Categories: Global variables · Console interaction
Terminates the Maxima session. Note that the function must be invoked as
quit(); or quit()$, not quit by itself.
To stop a lengthy computation, type control-C. The default action is to
return to the Maxima prompt. If *debugger-hook* is nil,
control-C opens the Lisp debugger. See also Debugging.
Categories: Console interaction
Prints expr_1, …, expr_n, then reads one expression from the
console and returns the evaluated expression. The expression is terminated with
a semicolon ; or dollar sign $.
See also readonly
Example:
(%i1) foo: 42$
(%i2) foo: read ("foo is", foo, " -- enter new value.")$
foo is 42 -- enter new value.
(a+b)^3;
(%i3) foo;
3
(%o3) (b + a)
Categories: Console interaction
Prints expr_1, …, expr_n, then reads one expression from the
console and returns the expression (without evaluation). The expression is
terminated with a ; (semicolon) or $ (dollar sign).
See also read.
Examples:
(%i1) aa: 7$
(%i2) foo: readonly ("Enter an expression:");
Enter an expression:
2^aa;
aa
(%o2) 2
(%i3) foo: read ("Enter an expression:");
Enter an expression:
2^aa;
(%o3) 128
Categories: Console interaction
Resets many global variables and options, and some other variables, to their default values.
reset processes the variables on the Lisp list
*variable-initial-values*. The Lisp macro defmvar puts variables
on this list (among other actions). Many, but not all, global variables and
options are defined by defmvar, and some variables defined by
defmvar are not global variables or options.
Categories: Session management
Default value: false
When showtime is true, the computation time and elapsed time is
printed with each output expression.
The computation time is always recorded, so time and playback can
display the computation time even when showtime is false.
See also timer.
Categories: Display flags and variables · Debugging
Enters the Lisp system under Maxima. (to-maxima) returns to Maxima.
Example:
Define a function and enter the Lisp system under Maxima. The definition is inspected on the property list, then the function definition is extracted, factored and stored in the variable $result. The variable can be used in Maxima after returning to Maxima.
(%i1) f(x):=x^2+x;
2
(%o1) f(x) := x + x
(%i2) to_lisp();
Type (to-maxima) to restart, ($quit) to quit Maxima.
MAXIMA> (symbol-plist '$f)
(MPROPS (NIL MEXPR ((LAMBDA) ((MLIST) $X)
((MPLUS) ((MEXPT) $X 2) $X))))
MAXIMA> (setq $result ($factor (caddr (mget '$f 'mexpr))))
((MTIMES SIMP FACTORED) $X ((MPLUS SIMP IRREDUCIBLE) 1 $X))
MAXIMA> (to-maxima)
Returning to Maxima
(%o2) true
(%i3) result;
(%o3) x (x + 1)
Categories: Console interaction
Initial value: []
values is a list of all bound user variables (not Maxima options or
switches). The list comprises symbols bound by :, or ::.
If the value of a variable is removed with the commands kill,
remove, or remvalue the variable is deleted from
values.
See functions for a list of user defined functions.
Examples:
First, values shows the symbols a, b, and c, but
not d, it is not bound to a value, and not the user function f.
The values are removed from the variables. values is the empty list.
(%i1) [a:99, b:: a-90, c:a-b, d, f(x):=x^2];
2
(%o1) [99, 9, 90, d, f(x) := x ]
(%i2) values;
(%o2) [a, b, c]
(%i3) [kill(a), remove(b,value), remvalue(c)];
(%o3) [done, done, [c]]
(%i4) values;
(%o4) []
Categories: Evaluation · Global variables
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Default value: false
When %edispflag is true, Maxima displays %e to a negative
exponent as a quotient. For example, %e^-x is displayed as
1/%e^x. See also exptdispflag.
Example:
(%i1) %e^-10;
- 10
(%o1) %e
(%i2) %edispflag:true$
(%i3) %e^-10;
1
(%o3) ----
10
%e
Default value: !
absboxchar is the character used to draw absolute value
signs around expressions which are more than one line tall.
Example:
(%i1) abs((x^3+1));
! 3 !
(%o1) !x + 1!
Categories: Display flags and variables
is like display but only the value of the arguments are displayed rather
than equations. This is useful for complicated arguments which don't have names
or where only the value of the argument is of interest and not the name.
Example:
(%i1) b[1,2]:x-x^2$
(%i2) x:123$
(%i3) disp(x, b[1,2], sin(1.0));
123
2
x - x
0.8414709848078965
(%o3) done
Categories: Display functions
Displays equations whose left side is expr_i unevaluated, and whose right
side is the value of the expression centered on the line. This function is
useful in blocks and for statements in order to have intermediate results
displayed. The arguments to display are usually atoms, subscripted
variables, or function calls.
See also ldisplay, disp, and ldisp.
Example:
(%i1) b[1,2]:x-x^2$
(%i2) x:123$
(%i3) display(x, b[1,2], sin(1.0));
x = 123
2
b = x - x
1, 2
sin(1.0) = 0.8414709848078965
(%o3) done
Categories: Display functions
Default value: true
When display2d is false, the console display is a string
(1-dimensional) form rather than a display (2-dimensional) form.
See also leftjust to switch between a left justified and a centered
display of equations.
Example:
(%i1) x/(x^2+1);
x
(%o1) ------
2
x + 1
(%i2) display2d:false$
(%i3) x/(x^2+1);
(%o3) x/(x^2+1)
Categories: Display flags and variables
Default value: false
When display_format_internal is true, expressions are displayed
without being transformed in ways that hide the internal mathematical
representation. The display then corresponds to what inpart returns
rather than part.
Examples:
User part inpart
a-b; a - b a + (- 1) b
a - 1
a/b; - a b
b
1/2
sqrt(x); sqrt(x) x
4 X 4
X*4/3; --- - X
3 3
Categories: Display flags and variables
Displays expr in parts one below the other. That is, first the operator
of expr is displayed, then each term in a sum, or factor in a product, or
part of a more general expression is displayed separately. This is useful if
expr is too large to be otherwise displayed. For example if P1,
P2, … are very large expressions then the display program may run
out of storage space in trying to display P1 + P2 + ... all at once.
However, dispterms (P1 + P2 + ...) displays P1, then below it
P2, etc. When not using dispterms, if an exponential expression
is too wide to be displayed as A^B it appears as expt (A, B) (or
as ncexpt (A, B) in the case of A^^B).
Example:
(%i1) dispterms(2*a*sin(x)+%e^x); + 2 a sin(x) x %e (%o1) done
Categories: Display functions
If an exponential expression is too wide to be displayed as
a^b it appears as expt (a, b) (or as
ncexpt (a, b) in the case of a^^b).
expt and ncexpt are not recognized in input.
Default value: true
When exptdispflag is true, Maxima displays expressions
with negative exponents using quotients. See also %edispflag.
Example:
(%i1) exptdispflag:true;
(%o1) true
(%i2) 10^-x;
1
(%o2) ---
x
10
(%i3) exptdispflag:false;
(%o3) false
(%i4) 10^-x;
- x
(%o4) 10
Categories: Expressions · Display flags and variables
The function grind prints expr to the console in a form suitable
for input to Maxima. grind always returns done.
When expr is the name of a function or macro, grind prints the
function or macro definition instead of just the name.
See also string, which returns a string instead of printing its
output. grind attempts to print the expression in a manner which makes
it slightly easier to read than the output of string.
grind evaluates its argument.
Examples:
(%i1) aa + 1729;
(%o1) aa + 1729
(%i2) grind (%);
aa+1729$
(%o2) done
(%i3) [aa, 1729, aa + 1729];
(%o3) [aa, 1729, aa + 1729]
(%i4) grind (%);
[aa,1729,aa+1729]$
(%o4) done
(%i5) matrix ([aa, 17], [29, bb]);
[ aa 17 ]
(%o5) [ ]
[ 29 bb ]
(%i6) grind (%);
matrix([aa,17],[29,bb])$
(%o6) done
(%i7) set (aa, 17, 29, bb);
(%o7) {17, 29, aa, bb}
(%i8) grind (%);
{17,29,aa,bb}$
(%o8) done
(%i9) exp (aa / (bb + 17)^29);
aa
-----------
29
(bb + 17)
(%o9) %e
(%i10) grind (%);
%e^(aa/(bb+17)^29)$
(%o10) done
(%i11) expr: expand ((aa + bb)^10);
10 9 2 8 3 7 4 6
(%o11) bb + 10 aa bb + 45 aa bb + 120 aa bb + 210 aa bb
5 5 6 4 7 3 8 2
+ 252 aa bb + 210 aa bb + 120 aa bb + 45 aa bb
9 10
+ 10 aa bb + aa
(%i12) grind (expr);
bb^10+10*aa*bb^9+45*aa^2*bb^8+120*aa^3*bb^7+210*aa^4*bb^6
+252*aa^5*bb^5+210*aa^6*bb^4+120*aa^7*bb^3+45*aa^8*bb^2
+10*aa^9*bb+aa^10$
(%o12) done
(%i13) string (expr);
(%o13) bb^10+10*aa*bb^9+45*aa^2*bb^8+120*aa^3*bb^7+210*aa^4*bb^6\
+252*aa^5*bb^5+210*aa^6*bb^4+120*aa^7*bb^3+45*aa^8*bb^2+10*aa^9*\
bb+aa^10
(%i14) cholesky (A):= block ([n : length (A), L : copymatrix (A),
p : makelist (0, i, 1, length (A))],
for i thru n do for j : i thru n do
(x : L[i, j], x : x - sum (L[j, k] * L[i, k], k, 1, i - 1),
if i = j then p[i] : 1 / sqrt(x) else L[j, i] : x * p[i]),
for i thru n do L[i, i] : 1 / p[i],
for i thru n do for j : i + 1 thru n do L[i, j] : 0, L)$
define: warning: redefining the built-in function cholesky
(%i15) grind (cholesky);
cholesky(A):=block(
[n:length(A),L:copymatrix(A),
p:makelist(0,i,1,length(A))],
for i thru n do
(for j from i thru n do
(x:L[i,j],x:x-sum(L[j,k]*L[i,k],k,1,i-1),
if i = j then p[i]:1/sqrt(x)
else L[j,i]:x*p[i])),
for i thru n do L[i,i]:1/p[i],
for i thru n do (for j from i+1 thru n do L[i,j]:0),L)$
(%o15) done
(%i16) string (fundef (cholesky));
(%o16) cholesky(A):=block([n:length(A),L:copymatrix(A),p:makelis\
t(0,i,1,length(A))],for i thru n do (for j from i thru n do (x:L\
[i,j],x:x-sum(L[j,k]*L[i,k],k,1,i-1),if i = j then p[i]:1/sqrt(x\
) else L[j,i]:x*p[i])),for i thru n do L[i,i]:1/p[i],for i thru \
n do (for j from i+1 thru n do L[i,j]:0),L)
Categories: Display functions
When the variable grind is true, the output of string and
stringout has the same format as that of grind; otherwise no
attempt is made to specially format the output of those functions. The default
value of the variable grind is false.
grind can also be specified as an argument of playback. When
grind is present, playback prints input expressions in the same
format as the grind function. Otherwise, no attempt is made to specially
format input expressions.
Categories: Display flags and variables
Default value: 10
ibase is the base for integers read by Maxima.
ibase may be assigned any integer between 2 and 36 (decimal), inclusive.
When ibase is greater than 10,
the numerals comprise the decimal numerals 0 through 9
plus letters of the alphabet A, B, C, …,
as needed to make ibase digits in all.
Letters are interpreted as digits only if the first digit is 0 through 9.
Uppercase and lowercase letters are not distinguished.
The numerals for base 36, the largest acceptable base,
comprise 0 through 9 and A through Z.
Whatever the value of ibase,
when an integer is terminated by a decimal point,
it is interpreted in base 10.
See also obase.
Examples:
ibase less than 10 (for example binary numbers).
(%i1) ibase : 2 $ (%i2) obase; (%o2) 10 (%i3) 1111111111111111; (%o3) 65535
ibase greater than 10.
Letters are interpreted as digits only if the first digit is 0
through 9 which means that hexadecimal numbers might need to
be prepended by a 0.
(%i1) ibase : 16 $ (%i2) obase; (%o2) 10 (%i3) 1000; (%o3) 4096 (%i4) abcd; (%o4) abcd (%i5) symbolp (abcd); (%o5) true (%i6) 0abcd; (%o6) 43981 (%i7) symbolp (0abcd); (%o7) false
When an integer is terminated by a decimal point, it is interpreted in base 10.
(%i1) ibase : 36 $ (%i2) obase; (%o2) 10 (%i3) 1234; (%o3) 49360 (%i4) 1234.; (%o4) 1234
Categories: Console interaction
Displays expressions expr_1, …, expr_n to the console as
printed output. ldisp assigns an intermediate expression label to each
argument and returns the list of labels.
See also disp, display, and ldisplay.
Examples:
(%i1) e: (a+b)^3;
3
(%o1) (b + a)
(%i2) f: expand (e);
3 2 2 3
(%o2) b + 3 a b + 3 a b + a
(%i3) ldisp (e, f);
3
(%t3) (b + a)
3 2 2 3
(%t4) b + 3 a b + 3 a b + a
(%o4) [%t3, %t4]
(%i4) %t3;
3
(%o4) (b + a)
(%i5) %t4;
3 2 2 3
(%o5) b + 3 a b + 3 a b + a
Categories: Display functions
Displays expressions expr_1, …, expr_n to the console as
printed output. Each expression is printed as an equation of the form
lhs = rhs in which lhs is one of the arguments of ldisplay
and rhs is its value. Typically each argument is a variable.
ldisp assigns an intermediate expression label to each equation and
returns the list of labels.
See also display, disp, and ldisp.
Examples:
(%i1) e: (a+b)^3;
3
(%o1) (b + a)
(%i2) f: expand (e);
3 2 2 3
(%o2) b + 3 a b + 3 a b + a
(%i3) ldisplay (e, f);
3
(%t3) e = (b + a)
3 2 2 3
(%t4) f = b + 3 a b + 3 a b + a
(%o4) [%t3, %t4]
(%i4) %t3;
3
(%o4) e = (b + a)
(%i5) %t4;
3 2 2 3
(%o5) f = b + 3 a b + 3 a b + a
Categories: Display functions
Default value: false
When leftjust is true, equations in 2D-display are drawn left
justified rather than centered.
See also display2d to switch between 1D- and 2D-display.
Example:
(%i1) expand((x+1)^3);
3 2
(%o1) x + 3 x + 3 x + 1
(%i2) leftjust:true$
(%i3) expand((x+1)^3);
3 2
(%o3) x + 3 x + 3 x + 1
Categories: Display flags and variables
Default value: 79
linel is the assumed width (in characters) of the console display for the
purpose of displaying expressions. linel may be assigned any value by
the user, although very small or very large values may be impractical. Text
printed by built-in Maxima functions, such as error messages and the output of
describe, is not affected by linel.
Categories: Display flags and variables
Default value: false
When lispdisp is true, Lisp symbols are displayed with a leading
question mark ?. Otherwise, Lisp symbols are displayed with no leading
mark. This has the same effect for 1-d and 2-d display.
Examples:
(%i1) lispdisp: false$ (%i2) ?foo + ?bar; (%o2) foo + bar (%i3) lispdisp: true$ (%i4) ?foo + ?bar; (%o4) ?foo + ?bar
Categories: Display flags and variables
Default value: true
When negsumdispflag is true, x - y displays as x - y
instead of as - y + x. Setting it to false causes the special
check in display for the difference of two expressions to not be done. One
application is that thus a + %i*b and a - %i*b may both be
displayed the same way.
Categories: Display flags and variables
Default value: 10
obase is the base for integers displayed by Maxima.
obase may be assigned any integer between 2 and 36 (decimal), inclusive.
When obase is greater than 10,
the numerals comprise the decimal numerals 0 through 9
plus capital letters of the alphabet A, B, C, …, as needed.
A leading 0 digit is displayed if the leading digit is otherwise a letter.
The numerals for base 36, the largest acceptable base,
comprise 0 through 9, and A through Z.
See also ibase.
Examples:
(%i1) obase : 2; (%o1) 10 (%i10) 2^8 - 1; (%o10) 11111111 (%i11) obase : 8; (%o3) 10 (%i4) 8^8 - 1; (%o4) 77777777 (%i5) obase : 16; (%o5) 10 (%i6) 16^8 - 1; (%o6) 0FFFFFFFF (%i7) obase : 36; (%o7) 10 (%i8) 36^8 - 1; (%o8) 0ZZZZZZZZ
Categories: Display flags and variables · Console interaction
Default value: false
When pfeformat is true, a ratio of integers is displayed with the
solidus (forward slash) character, and an integer denominator n is
displayed as a leading multiplicative term 1/n.
Examples:
(%i1) pfeformat: false$
(%i2) 2^16/7^3;
65536
(%o2) -----
343
(%i3) (a+b)/8;
b + a
(%o3) -----
8
(%i4) pfeformat: true$
(%i5) 2^16/7^3;
(%o5) 65536/343
(%i6) (a+b)/8;
(%o6) 1/8 (b + a)
Categories: Display flags and variables
Default value: false
When powerdisp is true,
a sum is displayed with its terms in order of increasing power.
Thus a polynomial is displayed as a truncated power series,
with the constant term first and the highest power last.
By default, terms of a sum are displayed in order of decreasing power.
Example:
(%i1) powerdisp:true;
(%o1) true
(%i2) x^2+x^3+x^4;
2 3 4
(%o2) x + x + x
(%i3) powerdisp:false;
(%o3) false
(%i4) x^2+x^3+x^4;
4 3 2
(%o4) x + x + x
Categories: Display flags and variables
Evaluates and displays expr_1, …, expr_n one after another, from left to right, starting at the left edge of the console display.
The value returned by print is the value of its last argument.
print does not generate intermediate expression labels.
See also display, disp, ldisplay, and
ldisp. Those functions display one expression per line, while
print attempts to display two or more expressions per line.
To display the contents of a file, see printfile.
Examples:
(%i1) r: print ("(a+b)^3 is", expand ((a+b)^3), "log (a^10/b) is",
radcan (log (a^10/b)))$
3 2 2 3
(a+b)^3 is b + 3 a b + 3 a b + a log (a^10/b) is
10 log(a) - log(b)
(%i2) r;
(%o2) 10 log(a) - log(b)
(%i3) disp ("(a+b)^3 is", expand ((a+b)^3), "log (a^10/b) is",
radcan (log (a^10/b)))$
(a+b)^3 is
3 2 2 3
b + 3 a b + 3 a b + a
log (a^10/b) is
10 log(a) - log(b)
Categories: Display functions
Default value: true
When sqrtdispflag is false, causes sqrt to display with
exponent 1/2.
Categories: Mathematical functions · Display flags and variables
Default value: false
When stardisp is true, multiplication is
displayed with an asterisk * between operands.
Categories: Display flags and variables
Default value: false
When ttyoff is true, output expressions are not displayed.
Output expressions are still computed and assigned labels. See labels.
Text printed by built-in Maxima functions, such as error messages and the output
of describe, is not affected by ttyoff.
Categories: Display flags and variables
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| 5.1 Numbers | ||
| 5.2 Strings | ||
| 5.3 Constants | ||
| 5.4 Lists | ||
| 5.5 Arrays | ||
| 5.6 Structures |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 5.1.1 Introduction to Numbers | ||
| 5.1.2 Functions and Variables for Numbers |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
A complex expression is specified in Maxima by adding the real part of the
expression to %i times the imaginary part. Thus the roots of the
equation x^2 - 4*x + 13 = 0 are 2 + 3*%i and 2 - 3*%i.
Note that simplification of products of complex expressions can be effected by
expanding the product. Simplification of quotients, roots, and other functions
of complex expressions can usually be accomplished by using the realpart,
imagpart, rectform, polarform, abs, carg
functions.
Categories: Complex variables
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Converts all numbers and functions of numbers in expr to bigfloat numbers.
The number of significant digits in the resulting bigfloats is specified by the
global variable fpprec.
When float2bf is false a warning message is printed when
a floating point number is converted into a bigfloat number (since
this may lead to loss of precision).
Categories: Numerical evaluation
Returns true if expr is a bigfloat number, otherwise false.
Categories: Numerical evaluation · Predicate functions
Default value: false
bftorat controls the conversion of bfloats to rational numbers. When
bftorat is false, ratepsilon will be used to control the
conversion (this results in relatively small rational numbers). When
bftorat is true, the rational number generated will accurately
represent the bfloat.
Note: bftorat has no effect on the transformation to rational numbers
with the function rationalize.
Example:
(%i1) ratepsilon:1e-4;
(%o1) 1.e-4
(%i2) rat(bfloat(11111/111111)), bftorat:false;
`rat' replaced 9.99990999991B-2 by 1/10 = 1.0B-1
1
(%o2)/R/ --
10
(%i3) rat(bfloat(11111/111111)), bftorat:true;
`rat' replaced 9.99990999991B-2 by 11111/111111 = 9.99990999991B-2
11111
(%o3)/R/ ------
111111
Categories: Numerical evaluation
Default value: true
bftrunc causes trailing zeroes in non-zero bigfloat numbers not to be
displayed. Thus, if bftrunc is false, bfloat (1)
displays as 1.000000000000000B0. Otherwise, this is displayed as
1.0B0.
Categories: Numerical evaluation
Returns true if expr is a literal even integer, otherwise
false.
evenp returns false if expr is a symbol, even if expr
is declared even.
Categories: Predicate functions
Converts integers, rational numbers and bigfloats in expr to floating
point numbers. It is also an evflag, float causes
non-integral rational numbers and bigfloat numbers to be converted to floating
point.
Categories: Numerical evaluation · Evaluation flags
Default value: true
When float2bf is false, a warning message is printed when
a floating point number is converted into a bigfloat number (since
this may lead to loss of precision).
Categories: Numerical evaluation
Returns true if expr is a floating point number, otherwise
false.
Categories: Numerical evaluation · Predicate functions
Default value: 16
fpprec is the number of significant digits for arithmetic on bigfloat
numbers. fpprec does not affect computations on ordinary floating point
numbers.
See also bfloat and fpprintprec.
Categories: Numerical evaluation
Default value: 0
fpprintprec is the number of digits to print when printing an ordinary
float or bigfloat number.
For ordinary floating point numbers,
when fpprintprec has a value between 2 and 16 (inclusive),
the number of digits printed is equal to fpprintprec.
Otherwise, fpprintprec is 0, or greater than 16,
and the number of digits printed is 16.
For bigfloat numbers,
when fpprintprec has a value between 2 and fpprec (inclusive),
the number of digits printed is equal to fpprintprec.
Otherwise, fpprintprec is 0, or greater than fpprec,
and the number of digits printed is equal to fpprec.
For both ordinary floats and bigfloats,
trailing zero digits are suppressed.
The actual number of digits printed is less than fpprintprec
if there are trailing zero digits.
fpprintprec cannot be 1.
Categories: Numerical evaluation · Display flags and variables
Returns true if expr is a literal numeric integer, otherwise
false.
integerp returns false if expr is a symbol, even if expr
is declared integer.
Examples:
(%i1) integerp (0); (%o1) true (%i2) integerp (1); (%o2) true (%i3) integerp (-17); (%o3) true (%i4) integerp (0.0); (%o4) false (%i5) integerp (1.0); (%o5) false (%i6) integerp (%pi); (%o6) false (%i7) integerp (n); (%o7) false (%i8) declare (n, integer); (%o8) done (%i9) integerp (n); (%o9) false
Categories: Predicate functions
Default value: false
m1pbranch is the principal branch for -1 to a power.
Quantities such as (-1)^(1/3) (that is, an "odd" rational exponent) and
(-1)^(1/4) (that is, an "even" rational exponent) are handled as follows:
domain:real
(-1)^(1/3): -1
(-1)^(1/4): (-1)^(1/4)
domain:complex
m1pbranch:false m1pbranch:true
(-1)^(1/3) 1/2+%i*sqrt(3)/2
(-1)^(1/4) sqrt(2)/2+%i*sqrt(2)/2
Categories: Expressions · Global flags
Return true if and only if n >= 0 and n is an integer.
Categories: Package linearalgebra · Predicate functions
Returns true if expr is a literal integer, rational number,
floating point number, or bigfloat, otherwise false.
numberp returns false if expr is a symbol, even if expr
is a symbolic number such as %pi or %i, or declared to be
even, odd, integer, rational, irrational,
real, imaginary, or complex.
Examples:
(%i1) numberp (42);
(%o1) true
(%i2) numberp (-13/19);
(%o2) true
(%i3) numberp (3.14159);
(%o3) true
(%i4) numberp (-1729b-4);
(%o4) true
(%i5) map (numberp, [%e, %pi, %i, %phi, inf, minf]);
(%o5) [false, false, false, false, false, false]
(%i6) declare (a, even, b, odd, c, integer, d, rational,
e, irrational, f, real, g, imaginary, h, complex);
(%o6) done
(%i7) map (numberp, [a, b, c, d, e, f, g, h]);
(%o7) [false, false, false, false, false, false, false, false]
Categories: Predicate functions
numer causes some mathematical functions (including exponentiation)
with numerical arguments to be evaluated in floating point. It causes
variables in expr which have been given numerals to be replaced by
their values. It also sets the float switch on.
See also %enumer.
Examples:
(%i1) [sqrt(2), sin(1), 1/(1+sqrt(3))];
1
(%o1) [sqrt(2), sin(1), -----------]
sqrt(3) + 1
(%i2) [sqrt(2), sin(1), 1/(1+sqrt(3))],numer;
(%o2) [1.414213562373095, 0.8414709848078965, 0.3660254037844387]
Categories: Numerical evaluation · Evaluation flags
Default value: false
The option variable numer_pbranch controls the numerical evaluation of
the power of a negative integer, rational, or floating point number. When
numer_pbranch is true and the exponent is a floating point number
or the option variable numer is true too, Maxima evaluates
the numerical result using the principal branch. Otherwise a simplified, but
not an evaluated result is returned.
Examples:
(%i1) (-2)^0.75;
0.75
(%o1) (- 2)
(%i2) (-2)^0.75,numer_pbranch:true;
(%o2) 1.189207115002721 %i - 1.189207115002721
(%i3) (-2)^(3/4);
3/4 3/4
(%o3) (- 1) 2
(%i4) (-2)^(3/4),numer;
0.75
(%o4) 1.681792830507429 (- 1)
(%i5) (-2)^(3/4),numer,numer_pbranch:true;
(%o5) 1.189207115002721 %i - 1.189207115002721
Categories: Numerical evaluation
Declares the variables x_1, …, x_n to have
numeric values equal to expr_1, …, expr_n.
The numeric value is evaluated and substituted for the variable
in any expressions in which the variable occurs if the numer flag is
true. See also ev.
The expressions expr_1, …, expr_n can be any expressions,
not necessarily numeric.
Categories: Declarations and inferences · Numerical evaluation
Returns true if expr is a literal odd integer, otherwise
false.
oddp returns false if expr is a symbol, even if expr
is declared odd.
Categories: Predicate functions
Default value: 2.0e-15
ratepsilon is the tolerance used in the conversion
of floating point numbers to rational numbers, when the option variable
bftorat has the value false. See bftorat for an example.
Categories: Numerical evaluation · Rational expressions
Convert all double floats and big floats in the Maxima expression expr to
their exact rational equivalents. If you are not familiar with the binary
representation of floating point numbers, you might be surprised that
rationalize (0.1) does not equal 1/10. This behavior isn't special to
Maxima - the number 1/10 has a repeating, not a terminating, binary
representation.
(%i1) rationalize (0.5);
1
(%o1) -
2
(%i2) rationalize (0.1);
3602879701896397
(%o2) -----------------
36028797018963968
(%i3) fpprec : 5$
(%i4) rationalize (0.1b0);
209715
(%o4) -------
2097152
(%i5) fpprec : 20$
(%i6) rationalize (0.1b0);
236118324143482260685
(%o6) ----------------------
2361183241434822606848
(%i7) rationalize (sin (0.1*x + 5.6));
3602879701896397 x 3152519739159347
(%o7) sin(------------------ + ----------------)
36028797018963968 562949953421312
Categories: Numerical evaluation
Returns true if expr is a literal integer or ratio of literal
integers, otherwise false.
Categories: Predicate functions · Rational expressions
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| 5.2.1 Introduction to Strings | ||
| 5.2.2 Functions and Variables for Strings |
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Strings (quoted character sequences) are enclosed in double quote marks "
for input, and displayed with or without the quote marks, depending on the
global variable stringdisp.
Strings may contain any characters, including embedded tab, newline, and
carriage return characters. The sequence \" is recognized as a literal
double quote, and \\ as a literal backslash. When backslash appears at
the end of a line, the backslash and the line termination (either newline or
carriage return and newline) are ignored, so that the string continues with the
next line. No other special combinations of backslash with another character
are recognized; when backslash appears before any character other than ",
\, or a line termination, the backslash is ignored. There is no way to
represent a special character (such as tab, newline, or carriage return)
except by embedding the literal character in the string.
There is no character type in Maxima; a single character is represented as a one-character string.
The stringproc add-on package contains many functions for working with
strings.
Examples:
(%i1) s_1 : "This is a string."; (%o1) This is a string. (%i2) s_2 : "Embedded \"double quotes\" and backslash \\ characters."; (%o2) Embedded "double quotes" and backslash \ characters. (%i3) s_3 : "Embedded line termination (%o3) Embedded line termination in this string. (%i4) in this string."; (%o4) Ignore the line termination characters in this string. (%i5) s_4 : "Ignore the \ (%o5) false (%i6) line termination \ (%o6) This is a string. (%i7) characters in \ (%o7) true (%i8) this string."; (%o8) "This is a string." (%i9) stringdisp : false;
Categories: Syntax
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Concatenates its arguments. The arguments must evaluate to atoms. The return value is a symbol if the first argument is a symbol and a string otherwise.
concat evaluates its arguments. The single quote ' prevents
evaluation.
(%i1) y: 7$
(%i2) z: 88$
(%i3) concat (y, z/2);
(%o3) 744
(%i4) concat ('y, z/2);
(%o4) y44
A symbol constructed by concat may be assigned a value and appear in
expressions. The :: (double colon) assignment operator evaluates its
left-hand side.
(%i5) a: concat ('y, z/2);
(%o5) y44
(%i6) a:: 123;
(%o6) 123
(%i7) y44;
(%o7) 123
(%i8) b^a;
y44
(%o8) b
(%i9) %, numer;
123
(%o9) b
Note that although concat (1, 2) looks like a number, it is a string.
(%i10) concat (1, 2) + 3; (%o10) 12 + 3
Categories: Expressions · Strings
Concatenates its arguments into a string. Unlike concat, the
arguments do not need to be atoms.
(%i1) sconcat ("xx[", 3, "]:", expand ((x+y)^3));
(%o1) xx[3]:y^3+3*x*y^2+3*x^2*y+x^3
Categories: Expressions · Strings
Converts expr to Maxima's linear notation just as if it had been typed
in.
The return value of string is a string, and thus it cannot be used in a
computation.
Categories: Strings
Default value: false
When stringdisp is true, strings are displayed enclosed in double
quote marks. Otherwise, quote marks are not displayed.
stringdisp is always true when displaying a function definition.
Examples:
(%i1) stringdisp: false$
(%i2) "This is an example string.";
(%o2) This is an example string.
(%i3) foo () :=
print ("This is a string in a function definition.");
(%o3) foo() :=
print("This is a string in a function definition.")
(%i4) stringdisp: true$
(%i5) "This is an example string.";
(%o5) "This is an example string."
Categories: Display flags and variables
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| 5.3.1 Functions and Variables for Constants |
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%e represents the base of the natural logarithm, also known as Euler's
number. The numeric value of %e is the double-precision floating-point
value 2.718281828459045d0.
Categories: Constants
%i represents the imaginary unit, sqrt(- 1).
Categories: Constants
false represents the Boolean constant of the same name.
Maxima implements false by the value NIL in Lisp.
Categories: Constants
The Euler-Mascheroni constant, 0.5772156649015329 ....
Categories: Constants
ind represents a bounded, indefinite result.
See also limit.
Example:
(%i1) limit (sin(1/x), x, 0); (%o1) ind
Categories: Constants
inf represents real positive infinity.
Categories: Constants
infinity represents complex infinity.
Categories: Constants
minf represents real minus (i.e., negative) infinity.
Categories: Constants
%phi represents the so-called golden mean, (1 + sqrt(5))/2.
The numeric value of %phi is the double-precision floating-point value
1.618033988749895d0.
fibtophi expresses Fibonacci numbers fib(n) in terms of
%phi.
By default, Maxima does not know the algebraic properties of %phi.
After evaluating tellrat(%phi^2 - %phi - 1) and algebraic: true,
ratsimp can simplify some expressions containing %phi.
Examples:
fibtophi expresses Fibonacci numbers fib(n) in terms of %phi.
(%i1) fibtophi (fib (n));
n n
%phi - (1 - %phi)
(%o1) -------------------
2 %phi - 1
(%i2) fib (n-1) + fib (n) - fib (n+1);
(%o2) - fib(n + 1) + fib(n) + fib(n - 1)
(%i3) fibtophi (%);
n + 1 n + 1 n n
%phi - (1 - %phi) %phi - (1 - %phi)
(%o3) - --------------------------- + -------------------
2 %phi - 1 2 %phi - 1
n - 1 n - 1
%phi - (1 - %phi)
+ ---------------------------
2 %phi - 1
(%i4) ratsimp (%);
(%o4) 0
By default, Maxima does not know the algebraic properties of %phi.
After evaluating tellrat (%phi^2 - %phi - 1) and algebraic: true,
ratsimp can simplify some expressions containing %phi.
(%i1) e : expand ((%phi^2 - %phi - 1) * (A + 1));
2 2
(%o1) %phi A - %phi A - A + %phi - %phi - 1
(%i2) ratsimp (e);
2 2
(%o2) (%phi - %phi - 1) A + %phi - %phi - 1
(%i3) tellrat (%phi^2 - %phi - 1);
2
(%o3) [%phi - %phi - 1]
(%i4) algebraic : true;
(%o4) true
(%i5) ratsimp (e);
(%o5) 0
Categories: Constants
%pi represents the ratio of the perimeter of a circle to its diameter.
The numeric value of %pi is the double-precision floating-point value
3.141592653589793d0.
Categories: Constants
true represents the Boolean constant of the same name.
Maxima implements true by the value T in Lisp.
Categories: Constants
und represents an undefined result.
See also limit.
Example:
(%i1) limit (x*sin(x), x, inf); (%o1) und
Categories: Constants
zeroa represents an infinitesimal above zero. zeroa can be used
in expressions. limit simplifies expressions which contain
infinitesimals.
Example:
limit simplifies expressions which contain infinitesimals:
(%i1) limit(zeroa); (%o1) 0 (%i2) limit(x+zeroa); (%o2) x
Categories: Constants
zerob represents an infinitesimal below zero. zerob can be used
in expressions. limit simplifies expressions which contain
infinitesimals.
Categories: Constants
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| 5.4.1 Introduction to Lists | ||
| 5.4.2 Functions and Variables for Lists | ||
| 5.4.3 Performance considerations for Lists |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Lists are the basic building block for Maxima and Lisp. All data types other than arrays, hash tables, numbers are represented as Lisp lists, These Lisp lists have the form
((MPLUS) $A 2)
to indicate an expression a+2. At Maxima level one would see
the infix notation a+2. Maxima also has lists which are printed
as
[1, 2, 7, x+y]
for a list with 4 elements. Internally this corresponds to a Lisp list of the form
((MLIST) 1 2 7 ((MPLUS) $X $Y))
The flag which denotes the type field of the Maxima expression is a list itself, since after it has been through the simplifier the list would become
((MLIST SIMP) 1 2 7 ((MPLUS SIMP) $X $Y))
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[ and ] mark the beginning and end, respectively, of a list.
[ and ] also enclose the subscripts of
a list, array, hash array, or array function. Note that
other than for arrays accessing the nth element of a list
needs an amount of time that is roughly proportional to n,
See section Performance considerations for Lists.
Examples:
(%i1) x: [a, b, c];
(%o1) [a, b, c]
(%i2) x[3];
(%o2) c
(%i3) array (y, fixnum, 3);
(%o3) y
(%i4) y[2]: %pi;
(%o4) %pi
(%i5) y[2];
(%o5) %pi
(%i6) z['foo]: 'bar;
(%o6) bar
(%i7) z['foo];
(%o7) bar
(%i8) g[k] := 1/(k^2+1);
1
(%o8) g := ------
k 2
k + 1
(%i9) g[10];
1
(%o9) ---
101
Returns a single list of the elements of list_1 followed
by the elements of list_2, … append also works on
general expressions, e.g. append (f(a,b), f(c,d,e)); yields
f(a,b,c,d,e).
See also join.
Do example(append); for an example.
Categories: Lists · Expressions
This function searches for key in the left hand side of the input list. The list argument should be a list, each of whose elements is an expression with exactly two parts. Most usually, the elements of list are themselves lists, each with two elements.
The assoc function iterates along list, checking the
first part of each element for equality with key. If an element
is found where the comparison is true, assoc returns the second
part of that element. If there is no such element in the list,
assoc returns either false or default, if given.
For example, in the expression assoc (y, [[x,1], [y,2],
[z,3]]), the assoc function searches for x in the left
hand side of the list [[y,1],[x,2]] and finds it at the second
term, returning 2. In assoc (z, [[x,1], [z,2], [z,3]]),
the search stops at the first term starting with z and returns
2. In assoc(x, [[y,1]]), there is no matching element,
so assoc returns false.
(%i1) assoc (y, [[x,1], [y,2],[z,3]]); (%o1) 2 (%i2) assoc (z, [[x,1], [z,2], [z,3]]); (%o2) 2 (%i3) assoc (x, [[y,1]]); (%o3) false
Categories: Lists · Expressions
cons (expr, list) returns a new list constructed of the element
expr as its first element, followed by the elements of list. This is
analogous to the Lisp language construction operation "cons".
The Maxima function cons can also be used where the second argument is other
than a list and this might be useful. In this case, cons (expr_1, expr_2)
returns an expression with same operator as expr_2 but with argument cons(expr_1, args(expr_2)).
Examples:
(%i1) cons(a,[b,c,d]); (%o1) [a, b, c, d] (%i2) cons(a,f(b,c,d)); (%o2) f(a, b, c, d)
In general, cons applied to a nonlist doesn't make sense. For instance, cons(a,b^c)
results in an illegal expression, since '^' cannot take three arguments.
When inflag is true, cons operates on the internal structure of an expression, otherwise
cons operates on the displayed form. Especially when inflag is true, cons applied
to a nonlist sometimes gives a surprising result; for example
(%i1) cons(a,-a), inflag : true;
2
(%o1) - a
(%i2) cons(a,-a), inflag : false;
(%o2) 0
Categories: Lists · Expressions
Returns a copy of the list list.
Categories: Lists
Create a list by evaluating form with x_1 bound to each element of list_1, and for each such binding bind x_2 to each element of list_2, … The number of elements in the result will be the product of the number of elements in each list. Each variable x_i must actually be a symbol - it will not be evaluated. The list arguments will be evaluated once at the beginning of the iteration.
(%i1) create_list (x^i, i, [1, 3, 7]);
3 7
(%o1) [x, x , x ]
With a double iteration:
(%i1) create_list ([i, j], i, [a, b], j, [e, f, h]); (%o1) [[a, e], [a, f], [a, h], [b, e], [b, f], [b, h]]
Instead of list_i two args may be supplied each of which should evaluate to a number. These will be the inclusive lower and upper bounds for the iteration.
(%i1) create_list ([i, j], i, [1, 2, 3], j, 1, i); (%o1) [[1, 1], [2, 1], [2, 2], [3, 1], [3, 2], [3, 3]]
Note that the limits or list for the j variable can
depend on the current value of i.
Categories: Lists
delete(expr_1, expr_2)
removes from expr_2 any arguments of its top-level operator
which are the same (as determined by "=") as expr_1.
Note that "=" tests for formal equality, not equivalence.
Note also that arguments of subexpressions are not affected.
expr_1 may be an atom or a non-atomic expression.
expr_2 may be any non-atomic expression.
delete returns a new expression;
it does not modify expr_2.
delete(expr_1, expr_2, n)
removes from expr_2 the first n arguments of the top-level operator
which are the same as expr_1.
If there are fewer than n such arguments,
then all such arguments are removed.
Examples:
Removing elements from a list.
(%i1) delete (y, [w, x, y, z, z, y, x, w]); (%o1) [w, x, z, z, x, w]
Removing terms from a sum.
(%i1) delete (sin(x), x + sin(x) + y); (%o1) y + x
Removing factors from a product.
(%i1) delete (u - x, (u - w)*(u - x)*(u - y)*(u - z)); (%o1) (u - w) (u - y) (u - z)
Removing arguments from an arbitrary expression.
(%i1) delete (a, foo (a, b, c, d, a)); (%o1) foo(b, c, d)
Limit the number of removed arguments.
(%i1) delete (a, foo (a, b, a, c, d, a), 2); (%o1) foo(b, c, d, a)
Whether arguments are the same as expr_1 is determined by "=".
Arguments which are equal but not "=" are not removed.
(%i1) [is (equal (0, 0)), is (equal (0, 0.0)), is (equal (0, 0b0))];
(%o1) [true, true, true]
(%i2) [is (0 = 0), is (0 = 0.0), is (0 = 0b0)];
(%o2) [true, false, false]
(%i3) delete (0, [0, 0.0, 0b0]);
(%o3) [0.0, 0.0b0]
(%i4) is (equal ((x + y)*(x - y), x^2 - y^2));
(%o4) true
(%i5) is ((x + y)*(x - y) = x^2 - y^2);
(%o5) false
(%i6) delete ((x + y)*(x - y), [(x + y)*(x - y), x^2 - y^2]);
2 2
(%o6) [x - y ]
Categories: Lists · Expressions
Returns the 8'th item of expression or list expr.
See first for more details.
Categories: Lists · Expressions
endcons (expr, list) returns a new list constructed of the elements of
list followed by expr. The Maxima function endcons can also be used where
the second argument is other than a list and this might be useful. In this case,
endcons (expr_1, expr_2) returns an expression with same operator as
expr_2 but with argument endcons(expr_1, args(expr_2)). Examples:
(%i1) endcons(a,[b,c,d]); (%o1) [b, c, d, a] (%i2) endcons(a,f(b,c,d)); (%o2) f(b, c, d, a)
In general, endcons applied to a nonlist doesn't make sense. For instance, endcons(a,b^c)
results in an illegal expression, since '^' cannot take three arguments.
When inflag is true, endcons operates on the internal structure of an expression, otherwise
endcons operates on the displayed form. Especially when inflag is true, endcons applied
to a nonlist sometimes gives a surprising result; for example
(%i1) endcons(a,-a), inflag : true;
2
(%o1) - a
(%i2) endcons(a,-a), inflag : false;
(%o2) 0
Categories: Lists · Expressions
Returns the 5'th item of expression or list expr.
See first for more details.
Categories: Lists · Expressions
Returns the first part of expr which may result in the first element of a list, the first row of a matrix, the first term of a sum, etc.:
(%i1) matrix([1,2],[3,4]);
[ 1 2 ]
(%o1) [ ]
[ 3 4 ]
(%i2) first(%);
(%o2) [1,2]
(%i3) first(%);
(%o3) 1
(%i4) first(a*b/c+d+e/x);
a b
(%o4) ---
c
(%i5) first(a=b/c+d+e/x);
(%o5) a
Note that
first and its related functions, rest and last, work
on the form of expr which is displayed not the form which is typed on
input. If the variable inflag is set to true however, these
functions will look at the internal form of expr. One reason why this may
make a difference is that the simplifier re-orders expressions:
(%i1) x+y; (%o1) y+1 (%i2) first(x+y),inflag : true; (%o2) x (%i3) first(x+y),inflag : false; (%o3) y
The functions second …
tenth yield the second through the tenth part of their input argument.
See also part.
Categories: Lists · Expressions
Returns the 4'th item of expression or list expr.
See first for more details.
Categories: Lists · Expressions
Creates a new list containing the elements of lists l and m,
interspersed. The result has elements [l[1], m[1],
l[2], m[2], ...]. The lists l and m may contain any
type of elements.
If the lists are different lengths, join ignores elements of the longer
list.
Maxima complains if l or m is not a list.
See also append.
Examples:
(%i1) L1: [a, sin(b), c!, d - 1]; (%o1) [a, sin(b), c!, d - 1] (%i2) join (L1, [1, 2, 3, 4]); (%o2) [a, 1, sin(b), 2, c!, 3, d - 1, 4] (%i3) join (L1, [aa, bb, cc, dd, ee, ff]); (%o3) [a, aa, sin(b), bb, c!, cc, d - 1, dd]
Categories: Lists
Returns the last part (term, row, element, etc.) of the expr.
Categories: Lists · Expressions
Returns (by default) the number of parts in the external
(displayed) form of expr. For lists this is the number of elements,
for matrices it is the number of rows, and for sums it is the number
of terms (see dispform).
The length command is affected by the inflag switch. So, e.g.
length(a/(b*c)); gives 2 if inflag is false (Assuming
exptdispflag is true), but 3 if inflag is true (the
internal representation is essentially a*b^-1*c^-1).
Determining a list's length typically needs an amount of time proportional to the number of elements in the list. If the length of a list is used inside a loop it therefore might drastically increase the performance if the length is calculated outside the loop instead.
Categories: Lists · Expressions
Default value: true
If false causes any arithmetic operations with lists to be suppressed;
when true, list-matrix operations are contagious causing lists to be
converted to matrices yielding a result which is always a matrix. However,
list-list operations should return lists.
Categories: Lists · Global flags
Returns true if expr is a list else false.
Categories: Lists · Predicate functions
The first form, makelist (), creates an empty list. The second form,
makelist (expr), creates a list with expr as its single
element. makelist (expr, n) creates a list of n
elements generated from expr.
The most general form, makelist (expr, i, i_0,
i_max, step), returns the list of elements obtained when
ev (expr, i=j) is applied to the elements
j of the sequence: i_0, i_0 + step, i_0 +
2*step, ..., with |j| less than or equal to |i_max|.
The increment step can be a number (positive or negative) or an expression. If it is omitted, the default value 1 will be used. If both i_0 and step are omitted, they will both have a default value of 1.
makelist (expr, x, list) returns a list, the
j'th element of which is equal to
ev (expr, x=list[j]) for j equal to 1 through
length (list).
Examples:
(%i1) makelist (concat (x,i), i, 6); (%o1) [x1, x2, x3, x4, x5, x6] (%i2) makelist (x=y, y, [a, b, c]); (%o2) [x = a, x = b, x = c] (%i3) makelist (x^2, x, 3, 2*%pi, 2); (%o3) [9, 25] (%i4) makelist (random(6), 4); (%o4) [2, 0, 2, 5] (%i5) flatten (makelist (makelist (i^2, 3), i, 4)); (%o5) [1, 1, 1, 4, 4, 4, 9, 9, 9, 16, 16, 16] (%i6) flatten (makelist (makelist (i^2, i, 3), 4)); (%o6) [1, 4, 9, 1, 4, 9, 1, 4, 9, 1, 4, 9]
Categories: Lists
Returns true if is(expr_1 = a)
for some element a in args(expr_2),
otherwise returns false.
expr_2 is typically a list, in which case
args(expr_2) = expr_2 and is(expr_1 = a)
for some element a in expr_2 is the test.
member does not inspect parts of the arguments of expr_2, so it
may return false even if expr_1 is a part of some argument of
expr_2.
See also elementp.
Examples:
(%i1) member (8, [8, 8.0, 8b0]);
(%o1) true
(%i2) member (8, [8.0, 8b0]);
(%o2) false
(%i3) member (b, [a, b, c]);
(%o3) true
(%i4) member (b, [[a, b], [b, c]]);
(%o4) false
(%i5) member ([b, c], [[a, b], [b, c]]);
(%o5) true
(%i6) F (1, 1/2, 1/4, 1/8);
1 1 1
(%o6) F(1, -, -, -)
2 4 8
(%i7) member (1/8, %);
(%o7) true
(%i8) member ("ab", ["aa", "ab", sin(1), a + b]);
(%o8) true
Categories: Lists · Expressions · Predicate functions
Returns the 9'th item of expression or list expr.
See first for more details.
Categories: Lists · Expressions
pop removes and returns the first element from the list list. The second argument
list must be a mapatom that is bound to a nonempty list. If the argument list is
not bound to a nonempty list, Maxima signals an error. For examples, see push.
Categories: Lists · Expressions
push prepends the item item to the list list and returns a copy of the new list.
The second argument list must be a mapatom that is bound to a list. The first argument item
can be any Maxima symbol or expression. If the argument list is not bound to a list, Maxima
signals an error.
To remove the first item from a list, see pop.
Examples:
(%i1) ll: [];
(%o1) []
(%i2) push (x, ll);
(%o2) [x]
(%i3) push (x^2+y, ll);
2
(%o3) [y + x , x]
(%i4) a: push ("string", ll);
2
(%o4) [string, y + x , x]
(%i5) pop (ll);
(%o5) string
(%i6) pop (ll);
2
(%o6) y + x
(%i7) pop (ll);
(%o7) x
(%i8) ll;
(%o8) []
(%i9) a;
2
(%o9) [string, y + x , x]
Categories: Lists · Expressions
Returns expr with its first n elements removed if n
is positive and its last - n elements removed if n
is negative. If n is 1 it may be omitted. The first argument
expr may be a list, matrix, or other expression. When expr
is a mapatom, rest signals an error; when expr is an empty
list and partswitch is false, rest signals an error. When
expr is an empty list and partswitch is true, rest
returns end.
Applying rest to expression such as f(a,b,c) returns
f(b,c). In general, applying rest to an nonlist doesn't
make sense. For example, because '^' requires two arguments,
rest(a^b) results in an error message. The functions
args and op may be useful as well, since args(a^b)
returns [a,b] and op(a^b) returns ^.
(%i1) rest(a+b+c); (%o1) b+a (%i2) rest(a+b+c,2); (%o2) a (%i3) rest(a+b+c,-2); (%o3) c
Categories: Lists · Expressions
Reverses the order of the members of the list (not
the members themselves). reverse also works on general expressions,
e.g. reverse(a=b); gives b=a.
Categories: Lists · Expressions
Returns the 2'nd item of expression or list expr.
See first for more details.
Categories: Lists · Expressions
Returns the 7'th item of expression or list expr.
See first for more details.
Categories: Lists · Expressions
Returns the 6'th item of expression or list expr.
See first for more details.
Categories: Lists · Expressions
sort(L, P) sorts a list L according to a predicate P of two arguments
which defines a strict weak order on the elements of L.
If P(a, b) is true, then a appears before b in the result.
If neither P(a, b) nor P(b, a) are true,
then a and b are equivalent, and appear in the result in the same order as in the input.
That is, sort is a stable sort.
If P(a, b) and P(b, a) are both true for some elements of L,
then P is not a valid sort predicate, and the result is undefined.
If P(a, b) is something other than true or false, sort signals an error.
The predicate may be specified as the name of a function
or binary infix operator, or as a lambda expression. If specified as
the name of an operator, the name must be enclosed in double quotes.
The sorted list is returned as a new object; the argument L is not modified.
sort(L) is equivalent to sort(L, orderlessp).
The default sorting order is ascending, as determined by orderlessp. The predicate ordergreatp sorts a list in descending order.
All Maxima atoms and expressions are comparable under orderlessp and ordergreatp.
Operators < and > order numbers, constants, and constant expressions by magnitude.
Note that orderlessp and ordergreatp do not order numbers, constants, and constant expressions by magnitude.
ordermagnitudep orders numbers, constants, and constant expressions the same as <,
and all other elements the same as orderlessp.
Examples:
sort sorts a list according to a predicate of two arguments
which defines a strict weak order on the elements of the list.
(%i1) sort ([1, a, b, 2, 3, c], 'orderlessp); (%o1) [1, 2, 3, a, b, c] (%i2) sort ([1, a, b, 2, 3, c], 'ordergreatp); (%o2) [c, b, a, 3, 2, 1]
The predicate may be specified as the name of a function
or binary infix operator, or as a lambda expression. If specified as
the name of an operator, the name must be enclosed in double quotes.
(%i1) L : [[1, x], [3, y], [4, w], [2, z]];
(%o1) [[1, x], [3, y], [4, w], [2, z]]
(%i2) foo (a, b) := a[1] > b[1];
(%o2) foo(a, b) := a > b
1 1
(%i3) sort (L, 'foo);
(%o3) [[4, w], [3, y], [2, z], [1, x]]
(%i4) infix (">>");
(%o4) >>
(%i5) a >> b := a[1] > b[1];
(%o5) (a >> b) := a > b
1 1
(%i6) sort (L, ">>");
(%o6) [[4, w], [3, y], [2, z], [1, x]]
(%i7) sort (L, lambda ([a, b], a[1] > b[1]));
(%o7) [[4, w], [3, y], [2, z], [1, x]]
sort(L) is equivalent to sort(L, orderlessp).
(%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi]; (%o1) [a, 2 b, - 5, 7, %e + 1, %pi] (%i2) sort (L); (%o2) [- 5, 7, %e + 1, %pi, a, 2 b] (%i3) sort (L, 'orderlessp); (%o3) [- 5, 7, %e + 1, %pi, a, 2 b]
The default sorting order is ascending, as determined by orderlessp. The predicate ordergreatp sorts a list in descending order.
(%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi]; (%o1) [a, 2 b, - 5, 7, %e + 1, %pi] (%i2) sort (L); (%o2) [- 5, 7, %e + 1, %pi, a, 2 b] (%i3) sort (L, 'ordergreatp); (%o3) [2 b, a, %pi, %e + 1, 7, - 5]
All Maxima atoms and expressions are comparable under orderlessp and ordergreatp.
(%i1) L : [11, -17, 29b0, 9*c, 7.55, foo(x, y), -5/2, b + a];
5
(%o1) [11, - 17, 2.9b1, 9 c, 7.55, foo(x, y), - -, b + a]
2
(%i2) sort (L, orderlessp);
5
(%o2) [- 17, - -, 7.55, 11, 2.9b1, b + a, 9 c, foo(x, y)]
2
(%i3) sort (L, ordergreatp);
5
(%o3) [foo(x, y), 9 c, b + a, 2.9b1, 11, 7.55, - -, - 17]
2
Operators < and > order numbers, constants, and constant expressions by magnitude.
Note that orderlessp and ordergreatp do not order numbers, constants, and constant expressions by magnitude.
(%i1) L : [%pi, 3, 4, %e, %gamma]; (%o1) [%pi, 3, 4, %e, %gamma] (%i2) sort (L, ">"); (%o2) [4, %pi, 3, %e, %gamma] (%i3) sort (L, ordergreatp); (%o3) [%pi, %gamma, %e, 4, 3]
ordermagnitudep orders numbers, constants, and constant expressions the same as <,
and all other elements the same as orderlessp.
(%i1) L : [%i, 1+%i, 2*x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0, 1.0b0];
(%o1) [%i, %i + 1, 2 x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0,
1.0b0]
(%i2) sort (L, ordermagnitudep);
(%o2) [minf, 0, sin(1), 1, 1.0, 1.0b0, 2, %e, 3, inf, %i,
%i + 1, 2 x]
(%i3) sort (L, orderlessp);
(%o3) [0, 1, 1.0, 2, 3, sin(1), 1.0b0, %e, %i, %i + 1, inf,
minf, 2 x]
Categories: Lists
Returns the list of elements of list for which the predicate p
returns true.
Example:
(%i1) L: [1, 2, 3, 4, 5, 6]; (%o1) [1, 2, 3, 4, 5, 6] (%i2) sublist (L, evenp); (%o2) [2, 4, 6]
Categories: Lists
Returns the indices of the elements x of the list L for which
the predicate maybe(P(x)) returns true;
this excludes unknown as well as false.
P may be the name of a function or a lambda expression.
L must be a literal list.
Examples:
(%i1) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b],
lambda ([x], x='b));
(%o1) [2, 3, 7, 9]
(%i2) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b], symbolp);
(%o2) [1, 2, 3, 4, 7, 9]
(%i3) sublist_indices ([1 > 0, 1 < 0, 2 < 1, 2 > 1, 2 > 0],
identity);
(%o3) [1, 4, 5]
(%i4) assume (x < -1);
(%o4) [x < - 1]
(%i5) map (maybe, [x > 0, x < 0, x < -2]);
(%o5) [false, true, unknown]
(%i6) sublist_indices ([x > 0, x < 0, x < -2], identity);
(%o6) [2]
Categories: Lists
Returns the unique elements of the list L.
When all the elements of L are unique,
unique returns a shallow copy of L,
not L itself.
If L is not a list, unique returns L.
Example:
(%i1) unique ([1, %pi, a + b, 2, 1, %e, %pi, a + b, [1]]); (%o1) [1, 2, %e, %pi, [1], b + a]
Returns the 10'th item of expression or list expr.
See first for more details.
Categories: Lists · Expressions
Returns the 3'rd item of expression or list expr.
See first for more details.
Categories: Lists · Expressions
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Lists provide efficient ways of appending and removing elements. They can be created without knowing their final dimensions. Lisp provides efficient means of copying and handling lists. Also nested lists do not need to be strictly rectangular. These advantages over declared arrays come with the drawback that the amount of time needed for accessing a random element within a list is roughly proportional to the element's distance from its beginning. Efficient traversal of lists is still possible, though, by using the list as a stack or a fifo:
(%i1) l:[Test,1,2,3,4];
(%o1) [Test, 1, 2, 3, 4]
(%i2) while l # [] do
disp(pop(l));
Test
1
2
3
4
(%o2) done
Another even faster example would be:
(%i1) l:[Test,1,2,3,4];
(%o1) [Test, 1, 2, 3, 4]
(%i2) for i in l do
disp(pop(l));
Test
1
2
3
4
(%o2) done
Beginning traversal with the last element of a list is possible after
reversing the list using reverse ().
If the elements of a long list need to be processed in a different
order performance might be increased by converting the list into a
declared array first.
It is also to note that the ending condition of for loops
is tested for every iteration which means that the result of a
length should be cached if it is used in the ending
condition:
(%i1) l:makelist(i,i,1,100000)$
(%i2) lngth:length(l);
(%o2) 100000
(%i3) x:1;
(%o3) 1
(%i4) for i:1 thru lngth do
x:x+1$
(%i5) x;
(%o5) 100001
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| 5.5.1 Functions and Variables for Arrays |
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Creates an n-dimensional array. n may be less than or equal to 5. The subscripts for the i'th dimension are the integers running from 0 to dim_i.
array (name, dim_1, ..., dim_n) creates a general
array.
array (name, type, dim_1, ..., dim_n) creates
an array, with elements of a specified type. type can be fixnum
for integers of limited size or flonum for floating-point numbers.
array ([name_1, ..., name_m], dim_1, ..., dim_n)
creates m arrays, all of the same dimensions.
If the user assigns to a subscripted variable before declaring the
corresponding array, an undeclared array is created.
Undeclared arrays, otherwise known as hashed arrays (because hash
coding is done on the subscripts), are more general than declared
arrays. The user does not declare their maximum size, and they grow
dynamically by hashing as more elements are assigned values. The
subscripts of undeclared arrays need not even be numbers. However,
unless an array is rather sparse, it is probably more efficient to
declare it when possible than to leave it undeclared. The array
function can be used to transform an undeclared array into a declared
array.
Categories: Arrays
Evaluates A [i_1, ..., i_n],
where A is an array and i_1, …, i_n are integers.
This is reminiscent of apply, except the first argument is an array
instead of a function.
Categories: Expressions · Arrays
Returns information about the array A. The argument A may be a declared array, an undeclared (hashed) array, an array function, or a subscripted function.
For declared arrays, arrayinfo returns a list comprising the atom
declared, the number of dimensions, and the size of each dimension.
The elements of the array, both bound and unbound, are returned by
listarray.
For undeclared arrays (hashed arrays), arrayinfo returns a list
comprising the atom hashed, the number of subscripts,
and the subscripts of every element which has a value.
The values are returned by listarray.
For array functions, arrayinfo returns a list comprising the atom
hashed, the number of subscripts,
and any subscript values for which there are stored function values.
The stored function values are returned by listarray.
For subscripted functions, arrayinfo returns a list comprising the atom
hashed, the number of subscripts,
and any subscript values for which there are lambda expressions.
The lambda expressions are returned by listarray.
See also listarray.
Examples:
arrayinfo and listarray applied to a declared array.
(%i1) array (aa, 2, 3);
(%o1) aa
(%i2) aa [2, 3] : %pi;
(%o2) %pi
(%i3) aa [1, 2] : %e;
(%o3) %e
(%i4) arrayinfo (aa);
(%o4) [declared, 2, [2, 3]]
(%i5) listarray (aa);
(%o5) [#####, #####, #####, #####, #####, #####, %e, #####,
#####, #####, #####, %pi]
arrayinfo and listarray applied to an undeclared (hashed) array.
(%i1) bb [FOO] : (a + b)^2;
2
(%o1) (b + a)
(%i2) bb [BAR] : (c - d)^3;
3
(%o2) (c - d)
(%i3) arrayinfo (bb);
(%o3) [hashed, 1, [BAR], [FOO]]
(%i4) listarray (bb);
3 2
(%o4) [(c - d) , (b + a) ]
arrayinfo and listarray applied to an array function.
(%i1) cc [x, y] := y / x;
y
(%o1) cc := -
x, y x
(%i2) cc [u, v];
v
(%o2) -
u
(%i3) cc [4, z];
z
(%o3) -
4
(%i4) arrayinfo (cc);
(%o4) [hashed, 2, [4, z], [u, v]]
(%i5) listarray (cc);
z v
(%o5) [-, -]
4 u
arrayinfo and listarray applied to a subscripted function.
(%i1) dd [x] (y) := y ^ x;
x
(%o1) dd (y) := y
x
(%i2) dd [a + b];
b + a
(%o2) lambda([y], y )
(%i3) dd [v - u];
v - u
(%o3) lambda([y], y )
(%i4) arrayinfo (dd);
(%o4) [hashed, 1, [b + a], [v - u]]
(%i5) listarray (dd);
b + a v - u
(%o5) [lambda([y], y ), lambda([y], y )]
Categories: Arrays
Returns the expression A[i_1, ..., i_n].
The result is an unevaluated array reference.
arraymake is reminiscent of funmake, except the return value
is an unevaluated array reference instead of an unevaluated function call.
Examples:
(%i1) arraymake (A, [1]);
(%o1) A
1
(%i2) arraymake (A, [k]);
(%o2) A
k
(%i3) arraymake (A, [i, j, 3]);
(%o3) A
i, j, 3
(%i4) array (A, fixnum, 10);
(%o4) A
(%i5) fillarray (A, makelist (i^2, i, 1, 11));
(%o5) A
(%i6) arraymake (A, [5]);
(%o6) A
5
(%i7) ''%;
(%o7) 36
(%i8) L : [a, b, c, d, e];
(%o8) [a, b, c, d, e]
(%i9) arraymake ('L, [n]);
(%o9) L
n
(%i10) ''%, n = 3;
(%o10) c
(%i11) A2 : make_array (fixnum, 10);
(%o11) {Array: #(0 0 0 0 0 0 0 0 0 0)}
(%i12) fillarray (A2, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
(%o12) {Array: #(1 2 3 4 5 6 7 8 9 10)}
(%i13) arraymake ('A2, [8]);
(%o13) A2
8
(%i14) ''%;
(%o14) 9
Categories: Expressions · Arrays
Default value: []
arrays is a list of arrays that have been allocated.
These comprise arrays declared by array, hashed arrays constructed by
implicit definition (assigning something to an array element),
and array functions defined by := and define.
Arrays defined by make_array are not included.
See also
array, arrayapply, arrayinfo,
arraymake, fillarray, listarray, and
rearray.
Examples:
(%i1) array (aa, 5, 7);
(%o1) aa
(%i2) bb [FOO] : (a + b)^2;
2
(%o2) (b + a)
(%i3) cc [x] := x/100;
x
(%o3) cc := ---
x 100
(%i4) dd : make_array ('any, 7);
(%o4) {Array: #(NIL NIL NIL NIL NIL NIL NIL)}
(%i5) arrays;
(%o5) [aa, bb, cc]
Categories: Arrays · Global variables
Assigns x to A[i_1, ..., i_n],
where A is an array and i_1, …, i_n are integers.
arraysetapply evaluates its arguments.
Categories: Expressions · Arrays
Fills array A from B, which is a list or an array.
If a specific type was declared for A when it was created, it can only be filled with elements of that same type; it is an error if an attempt is made to copy an element of a different type.
If the dimensions of the arrays A and B are different, A is filled in row-major order. If there are not enough elements in B the last element is used to fill out the rest of A. If there are too many, the remaining ones are ignored.
fillarray returns its first argument.
Examples:
Create an array of 9 elements and fill it from a list.
(%i1) array (a1, fixnum, 8); (%o1) a1 (%i2) listarray (a1); (%o2) [0, 0, 0, 0, 0, 0, 0, 0, 0] (%i3) fillarray (a1, [1, 2, 3, 4, 5, 6, 7, 8, 9]); (%o3) a1 (%i4) listarray (a1); (%o4) [1, 2, 3, 4, 5, 6, 7, 8, 9]
When there are too few elements to fill the array, the last element is repeated. When there are too many elements, the extra elements are ignored.
(%i1) a2 : make_array (fixnum, 8);
(%o1) {Array: #(0 0 0 0 0 0 0 0)}
(%i2) fillarray (a2, [1, 2, 3, 4, 5]);
(%o2) {Array: #(1 2 3 4 5 5 5 5)}
(%i3) fillarray (a2, [4]);
(%o3) {Array: #(4 4 4 4 4 4 4 4)}
(%i4) fillarray (a2, makelist (i, i, 1, 100));
(%o4) {Array: #(1 2 3 4 5 6 7 8)}
Multple-dimension arrays are filled in row-major order.
(%i1) a3 : make_array (fixnum, 2, 5);
(%o1) {Array: #2A((0 0 0 0 0) (0 0 0 0 0))}
(%i2) fillarray (a3, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
(%o2) {Array: #2A((1 2 3 4 5) (6 7 8 9 10))}
(%i3) a4 : make_array (fixnum, 5, 2);
(%o3) {Array: #2A((0 0) (0 0) (0 0) (0 0) (0 0))}
(%i4) fillarray (a4, a3);
(%o4) {Array: #2A((1 2) (3 4) (5 6) (7 8) (9 10))}
Categories: Arrays
Returns a list of the elements of the array A. The argument A may be a declared array, an undeclared (hashed) array, an array function, or a subscripted function.
Elements are listed in row-major order.
That is, elements are sorted according to the first index, then according to
the second index, and so on. The sorting order of index values is the same as
the order established by orderless.
For undeclared arrays, array functions, and subscripted functions,
the elements correspond to the index values returned by arrayinfo.
Unbound elements of declared general arrays (that is, not fixnum and not
flonum) are returned as #####.
Unbound elements of declared fixnum or flonum arrays
are returned as 0 or 0.0, respectively.
Unbound elements of undeclared arrays, array functions,
and subscripted functions are not returned.
Examples:
listarray and arrayinfo applied to a declared array.
(%i1) array (aa, 2, 3);
(%o1) aa
(%i2) aa [2, 3] : %pi;
(%o2) %pi
(%i3) aa [1, 2] : %e;
(%o3) %e
(%i4) listarray (aa);
(%o4) [#####, #####, #####, #####, #####, #####, %e, #####,
#####, #####, #####, %pi]
(%i5) arrayinfo (aa);
(%o5) [declared, 2, [2, 3]]
listarray and arrayinfo applied to an undeclared (hashed) array.
(%i1) bb [FOO] : (a + b)^2;
2
(%o1) (b + a)
(%i2) bb [BAR] : (c - d)^3;
3
(%o2) (c - d)
(%i3) listarray (bb);
3 2
(%o3) [(c - d) , (b + a) ]
(%i4) arrayinfo (bb);
(%o4) [hashed, 1, [BAR], [FOO]]
listarray and arrayinfo applied to an array function.
(%i1) cc [x, y] := y / x;
y
(%o1) cc := -
x, y x
(%i2) cc [u, v];
v
(%o2) -
u
(%i3) cc [4, z];
z
(%o3) -
4
(%i4) listarray (cc);
z v
(%o4) [-, -]
4 u
(%i5) arrayinfo (cc);
(%o5) [hashed, 2, [4, z], [u, v]]
listarray and arrayinfo applied to a subscripted function.
(%i1) dd [x] (y) := y ^ x;
x
(%o1) dd (y) := y
x
(%i2) dd [a + b];
b + a
(%o2) lambda([y], y )
(%i3) dd [v - u];
v - u
(%o3) lambda([y], y )
(%i4) listarray (dd);
b + a v - u
(%o4) [lambda([y], y ), lambda([y], y )]
(%i5) arrayinfo (dd);
(%o5) [hashed, 1, [b + a], [v - u]]
Categories: Arrays
Creates and returns a Lisp array. type may
be any, flonum, fixnum, hashed or
functional.
There are n indices,
and the i'th index runs from 0 to dim_i - 1.
The advantage of make_array over array is that the return value
doesn't have a name, and once a pointer to it goes away, it will also go away.
For example, if y: make_array (...) then y points to an object
which takes up space, but after y: false, y no longer
points to that object, so the object can be garbage collected.
Examples:
(%i1) A1 : make_array (fixnum, 10);
(%o1) {Array: #(0 0 0 0 0 0 0 0 0 0)}
(%i2) A1 [8] : 1729;
(%o2) 1729
(%i3) A1;
(%o3) {Array: #(0 0 0 0 0 0 0 0 1729 0)}
(%i4) A2 : make_array (flonum, 10);
(%o4) {Array: #(0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0)}
(%i5) A2 [2] : 2.718281828;
(%o5) 2.718281828
(%i6) A2;
(%o6)
{Array: #(0.0 0.0 2.718281828 0.0 0.0 0.0 0.0 0.0 0.0 0.0)}
(%i7) A3 : make_array (any, 10);
(%o7) {Array: #(NIL NIL NIL NIL NIL NIL NIL NIL NIL NIL)}
(%i8) A3 [4] : x - y - z;
(%o8) - z - y + x
(%i9) A3;
(%o9) {Array: #(NIL NIL NIL NIL ((MPLUS SIMP) $X ((MTIMES SIMP)\
-1 $Y) ((MTIMES SIMP) -1 $Z))
NIL NIL NIL NIL NIL)}
(%i10) A4 : make_array (fixnum, 2, 3, 5);
(%o10) {Array: #3A(((0 0 0 0 0) (0 0 0 0 0) (0 0 0 0 0)) ((0 0 \
0 0 0) (0 0 0 0 0) (0 0 0 0 0)))}
(%i11) fillarray (A4, makelist (i, i, 1, 2*3*5));
(%o11) {Array: #3A(((1 2 3 4 5) (6 7 8 9 10) (11 12 13 14 15))
((16 17 18 19 20) (21 22 23 24 25) (26 27 28 29 30)))}
(%i12) A4 [0, 2, 1];
(%o12) 12
Categories: Arrays
Changes the dimensions of an array.
The new array will be filled with the elements of the old one in
row-major order. If the old array was too small,
the remaining elements are filled with
false, 0.0 or 0,
depending on the type of the array. The type of the array cannot be
changed.
Categories: Arrays
Removes arrays and array associated functions and frees the storage occupied. The arguments may be declared arrays, undeclared (hashed) arrays, array functions, and subscripted functions.
remarray (all) removes all items in the global list arrays.
It may be necessary to use this function if it is desired to redefine the values in a hashed array.
remarray returns the list of arrays removed.
remarray quotes its arguments.
Categories: Arrays
Evaluates the subscripted expression x[i].
subvar evaluates its arguments.
arraymake (x, [i]) constructs the expression
x[i], but does not evaluate it.
Examples:
(%i1) x : foo $
(%i2) i : 3 $
(%i3) subvar (x, i);
(%o3) foo
3
(%i4) foo : [aa, bb, cc, dd, ee]$
(%i5) subvar (x, i);
(%o5) cc
(%i6) arraymake (x, [i]);
(%o6) foo
3
(%i7) ''%;
(%o7) cc
Categories: Expressions · Arrays
Returns true if expr is a subscripted variable, for example
a[i].
Categories: Predicate functions
Default value: false
When use_fast_arrays is true,
undeclared arrays and arrays declared by array are values instead of properties,
and undeclared arrays are implemented as Lisp hash tables.
When use_fast_arrays is false,
undeclared arrays and arrays declared by array are properties,
and undeclared arrays are implemented with Maxima's own hash table implementation.
Arrays created by make_array are not affected by use_fast_arrays.
See also translate_fast_arrays.
Categories: Arrays · Global flags
Default value: false
When translate_fast_arrays is true,
the Maxima-to-Lisp translator generates code that assumes arrays are values instead of properties,
as if use_fast_arrays were true.
When translate_fast_arrays is false,
the Maxima-to-Lisp translator generates code that assumes arrays are properties,
as if use_fast_arrays were false.
Categories: Arrays · Translation flags and variables
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| 5.6.1 Introduction to Structures | ||
| 5.6.2 Functions and Variables for Structures |
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Maxima provides a simple data aggregate called a structure. A structure is an expression in which arguments are identified by name (the field name) and the expression as a whole is identified by its operator (the structure name). A field value can be any expression.
A structure is defined by the defstruct function;
the global variable structures is the list of user-defined structures.
The function new creates instances of structures.
The @ operator refers to fields.
kill(S) removes the structure definition S,
and kill(x@ a) unbinds the field a of the structure instance x.
In the pretty-printing console display (with display2d equal to true),
structure instances are displayed with the value of each field
represented as an equation, with the field name on the left-hand side
and the value on the right-hand side.
(The equation is only a display construct; only the value is actually stored.)
In 1-dimensional display (via grind or with display2d equal to false),
structure instances are displayed without the field names.
There is no way to use a field name as a function name, although a field value can be a lambda expression. Nor can the values of fields be restricted to certain types; any field can be assigned any kind of expression. There is no way to make some fields accessible or inaccessible in different contexts; all fields are always visible.
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structures is the list of user-defined structures defined by defstruct.
Categories: Structures · Global variables
Define a structure, which is a list of named fields a_1, …,
a_n associated with a symbol S.
An instance of a structure is just an expression which has operator S
and exactly n arguments.
new(S) creates a new instance of structure S.
An argument which is just a symbol a specifies the name of a field.
An argument which is an equation a = v specifies the field name a
and its default value v.
The default value can be any expression.
defstruct puts S on the list of user-defined structures, structures.
kill(S) removes S from the list of user-defined structures,
and removes the structure definition.
Examples:
(%i1) defstruct (foo (a, b, c)); (%o1) [foo(a, b, c)] (%i2) structures; (%o2) [foo(a, b, c)] (%i3) new (foo); (%o3) foo(a, b, c) (%i4) defstruct (bar (v, w, x = 123, y = %pi)); (%o4) [bar(v, w, x = 123, y = %pi)] (%i5) structures; (%o5) [foo(a, b, c), bar(v, w, x = 123, y = %pi)] (%i6) new (bar); (%o6) bar(v, w, x = 123, y = %pi) (%i7) kill (foo); (%o7) done (%i8) structures; (%o8) [bar(v, w, x = 123, y = %pi)]
Categories: Structures
new creates new instances of structures.
new(S) creates a new instance of structure S
in which each field is assigned its default value, if any,
or no value at all if no default was specified in the structure definition.
new(S(v_1, ..., v_n)) creates a new instance of S
in which fields are assigned the values v_1, …, v_n.
Examples:
(%i1) defstruct (foo (w, x = %e, y = 42, z)); (%o1) [foo(w, x = %e, y = 42, z)] (%i2) new (foo); (%o2) foo(w, x = %e, y = 42, z) (%i3) new (foo (1, 2, 4, 8)); (%o3) foo(w = 1, x = 2, y = 4, z = 8)
Categories: Structures
@ is the structure field access operator.
The expression x@ a refers to the value of field a of the structure instance x.
The field name is not evaluated.
If the field a in x has not been assigned a value,
x@ a evaluates to itself.
kill(x@ a) removes the value of field a in x.
Examples:
(%i1) defstruct (foo (x, y, z)); (%o1) [foo(x, y, z)] (%i2) u : new (foo (123, a - b, %pi)); (%o2) foo(x = 123, y = a - b, z = %pi) (%i3) u@z; (%o3) %pi (%i4) u@z : %e; (%o4) %e (%i5) u; (%o5) foo(x = 123, y = a - b, z = %e) (%i6) kill (u@z); (%o6) done (%i7) u; (%o7) foo(x = 123, y = a - b, z) (%i8) u@z; (%o8) u@z
The field name is not evaluated.
(%i1) defstruct (bar (g, h)); (%o1) [bar(g, h)] (%i2) x : new (bar); (%o2) bar(g, h) (%i3) x@h : 42; (%o3) 42 (%i4) h : 123; (%o4) 123 (%i5) x@h; (%o5) 42 (%i6) x@h : 19; (%o6) 19 (%i7) x; (%o7) bar(g, h = 19) (%i8) h; (%o8) 123
Categories: Structures · Operators
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| 6.1 Introduction to Expressions | ||
| 6.2 Nouns and Verbs | ||
| 6.3 Identifiers | ||
| 6.4 Inequality | ||
| 6.5 Functions and Variables for Expressions |
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There are a number of reserved words which should not be used as variable names. Their use would cause a possibly cryptic syntax error.
integrate next from diff in at limit sum for and elseif then else do or if unless product while thru step
Most things in Maxima are expressions. A sequence of expressions can be made into an expression by separating them by commas and putting parentheses around them. This is similar to the C comma expression.
(%i1) x: 3$ (%i2) (x: x+1, x: x^2); (%o2) 16 (%i3) (if (x > 17) then 2 else 4); (%o3) 4 (%i4) (if (x > 17) then x: 2 else y: 4, y+x); (%o4) 20
Even loops in Maxima are expressions, although the value they
return is the not too useful done.
(%i1) y: (x: 1, for i from 1 thru 10 do (x: x*i))$ (%i2) y; (%o2) done
Whereas what you really want is probably to include a third term in the comma expression which actually gives back the value.
(%i3) y: (x: 1, for i from 1 thru 10 do (x: x*i), x)$ (%i4) y; (%o4) 3628800
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Maxima distinguishes between operators which are "nouns" and operators which are
"verbs". A verb is an operator which can be executed. A noun is an operator
which appears as a symbol in an expression, without being executed. By default,
function names are verbs. A verb can be changed into a noun by quoting the
function name or applying the nounify function. A noun can be changed
into a verb by applying the verbify function. The evaluation flag
nouns causes ev to evaluate nouns in an expression.
The verb form is distinguished by a leading dollar sign $ on the
corresponding Lisp symbol. In contrast, the noun form is distinguished by a
leading percent sign % on the corresponding Lisp symbol. Some nouns have
special display properties, such as 'integrate and 'derivative
(returned by diff), but most do not. By default, the noun and verb forms
of a function are identical when displayed. The global flag noundisp
causes Maxima to display nouns with a leading quote mark '.
See also noun, nouns, nounify, and
verbify.
Examples:
(%i1) foo (x) := x^2;
2
(%o1) foo(x) := x
(%i2) foo (42);
(%o2) 1764
(%i3) 'foo (42);
(%o3) foo(42)
(%i4) 'foo (42), nouns;
(%o4) 1764
(%i5) declare (bar, noun);
(%o5) done
(%i6) bar (x) := x/17;
x
(%o6) bar(x) := --
17
(%i7) bar (52);
(%o7) bar(52)
(%i8) bar (52), nouns;
(%o8) bar(52)
(%i9) integrate (1/x, x, 1, 42);
(%o9) log(42)
(%i10) 'integrate (1/x, x, 1, 42);
42
/
[ 1
(%o10) I - dx
] x
/
1
(%i11) ev (%, nouns);
(%o11) log(42)
Categories: Evaluation · Nouns and verbs
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Maxima identifiers may comprise alphabetic characters, plus the numerals 0
through 9, plus any special character preceded by the backslash \
character.
A numeral may be the first character of an identifier if it is preceded by a backslash. Numerals which are the second or later characters need not be preceded by a backslash.
Characters may be declared alphabetic by the declare function.
If so declared, they need not be preceded by a backslash in an identifier.
The alphabetic characters are initially A through Z, a
through z, %, and _.
Maxima is case-sensitive. The identifiers foo, FOO, and
Foo are distinct. See Lisp and Maxima for more on this point.
A Maxima identifier is a Lisp symbol which begins with a dollar sign $.
Any other Lisp symbol is preceded by a question mark ? when it appears
in Maxima. See Lisp and Maxima for more on this point.
Examples:
(%i1) %an_ordinary_identifier42;
(%o1) %an_ordinary_identifier42
(%i2) embedded\ spaces\ in\ an\ identifier;
(%o2) embedded spaces in an identifier
(%i3) symbolp (%);
(%o3) true
(%i4) [foo+bar, foo\+bar];
(%o4) [foo + bar, foo+bar]
(%i5) [1729, \1729];
(%o5) [1729, 1729]
(%i6) [symbolp (foo\+bar), symbolp (\1729)];
(%o6) [true, true]
(%i7) [is (foo\+bar = foo+bar), is (\1729 = 1729)];
(%o7) [false, false]
(%i8) baz\~quux;
(%o8) baz~quux
(%i9) declare ("~", alphabetic);
(%o9) done
(%i10) baz~quux;
(%o10) baz~quux
(%i11) [is (foo = FOO), is (FOO = Foo), is (Foo = foo)];
(%o11) [false, false, false]
(%i12) :lisp (defvar *my-lisp-variable* '$foo)
*MY-LISP-VARIABLE*
(%i12) ?\*my\-lisp\-variable\*;
(%o12) foo
Categories: Syntax
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Maxima has the inequality operators <, <=, >=, >,
#, and notequal. See if for a description of conditional
expressions.
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provides an alternate name for a (user or system) function, variable, array, etc. Any even number of arguments may be used.
Categories: Declarations and inferences
Default value: []
aliases is the list of atoms which have a user defined alias (set up by
the alias, ordergreat, orderless functions or by
declaring the atom a noun with declare.)
Categories: Declarations and inferences · Global variables
works with the part commands (i.e. part,
inpart, substpart, substinpart,
dpart, and lpart).
For example,
(%i1) expr : e + d + c + b + a; (%o1) e + d + c + b + a (%i2) part (expr, [2, 5]); (%o2) d + a
while
(%i1) expr : e + d + c + b + a; (%o1) e + d + c + b + a (%i2) part (expr, allbut (2, 5)); (%o2) e + c + b
allbut is also recognized by kill.
(%i1) [aa : 11, bb : 22, cc : 33, dd : 44, ee : 55]; (%o1) [11, 22, 33, 44, 55] (%i2) kill (allbut (cc, dd)); (%o0) done (%i1) [aa, bb, cc, dd]; (%o1) [aa, bb, 33, 44]
kill(allbut(a_1, a_2, ...)) has the effect of
kill(all) except that it does not kill the symbols a_1, a_2,
…
Returns the list of arguments of expr, which may be any kind of
expression other than an atom. Only the arguments of the top-level operator
are extracted; subexpressions of expr appear as elements or
subexpressions of elements of the list of arguments.
The order of the items in the list may depend on the global flag
inflag.
args (expr) is equivalent to substpart ("[", expr, 0).
See also substpart, apply, funmake, and op.
How to convert a matrix to a nested list:
(%i1) M:matrix([1,2],[3,4]);
[ 1 2 ]
(%o1) [ ]
[ 3 4 ]
(%i2) args(M);
(%o2) [[1, 2], [3, 4]]
Since maxima internally treats a sum of n terms as a summation command
with n arguments args() can extract the list of terms in a sum:
(%i1) a+b+c; (%o1) c + b + a (%i2) args(%); (%o2) [c, b, a]
Categories: Expressions
Returns true if expr is atomic (i.e. a number, name or string) else
false. Thus atom(5) is true while atom(a[1]) and
atom(sin(x)) are false (assuming a[1] and x are
unbound).
Categories: Expressions · Predicate functions
Returns expr enclosed in a box. The return value is an expression with
box as the operator and expr as the argument. A box is drawn on
the display when display2d is true.
box (expr, a) encloses expr in a box labelled by the
symbol a. The label is truncated if it is longer than the width of the
box.
box evaluates its argument. However, a boxed expression does not
evaluate to its content, so boxed expressions are effectively excluded from
computations.
boxchar is the character used to draw the box in box and in the
dpart and lpart functions.
Examples:
(%i1) box (a^2 + b^2);
"""""""""
" 2 2"
(%o1) "b + a "
"""""""""
(%i2) a : 1234;
(%o2) 1234
(%i3) b : c - d;
(%o3) c - d
(%i4) box (a^2 + b^2);
""""""""""""""""""""
" 2 "
(%o4) "(c - d) + 1522756"
""""""""""""""""""""
(%i5) box (a^2 + b^2, term_1);
term_1""""""""""""""
" 2 "
(%o5) "(c - d) + 1522756"
""""""""""""""""""""
(%i6) 1729 - box (1729);
""""""
(%o6) 1729 - "1729"
""""""
(%i7) boxchar: "-";
(%o7) -
(%i8) box (sin(x) + cos(y));
-----------------
(%o8) -cos(y) + sin(x)-
-----------------
Categories: Expressions
Default value: "
boxchar is the character used to draw the box in the box
and in the dpart and lpart functions.
All boxes in an expression are drawn with the current value of boxchar;
the drawing character is not stored with the box expression.
Categories: Expressions
Collapses expr by causing all of its common (i.e., equal) subexpressions
to share (i.e., use the same cells), thereby saving space. (collapse is
a subroutine used by the optimize command.) Thus, calling
collapse may be useful after loading in a save file. You can
collapse several expressions together by using
collapse ([expr_1, ..., expr_n]). Similarly, you can
collapse the elements of the array A by doing
collapse (listarray ('A)).
Categories: Expressions
is similar to isolate (expr, x) except that it enables the
user to isolate more than one variable simultaneously. This might be useful,
for example, if one were attempting to change variables in a multiple
integration, and that variable change involved two or more of the integration
variables. This function is autoloaded from `simplification/disol.mac'.
A demo is available by demo("disol")$.
Categories: Expressions
Returns the external representation of expr.
dispform(expr) returns the external representation with respect to
the main (top-level) operator. dispform(expr, all) returns the
external representation with respect to all operators in expr.
See also part, inpart, and inflag.
Examples:
The internal representation of - x is "negative one times x"
while the external representation is "minus x".
(%i1) - x; (%o1) - x (%i2) ?format (true, "~S~%", %); ((MTIMES SIMP) -1 $X) (%o2) false (%i3) dispform (- x); (%o3) - x (%i4) ?format (true, "~S~%", %); ((MMINUS SIMP) $X) (%o4) false
The internal representation of sqrt(x) is "x to the power 1/2"
while the external representation is "square root of x".
(%i1) sqrt (x); (%o1) sqrt(x) (%i2) ?format (true, "~S~%", %); ((MEXPT SIMP) $X ((RAT SIMP) 1 2)) (%o2) false (%i3) dispform (sqrt (x)); (%o3) sqrt(x) (%i4) ?format (true, "~S~%", %); ((%SQRT SIMP) $X) (%o4) false
Use of the optional argument all.
(%i1) expr : sin (sqrt (x)); (%o1) sin(sqrt(x)) (%i2) freeof (sqrt, expr); (%o2) true (%i3) freeof (sqrt, dispform (expr)); (%o3) true (%i4) freeof (sqrt, dispform (expr, all)); (%o4) false
Categories: Expressions
Selects the same subexpression as part, but instead of just returning
that subexpression as its value, it returns the whole expression with the
selected subexpression displayed inside a box. The box is actually part of the
expression.
(%i1) dpart (x+y/z^2, 1, 2, 1);
y
(%o1) ---- + x
2
"""
"z"
"""
Categories: Expressions
Default value: false
exptisolate, when true, causes isolate (expr, var) to
examine exponents of atoms (such as %e) which contain var.
Categories: Expressions
Default value: false
exptsubst, when true, permits substitutions such as y
for %e^x in %e^(a x).
(%i1) %e^(a*x);
a x
(%o1) %e
(%i2) exptsubst;
(%o2) false
(%i3) subst(y, %e^x, %e^(a*x));
a x
(%o3) %e
(%i4) exptsubst: not exptsubst;
(%o4) true
(%i5) subst(y, %e^x, %e^(a*x));
a
(%o5) y
Categories: Exponential and logarithm functions · Expressions
freeof (x_1, expr) returns true if no subexpression of
expr is equal to x_1 or if x_1 occurs only as a dummy variable
in expr, or if x_1 is neither the noun nor verb form of any operator
in expr, and returns false otherwise.
freeof (x_1, ..., x_n, expr) is equivalent to
freeof (x_1, expr) and ... and freeof (x_n,
expr).
The arguments x_1, …, x_n may be names of functions and
variables, subscripted names, operators (enclosed in double quotes), or general
expressions. freeof evaluates its arguments.
freeof operates only on expr as it stands (after simplification and
evaluation) and does not attempt to determine if some equivalent expression
would give a different result. In particular, simplification may yield an
equivalent but different expression which comprises some different elements than
the original form of expr.
A variable is a dummy variable in an expression if it has no binding outside of
the expression. Dummy variables recognized by freeof are the index of a
sum or product, the limit variable in limit, the integration variable
in the definite integral form of integrate, the original variable in
laplace, formal variables in at expressions, and arguments in
lambda expressions.
The indefinite form of integrate is not free of its variable of
integration.
Examples:
Arguments are names of functions, variables, subscripted names, operators, and
expressions. freeof (a, b, expr) is equivalent to
freeof (a, expr) and freeof (b, expr).
(%i1) expr: z^3 * cos (a[1]) * b^(c+d);
d + c 3
(%o1) cos(a ) b z
1
(%i2) freeof (z, expr);
(%o2) false
(%i3) freeof (cos, expr);
(%o3) false
(%i4) freeof (a[1], expr);
(%o4) false
(%i5) freeof (cos (a[1]), expr);
(%o5) false
(%i6) freeof (b^(c+d), expr);
(%o6) false
(%i7) freeof ("^", expr);
(%o7) false
(%i8) freeof (w, sin, a[2], sin (a[2]), b*(c+d), expr);
(%o8) true
freeof evaluates its arguments.
(%i1) expr: (a+b)^5$ (%i2) c: a$ (%i3) freeof (c, expr); (%o3) false
freeof does not consider equivalent expressions.
Simplification may yield an equivalent but different expression.
(%i1) expr: (a+b)^5$
(%i2) expand (expr);
5 4 2 3 3 2 4 5
(%o2) b + 5 a b + 10 a b + 10 a b + 5 a b + a
(%i3) freeof (a+b, %);
(%o3) true
(%i4) freeof (a+b, expr);
(%o4) false
(%i5) exp (x);
x
(%o5) %e
(%i6) freeof (exp, exp (x));
(%o6) true
A summation or definite integral is free of its dummy variable. An indefinite integral is not free of its variable of integration.
(%i1) freeof (i, 'sum (f(i), i, 0, n)); (%o1) true (%i2) freeof (x, 'integrate (x^2, x, 0, 1)); (%o2) true (%i3) freeof (x, 'integrate (x^2, x)); (%o3) false
Categories: Expressions
Default value: false
When inflag is true, functions for part extraction inspect the
internal form of expr.
Note that the simplifier re-orders expressions. Thus first (x + y)
returns x if inflag is true and y if inflag
is false. (first (y + x) gives the same results.)
Also, setting inflag to true and calling part or
substpart is the same as calling inpart or substinpart.
Functions affected by the setting of inflag are: part,
substpart, first, rest, last,
length, the for … in construct,
map, fullmap, maplist, reveal and
pickapart.
Categories: Expressions
is similar to part but works on the internal representation of the
expression rather than the displayed form and thus may be faster since no
formatting is done. Care should be taken with respect to the order of
subexpressions in sums and products (since the order of variables in the
internal form is often different from that in the displayed form) and in dealing
with unary minus, subtraction, and division (since these operators are removed
from the expression). part (x+y, 0) or inpart (x+y, 0) yield
+, though in order to refer to the operator it must be enclosed in "s.
For example ... if inpart (%o9,0) = "+" then ....
Examples:
(%i1) x + y + w*z;
(%o1) w z + y + x
(%i2) inpart (%, 3, 2);
(%o2) z
(%i3) part (%th (2), 1, 2);
(%o3) z
(%i4) 'limit (f(x)^g(x+1), x, 0, minus);
g(x + 1)
(%o4) limit f(x)
x -> 0-
(%i5) inpart (%, 1, 2);
(%o5) g(x + 1)
Categories: Expressions
Returns expr with subexpressions which are sums and which do not contain
var replaced by intermediate expression labels (these being atomic symbols
like %t1, %t2, …). This is often useful to avoid
unnecessary expansion of subexpressions which don't contain the variable of
interest. Since the intermediate labels are bound to the subexpressions they
can all be substituted back by evaluating the expression in which they occur.
exptisolate (default value: false) if true will cause
isolate to examine exponents of atoms (like %e) which contain
var.
isolate_wrt_times if true, then isolate will also isolate
with respect to products. See isolate_wrt_times.
Do example (isolate) for examples.
Categories: Expressions
Default value: false
When isolate_wrt_times is true, isolate will also isolate
with respect to products. E.g. compare both settings of the switch on
(%i1) isolate_wrt_times: true$
(%i2) isolate (expand ((a+b+c)^2), c);
(%t2) 2 a
(%t3) 2 b
2 2
(%t4) b + 2 a b + a
2
(%o4) c + %t3 c + %t2 c + %t4
(%i4) isolate_wrt_times: false$
(%i5) isolate (expand ((a+b+c)^2), c);
2
(%o5) c + 2 b c + 2 a c + %t4
Categories: Expressions
Default value: false
When listconstvars is true the list returned by
listofvars contains constant variables, such as %e,
%pi, %i or any variables declared as constant that
occur in expr. A variable is declared as constant
type via declare, and constantp returns true
for all variables declared as constant. The default is to
omit constant variables from listofvars return value.
Categories: Expressions
Default value: true
When listdummyvars is false, "dummy variables" in the expression
will not be included in the list returned by listofvars. (The meaning
of "dummy variables" is as given in freeof. "Dummy variables" are
mathematical things like the index of a sum or product, the limit variable,
and the definite integration variable.)
Example:
(%i1) listdummyvars: true$
(%i2) listofvars ('sum(f(i), i, 0, n));
(%o2) [i, n]
(%i3) listdummyvars: false$
(%i4) listofvars ('sum(f(i), i, 0, n));
(%o4) [n]
Categories: Expressions
Returns a list of the variables in expr.
listconstvars if true causes listofvars to include
%e, %pi, %i, and any variables declared constant in the
list it returns if they appear in expr. The default is to omit these.
See also the option variable listdummyvars to exclude or include
"dummy variables" in the list of variables.
(%i1) listofvars (f (x[1]+y) / g^(2+a));
(%o1) [g, a, x , y]
1
Categories: Expressions
For each member m of list, calls
freeof (m, expr). It returns false if any call to
freeof does and true otherwise.
Example:
(%i1) lfreeof ([ a, x], x^2+b); (%o1) false (%i2) lfreeof ([ b, x], x^2+b); (%o2) false (%i3) lfreeof ([ a, y], x^2+b); (%o3) true
Categories: Expressions
is similar to dpart but uses a labelled box. A labelled box is similar
to the one produced by dpart but it has a name in the top line.
Categories: Expressions
You may declare variables to be mainvar. The ordering scale for atoms is
essentially: numbers < constants (e.g., %e, %pi) < scalars < other
variables < mainvars. E.g., compare expand ((X+Y)^4) with
(declare (x, mainvar), expand ((x+y)^4)). (Note: Care should be taken if
you elect to use the above feature. E.g., if you subtract an expression in
which x is a mainvar from one in which x isn't a
mainvar, resimplification e.g. with ev (expr, simp) may be
necessary if cancellation is to occur. Also, if you save an expression in which
x is a mainvar, you probably should also save x.)
Categories: Declarations and inferences · Expressions
noun is one of the options of the declare command. It makes a
function so declared a "noun", meaning that it won't be evaluated
automatically.
Example:
(%i1) factor (12345678);
2
(%o1) 2 3 47 14593
(%i2) declare (factor, noun);
(%o2) done
(%i3) factor (12345678);
(%o3) factor(12345678)
(%i4) ''%, nouns;
2
(%o4) 2 3 47 14593
Categories: Nouns and verbs
Default value: false
When noundisp is true, nouns display with
a single quote. This switch is always true when displaying function
definitions.
Categories: Display flags and variables · Nouns and verbs
Returns the noun form of the function name f. This is needed if one wishes to refer to the name of a verb function as if it were a noun. Note that some verb functions will return their noun forms if they can't be evaluated for certain arguments. This is also the form returned if a function call is preceded by a quote.
See also verbify.
Categories: Nouns and verbs
Returns the number of terms that expr would have if it were fully
expanded out and no cancellations or combination of terms occurred.
Note that expressions like sin (expr), sqrt (expr),
exp (expr), etc. count as just one term regardless of how many
terms expr has (if it is a sum).
Categories: Expressions
Returns the main operator of the expression expr.
op (expr) is equivalent to part (expr, 0).
op returns a string if the main operator is a built-in or user-defined
prefix, binary or n-ary infix, postfix, matchfix, or nofix operator.
Otherwise, if expr is a subscripted function expression, op
returns the subscripted function; in this case the return value is not an atom.
Otherwise, expr is an array function or ordinary function expression,
and op returns a symbol.
op observes the value of the global flag inflag.
op evaluates it argument.
See also args.
Examples:
(%i1) stringdisp: true$
(%i2) op (a * b * c);
(%o2) "*"
(%i3) op (a * b + c);
(%o3) "+"
(%i4) op ('sin (a + b));
(%o4) sin
(%i5) op (a!);
(%o5) "!"
(%i6) op (-a);
(%o6) "-"
(%i7) op ([a, b, c]);
(%o7) "["
(%i8) op ('(if a > b then c else d));
(%o8) "if"
(%i9) op ('foo (a));
(%o9) foo
(%i10) prefix (foo);
(%o10) "foo"
(%i11) op (foo a);
(%o11) "foo"
(%i12) op (F [x, y] (a, b, c));
(%o12) F
x, y
(%i13) op (G [u, v, w]);
(%o13) G
Categories: Expressions · Operators
operatorp (expr, op) returns true
if op is equal to the operator of expr.
operatorp (expr, [op_1, ..., op_n]) returns
true if some element op_1, …, op_n is equal to the
operator of expr.
Categories: Operators · Predicate functions
Default value: true
When opsubst is false, subst does not attempt to
substitute into the operator of an expression. E.g.,
(opsubst: false, subst (x^2, r, r+r[0])) will work.
(%i1) r+r[0];
(%o1) r + r
0
(%i2) opsubst;
(%o2) true
(%i3) subst (x^2, r, r+r[0]);
2 2
(%o3) x + (x )
0
(%i4) opsubst: not opsubst;
(%o4) false
(%i5) subst (x^2, r, r+r[0]);
2
(%o5) x + r
0
Categories: Expressions
Returns an expression that produces the same value and
side effects as expr but does so more efficiently by avoiding the
recomputation of common subexpressions. optimize also has the side
effect of "collapsing" its argument so that all common subexpressions
are shared. Do example (optimize) for examples.
Categories: Expressions
Default value: %
optimprefix is the prefix used for generated symbols by
the optimize command.
Categories: Expressions
ordergreat changes the canonical ordering of Maxima expressions
such that v_1 succeeds v_2 succeeds … succeeds v_n,
and v_n succeeds any other symbol not mentioned as an argument.
orderless changes the canonical ordering of Maxima expressions
such that v_1 precedes v_2 precedes … precedes v_n,
and v_n precedes any other variable not mentioned as an argument.
The order established by ordergreat and orderless is dissolved
by unorder. ordergreat and orderless can be called only
once each, unless unorder is called; only the last call to
ordergreat and orderless has any effect.
See also ordergreatp.
Categories: Expressions
ordergreatp returns true if expr_1 succeeds expr_2 in
the canonical ordering of Maxima expressions, and false otherwise.
orderlessp returns true if expr_1 precedes expr_2 in
the canonical ordering of Maxima expressions, and false otherwise.
All Maxima atoms and expressions are comparable under ordergreatp and
orderlessp, although there are isolated examples of expressions for which
these predicates are not transitive; that is a bug.
The canonical ordering of atoms (symbols, literal numbers, and strings) is the following.
(integers and floats) precede (bigfloats) precede
(declared constants) precede (strings) precede (declared scalars)
precede (first argument to orderless) precedes … precedes
(last argument to orderless) precedes (other symbols) precede
(last argument to ordergreat) precedes … precedes
(first argument to ordergreat) precedes (declared main variables)
For non-atomic expressions, the canonical ordering is derived from the ordering
for atoms. For the built-in + * and ^ operators,
the ordering is not easily summarized. For other built-in operators and all
other functions and operators, expressions are ordered by their arguments
(beginning with the first argument), then by the name of the operator or
function. In the case of subscripted expressions, the subscripted symbol is
considered the operator and the subscript is considered an argument.
The canonical ordering of expressions is modified by the functions
ordergreat and orderless, and the mainvar,
constant, and scalar declarations.
See also sort.
Examples:
Ordering ordinary symbols and constants.
Note that %pi is not ordered according to its numerical value.
(%i1) stringdisp : true; (%o1) true (%i2) sort ([%pi, 3b0, 3.0, x, X, "foo", 3, a, 4, "bar", 4.0, 4b0]); (%o2) [3, 3.0, 4, 4.0, 3.0b0, 4.0b0, %pi, "bar", "foo", X, a, x]
Effect of ordergreat and orderless functions.
(%i1) sort ([M, H, K, T, E, W, G, A, P, J, S]); (%o1) [A, E, G, H, J, K, M, P, S, T, W] (%i2) ordergreat (S, J); (%o2) done (%i3) orderless (M, H); (%o3) done (%i4) sort ([M, H, K, T, E, W, G, A, P, J, S]); (%o4) [M, H, A, E, G, K, P, T, W, J, S]
Effect of mainvar, constant, and scalar declarations.
(%i1) sort ([aa, foo, bar, bb, baz, quux, cc, dd, A1, B1, C1]); (%o1) [A1, B1, C1, aa, bar, baz, bb, cc, dd, foo, quux] (%i2) declare (aa, mainvar); (%o2) done (%i3) declare ([baz, quux], constant); (%o3) done (%i4) declare ([A1, B1], scalar); (%o4) done (%i5) sort ([aa, foo, bar, bb, baz, quux, cc, dd, A1, B1, C1]); (%o5) [baz, quux, A1, B1, C1, bar, bb, cc, dd, foo, aa]
Ordering non-atomic expressions.
(%i1) sort ([1, 2, n, f(1), f(2), f(2, 1), g(1), g(1, 2), g(n),
f(n, 1)]);
(%o1) [1, 2, f(1), g(1), g(1, 2), f(2), f(2, 1), n, g(n),
f(n, 1)]
(%i2) sort ([foo(1), X[1], X[k], foo(k), 1, k]);
(%o2) [1, X , foo(1), k, X , foo(k)]
1 k
Categories: Expressions · Predicate functions
Returns parts of the displayed form of expr. It obtains the part of
expr as specified by the indices n_1, …, n_k. First
part n_1 of expr is obtained, then part n_2 of that, etc.
The result is part n_k of … part n_2 of part n_1 of
expr. If no indices are specified expr is returned.
part can be used to obtain an element of a list, a row of a matrix, etc.
If the last argument to a part function is a list of indices then
several subexpressions are picked out, each one corresponding to an
index of the list. Thus part (x + y + z, [1, 3]) is z+x.
piece holds the last expression selected when using the part
functions. It is set during the execution of the function and thus
may be referred to in the function itself as shown below.
If partswitch is set to true then end is returned when a
selected part of an expression doesn't exist, otherwise an error message is
given.
See also inpart, substpart, substinpart,
dpart, and lpart.
Examples:
(%i1) part(z+2*y+a,2); (%o1) 2 y (%i2) part(z+2*y+a,[1,3]); (%o2) z + a (%i3) part(z+2*y+a,2,1); (%o3) 2
example (part) displays additional examples.
Categories: Expressions
Returns a list of two expressions. They are (1) the factors of expr (if it is a product), the terms of expr (if it is a sum), or the list (if it is a list) which don't contain x and, (2) the factors, terms, or list which do.
Examples:
(%i1) partition (2*a*x*f(x), x); (%o1) [2 a, x f(x)] (%i2) partition (a+b, x); (%o2) [b + a, 0] (%i3) partition ([a, b, f(a), c], a); (%o3) [[b, c], [a, f(a)]]
Categories: Expressions
Default value: false
When partswitch is true, end is returned
when a selected part of an expression doesn't exist, otherwise an
error message is given.
Categories: Expressions
Assigns intermediate expression labels to subexpressions of expr at depth
n, an integer. Subexpressions at greater or lesser depths are not
assigned labels. pickapart returns an expression in terms of
intermediate expressions equivalent to the original expression expr.
See also part, dpart, lpart,
inpart, and reveal.
Examples:
(%i1) expr: (a+b)/2 + sin (x^2)/3 - log (1 + sqrt(x+1));
2
sin(x ) b + a
(%o1) - log(sqrt(x + 1) + 1) + ------- + -----
3 2
(%i2) pickapart (expr, 0);
2
sin(x ) b + a
(%t2) - log(sqrt(x + 1) + 1) + ------- + -----
3 2
(%o2) %t2
(%i3) pickapart (expr, 1);
(%t3) - log(sqrt(x + 1) + 1)
2
sin(x )
(%t4) -------
3
b + a
(%t5) -----
2
(%o5) %t5 + %t4 + %t3
(%i5) pickapart (expr, 2);
(%t6) log(sqrt(x + 1) + 1)
2
(%t7) sin(x )
(%t8) b + a
%t8 %t7
(%o8) --- + --- - %t6
2 3
(%i8) pickapart (expr, 3);
(%t9) sqrt(x + 1) + 1
2
(%t10) x
b + a sin(%t10)
(%o10) ----- - log(%t9) + ---------
2 3
(%i10) pickapart (expr, 4);
(%t11) sqrt(x + 1)
2
sin(x ) b + a
(%o11) ------- + ----- - log(%t11 + 1)
3 2
(%i11) pickapart (expr, 5);
(%t12) x + 1
2
sin(x ) b + a
(%o12) ------- + ----- - log(sqrt(%t12) + 1)
3 2
(%i12) pickapart (expr, 6);
2
sin(x ) b + a
(%o12) ------- + ----- - log(sqrt(x + 1) + 1)
3 2
Categories: Expressions
Holds the last expression selected when using the part functions.
It is set during the execution of the function and thus may be referred to in
the function itself.
Categories: Expressions
psubst(a, b, expr) is simliar to subst. See
subst.
In distinction from subst the function psubst makes parallel
substitutions, if the first argument list is a list of equations.
See also sublis for making parallel substitutions.
Example:
The first example shows parallel substitution with psubst. The second
example shows the result for the function subst, which does a serial
substitution.
(%i1) psubst ([a^2=b, b=a], sin(a^2) + sin(b)); (%o1) sin(b) + sin(a) (%i2) subst ([a^2=b, b=a], sin(a^2) + sin(b)); (%o2) 2 sin(a)
Categories: Expressions
Removes boxes from expr.
rembox (expr, unlabelled) removes all unlabelled boxes from
expr.
rembox (expr, label) removes only boxes bearing label.
rembox (expr) removes all boxes, labelled and unlabelled.
Boxes are drawn by the box, dpart, and lpart
functions.
Examples:
(%i1) expr: (a*d - b*c)/h^2 + sin(%pi*x);
a d - b c
(%o1) sin(%pi x) + ---------
2
h
(%i2) dpart (dpart (expr, 1, 1), 2, 2);
dpart: fell off the end.
-- an error. To debug this try: debugmode(true);
(%i3) expr2: lpart (BAR, lpart (FOO, %, 1), 2);
BAR""""""""
FOO""""""""" "a d - b c"
(%o3) "sin(%pi x)" + "---------"
"""""""""""" " 2 "
" h "
"""""""""""
(%i4) rembox (expr2, unlabelled);
BAR""""""""
FOO""""""""" "a d - b c"
(%o4) "sin(%pi x)" + "---------"
"""""""""""" " 2 "
" h "
"""""""""""
(%i5) rembox (expr2, FOO);
BAR""""""""
"a d - b c"
(%o5) sin(%pi x) + "---------"
" 2 "
" h "
"""""""""""
(%i6) rembox (expr2, BAR);
FOO""""""""" a d - b c
(%o6) "sin(%pi x)" + ---------
"""""""""""" 2
h
(%i7) rembox (expr2);
a d - b c
(%o7) sin(%pi x) + ---------
2
h
Categories: Expressions
Replaces parts of expr at the specified integer depth with descriptive summaries.
Sum(n)
where n is the number of operands of the sum.
Product(n)
where n is the number of operands of the product.
Expt.
Quotient.
Negterm.
List(n) where n ist the number of
elements of the list.
When depth is greater than or equal to the maximum depth of expr,
reveal (expr, depth) returns expr unmodified.
reveal evaluates its arguments.
reveal returns the summarized expression.
Example:
(%i1) e: expand ((a - b)^2)/expand ((exp(a) + exp(b))^2);
2 2
b - 2 a b + a
(%o1) -------------------------
b + a 2 b 2 a
2 %e + %e + %e
(%i2) reveal (e, 1);
(%o2) Quotient
(%i3) reveal (e, 2);
Sum(3)
(%o3) ------
Sum(3)
(%i4) reveal (e, 3);
Expt + Negterm + Expt
(%o4) ------------------------
Product(2) + Expt + Expt
(%i5) reveal (e, 4);
2 2
b - Product(3) + a
(%o5) ------------------------------------
Product(2) Product(2)
2 Expt + %e + %e
(%i6) reveal (e, 5);
2 2
b - 2 a b + a
(%o6) --------------------------
Sum(2) 2 b 2 a
2 %e + %e + %e
(%i7) reveal (e, 6);
2 2
b - 2 a b + a
(%o7) -------------------------
b + a 2 b 2 a
2 %e + %e + %e
Categories: Expressions · Display functions
Makes multiple parallel substitutions into an expression. list is a list of equations. The left hand side of the equations must be an atom.
The variable sublis_apply_lambda controls simplification after
sublis.
See also psubst for making parallel substitutions.
Example:
(%i1) sublis ([a=b, b=a], sin(a) + cos(b)); (%o1) sin(b) + cos(a)
Categories: Expressions
Default value: true
Controls whether lambda's substituted are applied in simplification after
sublis is used or whether you have to do an ev to get things to
apply. true means do the application.
Categories: Expressions
Default value: false
If true then the functions subst and psubst can substitute
a subscripted variable f[x] with a number, when only the symbol f
is given.
See also subst.
(%i1) subst(100,g,g[x]+2);
subst: cannot substitute 100 for operator g in expression g
x
-- an error. To debug this try: debugmode(true);
(%i2) subst(100,g,g[x]+2),subnumsimp:true;
(%o2) 102
Categories: Expressions
Substitutes a for b in c. b must be an atom or a
complete subexpression of c. For example, x+y+z is a complete
subexpression of 2*(x+y+z)/w while x+y is not. When b does
not have these characteristics, one may sometimes use substpart or
ratsubst (see below). Alternatively, if b is of the form
e/f then one could use subst (a*f, e, c) while if b is of
the form e^(1/f) then one could use subst (a^f, e, c). The
subst command also discerns the x^y in x^-y so that
subst (a, sqrt(x), 1/sqrt(x)) yields 1/a. a and b
may also be operators of an expression enclosed in double-quotes " or
they may be function names. If one wishes to substitute for the independent
variable in derivative forms then the at function (see below) should be
used.
subst is an alias for substitute.
The commands subst (eq_1, expr) or
subst ([eq_1, ..., eq_k], expr) are other permissible
forms. The eq_i are equations indicating substitutions to be made.
For each equation, the right side will be substituted for the left in the
expression expr. The equations are substituted in serial from left to
right in expr. See the functions sublis and psubst for
making parallel substitutions.
exptsubst if true permits substitutions
like y for %e^x in %e^(a*x) to take place.
When opsubst is false,
subst will not attempt to substitute into the operator of an expression.
E.g. (opsubst: false, subst (x^2, r, r+r[0])) will work.
Examples:
(%i1) subst (a, x+y, x + (x+y)^2 + y);
2
(%o1) y + x + a
(%i2) subst (-%i, %i, a + b*%i);
(%o2) a - %i b
The substitution is done in serial for a list of equations. Compare this with a parallel substitution:
(%i1) subst([a=b, b=c], a+b); (%o1) 2 c (%i2) sublis([a=b, b=c], a+b); (%o2) c + b
Single-character Operators like + and - have to be quoted in
order to be replaced by subst. It is to note, though, that a+b-c
might be expressed as a+b+(-1*c) internally.
(%i3) subst(["+"="-"],a+b-c); (%o3) c-b+a
For further examples, do example (subst).
Categories: Expressions
Similar to substpart, but substinpart works on the
internal representation of expr.
Examples:
(%i1) x . 'diff (f(x), x, 2);
2
d
(%o1) x . (--- (f(x)))
2
dx
(%i2) substinpart (d^2, %, 2);
2
(%o2) x . d
(%i3) substinpart (f1, f[1](x + 1), 0);
(%o3) f1(x + 1)
If the last argument to a part function is a list of indices then
several subexpressions are picked out, each one corresponding to an
index of the list. Thus
(%i1) part (x + y + z, [1, 3]); (%o1) z + x
piece holds the value of the last expression selected when using the
part functions. It is set during the execution of the function and
thus may be referred to in the function itself as shown below.
If partswitch is set to true then end is returned when a
selected part of an expression doesn't exist, otherwise an error
message is given.
(%i1) expr: 27*y^3 + 54*x*y^2 + 36*x^2*y + y + 8*x^3 + x + 1;
3 2 2 3
(%o1) 27 y + 54 x y + 36 x y + y + 8 x + x + 1
(%i2) part (expr, 2, [1, 3]);
2
(%o2) 54 y
(%i3) sqrt (piece/54);
(%o3) abs(y)
(%i4) substpart (factor (piece), expr, [1, 2, 3, 5]);
3
(%o4) (3 y + 2 x) + y + x + 1
(%i5) expr: 1/x + y/x - 1/z;
1 y 1
(%o5) (- -) + - + -
z x x
(%i6) substpart (xthru (piece), expr, [2, 3]);
y + 1 1
(%o6) ----- - -
x z
Also, setting the option inflag to true and calling part
or substpart is the same as calling inpart or substinpart.
Categories: Expressions
Substitutes x for the subexpression picked out by the rest of the
arguments as in part. It returns the new value of expr. x
may be some operator to be substituted for an operator of expr. In some
cases x needs to be enclosed in double-quotes " (e.g.
substpart ("+", a*b, 0) yields b + a).
Example:
(%i1) 1/(x^2 + 2);
1
(%o1) ------
2
x + 2
(%i2) substpart (3/2, %, 2, 1, 2);
1
(%o2) --------
3/2
x + 2
(%i3) a*x + f(b, y);
(%o3) a x + f(b, y)
(%i4) substpart ("+", %, 1, 0);
(%o4) x + f(b, y) + a
Also, setting the option inflag to true and calling part
or substpart is the same as calling inpart or
substinpart.
Categories: Expressions
Returns true if expr is a symbol, else false. In effect,
symbolp(x) is equivalent to the predicate
atom(x) and not numberp(x).
See also Identifiers.
Categories: Predicate functions
Disables the aliasing created by the last use of the ordering commands
ordergreat and orderless. ordergreat and orderless
may not be used more than one time each without calling unorder.
unorder does not substitute back in expressions the original symbols for
the aliases introduced by ordergreat and orderless. Therefore,
after execution of unorder the aliases appear in previous expressions.
See also ordergreat and orderless.
Examples:
ordergreat(a) introduces an alias for the symbol a. Therefore,
the difference of %o2 and %o4 does not vanish. unorder
does not substitute back the symbol a and the alias appears in the
output %o7.
(%i1) unorder();
(%o1) []
(%i2) b*x + a^2;
2
(%o2) b x + a
(%i3) ordergreat (a);
(%o3) done
(%i4) b*x + a^2;
%th(1) - %th(3);
2
(%o4) a + b x
(%i5) unorder();
2 2
(%o5) a - a
(%i6) %th(2);
(%o6) [a]
Categories: Expressions
Returns the verb form of the function name f.
See also verb, noun, and nounify.
Examples:
(%i1) verbify ('foo);
(%o1) foo
(%i2) :lisp $%
$FOO
(%i2) nounify (foo);
(%o2) foo
(%i3) :lisp $%
%FOO
Categories: Nouns and verbs
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| 7.1 Introduction to operators | ||
| 7.2 Arithmetic operators | ||
| 7.3 Relational operators | ||
| 7.4 Logical operators | ||
| 7.5 Operators for Equations | ||
| 7.6 Assignment operators | ||
| 7.7 User defined operators |
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It is possible to define new operators with specified precedence, to undefine existing operators, or to redefine the precedence of existing operators. An operator may be unary prefix or unary postfix, binary infix, n-ary infix, matchfix, or nofix. "Matchfix" means a pair of symbols which enclose their argument or arguments, and "nofix" means an operator which takes no arguments. As examples of the different types of operators, there are the following.
negation - a
factorial a!
exponentiation a^b
addition a + b
list construction [a, b]
(There are no built-in nofix operators; for an example of such an operator,
see nofix.)
The mechanism to define a new operator is straightforward. It is only necessary to declare a function as an operator; the operator function might or might not be defined.
An example of user-defined operators is the following. Note that the explicit
function call "dd" (a) is equivalent to dd a, likewise
"<-" (a, b) is equivalent to a <- b. Note also that the functions
"dd" and "<-" are undefined in this example.
(%i1) prefix ("dd");
(%o1) dd
(%i2) dd a;
(%o2) dd a
(%i3) "dd" (a);
(%o3) dd a
(%i4) infix ("<-");
(%o4) <-
(%i5) a <- dd b;
(%o5) a <- dd b
(%i6) "<-" (a, "dd" (b));
(%o6) a <- dd b
The Maxima functions which define new operators are summarized in this table, stating the default left and right binding powers (lbp and rbp, respectively). (Binding power determines operator precedence. However, since left and right binding powers can differ, binding power is somewhat more complicated than precedence.) Some of the operation definition functions take additional arguments; see the function descriptions for details.
prefixrbp=180
postfixlbp=180
infixlbp=180, rbp=180
narylbp=180, rbp=180
matchfix(binding power not applicable)
nofix(binding power not applicable)
For comparison, here are some built-in operators and their left and right binding powers.
Operator lbp rbp : 180 20 :: 180 20 := 180 20 ::= 180 20 ! 160 !! 160 ^ 140 139 . 130 129 * 120 / 120 120 + 100 100 - 100 134 = 80 80 # 80 80 > 80 80 >= 80 80 < 80 80 <= 80 80 not 70 and 65 or 60 , 10 $ -1 ; -1
remove and kill remove operator properties from an atom.
remove ("a", op) removes only the operator properties of a.
kill ("a") removes all properties of a, including the
operator properties. Note that the name of the operator must be enclosed in
quotation marks.
(%i1) infix ("##");
(%o1) ##
(%i2) "##" (a, b) := a^b;
b
(%o2) a ## b := a
(%i3) 5 ## 3;
(%o3) 125
(%i4) remove ("##", op);
(%o4) done
(%i5) 5 ## 3;
Incorrect syntax: # is not a prefix operator
5 ##
^
(%i5) "##" (5, 3);
(%o5) 125
(%i6) infix ("##");
(%o6) ##
(%i7) 5 ## 3;
(%o7) 125
(%i8) kill ("##");
(%o8) done
(%i9) 5 ## 3;
Incorrect syntax: # is not a prefix operator
5 ##
^
(%i9) "##" (5, 3);
(%o9) ##(5, 3)
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The symbols + * / and ^ represent addition,
multiplication, division, and exponentiation, respectively. The names of these
operators are "+" "*" "/" and "^", which may appear
where the name of a function or operator is required.
The symbols + and - represent unary addition and negation,
respectively, and the names of these operators are "+" and "-",
respectively.
Subtraction a - b is represented within Maxima as addition,
a + (- b). Expressions such as a + (- b) are displayed as
subtraction. Maxima recognizes "-" only as the name of the unary
negation operator, and not as the name of the binary subtraction operator.
Division a / b is represented within Maxima as multiplication,
a * b^(- 1). Expressions such as a * b^(- 1) are displayed as
division. Maxima recognizes "/" as the name of the division operator.
Addition and multiplication are n-ary, commutative operators. Division and exponentiation are binary, noncommutative operators.
Maxima sorts the operands of commutative operators to construct a canonical
representation. For internal storage, the ordering is determined by
orderlessp. For display, the ordering for addition is determined by
ordergreatp, and for multiplication, it is the same as the internal
ordering.
Arithmetic computations are carried out on literal numbers (integers, rationals,
ordinary floats, and bigfloats). Except for exponentiation, all arithmetic
operations on numbers are simplified to numbers. Exponentiation is simplified
to a number if either operand is an ordinary float or bigfloat or if the result
is an exact integer or rational; otherwise an exponentiation may be simplified
to sqrt or another exponentiation or left unchanged.
Floating-point contagion applies to arithmetic computations: if any operand is a bigfloat, the result is a bigfloat; otherwise, if any operand is an ordinary float, the result is an ordinary float; otherwise, the operands are rationals or integers and the result is a rational or integer.
Arithmetic computations are a simplification, not an evaluation. Thus arithmetic is carried out in quoted (but simplified) expressions.
Arithmetic operations are applied element-by-element to lists when the global
flag listarith is true, and always applied element-by-element to
matrices. When one operand is a list or matrix and another is an operand of
some other type, the other operand is combined with each of the elements of the
list or matrix.
Examples:
Addition and multiplication are n-ary, commutative operators.
Maxima sorts the operands to construct a canonical representation.
The names of these operators are "+" and "*".
(%i1) c + g + d + a + b + e + f;
(%o1) g + f + e + d + c + b + a
(%i2) [op (%), args (%)];
(%o2) [+, [g, f, e, d, c, b, a]]
(%i3) c * g * d * a * b * e * f;
(%o3) a b c d e f g
(%i4) [op (%), args (%)];
(%o4) [*, [a, b, c, d, e, f, g]]
(%i5) apply ("+", [a, 8, x, 2, 9, x, x, a]);
(%o5) 3 x + 2 a + 19
(%i6) apply ("*", [a, 8, x, 2, 9, x, x, a]);
2 3
(%o6) 144 a x
Division and exponentiation are binary, noncommutative operators.
The names of these operators are "/" and "^".
(%i1) [a / b, a ^ b];
a b
(%o1) [-, a ]
b
(%i2) [map (op, %), map (args, %)];
(%o2) [[/, ^], [[a, b], [a, b]]]
(%i3) [apply ("/", [a, b]), apply ("^", [a, b])];
a b
(%o3) [-, a ]
b
Subtraction and division are represented internally in terms of addition and multiplication, respectively.
(%i1) [inpart (a - b, 0), inpart (a - b, 1), inpart (a - b, 2)];
(%o1) [+, a, - b]
(%i2) [inpart (a / b, 0), inpart (a / b, 1), inpart (a / b, 2)];
1
(%o2) [*, a, -]
b
Computations are carried out on literal numbers. Floating-point contagion applies.
(%i1) 17 + b - (1/2)*29 + 11^(2/4);
5
(%o1) b + sqrt(11) + -
2
(%i2) [17 + 29, 17 + 29.0, 17 + 29b0];
(%o2) [46, 46.0, 4.6b1]
Arithmetic computations are a simplification, not an evaluation.
(%i1) simp : false;
(%o1) false
(%i2) '(17 + 29*11/7 - 5^3);
29 11 3
(%o2) 17 + ----- - 5
7
(%i3) simp : true;
(%o3) true
(%i4) '(17 + 29*11/7 - 5^3);
437
(%o4) - ---
7
Arithmetic is carried out element-by-element for lists (depending on
listarith) and matrices.
(%i1) matrix ([a, x], [h, u]) - matrix ([1, 2], [3, 4]);
[ a - 1 x - 2 ]
(%o1) [ ]
[ h - 3 u - 4 ]
(%i2) 5 * matrix ([a, x], [h, u]);
[ 5 a 5 x ]
(%o2) [ ]
[ 5 h 5 u ]
(%i3) listarith : false;
(%o3) false
(%i4) [a, c, m, t] / [1, 7, 2, 9];
[a, c, m, t]
(%o4) ------------
[1, 7, 2, 9]
(%i5) [a, c, m, t] ^ x;
x
(%o5) [a, c, m, t]
(%i6) listarith : true;
(%o6) true
(%i7) [a, c, m, t] / [1, 7, 2, 9];
c m t
(%o7) [a, -, -, -]
7 2 9
(%i8) [a, c, m, t] ^ x;
x x x x
(%o8) [a , c , m , t ]
Categories: Operators
Exponentiation operator.
Maxima recognizes ** as the same operator as ^ in input,
and it is displayed as ^ in 1-dimensional output,
or by placing the exponent as a superscript in 2-dimensional output.
The fortran function displays the exponentiation operator as **,
whether it was input as ** or ^.
Examples:
(%i1) is (a**b = a^b);
(%o1) true
(%i2) x**y + x^z;
z y
(%o2) x + x
(%i3) string (x**y + x^z);
(%o3) x^z+x^y
(%i4) fortran (x**y + x^z);
x**z+x**y
(%o4) done
Categories: Operators
Noncommutative exponentiation operator.
^^ is the exponentiation operator corresponding to noncommutative
multiplication ., just as the ordinary exponentiation operator ^
corresponds to commutative multiplication *.
Noncommutative exponentiation is displayed by ^^ in 1-dimensional output,
and by placing the exponent as a superscript within angle brackets < >
in 2-dimensional output.
Examples:
(%i1) a . a . b . b . b + a * a * a * b * b;
3 2 <2> <3>
(%o1) a b + a . b
(%i2) string (a . a . b . b . b + a * a * a * b * b);
(%o2) a^3*b^2+a^^2 . b^^3
Categories: Operators
The dot operator, for matrix (non-commutative) multiplication.
When "." is used in this way, spaces should be left on both sides of
it, e.g. A . B This distinguishes it plainly from a decimal point in
a floating point number.
See also
Dot,
dot0nscsimp,
dot0simp,
dot1simp,
dotassoc,
dotconstrules,
dotdistrib,
dotexptsimp,
dotident,
and
dotscrules.
Categories: Operators
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The symbols < <= >= and > represent less than, less
than or equal, greater than or equal, and greater than, respectively. The names
of these operators are "<" "<=" ">=" and ">", which
may appear where the name of a function or operator is required.
These relational operators are all binary operators; constructs such as
a < b < c are not recognized by Maxima.
Relational expressions are evaluated to Boolean values by the functions
is and maybe, and the programming constructs
if, while, and unless. Relational expressions
are not otherwise evaluated or simplified to Boolean values, although the
arguments of relational expressions are evaluated (when evaluation is not
otherwise prevented by quotation).
When a relational expression cannot be evaluated to true or false,
the behavior of is and if are governed by the global flag
prederror. When prederror is true, is and
if trigger an error. When prederror is false, is
returns unknown, and if returns a partially-evaluated conditional
expression.
maybe always behaves as if prederror were false, and
while and unless always behave as if prederror were
true.
Relational operators do not distribute over lists or other aggregates.
See also =, #, equal, and notequal.
Examples:
Relational expressions are evaluated to Boolean values by some functions and programming constructs.
(%i1) [x, y, z] : [123, 456, 789];
(%o1) [123, 456, 789]
(%i2) is (x < y);
(%o2) true
(%i3) maybe (y > z);
(%o3) false
(%i4) if x >= z then 1 else 0;
(%o4) 0
(%i5) block ([S], S : 0, for i:1 while i <= 100 do S : S + i,
return (S));
(%o5) 5050
Relational expressions are not otherwise evaluated or simplified to Boolean values, although the arguments of relational expressions are evaluated.
(%o1) [123, 456, 789] (%i2) [x < y, y <= z, z >= y, y > z]; (%o2) [123 < 456, 456 <= 789, 789 >= 456, 456 > 789] (%i3) map (is, %); (%o3) [true, true, true, false]
Categories: Operators
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The logical conjunction operator. and is an n-ary infix operator;
its operands are Boolean expressions, and its result is a Boolean value.
and forces evaluation (like is) of one or more operands,
and may force evaluation of all operands.
Operands are evaluated in the order in which they appear. and evaluates
only as many of its operands as necessary to determine the result. If any
operand is false, the result is false and no further operands are
evaluated.
The global flag prederror governs the behavior of and when an
evaluated operand cannot be determined to be true or false.
and prints an error message when prederror is true.
Otherwise, operands which do not evaluate to true or false are
accepted, and the result is a Boolean expression.
and is not commutative: a and b might not be equal to
b and a due to the treatment of indeterminate operands.
Categories: Operators
The logical negation operator. not is a prefix operator;
its operand is a Boolean expression, and its result is a Boolean value.
not forces evaluation (like is) of its operand.
The global flag prederror governs the behavior of not when its
operand cannot be determined to be true or false. not
prints an error message when prederror is true. Otherwise,
operands which do not evaluate to true or false are accepted,
and the result is a Boolean expression.
Categories: Operators
The logical disjunction operator. or is an n-ary infix operator;
its operands are Boolean expressions, and its result is a Boolean value.
or forces evaluation (like is) of one or more operands,
and may force evaluation of all operands.
Operands are evaluated in the order in which they appear. or evaluates
only as many of its operands as necessary to determine the result. If any
operand is true, the result is true and no further operands are
evaluated.
The global flag prederror governs the behavior of or when an
evaluated operand cannot be determined to be true or false.
or prints an error message when prederror is true.
Otherwise, operands which do not evaluate to true or false are
accepted, and the result is a Boolean expression.
or is not commutative: a or b might not be equal to b or a
due to the treatment of indeterminate operands.
Categories: Operators
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Represents the negation of syntactic equality =.
Note that because of the rules for evaluation of predicate expressions
(in particular because not expr causes evaluation of expr),
not a = b is equivalent to is(a # b),
instead of a # b.
Examples:
(%i1) a = b; (%o1) a = b (%i2) is (a = b); (%o2) false (%i3) a # b; (%o3) a # b (%i4) not a = b; (%o4) true (%i5) is (a # b); (%o5) true (%i6) is (not a = b); (%o6) true
Categories: Operators
The equation operator.
An expression a = b, by itself, represents an unevaluated
equation, which might or might not hold. Unevaluated equations may appear as
arguments to solve and algsys or some other functions.
The function is evaluates = to a Boolean value.
is(a = b) evaluates a = b to true
when a and b are identical. That is, a and b are atoms
which are identical, or they are not atoms and their operators are identical and
their arguments are identical. Otherwise, is(a = b)
evaluates to false; it never evaluates to unknown. When
is(a = b) is true, a and b are said to be
syntactically equal, in contrast to equivalent expressions, for which
is(equal(a, b)) is true. Expressions can be
equivalent and not syntactically equal.
The negation of = is represented by #.
As with =, an expression a # b, by itself, is not
evaluated. is(a # b) evaluates a # b to
true or false.
In addition to is, some other operators evaluate = and #
to true or false, namely if, and,
or, and not.
Note that because of the rules for evaluation of predicate expressions
(in particular because not expr causes evaluation of expr),
not a = b is equivalent to is(a # b),
instead of a # b.
rhs and lhs return the right-hand and left-hand sides,
respectively, of an equation or inequation.
Examples:
An expression a = b, by itself, represents
an unevaluated equation, which might or might not hold.
(%i1) eq_1 : a * x - 5 * y = 17;
(%o1) a x - 5 y = 17
(%i2) eq_2 : b * x + 3 * y = 29;
(%o2) 3 y + b x = 29
(%i3) solve ([eq_1, eq_2], [x, y]);
196 29 a - 17 b
(%o3) [[x = ---------, y = -----------]]
5 b + 3 a 5 b + 3 a
(%i4) subst (%, [eq_1, eq_2]);
196 a 5 (29 a - 17 b)
(%o4) [--------- - --------------- = 17,
5 b + 3 a 5 b + 3 a
196 b 3 (29 a - 17 b)
--------- + --------------- = 29]
5 b + 3 a 5 b + 3 a
(%i5) ratsimp (%);
(%o5) [17 = 17, 29 = 29]
is(a = b) evaluates a = b to true
when a and b are syntactically equal (that is, identical).
Expressions can be equivalent and not syntactically equal.
(%i1) a : (x + 1) * (x - 1);
(%o1) (x - 1) (x + 1)
(%i2) b : x^2 - 1;
2
(%o2) x - 1
(%i3) [is (a = b), is (a # b)];
(%o3) [false, true]
(%i4) [is (equal (a, b)), is (notequal (a, b))];
(%o4) [true, false]
Some operators evaluate = and # to true or false.
(%i1) if expand ((x + y)^2) = x^2 + 2 * x * y + y^2 then FOO else
BAR;
(%o1) FOO
(%i2) eq_3 : 2 * x = 3 * x;
(%o2) 2 x = 3 x
(%i3) eq_4 : exp (2) = %e^2;
2 2
(%o3) %e = %e
(%i4) [eq_3 and eq_4, eq_3 or eq_4, not eq_3];
(%o4) [false, true, true]
Because not expr causes evaluation of expr,
not a = b is equivalent to is(a # b).
(%i1) [2 * x # 3 * x, not (2 * x = 3 * x)]; (%o1) [2 x # 3 x, true] (%i2) is (2 * x # 3 * x); (%o2) true
Categories: Operators
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Assignment operator.
When the left-hand side is a simple variable (not subscripted), :
evaluates its right-hand side and associates that value with the left-hand side.
When the left-hand side is a subscripted element of a list, matrix, declared Maxima array, or Lisp array, the right-hand side is assigned to that element. The subscript must name an existing element; such objects cannot be extended by naming nonexistent elements.
When the left-hand side is a subscripted element of an undeclared Maxima array, the right-hand side is assigned to that element, if it already exists, or a new element is allocated, if it does not already exist.
When the left-hand side is a list of simple and/or subscripted variables, the right-hand side must evaluate to a list, and the elements of the right-hand side are assigned to the elements of the left-hand side, in parallel.
See also kill and remvalue, which undo the association between
the left-hand side and its value.
Examples:
Assignment to a simple variable.
(%i1) a; (%o1) a (%i2) a : 123; (%o2) 123 (%i3) a; (%o3) 123
Assignment to an element of a list.
(%i1) b : [1, 2, 3]; (%o1) [1, 2, 3] (%i2) b[3] : 456; (%o2) 456 (%i3) b; (%o3) [1, 2, 456]
Assignment creates an undeclared array.
(%i1) c[99] : 789; (%o1) 789 (%i2) c[99]; (%o2) 789 (%i3) c; (%o3) c (%i4) arrayinfo (c); (%o4) [hashed, 1, [99]] (%i5) listarray (c); (%o5) [789]
Multiple assignment.
(%i1) [a, b, c] : [45, 67, 89]; (%o1) [45, 67, 89] (%i2) a; (%o2) 45 (%i3) b; (%o3) 67 (%i4) c; (%o4) 89
Multiple assignment is carried out in parallel.
The values of a and b are exchanged in this example.
(%i1) [a, b] : [33, 55]; (%o1) [33, 55] (%i2) [a, b] : [b, a]; (%o2) [55, 33] (%i3) a; (%o3) 55 (%i4) b; (%o4) 33
Categories: Evaluation · Operators
Assignment operator.
:: is the same as : (which see) except that :: evaluates
its left-hand side as well as its right-hand side.
Examples:
(%i1) x : 'foo; (%o1) foo (%i2) x :: 123; (%o2) 123 (%i3) foo; (%o3) 123 (%i4) x : '[a, b, c]; (%o4) [a, b, c] (%i5) x :: [11, 22, 33]; (%o5) [11, 22, 33] (%i6) a; (%o6) 11 (%i7) b; (%o7) 22 (%i8) c; (%o8) 33
Categories: Evaluation · Operators
Macro function definition operator.
::= defines a function (called a "macro" for historical reasons) which
quotes its arguments, and the expression which it returns (called the "macro
expansion") is evaluated in the context from which the macro was called.
A macro function is otherwise the same as an ordinary function.
macroexpand returns a macro expansion (without evaluating it).
macroexpand (foo (x)) followed by ''% is equivalent to
foo (x) when foo is a macro function.
::= puts the name of the new macro function onto the global list
macros. kill, remove, and remfunction
unbind macro function definitions and remove names from macros.
fundef or dispfun return a macro function definition or assign it
to a label, respectively.
Macro functions commonly contain buildq and splice expressions to
construct an expression, which is then evaluated.
Examples
A macro function quotes its arguments, so message (1) shows y - z, not
the value of y - z. The macro expansion (the quoted expression
'(print ("(2) x is equal to", x))) is evaluated in the context from which
the macro was called, printing message (2).
(%i1) x: %pi$
(%i2) y: 1234$
(%i3) z: 1729 * w$
(%i4) printq1 (x) ::= block (print ("(1) x is equal to", x),
'(print ("(2) x is equal to", x)))$
(%i5) printq1 (y - z);
(1) x is equal to y - z
(2) x is equal to %pi
(%o5) %pi
An ordinary function evaluates its arguments, so message (1) shows the value of
y - z. The return value is not evaluated, so message (2) is not printed
until the explicit evaluation ''%.
(%i1) x: %pi$
(%i2) y: 1234$
(%i3) z: 1729 * w$
(%i4) printe1 (x) := block (print ("(1) x is equal to", x),
'(print ("(2) x is equal to", x)))$
(%i5) printe1 (y - z);
(1) x is equal to 1234 - 1729 w
(%o5) print((2) x is equal to, x)
(%i6) ''%;
(2) x is equal to %pi
(%o6) %pi
macroexpand returns a macro expansion.
macroexpand (foo (x)) followed by ''% is equivalent to
foo (x) when foo is a macro function.
(%i1) x: %pi$
(%i2) y: 1234$
(%i3) z: 1729 * w$
(%i4) g (x) ::= buildq ([x], print ("x is equal to", x))$
(%i5) macroexpand (g (y - z));
(%o5) print(x is equal to, y - z)
(%i6) ''%;
x is equal to 1234 - 1729 w
(%o6) 1234 - 1729 w
(%i7) g (y - z);
x is equal to 1234 - 1729 w
(%o7) 1234 - 1729 w
Categories: Function definition · Operators
The function definition operator.
f(x_1, ..., x_n) := expr defines a function named
f with arguments x_1, …, x_n and function body
expr. := never evaluates the function body (unless explicitly
evaluated by quote-quote '').
The function body is evaluated every time the function is called.
f[x_1, ..., x_n] := expr defines a so-called array function.
Its function body is evaluated just once for each distinct value of its arguments,
and that value is returned, without evaluating the function body,
whenever the arguments have those values again.
(A function of this kind is commonly known as a "memoizing function".)
f[x_1, ..., x_n](y_1, ..., y_m) := expr
is a special case of an array function.
f[x_1, ..., x_n] is an array function which returns a lambda expression
with arguments y_1, ..., y_m.
The function body is evaluated once for each distinct value of x_1, ..., x_n,
and the body of the lambda expression is that value.
When the last or only function argument x_n is a list of one element, the
function defined by := accepts a variable number of arguments. Actual
arguments are assigned one-to-one to formal arguments x_1, …,
x_(n - 1), and any further actual arguments, if present, are assigned to
x_n as a list.
All function definitions appear in the same namespace; defining a function
f within another function g does not automatically limit the scope
of f to g. However, local(f) makes the definition of
function f effective only within the block or other compound expression
in which local appears.
If some formal argument x_k is a quoted symbol, the function defined by
:= does not evaluate the corresponding actual argument. Otherwise all
actual arguments are evaluated.
Examples:
:= never evaluates the function body (unless explicitly evaluated by
quote-quote).
(%i1) expr : cos(y) - sin(x); (%o1) cos(y) - sin(x) (%i2) F1 (x, y) := expr; (%o2) F1(x, y) := expr (%i3) F1 (a, b); (%o3) cos(y) - sin(x) (%i4) F2 (x, y) := ''expr; (%o4) F2(x, y) := cos(y) - sin(x) (%i5) F2 (a, b); (%o5) cos(b) - sin(a)
f(x_1, ..., x_n) := ... defines an ordinary function.
(%i1) G1(x, y) := (print ("Evaluating G1 for x=", x, "and y=", y), x.y - y.x);
(%o1) G1(x, y) := (print("Evaluating G1 for x=", x, "and y=",
y), x . y - y . x)
(%i2) G1([1, a], [2, b]);
Evaluating G1 for x= [1, a] and y= [2, b]
(%o2) 0
(%i3) G1([1, a], [2, b]);
Evaluating G1 for x= [1, a] and y= [2, b]
(%o3) 0
f[x_1, ..., x_n] := ... defines an array function.
(%i1) G2[a] := (print ("Evaluating G2 for a=", a), a^2);
2
(%o1) G2 := (print("Evaluating G2 for a=", a), a )
a
(%i2) G2[1234];
Evaluating G2 for a= 1234
(%o2) 1522756
(%i3) G2[1234];
(%o3) 1522756
(%i4) G2[2345];
Evaluating G2 for a= 2345
(%o4) 5499025
(%i5) arrayinfo (G2);
(%o5) [hashed, 1, [1234], [2345]]
(%i6) listarray (G2);
(%o6) [1522756, 5499025]
f[x_1, ..., x_n](y_1, ..., y_m) := expr
is a special case of an array function.
(%i1) G3[n](x) := (print ("Evaluating G3 for n=", n), diff (sin(x)^2, x, n));
(%o1) G3 (x) := (print("Evaluating G3 for n=", n),
n
2
diff(sin (x), x, n))
(%i2) G3[2];
Evaluating G3 for n= 2
2 2
(%o2) lambda([x], 2 cos (x) - 2 sin (x))
(%i3) G3[2];
2 2
(%o3) lambda([x], 2 cos (x) - 2 sin (x))
(%i4) G3[2](1);
2 2
(%o4) 2 cos (1) - 2 sin (1)
(%i5) arrayinfo (G3);
(%o5) [hashed, 1, [2]]
(%i6) listarray (G3);
2 2
(%o6) [lambda([x], 2 cos (x) - 2 sin (x))]
When the last or only function argument x_n is a list of one element,
the function defined by := accepts a variable number of arguments.
(%i1) H ([L]) := apply ("+", L);
(%o1) H([L]) := apply("+", L)
(%i2) H (a, b, c);
(%o2) c + b + a
local makes a local function definition.
(%i1) foo (x) := 1 - x; (%o1) foo(x) := 1 - x (%i2) foo (100); (%o2) - 99 (%i3) block (local (foo), foo (x) := 2 * x, foo (100)); (%o3) 200 (%i4) foo (100); (%o4) - 99
Categories: Function definition · Operators
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Declares op to be an infix operator. An infix operator is a function of
two arguments, with the name of the function written between the arguments.
For example, the subtraction operator - is an infix operator.
infix (op) declares op to be an infix operator with default
binding powers (left and right both equal to 180) and parts of speech (left and
right both equal to any).
infix (op, lbp, rbp) declares op to be an infix
operator with stated left and right binding powers and default parts of speech
(left and right both equal to any).
infix (op, lbp, rbp, lpos, rpos, pos)
declares op to be an infix operator with stated left and right binding
powers and parts of speech lpos, rpos, and pos for the left
operand, the right operand, and the operator result, respectively.
"Part of speech", in reference to operator declarations, means expression type.
Three types are recognized: expr, clause, and any,
indicating an algebraic expression, a Boolean expression, or any kind of
expression, respectively. Maxima can detect some syntax errors by comparing the
declared part of speech to an actual expression.
The precedence of op with respect to other operators derives from the left and right binding powers of the operators in question. If the left and right binding powers of op are both greater the left and right binding powers of some other operator, then op takes precedence over the other operator. If the binding powers are not both greater or less, some more complicated relation holds.
The associativity of op depends on its binding powers. Greater left binding power (lbp) implies an instance of op is evaluated before other operators to its left in an expression, while greater right binding power (rbp) implies an instance of op is evaluated before other operators to its right in an expression. Thus greater lbp makes op right-associative, while greater rbp makes op left-associative. If lbp is equal to rbp, op is left-associative.
See also Introduction to operators.
Examples:
If the left and right binding powers of op are both greater the left and right binding powers of some other operator, then op takes precedence over the other operator.
(%i1) :lisp (get '$+ 'lbp)
100
(%i1) :lisp (get '$+ 'rbp)
100
(%i1) infix ("##", 101, 101);
(%o1) ##
(%i2) "##"(a, b) := sconcat("(", a, ",", b, ")");
(%o2) (a ## b) := sconcat("(", a, ",", b, ")")
(%i3) 1 + a ## b + 2;
(%o3) (a,b) + 3
(%i4) infix ("##", 99, 99);
(%o4) ##
(%i5) 1 + a ## b + 2;
(%o5) (a+1,b+2)
Greater lbp makes op right-associative, while greater rbp makes op left-associative.
(%i1) infix ("##", 100, 99);
(%o1) ##
(%i2) "##"(a, b) := sconcat("(", a, ",", b, ")")$
(%i3) foo ## bar ## baz;
(%o3) (foo,(bar,baz))
(%i4) infix ("##", 100, 101);
(%o4) ##
(%i5) foo ## bar ## baz;
(%o5) ((foo,bar),baz)
Maxima can detect some syntax errors by comparing the declared part of speech to an actual expression.
(%i1) infix ("##", 100, 99, expr, expr, expr);
(%o1) ##
(%i2) if x ## y then 1 else 0;
Incorrect syntax: Found algebraic expression where logical
expression expected
if x ## y then
^
(%i2) infix ("##", 100, 99, expr, expr, clause);
(%o2) ##
(%i3) if x ## y then 1 else 0;
(%o3) if x ## y then 1 else 0
Categories: Operators · Declarations and inferences · Syntax
Declares a matchfix operator with left and right delimiters ldelimiter and rdelimiter. The delimiters are specified as strings.
A "matchfix" operator is a function of any number of arguments,
such that the arguments occur between matching left and right delimiters.
The delimiters may be any strings, so long as the parser can
distinguish the delimiters from the operands
and other expressions and operators.
In practice this rules out unparseable delimiters such as
%, ,, $ and ;,
and may require isolating the delimiters with white space.
The right delimiter can be the same or different from the left delimiter.
A left delimiter can be associated with only one right delimiter; two different matchfix operators cannot have the same left delimiter.
An existing operator may be redeclared as a matchfix operator
without changing its other properties.
In particular, built-in operators such as addition + can
be declared matchfix,
but operator functions cannot be defined for built-in operators.
The command matchfix (ldelimiter, rdelimiter, arg_pos,
pos) declares the argument part-of-speech arg_pos and result
part-of-speech pos, and the delimiters ldelimiter and
rdelimiter.
"Part of speech", in reference to operator declarations, means expression type.
Three types are recognized: expr, clause, and any,
indicating an algebraic expression, a Boolean expression, or any kind of
expression, respectively.
Maxima can detect some syntax errors by comparing the
declared part of speech to an actual expression.
The function to carry out a matchfix operation is an ordinary
user-defined function.
The operator function is defined
in the usual way
with the function definition operator := or define.
The arguments may be written between the delimiters,
or with the left delimiter as a quoted string and the arguments
following in parentheses.
dispfun (ldelimiter) displays the function definition.
The only built-in matchfix operator is the list constructor [ ].
Parentheses ( ) and double-quotes " "
act like matchfix operators,
but are not treated as such by the Maxima parser.
matchfix evaluates its arguments.
matchfix returns its first argument, ldelimiter.
Examples:
Delimiters may be almost any strings.
(%i1) matchfix ("@@", "~");
(%o1) @@
(%i2) @@ a, b, c ~;
(%o2) @@a, b, c~
(%i3) matchfix (">>", "<<");
(%o3) >>
(%i4) >> a, b, c <<;
(%o4) >>a, b, c<<
(%i5) matchfix ("foo", "oof");
(%o5) foo
(%i6) foo a, b, c oof;
(%o6) fooa, b, coof
(%i7) >> w + foo x, y oof + z << / @@ p, q ~;
>>z + foox, yoof + w<<
(%o7) ----------------------
@@p, q~
Matchfix operators are ordinary user-defined functions.
(%i1) matchfix ("!-", "-!");
(%o1) "!-"
(%i2) !- x, y -! := x/y - y/x;
x y
(%o2) !-x, y-! := - - -
y x
(%i3) define (!-x, y-!, x/y - y/x);
x y
(%o3) !-x, y-! := - - -
y x
(%i4) define ("!-" (x, y), x/y - y/x);
x y
(%o4) !-x, y-! := - - -
y x
(%i5) dispfun ("!-");
x y
(%t5) !-x, y-! := - - -
y x
(%o5) done
(%i6) !-3, 5-!;
16
(%o6) - --
15
(%i7) "!-" (3, 5);
16
(%o7) - --
15
An nary operator is used to denote a function of any number of arguments,
each of which is separated by an occurrence of the operator, e.g. A+B or A+B+C.
The nary("x") function is a syntax extension function to declare x
to be an nary operator. Functions may be declared to be nary. If
declare(j,nary); is done, this tells the simplifier to simplify, e.g.
j(j(a,b),j(c,d)) to j(a, b, c, d).
See also Introduction to operators.
nofix operators are used to denote functions of no arguments.
The mere presence of such an operator in a command will cause the
corresponding function to be evaluated. For example, when one types
"exit;" to exit from a Maxima break, "exit" is behaving similar to a
nofix operator. The function nofix("x") is a syntax extension
function which declares x to be a nofix operator.
See also Introduction to operators.
postfix operators like the prefix variety denote functions of a
single argument, but in this case the argument immediately precedes an
occurrence of the operator in the input string, e.g. 3!. The
postfix("x") function is a syntax extension function to declare x
to be a postfix operator.
See also Introduction to operators.
A prefix operator is one which signifies a function of one argument,
which argument immediately follows an occurrence of the operator.
prefix("x") is a syntax extension function to declare x to be a
prefix operator.
See also Introduction to operators.
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| 8.1 Functions and Variables for Evaluation |
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The single quote operator ' prevents evaluation.
Applied to a symbol, the single quote prevents evaluation of the symbol.
Applied to a function call, the single quote prevents evaluation of the function call, although the arguments of the function are still evaluated (if evaluation is not otherwise prevented). The result is the noun form of the function call.
Applied to a parenthesized expression, the single quote prevents evaluation of
all symbols and function calls in the expression.
E.g., '(f(x)) means do not evaluate the expression f(x).
'f(x) (with the single quote applied to f instead of f(x))
means return the noun form of f applied to [x].
The single quote does not prevent simplification.
When the global flag noundisp is true, nouns display with a single
quote. This switch is always true when displaying function definitions.
See also the quote-quote operator '' and nouns.
Examples:
Applied to a symbol, the single quote prevents evaluation of the symbol.
(%i1) aa: 1024;
(%o1) 1024
(%i2) aa^2;
(%o2) 1048576
(%i3) 'aa^2;
2
(%o3) aa
(%i4) ''%;
(%o4) 1048576
Applied to a function call, the single quote prevents evaluation of the function call. The result is the noun form of the function call.
(%i1) x0: 5;
(%o1) 5
(%i2) x1: 7;
(%o2) 7
(%i3) integrate (x^2, x, x0, x1);
218
(%o3) ---
3
(%i4) 'integrate (x^2, x, x0, x1);
7
/
[ 2
(%o4) I x dx
]
/
5
(%i5) %, nouns;
218
(%o5) ---
3
Applied to a parenthesized expression, the single quote prevents evaluation of all symbols and function calls in the expression.
(%i1) aa: 1024; (%o1) 1024 (%i2) bb: 19; (%o2) 19 (%i3) sqrt(aa) + bb; (%o3) 51 (%i4) '(sqrt(aa) + bb); (%o4) bb + sqrt(aa) (%i5) ''%; (%o5) 51
The single quote does not prevent simplification.
(%i1) sin (17 * %pi) + cos (17 * %pi); (%o1) - 1 (%i2) '(sin (17 * %pi) + cos (17 * %pi)); (%o2) - 1
Maxima considers floating point operations by its in-built mathematical functions to be a simplification.
(%i1) sin(1.0); (%o1) .8414709848078965 (%i2) '(sin(1.0)); (%o2) .8414709848078965
When the global flag noundisp is true, nouns display with a single
quote.
(%i1) x:%pi;
(%o1) %pi
(%i2) bfloat(x);
(%o2) 3.141592653589793b0
(%i3) sin(x);
(%o3) 0
(%i4) noundisp;
(%o4) false
(%i5) 'bfloat(x);
(%o5) bfloat(%pi)
(%i6) bfloat('x);
(%o6) x
(%i7) 'sin(x);
(%o7) 0
(%i8) sin('x);
(%o8) sin(x)
(%i9) noundisp : not noundisp;
(%o9) true
(%i10) 'bfloat(x);
(%o10) 'bfloat(%pi)
(%i11) bfloat('x);
(%o11) x
(%i12) 'sin(x);
(%o12) 0
(%i13) sin('x);
(%o13) sin(x)
(%i14)
Categories: Evaluation · Operators
The quote-quote operator '' (two single quote marks) modifies
evaluation in input expressions.
Applied to a general expression expr, quote-quote causes the value of expr to be substituted for expr in the input expression.
Applied to the operator of an expression, quote-quote changes the operator from a noun to a verb (if it is not already a verb).
The quote-quote operator is applied by the input parser; it is not stored as
part of a parsed input expression. The quote-quote operator is always applied
as soon as it is parsed, and cannot be quoted. Thus quote-quote causes
evaluation when evaluation is otherwise suppressed, such as in function
definitions, lambda expressions, and expressions quoted by single quote
'.
Quote-quote is recognized by batch and load.
See also the single-quote operator ' and nouns.
Examples:
Applied to a general expression expr, quote-quote causes the value of expr to be substituted for expr in the input expression.
(%i1) expand ((a + b)^3);
3 2 2 3
(%o1) b + 3 a b + 3 a b + a
(%i2) [_, ''_];
3 3 2 2 3
(%o2) [expand((b + a) ), b + 3 a b + 3 a b + a ]
(%i3) [%i1, ''%i1];
3 3 2 2 3
(%o3) [expand((b + a) ), b + 3 a b + 3 a b + a ]
(%i4) [aa : cc, bb : dd, cc : 17, dd : 29];
(%o4) [cc, dd, 17, 29]
(%i5) foo_1 (x) := aa - bb * x;
(%o5) foo_1(x) := aa - bb x
(%i6) foo_1 (10);
(%o6) cc - 10 dd
(%i7) ''%;
(%o7) - 273
(%i8) ''(foo_1 (10));
(%o8) - 273
(%i9) foo_2 (x) := ''aa - ''bb * x;
(%o9) foo_2(x) := cc - dd x
(%i10) foo_2 (10);
(%o10) - 273
(%i11) [x0 : x1, x1 : x2, x2 : x3];
(%o11) [x1, x2, x3]
(%i12) x0;
(%o12) x1
(%i13) ''x0;
(%o13) x2
(%i14) '' ''x0;
(%o14) x3
Applied to the operator of an expression, quote-quote changes the operator from a noun to a verb (if it is not already a verb).
(%i1) declare (foo, noun); (%o1) done (%i2) foo (x) := x - 1729; (%o2) ''foo(x) := x - 1729 (%i3) foo (100); (%o3) foo(100) (%i4) ''foo (100); (%o4) - 1629
The quote-quote operator is applied by the input parser; it is not stored as part of a parsed input expression.
(%i1) [aa : bb, cc : dd, bb : 1234, dd : 5678];
(%o1) [bb, dd, 1234, 5678]
(%i2) aa + cc;
(%o2) dd + bb
(%i3) display (_, op (_), args (_));
_ = cc + aa
op(cc + aa) = +
args(cc + aa) = [cc, aa]
(%o3) done
(%i4) ''(aa + cc);
(%o4) 6912
(%i5) display (_, op (_), args (_));
_ = dd + bb
op(dd + bb) = +
args(dd + bb) = [dd, bb]
(%o5) done
Quote-quote causes evaluation when evaluation is otherwise suppressed, such as
in function definitions, lambda expressions, and expressions quoted by single
quote '.
(%i1) foo_1a (x) := ''(integrate (log (x), x));
(%o1) foo_1a(x) := x log(x) - x
(%i2) foo_1b (x) := integrate (log (x), x);
(%o2) foo_1b(x) := integrate(log(x), x)
(%i3) dispfun (foo_1a, foo_1b);
(%t3) foo_1a(x) := x log(x) - x
(%t4) foo_1b(x) := integrate(log(x), x)
(%o4) [%t3, %t4]
(%i5) integrate (log (x), x);
(%o5) x log(x) - x
(%i6) foo_2a (x) := ''%;
(%o6) foo_2a(x) := x log(x) - x
(%i7) foo_2b (x) := %;
(%o7) foo_2b(x) := %
(%i8) dispfun (foo_2a, foo_2b);
(%t8) foo_2a(x) := x log(x) - x
(%t9) foo_2b(x) := %
(%o9) [%t7, %t8]
(%i10) F : lambda ([u], diff (sin (u), u));
(%o10) lambda([u], diff(sin(u), u))
(%i11) G : lambda ([u], ''(diff (sin (u), u)));
(%o11) lambda([u], cos(u))
(%i12) '(sum (a[k], k, 1, 3) + sum (b[k], k, 1, 3));
(%o12) sum(b , k, 1, 3) + sum(a , k, 1, 3)
k k
(%i13) '(''(sum (a[k], k, 1, 3)) + ''(sum (b[k], k, 1, 3)));
(%o13) b + a + b + a + b + a
3 3 2 2 1 1
Categories: Evaluation · Operators
Evaluates the expression expr in the environment specified by the
arguments arg_1, …, arg_n. The arguments are switches
(Boolean flags), assignments, equations, and functions. ev returns the
result (another expression) of the evaluation.
The evaluation is carried out in steps, as follows.
simp causes expr to be simplified regardless of the setting of the
switch simp which inhibits simplification if false.
noeval suppresses the evaluation phase of ev (see step (4) below).
This is useful in conjunction with the other switches and in causing
expr to be resimplified without being reevaluated.
nouns causes the evaluation of noun forms (typically unevaluated
functions such as 'integrate or 'diff) in expr.
expand causes expansion.
expand (m, n) causes expansion, setting the values of
maxposex and maxnegex to m and n respectively.
detout causes any matrix inverses computed in expr to have their
determinant kept outside of the inverse rather than dividing through
each element.
diff causes all differentiations indicated in expr to be performed.
derivlist (x, y, z, ...) causes only differentiations
with respect to the indicated variables. See also derivlist.
risch causes integrals in expr to be evaluated using the Risch
algorithm. See risch. The standard integration routine is invoked
when using the special symbol nouns.
float causes non-integral rational numbers to be converted to floating
point.
numer causes some mathematical functions (including exponentiation)
with numerical arguments to be evaluated in floating point. It causes
variables in expr which have been given numervals to be replaced by
their values. It also sets the float switch on.
pred causes predicates (expressions which evaluate to true or
false) to be evaluated.
eval causes an extra post-evaluation of expr to occur.
(See step (5) below.)
eval may occur multiple times. For each instance of eval, the
expression is evaluated again.
A where A is an atom declared to be an evaluation flag
evflag causes A to be bound to true during the evaluation
of expr.
V: expression (or alternately V=expression) causes V to be
bound to the value of expression during the evaluation of expr.
Note that if V is a Maxima option, then expression is used for
its value during the evaluation of expr. If more than one argument to
ev is of this type then the binding is done in parallel. If V is
a non-atomic expression then a substitution rather than a binding is performed.
F where F, a function name, has been declared to be an evaluation
function evfun causes F to be applied to expr.
sum, cause evaluation of occurrences
of those names in expr as though they were verbs.
F(x)) may be defined
locally for the purpose of this evaluation of expr by giving
F(x) := expression as an argument to ev.
ev. This permits a list of equations to be
given (e.g. [X=1, Y=A**2]) or a list of names of equations (e.g.,
[%t1, %t2] where %t1 and %t2 are equations) such as that
returned by solve.
The arguments of ev may be given in any order with the exception of
substitution equations which are handled in sequence, left to right, and
evaluation functions which are composed, e.g., ev (expr, ratsimp,
realpart) is handled as realpart (ratsimp (expr)).
The simp, numer, and float switches may also be set
locally in a block, or globally in Maxima so that they will remain in effect
until being reset.
If expr is a canonical rational expression (CRE), then the expression
returned by ev is also a CRE, provided the numer and float
switches are not both true.
% (as in
%i2 in the example below), so this step simply retrieves the expression
named by the label, so that ev may work on it.
noeval) and simplified according to the arguments. Note that any
function calls in expr will be carried out after the variables in it are
evaluated and that ev(F(x)) thus may behave like F(ev(x)).
eval in the arguments, steps (3) and (4) are
repeated.
Examples:
(%i1) sin(x) + cos(y) + (w+1)^2 + 'diff (sin(w), w);
d 2
(%o1) cos(y) + sin(x) + -- (sin(w)) + (w + 1)
dw
(%i2) ev (%, numer, expand, diff, x=2, y=1);
2
(%o2) cos(w) + w + 2 w + 2.449599732693821
An alternate top level syntax has been provided for ev, whereby one
may just type in its arguments, without the ev(). That is, one may
write simply
expr, arg_1, ..., arg_n
This is not permitted as part of another expression, e.g., in functions, blocks, etc.
Notice the parallel binding process in the following example.
(%i3) programmode: false;
(%o3) false
(%i4) x+y, x: a+y, y: 2;
(%o4) y + a + 2
(%i5) 2*x - 3*y = 3$
(%i6) -3*x + 2*y = -4$
(%i7) solve ([%o5, %o6]);
Solution
1
(%t7) y = - -
5
6
(%t8) x = -
5
(%o8) [[%t7, %t8]]
(%i8) %o6, %o8;
(%o8) - 4 = - 4
(%i9) x + 1/x > gamma (1/2);
1
(%o9) x + - > sqrt(%pi)
x
(%i10) %, numer, x=1/2;
(%o10) 2.5 > 1.772453850905516
(%i11) %, pred;
(%o11) true
Categories: Evaluation
As an argument in a call to ev (expr), eval causes an extra
evaluation of expr. See ev.
Example:
(%i1) [a:b,b:c,c:d,d:e]; (%o1) [b, c, d, e] (%i2) a; (%o2) b (%i3) ev(a); (%o3) c (%i4) ev(a),eval; (%o4) e (%i5) a,eval,eval; (%o5) e
Categories: Evaluation flags
When a symbol x has the evflag property, the expressions
ev(expr, x) and expr, x (at the
interactive prompt) are equivalent to ev(expr, x = true).
That is, x is bound to true while expr is evaluated.
The expression declare(x, evflag) gives the evflag property
to the variable x.
The flags which have the evflag property by default are the following:
algebraic cauchysum demoivre dotscrules %emode %enumer exponentialize exptisolate factorflag float halfangles infeval isolate_wrt_times keepfloat letrat listarith logabs logarc logexpand lognegint m1pbranch numer_pbranch programmode radexpand ratalgdenom ratfac ratmx ratsimpexpons simp simpproduct simpsum sumexpand trigexpand
Examples:
(%i1) sin (1/2);
1
(%o1) sin(-)
2
(%i2) sin (1/2), float;
(%o2) 0.479425538604203
(%i3) sin (1/2), float=true;
(%o3) 0.479425538604203
(%i4) simp : false;
(%o4) false
(%i5) 1 + 1;
(%o5) 1 + 1
(%i6) 1 + 1, simp;
(%o6) 2
(%i7) simp : true;
(%o7) true
(%i8) sum (1/k^2, k, 1, inf);
inf
====
\ 1
(%o8) > --
/ 2
==== k
k = 1
(%i9) sum (1/k^2, k, 1, inf), simpsum;
2
%pi
(%o9) ----
6
(%i10) declare (aa, evflag);
(%o10) done
(%i11) if aa = true then YES else NO;
(%o11) NO
(%i12) if aa = true then YES else NO, aa;
(%o12) YES
Categories: Evaluation flags · Simplification flags and variables
When a function F has the evfun property, the expressions
ev(expr, F) and expr, F (at the
interactive prompt) are equivalent to F(ev(expr)).
If two or more evfun functions F, G, etc., are specified,
the functions are applied in the order that they are specified.
The expression declare(F, evfun) gives the evfun property
to the function F. The functions which have the evfun property by
default are the following:
bfloat factor fullratsimp logcontract polarform radcan ratexpand ratsimp rectform rootscontract trigexpand trigreduce
Examples:
(%i1) x^3 - 1;
3
(%o1) x - 1
(%i2) x^3 - 1, factor;
2
(%o2) (x - 1) (x + x + 1)
(%i3) factor (x^3 - 1);
2
(%o3) (x - 1) (x + x + 1)
(%i4) cos(4 * x) / sin(x)^4;
cos(4 x)
(%o4) --------
4
sin (x)
(%i5) cos(4 * x) / sin(x)^4, trigexpand;
4 2 2 4
sin (x) - 6 cos (x) sin (x) + cos (x)
(%o5) -------------------------------------
4
sin (x)
(%i6) cos(4 * x) / sin(x)^4, trigexpand, ratexpand;
2 4
6 cos (x) cos (x)
(%o6) - --------- + ------- + 1
2 4
sin (x) sin (x)
(%i7) ratexpand (trigexpand (cos(4 * x) / sin(x)^4));
2 4
6 cos (x) cos (x)
(%o7) - --------- + ------- + 1
2 4
sin (x) sin (x)
(%i8) declare ([F, G], evfun);
(%o8) done
(%i9) (aa : bb, bb : cc, cc : dd);
(%o9) dd
(%i10) aa;
(%o10) bb
(%i11) aa, F;
(%o11) F(cc)
(%i12) F (aa);
(%o12) F(bb)
(%i13) F (ev (aa));
(%o13) F(cc)
(%i14) aa, F, G;
(%o14) G(F(cc))
(%i15) G (F (ev (aa)));
(%o15) G(F(cc))
Categories: Evaluation flags
Enables "infinite evaluation" mode. ev repeatedly evaluates an
expression until it stops changing. To prevent a variable, say X, from
being evaluated away in this mode, simply include X='X as an argument to
ev. Of course expressions such as ev (X, X=X+1, infeval) will
generate an infinite loop.
Categories: Evaluation flags
noeval suppresses the evaluation phase of ev. This is useful in
conjunction with other switches and in causing expressions
to be resimplified without being reevaluated.
Categories: Evaluation flags
nouns is an evflag. When used as an option to the ev
command, nouns converts all "noun" forms occurring in the expression
being ev'd to "verbs", i.e., evaluates them. See also
noun, nounify, verb, and verbify.
Categories: Evaluation flags · Nouns and verbs
As an argument in a call to ev (expr), pred causes
predicates (expressions which evaluate to true or false) to be
evaluated. See ev.
Example:
(%i1) 1<2; (%o1) 1 < 2 (%i2) 1<2,pred; (%o2) true
Categories: Evaluation flags
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If declare(f,additive) has been executed, then:
(1) If f is univariate, whenever the simplifier encounters f
applied to a sum, f will be distributed over that sum. I.e.
f(y+x) will simplify to f(y)+f(x).
(2) If f is a function of 2 or more arguments, additivity is defined as
additivity in the first argument to f, as in the case of sum or
integrate, i.e. f(h(x)+g(x),x) will simplify to
f(h(x),x)+f(g(x),x). This simplification does not occur when f is
applied to expressions of the form sum(x[i],i,lower-limit,upper-limit).
Example:
(%i1) F3 (a + b + c); (%o1) F3(c + b + a) (%i2) declare (F3, additive); (%o2) done (%i3) F3 (a + b + c); (%o3) F3(c) + F3(b) + F3(a)
Categories: Operators · Declarations and inferences
If declare(h,antisymmetric) is done, this tells the simplifier that
h is antisymmetric. E.g. h(x,z,y) will simplify to
- h(x, y, z). That is, it will give (-1)^n times the result given by
symmetric or commutative, where n is the number of interchanges
of two arguments necessary to convert it to that form.
Examples:
(%i1) S (b, a); (%o1) S(b, a) (%i2) declare (S, symmetric); (%o2) done (%i3) S (b, a); (%o3) S(a, b) (%i4) S (a, c, e, d, b); (%o4) S(a, b, c, d, e) (%i5) T (b, a); (%o5) T(b, a) (%i6) declare (T, antisymmetric); (%o6) done (%i7) T (b, a); (%o7) - T(a, b) (%i8) T (a, c, e, d, b); (%o8) T(a, b, c, d, e)
Categories: Operators · Declarations and inferences
Simplifies the sum expr by combining terms with the same denominator into a single term.
Example:
(%i1) 1*f/2*b + 2*c/3*a + 3*f/4*b +c/5*b*a;
5 b f a b c 2 a c
(%o1) ----- + ----- + -----
4 5 3
(%i2) combine (%);
75 b f + 4 (3 a b c + 10 a c)
(%o2) -----------------------------
60
Categories: Expressions
If declare(h, commutative) is done, this tells the simplifier that
h is a commutative function. E.g. h(x, z, y) will simplify to
h(x, y, z). This is the same as symmetric.
Exemplo:
(%i1) S (b, a); (%o1) S(b, a) (%i2) S (a, b) + S (b, a); (%o2) S(b, a) + S(a, b) (%i3) declare (S, commutative); (%o3) done (%i4) S (b, a); (%o4) S(a, b) (%i5) S (a, b) + S (b, a); (%o5) 2 S(a, b) (%i6) S (a, c, e, d, b); (%o6) S(a, b, c, d, e)
Categories: Operators · Declarations and inferences
The function demoivre (expr) converts one expression
without setting the global variable demoivre.
When the variable demoivre is true, complex exponentials are
converted into equivalent expressions in terms of circular functions:
exp (a + b*%i) simplifies to %e^a * (cos(b) + %i*sin(b))
if b is free of %i. a and b are not expanded.
The default value of demoivre is false.
exponentialize converts circular and hyperbolic functions to exponential
form. demoivre and exponentialize cannot both be true at the same
time.
Categories: Complex variables · Trigonometric functions · Hyperbolic functions
Distributes sums over products. It differs from expand in that it works
at only the top level of an expression, i.e., it doesn't recurse and it is
faster than expand. It differs from multthru in that it expands
all sums at that level.
Examples:
(%i1) distrib ((a+b) * (c+d));
(%o1) b d + a d + b c + a c
(%i2) multthru ((a+b) * (c+d));
(%o2) (b + a) d + (b + a) c
(%i3) distrib (1/((a+b) * (c+d)));
1
(%o3) ---------------
(b + a) (d + c)
(%i4) expand (1/((a+b) * (c+d)), 1, 0);
1
(%o4) ---------------------
b d + a d + b c + a c
Categories: Expressions
Default value: true
distribute_over controls the mapping of functions over bags like lists,
matrices, and equations. At this time not all Maxima functions have this
property. It is possible to look up this property with the command
properties.
The mapping of functions is switched off, when setting distribute_over
to the value false.
Examples:
The sin function maps over a list:
(%i1) sin([x,1,1.0]); (%o1) [sin(x), sin(1), .8414709848078965]
mod is a function with two arguments which maps over lists. Mapping over
nested lists is possible too:
(%i2) mod([x,11,2*a],10); (%o2) [mod(x, 10), 1, 2 mod(a, 5)] (%i3) mod([[x,y,z],11,2*a],10); (%o3) [[mod(x, 10), mod(y, 10), mod(z, 10)], 1, 2 mod(a, 5)]
Mapping of the floor function over a matrix and an equation:
(%i4) floor(matrix([a,b],[c,d]));
[ floor(a) floor(b) ]
(%o4) [ ]
[ floor(c) floor(d) ]
(%i5) floor(a=b);
(%o5) floor(a) = floor(b)
Functions with more than one argument map over any of the arguments or all arguments:
(%i6) expintegral_e([1,2],[x,y]);
(%o6) [[expintegral_e(1, x), expintegral_e(1, y)],
[expintegral_e(2, x), expintegral_e(2, y)]]
Check if a function has the property distribute_over:
(%i7) properties(abs); (%o7) [integral, distributes over bags, noun, rule, gradef]
The mapping of functions is switched off, when setting distribute_over
to the value false.
(%i1) distribute_over; (%o1) true (%i2) sin([x,1,1.0]); (%o2) [sin(x), sin(1), 0.8414709848078965] (%i3) distribute_over : not distribute_over; (%o3) false (%i4) sin([x,1,1.0]); (%o4) sin([x, 1, 1.0]) (%i5)
Categories: Simplification flags and variables
Default value: real
When domain is set to complex, sqrt (x^2) will remain
sqrt (x^2) instead of returning abs(x).
Categories: Simplification flags and variables
declare(f, evenfun) or declare(f, oddfun) tells Maxima to recognize
the function f as an even or odd function.
Examples:
(%i1) o (- x) + o (x); (%o1) o(x) + o(- x) (%i2) declare (o, oddfun); (%o2) done (%i3) o (- x) + o (x); (%o3) 0 (%i4) e (- x) - e (x); (%o4) e(- x) - e(x) (%i5) declare (e, evenfun); (%o5) done (%i6) e (- x) - e (x); (%o6) 0
Expand expression expr. Products of sums and exponentiated sums are multiplied out, numerators of rational expressions which are sums are split into their respective terms, and multiplication (commutative and non-commutative) are distributed over addition at all levels of expr.
For polynomials one should usually use ratexpand which uses a
more efficient algorithm.
maxnegex and maxposex control the maximum negative and
positive exponents, respectively, which will expand.
expand (expr, p, n) expands expr,
using p for maxposex and n for maxnegex.
This is useful in order to expand part but not all of an expression.
expon - the exponent of the largest negative power which is
automatically expanded (independent of calls to expand). For example
if expon is 4 then (x+1)^(-5) will not be automatically expanded.
expop - the highest positive exponent which is automatically expanded.
Thus (x+1)^3, when typed, will be automatically expanded only if
expop is greater than or equal to 3. If it is desired to have
(x+1)^n expanded where n is greater than expop then
executing expand ((x+1)^n) will work only if maxposex is not
less than n.
expand(expr, 0, 0) causes a resimplification of expr. expr
is not reevaluated. In distinction from ev(expr, noeval) a special
representation (e. g. a CRE form) is removed. See also ev.
The expand flag used with ev causes expansion.
The file `share/simplification/facexp.mac'
contains several related functions (in particular facsum,
factorfacsum and collectterms, which are autoloaded) and variables
(nextlayerfactor and facsum_combine) that provide the user with
the ability to structure expressions by controlled expansion.
Brief function descriptions are available in `simplification/facexp.usg'.
A demo is available by doing demo("facexp").
Examples:
(%i1) expr:(x+1)^2*(y+1)^3;
2 3
(%o1) (x + 1) (y + 1)
(%i2) expand(expr);
2 3 3 3 2 2 2 2 2
(%o2) x y + 2 x y + y + 3 x y + 6 x y + 3 y + 3 x y
2
+ 6 x y + 3 y + x + 2 x + 1
(%i3) expand(expr,2);
2 3 3 3
(%o3) x (y + 1) + 2 x (y + 1) + (y + 1)
(%i4) expr:(x+1)^-2*(y+1)^3;
3
(y + 1)
(%o4) --------
2
(x + 1)
(%i5) expand(expr);
3 2
y 3 y 3 y 1
(%o5) ------------ + ------------ + ------------ + ------------
2 2 2 2
x + 2 x + 1 x + 2 x + 1 x + 2 x + 1 x + 2 x + 1
(%i6) expand(expr,2,2);
3
(y + 1)
(%o6) ------------
2
x + 2 x + 1
Resimplify an expression without expansion:
(%i7) expr:(1+x)^2*sin(x);
2
(%o7) (x + 1) sin(x)
(%i8) exponentialize:true;
(%o8) true
(%i9) expand(expr,0,0);
2 %i x - %i x
%i (x + 1) (%e - %e )
(%o9) - -------------------------------
2
Categories: Expressions
Expands expression expr with respect to the
variables x_1, …, x_n.
All products involving the variables appear explicitly. The form returned
will be free of products of sums of expressions that are not free of
the variables. x_1, …, x_n
may be variables, operators, or expressions.
By default, denominators are not expanded, but this can be controlled by
means of the switch expandwrt_denom.
This function is autoloaded from `simplification/stopex.mac'.
Categories: Expressions
Default value: false
expandwrt_denom controls the treatment of rational
expressions by expandwrt. If true, then both the numerator and
denominator of the expression will be expanded according to the
arguments of expandwrt, but if expandwrt_denom is false,
then only the numerator will be expanded in that way.
Categories: Expressions
is similar to expandwrt, but treats expressions that are products
somewhat differently. expandwrt_factored expands only on those factors
of expr that contain the variables x_1, …, x_n.
This function is autoloaded from `simplification/stopex.mac'.
Categories: Expressions
Default value: 0
expon is the exponent of the largest negative power which
is automatically expanded (independent of calls to expand). For
example, if expon is 4 then (x+1)^(-5) will not be automatically
expanded.
Categories: Expressions
The function exponentialize (expr) converts
circular and hyperbolic functions in expr to exponentials,
without setting the global variable exponentialize.
When the variable exponentialize is true,
all circular and hyperbolic functions are converted to exponential form.
The default value is false.
demoivre converts complex exponentials into circular functions.
exponentialize and demoivre cannot
both be true at the same time.
Categories: Complex variables · Trigonometric functions · Hyperbolic functions
Default value: 0
expop is the highest positive exponent which is automatically expanded.
Thus (x + 1)^3, when typed, will be automatically expanded only if
expop is greater than or equal to 3. If it is desired to have
(x + 1)^n expanded where n is greater than expop then
executing expand ((x + 1)^n) will work only if maxposex is not
less than n.
Categories: Expressions
declare (g, lassociative) tells the Maxima simplifier that g is
left-associative. E.g., g (g (a, b), g (c, d)) will simplify to
g (g (g (a, b), c), d).
Categories: Declarations and inferences · Operators · Simplification
One of Maxima's operator properties. For univariate f so
declared, "expansion" f(x + y) yields f(x) + f(y),
f(a*x) yields a*f(x) takes
place where a is a "constant". For functions of two or more arguments,
"linearity" is defined to be as in the case of sum or integrate,
i.e., f (a*x + b, x) yields a*f(x,x) + b*f(1,x)
for a and b free of x.
Example:
(%i1) declare (f, linear); (%o1) done (%i2) f(x+y); (%o2) f(y) + f(x) (%i3) declare (a, constant); (%o3) done (%i4) f(a*x); (%o4) a f(x)
linear is equivalent to additive and outative.
See also opproperties.
Example:
(%i1) 'sum (F(k) + G(k), k, 1, inf);
inf
====
\
(%o1) > (G(k) + F(k))
/
====
k = 1
(%i2) declare (nounify (sum), linear);
(%o2) done
(%i3) 'sum (F(k) + G(k), k, 1, inf);
inf inf
==== ====
\ \
(%o3) > G(k) + > F(k)
/ /
==== ====
k = 1 k = 1
Categories: Declarations and inferences · Operators · Simplification
Default value: 1000
maxnegex is the largest negative exponent which will
be expanded by the expand command, see also maxposex.
Categories: Expressions
Default value: 1000
maxposex is the largest exponent which will be
expanded with the expand command, see also maxnegex.
Categories: Expressions
declare(f, multiplicative) tells the Maxima simplifier that f
is multiplicative.
f is univariate, whenever the simplifier encounters f applied
to a product, f distributes over that product. E.g., f(x*y)
simplifies to f(x)*f(y).
This simplification is not applied to expressions of the form f('product(...)).
f is a function of 2 or more arguments, multiplicativity is
defined as multiplicativity in the first argument to f, e.g.,
f (g(x) * h(x), x) simplifies to f (g(x) ,x) * f (h(x), x).
declare(nounify(product), multiplicative) tells Maxima to simplify symbolic products.
Example:
(%i1) F2 (a * b * c); (%o1) F2(a b c) (%i2) declare (F2, multiplicative); (%o2) done (%i3) F2 (a * b * c); (%o3) F2(a) F2(b) F2(c)
declare(nounify(product), multiplicative) tells Maxima to simplify symbolic products.
(%i1) product (a[i] * b[i], i, 1, n);
n
/===\
! !
(%o1) ! ! a b
! ! i i
i = 1
(%i2) declare (nounify (product), multiplicative);
(%o2) done
(%i3) product (a[i] * b[i], i, 1, n);
n n
/===\ /===\
! ! ! !
(%o3) ( ! ! a ) ! ! b
! ! i ! ! i
i = 1 i = 1
Categories: Declarations and inferences · Expressions · Simplification
Multiplies a factor (which should be a sum) of expr by the other factors
of expr. That is, expr is f_1 f_2 ... f_n
where at least one factor, say f_i, is a sum of terms. Each term in that
sum is multiplied by the other factors in the product. (Namely all the factors
except f_i). multthru does not expand exponentiated sums.
This function is the fastest way to distribute products (commutative or
noncommutative) over sums. Since quotients are represented as products
multthru can be used to divide sums by products as well.
multthru (expr_1, expr_2) multiplies each term in
expr_2 (which should be a sum or an equation) by expr_1. If
expr_1 is not itself a sum then this form is equivalent to
multthru (expr_1*expr_2).
(%i1) x/(x-y)^2 - 1/(x-y) - f(x)/(x-y)^3;
1 x f(x)
(%o1) - ----- + -------- - --------
x - y 2 3
(x - y) (x - y)
(%i2) multthru ((x-y)^3, %);
2
(%o2) - (x - y) + x (x - y) - f(x)
(%i3) ratexpand (%);
2
(%o3) - y + x y - f(x)
(%i4) ((a+b)^10*s^2 + 2*a*b*s + (a*b)^2)/(a*b*s^2);
10 2 2 2
(b + a) s + 2 a b s + a b
(%o4) ------------------------------
2
a b s
(%i5) multthru (%); /* note that this does not expand (b+a)^10 */
10
2 a b (b + a)
(%o5) - + --- + ---------
s 2 a b
s
(%i6) multthru (a.(b+c.(d+e)+f));
(%o6) a . f + a . c . (e + d) + a . b
(%i7) expand (a.(b+c.(d+e)+f));
(%o7) a . f + a . c . e + a . c . d + a . b
Categories: Expressions
declare(f, nary) tells Maxima to recognize the function f as an
n-ary function.
The nary declaration is not the same as calling the
nary function. The sole effect of
declare(f, nary) is to instruct the Maxima simplifier to flatten nested
expressions, for example, to simplify foo(x, foo(y, z)) to
foo(x, y, z). See also declare.
Example:
(%i1) H (H (a, b), H (c, H (d, e))); (%o1) H(H(a, b), H(c, H(d, e))) (%i2) declare (H, nary); (%o2) done (%i3) H (H (a, b), H (c, H (d, e))); (%o3) H(a, b, c, d, e)
Default value: true
When negdistrib is true, -1 distributes over an expression.
E.g., -(x + y) becomes - y - x. Setting it to false
will allow - (x + y) to be displayed like that. This is sometimes useful
but be very careful: like the simp flag, this is one flag you do not
want to set to false as a matter of course or necessarily for other
than local use in your Maxima.
Example:
(%i1) negdistrib; (%o1) true (%i2) -(x+y); (%o2) - y - x (%i3) negdistrib : not negdistrib ; (%o3) false (%i4) -(x+y); (%o4) - (y + x)
Categories: Simplification flags and variables
opproperties is the list of the special operator properties recognized
by the Maxima simplifier:
Example:
(%i1) opproperties;
(%o1) [linear, additive, multiplicative, outative, evenfun, oddfun,.
commutative, symmetric, antisymmetric, nary, lassociative, rassociative]
Categories: Global variables · Operators
declare(f, outative) tells the Maxima simplifier that constant factors
in the argument of f can be pulled out.
f is univariate, whenever the simplifier encounters f applied
to a product, that product will be partitioned into factors that are constant
and factors that are not and the constant factors will be pulled out. E.g.,
f(a*x) will simplify to a*f(x) where a is a constant.
Non-atomic constant factors will not be pulled out.
f is a function of 2 or more arguments, outativity is defined as in
the case of sum or integrate, i.e., f (a*g(x), x) will
simplify to a * f(g(x), x) for a free of x.
sum, integrate, and limit are all outative.
Example:
(%i1) F1 (100 * x); (%o1) F1(100 x) (%i2) declare (F1, outative); (%o2) done (%i3) F1 (100 * x); (%o3) 100 F1(x) (%i4) declare (zz, constant); (%o4) done (%i5) F1 (zz * y); (%o5) zz F1(y)
Categories: Declarations and inferences · Operators
Simplifies expr, which can contain logs, exponentials, and radicals, by
converting it into a form which is canonical over a large class of expressions
and a given ordering of variables; that is, all functionally equivalent forms
are mapped into a unique form. For a somewhat larger class of expressions,
radcan produces a regular form. Two equivalent expressions in this class
do not necessarily have the same appearance, but their difference can be
simplified by radcan to zero.
For some expressions radcan is quite time consuming. This is the cost
of exploring certain relationships among the components of the expression for
simplifications based on factoring and partial-fraction expansions of exponents.
Examples:
(%i1) radcan((log(x+x^2)-log(x))^a/log(1+x)^(a/2));
a/2
(%o1) log(x + 1)
(%i2) radcan((log(1+2*a^x+a^(2*x))/log(1+a^x)));
(%o2) 2
(%i3) radcan((%e^x-1)/(1+%e^(x/2)));
x/2
(%o3) %e - 1
Categories: Simplification functions
Default value: true
radexpand controls some simplifications of radicals.
When radexpand is all, causes nth roots of factors of a product
which are powers of n to be pulled outside of the radical. E.g. if
radexpand is all, sqrt (16*x^2) simplifies to 4*x.
More particularly, consider sqrt (x^2).
radexpand is all or assume (x > 0) has been executed,
sqrt(x^2) simplifies to x.
radexpand is true and domain is real
(its default), sqrt(x^2) simplifies to abs(x).
radexpand is false, or radexpand is true and
domain is complex, sqrt(x^2) is not simplified.
Note that domain only matters when radexpand is true.
Categories: Simplification flags and variables
declare (g, rassociative) tells the Maxima
simplifier that g is right-associative. E.g.,
g(g(a, b), g(c, d)) simplifies to g(a, g(b, g(c, d))).
Categories: Declarations and inferences · Operators
Sequential Comparative Simplification (method due to Stoute).
scsimp attempts to simplify expr
according to the rules rule_1, …, rule_n.
If a smaller expression is obtained, the process repeats. Otherwise after all
simplifications are tried, it returns the original answer.
example (scsimp) displays some examples.
Categories: Simplification functions
Default value: true
simp enables simplification. This is the standard. simp is also
an evflag, which is recognized by the function ev. See ev.
When simp is used as an evflag with a value false, the
simplification is suppressed only during the evaluation phase of an expression.
The flag can not suppress the simplification which follows the evaluation
phase.
Examples:
The simplification is switched off globally. The expression sin(1.0) is
not simplified to its numerical value. The simp-flag switches the
simplification on.
(%i1) simp:false; (%o1) false (%i2) sin(1.0); (%o2) sin(1.0) (%i3) sin(1.0),simp; (%o3) .8414709848078965
The simplification is switched on again. The simp-flag cannot suppress
the simplification completely. The output shows a simplified expression, but
the variable x has an unsimplified expression as a value, because the
assignment has occurred during the evaluation phase of the expression.
(%i4) simp:true; (%o4) true (%i5) x:sin(1.0),simp:false; (%o5) .8414709848078965 (%i6) :lisp $X ((%SIN) 1.0)
Categories: Evaluation flags
declare (h, symmetric) tells the Maxima
simplifier that h is a symmetric function. E.g., h (x, z, y)
simplifies to h (x, y, z).
commutative is synonymous with symmetric.
Categories: Declarations and inferences · Operators
Combines all terms of expr (which should be a sum) over a common
denominator without expanding products and exponentiated sums as ratsimp
does. xthru cancels common factors in the numerator and denominator of
rational expressions but only if the factors are explicit.
Sometimes it is better to use xthru before ratsimping an
expression in order to cause explicit factors of the gcd of the numerator and
denominator to be canceled thus simplifying the expression to be
ratsimped.
Examples:
(%i1) ((x+2)^20 - 2*y)/(x+y)^20 + (x+y)^(-19) - x/(x+y)^20;
20
1 (x + 2) - 2 y x
(%o1) --------- + --------------- - ---------
19 20 20
(y + x) (y + x) (y + x)
(%i2) xthru (%);
20
(x + 2) - y
(%o2) -------------
20
(y + x)
Categories: Expressions
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 10.1 Functions for Numbers | ||
| 10.2 Functions for Complex Numbers | ||
| 10.3 Combinatorial Functions | ||
| 10.4 Root, Exponential and Logarithmic Functions | ||
| 10.5 Trigonometric Functions | ||
| 10.6 Random Numbers |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
The abs function represents the mathematical absolute value function and
works for both numerical and symbolic values. If the argument, z, is a
real or complex number, abs returns the absolute value of z. If
possible, symbolic expressions using the absolute value function are
also simplified.
Maxima can differentiate, integrate and calculate limits for expressions
containing abs. The abs_integrate package further extends
Maxima's ability to calculate integrals involving the abs function. See
(%i12) in the examples below.
When applied to a list or matrix, abs automatically distributes over
the terms. Similarly, it distributes over both sides of an
equation. To alter this behaviour, see the variable distribute_over.
Examples:
Calculation of abs for real and complex numbers, including numerical
constants and various infinities. The first example shows how abs
distributes over the elements of a list.
(%i1) abs([-4, 0, 1, 1+%i]);
(%o1) [4, 0, 1, sqrt(2)]
(%i2) abs((1+%i)*(1-%i));
(%o2) 2
(%i3) abs(%e+%i);
2
(%o3) sqrt(%e + 1)
(%i4) abs([inf, infinity, minf]);
(%o4) [inf, inf, inf]
Simplification of expressions containing abs:
(%i5) abs(x^2);
2
(%o5) x
(%i6) abs(x^3);
2
(%o6) x abs(x)
(%i7) abs(abs(x));
(%o7) abs(x)
(%i8) abs(conjugate(x));
(%o8) abs(x)
Integrating and differentiating with the abs function. Note that more
integrals involving the abs function can be performed, if the
abs_integrate package is loaded. The last example shows the Laplace
transform of abs: see laplace.
(%i9) diff(x*abs(x),x),expand;
(%o9) 2 abs(x)
(%i10) integrate(abs(x),x);
x abs(x)
(%o10) --------
2
(%i11) integrate(x*abs(x),x);
/
[
(%o11) I x abs(x) dx
]
/
(%i12) load(abs_integrate)$
(%i13) integrate(x*abs(x),x);
2 3
x abs(x) x signum(x)
(%o13) --------- - ------------
2 6
(%i14) integrate(abs(x),x,-2,%pi);
2
%pi
(%o14) ---- + 2
2
(%i15) laplace(abs(x),x,s);
1
(%o15) --
2
s
Categories: Mathematical functions
When x is a real number, return the least integer that is greater than or equal to x.
If x is a constant expression (10 * %pi, for example),
ceiling evaluates x using big floating point numbers, and
applies ceiling to the resulting big float. Because ceiling uses
floating point evaluation, it's possible, although unlikely, that ceiling
could return an erroneous value for constant inputs. To guard against errors,
the floating point evaluation is done using three values for fpprec.
For non-constant inputs, ceiling tries to return a simplified value.
Here are examples of the simplifications that ceiling knows about:
(%i1) ceiling (ceiling (x)); (%o1) ceiling(x) (%i2) ceiling (floor (x)); (%o2) floor(x) (%i3) declare (n, integer)$ (%i4) [ceiling (n), ceiling (abs (n)), ceiling (max (n, 6))]; (%o4) [n, abs(n), max(6, n)] (%i5) assume (x > 0, x < 1)$ (%i6) ceiling (x); (%o6) 1 (%i7) tex (ceiling (a)); $$\left \lceil a \right \rceil$$ (%o7) false
The ceiling function distributes over lists, matrices and equations.
See distribute_over.
Finally, for all inputs that are manifestly complex, ceiling returns
a noun form.
If the range of a function is a subset of the integers, it can be declared to
be integervalued. Both the ceiling and floor functions
can use this information; for example:
(%i1) declare (f, integervalued)$ (%i2) floor (f(x)); (%o2) f(x) (%i3) ceiling (f(x) - 1); (%o3) f(x) - 1
Example use:
(%i1) unitfrac(r) := block([uf : [], q],
if not(ratnump(r)) then
error("unitfrac: argument must be a rational number"),
while r # 0 do (
uf : cons(q : 1/ceiling(1/r), uf),
r : r - q),
reverse(uf));
(%o1) unitfrac(r) := block([uf : [], q],
if not ratnump(r) then error("unitfrac: argument must be a rational number"
1
), while r # 0 do (uf : cons(q : ----------, uf), r : r - q),
1
ceiling(-)
r
reverse(uf))
(%i2) unitfrac (9/10);
1 1 1
(%o2) [-, -, --]
2 3 15
(%i3) apply ("+", %);
9
(%o3) --
10
(%i4) unitfrac (-9/10);
1
(%o4) [- 1, --]
10
(%i5) apply ("+", %);
9
(%o5) - --
10
(%i6) unitfrac (36/37);
1 1 1 1 1
(%o6) [-, -, -, --, ----]
2 3 8 69 6808
(%i7) apply ("+", %);
36
(%o7) --
37
Categories: Mathematical functions
Returns the largest integer less than or equal to x where x is
numeric. fix (as in fixnum) is a synonym for this, so
fix(x) is precisely the same.
Categories: Mathematical functions
When x is a real number, return the largest integer that is less than or equal to x.
If x is a constant expression (10 * %pi, for example), floor
evaluates x using big floating point numbers, and applies floor to
the resulting big float. Because floor uses floating point evaluation,
it's possible, although unlikely, that floor could return an erroneous
value for constant inputs. To guard against errors, the floating point
evaluation is done using three values for fpprec.
For non-constant inputs, floor tries to return a simplified value. Here
are examples of the simplifications that floor knows about:
(%i1) floor (ceiling (x)); (%o1) ceiling(x) (%i2) floor (floor (x)); (%o2) floor(x) (%i3) declare (n, integer)$ (%i4) [floor (n), floor (abs (n)), floor (min (n, 6))]; (%o4) [n, abs(n), min(6, n)] (%i5) assume (x > 0, x < 1)$ (%i6) floor (x); (%o6) 0 (%i7) tex (floor (a)); $$\left \lfloor a \right \rfloor$$ (%o7) false
The floor function distributes over lists, matrices and equations.
See distribute_over.
Finally, for all inputs that are manifestly complex, floor returns
a noun form.
If the range of a function is a subset of the integers, it can be declared to
be integervalued. Both the ceiling and floor functions
can use this information; for example:
(%i1) declare (f, integervalued)$ (%i2) floor (f(x)); (%o2) f(x) (%i3) ceiling (f(x) - 1); (%o3) f(x) - 1
Categories: Mathematical functions
A synonym for entier (x).
Categories: Mathematical functions
When L is a list or a set, return apply ('max, args (L)).
When L is not a list or a set, signal an error.
See also lmin and max.
Categories: Mathematical functions · Lists · Sets
When L is a list or a set, return apply ('min, args (L)).
When L is not a list or a set, signal an error.
See also lmax and min.
Categories: Mathematical functions · Lists · Sets
Return a simplified value for the maximum of the expressions x_1 through
x_n. When get (trylevel, maxmin), is 2 or greater, max
uses the simplification max (e, -e) --> |e|. When
get (trylevel, maxmin) is 3 or greater, max tries to eliminate
expressions that are between two other arguments; for example,
max (x, 2*x, 3*x) --> max (x, 3*x). To set the value of trylevel
to 2, use put (trylevel, 2, maxmin).
Categories: Mathematical functions
Return a simplified value for the minimum of the expressions x_1 through
x_n. When get (trylevel, maxmin), is 2 or greater, min
uses the simplification min (e, -e) --> -|e|. When
get (trylevel, maxmin) is 3 or greater, min tries to eliminate
expressions that are between two other arguments; for example,
min (x, 2*x, 3*x) --> min (x, 3*x). To set the value of trylevel
to 2, use put (trylevel, 2, maxmin).
Categories: Mathematical functions
When x is a real number, returns the closest integer to x.
Multiples of 1/2 are rounded to the nearest even integer. Evaluation of
x is similar to floor and ceiling.
The round function distributes over lists, matrices and equations.
See distribute_over.
Categories: Mathematical functions
For either real or complex numbers x, the signum function returns
0 if x is zero; for a nonzero numeric input x, the signum function
returns x/abs(x).
For non-numeric inputs, Maxima attempts to determine the sign of the input.
When the sign is negative, zero, or positive, signum returns -1,0, 1,
respectively. For all other values for the sign, signum a simplified but
equivalent form. The simplifications include reflection (signum(-x)
gives -signum(x)) and multiplicative identity (signum(x*y) gives
signum(x) * signum(y)).
The signum function distributes over a list, a matrix, or an
equation. See distribute_over.
Categories: Mathematical functions
When x is a real number, return the closest integer to x not
greater in absolute value than x. Evaluation of x is similar
to floor and ceiling.
The truncate function distributes over lists, matrices and equations.
See distribute_over.
Categories: Mathematical functions
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Calculates the absolute value of an expression representing a complex
number. Unlike the function abs, the cabs function always
decomposes its argument into a real and an imaginary part. If x and
y represent real variables or expressions, the cabs function
calculates the absolute value of x + %i*y as
(%i1) cabs (1); (%o1) 1 (%i2) cabs (1 + %i); (%o2) sqrt(2) (%i3) cabs (exp (%i)); (%o3) 1 (%i4) cabs (exp (%pi * %i)); (%o4) 1 (%i5) cabs (exp (3/2 * %pi * %i)); (%o5) 1 (%i6) cabs (17 * exp (2 * %i)); (%o6) 17
If cabs returns a noun form this most commonly is caused by
some properties of the variables involved not being known:
(%i1) cabs (a+%i*b);
2 2
(%o1) sqrt(b + a )
(%i2) declare(a,real,b,real);
(%o2) done
(%i3) cabs (a+%i*b);
2 2
(%o3) sqrt(b + a )
(%i4) assume(a>0,b>0);
(%o4) [a > 0, b > 0]
(%i5) cabs (a+%i*b);
2 2
(%o5) sqrt(b + a )
The cabs function can use known properties like symmetry properties of
complex functions to help it calculate the absolute value of an expression. If
such identities exist, they can be advertised to cabs using function
properties. The symmetries that cabs understands are: mirror symmetry,
conjugate function and complex characteristic.
cabs is a verb function and is not suitable for symbolic
calculations. For such calculations (including integration,
differentiation and taking limits of expressions containing absolute
values), use abs.
The result of cabs can include the absolute value function,
abs, and the arc tangent, atan2.
When applied to a list or matrix, cabs automatically distributes over
the terms. Similarly, it distributes over both sides of an equation.
For further ways to compute with complex numbers, see the functions
rectform, realpart, imagpart,
carg, conjugate and polarform.
Examples:
(%i1) cabs(sqrt(1+%i*x));
2 1/4
(%o1) (x + 1)
(%i2) cabs(sin(x+%i*y));
2 2 2 2
(%o2) sqrt(cos (x) sinh (y) + sin (x) cosh (y))
The error function, erf, has mirror symmetry, which is used here in
the calculation of the absolute value with a complex argument:
(%i3) cabs(erf(x+%i*y));
2
(erf(%i y + x) - erf(%i y - x))
(%o3) sqrt(--------------------------------
4
2
(erf(%i y + x) + erf(%i y - x))
- --------------------------------)
4
Maxima knows complex identities for the Bessel functions, which allow
it to compute the absolute value for complex arguments. Here is an
example for bessel_j.
(%i4) cabs(bessel_j(1,%i)); (%o4) abs(bessel_j(1, %i))
Categories: Complex variables
Returns the complex argument of z. The complex argument is an angle
theta in (-%pi, %pi] such that r exp (theta %i) = z
where r is the magnitude of z.
carg is a computational function, not a simplifying function.
See also abs (complex magnitude), polarform,
rectform, realpart, and imagpart.
Examples:
(%i1) carg (1);
(%o1) 0
(%i2) carg (1 + %i);
%pi
(%o2) ---
4
(%i3) carg (exp (%i));
sin(1)
(%o3) atan(------)
cos(1)
(%i4) carg (exp (%pi * %i));
(%o4) %pi
(%i5) carg (exp (3/2 * %pi * %i));
%pi
(%o5) - ---
2
(%i6) carg (17 * exp (2 * %i));
sin(2)
(%o6) atan(------) + %pi
cos(2)
If carg returns a noun form this most communly is caused by
some properties of the variables involved not being known:
(%i1) carg (a+%i*b);
(%o1) atan2(b, a)
(%i2) declare(a,real,b,real);
(%o2) done
(%i3) carg (a+%i*b);
(%o3) atan2(b, a)
(%i4) assume(a>0,b>0);
(%o4) [a > 0, b > 0]
(%i5) carg (a+%i*b);
b
(%o5) atan(-)
a
Categories: Complex variables
Returns the complex conjugate of x.
(%i1) declare ([aa, bb], real, cc, complex, ii, imaginary); (%o1) done (%i2) conjugate (aa + bb*%i); (%o2) aa - %i bb (%i3) conjugate (cc); (%o3) conjugate(cc) (%i4) conjugate (ii); (%o4) - ii (%i5) conjugate (xx + yy); (%o5) yy + xx
Categories: Complex variables
Returns the imaginary part of the expression expr.
imagpart is a computational function, not a simplifying function.
See also abs, carg, polarform,
rectform, and realpart.
Example:
(%i1) imagpart (a+b*%i); (%o1) b (%i2) imagpart (1+sqrt(2)*%i); (%o2) sqrt(2) (%i3) imagpart (1); (%o3) 0 (%i4) imagpart (sqrt(2)*%i); (%o4) sqrt(2)
Categories: Complex variables
Returns an expression r %e^(%i theta) equivalent to expr,
such that r and theta are purely real.
Example:
(%i1) polarform(a+b*%i);
2 2 %i atan2(b, a)
(%o1) sqrt(b + a ) %e
(%i2) polarform(1+%i);
%i %pi
------
4
(%o2) sqrt(2) %e
(%i3) polarform(1+2*%i);
%i atan(2)
(%o3) sqrt(5) %e
Categories: Complex variables · Exponential and logarithm functions
Returns the real part of expr. realpart and imagpart will
work on expressions involving trigonometric and hyperbolic functions,
as well as square root, logarithm, and exponentiation.
Example:
(%i1) realpart (a+b*%i); (%o1) a (%i2) realpart (1+sqrt(2)*%i); (%o2) 1 (%i3) realpart (sqrt(2)*%i); (%o3) 0 (%i4) realpart (1); (%o4) 1
Categories: Complex variables
Returns an expression a + b %i equivalent to expr,
such that a and b are purely real.
Example:
(%i1) rectform(sqrt(2)*%e^(%i*%pi/4)); (%o1) %i + 1 (%i2) rectform(sqrt(b^2+a^2)*%e^(%i*atan2(b, a))); (%o2) %i b + a (%i3) rectform(sqrt(5)*%e^(%i*atan(2))); (%o3) 2 %i + 1
Categories: Complex variables
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The double factorial operator.
For an integer, float, or rational number n, n!! evaluates to the
product n (n-2) (n-4) (n-6) ... (n - 2 (k-1)) where k is equal to
entier (n/2), that is, the largest integer less than or equal to
n/2. Note that this definition does not coincide with other published
definitions for arguments which are not integers.
For an even (or odd) integer n, n!! evaluates to the product of
all the consecutive even (or odd) integers from 2 (or 1) through n
inclusive.
For an argument n which is not an integer, float, or rational, n!!
yields a noun form genfact (n, n/2, 2).
Categories: Gamma and factorial functions · Operators
The binomial coefficient x!/(y! (x - y)!).
If x and y are integers, then the numerical value of the binomial
coefficient is computed. If y, or x - y, is an integer, the
binomial coefficient is expressed as a polynomial.
Examples:
(%i1) binomial (11, 7);
(%o1) 330
(%i2) 11! / 7! / (11 - 7)!;
(%o2) 330
(%i3) binomial (x, 7);
(x - 6) (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) x
(%o3) -------------------------------------------------
5040
(%i4) binomial (x + 7, x);
(x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x + 6) (x + 7)
(%o4) -------------------------------------------------------
5040
(%i5) binomial (11, y);
(%o5) binomial(11, y)
Categories: Number theory
Tries to combine the coefficients of factorials in expr
with the factorials themselves by converting, for example, (n + 1)*n!
into (n + 1)!.
sumsplitfact if set to false will cause minfactorial to be
applied after a factcomb.
Example:
(%i1) sumsplitfact;
(%o1) true
(%i2) (n + 1)*(n + 1)*n!;
2
(%o2) (n + 1) n!
(%i3) factcomb (%);
(%o3) (n + 2)! - (n + 1)!
(%i4) sumsplitfact: not sumsplitfact;
(%o4) false
(%i5) (n + 1)*(n + 1)*n!;
2
(%o5) (n + 1) n!
(%i6) factcomb (%);
(%o6) n (n + 1)! + (n + 1)!
Categories: Gamma and factorial functions
Represents the factorial function. Maxima treats factorial (x)
the same as x!.
For any complex number x, except for negative integers, x! is
defined as gamma(x+1).
For an integer x, x! simplifies to the product of the integers
from 1 to x inclusive. 0! simplifies to 1. For a real or complex
number in float or bigfloat precision x, x! simplifies to the
value of gamma (x+1). For x equal to n/2 where n is
an odd integer, x! simplifies to a rational factor times
sqrt (%pi) (since gamma (1/2) is equal to sqrt (%pi)).
The option variables factlim and gammalim control the numerical
evaluation of factorials for integer and rational arguments. The functions
minfactorial and factcomb simplifies expressions containing
factorials.
The functions gamma, bffac, and cbffac are
varieties of the gamma function. bffac and cbffac are called
internally by gamma to evaluate the gamma function for real and complex
numbers in bigfloat precision.
makegamma substitutes gamma for factorials and related functions.
Maxima knows the derivative of the factorial function and the limits for specific values like negative integers.
The option variable factorial_expand controls the simplification of
expressions like (n+x)!, where n is an integer.
See also binomial.
The factorial of an integer is simplified to an exact number unless the operand
is greater than factlim. The factorial for real and complex numbers is
evaluated in float or bigfloat precision.
(%i1) factlim : 10;
(%o1) 10
(%i2) [0!, (7/2)!, 8!, 20!];
105 sqrt(%pi)
(%o2) [1, -------------, 40320, 20!]
16
(%i3) [4,77!, (1.0+%i)!];
(%o3) [4, 77!, 0.3430658398165453 %i + 0.6529654964201667]
(%i4) [2.86b0!, 1.0b0+%i)!];
incorrect syntax: Missing ]
[2.86b0!, 1.0b0+%i)
^
The factorial of a known constant, or general expression is not simplified. Even so it may be possible to simplify the factorial after evaluating the operand.
(%i1) [(%i + 1)!, %pi!, %e!, (cos(1) + sin(1))!];
(%o1) [(%i + 1)!, %pi!, %e!, (sin(1) + cos(1))!]
(%i2) ev (%, numer, %enumer);
(%o2) [0.3430658398165453 %i + 0.6529654964201667,
7.188082728976031, 4.260820476357003, 1.227580202486819]
Factorials are simplified, not evaluated.
Thus x! may be replaced even in a quoted expression.
(%i1) '([0!, (7/2)!, 4.77!, 8!, 20!]);
105 sqrt(%pi)
(%o1) [1, -------------, 81.44668037931197, 40320,
16
2432902008176640000]
Maxima knows the derivative of the factorial function.
(%i1) diff(x!,x);
(%o1) x! psi (x + 1)
0
The option variable factorial_expand controls expansion and
simplification of expressions with the factorial function.
(%i1) (n+1)!/n!,factorial_expand:true; (%o1) n + 1
Categories: Gamma and factorial functions · Operators
Default value: 100000
factlim specifies the highest factorial which is
automatically expanded. If it is -1 then all integers are expanded.
Categories: Gamma and factorial functions
Default value: false
The option variable factorial_expand controls the simplification of
expressions like (n+1)!, where n is an integer.
See factorial for an example.
Categories: Gamma and factorial functions
Returns the generalized factorial, defined as
x (x-z) (x - 2 z) ... (x - (y - 1) z). Thus, when x is an integer,
genfact (x, x, 1) = x! and genfact (x, x/2, 2) = x!!.
Categories: Gamma and factorial functions
Examines expr for occurrences of two factorials
which differ by an integer.
minfactorial then turns one into a polynomial times the other.
(%i1) n!/(n+2)!;
n!
(%o1) --------
(n + 2)!
(%i2) minfactorial (%);
1
(%o2) ---------------
(n + 1) (n + 2)
Categories: Number theory
Default value: true
When sumsplitfact is false,
minfactorial is applied after a factcomb.
(%i1) sumsplitfact;
(%o1) true
(%i2) n!/(n+2)!;
n!
(%o2) --------
(n + 2)!
(%i3) factcomb(%);
n!
(%o3) --------
(n + 2)!
(%i4) sumsplitfact: not sumsplitfact ;
(%o4) false
(%i5) n!/(n+2)!;
n!
(%o5) --------
(n + 2)!
(%i6) factcomb(%);
1
(%o6) ---------------
(n + 1) (n + 2)
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Default value: false
When true, r some rational number, and x some expression,
%e^(r*log(x)) will be simplified into x^r . It should be noted
that the radcan command also does this transformation, and more
complicated transformations of this ilk as well. The logcontract
command "contracts" expressions containing log.
Default value: true
When %emode is true, %e^(%pi %i x) is simplified as
follows.
%e^(%pi %i x) simplifies to cos (%pi x) + %i sin (%pi x) if
x is a floating point number, an integer, or a multiple of 1/2, 1/3, 1/4,
or 1/6, and then further simplified.
For other numerical x, %e^(%pi %i x) simplifies to
%e^(%pi %i y) where y is x - 2 k for some integer k
such that abs(y) < 1.
When %emode is false, no special simplification of
%e^(%pi %i x) is carried out.
(%i1) %emode;
(%o1) true
(%i2) %e^(%pi*%i*1);
(%o2) - 1
(%i3) %e^(%pi*%i*216/144);
(%o3) - %i
(%i4) %e^(%pi*%i*192/144);
sqrt(3) %i 1
(%o4) (- ----------) - -
2 2
(%i5) %e^(%pi*%i*180/144);
%i 1
(%o5) (- -------) - -------
sqrt(2) sqrt(2)
(%i6) %e^(%pi*%i*120/144);
%i sqrt(3)
(%o6) -- - -------
2 2
(%i7) %e^(%pi*%i*121/144);
121 %i %pi
----------
144
(%o7) %e
Default value: false
When %enumer is true, %e is replaced by its numeric value
2.718… whenever numer is true.
When %enumer is false, this substitution is carried out
only if the exponent in %e^x evaluates to a number.
(%i1) %enumer;
(%o1) false
(%i2) numer;
(%o2) false
(%i3) 2*%e;
(%o3) 2 %e
(%i4) %enumer: not %enumer;
(%o4) true
(%i5) 2*%e;
(%o5) 2 %e
(%i6) numer: not numer;
(%o6) true
(%i7) 2*%e;
(%o7) 5.43656365691809
(%i8) 2*%e^1;
(%o8) 5.43656365691809
(%i9) 2*%e^x;
x
(%o9) 2 2.718281828459045
Categories: Exponential and logarithm functions · Evaluation flags
Represents the exponential function. Instances of exp (x) in input
are simplified to %e^x; exp does not appear in simplified
expressions.
demoivre if true causes %e^(a + b %i) to simplify to
%e^(a (cos(b) + %i sin(b))) if b is free of %i.
See demoivre.
%emode, when true, causes %e^(%pi %i x) to be simplified.
See %emode.
%enumer, when true causes %e to be replaced by
2.718… whenever numer is true. See %enumer.
(%i1) demoivre;
(%o1) false
(%i2) %e^(a + b*%i);
%i b + a
(%o2) %e
(%i3) demoivre: not demoivre;
(%o3) true
(%i4) %e^(a + b*%i);
a
(%o4) %e (%i sin(b) + cos(b))
Categories: Exponential and logarithm functions
Represents the polylogarithm function of order s and argument z, defined by the infinite series
inf
==== k
\ z
Li (z) = > --
s / s
==== k
k = 1
li [1] is - log (1 - z). li [2] and li [3] are the
dilogarithm and trilogarithm functions, respectively.
When the order is 1, the polylogarithm simplifies to - log (1 - z), which
in turn simplifies to a numerical value if z is a real or complex floating
point number or the numer evaluation flag is present.
When the order is 2 or 3,
the polylogarithm simplifies to a numerical value
if z is a real floating point number
or the numer evaluation flag is present.
Examples:
(%i1) assume (x > 0);
(%o1) [x > 0]
(%i2) integrate ((log (1 - t)) / t, t, 0, x);
(%o2) - li (x)
2
(%i3) li [2] (7);
(%o3) li (7)
2
(%i4) li [2] (7), numer;
(%o4) 1.248273182099423 - 6.113257028817991 %i
(%i5) li [3] (7);
(%o5) li (7)
3
(%i6) li [2] (7), numer;
(%o6) 1.248273182099423 - 6.113257028817991 %i
(%i7) L : makelist (i / 4.0, i, 0, 8);
(%o7) [0.0, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0]
(%i8) map (lambda ([x], li [2] (x)), L);
(%o8) [0.0, 0.2676526390827326, 0.5822405264650125,
0.978469392930306, 1.644934066848226,
2.190177011441645 - 0.7010261415046585 %i,
2.37439527027248 - 1.2738062049196 %i,
2.448686765338205 - 1.758084848210787 %i,
2.467401100272339 - 2.177586090303601 %i]
(%i9) map (lambda ([x], li [3] (x)), L);
(%o9) [0.0, 0.2584613953442624, 0.537213192678042,
0.8444258046482203, 1.2020569, 1.642866878950322
- 0.07821473130035025 %i, 2.060877505514697
- 0.2582419849982037 %i, 2.433418896388322
- 0.4919260182322965 %i, 2.762071904015935
- 0.7546938285978846 %i]
Categories: Exponential and logarithm functions
Represents the natural (base e) logarithm of x.
Maxima does not have a built-in function for the base 10 logarithm or other
bases. log10(x) := log(x) / log(10) is a useful definition.
Simplification and evaluation of logarithms is governed by several global flags:
logexpandcauses log(a^b) to become b*log(a). If it is
set to all, log(a*b) will also simplify to log(a)+log(b).
If it is set to super, then log(a/b) will also simplify to
log(a)-log(b) for rational numbers a/b, a#1.
(log(1/b), for b integer, always simplifies.) If it is set to
false, all of these simplifications will be turned off.
logsimpif false then no simplification of %e to a power containing
log's is done.
lognegintif true implements the rule log(-n) -> log(n)+%i*%pi for
n a positive integer.
%e_to_numlogwhen true, r some rational number, and x some expression,
the expression %e^(r*log(x)) will be simplified into x^r. It
should be noted that the radcan command also does this transformation,
and more complicated transformations of this as well. The logcontract
command "contracts" expressions containing log.
Categories: Exponential and logarithm functions
Default value: false
When doing indefinite integration where logs are generated, e.g.
integrate(1/x,x), the answer is given in terms of log(abs(...))
if logabs is true, but in terms of log(...) if
logabs is false. For definite integration, the logabs:true
setting is used, because here "evaluation" of the indefinite integral at the
endpoints is often needed.
Categories: Exponential and logarithm functions · Integral calculus · Global flags
When the global variable logarc is true,
inverse circular and hyperbolic functions are replaced by
equivalent logarithmic functions.
The default value of logarc is false.
The function logarc(expr) carries out that replacement for
an expression expr
without setting the global variable logarc.
Default value: false
Controls which coefficients are
contracted when using logcontract. It may be set to the name of a
predicate function of one argument. E.g. if you like to generate
SQRTs, you can do logconcoeffp:'logconfun$
logconfun(m):=featurep(m,integer) or ratnump(m)$ . Then
logcontract(1/2*log(x)); will give log(sqrt(x)).
Recursively scans the expression expr, transforming
subexpressions of the form a1*log(b1) + a2*log(b2) + c into
log(ratsimp(b1^a1 * b2^a2)) + c
(%i1) 2*(a*log(x) + 2*a*log(y))$
(%i2) logcontract(%);
2 4
(%o2) a log(x y )
The declaration declare(n,integer) causes
logcontract(2*a*n*log(x)) to simplify to a*log(x^(2*n)). The
coefficients that "contract" in this manner are those such as the 2 and the
n here which satisfy featurep(coeff,integer). The user can
control which coefficients are contracted by setting the option
logconcoeffp to the name of a predicate function of one argument.
E.g. if you like to generate SQRTs, you can do logconcoeffp:'logconfun$
logconfun(m):=featurep(m,integer) or ratnump(m)$ . Then
logcontract(1/2*log(x)); will give log(sqrt(x)).
Categories: Exponential and logarithm functions
Default value: true
If true, that is the default value, causes log(a^b) to become
b*log(a). If it is set to all, log(a*b) will also simplify
to log(a)+log(b). If it is set to super, then log(a/b)
will also simplify to log(a)-log(b) for rational numbers a/b,
a#1. (log(1/b), for integer b, always simplifies.) If it
is set to false, all of these simplifications will be turned off.
Default value: false
If true implements the rule
log(-n) -> log(n)+%i*%pi for n a positive integer.
Default value: true
If false then no simplification of %e to a
power containing log's is done.
Represents the principal branch of the complex-valued natural
logarithm with -%pi < carg(x) <= +%pi .
Categories: Exponential and logarithm functions · Complex variables
The square root of x. It is represented internally by
x^(1/2). See also rootscontract and radexpand.
Categories: Mathematical functions
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| 10.5.1 Introduction to Trigonometric | ||
| 10.5.2 Functions and Variables for Trigonometric |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Maxima has many trigonometric functions defined. Not all trigonometric
identities are programmed, but it is possible for the user to add many
of them using the pattern matching capabilities of the system. The
trigonometric functions defined in Maxima are: acos,
acosh, acot, acoth, acsc,
acsch, asec, asech, asin,
asinh, atan, atanh, cos,
cosh, cot, coth, csc, csch,
sec, sech, sin, sinh, tan,
and tanh. There are a number of commands especially for
handling trigonometric functions, see trigexpand,
trigreduce, and the switch trigsign. Two share
packages extend the simplification rules built into Maxima,
ntrig and atrig1. Do describe(command)
for details.
Categories: Trigonometric functions
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Default value: true
When %piargs is true,
trigonometric functions are simplified to algebraic constants
when the argument is an integer multiple of
%pi, %pi/2, %pi/3, %pi/4, or %pi/6.
Maxima knows some identities which can be applied when %pi, etc., are multiplied by an integer variable (that is, a symbol declared to be integer).
Examples:
(%i1) %piargs : false$
(%i2) [sin (%pi), sin (%pi/2), sin (%pi/3)];
%pi %pi
(%o2) [sin(%pi), sin(---), sin(---)]
2 3
(%i3) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
%pi %pi %pi
(%o3) [sin(---), sin(---), sin(---)]
4 5 6
(%i4) %piargs : true$
(%i5) [sin (%pi), sin (%pi/2), sin (%pi/3)];
sqrt(3)
(%o5) [0, 1, -------]
2
(%i6) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
1 %pi 1
(%o6) [-------, sin(---), -]
sqrt(2) 5 2
(%i7) [cos (%pi/3), cos (10*%pi/3), tan (10*%pi/3),
cos (sqrt(2)*%pi/3)];
1 1 sqrt(2) %pi
(%o7) [-, - -, sqrt(3), cos(-----------)]
2 2 3
Some identities are applied when %pi and %pi/2 are multiplied by an integer variable.
(%i1) declare (n, integer, m, even)$
(%i2) [sin (%pi * n), cos (%pi * m), sin (%pi/2 * m),
cos (%pi/2 * m)];
m/2
(%o2) [0, 1, 0, (- 1) ]
Categories: Trigonometric functions · Simplification flags and variables
Default value: true
When %iargs is true,
trigonometric functions are simplified to hyperbolic functions
when the argument is apparently a multiple of the imaginary unit %i.
Even when the argument is demonstrably real, the simplification is applied; Maxima considers only whether the argument is a literal multiple of %i.
Examples:
(%i1) %iargs : false$ (%i2) [sin (%i * x), cos (%i * x), tan (%i * x)]; (%o2) [sin(%i x), cos(%i x), tan(%i x)] (%i3) %iargs : true$ (%i4) [sin (%i * x), cos (%i * x), tan (%i * x)]; (%o4) [%i sinh(x), cosh(x), %i tanh(x)]
Even when the argument is demonstrably real, the simplification is applied.
(%i1) declare (x, imaginary)$ (%i2) [featurep (x, imaginary), featurep (x, real)]; (%o2) [true, false] (%i3) sin (%i * x); (%o3) %i sinh(x)
- Arc Cosine.
Categories: Trigonometric functions
- Hyperbolic Arc Cosine.
Categories: Hyperbolic functions
- Arc Cotangent.
Categories: Trigonometric functions
- Hyperbolic Arc Cotangent.
Categories: Hyperbolic functions
- Arc Cosecant.
Categories: Trigonometric functions
- Hyperbolic Arc Cosecant.
Categories: Hyperbolic functions
- Arc Secant.
Categories: Trigonometric functions
- Hyperbolic Arc Secant.
Categories: Hyperbolic functions
- Arc Sine.
Categories: Trigonometric functions
- Hyperbolic Arc Sine.
Categories: Hyperbolic functions
- Arc Tangent.
Categories: Trigonometric functions
- yields the value of atan(y/x) in the interval -%pi
to %pi.
Categories: Trigonometric functions
- Hyperbolic Arc Tangent.
Categories: Hyperbolic functions
The atrig1 package contains several additional simplification rules
for inverse trigonometric functions. Together with rules
already known to Maxima, the following angles are fully implemented:
0, %pi/6, %pi/4, %pi/3, and %pi/2.
Corresponding angles in the other three quadrants are also available.
Do load(atrig1); to use them.
Categories: Trigonometric functions · Package atrig1
- Cosine.
Categories: Trigonometric functions
- Hyperbolic Cosine.
Categories: Hyperbolic functions
- Cotangent.
Categories: Trigonometric functions
- Hyperbolic Cotangent.
Categories: Hyperbolic functions
- Cosecant.
Categories: Trigonometric functions
- Hyperbolic Cosecant.
Categories: Hyperbolic functions
Default value: false
When halfangles is true, trigonometric functions of arguments
expr/2 are simplified to functions of expr.
For a real argument x in the interval 0 < x < 2*%pi the sine of
the half-angle simplifies to a simple formula:
sqrt(1 - cos(x))
----------------
sqrt(2)
A complicated factor is needed to make this formula correct for all complex arguments z:
realpart(z)
floor(-----------)
2 %pi
(- 1) (1 - unit_step(- imagpart(z))
realpart(z) realpart(z)
floor(-----------) - ceiling(-----------)
2 %pi 2 %pi
((- 1) + 1))
Maxima knows this factor and similar factors for the functions sin,
cos, sinh, and cosh. For special values of the argument
z these factors simplify accordingly.
Examples:
(%i1) halfangles : false$
(%i2) sin (x / 2);
x
(%o2) sin(-)
2
(%i3) halfangles : true$
(%i4) sin (x / 2);
x
floor(-----)
2 %pi
(- 1) sqrt(1 - cos(x))
(%o4) ----------------------------------
sqrt(2)
(%i5) assume(x>0, x<2*%pi)$
(%i6) sin(x / 2);
sqrt(1 - cos(x))
(%o6) ----------------
sqrt(2)
Categories: Trigonometric functions · Simplification flags and variables
The ntrig package contains a set of simplification rules that are
used to simplify trigonometric function whose arguments are of the form
f(n %pi/10) where f is any of the functions
sin, cos, tan, csc, sec and cot.
Categories: Trigonometric functions · Package ntrig
- Secant.
Categories: Trigonometric functions
- Hyperbolic Secant.
Categories: Hyperbolic functions
- Sine.
Categories: Trigonometric functions
- Hyperbolic Sine.
Categories: Hyperbolic functions
- Tangent.
Categories: Trigonometric functions
- Hyperbolic Tangent.
Categories: Hyperbolic functions
Expands trigonometric and hyperbolic functions of
sums of angles and of multiple angles occurring in expr. For best
results, expr should be expanded. To enhance user control of
simplification, this function expands only one level at a time,
expanding sums of angles or multiple angles. To obtain full expansion
into sines and cosines immediately, set the switch trigexpand: true.
trigexpand is governed by the following global flags:
trigexpandIf true causes expansion of all
expressions containing sin's and cos's occurring subsequently.
halfanglesIf true causes half-angles to be simplified
away.
trigexpandplusControls the "sum" rule for trigexpand,
expansion of sums (e.g. sin(x + y)) will take place only if
trigexpandplus is true.
trigexpandtimesControls the "product" rule for trigexpand,
expansion of products (e.g. sin(2 x)) will take place only if
trigexpandtimes is true.
Examples:
(%i1) x+sin(3*x)/sin(x),trigexpand=true,expand;
2 2
(%o1) (- sin (x)) + 3 cos (x) + x
(%i2) trigexpand(sin(10*x+y));
(%o2) cos(10 x) sin(y) + sin(10 x) cos(y)
Categories: Trigonometric functions · Simplification functions
Default value: true
trigexpandplus controls the "sum" rule for
trigexpand. Thus, when the trigexpand command is used or the
trigexpand switch set to true, expansion of sums
(e.g. sin(x+y)) will take place only if trigexpandplus is
true.
Categories: Trigonometric functions · Simplification flags and variables
Default value: true
trigexpandtimes controls the "product" rule for trigexpand.
Thus, when the trigexpand command is used or the trigexpand
switch set to true, expansion of products (e.g. sin(2*x))
will take place only if trigexpandtimes is true.
Categories: Trigonometric functions · Simplification flags and variables
Default value: true
triginverses controls the simplification of the
composition of trigonometric and hyperbolic functions with their inverse
functions.
If all, both e.g. atan(tan(x))
and tan(atan(x)) simplify to x.
If true, the arcfun(fun(x))
simplification is turned off.
If false, both the
arcfun(fun(x)) and
fun(arcfun(x))
simplifications are turned off.
Categories: Trigonometric functions · Simplification flags and variables
Combines products and powers of trigonometric and hyperbolic sin's and cos's of x into those of multiples of x. It also tries to eliminate these functions when they occur in denominators. If x is omitted then all variables in expr are used.
See also poissimp.
(%i1) trigreduce(-sin(x)^2+3*cos(x)^2+x);
cos(2 x) cos(2 x) 1 1
(%o1) -------- + 3 (-------- + -) + x - -
2 2 2 2
Categories: Trigonometric functions · Simplification functions
Default value: true
When trigsign is true, it permits simplification of negative
arguments to trigonometric functions. E.g., sin(-x) will become
-sin(x) only if trigsign is true.
Categories: Trigonometric functions · Simplification flags and variables
Employs the identities sin(x)^2 + cos(x)^2 = 1 and
cosh(x)^2 - sinh(x)^2 = 1 to simplify expressions containing tan,
sec, etc., to sin, cos, sinh, cosh.
trigreduce, ratsimp, and radcan may be
able to further simplify the result.
demo ("trgsmp.dem") displays some examples of trigsimp.
Categories: Trigonometric functions · Simplification functions
Gives a canonical simplified quasilinear form of a trigonometrical expression;
expr is a rational fraction of several sin, cos or
tan, the arguments of them are linear forms in some variables (or
kernels) and %pi/n (n integer) with integer coefficients.
The result is a simplified fraction with numerator and denominator linear in
sin and cos. Thus trigrat linearize always when it is
possible.
(%i1) trigrat(sin(3*a)/sin(a+%pi/3)); (%o1) sqrt(3) sin(2 a) + cos(2 a) - 1
The following example is taken from Davenport, Siret, and Tournier, Calcul Formel, Masson (or in English, Addison-Wesley), section 1.5.5, Morley theorem.
(%i1) c : %pi/3 - a - b$
(%i2) bc : sin(a)*sin(3*c)/sin(a+b);
%pi
sin(a) sin(3 ((- b) - a + ---))
3
(%o2) -------------------------------
sin(b + a)
(%i3) ba : bc, c=a, a=c;
%pi
sin(3 a) sin(b + a - ---)
3
(%o3) -------------------------
%pi
sin(a - ---)
3
(%i4) ac2 : ba^2 + bc^2 - 2*bc*ba*cos(b);
2 2 %pi
sin (3 a) sin (b + a - ---)
3
(%o4) ---------------------------
2 %pi
sin (a - ---)
3
%pi
- (2 sin(a) sin(3 a) sin(3 ((- b) - a + ---)) cos(b)
3
%pi %pi
sin(b + a - ---))/(sin(a - ---) sin(b + a))
3 3
2 2 %pi
sin (a) sin (3 ((- b) - a + ---))
3
+ ---------------------------------
2
sin (b + a)
(%i5) trigrat (ac2);
(%o5) - (sqrt(3) sin(4 b + 4 a) - cos(4 b + 4 a)
- 2 sqrt(3) sin(4 b + 2 a) + 2 cos(4 b + 2 a)
- 2 sqrt(3) sin(2 b + 4 a) + 2 cos(2 b + 4 a)
+ 4 sqrt(3) sin(2 b + 2 a) - 8 cos(2 b + 2 a) - 4 cos(2 b - 2 a)
+ sqrt(3) sin(4 b) - cos(4 b) - 2 sqrt(3) sin(2 b) + 10 cos(2 b)
+ sqrt(3) sin(4 a) - cos(4 a) - 2 sqrt(3) sin(2 a) + 10 cos(2 a)
- 9)/4
Categories: Trigonometric functions · Simplification functions
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A random state object represents the state of the random number generator. The state comprises 627 32-bit words.
make_random_state (n) returns a new random state object
created from an integer seed value equal to n modulo 2^32.
n may be negative.
make_random_state (s) returns a copy of the random state s.
make_random_state (true) returns a new random state object,
using the current computer clock time as the seed.
make_random_state (false) returns a copy of the current state
of the random number generator.
Categories: Random numbers
Copies s to the random number generator state.
set_random_state always returns done.
Categories: Random numbers
Returns a pseudorandom number. If x is an integer,
random (x) returns an integer from 0 through x - 1
inclusive. If x is a floating point number, random (x)
returns a nonnegative floating point number less than x. random
complains with an error if x is neither an integer nor a float, or if
x is not positive.
The functions make_random_state and set_random_state
maintain the state of the random number generator.
The Maxima random number generator is an implementation of the Mersenne twister MT 19937.
Examples:
(%i1) s1: make_random_state (654321)$ (%i2) set_random_state (s1); (%o2) done (%i3) random (1000); (%o3) 768 (%i4) random (9573684); (%o4) 7657880 (%i5) random (2^75); (%o5) 11804491615036831636390 (%i6) s2: make_random_state (false)$ (%i7) random (1.0); (%o7) 0.2310127244107132 (%i8) random (10.0); (%o8) 4.394553645870825 (%i9) random (100.0); (%o9) 32.28666704056853 (%i10) set_random_state (s2); (%o10) done (%i11) random (1.0); (%o11) 0.2310127244107132 (%i12) random (10.0); (%o12) 4.394553645870825 (%i13) random (100.0); (%o13) 32.28666704056853
Categories: Random numbers · Numerical methods
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| 11.1 Introduction to Maximas Database | ||
| 11.2 Functions and Variables for Properties | ||
| 11.3 Functions and Variables for Facts | ||
| 11.4 Functions and Variables for Predicates |
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alphabetic is a property type recognized by declare.
The expression declare(s, alphabetic) tells Maxima to recognize
as alphabetic all of the characters in s, which must be a string.
See also Identifiers.
Example:
(%i1) xx\~yy\`\@ : 1729;
(%o1) 1729
(%i2) declare ("~`@", alphabetic);
(%o2) done
(%i3) xx~yy`@ + @yy`xx + `xx@@yy~;
(%o3) `xx@@yy~ + @yy`xx + 1729
(%i4) listofvars (%);
(%o4) [@yy`xx, `xx@@yy~]
Categories: Declarations and inferences
The command declare(x, bindtest) tells Maxima to trigger an error
when the symbol x is evaluated unbound.
(%i1) aa + bb; (%o1) bb + aa (%i2) declare (aa, bindtest); (%o2) done (%i3) aa + bb; aa unbound variable -- an error. Quitting. To debug this try debugmode(true); (%i4) aa : 1234; (%o4) 1234 (%i5) aa + bb; (%o5) bb + 1234
declare(a, constant) declares a to be a constant. The
declaration of a symbol to be constant does not prevent the assignment of a
nonconstant value to the symbol.
Example:
(%i1) declare(c, constant); (%o1) done (%i2) constantp(c); (%o2) true (%i3) c : x; (%o3) x (%i4) constantp(c); (%o4) false
Categories: Declarations and inferences · Constants
Returns true if expr is a constant expression, otherwise returns
false.
An expression is considered a constant expression if its arguments are
numbers (including rational numbers, as displayed with /R/),
symbolic constants such as %pi, %e, and %i,
variables bound to a constant or declared constant by declare,
or functions whose arguments are constant.
constantp evaluates its arguments.
See the property constant which declares a symbol to be constant.
Examples:
(%i1) constantp (7 * sin(2)); (%o1) true (%i2) constantp (rat (17/29)); (%o2) true (%i3) constantp (%pi * sin(%e)); (%o3) true (%i4) constantp (exp (x)); (%o4) false (%i5) declare (x, constant); (%o5) done (%i6) constantp (exp (x)); (%o6) true (%i7) constantp (foo (x) + bar (%e) + baz (2)); (%o7) false (%i8)
Categories: Predicate functions · Constants
Assigns the atom or list of atoms a_i the property or list of properties p_i. When a_i and/or p_i are lists, each of the atoms gets all of the properties.
declare quotes its arguments. declare always returns done.
As noted in the description for each declaration flag, for some flags
featurep(object, feature) returns true if object
has been declared to have feature.
For more information about the features system, see features. To
remove a property from an atom, use remove.
declare recognizes the following properties:
additiveTells Maxima to simplify a_i expressions by the substitution
a_i(x + y + z + ...) -->
a_i(x) + a_i(y) + a_i(z) + ....
The substitution is carried out on the first argument only.
alphabeticTells Maxima to recognize all characters in a_i (which must be a string) as alphabetic characters.
antisymmetric, commutative, symmetricTells Maxima to recognize a_i as a symmetric or antisymmetric
function. commutative is the same as symmetric.
bindtestTells Maxima to trigger an error when a_i is evaluated unbound.
constantTells Maxima to consider a_i a symbolic constant.
even, oddTells Maxima to recognize a_i as an even or odd integer variable.
evenfun, oddfunTells Maxima to recognize a_i as an odd or even function.
evflagMakes a_i known to the ev function so that a_i is bound
to true during the execution of ev when a_i appears as
a flag argument of ev. See evflag.
evfunMakes a_i known to ev so that the function named by a_i
is applied when a_i appears as a flag argument of ev.
See evfun.
featureTells Maxima to recognize a_i as the name of a feature. Other atoms may then be declared to have the a_i property.
increasing, decreasingTells Maxima to recognize a_i as an increasing or decreasing function.
integer, nonintegerTells Maxima to recognize a_i as an integer or noninteger variable.
integervaluedTells Maxima to recognize a_i as an integer-valued function.
lassociative, rassociativeTells Maxima to recognize a_i as a right-associative or left-associative function.
linearEquivalent to declaring a_i both outative and
additive.
mainvarTells Maxima to consider a_i a "main variable". A main variable
succeeds all other constants and variables in the canonical ordering of
Maxima expressions, as determined by ordergreatp.
multiplicativeTells Maxima to simplify a_i expressions by the substitution
a_i(x * y * z * ...) -->
a_i(x) * a_i(y) * a_i(z) * ....
The substitution is carried out on the first argument only.
naryTells Maxima to recognize a_i as an n-ary function.
The nary declaration is not the same as calling the nary
function. The sole effect of declare(foo, nary) is to instruct the
Maxima simplifier to flatten nested expressions, for example, to simplify
foo(x, foo(y, z)) to foo(x, y, z).
nonarrayTells Maxima to consider a_i not an array. This declaration prevents multiple evaluation of a subscripted variable name.
nonscalarTells Maxima to consider a_i a nonscalar variable. The usual application is to declare a variable as a symbolic vector or matrix.
nounTells Maxima to parse a_i as a noun. The effect of this is to
replace instances of a_i with 'a_i or
nounify(a_i), depending on the context.
outativeTells Maxima to simplify a_i expressions by pulling constant factors out of the first argument.
When a_i has one argument, a factor is considered constant if it is a literal or declared constant.
When a_i has two or more arguments, a factor is considered constant if the second argument is a symbol and the factor is free of the second argument.
posfunTells Maxima to recognize a_i as a positive function.
rational, irrationalTells Maxima to recognize a_i as a rational or irrational real variable.
real, imaginary, complexTells Maxima to recognize a_i as a real, pure imaginary, or complex variable.
scalarTells Maxima to consider a_i a scalar variable.
Examples of the usage of the properties are available in the documentation for each separate description of a property.
Categories: Declarations and inferences
The commands declare(f, decreasing) or
declare(f, increasing) tell Maxima to recognize the function
f as an decreasing or increasing function.
See also declare for more properties.
Example:
(%i1) assume(a > b); (%o1) [a > b] (%i2) is(f(a) > f(b)); (%o2) unknown (%i3) declare(f, increasing); (%o3) done (%i4) is(f(a) > f(b)); (%o4) true
Categories: Declarations and inferences
declare(a, even) or declare(a, odd) tells Maxima to
recognize the symbol a as an even or odd integer variable. The
properties even and odd are not recognized by the functions
evenp, oddp, and integerp.
See also declare and askinteger.
Example:
(%i1) declare(n, even); (%o1) done (%i2) askinteger(n, even); (%o2) yes (%i3) askinteger(n); (%o3) yes (%i4) evenp(n); (%o4) false
Categories: Declarations and inferences
Maxima understands two distinct types of features, system features and features
which apply to mathematical expressions. See also status for information
about system features. See also features and featurep for
information about mathematical features.
feature itself is not the name of a function or variable.
Attempts to determine whether the object a has the feature f on the
basis of the facts in the current database. If so, it returns true,
else false.
Note that featurep returns false when neither f
nor the negation of f can be established.
featurep evaluates its argument.
See also declare and features.
(%i1) declare (j, even)$ (%i2) featurep (j, integer); (%o2) true
Categories: Predicate functions · Declarations and inferences
Maxima recognizes certain mathematical properties of functions and variables. These are called "features".
declare (x, foo) gives the property foo
to the function or variable x.
declare (foo, feature) declares a new feature foo.
For example,
declare ([red, green, blue], feature)
declares three new features, red, green, and blue.
The predicate featurep (x, foo)
returns true if x has the foo property,
and false otherwise.
The infolist features is a list of known features. These are
integer noninteger even odd rational irrational real imaginary complex analytic increasing decreasing oddfun evenfun posfun constant commutative lassociative rassociative symmetric antisymmetric integervalued
plus any user-defined features.
features is a list of mathematical features. There is also a list of
non-mathematical, system-dependent features. See status.
Example:
(%i1) declare (FOO, feature); (%o1) done (%i2) declare (x, FOO); (%o2) done (%i3) featurep (x, FOO); (%o3) true
Categories: Declarations and inferences
Retrieves the user property indicated by i associated with
atom a or returns false if a doesn't have property i.
get evaluates its arguments.
(%i1) put (%e, 'transcendental, 'type);
(%o1) transcendental
(%i2) put (%pi, 'transcendental, 'type)$
(%i3) put (%i, 'algebraic, 'type)$
(%i4) typeof (expr) := block ([q],
if numberp (expr)
then return ('algebraic),
if not atom (expr)
then return (maplist ('typeof, expr)),
q: get (expr, 'type),
if q=false
then errcatch (error(expr,"is not numeric.")) else q)$
(%i5) typeof (2*%e + x*%pi);
x is not numeric.
(%o5) [[transcendental, []], [algebraic, transcendental]]
(%i6) typeof (2*%e + %pi);
(%o6) [transcendental, [algebraic, transcendental]]
Categories: Declarations and inferences
declare(a, integer) or declare(a, noninteger) tells
Maxima to recognize a as an integer or noninteger variable.
See also declare.
Example:
(%i1) declare(n, integer, x, noninteger); (%o1) done (%i2) askinteger(n); (%o2) yes (%i3) askinteger(x); (%o3) no
Categories: Declarations and inferences
declare(f, integervalued) tells Maxima to recognize f as an
integer-valued function.
See also declare.
Example:
(%i1) exp(%i)^f(x);
%i f(x)
(%o1) (%e )
(%i2) declare(f, integervalued);
(%o2) done
(%i3) exp(%i)^f(x);
%i f(x)
(%o3) %e
Categories: Declarations and inferences
The command declare(a, nonarray) tells Maxima to consider a not
an array. This declaration prevents multiple evaluation, if a is a
subscripted variable.
See also declare.
Example:
(%i1) a:'b$ b:'c$ c:'d$
(%i4) a[x];
(%o4) d
x
(%i5) declare(a, nonarray);
(%o5) done
(%i6) a[x];
(%o6) a
x
Categories: Expressions
Makes atoms behave as does a list or matrix with respect to the dot operator.
See also declare.
Categories: Declarations and inferences · Vectors · Matrices
Returns true if expr is a non-scalar, i.e., it contains
atoms declared as non-scalars, lists, or matrices.
See also the predicate function scalarp and declare.
Categories: Predicate functions · Vectors · Matrices
declare (f, posfun) declares f to be a positive function.
is (f(x) > 0) yields true.
See also declare.
Categories: Declarations and inferences · Operators
Displays the property with the indicator i associated with the atom
a. a may also be a list of atoms or the atom all in which
case all of the atoms with the given property will be used. For example,
printprops ([f, g], atvalue). printprops is for properties that
cannot otherwise be displayed, i.e. for atvalue,
atomgrad, gradef, and matchdeclare.
Categories: Declarations and inferences · Display functions
Returns a list of the names of all the properties associated with the atom a.
Categories: Declarations and inferences
Default value: []
props are atoms which have any property other than those explicitly
mentioned in infolists, such as specified by atvalue,
matchdeclare, etc., as well as properties specified in the
declare function.
Categories: Declarations and inferences · Global variables
Returns a list of those atoms on the props list which
have the property indicated by prop. Thus propvars (atvalue)
returns a list of atoms which have atvalues.
Categories: Declarations and inferences
Assigns value to the property (specified by indicator) of atom. indicator may be the name of any property, not just a system-defined property.
rem reverses the effect of put.
put evaluates its arguments.
put returns value.
Examples:
(%i1) put (foo, (a+b)^5, expr);
5
(%o1) (b + a)
(%i2) put (foo, "Hello", str);
(%o2) Hello
(%i3) properties (foo);
(%o3) [[user properties, str, expr]]
(%i4) get (foo, expr);
5
(%o4) (b + a)
(%i5) get (foo, str);
(%o5) Hello
Categories: Declarations and inferences
Assigns value to the property (specified by indicator) of
atom. This is the same as put, except that the arguments are
quoted.
See also get.
Example:
(%i1) foo: aa$
(%i2) bar: bb$
(%i3) baz: cc$
(%i4) put (foo, bar, baz);
(%o4) bb
(%i5) properties (aa);
(%o5) [[user properties, cc]]
(%i6) get (aa, cc);
(%o6) bb
(%i7) qput (foo, bar, baz);
(%o7) bar
(%i8) properties (foo);
(%o8) [value, [user properties, baz]]
(%i9) get ('foo, 'baz);
(%o9) bar
Categories: Declarations and inferences
declare(a, rational) or declare(a, irrational) tells
Maxima to recognize a as a rational or irrational real variable.
See also declare.
Categories: Declarations and inferences
declare(a, real), declare(a, imaginary), or
declare(a, complex) tells Maxima to recognize a as a real,
pure imaginary, or complex variable.
See also declare.
Categories: Declarations and inferences
Removes the property indicated by indicator from atom.
rem reverses the effect of put.
rem returns done if atom had an indicator property
when rem was called, or false if it had no such property.
Categories: Declarations and inferences
Removes properties associated with atoms.
remove (a_1, p_1, ..., a_n, p_n)
removes property p_k from atom a_k.
remove ([a_1, ..., a_m], [p_1, ..., p_n], ...)
removes properties p_1, ..., p_n
from atoms a_1, …, a_m.
There may be more than one pair of lists.
remove (all, p) removes the property p from all atoms which
have it.
The removed properties may be system-defined properties such as
function, macro, or mode_declare.
remove does not remove properties defined by put.
A property may be transfun to remove
the translated Lisp version of a function.
After executing this, the Maxima version of the function is executed
rather than the translated version.
remove ("a", operator) or, equivalently,
remove ("a", op) removes from a the operator properties
declared by prefix, infix,
nary, postfix, matchfix, or
nofix. Note that the name of the operator must be written as a quoted
string.
remove always returns done whether or not an atom has a specified
property. This behavior is unlike the more specific remove functions
remvalue, remarray, remfunction, and
remrule.
remove quotes its arguments.
Categories: Declarations and inferences
declare(a, scalar) tells Maxima to consider a a scalar
variable.
See also declare.
Categories: Declarations and inferences
Returns true if expr is a number, constant, or variable declared
scalar with declare, or composed entirely of numbers,
constants, and such variables, but not containing matrices or lists.
See also the predicate function nonscalarp.
Categories: Predicate functions · Vectors · Matrices
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Activates the contexts context_1, …, context_n.
The facts in these contexts are then available to
make deductions and retrieve information.
The facts in these contexts are not listed by facts ().
The variable activecontexts is the list
of contexts which are active by way of the activate function.
Categories: Declarations and inferences
Default value: []
activecontexts is a list of the contexts which are active
by way of the activate function, as opposed to being active because
they are subcontexts of the current context.
Categories: Declarations and inferences
askinteger (expr, integer) attempts to determine from the
assume database whether expr is an integer.
askinteger prompts the user if it cannot tell otherwise,
and attempt to install the information in the database if possible.
askinteger (expr) is equivalent to
askinteger (expr, integer).
askinteger (expr, even) and askinteger (expr, odd)
likewise attempt to determine if expr is an even integer or odd integer,
respectively.
Categories: Declarations and inferences
First attempts to determine whether the specified
expression is positive, negative, or zero. If it cannot, it asks the
user the necessary questions to complete its deduction. The user's
answer is recorded in the data base for the duration of the current
computation. The return value of asksign is one of pos,
neg, or zero.
Categories: Declarations and inferences
Adds predicates pred_1, …, pred_n to the current context.
If a predicate is inconsistent or redundant with the predicates in the current
context, it is not added to the context. The context accumulates predicates
from each call to assume.
assume returns a list whose elements are the predicates added to the
context or the atoms redundant or inconsistent where applicable.
The predicates pred_1, …, pred_n can only be expressions
with the relational operators < <= equal notequal >= and >.
Predicates cannot be literal equality = or literal inequality #
expressions, nor can they be predicate functions such as integerp.
Compound predicates of the form pred_1 and ... and pred_n
are recognized, but not pred_1 or ... or pred_n.
not pred_k is recognized if pred_k is a relational predicate.
Expressions of the form not (pred_1 and pred_2)
and not (pred_1 or pred_2) are not recognized.
Maxima's deduction mechanism is not very strong;
there are many obvious consequences which cannot be determined by is.
This is a known weakness.
assume does not handle predicates with complex numbers. If a predicate
contains a complex number assume returns inconsistent or
redunant.
assume evaluates its arguments.
See also is, facts, forget,
context, and declare.
Examples:
(%i1) assume (xx > 0, yy < -1, zz >= 0); (%o1) [xx > 0, yy < - 1, zz >= 0] (%i2) assume (aa < bb and bb < cc); (%o2) [bb > aa, cc > bb] (%i3) facts (); (%o3) [xx > 0, - 1 > yy, zz >= 0, bb > aa, cc > bb] (%i4) is (xx > yy); (%o4) true (%i5) is (yy < -yy); (%o5) true (%i6) is (sinh (bb - aa) > 0); (%o6) true (%i7) forget (bb > aa); (%o7) [bb > aa] (%i8) prederror : false; (%o8) false (%i9) is (sinh (bb - aa) > 0); (%o9) unknown (%i10) is (bb^2 < cc^2); (%o10) unknown
Categories: Declarations and inferences
Default value: true
assumescalar helps govern whether expressions expr
for which nonscalarp (expr) is false
are assumed to behave like scalars for certain transformations.
Let expr represent any expression other than a list or a matrix,
and let [1, 2, 3] represent any list or matrix.
Then expr . [1, 2, 3] yields [expr, 2 expr, 3 expr]
if assumescalar is true, or scalarp (expr) is
true, or constantp (expr) is true.
If assumescalar is true, such
expressions will behave like scalars only for commutative
operators, but not for noncommutative multiplication ..
When assumescalar is false, such
expressions will behave like non-scalars.
When assumescalar is all, such expressions will behave like
scalars for all the operators listed above.
Categories: Declarations and inferences
Default value: false
When assume_pos is true and the sign of a parameter x
cannot be determined from the current context
or other considerations,
sign and asksign (x) return true.
This may forestall some automatically-generated asksign queries,
such as may arise from integrate or other computations.
By default, a parameter is x such that symbolp (x)
or subvarp (x).
The class of expressions considered parameters can be modified to some extent
via the variable assume_pos_pred.
sign and asksign attempt to deduce the sign of expressions
from the sign of operands within the expression.
For example, if a and b are both positive,
then a + b is also positive.
However, there is no way to bypass all asksign queries.
In particular, when the asksign argument is a
difference x - y or a logarithm log(x),
asksign always requests an input from the user,
even when assume_pos is true and assume_pos_pred is
a function which returns true for all arguments.
Categories: Declarations and inferences
Default value: false
When assume_pos_pred is assigned the name of a function
or a lambda expression of one argument x,
that function is called to determine
whether x is considered a parameter for the purpose of assume_pos.
assume_pos_pred is ignored when assume_pos is false.
The assume_pos_pred function is called by sign and asksign
with an argument x
which is either an atom, a subscripted variable, or a function call expression.
If the assume_pos_pred function returns true,
x is considered a parameter for the purpose of assume_pos.
By default, a parameter is x such that symbolp (x)
or subvarp (x).
See also assume and assume_pos.
Examples:
(%i1) assume_pos: true$
(%i2) assume_pos_pred: symbolp$
(%i3) sign (a);
(%o3) pos
(%i4) sign (a[1]);
(%o4) pnz
(%i5) assume_pos_pred: lambda ([x], display (x), true)$
(%i6) asksign (a);
x = a
(%o6) pos
(%i7) asksign (a[1]);
x = a
1
(%o7) pos
(%i8) asksign (foo (a));
x = foo(a)
(%o8) pos
(%i9) asksign (foo (a) + bar (b));
x = foo(a)
x = bar(b)
(%o9) pos
(%i10) asksign (log (a));
x = a
Is a - 1 positive, negative, or zero?
p;
(%o10) pos
(%i11) asksign (a - b);
x = a
x = b
x = a
x = b
Is b - a positive, negative, or zero?
p;
(%o11) neg
Categories: Declarations and inferences
Default value: initial
context names the collection of facts maintained by assume and
forget. assume adds facts to the collection named by
context, while forget removes facts.
Binding context to a name foo changes the current context to
foo. If the specified context foo does not yet exist,
it is created automatically by a call to newcontext.
The specified context is activated automatically.
See contexts for a general description of the context mechanism.
Categories: Declarations and inferences
Default value: [initial, global]
contexts is a list of the contexts which
currently exist, including the currently active context.
The context mechanism makes it possible for a user to bind together and name a collection of facts, called a context. Once this is done, the user can have Maxima assume or forget large numbers of facts merely by activating or deactivating their context.
Any symbolic atom can be a context, and the facts contained in that
context will be retained in storage until destroyed one by one
by calling forget or destroyed as a whole by calling kill
to destroy the context to which they belong.
Contexts exist in a hierarchy, with the root always being
the context global, which contains information about Maxima that some
functions need. When in a given context, all the facts in that
context are "active" (meaning that they are used in deductions and
retrievals) as are all the facts in any context which is a subcontext
of the active context.
When a fresh Maxima is started up, the user is in a
context called initial, which has global as a subcontext.
See also facts, newcontext, supcontext,
killcontext, activate, deactivate,
assume, and forget.
Categories: Declarations and inferences
Deactivates the specified contexts context_1, …, context_n.
Categories: Declarations and inferences
If item is the name of a context, facts (item) returns a
list of the facts in the specified context.
If item is not the name of a context, facts (item) returns a
list of the facts known about item in the current context. Facts that
are active, but in a different context, are not listed.
facts () (i.e., without an argument) lists the current context.
Categories: Declarations and inferences
Removes predicates established by assume.
The predicates may be expressions equivalent to (but not necessarily identical
to) those previously assumed.
forget (L), where L is a list of predicates,
forgets each item on the list.
Categories: Declarations and inferences
Attempts to determine whether the predicate expr is provable from the
facts in the assume database.
If the predicate is provably true or false, is returns
true or false, respectively. Otherwise, the return value is
governed by the global flag prederror. When prederror is
true, is complains with an error message. Otherwise, is
returns unknown.
ev(expr, pred) (which can be written expr, pred at
the interactive prompt) is equivalent to is(expr).
See also assume, facts, and maybe.
Examples:
is causes evaluation of predicates.
(%i1) %pi > %e; (%o1) %pi > %e (%i2) is (%pi > %e); (%o2) true
is attempts to derive predicates from the assume database.
(%i1) assume (a > b); (%o1) [a > b] (%i2) assume (b > c); (%o2) [b > c] (%i3) is (a < b); (%o3) false (%i4) is (a > c); (%o4) true (%i5) is (equal (a, c)); (%o5) false
If is can neither prove nor disprove a predicate from the assume
database, the global flag prederror governs the behavior of is.
(%i1) assume (a > b); (%o1) [a > b] (%i2) prederror: true$ (%i3) is (a > 0); Maxima was unable to evaluate the predicate: a > 0 -- an error. Quitting. To debug this try debugmode(true); (%i4) prederror: false$ (%i5) is (a > 0); (%o5) unknown
Categories: Predicate functions · Declarations and inferences
Kills the contexts context_1, …, context_n.
If one of the contexts is the current context, the new current context will
become the first available subcontext of the current context which has not been
killed. If the first available unkilled context is global then
initial is used instead. If the initial context is killed, a
new, empty initial context is created.
killcontext refuses to kill a context which is
currently active, either because it is a subcontext of the current
context, or by use of the function activate.
killcontext evaluates its arguments.
killcontext returns done.
Categories: Declarations and inferences
Attempts to determine whether the predicate expr is provable from the
facts in the assume database.
If the predicate is provably true or false, maybe returns
true or false, respectively. Otherwise, maybe returns
unknown.
maybe is functionally equivalent to is with
prederror: false, but the result is computed without actually assigning
a value to prederror.
See also assume, facts, and is.
Examples:
(%i1) maybe (x > 0); (%o1) unknown (%i2) assume (x > 1); (%o2) [x > 1] (%i3) maybe (x > 0); (%o3) true
Categories: Predicate functions · Declarations and inferences
Creates a new, empty context, called name, which
has global as its only subcontext. The newly-created context
becomes the currently active context.
If name is not specified, a new name is created (via gensym) and returned.
newcontext evaluates its argument.
newcontext returns name (if specified) or the new context name.
Categories: Declarations and inferences
Attempts to determine the sign of expr on the basis of the facts in the
current data base. It returns one of the following answers: pos
(positive), neg (negative), zero, pz (positive or zero),
nz (negative or zero), pn (positive or negative), or pnz
(positive, negative, or zero, i.e. nothing known).
Categories: Declarations and inferences
Creates a new context, called name, which has context as a subcontext. context must exist.
If context is not specified, the current context is assumed.
If name is not specified, a new name is created (via gensym) and returned.
supcontext evaluates its argument.
supcontext returns name (if specified) or the new context name.
Categories: Declarations and inferences
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Return 0 when the predicate p evaluates to false; return 1 when
the predicate evaluates to true. When the predicate evaluates to
something other than true or false (unknown), return a noun form.
Examples:
(%i1) charfun (x < 1);
(%o1) charfun(x < 1)
(%i2) subst (x = -1, %);
(%o2) 1
(%i3) e : charfun ('"and" (-1 < x, x < 1))$
(%i4) [subst (x = -1, e), subst (x = 0, e), subst (x = 1, e)];
(%o4) [0, 1, 0]
Categories: Mathematical functions
Return a comparison operator op (<, <=, >, >=,
=, or #) such that is (x op y) evaluates
to true; when either x or y depends on %i and
x # y, return notcomparable; when there is no such
operator or Maxima isn't able to determine the operator, return unknown.
Examples:
(%i1) compare (1, 2); (%o1) < (%i2) compare (1, x); (%o2) unknown (%i3) compare (%i, %i); (%o3) = (%i4) compare (%i, %i + 1); (%o4) notcomparable (%i5) compare (1/x, 0); (%o5) # (%i6) compare (x, abs(x)); (%o6) <=
The function compare doesn't try to determine whether the real domains of
its arguments are nonempty; thus
(%i1) compare (acos (x^2 + 1), acos (x^2 + 1) + 1); (%o1) <
The real domain of acos (x^2 + 1) is empty.
Categories: Declarations and inferences
Represents equivalence, that is, equal value.
By itself, equal does not evaluate or simplify.
The function is attempts to evaluate equal to a Boolean value.
is(equal(a, b)) returns true (or false) if
and only if a and b are equal (or not equal) for all possible
values of their variables, as determined by evaluating
ratsimp(a - b); if ratsimp returns 0, the two
expressions are considered equivalent. Two expressions may be equivalent even
if they are not syntactically equal (i.e., identical).
When is fails to reduce equal to true or false, the
result is governed by the global flag prederror. When prederror
is true, is complains with an error message. Otherwise, is
returns unknown.
In addition to is, some other operators evaluate equal and
notequal to true or false, namely if,
and, or, and not.
The negation of equal is notequal.
Examples:
By itself, equal does not evaluate or simplify.
(%i1) equal (x^2 - 1, (x + 1) * (x - 1));
2
(%o1) equal(x - 1, (x - 1) (x + 1))
(%i2) equal (x, x + 1);
(%o2) equal(x, x + 1)
(%i3) equal (x, y);
(%o3) equal(x, y)
The function is attempts to evaluate equal to a Boolean value.
is(equal(a, b)) returns true when
ratsimp(a - b) returns 0. Two expressions may be equivalent
even if they are not syntactically equal (i.e., identical).
(%i1) ratsimp (x^2 - 1 - (x + 1) * (x - 1)); (%o1) 0 (%i2) is (equal (x^2 - 1, (x + 1) * (x - 1))); (%o2) true (%i3) is (x^2 - 1 = (x + 1) * (x - 1)); (%o3) false (%i4) ratsimp (x - (x + 1)); (%o4) - 1 (%i5) is (equal (x, x + 1)); (%o5) false (%i6) is (x = x + 1); (%o6) false (%i7) ratsimp (x - y); (%o7) x - y (%i8) is (equal (x, y)); (%o8) unknown (%i9) is (x = y); (%o9) false
When is fails to reduce equal to true or false,
the result is governed by the global flag prederror.
(%i1) [aa : x^2 + 2*x + 1, bb : x^2 - 2*x - 1];
2 2
(%o1) [x + 2 x + 1, x - 2 x - 1]
(%i2) ratsimp (aa - bb);
(%o2) 4 x + 2
(%i3) prederror : true;
(%o3) true
(%i4) is (equal (aa, bb));
Maxima was unable to evaluate the predicate:
2 2
equal(x + 2 x + 1, x - 2 x - 1)
-- an error. Quitting. To debug this try debugmode(true);
(%i5) prederror : false;
(%o5) false
(%i6) is (equal (aa, bb));
(%o6) unknown
Some operators evaluate equal and notequal to true or
false.
(%i1) if equal (y, y - 1) then FOO else BAR;
(%o1) BAR
(%i2) eq_1 : equal (x, x + 1);
(%o2) equal(x, x + 1)
(%i3) eq_2 : equal (y^2 + 2*y + 1, (y + 1)^2);
2 2
(%o3) equal(y + 2 y + 1, (y + 1) )
(%i4) [eq_1 and eq_2, eq_1 or eq_2, not eq_1];
(%o4) [false, true, true]
Because not expr causes evaluation of expr,
not equal(a, b) is equivalent to
is(notequal(a, b)).
(%i1) [notequal (2*z, 2*z - 1), not equal (2*z, 2*z - 1)]; (%o1) [notequal(2 z, 2 z - 1), true] (%i2) is (notequal (2*z, 2*z - 1)); (%o2) true
Categories: Operators
Represents the negation of equal(a, b).
Examples:
(%i1) equal (a, b); (%o1) equal(a, b) (%i2) maybe (equal (a, b)); (%o2) unknown (%i3) notequal (a, b); (%o3) notequal(a, b) (%i4) not equal (a, b); (%o4) notequal(a, b) (%i5) maybe (notequal (a, b)); (%o5) unknown (%i6) assume (a > b); (%o6) [a > b] (%i7) equal (a, b); (%o7) equal(a, b) (%i8) maybe (equal (a, b)); (%o8) false (%i9) notequal (a, b); (%o9) notequal(a, b) (%i10) maybe (notequal (a, b)); (%o10) true
Categories: Operators
Returns true if and only if expr contains an operator or function
not recognized by the Maxima simplifier.
Categories: Predicate functions · Simplification functions
Tests whether the expression expr in the variable v is equivalent
to zero, returning true, false, or dontknow.
zeroequiv has these restrictions:
For example zeroequiv (sin(2 * x) - 2 * sin(x) * cos(x), x) returns
true and zeroequiv (%e^x + x, x) returns false.
On the other hand zeroequiv (log(a * b) - log(a) - log(b), a) returns
dontknow because of the presence of an extra parameter b.
Categories: Predicate functions
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| 12.1 Introduction to Plotting | ||
| 12.2 Plotting Formats | ||
| 12.3 Functions and Variables for Plotting | ||
| 12.4 Plotting Options | ||
| 12.5 Gnuplot Options | ||
| 12.6 Gnuplot_pipes Format Functions |
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Maxima uses an external plotting package to make the plots (see the
section on Plotting Formats). The plotting functions calculate a
set of points and pass them to the plotting package together with a set
of commands. That information can be passed to the external program
either through a pipe or by calling the program with the name of a file
where the data has been saved. The data file is given the name
maxout_xxx.format, where xxx is a number that is unique
to every concurrently-running instance of maxima and format is
the name of the plotting format being used (gnuplot,
xmaxima, mgnuplot or gnuplot_pipes).
There are to save the plot in a graphic format file. In those cases, the
file maxout_xxx.format created by Maxima includes commands that will
make the external plotting program save the result in a graphic
file. The default name for that graphic file is
maxplot.extension, where extension is the extension
normally used for the kind of graphic file selected.
The maxout_xxx.format and maxplot.extension files are created
in the directory specified by the system variable
maxima_tempdir. That location can be changed by assigning to
that variable (or to the environment variable MAXIMA_TEMPDIR) a string
that represents a valid directory where Maxima can create new files. The
output of the Maxima plotting command will be a list with the names of
the file(s) created, including their complete path.
If the format used is either gnuplot or xmaxima, the
external programs gnuplot or xmaxima can be run, giving it
the file maxout_xxx.format as argument, in order to view again a plot
previously created in Maxima. Thus, when a Maxima plotting command
fails, the format can be set to gnuplot or xmaxima and the
plain-text file maxout_xxx.gnuplot (or maxout_xxx.xmaxima) can be
inspected to look for the source of the problem.
The additional package draw provides functions similar to the ones
described in this section with some extra features. Note that some
plotting options have the same name in both plotting packages, but their
syntax and behavior is different. To view the documentation for a
graphic option opt, type ?? opt in order to choose the
information for either of those two packages.
Categories: Plotting
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Maxima can use either Gnuplot or Xmaxima as graphics program. Gnuplot is
an external program that has to be installed separately, while Xmaxima
is distributed with Maxima. There are various different formats for
those programs, which can be selected with the option plot_format
(see also the Plotting Options section).
The plotting formats are the following:
Used to launch the external program gnuplot, which must be installed in
your system. All plotting commands and data are saved into the file
maxout_xxx.gnuplot.
This format is not available in Windows platforms.
It is similar to the gnuplot format except that the commands are sent
to gnuplot through a pipe, while the data are saved into the file
maxout_xxx.gnuplot_pipes. A single gnuplot process is kept open
and subsequent plot commands will be sent to the same process, replacing
previous plots, unless the gnuplot pipe is closed with the function
gnuplot_close. When this format is used, the function
gnuplot_replot can be used to modify a plot that has already
displayed on the screen.
This format is only used to plot to the screen; whenever graphic files are
created, the format is silently switched to gnuplot and
the commands needed to create the graphic file are saved with the data
in file maxout_xxx.gnuplot.
Mgnuplot is a Tk-based wrapper around gnuplot. It is included in the Maxima distribution. Mgnuplot offers a rudimentary GUI for gnuplot, but has fewer overall features than the plain gnuplot interface. Mgnuplot requires an external gnuplot installation and, in Unix systems, the Tcl/Tk system.
Xmaxima is a Tcl/Tk graphical interface for Maxima that can also be used
to display plots created when Maxima is run from the console or from
other graphical interfaces. To use this format, the xmaxima program,
which is distributed together with Maxima, should be installed. If
Maxima is being run from the Xmaxima console, the data and commands are
passed to xmaxima through the same socket used for the communication
between Maxima and the Xmaxima console. When used from a terminal or
from graphical interfaces different from Xmaxima, the commands and data
are saved in the file maxout_xxx.xmaxima and xmaxima is run with the
name of that file as argument.
In previous versions this format used to be called openmath; that
old name still works as a synonym for xmaxima.
Categories: Plotting
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It plots the contours (curves of equal value) of expr
over the region x_range by y_range.
Any additional arguments are treated the same as in plot3d.
This function only works when the plot format is either gnuplot
or gnuplot_pipes. The additional package
implicit_plot, which works in any graphic format, can also be
used to plot contours but a separate expression must be given for each
contour.
Examples:
(%i1) contour_plot (x^2 + y^2, [x, -4, 4], [y, -4, 4])$
You can add any options accepted by plot3d; for instance, the
option legend with a value of false, to remove the
legend. By default, Gnuplot chooses and displays 3 contours. To increase the
number of contours, it is necessary to use a custom
gnuplot_preamble, as in the next example:
(%i1) contour_plot (u^3 + v^2, [u, -4, 4], [v, -4, 4],
[legend,false],
[gnuplot_preamble, "set cntrparam levels 12"])$
Categories: Plotting
Returns the current default value of the option named keyword, which is a list. The optional argument index must be a positive integer which can be used to extract only one element from the list (element 1 is the name of the option).
See also set_plot_option, remove_plot_option and the
section on Plotting Options.
This variable stores the name of the command used to run the gnuplot
program when the plot format is gnuplot. Its default value is
"wgnuplot" in Windows and "gnuplot" in other systems. If the gnuplot
program is not found unless you give its complete path or if you want
to try a different version of it, you may change the value of this
variable. For instance,
(%i1) gnuplot_command: "/usr/local/bin/my_gnuplot"$
Categories: Plotting
When a graphic file is going to be created using gnuplot, this
variable is used to specify the way the file name should be passed to
gnuplot. Its default value is "~s", which means that the name of the
file will be passed directly. The contents of this variable can be
changed in order to add options for the gnuplot program, adding those
options before the format directive "~s".
Categories: Plotting
This variable is used to parse the argument that will be passed to the
gnuplot program when the plot format is gnuplot. Its default
value is "-persist ~s", where "~s" will be replaced with the name of the
file where the gnuplot commands have been written (usually
"maxout_xxx.gnuplot"). The option -persist tells gnuplot to exit
after the commands in the file have been executed, without closing the
window that displays the plot.
Those familiar with gnuplot, might want to change the value of this variable. For example, by changing it to:
(%i1) gnuplot_view_args: "~s -"$
gnuplot will not be closed after the commands in the file have been executed; thus, the window with the plot will remain, as well as the gnuplot interactive shell where other commands can be issued in order to modify the plot.
In Windows versions of Gnuplot older than 4.6.3 the behavior of "~s -"
and "-persist ~s" were the opposite; namely, "-persist ~s" made the plot
window and the gnuplot interactive shell remain, while "~s -" closed the
gnuplot shell keeping the plot window. Therefore, when older gnuplot
versions are used in Windows, it might be necessary to adjust the value
of gnuplot_view_args.
Categories: Plotting
Displays a plot of a function on the real plane, defined implicitly by
the expression expr. The domain in the plane is defined by
x_range and y_range. Several functions can be represented on
the same plot, giving a list [expr_1, …, expr_n] of
expressions that define them. This function uses the global format
options set up with the set_plot_option. Additional options can
also be given as extra arguments for the implicit_plot command.
The method used by implicit_plot consists of tracking sign
changes on the domain given and it can fail for complicated expressions.
load(implicit_plot) loads this function.
Example:
(%i1) load(implicit_plot)$ (%i2) implicit_plot (x^2 = y^3 - 3*y + 1, [x, -4, 4], [y, -4, 4])$
Categories: Plotting · Share packages · Package implicit_plot
Creates a graphic representation of the Julia set for the complex number
(x + i y). The two mandatory parameters x and y
must be real. This program is part of the additional package
dynamics, but that package does not have to be loaded; the first
time julia is used, it will be loaded automatically.
Each pixel in the grid is given a color corresponding to the number of
iterations it takes the sequence that starts at that point to move out
of the convergence circle of radius 2 centered at the origin. The number
of pixels in the grid is controlled by the grid plot option
(default 30 by 30). The maximum number of iterations is set with the
option iterations. The program uses its own default palette:
magenta,violet, blue, cyan, green, yellow, orange, red, brown and black,
but it can be changed by adding an explicit palette option in the
command.
The default domain used goes from -2 to 2 in both axes and can be
changed with the x and y options. By default, the two axes
are shown with the same scale, unless the option yx_ratio is used
or the option same_xy is disabled. Other general plot options are
also accepted.
The following example shows a region of the Julia set for the number
-0.55 + i0.6. The option color_bar_tics is used to prevent
Gnuplot from adjusting the color box up to 40, in which case the points
corresponding the maximum 36 iterations would not be black.
(%i1) julia (-0.55, 0.6, [iterations, 36], [x, -0.3, 0.2],
[y, 0.3, 0.9], [grid, 400, 400], [color_bar_tics, 0, 6, 36])$
Categories: Package dynamics · Plotting
Returns a function suitable to be used in the option transform_xy
of plot3d. The three variables var1, var2, var3 are
three dummy variable names, which represent the 3 variables given by the
plot3d command (first the two independent variables and then the
function that depends on those two variables). The three functions
fx, fy, fz must depend only on those 3 variables, and
will give the corresponding x, y and z coordinates that should be
plotted. There are two transformations defined by default:
polar_to_xy and spherical_to_xyz. See the documentation
for those two transformations.
Categories: Plotting
Creates a graphic representation of the Mandelbrot set. This program is
part of the additional package dynamics, but that package does
not have to be loaded; the first time mandelbrot is used, the package
will be loaded automatically.
This program can be called without any arguments, in which case it will
use a default value of 9 iterations per point, a grid with dimensions
set by the grid plot option (default 30 by 30) and a region
that extends from -2 to 2 in both axes. The options are all the same
that plot2d accepts, plus an option iterations to change the
number of iterations.
Each pixel in the grid is given a color corresponding to the number of
iterations it takes the sequence starting at zero to move out
of the convergence circle of radius 2, centered at the origin. The
maximum number of iterations is set by the option iterations.
The program uses its own default palette: magenta,violet, blue, cyan,
green, yellow, orange, red, brown and black, but it can be changed by
adding an explicit palette option in the command. By default, the
two axes are shown with the same scale, unless the option yx_ratio
is used or the option same_xy is disabled.
Example:
[grid,400,400])$
(%i1) mandelbrot ([iterations, 30], [x, -2, 1], [y, -1.2, 1.2],
[grid,400,400])$
Categories: Package dynamics · Plotting
It can be given as value for the transform_xy option of
plot3d. Its effect will be to interpret the two independent variables in
plot3d as the distance from the z axis and the azimuthal angle (polar
coordinates), and transform them into x and y coordinates.
Categories: Plotting
Where plot, plot_1, …, plot_n can be either
expressions, function names or a list with
the any of the forms: [discrete, [x1, ..., xn],
[y1, ..., yn]], [discrete, [[x1, y1],
..., [xn, ..., yn]]] or [parametric, x_expr,
y_expr, t_range].
Displays a plot of one or more expressions as a function of one variable or parameter.
plot2d displays one or several plots in two dimensions. When
expressions or function name are used to define the plots,
they should all depend on only one variable var and the use of
x_range will be mandatory, to provide the name of the variable and
its minimum and maximum values; the syntax for x_range is:
[variable, min, max].
A plot can also be defined in the discrete or parametric forms. The discrete form is used to plot a set of points with given coordinates. A discrete plot is defined by a list starting with the keyword discrete, followed by one or two lists of values. If two lists are given, they must have the same length; the first list will be interpreted as the x coordinates of the points to be plotted and the second list as the y coordinates. If only one list is given after the discrete keyword, each element on the list could also be a list with two values that correspond to the x and y coordinates of a point, or it could be a sequence of numerical values which will be plotted at consecutive integer values (1,2,3,...) on the x axis.
A parametric plot is defined by a list starting with the keyword
parametric, followed by two expressions or function names and a
range for the parameter. The range for the parameter must be a list with
the name of the parameter followed by its minimum and maximum values:
[param, min, max]. The plot will show the path
traced out by the point with coordinates given by the two expressions or
functions, as param increases from min to max.
A range for the vertical axis is an optional argument with the form:
[y, min, max] (the keyword y is always used for
the vertical axis). If that option is used, the plot will show that
exact vertical range, independently of the values reached by the plot.
If the vertical range is not specified, it will be set up according to
the minimum and maximum values of the second coordinate of the plot
points.
All other options should also be lists, starting with a keyword and
followed by one or more values. See plot_options.
If there are several plots to be plotted, a legend will be
written to identity each of the expressions. The labels that should be
used in that legend can be given with the option legend. If that
option is not used, Maxima will create labels from the expressions or
function names.
Examples:
Plot of a common function:
(%i1) plot2d (sin(x), [x, -%pi, %pi])$
If the function grows too fast, it might be necessary to limit the
values in the vertical axis using the y option:
(%i1) plot2d (sec(x), [x, -2, 2], [y, -20, 20])$
When the plot box is disabled, no labels are created for the axes. In
that case, instead of using xlabel and ylabel to set the
names of the axes, it is better to use option label, which
allows more flexibility. Option yx_ratio is used to change the
default rectangular shape of the plot; in this example the plot will
fill a square.
(%i1) plot2d ( x^2 - 1, [x, -3, 3], [box, false], grid2d,
[yx_ratio, 1], [axes, solid], [xtics, -2, 4, 2],
[ytics, 2, 2, 6], [label, ["x", 2.9, -0.3],
["x^2-1", 0.1, 8]], [title, "A parabola"])$
A plot with a logarithmic scale in the vertical axis:
(%i1) plot2d (exp(3*s), [s, -2, 2], logy)$
Plotting functions by name:
(%i1) F(x) := x^2 $ (%i2) :lisp (defun |$g| (x) (m* x x x)) $g (%i2) H(x) := if x < 0 then x^4 - 1 else 1 - x^5 $ (%i3) plot2d ([F, G, H], [u, -1, 1], [y, -1.5, 1.5])$
A plot of the butterfly curve, defined parametrically:
(%i1) r: (exp(cos(t))-2*cos(4*t)-sin(t/12)^5)$ (%i2) plot2d([parametric, r*sin(t), r*cos(t), [t, -8*%pi, 8*%pi]])$
Plot of a circle, using its parametric representation, together with the
function -|x|. The circle will only look like a circle if the scale in the
two axes is the same, which is done with the option same_xy.
(%i1) plot2d([[parametric, cos(t), sin(t), [t,0,2*%pi]], -abs(x)],
[x, -sqrt(2), sqrt(2)], same_xy)$
A plot of 200 random numbers between 0 and 9:
(%i1) plot2d ([discrete, makelist ( random(10), 200)])$
A plot of a discrete set of points, defining x and y coordinates separately:
(%i1) plot2d ([discrete, makelist(i*%pi, i, 1, 5),
[0.6, 0.9, 0.2, 1.3, 1]])$
In the next example a table with three columns is saved in a file "data.txt" which is then read and the second and third column are plotted on the two axes:
(%i1) with_stdout ("data.txt", for x:0 thru 10 do
print (x, x^2, x^3))$
(%i2) data: read_matrix ("data.txt")$
(%i3) plot2d ([discrete, transpose(data)[2], transpose(data)[3]],
[style,points], [point_type,diamond], [color,red])$
A plot of discrete data points together with a continuous function:
(%i1) xy: [[10, .6], [20, .9], [30, 1.1], [40, 1.3], [50, 1.4]]$
(%i2) plot2d([[discrete, xy], 2*%pi*sqrt(l/980)], [l,0,50],
[style, points, lines], [color, red, blue],
[point_type, asterisk],
[legend, "experiment", "theory"],
[xlabel, "pendulum's length (cm)"],
[ylabel, "period (s)"])$
See also the section about Plotting Options.
Categories: Plotting
Displays a plot of one or more surfaces defined as functions of two variables or in parametric form.
The functions to be plotted may be specified as expressions or function names. The mouse can be used to rotate the plot looking at the surface from different sides.
Examples:
Plot of a function of two variables:
(%i1) plot3d (u^2 - v^2, [u, -2, 2], [v, -3, 3], [grid, 100, 100],
[mesh_lines_color,false])$
Use of the z option to limit a function that goes to infinity
(in this case the function is minus infinity on the x and y axes); this also
shows how to plot with only lines and no shading:
(%i1) plot3d ( log ( x^2*y^2 ), [x, -2, 2], [y, -2, 2], [z, -8, 4],
[palette, false], [color, magenta])$
The infinite values of z can also be avoided by choosing a grid that does not fall on any points where the function is undefined, as in the next example, which also shows how to change the palette and how to include a color bar that relates colors to values of the z variable:
(%i1) plot3d (log (x^2*y^2), [x, -2, 2], [y, -2, 2],[grid, 29, 29],
[palette, [gradient, red, orange, yellow, green]],
color_bar, [xtics, 1], [ytics, 1], [ztics, 4],
[color_bar_tics, 4])$
Two surfaces in the same plot. Ranges specific to one of the surfaces can be given by placing each expression and its ranges in a separate list; global ranges for the complete plot are also given after the functions definitions.
(%i1) plot3d ([[-3*x - y, [x, -2, 2], [y, -2, 2]], 4*sin(3*(x^2 + y^2))/(x^2 + y^2), [x, -3, 3], [y, -3, 3]], [x, -4, 4], [y, -4, 4])$
Plot of a Klein bottle, defined parametrically:
(%i1) expr_1: 5*cos(x)*(cos(x/2)*cos(y)+sin(x/2)*sin(2*y)+3)-10$
(%i2) expr_2: -5*sin(x)*(cos(x/2)*cos(y)+sin(x/2)*sin(2*y)+3)$
(%i3) expr_3: 5*(-sin(x/2)*cos(y)+cos(x/2)*sin(2*y))$
(%i4) plot3d ([expr_1, expr_2, expr_3], [x, -%pi, %pi],
[y, -%pi, %pi], [grid, 50, 50])$
Plot of a "spherical harmonic" function, using the predefined
transformation, spherical_to_xyz to transform from spherical
coordinates to rectangular coordinates. See the documentation for
spherical_to_xyz.
(%i1) plot3d (sin(2*theta)*cos(phi), [theta, 0, %pi],
[phi, 0, 2*%pi],
[transform_xy, spherical_to_xyz], [grid,30,60],
[legend,false])$
Use of the pre-defined function polar_to_xy to transform from
cylindrical to rectangular coordinates. See the documentation for
polar_to_xy.
(%i1) plot3d (r^.33*cos(th/3), [r,0,1], [th,0,6*%pi], [box, false], [grid, 12, 80], [transform_xy, polar_to_xy], [legend, false])$
Plot of a sphere using the transformation from spherical to rectangular
coordinates. Option same_xyz is used to get the three axes
scaled in the same proportion. When transformations are used, it is not
convenient to eliminate the mesh lines, because Gnuplot will not show the
surface correctly.
(%i1) plot3d ( 5, [theta, 0, %pi], [phi, 0, 2*%pi], same_xyz, [transform_xy, spherical_to_xyz], [mesh_lines_color,blue], [palette,[gradient,"#1b1b4e", "#8c8cf8"]], [legend, false])$
Definition of a function of two-variables using a matrix. Notice the
single quote in the definition of the function, to prevent plot3d
from failing when it realizes that the matrix will require integer
indices.
(%i1) M: matrix([1,2,3,4], [1,2,3,2], [1,2,3,4], [1,2,3,3])$
(%i2) f(x, y) := float('M [round(x), round(y)])$
(%i3) plot3d (f(x,y), [x,1,4],[y,1,4],[grid,3,3],[legend,false])$
By setting the elevation equal to zero, a surface can be seen as a map in which each color represents a different level.
(%i1) plot3d (cos (-x^2 + y^3/4), [x,-4,4], [y,-4,4], [zlabel,""],
[mesh_lines_color,false], [elevation,0], [azimuth,0],
color_bar, [grid,80,80], [ztics,false], [color_bar_tics,1])$
See also the section about Plotting Options.
Categories: Plotting
This option is being kept for compatibility with older versions, but its
use is deprecated. To set global plotting options, see their current
values or remove options, use set_plot_option,
get_plot_option and remove_plot_option.
Categories: Plotting
Removes the default value of an option. The name of the option must be given.
See also set_plot_option, get_plot_option and the section
on Plotting Options.
Categories: Plotting
Accepts any of the options listed in the section Plotting Options, and saves them for use in plotting commands. The values of the options set in each plotting command will have precedence, but if those options are not given, the default values set with this function will be used.
set_plot_option evaluates its argument and returns the complete
list of options (after modifying the option given). If called without
any arguments, it will simply show the list of current default options.
See also remove_plot_option, get_plot_option and the section
on Plotting Options.
Example:
Modification of the grid values.
(%i1) set_plot_option ([grid, 30, 40]); (%o1) [[plot_format, gnuplot_pipes], [grid, 30, 40], [run_viewer, true], [axes, true], [nticks, 29], [adapt_depth, 5], [color, blue, red, green, magenta, black, cyan], [point_type, bullet, box, triangle, plus, times, asterisk], [palette, [gradient, green, cyan, blue, violet], [gradient, magenta, violet, blue, cyan, green, yellow, orange, red, brown, black]], [gnuplot_preamble, ], [gnuplot_term, default]]
Categories: Plotting
It can be given as value for the transform_xy option of
plot3d. Its effect will be to interpret the two independent
variables and the function in plot3d as the spherical coordinates
of a point (first, the angle with the z axis, then the angle of the xy
projection with the x axis and finally the distance from the origin) and
transform them into x, y and z coordinates.
Categories: Plotting
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All options consist of a list starting with one of the keywords in this section, followed by one or more values. Some of the options may have different effects in different plotting commands as it will be pointed out in the following list. The options that accept among their possible values true or false, can also be set to true by simply writing their names. For instance, typing logx as an option is equivalent to writing [logx, true].
Default value: 5
The maximum number of splittings used by the adaptive plotting routine.
Categories: Plotting
Default value: true
Where symbol can be either true, false, x,
y or solid. If false, no axes are shown; if equal
to x or y only the x or y axis will be shown; if it is
equal to true, both axes will be shown and solid will show
the two axes with a solid line, rather than the default broken
line. This option does not have any effect in the 3 dimensional plots.
Categories: Plotting
Default value: 30
A plot3d plot can be thought of as starting with the x and y axis in the
horizontal and vertical axis, as in plot2d, and the z axis coming out of
the screen. The z axis is then rotated around the x axis through an
angle equal to elevation and then the new xy plane is rotated
around the new z axis through an angle azimuth. This option sets
the value for the azimuth, in degrees.
See also elevation.
Categories: Plotting
Default value: true
If set to true, a bounding box will be drawn for the plot; if set
to false, no box will be drawn.
Categories: Plotting
In 2d plots it defines the color (or colors) for the various curves. In
plot3d, it defines the colors used for the mesh lines of the
surfaces, when no palette is being used.
If there are more curves or surfaces than colors, the colors will be
repeated in sequence. The valid colors are red, green,
blue, magenta, cyan, yellow, orange,
violet, brown, gray, black, white, or
a string starting with the character # and followed by six hexadecimal
digits: two for the red component, two for green component and two for
the blue component. If the name of a given color is unknown color, black
will be used instead.
Categories: Plotting
Default value: false in plot3d, true in mandelbrot and julia
Where symbol can be either true or false. If
true, whenever plot3d, mandelbrot or
julia use a palette to represent different values, a box will be
shown on the right, showing the corresponding between colors and values.
Categories: Plotting
Defines the values at which a mark and a number will be placed in the color bar. The first number is the initial value, the second the increments and the third is the last value where a mark is placed. The second and third numbers can be omitted. When only one number is given, it will be used as the increment from an initial value that will be chosen automatically.
Categories: Plotting
Default value: 60
A plot3d plot can be thought of as starting with the x and y axis in the
horizontal and vertical axis, as in plot2d, and the z axis coming out of
the screen. The z axis is then rotated around the x axis through an
angle equal to elevation and then the new xy plane is rotated
around the new z axis through an angle azimuth. This option sets
the value for the azimuth, in degrees.
See also azimuth.
Categories: Plotting
Default value: 30, 30
Sets the number of grid points to use in the x- and y-directions for
three-dimensional plotting or for the julia and mandelbrot
programs.
Categories: Plotting
Default value: false
Shows a grid of lines on the xy plane. The points where the grid lines
are placed are the same points where tics are marked in the x and y
axes, which can be controlled with the xtics and ytics
options.
Categories: Plotting
Default value: 9
Number of iterations made by the programs mandelbrot and julia.
Categories: Plotting
Writes one or several labels in the points with x, y coordinates indicated after each label.
Categories: Plotting
It specifies the labels for the plots when various plots are shown. If
there are more plots than the number of labels given, they will be
repeated. If given the value false, no legends will be shown. By
default, the names of the expressions or functions will be used, or the
words discrete1, discrete2, …, for discrete sets of points.
Categories: Plotting
Makes the horizontal axes to be scaled logarithmically. It can be either true or false.
Categories: Plotting
Makes the vertical axes to be scaled logarithmically. It can be either true or false.
Categories: Plotting
Default value: black
It sets the color used by plot3d to draw the mesh lines, when a palette is
being used. It accepts the same colors as for the option color
(see the list of allowed colors in color). It can also be given a
value false to eliminate completely the mesh lines.
Categories: Plotting
Default value: 29
When plotting functions with plot2d, it is gives the initial
number of points used by the adaptive plotting routine for plotting
functions. When plotting parametric functions with plot3d,
it sets the number of points that will be shown for the plot.
Categories: Plotting
It can consist of one palette or a list of several palettes. Each palette is a list with a keyword followed by values. If the keyword is gradient, it should be followed by a list of valid colors.
If the keyword is hue, saturation or value, it must be followed by 4 numbers. The first three numbers, which must be between 0 and 1, define the hue, saturation and value of a basic color to be assigned to the minimum value of z. The keyword specifies which of the three attributes (hue, saturation or value) will be increased according to the values of z. The last number indicates the increase corresponding to the maximum value of z. That last number can be bigger than 1 or negative; the corresponding values of the modified attribute will be rounded modulo 1.
Gnuplot only uses the first palette in the list; xmaxima will use the palettes in the list sequentially, when several surfaces are plotted together; if the number of palettes is exhausted, they will be repeated sequentially.
The color of the mesh lines will be given by the option
mesh_lines_color. If palette is given the value
false, the surfaces will not be shaded but represented with a
mesh of curves only. In that case, the colors of the lines will be
determined by the option color.
Categories: Plotting
Default value: gnuplot, in Windows systems, or gnuplot_pipes in
other systems.
Where format is one of the following: gnuplot, xmaxima, mgnuplot or gnuplot_pipes.
It sets the format to be used for plotting.
Categories: Plotting
Default value: false
If set to true, the functions to be plotted will be considered
as complex functions whose real value should be plotted; this is
equivalent to plotting realpart(function). If set to
false, nothing will be plotted when the function does not give a
real value. For instance, when x is negative, log(x) gives
a complex value, with real value equal to log(abs(x)); if
plot_realpart were true, log(-5) would be plotted
as log(5), while nothing would be plotted if
plot_realpart were false.
Categories: Plotting
In gnuplot, each set of points to be plotted with the style "points"
or "linespoints" will be represented with objects taken from this
list, in sequential order. If there are more sets of points than objects
in this list, they will be repeated sequentially.
The possible objects that can be used are: bullet, circle,
plus, times, asterisk, box, square,
triangle, delta, wedge, nabla, diamond,
lozenge.
Categories: Plotting
Saves the plot into a PDF file with name equal to file_name,
rather than showing it in the screen. By default, the file will be
created in the directory defined by the variable
maxima_tempdir, unless file_name contains the character
"/", in which case it will be assumed to contain the complete path where
the file should be created. The value of maxima_tempdir can be changed
to save the file in a different directory. When the option
gnuplot_pdf_term_command is also given, it will be used to set up
Gnuplot's PDF terminal; otherwise, Gnuplot's pdfcairo terminal
will be used with solid colored lines of width 3, plot
size of 17.2 cm by 12.9 cm and font of 18 points.
Categories: Plotting
Saves the plot into a PNG graphics file with name equal to file_name,
rather than showing it in the screen. By default, the file will be
created in the directory defined by the variable
maxima_tempdir, unless file_name contains the character
"/", in which case it will be assumed to contain the complete path where
the file should be created. The value of maxima_tempdir can be changed
to save the file in a different directory. When the option
gnuplot_png_term_command is also given, it will be used to set up
Gnuplot's PNG terminal; otherwise, Gnuplot's pngcairo terminal
will be used, with a font of size 12.
Categories: Plotting
Saves the plot into a Postscript file with name equal to file_name,
rather than showing it in the screen. By default, the file will be
created in the directory defined by the variable
maxima_tempdir, unless file_name contains the character
"/", in which case it will be assumed to contain the complete path where
the file should be created. The value of maxima_tempdir can be changed
to save the file in a different directory. When the option
gnuplot_ps_term_command is also given, it will be used to set up
Gnuplot's Postscript terminal; otherwise, Gnuplot's postscript terminal
will be used with the EPS option, solid colored lines of width 2, plot
size of 16.4 cm by 12.3 cm and font of 24 points.
Categories: Plotting
This option is only used when the plot format is gnuplot and the
terminal is default or when the Gnuplot terminal is set to
dumb (see gnuplot_term) and can have a true or false
value.
If the terminal is default, a file maxout_xxx.gnuplot (or
other name specified with gnuplot_out_file) is created with the
gnuplot commands necessary to generate the plot. Option run_viewer
controls whether or not Gnuplot will be launched to execute those
commands and show the plot.
If the terminal is default, gnuplot is run to execute the
commands in maxout_xxx.gnuplot, producing another file
maxplot.txt (or other name specified with
gnuplot_out_file). Option run_viewer controls whether or
not that file, with an ASCII representation of the plot, will be shown
in the Maxima or Xmaxima console.
The default value for this option is true, making the plots to be shown in either the console or a separate graphics window.
Categories: Plotting
It can be either true or false. If true, the scales used in the x and y
axes will be the same, in either 2d or 3d plots. See also yx_ratio.
Categories: Plotting
It can be either true or false. If true, the scales used in the 3 axes of a 3d plot will be the same.
Categories: Plotting
The styles that will be used for the various functions or sets of data in a 2d plot. The word style must be followed by one or more styles. If there are more functions and data sets than the styles given, the styles will be repeated. Each style can be either lines for line segments, points for isolated points, linespoints for segments and points, or dots for small isolated dots. Gnuplot accepts also an impulses style.
Each of the styles can be enclosed inside a list with some additional parameters. lines accepts one or two numbers: the width of the line and an integer that identifies a color. The default color codes are: 1: blue, 2: red, 3: magenta, 4: orange, 5: brown, 6: lime and 7: aqua. If you use Gnuplot with a terminal different than X11, those colors might be different; for example, if you use the option [gnuplot_term, ps], color index 4 will correspond to black, instead of orange.
points accepts one two or three parameters; the first parameter is the radius of the points, the second parameter is an integer that selects the color, using the same code used for lines and the third parameter is currently used only by Gnuplot and it corresponds to several objects instead of points. The default types of objects are: 1: filled circles, 2: open circles, 3: plus signs, 4: x, 5: *, 6: filled squares, 7: open squares, 8: filled triangles, 9: open triangles, 10: filled inverted triangles, 11: open inverted triangles, 12: filled lozenges and 13: open lozenges.
linespoints accepts up to four parameters: line width, points radius, color and type of object to replace the points.
See also color and point_type.
Categories: Plotting
Saves the plot into an SVG file with name equal to file_name,
rather than showing it in the screen. By default, the file will be
created in the directory defined by the variable
maxima_tempdir, unless file_name contains the character
"/", in which case it will be assumed to contain the complete path where
the file should be created. The value of maxima_tempdir can be changed
to save the file in a different directory. When the option
gnuplot_svg_term_command is also given, it will be used to set up
Gnuplot's SVG terminal; otherwise, Gnuplot's svg terminal
will be used with font of 14 points.
Categories: Plotting
Default range for parametric plots.
Categories: Plotting
Defines a title that will be written at the top of the plot.
Categories: Plotting
Where symbol is either false or the result obtained by
using the function transform_xy. If different from false,
it will be used to transform the 3 coordinates in plot3d.
See make_transform, polar_to_xy and
spherical_to_xyz.
Categories: Plotting
When used as the first option in a plot2d command (or any of the
first two in plot3d), it indicates that the first independent variable
is x and it sets its range. It can also be used again after the first
option (or after the second option in plot3d) to define the effective
horizontal domain that will be shown in the plot.
Categories: Plotting
Specifies the string that will label the first axis; if this option is
not used, that label will be the name of the independent variable, when plotting
functions with plot2d or implicit_plot, or the name of the
first variable, when plotting surfaces with plot3d or contours with
contour_plot, or the first expression in the case of a parametric
plot. It can not be used with set_plot_option.
Categories: Plotting
Defines the values at which a mark and a number will be placed in the x axis. The first number is the initial value, the second the increments and the third is the last value where a mark is placed. The second and third numbers can be omitted. When only one number is given, it will be used as the increment from an initial value that will be chosen automatically.
Categories: Plotting
In a 2d plot, it defines the ratio of the total size of the Window to the size that will be used for the plot. The two numbers given as arguments are the scale factors for the x and y axes.
Categories: Plotting
When used as one of the first two options in plot3d, it indicates
that one of the independent variables is y and it sets its range. Otherwise,
it defines the effective domain of the second variable that will be
shown in the plot.
Categories: Plotting
Specifies the string that will label the second axis; if this
option is not used, that label will be "y", when plotting functions
with plot2d or implicit_plot, or the name of the second
variable, when plotting surfaces with plot3d or contours with
contour_plot, or the second expression in the case of a parametric
plot. It can not be used with set_plot_option.
Categories: Plotting
Defines the values at which a mark and a number will be placed in the y axis. The first number is the initial value, the second the increments and the third is the last value where a mark is placed. The second and third numbers can be omitted. When only one number is given, it will be used as the increment from an initial value that will be chosen automatically
Categories: Plotting
In a 2d plot, the ratio between the vertical and the horizontal sides of
the rectangle used to make the plot. See also same_xy.
Categories: Plotting
Used in plot3d to set the effective range of values of z that will be
shown in the plot.
Categories: Plotting
Specifies the string that will label the third axis, when using
plot3d. If this option is not used, that label will be "z", when
plotting surfaces, or the third expression in the case of a parametric
plot. It can not be used with set_plot_option and it will be
ignored by plot2d and implicit_plot.
Categories: Plotting
In 3d plots, the value of z that will be at the bottom of the plot box.
Categories: Plotting
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There are several plot options specific to gnuplot. All of them consist of a keyword (the name of the option), followed by a string that should be a valid gnuplot command, to be passed directly to gnuplot. In most cases, there exist a corresponding plotting option that will produce a similar result and whose use is more recommended than the gnuplot specific option.
Sets the output terminal type for gnuplot. The argument terminal_name can be a string or one of the following 3 special symbols
Gnuplot output is displayed in a separate graphical window and the
gnuplot terminal used will be specified by the value of the option
gnuplot_default_term_command.
Gnuplot output is saved to a file maxout_xxx.gnuplot using "ASCII
art" approximation to graphics. If the option gnuplot_out_file is
set to filename, the plot will be saved there, instead of the
default maxout_xxx.gnuplot. The settings for the "dumb" terminal of
Gnuplot are given by the value of option
gnuplot_dumb_term_command. If option run_viewer is set
to true and the plot_format is gnuplot that ASCII representation
will also be shown in the Maxima or Xmaxima console.
Gnuplot generates commands in the PostScript page description language.
If the option gnuplot_out_file is set to filename, gnuplot
writes the PostScript commands to filename. Otherwise, it is
saved as maxplot.ps file. The settings for this terminal are given by the value of the option gnuplot_dumb_term_command.
Gnuplot can generate output in many other graphical formats such as png,
jpeg, svg etc. To use those formats, option gnuplot_term can be
set to any supported gnuplot term name (which must be a symbol) or even a
full gnuplot term specification with any valid options (which must be a string). For
example [gnuplot_term, png] creates output in PNG (Portable
Network Graphics) format while [gnuplot_term, "png size
1000,1000"] creates PNG of 1000 x 1000 pixels size. If the option
gnuplot_out_file is set to filename, gnuplot writes the
output to filename. Otherwise, it is saved as
maxplot.term file, where term is gnuplot terminal
name.
Categories: Plotting
It can be used to replace the default name for the file that contains
the commands that will interpreted by gnuplot, when the terminal is set
to default, or to replace the default name of the graphic file
that gnuplot creates, when the terminal is different from
default. If it contains one or more slashes, "/", the name of
the file will be left as it is; otherwise, it will be appended to the
path of the temporary directory. The complete name of the files created
by the plotting commands is always sent as output of those commands so
they can be seen if the command is ended by semi-colon.
When used in conjunction with the gnuplot_term option, it can be
used to save the plot in a file, in one of the graphic formats supported
by Gnuplot. To create PNG, PDF, Postscript or SVG, it is easier to use
options png_file, pdf_file, ps_file,
or svg_file.
Categories: Plotting
With a value of false, it can be used to disable the use of PM3D
mode, which is enabled by default.
Categories: Plotting
This option inserts gnuplot commands before any other commands sent to
Gnuplot. Any valid gnuplot commands may be used. Multiple commands should
be separated with a semi-colon. See also gnuplot_postamble.
Categories: Plotting
This option inserts gnuplot commands after other commands sent to
Gnuplot and right before the plot command is sent. Any valid gnuplot
commands may be used. Multiple commands should be separated with a
semi-colon. See also gnuplot_preamble.
Categories: Plotting
[gnuplot_default_term_command, command]
The gnuplot command to set the terminal type for the default
terminal. It this option is not set, the command used will be: "set term wxt size 640,480 font \",12\"; set term pop".
Categories: Plotting
[gnuplot_dumb_term_command, command]
The gnuplot command to set the terminal type for the dumb terminal. It
this option is not set, the command used will be: "set term dumb
79 22", which makes the text output 79 characters by 22 characters.
Categories: Plotting
The gnuplot command to set the terminal type for the PDF
terminal. If this option is not set, the command used will be: "set term pdfcairo color solid lw 3 size 17.2 cm, 12.9 cm font \",18\"". See the gnuplot documentation for more information.
Categories: Plotting
The gnuplot command to set the terminal type for the PNG terminal. If
this option is not set, the command used will be:
"set term pngcairo font \",12\"". See the gnuplot documentation
for more information.
Categories: Plotting
The gnuplot command to set the terminal type for the PostScript
terminal. If this option is not set, the command used will be: "set term postscript eps color solid lw 2 size 16.4 cm, 12.3 cm font \",24\"". See the gnuplot documentation for set term postscript for
more information.
Categories: Plotting
The gnuplot command to set the terminal type for the SVG
terminal. If this option is not set, the command used will be:
"set term svg font \",14\"". See the gnuplot documentation for
more information.
Categories: Plotting
This is an obsolete option that has been replaced legend described
above.
Categories: Plotting
This is an obsolete option that has been replaced by style.
Categories: Plotting
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Opens the pipe to gnuplot used for plotting with the gnuplot_pipes
format. Is not necessary to manually open the pipe before plotting.
Categories: Plotting
Closes the pipe to gnuplot which is used with the gnuplot_pipes format.
Categories: Plotting
Closes the pipe to gnuplot which is used with the gnuplot_pipes
format and opens a new pipe.
Categories: Plotting
Updates the gnuplot window. If gnuplot_replot is called with a
gnuplot command in a string s, then s is sent to gnuplot
before reploting the window.
Categories: Plotting
Resets the state of gnuplot used with the gnuplot_pipes format. To
update the gnuplot window call gnuplot_replot after gnuplot_reset.
Categories: Plotting
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| 13.1 Comments | ||
| 13.2 Files | ||
| 13.3 Functions and Variables for File Input and Output | ||
| 13.4 Functions and Variables for TeX Output | ||
| 13.5 Functions and Variables for Fortran Output |
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A comment in Maxima input is any text between /* and */.
The Maxima parser treats a comment as whitespace for the purpose of finding
tokens in the input stream; a token always ends at a comment. An input such as
a/* foo */b contains two tokens, a and b,
and not a single token ab. Comments are otherwise ignored by Maxima;
neither the content nor the location of comments is stored in parsed input
expressions.
Comments can be nested to arbitrary depth. The /* and */
delimiters form matching pairs. There must be the same number of /*
as there are */.
Examples:
(%i1) /* aa is a variable of interest */ aa : 1234;
(%o1) 1234
(%i2) /* Value of bb depends on aa */ bb : aa^2;
(%o2) 1522756
(%i3) /* User-defined infix operator */ infix ("b");
(%o3) b
(%i4) /* Parses same as a b c, not abc */ a/* foo */b/* bar */c;
(%o4) a b c
(%i5) /* Comments /* can be nested /* to any depth */ */ */ 1 + xyz;
(%o5) xyz + 1
Categories: Syntax
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A file is simply an area on a particular storage device which contains data or text. Files on the disks are figuratively grouped into "directories". A directory is just a list of files. Commands which deal with files are:
appendfile batch batchload closefile file_output_append filename_merge file_search file_search_maxima file_search_lisp file_search_demo file_search_usage file_search_tests file_type file_type_lisp file_type_maxima load load_pathname loadfile loadprint pathname_directory pathname_name pathname_type printfile save stringout with_stdout writefile
When a file name is passed to functions like plot2d,
save, or writefile and the file name does not include a path,
Maxima stores the file in the current working directory. The current working
directory depends on the system like Windows or Linux and on the installation.
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Appends a console transcript to filename. appendfile is the same
as writefile, except that the transcript file, if it exists, is
always appended.
closefile closes the transcript file opened by appendfile or
writefile.
Categories: File output · Console interaction
option)
batch(filename) reads Maxima expressions from filename and
evaluates them. batch searches for filename in the list
file_search_maxima. See also file_search.
batch(filename, is like demo)demo(filename).
In this case batch searches for filename in the list
file_search_demo. See demo.
batch(filename, is like test)run_testsuite with the
option display_all=true. For this case batch searches
filename in the list file_search_maxima and not in the list
file_search_tests like run_testsuite. Furthermore,
run_testsuite runs tests which are in the list
testsuite_files. With batch it is possible to run any file in
a test mode, which can be found in the list file_search_maxima. This is
useful, when writing a test file.
filename comprises a sequence of Maxima expressions, each terminated with
; or $. The special variable % and the function
%th refer to previous results within the file. The file may include
:lisp constructs. Spaces, tabs, and newlines in the file are ignored.
A suitable input file may be created by a text editor or by the
stringout function.
batch reads each input expression from filename, displays the input
to the console, computes the corresponding output expression, and displays the
output expression. Input labels are assigned to the input expressions and
output labels are assigned to the output expressions. batch evaluates
every input expression in the file unless there is an error. If user input is
requested (by asksign or askinteger, for example) batch
pauses to collect the requisite input and then continue.
It may be possible to halt batch by typing control-C at the
console. The effect of control-C depends on the underlying Lisp
implementation.
batch has several uses, such as to provide a reservoir for working
command lines, to give error-free demonstrations, or to help organize one's
thinking in solving complex problems.
batch evaluates its argument. batch returns the path of
filename as a string, when called with no second argument or with the
option demo. When called with the option test, the return value
is a an empty list [] or a list with filename and the numbers of
the tests which have failed.
See also load, batchload, and demo.
Categories: Session management · File input
Reads Maxima expressions from filename and evaluates them, without
displaying the input or output expressions and without assigning labels to
output expressions. Printed output (such as produced by print or
describe)) is displayed, however.
The special variable % and the function %th refer to previous
results from the interactive interpreter, not results within the file.
The file cannot include :lisp constructs.
batchload returns the path of filename, as a string.
batchload evaluates its argument.
Categories: Session management · File input
Closes the transcript file opened by writefile or appendfile.
Categories: File output · Console interaction
Default value: false
file_output_append governs whether file output functions append or
truncate their output file. When file_output_append is true, such
functions append to their output file. Otherwise, the output file is truncated.
save, stringout, and with_stdout respect
file_output_append. Other functions which write output files do not
respect file_output_append. In particular, plotting and translation
functions always truncate their output file, and tex and
appendfile always append.
Categories: File output · Global flags
Constructs a modified path from path and filename. If the final
component of path is of the form ###.something, the component
is replaced with filename.something. Otherwise, the final
component is simply replaced by filename.
The result is a Lisp pathname object.
Categories: File input · File output
file_search searches for the file filename and returns the path to
the file (as a string) if it can be found; otherwise file_search returns
false. file_search (filename) searches in the default
search directories, which are specified by the
file_search_maxima, file_search_lisp, and
file_search_demo variables.
file_search first checks if the actual name passed exists,
before attempting to match it to "wildcard" file search patterns.
See file_search_maxima concerning file search patterns.
The argument filename can be a path and file name, or just a file name, or, if a file search directory includes a file search pattern, just the base of the file name (without an extension). For example,
file_search ("/home/wfs/special/zeta.mac");
file_search ("zeta.mac");
file_search ("zeta");
all find the same file, assuming the file exists and
/home/wfs/special/###.mac is in file_search_maxima.
file_search (filename, pathlist) searches only in the
directories specified by pathlist, which is a list of strings. The
argument pathlist supersedes the default search directories, so if the
path list is given, file_search searches only the ones specified, and not
any of the default search directories. Even if there is only one directory in
pathlist, it must still be given as a one-element list.
The user may modify the default search directories.
See file_search_maxima.
file_search is invoked by load with file_search_maxima and
file_search_lisp as the search directories.
Categories: File input
These variables specify lists of directories to be searched by
load, demo, and some other Maxima functions. The default
values of these variables name various directories in the Maxima installation.
The user can modify these variables, either to replace the default values or to append additional directories. For example,
file_search_maxima: ["/usr/local/foo/###.mac",
"/usr/local/bar/###.mac"]$
replaces the default value of file_search_maxima, while
file_search_maxima: append (file_search_maxima,
["/usr/local/foo/###.mac", "/usr/local/bar/###.mac"])$
appends two additional directories. It may be convenient to put such an
expression in the file maxima-init.mac so that the file search path is
assigned automatically when Maxima starts.
See also Introduction for Runtime Environment.
Multiple filename extensions and multiple paths can be specified by special
"wildcard" constructions. The string ### expands into the sought-after
name, while a comma-separated list enclosed in curly braces
{foo,bar,baz} expands into multiple strings. For example, supposing
the sought-after name is neumann,
"/home/{wfs,gcj}/###.{lisp,mac}"
expands into /home/wfs/neumann.lisp, /home/gcj/neumann.lisp,
/home/wfs/neumann.mac, and /home/gcj/neumann.mac.
Categories: File input · Global variables
Returns a guess about the content of filename, based on the filename extension. filename need not refer to an actual file; no attempt is made to open the file and inspect the content.
The return value is a symbol, either object, lisp, or
maxima. If the extension is matches one of the values in
file_type_maxima, file_type returns maxima. If the
extension matches one of the values in file_type_lisp, file_type
returns lisp. If none of the above, file_type returns
object.
See also pathname_type.
See file_type_maxima and file_type_lisp for the default values.
Examples:
(%i2) map('file_type,
["test.lisp", "test.mac", "test.dem", "test.txt"]);
(%o2) [lisp, maxima, maxima, object]
Categories: File input
Default value: [l, lsp, lisp]
file_type_lisp is a list of file extensions that maxima recognizes
as denoting a Lisp source file.
See also file_type.
Default value: [mac, mc, demo, dem, dm1, dm2, dm3, dmt]
file_type_maxima is a list of file extensions that maxima recognizes
as denoting a Maxima source file.
See also file_type.
Evaluates expressions in filename, thus bringing variables, functions, and
other objects into Maxima. The binding of any existing object is clobbered by
the binding recovered from filename. To find the file, load calls
file_search with file_search_maxima and
file_search_lisp as the search directories. If load succeeds, it
returns the name of the file. Otherwise load prints an error message.
load works equally well for Lisp code and Maxima code. Files created by
save, translate_file, and compile_file, which
create Lisp code, and stringout, which creates Maxima code, can all
be processed by load. load calls loadfile to load Lisp
files and batchload to load Maxima files.
load does not recognize :lisp constructs in Maxima files, and
while processing filename, the global variables _, __,
%, and %th have whatever bindings they had when load was
called.
It is also to note that structures will only be read back as structures if
they have been defined by defstruct before the load command
is called.
See also loadfile, batch, batchload, and
demo. loadfile processes Lisp files; batch,
batchload, and demo process Maxima files.
See file_search for more detail about the file search mechanism.
load evaluates its argument.
Categories: Session management · File input
Default value: false
When a file is loaded with the functions load, loadfile or
batchload the system variable load_pathname is bound to the
pathname of the file which is processed.
The variable load_pathname can be accessed from the file during the
loading.
Example:
Suppose we have a batchfile test.mac in the directory
"/home/dieter/workspace/mymaxima/temp/" with the following commands
print("The value of load_pathname is: ", load_pathname)$
print("End of batchfile")$
then we get the following output
(%i1) load("/home/dieter/workspace/mymaxima/temp/test.mac")$
The value of load_pathname is:
/home/dieter/workspace/mymaxima/temp/test.mac
End of batchfile
Categories: File input
Evaluates Lisp expressions in filename. loadfile does not invoke
file_search, so filename must include the file extension and
as much of the path as needed to find the file.
loadfile can process files created by save,
translate_file, and compile_file. The user may find it
more convenient to use load instead of loadfile.
Categories: Session management · File input
Default value: true
loadprint tells whether to print a message when a file is loaded.
loadprint is true, always print a message.
loadprint is 'loadfile, print a message only if
a file is loaded by the function loadfile.
loadprint is 'autoload,
print a message only if a file is automatically loaded.
See setup_autoload.
loadprint is false, never print a message.
Categories: File input · Global flags
Returns a list of the files and directories found in path in the file system.
path may contain wildcard characters (i.e., characters which represent unspecified parts of the path), which include at least the asterisk on most systems, and possibly other characters, depending on the system.
directory relies on the Lisp function DIRECTORY,
which may have implementation-specific behavior.
Categories: File input
These functions return the components of pathname.
Examples:
(%i1) pathname_directory("/home/dieter/maxima/changelog.txt");
(%o1) /home/dieter/maxima/
(%i2) pathname_name("/home/dieter/maxima/changelog.txt");
(%o2) changelog
(%i3) pathname_type("/home/dieter/maxima/changelog.txt");
(%o3) txt
Categories: File input
Prints the file named by path to the console. path may be a string or a symbol; if it is a symbol, it is converted to a string.
If path names a file which is accessible from the current working
directory, that file is printed to the console. Otherwise, printfile
attempts to locate the file by appending path to each of the elements of
file_search_usage via filename_merge.
printfile returns path if it names an existing file,
or otherwise the result of a successful filename merge.
Categories: File input · Console interaction
Stores the current values of name_1, name_2, name_3, …,
in filename. The arguments are the names of variables, functions, or
other objects. If a name has no value or function associated with it, it is
ignored. save returns filename.
save stores data in the form of Lisp expressions.
If filename ends in .lisp the
data stored by save may be recovered by load (filename).
See load.
The global flag file_output_append governs whether save appends or
truncates the output file. When file_output_append is true,
save appends to the output file. Otherwise, save truncates the
output file. In either case, save creates the file if it does not yet
exist.
The special form save (filename, values, functions, labels, ...)
stores the items named by values, functions,
labels, etc. The names may be any specified by the variable
infolists. values comprises all user-defined variables.
The special form save (filename, [m, n]) stores the
values of input and output labels m through n. Note that m
and n must be literal integers. Input and output labels may also be
stored one by one, e.g., save ("foo.1", %i42, %o42).
save (filename, labels) stores all input and output labels.
When the stored labels are recovered, they clobber existing labels.
The special form save (filename, name_1=expr_1,
name_2=expr_2, ...) stores the values of expr_1,
expr_2, …, with names name_1, name_2, …
It is useful to apply this form to input and output labels, e.g.,
save ("foo.1", aa=%o88). The right-hand side of the equality in this
form may be any expression, which is evaluated. This form does not introduce
the new names into the current Maxima environment, but only stores them in
filename.
These special forms and the general form of save may be mixed at will.
For example, save (filename, aa, bb, cc=42, functions, [11, 17]).
The special form save (filename, all) stores the current state of
Maxima. This includes all user-defined variables, functions, arrays, etc., as
well as some automatically defined items. The saved items include system
variables, such as file_search_maxima or showtime, if they
have been assigned new values by the user; see myoptions.
save evaluates filename and quotes all other arguments.
Categories: Session management · File output
stringout writes expressions to a file in the same form the expressions
would be typed for input. The file can then be used as input for the
batch or demo commands, and it may be edited for any purpose.
stringout can be executed while writefile is in progress.
The global flag file_output_append governs whether stringout
appends or truncates the output file. When file_output_append is
true, stringout appends to the output file. Otherwise,
stringout truncates the output file. In either case, stringout
creates the file if it does not yet exist.
The general form of stringout writes the values of one or more
expressions to the output file. Note that if an expression is a
variable, only the value of the variable is written and not the name
of the variable. As a useful special case, the expressions may be
input labels (%i1, %i2, %i3, …) or output labels
(%o1, %o2, %o3, …).
If grind is true, stringout formats the output using the
grind format. Otherwise the string format is used. See
grind and string.
The special form stringout (filename, [m, n]) writes
the values of input labels m through n, inclusive.
The special form stringout (filename, input) writes all
input labels to the file.
The special form stringout (filename, functions) writes all
user-defined functions (named by the global list functions)) to the
file.
The special form stringout (filename, values) writes all
user-assigned variables (named by the global list values)) to the file.
Each variable is printed as an assignment statement, with the name of the
variable, a colon, and its value. Note that the general form of
stringout does not print variables as assignment statements.
Categories: Session management · File output
Evaluates expr_1, expr_2, expr_3, … and writes any
output thus generated to a file f or output stream s. The evaluated
expressions are not written to the output. Output may be generated by
print, display, grind, among other functions.
The global flag file_output_append governs whether with_stdout
appends or truncates the output file f. When file_output_append
is true, with_stdout appends to the output file. Otherwise,
with_stdout truncates the output file. In either case,
with_stdout creates the file if it does not yet exist.
with_stdout returns the value of its final argument.
See also writefile.
(%i1) with_stdout ("tmp.out", for i:5 thru 10 do
print (i, "! yields", i!))$
(%i2) printfile ("tmp.out")$
5 ! yields 120
6 ! yields 720
7 ! yields 5040
8 ! yields 40320
9 ! yields 362880
10 ! yields 3628800
Categories: File output
Begins writing a transcript of the Maxima session to filename. All interaction between the user and Maxima is then recorded in this file, just as it appears on the console.
As the transcript is printed in the console output format, it cannot be reloaded
into Maxima. To make a file containing expressions which can be reloaded,
see save and stringout. save stores expressions in Lisp
form, while stringout stores expressions in Maxima form.
The effect of executing writefile when filename already exists
depends on the underlying Lisp implementation; the transcript file may be
clobbered, or the file may be appended. appendfile always appends to
the transcript file.
It may be convenient to execute playback after writefile to save
the display of previous interactions. As playback displays only the
input and output variables (%i1, %o1, etc.), any output generated
by a print statement in a function (as opposed to a return value) is not
displayed by playback.
closefile closes the transcript file opened by writefile or
appendfile.
Categories: File output · Console interaction
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Note that the built-in TeX output functionality of wxMaxima makes no use of the functions described here but uses its own implementation instead.
Prints a representation of an expression suitable for the TeX document preparation system. The result is a fragment of a document, which can be copied into a larger document but not processed by itself.
tex (expr) prints a TeX representation of expr on the
console.
tex (label) prints a TeX representation of the expression named by
label and assigns it an equation label (to be displayed to the left of the
expression). The TeX equation label is the same as the Maxima label.
destination may be an output stream or file name. When destination
is a file name, tex appends its output to the file. The functions
openw and opena create output streams.
tex (expr, false) and tex (label, false)
return their TeX output as a string.
tex evaluates its first argument after testing it to see if it is a
label. Quote-quote '' forces evaluation of the argument, thereby
defeating the test and preventing the label.
See also texput.
Examples:
(%i1) integrate (1/(1+x^3), x);
2 x - 1
2 atan(-------)
log(x - x + 1) sqrt(3) log(x + 1)
(%o1) - --------------- + ------------- + ----------
6 sqrt(3) 3
(%i2) tex (%o1);
$$-{{\log \left(x^2-x+1\right)}\over{6}}+{{\arctan \left({{2\,x-1
}\over{\sqrt{3}}}\right)}\over{\sqrt{3}}}+{{\log \left(x+1\right)
}\over{3}}\leqno{\tt (\%o1)}$$
(%o2) (\%o1)
(%i3) tex (integrate (sin(x), x));
$$-\cos x$$
(%o3) false
(%i4) tex (%o1, "foo.tex");
(%o4) (\%o1)
tex (expr, false) returns its TeX output as a string.
(%i1) S : tex (x * y * z, false); (%o1) $$x\,y\,z$$ (%i2) S; (%o2) $$x\,y\,z$$
Categories: TeX output · File output
Returns a string which represents the TeX output for the expressions e. The TeX output is not enclosed in delimiters for an equation or any other environment.
Examples:
(%i1) tex1 (sin(x) + cos(x)); (%o1) \sin x+\cos x
Assign the TeX output for the atom a, which can be a symbol or the name of an operator.
texput (a, s) causes the tex function to interpolate
the string s into the TeX output in place of a.
texput (a, f) causes the tex function to call the
function f to generate TeX output. f must accept one argument,
which is an expression which has operator a, and must return a string
(the TeX output). f may call tex1 to generate TeX output for the
arguments of the input expression.
texput (a, s, operator_type), where operator_type
is prefix, infix, postfix, nary, or nofix,
causes the tex function to interpolate s into the TeX output in
place of a, and to place the interpolated text in the appropriate
position.
texput (a, [s_1, s_2], matchfix) causes the tex
function to interpolate s_1 and s_2 into the TeX output on either
side of the arguments of a. The arguments (if more than one) are
separated by commas.
texput (a, [s_1, s_2, s_3], matchfix) causes the
tex function to interpolate s_1 and s_2 into the TeX output
on either side of the arguments of a, with s_3 separating the
arguments.
Examples:
Assign TeX output for a variable.
(%i1) texput (me,"\\mu_e"); (%o1) \mu_e (%i2) tex (me); $$\mu_e$$ (%o2) false
Assign TeX output for an ordinary function (not an operator).
(%i1) texput (lcm, "\\mathrm{lcm}");
(%o1) \mathrm{lcm}
(%i2) tex (lcm (a, b));
$$\mathrm{lcm}\left(a , b\right)$$
(%o2) false
Call a function to generate TeX output.
(%i1) texfoo (e) := block ([a, b], [a, b] : args (e),
concat("\\left[\\stackrel{",tex1(b),"}{",tex1(a),"}\\right]"))$
(%i2) texput (foo, texfoo);
(%o2) texfoo
(%i3) tex (foo (2^x, %pi));
$$\left[\stackrel{\pi}{2^{x}}\right]$$
(%o3) false
Assign TeX output for a prefix operator.
(%i1) prefix ("grad");
(%o1) grad
(%i2) texput ("grad", " \\nabla ", prefix);
(%o2) \nabla
(%i3) tex (grad f);
$$ \nabla f$$
(%o3) false
Assign TeX output for an infix operator.
(%i1) infix ("~");
(%o1) ~
(%i2) texput ("~", " \\times ", infix);
(%o2) \times
(%i3) tex (a ~ b);
$$a \times b$$
(%o3) false
Assign TeX output for a postfix operator.
(%i1) postfix ("##");
(%o1) ##
(%i2) texput ("##", "!!", postfix);
(%o2) !!
(%i3) tex (x ##);
$$x!!$$
(%o3) false
Assign TeX output for a nary operator.
(%i1) nary ("@@");
(%o1) @@
(%i2) texput ("@@", " \\circ ", nary);
(%o2) \circ
(%i3) tex (a @@ b @@ c @@ d);
$$a \circ b \circ c \circ d$$
(%o3) false
Assign TeX output for a nofix operator.
(%i1) nofix ("foo");
(%o1) foo
(%i2) texput ("foo", "\\mathsc{foo}", nofix);
(%o2) \mathsc{foo}
(%i3) tex (foo);
$$\mathsc{foo}$$
(%o3) false
Assign TeX output for a matchfix operator.
(%i1) matchfix ("<<", ">>");
(%o1) <<
(%i2) texput ("<<", [" \\langle ", " \\rangle "], matchfix);
(%o2) [ \langle , \rangle ]
(%i3) tex (<<a>>);
$$ \langle a \rangle $$
(%o3) false
(%i4) tex (<<a, b>>);
$$ \langle a , b \rangle $$
(%o4) false
(%i5) texput ("<<", [" \\langle ", " \\rangle ", " \\, | \\,"],
matchfix);
(%o5) [ \langle , \rangle , \, | \,]
(%i6) tex (<<a>>);
$$ \langle a \rangle $$
(%o6) false
(%i7) tex (<<a, b>>);
$$ \langle a \, | \,b \rangle $$
(%o7) false
Categories: TeX output
Customize the TeX environment output by tex.
As maintained by these functions, the TeX environment comprises two strings:
one is printed before any other TeX output, and the other is printed after.
Only the TeX environment of the top-level operator in an expression is output; TeX environments associated with other operators are ignored.
get_tex_environment returns the TeX environment which is applied
to the operator op; returns the default if no other environment
has been assigned.
set_tex_environment assigns the TeX environment for the operator
op.
Examples:
(%i1) get_tex_environment (":=");
(%o1) [
\begin{verbatim}
, ;
\end{verbatim}
]
(%i2) tex (f (x) := 1 - x);
\begin{verbatim}
f(x):=1-x;
\end{verbatim}
(%o2) false
(%i3) set_tex_environment (":=", "$$", "$$");
(%o3) [$$, $$]
(%i4) tex (f (x) := 1 - x);
$$f(x):=1-x$$
(%o4) false
Categories: TeX output
Customize the TeX environment output by tex.
As maintained by these functions, the TeX environment comprises two strings:
one is printed before any other TeX output, and the other is printed after.
get_tex_environment_default returns the TeX environment which is
applied to expressions for which the top-level operator has no
specific TeX environment (as assigned by set_tex_environment).
set_tex_environment_default assigns the default TeX environment.
Examples:
(%i1) get_tex_environment_default ();
(%o1) [$$, $$]
(%i2) tex (f(x) + g(x));
$$g\left(x\right)+f\left(x\right)$$
(%o2) false
(%i3) set_tex_environment_default ("\\begin{equation}
", "
\\end{equation}");
(%o3) [\begin{equation}
,
\end{equation}]
(%i4) tex (f(x) + g(x));
\begin{equation}
g\left(x\right)+f\left(x\right)
\end{equation}
(%o4) false
Categories: TeX output
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Default value: 0
fortindent controls the left margin indentation of
expressions printed out by the fortran command. 0 gives normal
printout (i.e., 6 spaces), and positive values will causes the
expressions to be printed farther to the right.
Categories: Translation and compilation
Prints expr as a Fortran statement.
The output line is indented with spaces.
If the line is too long, fortran prints continuation lines.
fortran prints the exponentiation operator ^ as **,
and prints a complex number a + b %i in the form (a,b).
expr may be an equation. If so, fortran prints an assignment
statement, assigning the right-hand side of the equation to the left-hand side.
In particular, if the right-hand side of expr is the name of a matrix,
then fortran prints an assignment statement for each element of the
matrix.
If expr is not something recognized by fortran,
the expression is printed in grind format without complaint.
fortran does not know about lists, arrays, or functions.
fortindent controls the left margin of the printed lines.
0 is the normal margin (i.e., indented 6 spaces). Increasing
fortindent causes expressions to be printed further to the right.
When fortspaces is true, fortran fills out
each printed line with spaces to 80 columns.
fortran evaluates its arguments; quoting an argument defeats evaluation.
fortran always returns done.
See also the function f90 for printing one or more
expressions as a Fortran 90 program.
Examples:
(%i1) expr: (a + b)^12$
(%i2) fortran (expr);
(b+a)**12
(%o2) done
(%i3) fortran ('x=expr);
x = (b+a)**12
(%o3) done
(%i4) fortran ('x=expand (expr));
x = b**12+12*a*b**11+66*a**2*b**10+220*a**3*b**9+495*a**4*b**8+792
1 *a**5*b**7+924*a**6*b**6+792*a**7*b**5+495*a**8*b**4+220*a**9*b
2 **3+66*a**10*b**2+12*a**11*b+a**12
(%o4) done
(%i5) fortran ('x=7+5*%i);
x = (7,5)
(%o5) done
(%i6) fortran ('x=[1,2,3,4]);
x = [1,2,3,4]
(%o6) done
(%i7) f(x) := x^2$
(%i8) fortran (f);
f
(%o8) done
Categories: Translation and compilation
Default value: false
When fortspaces is true, fortran fills out
each printed line with spaces to 80 columns.
Categories: Translation and compilation
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| 14.1 Introduction to Polynomials | ||
| 14.2 Functions and Variables for Polynomials |
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Polynomials are stored in Maxima either in General Form or as Canonical Rational Expressions (CRE) form. The latter is a standard form, and is used internally by operations such as factor, ratsimp, and so on.
Canonical Rational Expressions constitute a kind of representation
which is especially suitable for expanded polynomials and rational
functions (as well as for partially factored polynomials and rational
functions when RATFAC is set to true). In this CRE form an
ordering of variables (from most to least main) is assumed for each
expression. Polynomials are represented recursively by a list
consisting of the main variable followed by a series of pairs of
expressions, one for each term of the polynomial. The first member of
each pair is the exponent of the main variable in that term and the
second member is the coefficient of that term which could be a number or
a polynomial in another variable again represented in this form. Thus
the principal part of the CRE form of 3*X^2-1 is (X 2 3 0 -1) and that
of 2*X*Y+X-3 is (Y 1 (X 1 2) 0 (X 1 1 0 -3)) assuming Y is the main
variable, and is (X 1 (Y 1 2 0 1) 0 -3) assuming X is the main
variable. "Main"-ness is usually determined by reverse alphabetical
order. The "variables" of a CRE expression needn't be atomic. In fact
any subexpression whose main operator is not + - * / or ^ with integer
power will be considered a "variable" of the expression (in CRE form) in
which it occurs. For example the CRE variables of the expression
X+SIN(X+1)+2*SQRT(X)+1 are X, SQRT(X), and SIN(X+1). If the user does
not specify an ordering of variables by using the RATVARS function
Maxima will choose an alphabetic one. In general, CRE's represent
rational expressions, that is, ratios of polynomials, where the
numerator and denominator have no common factors, and the denominator is
positive. The internal form is essentially a pair of polynomials (the
numerator and denominator) preceded by the variable ordering list. If
an expression to be displayed is in CRE form or if it contains any
subexpressions in CRE form, the symbol /R/ will follow the line label.
See the RAT function for converting an expression to CRE form. An
extended CRE form is used for the representation of Taylor series. The
notion of a rational expression is extended so that the exponents of the
variables can be positive or negative rational numbers rather than just
positive integers and the coefficients can themselves be rational
expressions as described above rather than just polynomials. These are
represented internally by a recursive polynomial form which is similar
to and is a generalization of CRE form, but carries additional
information such as the degree of truncation. As with CRE form, the
symbol /T/ follows the line label of such expressions.
Categories: Polynomials · Rational expressions
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Default value: false
algebraic must be set to true in order for the simplification of
algebraic integers to take effect.
Categories: Simplification flags and variables
Default value: true
When berlefact is false then the Kronecker factoring
algorithm will be used otherwise the Berlekamp algorithm, which is the
default, will be used.
Categories: Polynomials
an alternative to the resultant command. It
returns a matrix. determinant of this matrix is the desired resultant.
Examples:
(%i1) bezout(a*x+b, c*x^2+d, x);
[ b c - a d ]
(%o1) [ ]
[ a b ]
(%i2) determinant(%);
2 2
(%o2) a d + b c
(%i3) resultant(a*x+b, c*x^2+d, x);
2 2
(%o3) a d + b c
Categories: Polynomials
Returns a list whose first member is the coefficient of x in expr
(as found by ratcoef if expr is in CRE form
otherwise by coeff) and whose second member is the remaining part of
expr. That is, [A, B] where expr = A*x + B.
Example:
(%i1) islinear (expr, x) := block ([c],
c: bothcoef (rat (expr, x), x),
is (freeof (x, c) and c[1] # 0))$
(%i2) islinear ((r^2 - (x - r)^2)/x, x);
(%o2) true
Categories: Polynomials
Returns the coefficient of x^n in expr,
where expr is a polynomial or a monomial term in x.
Other than ratcoef coeff is a strictly syntactical
operation and will only find literal instances of
x^n in the internal representation of expr.
coeff(expr, x^n) is equivalent
to coeff(expr, x, n).
coeff(expr, x, 0) returns the remainder of expr
which is free of x.
If omitted, n is assumed to be 1.
x may be a simple variable or a subscripted variable, or a subexpression of expr which comprises an operator and all of its arguments.
It may be possible to compute coefficients of expressions which are equivalent
to expr by applying expand or factor. coeff itself
does not apply expand or factor or any other function.
coeff distributes over lists, matrices, and equations.
See also ratcoef.
Examples:
coeff returns the coefficient x^n in expr.
(%i1) coeff (b^3*a^3 + b^2*a^2 + b*a + 1, a^3);
3
(%o1) b
coeff(expr, x^n) is equivalent
to coeff(expr, x, n).
(%i1) coeff (c[4]*z^4 - c[3]*z^3 - c[2]*z^2 + c[1]*z, z, 3);
(%o1) - c
3
(%i2) coeff (c[4]*z^4 - c[3]*z^3 - c[2]*z^2 + c[1]*z, z^3);
(%o2) - c
3
coeff(expr, x, 0) returns the remainder of expr
which is free of x.
(%i1) coeff (a*u + b^2*u^2 + c^3*u^3, b, 0);
3 3
(%o1) c u + a u
x may be a simple variable or a subscripted variable, or a subexpression of expr which comprises an operator and all of its arguments.
(%i1) coeff (h^4 - 2*%pi*h^2 + 1, h, 2);
(%o1) - 2 %pi
(%i2) coeff (v[1]^4 - 2*%pi*v[1]^2 + 1, v[1], 2);
(%o2) - 2 %pi
(%i3) coeff (sin(1+x)*sin(x) + sin(1+x)^3*sin(x)^3, sin(1+x)^3);
3
(%o3) sin (x)
(%i4) coeff ((d - a)^2*(b + c)^3 + (a + b)^4*(c - d), a + b, 4);
(%o4) c - d
coeff itself does not apply expand or factor or any other
function.
(%i1) coeff (c*(a + b)^3, a);
(%o1) 0
(%i2) expand (c*(a + b)^3);
3 2 2 3
(%o2) b c + 3 a b c + 3 a b c + a c
(%i3) coeff (%, a);
2
(%o3) 3 b c
(%i4) coeff (b^3*c + 3*a*b^2*c + 3*a^2*b*c + a^3*c, (a + b)^3);
(%o4) 0
(%i5) factor (b^3*c + 3*a*b^2*c + 3*a^2*b*c + a^3*c);
3
(%o5) (b + a) c
(%i6) coeff (%, (a + b)^3);
(%o6) c
coeff distributes over lists, matrices, and equations.
(%i1) coeff ([4*a, -3*a, 2*a], a);
(%o1) [4, - 3, 2]
(%i2) coeff (matrix ([a*x, b*x], [-c*x, -d*x]), x);
[ a b ]
(%o2) [ ]
[ - c - d ]
(%i3) coeff (a*u - b*v = 7*u + 3*v, u);
(%o3) a = 7
Categories: Polynomials
Returns a list whose first element is the greatest common divisor of the coefficients of the terms of the polynomial p_1 in the variable x_n (this is the content) and whose second element is the polynomial p_1 divided by the content.
Examples:
(%i1) content (2*x*y + 4*x^2*y^2, y);
2
(%o1) [2 x, 2 x y + y]
Categories: Polynomials
Returns the denominator of the rational expression expr.
See also num
(%i1) g1:(x+2)*(x+1)/((x+3)^2);
(x + 1) (x + 2)
(%o1) ---------------
2
(x + 3)
(%i2) denom(g1);
2
(%o2) (x + 3)
(%i3) g2:sin(x)/10*cos(x)/y;
cos(x) sin(x)
(%o3) -------------
10 y
(%i4) denom(g2);
(%o4) 10 y
Categories: Expressions
computes the quotient and remainder
of the polynomial p_1 divided by the polynomial p_2, in a main
polynomial variable, x_n.
The other variables are as in the ratvars function.
The result is a list whose first element is the quotient
and whose second element is the remainder.
Examples:
(%i1) divide (x + y, x - y, x); (%o1) [1, 2 y] (%i2) divide (x + y, x - y); (%o2) [- 1, 2 x]
Note that y is the main variable in the second example.
Categories: Polynomials
Eliminates variables from equations (or expressions assumed equal to zero) by
taking successive resultants. This returns a list of n - k
expressions with the k variables x_1, …, x_k eliminated.
First x_1 is eliminated yielding n - 1 expressions, then
x_2 is eliminated, etc. If k = n then a single
expression in a list is returned free of the variables x_1, …,
x_k. In this case solve is called to solve the last resultant for
the last variable.
Example:
(%i1) expr1: 2*x^2 + y*x + z;
2
(%o1) z + x y + 2 x
(%i2) expr2: 3*x + 5*y - z - 1;
(%o2) - z + 5 y + 3 x - 1
(%i3) expr3: z^2 + x - y^2 + 5;
2 2
(%o3) z - y + x + 5
(%i4) eliminate ([expr3, expr2, expr1], [y, z]);
8 7 6 5 4
(%o4) [7425 x - 1170 x + 1299 x + 12076 x + 22887 x
3 2
- 5154 x - 1291 x + 7688 x + 15376]
Categories: Polynomials · Algebraic equations
Returns a list whose first element is the greatest common divisor of the
polynomials p_1, p_2, p_3, … and whose remaining
elements are the polynomials divided by the greatest common divisor. This
always uses the ezgcd algorithm.
See also gcd, gcdex, gcdivide, and
poly_gcd.
Examples:
The three polynomials have the greatest common divisor 2*x-3. The
gcd is first calculated with the function gcd and then with the function
ezgcd.
(%i1) p1 : 6*x^3-17*x^2+14*x-3;
3 2
(%o1) 6 x - 17 x + 14 x - 3
(%i2) p2 : 4*x^4-14*x^3+12*x^2+2*x-3;
4 3 2
(%o2) 4 x - 14 x + 12 x + 2 x - 3
(%i3) p3 : -8*x^3+14*x^2-x-3;
3 2
(%o3) - 8 x + 14 x - x - 3
(%i4) gcd(p1, gcd(p2, p3));
(%o4) 2 x - 3
(%i5) ezgcd(p1, p2, p3);
2 3 2 2
(%o5) [2 x - 3, 3 x - 4 x + 1, 2 x - 4 x + 1, - 4 x + x + 1]
Categories: Polynomials
Default value: true
facexpand controls whether the irreducible factors returned by
factor are in expanded (the default) or recursive (normal CRE) form.
Categories: Polynomials
Factors the expression expr, containing any number of variables or
functions, into factors irreducible over the integers.
factor (expr, p) factors expr over the field of
rationals with an element adjoined whose minimum polynomial is p.
factor uses ifactors function for factoring integers.
factorflag if false suppresses the factoring of integer factors
of rational expressions.
dontfactor may be set to a list of variables with respect to which
factoring is not to occur. (It is initially empty). Factoring also
will not take place with respect to any variables which are less
important (using the variable ordering assumed for CRE form) than
those on the dontfactor list.
savefactors if true causes the factors of an expression which
is a product of factors to be saved by certain functions in order to
speed up later factorizations of expressions containing some of the
same factors.
berlefact if false then the Kronecker factoring algorithm will
be used otherwise the Berlekamp algorithm, which is the default, will
be used.
intfaclim if true maxima will give up factorization of
integers if no factor is found after trial divisions and Pollard's rho
method. If set to false (this is the case when the user calls
factor explicitly), complete factorization of the integer will be
attempted. The user's setting of intfaclim is used for internal
calls to factor. Thus, intfaclim may be reset to prevent
Maxima from taking an inordinately long time factoring large integers.
Examples:
(%i1) factor (2^63 - 1);
2
(%o1) 7 73 127 337 92737 649657
(%i2) factor (-8*y - 4*x + z^2*(2*y + x));
(%o2) (2 y + x) (z - 2) (z + 2)
(%i3) -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2;
2 2 2 2 2
(%o3) x y + 2 x y + y - x - 2 x - 1
(%i4) block ([dontfactor: [x]], factor (%/36/(1 + 2*y + y^2)));
2
(x + 2 x + 1) (y - 1)
(%o4) ----------------------
36 (y + 1)
(%i5) factor (1 + %e^(3*x));
x 2 x x
(%o5) (%e + 1) (%e - %e + 1)
(%i6) factor (1 + x^4, a^2 - 2);
2 2
(%o6) (x - a x + 1) (x + a x + 1)
(%i7) factor (-y^2*z^2 - x*z^2 + x^2*y^2 + x^3);
2
(%o7) - (y + x) (z - x) (z + x)
(%i8) (2 + x)/(3 + x)/(b + x)/(c + x)^2;
x + 2
(%o8) ------------------------
2
(x + 3) (x + b) (x + c)
(%i9) ratsimp (%);
4 3
(%o9) (x + 2)/(x + (2 c + b + 3) x
2 2 2 2
+ (c + (2 b + 6) c + 3 b) x + ((b + 3) c + 6 b c) x + 3 b c )
(%i10) partfrac (%, x);
2 4 3
(%o10) - (c - 4 c - b + 6)/((c + (- 2 b - 6) c
2 2 2 2
+ (b + 12 b + 9) c + (- 6 b - 18 b) c + 9 b ) (x + c))
c - 2
- ---------------------------------
2 2
(c + (- b - 3) c + 3 b) (x + c)
b - 2
+ -------------------------------------------------
2 2 3 2
((b - 3) c + (6 b - 2 b ) c + b - 3 b ) (x + b)
1
- ----------------------------------------------
2
((b - 3) c + (18 - 6 b) c + 9 b - 27) (x + 3)
(%i11) map ('factor, %);
2
c - 4 c - b + 6 c - 2
(%o11) - ------------------------- - ------------------------
2 2 2
(c - 3) (c - b) (x + c) (c - 3) (c - b) (x + c)
b - 2 1
+ ------------------------ - ------------------------
2 2
(b - 3) (c - b) (x + b) (b - 3) (c - 3) (x + 3)
(%i12) ratsimp ((x^5 - 1)/(x - 1));
4 3 2
(%o12) x + x + x + x + 1
(%i13) subst (a, x, %);
4 3 2
(%o13) a + a + a + a + 1
(%i14) factor (%th(2), %);
2 3 3 2
(%o14) (x - a) (x - a ) (x - a ) (x + a + a + a + 1)
(%i15) factor (1 + x^12);
4 8 4
(%o15) (x + 1) (x - x + 1)
(%i16) factor (1 + x^99);
2 6 3
(%o16) (x + 1) (x - x + 1) (x - x + 1)
10 9 8 7 6 5 4 3 2
(x - x + x - x + x - x + x - x + x - x + 1)
20 19 17 16 14 13 11 10 9 7 6
(x + x - x - x + x + x - x - x - x + x + x
4 3 60 57 51 48 42 39 33
- x - x + x + 1) (x + x - x - x + x + x - x
30 27 21 18 12 9 3
- x - x + x + x - x - x + x + 1)
Categories: Polynomials
Default value: false
When factorflag is false, suppresses the factoring of
integer factors of rational expressions.
Categories: Polynomials
Rearranges the sum expr into a sum of terms of the form
f (x_1, x_2, …)*g where g is a product of
expressions not containing any x_i and f is factored.
Note that the option variable keepfloat is ignored by factorout.
Example:
(%i1) expand (a*(x+1)*(x-1)*(u+1)^2);
2 2 2 2 2
(%o1) a u x + 2 a u x + a x - a u - 2 a u - a
(%i2) factorout(%,x);
2
(%o2) a u (x - 1) (x + 1) + 2 a u (x - 1) (x + 1)
+ a (x - 1) (x + 1)
Categories: Expressions
Tries to group terms in factors of expr which are sums into groups of
terms such that their sum is factorable. factorsum can recover the
result of expand ((x + y)^2 + (z + w)^2) but it can't recover
expand ((x + 1)^2 + (x + y)^2) because the terms have variables in
common.
Example:
(%i1) expand ((x + 1)*((u + v)^2 + a*(w + z)^2));
2 2 2 2
(%o1) a x z + a z + 2 a w x z + 2 a w z + a w x + v x
2 2 2 2
+ 2 u v x + u x + a w + v + 2 u v + u
(%i2) factorsum (%);
2 2
(%o2) (x + 1) (a (z + w) + (v + u) )
Categories: Expressions
Returns the product of the polynomials p_1 and p_2 by using a
special algorithm for multiplication of polynomials. p_1 and p_2
should be multivariate, dense, and nearly the same size. Classical
multiplication is of order n_1 n_2 where
n_1 is the degree of p_1
and n_2 is the degree of p_2.
fasttimes is of order max (n_1, n_2)^1.585.
Categories: Polynomials
fullratsimp repeatedly
applies ratsimp followed by non-rational simplification to an
expression until no further change occurs,
and returns the result.
When non-rational expressions are involved, one call
to ratsimp followed as is usual by non-rational ("general")
simplification may not be sufficient to return a simplified result.
Sometimes, more than one such call may be necessary.
fullratsimp makes this process convenient.
fullratsimp (expr, x_1, ..., x_n) takes one or more
arguments similar to ratsimp and rat.
Example:
(%i1) expr: (x^(a/2) + 1)^2*(x^(a/2) - 1)^2/(x^a - 1);
a/2 2 a/2 2
(x - 1) (x + 1)
(%o1) -----------------------
a
x - 1
(%i2) ratsimp (expr);
2 a a
x - 2 x + 1
(%o2) ---------------
a
x - 1
(%i3) fullratsimp (expr);
a
(%o3) x - 1
(%i4) rat (expr);
a/2 4 a/2 2
(x ) - 2 (x ) + 1
(%o4)/R/ -----------------------
a
x - 1
Categories: Simplification functions · Rational expressions
is the same as ratsubst except that it calls
itself recursively on its result until that result stops changing.
This function is useful when the replacement expression and the
replaced expression have one or more variables in common.
fullratsubst will also accept its arguments in the format of
lratsubst. That is, the first argument may be a single substitution
equation or a list of such equations, while the second argument is the
expression being processed.
load ("lrats") loads fullratsubst and lratsubst.
Examples:
(%i1) load ("lrats")$
subst can carry out multiple substitutions.
lratsubst is analogous to subst.
(%i2) subst ([a = b, c = d], a + c); (%o2) d + b (%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c)); (%o3) (d + a c) e + a d + b c
(%i4) lratsubst (a^2 = b, a^3); (%o4) a b
fullratsubst is equivalent to ratsubst
except that it recurses until its result stops changing.
(%i5) ratsubst (b*a, a^2, a^3);
2
(%o5) a b
(%i6) fullratsubst (b*a, a^2, a^3);
2
(%o6) a b
fullratsubst also accepts a list of equations or a single
equation as first argument.
(%i7) fullratsubst ([a^2 = b, b^2 = c, c^2 = a], a^3*b*c);
(%o7) b
(%i8) fullratsubst (a^2 = b*a, a^3);
2
(%o8) a b
fullratsubst may cause an indefinite recursion.
(%i9) errcatch (fullratsubst (b*a^2, a^2, a^3)); *** - Lisp stack overflow. RESET
Categories: Rational expressions
Returns the greatest common divisor of p_1 and p_2. The flag
gcd determines which algorithm is employed. Setting gcd to
ez, subres, red, or spmod selects the ezgcd,
subresultant prs, reduced, or modular algorithm, respectively. If
gcd false then gcd (p_1, p_2, x) always
returns 1 for all x. Many functions (e.g. ratsimp,
factor, etc.) cause gcd's to be taken implicitly. For homogeneous
polynomials it is recommended that gcd equal to subres be used.
To take the gcd when an algebraic is present, e.g.,
gcd (x^2 - 2*sqrt(2)* x + 2, x - sqrt(2)), the option
variable algebraic must be true and gcd must not be
ez.
The gcd flag, default: spmod, if false will also prevent
the greatest common divisor from being taken when expressions are converted to
canonical rational expression (CRE) form. This will sometimes speed the
calculation if gcds are not required.
See also ezgcd, gcdex, gcdivide, and
poly_gcd.
Example:
(%i1) p1:6*x^3+19*x^2+19*x+6;
3 2
(%o1) 6 x + 19 x + 19 x + 6
(%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
5 4 3 2
(%o2) 6 x + 13 x + 12 x + 13 x + 6 x
(%i3) gcd(p1, p2);
2
(%o3) 6 x + 13 x + 6
(%i4) p1/gcd(p1, p2), ratsimp;
(%o4) x + 1
(%i5) p2/gcd(p1, p2), ratsimp;
3
(%o5) x + x
ezgcd returns a list whose first element is the greatest common divisor
of the polynomials p_1 and p_2, and whose remaining elements are
the polynomials divided by the greatest common divisor.
(%i6) ezgcd(p1, p2);
2 3
(%o6) [6 x + 13 x + 6, x + 1, x + x]
Categories: Polynomials · Rational expressions
Returns a list [a, b, u] where u is the greatest
common divisor (gcd) of f and g, and u is equal to
a f + b g. The arguments f and g
should be univariate polynomials, or else polynomials in x a supplied
main variable since we need to be in a principal ideal domain for this to
work. The gcd means the gcd regarding f and g as univariate
polynomials with coefficients being rational functions in the other variables.
gcdex implements the Euclidean algorithm, where we have a sequence of
L[i]: [a[i], b[i], r[i]] which are all perpendicular to [f, g, -1]
and the next one is built as if q = quotient(r[i]/r[i+1]) then
L[i+2]: L[i] - q L[i+1], and it terminates at L[i+1] when the
remainder r[i+2] is zero.
The arguments f and g can be integers. For this case the function
igcdex is called by gcdex.
See also ezgcd, gcd, gcdivide, and
poly_gcd.
Examples:
(%i1) gcdex (x^2 + 1, x^3 + 4);
2
x + 4 x - 1 x + 4
(%o1)/R/ [- ------------, -----, 1]
17 17
(%i2) % . [x^2 + 1, x^3 + 4, -1];
(%o2)/R/ 0
Note that the gcd in the following is 1 since we work in k(y)[x],
not the y+1 we would expect in k[y, x].
(%i1) gcdex (x*(y + 1), y^2 - 1, x);
1
(%o1)/R/ [0, ------, 1]
2
y - 1
Categories: Polynomials · Rational expressions
Factors the Gaussian integer n over the Gaussian integers, i.e., numbers
of the form a + b where a and b are
rational integers (i.e., ordinary integers). Factors are normalized by making
a and b non-negative.
%i
Categories: Integers
Factors the polynomial expr over the Gaussian integers
(that is, the integers with the imaginary unit %i adjoined).
This is like factor (expr, a^2+1) where a is %i.
Example:
(%i1) gfactor (x^4 - 1); (%o1) (x - 1) (x + 1) (x - %i) (x + %i)
Categories: Polynomials
is similar to factorsum but applies gfactor instead
of factor.
Categories: Expressions
Returns the highest explicit exponent of x in expr.
x may be a variable or a general expression.
If x does not appear in expr,
hipow returns 0.
hipow does not consider expressions equivalent to expr. In
particular, hipow does not expand expr, so
hipow (expr, x) and
hipow (expand (expr, x)) may yield different results.
Examples:
(%i1) hipow (y^3 * x^2 + x * y^4, x); (%o1) 2 (%i2) hipow ((x + y)^5, x); (%o2) 1 (%i3) hipow (expand ((x + y)^5), x); (%o3) 5 (%i4) hipow ((x + y)^5, x + y); (%o4) 5 (%i5) hipow (expand ((x + y)^5), x + y); (%o5) 0
Categories: Expressions
Default value: true
If true, maxima will give up factorization of
integers if no factor is found after trial divisions and Pollard's rho
method and factorization will not be complete.
When intfaclim is false (this is the case when the user
calls factor explicitly), complete factorization will be
attempted. intfaclim is set to false when factors are
computed in divisors, divsum and totient.
Internal calls to factor respect the user-specified value of
intfaclim. Setting intfaclim to true may reduce
the time spent factoring large integers.
Categories: Integers
Default value: false
When keepfloat is true, prevents floating
point numbers from being rationalized when expressions which contain
them are converted to canonical rational expression (CRE) form.
Note that the function solve and those functions calling it
(eigenvalues, for example) currently ignore this flag, converting
floating point numbers anyway.
Examples:
(%i1) rat(x/2.0);
rat: replaced 0.5 by 1/2 = 0.5
x
(%o1)/R/ -
2
(%i2) rat(x/2.0), keepfloat;
(%o2)/R/ 0.5 x
solve ignores keepfloat:
(%i1) solve(1.0-x,x), keepfloat; rat: replaced 1.0 by 1/1 = 1.0 (%o1) [x = 1]
Categories: Numerical evaluation
Returns the lowest exponent of x which explicitly appears in expr. Thus
(%i1) lopow ((x+y)^2 + (x+y)^a, x+y); (%o1) min(a, 2)
Categories: Expressions
is analogous to subst (L, expr)
except that it uses ratsubst instead of subst.
The first argument of
lratsubst is an equation or a list of equations identical in
format to that accepted by subst. The
substitutions are made in the order given by the list of equations,
that is, from left to right.
load ("lrats") loads fullratsubst and lratsubst.
Examples:
(%i1) load ("lrats")$
subst can carry out multiple substitutions.
lratsubst is analogous to subst.
(%i2) subst ([a = b, c = d], a + c); (%o2) d + b (%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c)); (%o3) (d + a c) e + a d + b c
(%i4) lratsubst (a^2 = b, a^3); (%o4) a b
Categories: Polynomials · Rational expressions
Default value: false
When modulus is a positive number p, operations on rational numbers
(as returned by rat and related functions) are carried out modulo
p, using the so-called "balanced" modulus system in which n
modulo p is defined as an integer k in
[-(p-1)/2, ..., 0, ..., (p-1)/2] when p is odd, or
[-(p/2 - 1), ..., 0, ...., p/2] when p is even, such
that a p + k equals n for some integer a.
If expr is already in canonical rational expression (CRE) form when
modulus is reset, then you may need to re-rat expr, e.g.,
expr: rat (ratdisrep (expr)), in order to get correct results.
Typically modulus is set to a prime number. If modulus is set to
a positive non-prime integer, this setting is accepted, but a warning message is
displayed. Maxima signals an error, when zero or a negative integer is
assigned to modulus.
Examples:
(%i1) modulus:7;
(%o1) 7
(%i2) polymod([0,1,2,3,4,5,6,7]);
(%o2) [0, 1, 2, 3, - 3, - 2, - 1, 0]
(%i3) modulus:false;
(%o3) false
(%i4) poly:x^6+x^2+1;
6 2
(%o4) x + x + 1
(%i5) factor(poly);
6 2
(%o5) x + x + 1
(%i6) modulus:13;
(%o6) 13
(%i7) factor(poly);
2 4 2
(%o7) (x + 6) (x - 6 x - 2)
(%i8) polymod(%);
6 2
(%o8) x + x + 1
Categories: Integers
Returns the numerator of expr if it is a ratio. If expr is not a ratio, expr is returned.
num evaluates its argument.
See also denom
(%i1) g1:(x+2)*(x+1)/((x+3)^2);
(x + 1) (x + 2)
(%o1) ---------------
2
(x + 3)
(%i2) num(g1);
(%o2) (x + 1) (x + 2)
(%i3) g2:sin(x)/10*cos(x)/y;
cos(x) sin(x)
(%o3) -------------
10 y
(%i4) num(g2);
(%o4) cos(x) sin(x)
Categories: Expressions
Decomposes the polynomial p in the variable x
into the functional composition of polynomials in x.
polydecomp returns a list [p_1, ..., p_n] such that
lambda ([x], p_1) (lambda ([x], p_2) (... (lambda ([x], p_n) (x)) ...))
is equal to p. The degree of p_i is greater than 1 for i less than n.
Such a decomposition is not unique.
Examples:
(%i1) polydecomp (x^210, x);
7 5 3 2
(%o1) [x , x , x , x ]
(%i2) p : expand (subst (x^3 - x - 1, x, x^2 - a));
6 4 3 2
(%o2) x - 2 x - 2 x + x + 2 x - a + 1
(%i3) polydecomp (p, x);
2 3
(%o3) [x - a, x - x - 1]
The following function composes L = [e_1, ..., e_n] as functions in
x; it is the inverse of polydecomp:
(%i1) compose (L, x) := block ([r : x], for e in L do r : subst (e, x, r), r) $
Re-express above example using compose:
(%i1) polydecomp (compose ([x^2 - a, x^3 - x - 1], x), x);
2 3
(%o1) [compose([x - a, x - x - 1], x)]
Note that though compose (polydecomp (p, x), x) always
returns p (unexpanded), polydecomp (compose ([p_1, ...,
p_n], x), x) does not necessarily return
[p_1, ..., p_n]:
(%i1) polydecomp (compose ([x^2 + 2*x + 3, x^2], x), x);
2 2
(%o1) [compose([x + 2 x + 3, x ], x)]
(%i2) polydecomp (compose ([x^2 + x + 1, x^2 + x + 1], x), x);
2 2
(%o2) [compose([x + x + 1, x + x + 1], x)]
Categories: Polynomials
Converts the polynomial p to a modular representation with respect to the
current modulus which is the value of the variable modulus.
polymod (p, m) specifies a modulus m to be used
instead of the current value of modulus.
See modulus.
Categories: Polynomials
Returns the polynomial p_1 divided by the polynomial p_2. The
arguments x_1, …, x_n are interpreted as in ratvars.
quotient returns the first element of the two-element list returned by
divide.
Categories: Polynomials
Converts expr to canonical rational expression (CRE) form by expanding and
combining all terms over a common denominator and cancelling out the
greatest common divisor of the numerator and denominator, as well as
converting floating point numbers to rational numbers within a
tolerance of ratepsilon.
The variables are ordered according
to the x_1, …, x_n, if specified, as in ratvars.
rat does not generally simplify functions other than addition +,
subtraction -, multiplication *, division /, and
exponentiation to an integer power,
whereas ratsimp does handle those cases.
Note that atoms (numbers and variables) in CRE form are not the
same as they are in the general form.
For example, rat(x)- x yields
rat(0) which has a different internal representation than 0.
When ratfac is true, rat yields a partially factored
form for CRE. During rational operations the expression is
maintained as fully factored as possible without an actual call to the
factor package. This should always save space and may save some time
in some computations. The numerator and denominator are still made
relatively prime
(e.g., rat((x^2 - 1)^4/(x + 1)^2) yields (x - 1)^4 (x + 1)^2
when ratfac is true),
but the factors within each part may not be relatively prime.
ratprint if false suppresses the printout of the message
informing the user of the conversion of floating point numbers to
rational numbers.
keepfloat if true prevents floating point numbers from being
converted to rational numbers.
See also ratexpand and ratsimp.
Examples:
(%i1) ((x - 2*y)^4/(x^2 - 4*y^2)^2 + 1)*(y + a)*(2*y + x) /
(4*y^2 + x^2);
4
(x - 2 y)
(y + a) (2 y + x) (------------ + 1)
2 2 2
(x - 4 y )
(%o1) ------------------------------------
2 2
4 y + x
(%i2) rat (%, y, a, x);
2 a + 2 y
(%o2)/R/ ---------
x + 2 y
Categories: Rational expressions
Default value: true
When ratalgdenom is true, allows rationalization of denominators
with respect to radicals to take effect. ratalgdenom has an effect only
when canonical rational expressions (CRE) are used in algebraic mode.
Categories: Simplification flags and variables
Returns the coefficient of the expression x^n
in the expression expr.
If omitted, n is assumed to be 1.
The return value is free (except possibly in a non-rational sense) of the variables in x. If no coefficient of this type exists, 0 is returned.
ratcoef
expands and rationally simplifies its first argument and thus it may
produce answers different from those of coeff which is purely
syntactic.
Thus ratcoef ((x + 1)/y + x, x) returns (y + 1)/y whereas
coeff returns 1.
ratcoef (expr, x, 0), viewing expr as a sum,
returns a sum of those terms which do not contain x.
Therefore if x occurs to any negative powers, ratcoef should not
be used.
Since expr is rationally simplified before it is examined, coefficients may not appear quite the way they were envisioned.
Example:
(%i1) s: a*x + b*x + 5$ (%i2) ratcoef (s, a + b); (%o2) x
Categories: Polynomials · Rational expressions
Returns the denominator of expr, after coercing expr to a canonical rational expression (CRE). The return value is a CRE.
expr is coerced to a CRE by rat
if it is not already a CRE.
This conversion may change the form of expr by putting all terms
over a common denominator.
denom is similar, but returns an ordinary expression instead of a CRE.
Also, denom does not attempt to place all terms over a common
denominator, and thus some expressions which are considered ratios by
ratdenom are not considered ratios by denom.
Categories: Rational expressions
Default value: true
When ratdenomdivide is true,
ratexpand expands a ratio in which the numerator is a sum
into a sum of ratios,
all having a common denominator.
Otherwise, ratexpand collapses a sum of ratios into a single ratio,
the numerator of which is the sum of the numerators of each ratio.
Examples:
(%i1) expr: (x^2 + x + 1)/(y^2 + 7);
2
x + x + 1
(%o1) ----------
2
y + 7
(%i2) ratdenomdivide: true$
(%i3) ratexpand (expr);
2
x x 1
(%o3) ------ + ------ + ------
2 2 2
y + 7 y + 7 y + 7
(%i4) ratdenomdivide: false$
(%i5) ratexpand (expr);
2
x + x + 1
(%o5) ----------
2
y + 7
(%i6) expr2: a^2/(b^2 + 3) + b/(b^2 + 3);
2
b a
(%o6) ------ + ------
2 2
b + 3 b + 3
(%i7) ratexpand (expr2);
2
b + a
(%o7) ------
2
b + 3
Categories: Simplification flags and variables · Rational expressions
Differentiates the rational expression expr with respect to x. expr must be a ratio of polynomials or a polynomial in x. The argument x may be a variable or a subexpression of expr.
The result is equivalent to diff, although perhaps in a different form.
ratdiff may be faster than diff, for rational expressions.
ratdiff returns a canonical rational expression (CRE) if expr is
a CRE. Otherwise, ratdiff returns a general expression.
ratdiff considers only the dependence of expr on x,
and ignores any dependencies established by depends.
Example:
(%i1) expr: (4*x^3 + 10*x - 11)/(x^5 + 5);
3
4 x + 10 x - 11
(%o1) ----------------
5
x + 5
(%i2) ratdiff (expr, x);
7 5 4 2
8 x + 40 x - 55 x - 60 x - 50
(%o2) - ---------------------------------
10 5
x + 10 x + 25
(%i3) expr: f(x)^3 - f(x)^2 + 7;
3 2
(%o3) f (x) - f (x) + 7
(%i4) ratdiff (expr, f(x));
2
(%o4) 3 f (x) - 2 f(x)
(%i5) expr: (a + b)^3 + (a + b)^2;
3 2
(%o5) (b + a) + (b + a)
(%i6) ratdiff (expr, a + b);
2 2
(%o6) 3 b + (6 a + 2) b + 3 a + 2 a
Categories: Rational expressions
Returns its argument as a general expression. If expr is a general expression, it is returned unchanged.
Typically ratdisrep is called to convert a canonical rational expression
(CRE) into a general expression.
This is sometimes convenient if one wishes to stop the "contagion", or
use rational functions in non-rational contexts.
See also totaldisrep.
Categories: Rational expressions
Expands expr by multiplying out products of sums and exponentiated sums, combining fractions over a common denominator, cancelling the greatest common divisor of the numerator and denominator, then splitting the numerator (if a sum) into its respective terms divided by the denominator.
The return value of ratexpand is a general expression,
even if expr is a canonical rational expression (CRE).
The switch ratexpand if true will cause CRE
expressions to be fully expanded when they are converted back to
general form or displayed, while if it is false then they will be put
into a recursive form.
See also ratsimp.
When ratdenomdivide is true,
ratexpand expands a ratio in which the numerator is a sum
into a sum of ratios,
all having a common denominator.
Otherwise, ratexpand collapses a sum of ratios into a single ratio,
the numerator of which is the sum of the numerators of each ratio.
When keepfloat is true, prevents floating
point numbers from being rationalized when expressions which contain
them are converted to canonical rational expression (CRE) form.
Examples:
(%i1) ratexpand ((2*x - 3*y)^3);
3 2 2 3
(%o1) - 27 y + 54 x y - 36 x y + 8 x
(%i2) expr: (x - 1)/(x + 1)^2 + 1/(x - 1);
x - 1 1
(%o2) -------- + -----
2 x - 1
(x + 1)
(%i3) expand (expr);
x 1 1
(%o3) ------------ - ------------ + -----
2 2 x - 1
x + 2 x + 1 x + 2 x + 1
(%i4) ratexpand (expr);
2
2 x 2
(%o4) --------------- + ---------------
3 2 3 2
x + x - x - 1 x + x - x - 1
Categories: Rational expressions
Default value: false
When ratfac is true, canonical rational expressions (CRE) are
manipulated in a partially factored form.
During rational operations the expression is maintained as fully factored as
possible without calling factor.
This should always save space and may save time in some computations.
The numerator and denominator are made relatively prime, for example
factor ((x^2 - 1)^4/(x + 1)^2) yields (x - 1)^4 (x + 1)^2,
but the factors within each part may not be relatively prime.
In the ctensr (Component Tensor Manipulation) package,
Ricci, Einstein, Riemann, and Weyl tensors and the scalar curvature
are factored automatically when ratfac is true.
ratfac should only be
set for cases where the tensorial components are known to consist of
few terms.
The ratfac and ratweight schemes are incompatible and may not
both be used at the same time.
Categories: Rational expressions
Returns the numerator of expr, after coercing expr to a canonical rational expression (CRE). The return value is a CRE.
expr is coerced to a CRE by rat
if it is not already a CRE.
This conversion may change the form of expr by putting all terms
over a common denominator.
num is similar, but returns an ordinary expression instead of a CRE.
Also, num does not attempt to place all terms over a common denominator,
and thus some expressions which are considered ratios by ratnumer
are not considered ratios by num.
Categories: Rational expressions
Returns true if expr is a canonical rational expression (CRE) or
extended CRE, otherwise false.
CRE are created by rat and related functions.
Extended CRE are created by taylor and related functions.
Categories: Predicate functions · Rational expressions
Default value: true
When ratprint is true,
a message informing the user of the conversion of floating point numbers
to rational numbers is displayed.
Categories: Rational expressions · Numerical evaluation · Console interaction
Simplifies the expression expr and all of its subexpressions, including
the arguments to non-rational functions. The result is returned as the quotient
of two polynomials in a recursive form, that is, the coefficients of the main
variable are polynomials in the other variables. Variables may include
non-rational functions (e.g., sin (x^2 + 1)) and the arguments to any
such functions are also rationally simplified.
ratsimp (expr, x_1, ..., x_n)
enables rational simplification with the
specification of variable ordering as in ratvars.
When ratsimpexpons is true,
ratsimp is applied to the exponents of expressions during simplification.
See also ratexpand.
Note that ratsimp is affected by some of the
flags which affect ratexpand.
Examples:
(%i1) sin (x/(x^2 + x)) = exp ((log(x) + 1)^2 - log(x)^2);
2 2
x (log(x) + 1) - log (x)
(%o1) sin(------) = %e
2
x + x
(%i2) ratsimp (%);
1 2
(%o2) sin(-----) = %e x
x + 1
(%i3) ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1));
3/2
(x - 1) - sqrt(x - 1) (x + 1)
(%o3) --------------------------------
sqrt((x - 1) (x + 1))
(%i4) ratsimp (%);
2 sqrt(x - 1)
(%o4) - -------------
2
sqrt(x - 1)
(%i5) x^(a + 1/a), ratsimpexpons: true;
2
a + 1
------
a
(%o5) x
Categories: Simplification functions · Rational expressions
Default value: false
When ratsimpexpons is true,
ratsimp is applied to the exponents of expressions during simplification.
Categories: Simplification flags and variables · Rational expressions
Default value: false
radsubstflag, if true, permits ratsubst to make
substitutions such as u for sqrt (x) in x.
Categories: Simplification flags and variables
Substitutes a for b in c and returns the resulting expression. b may be a sum, product, power, etc.
ratsubst knows something of the meaning of expressions
whereas subst does a purely syntactic substitution.
Thus subst (a, x + y, x + y + z) returns x + y + z
whereas ratsubst returns z + a.
When radsubstflag is true,
ratsubst makes substitutions for radicals in expressions
which don't explicitly contain them.
ratsubst ignores the value true of the option variable
keepfloat.
Examples:
(%i1) ratsubst (a, x*y^2, x^4*y^3 + x^4*y^8);
3 4
(%o1) a x y + a
(%i2) cos(x)^4 + cos(x)^3 + cos(x)^2 + cos(x) + 1;
4 3 2
(%o2) cos (x) + cos (x) + cos (x) + cos(x) + 1
(%i3) ratsubst (1 - sin(x)^2, cos(x)^2, %);
4 2 2
(%o3) sin (x) - 3 sin (x) + cos(x) (2 - sin (x)) + 3
(%i4) ratsubst (1 - cos(x)^2, sin(x)^2, sin(x)^4);
4 2
(%o4) cos (x) - 2 cos (x) + 1
(%i5) radsubstflag: false$
(%i6) ratsubst (u, sqrt(x), x);
(%o6) x
(%i7) radsubstflag: true$
(%i8) ratsubst (u, sqrt(x), x);
2
(%o8) u
Categories: Rational expressions
Declares main variables x_1, …, x_n for rational expressions. x_n, if present in a rational expression, is considered the main variable. Otherwise, x_[n-1] is considered the main variable if present, and so on through the preceding variables to x_1, which is considered the main variable only if none of the succeeding variables are present.
If a variable in a rational expression is not present in the ratvars
list, it is given a lower priority than x_1.
The arguments to ratvars can be either variables or non-rational
functions such as sin(x).
The variable ratvars is a list of the arguments of
the function ratvars when it was called most recently.
Each call to the function ratvars resets the list.
ratvars () clears the list.
Categories: Rational expressions
Default value: true
Maxima keeps an internal list in the Lisp variable VARLIST of the main
variables for rational expressions. If ratvarswitch is true,
every evaluation starts with a fresh list VARLIST. This is the default
behavior. Otherwise, the main variables from previous evaluations are not
removed from the internal list VARLIST.
The main variables, which are declared with the function ratvars are
not affected by the option variable ratvarswitch.
Examples:
If ratvarswitch is true, every evaluation starts with a fresh
list VARLIST.
(%i1) ratvarswitch:true$
(%i2) rat(2*x+y^2);
2
(%o2)/R/ y + 2 x
(%i3) :lisp varlist
($X $Y)
(%i3) rat(2*a+b^2);
2
(%o3)/R/ b + 2 a
(%i4) :lisp varlist
($A $B)
If ratvarswitch is false, the main variables from the last
evaluation are still present.
(%i4) ratvarswitch:false$
(%i5) rat(2*x+y^2);
2
(%o5)/R/ y + 2 x
(%i6) :lisp varlist
($X $Y)
(%i6) rat(2*a+b^2);
2
(%o6)/R/ b + 2 a
(%i7) :lisp varlist
($A $B $X $Y)
Categories: Rational expressions · Global flags
Assigns a weight w_i to the variable x_i.
This causes a term to be replaced by 0 if its weight exceeds the
value of the variable ratwtlvl (default yields no truncation).
The weight of a term is the sum of the products of the
weight of a variable in the term times its power.
For example, the weight of 3 x_1^2 x_2 is 2 w_1 + w_2.
Truncation according to ratwtlvl is carried out only when multiplying
or exponentiating canonical rational expressions (CRE).
ratweight () returns the cumulative list of weight assignments.
Note: The ratfac and ratweight schemes are incompatible and may
not both be used at the same time.
Examples:
(%i1) ratweight (a, 1, b, 1);
(%o1) [a, 1, b, 1]
(%i2) expr1: rat(a + b + 1)$
(%i3) expr1^2;
2 2
(%o3)/R/ b + (2 a + 2) b + a + 2 a + 1
(%i4) ratwtlvl: 1$
(%i5) expr1^2;
(%o5)/R/ 2 b + 2 a + 1
Categories: Rational expressions
Default value: []
ratweights is the list of weights assigned by ratweight.
The list is cumulative:
each call to ratweight places additional items in the list.
kill (ratweights) and save (ratweights) both work as expected.
Categories: Rational expressions
Default value: false
ratwtlvl is used in combination with the ratweight
function to control the truncation of canonical rational expressions (CRE).
For the default value of false, no truncation occurs.
Categories: Rational expressions
Returns the remainder of the polynomial p_1 divided by the polynomial
p_2. The arguments x_1, …, x_n are interpreted as in
ratvars.
remainder returns the second element
of the two-element list returned by divide.
Categories: Polynomials
The function resultant computes the resultant of the two polynomials
p_1 and p_2, eliminating the variable x. The resultant is a
determinant of the coefficients of x in p_1 and p_2, which
equals zero if and only if p_1 and p_2 have a non-constant factor
in common.
If p_1 or p_2 can be factored, it may be desirable to call
factor before calling resultant.
The option variable resultant controls which algorithm will be used to
compute the resultant. See the option variable
resultant.
The function bezout takes the same arguments as resultant and
returns a matrix. The determinant of the return value is the desired resultant.
Examples:
(%i1) resultant(2*x^2+3*x+1, 2*x^2+x+1, x);
(%o1) 8
(%i2) resultant(x+1, x+1, x);
(%o2) 0
(%i3) resultant((x+1)*x, (x+1), x);
(%o3) 0
(%i4) resultant(a*x^2+b*x+1, c*x + 2, x);
2
(%o4) c - 2 b c + 4 a
(%i5) bezout(a*x^2+b*x+1, c*x+2, x);
[ 2 a 2 b - c ]
(%o5) [ ]
[ c 2 ]
(%i6) determinant(%);
(%o6) 4 a - (2 b - c) c
Categories: Polynomials
Default value: subres
The option variable resultant controls which algorithm will be used to
compute the resultant with the function resultant. The possible
values are:
subresfor the subresultant polynomial remainder sequence (PRS) algorithm,
modfor the modular resultant algorithm, and
redfor the reduced polynomial remainder sequence (PRS) algorithm.
On most problems the default value subres should be best. On some
large degree univariate or bivariate problems mod may be better.
Categories: Polynomials
Default value: false
When savefactors is true, causes the factors of an
expression which is a product of factors to be saved by certain
functions in order to speed up later factorizations of expressions
containing some of the same factors.
Categories: Polynomials
Returns a list of the canonical rational expression (CRE) variables in
expression expr.
See also ratvars.
Categories: Rational expressions · Display functions
is similar to factor except that the polynomial factors are
"square-free." That is, they have factors only of degree one.
This algorithm, which is also used by the first stage of factor, utilizes
the fact that a polynomial has in common with its n'th derivative all
its factors of degree greater than n. Thus by taking greatest common divisors
with the polynomial of
the derivatives with respect to each variable in the polynomial, all
factors of degree greater than 1 can be found.
Example:
(%i1) sqfr (4*x^4 + 4*x^3 - 3*x^2 - 4*x - 1);
2 2
(%o1) (2 x + 1) (x - 1)
Categories: Polynomials
Adds to the ring of algebraic integers known to Maxima the elements which are the solutions of the polynomials p_1, …, p_n. Each argument p_i is a polynomial with integer coefficients.
tellrat (x) effectively means substitute 0 for x in rational
functions.
tellrat () returns a list of the current substitutions.
algebraic must be set to true in order for the simplification of
algebraic integers to take effect.
Maxima initially knows about the imaginary unit %i
and all roots of integers.
There is a command untellrat which takes kernels and
removes tellrat properties.
When tellrat'ing a multivariate
polynomial, e.g., tellrat (x^2 - y^2), there would be an ambiguity as to
whether to substitute y^2 for x^2
or vice versa.
Maxima picks a particular ordering, but if the user wants to specify which, e.g.
tellrat (y^2 = x^2) provides a syntax which says replace
y^2 by x^2.
Examples:
(%i1) 10*(%i + 1)/(%i + 3^(1/3));
10 (%i + 1)
(%o1) -----------
1/3
%i + 3
(%i2) ev (ratdisrep (rat(%)), algebraic);
2/3 1/3 2/3 1/3
(%o2) (4 3 - 2 3 - 4) %i + 2 3 + 4 3 - 2
(%i3) tellrat (1 + a + a^2);
2
(%o3) [a + a + 1]
(%i4) 1/(a*sqrt(2) - 1) + a/(sqrt(3) + sqrt(2));
1 a
(%o4) ------------- + -----------------
sqrt(2) a - 1 sqrt(3) + sqrt(2)
(%i5) ev (ratdisrep (rat(%)), algebraic);
(7 sqrt(3) - 10 sqrt(2) + 2) a - 2 sqrt(2) - 1
(%o5) ----------------------------------------------
7
(%i6) tellrat (y^2 = x^2);
2 2 2
(%o6) [y - x , a + a + 1]
Categories: Polynomials · Rational expressions
Converts every subexpression of expr from canonical rational expressions
(CRE) to general form and returns the result.
If expr is itself in CRE form then totaldisrep is identical to
ratdisrep.
totaldisrep may be useful for
ratdisrepping expressions such as equations, lists, matrices, etc., which
have some subexpressions in CRE form.
Categories: Rational expressions
Removes tellrat properties from x_1, …, x_n.
Categories: Polynomials · Rational expressions
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Special function notation follows:
bessel_j (index, expr) Bessel function, 1st kind
bessel_y (index, expr) Bessel function, 2nd kind
bessel_i (index, expr) Modified Bessel function, 1st kind
bessel_k (index, expr) Modified Bessel function, 2nd kind
hankel_1 (v,z) Hankel function of the 1st kind
hankel_2 (v,z) Hankel function of the 2nd kind
struve_h (v,z) Struve H function
struve_l (v,z) Struve L function
assoc_legendre_p[v,u] (z) Legendre function of degree v and order u
assoc_legendre_q[v,u] (z) Legendre function, 2nd kind
%f[p,q] ([], [], expr) Generalized Hypergeometric function
gamma (z) Gamma function
gamma_greek (a,z) Incomplete gamma function
gamma_incomplete (a,z) Tail of incomplete gamma function
hypergeometric (l1, l2, z) Hypergeometric function
slommel
%m[u,k] (z) Whittaker function, 1st kind
%w[u,k] (z) Whittaker function, 2nd kind
erfc (z) Complement of the erf function
expintegral_e (v,z) Exponential integral E
expintegral_e1 (z) Exponential integral E1
expintegral_ei (z) Exponential integral Ei
expintegral_li (z) Logarithmic integral Li
expintegral_si (z) Exponential integral Si
expintegral_ci (z) Exponential integral Ci
expintegral_shi (z) Exponential integral Shi
expintegral_chi (z) Exponential integral Chi
kelliptic (z) Complete elliptic integral of the first
kind (K)
parabolic_cylinder_d (v,z) Parabolic cylinder D function
Categories: Bessel functions · Airy functions · Special functions
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The Bessel function of the first kind of order v and argument z.
bessel_j is defined as
inf
==== k - v - 2 k v + 2 k
\ (- 1) 2 z
> --------------------------
/ k! gamma(v + k + 1)
====
k = 0
although the infinite series is not used for computations.
Categories: Bessel functions · Special functions
The Bessel function of the second kind of order v and argument z.
bessel_y is defined as
cos(%pi v) bessel_j(v, z) - bessel_j(-v, z)
-------------------------------------------
sin(%pi v)
when v is not an integer. When v is an integer n, the limit as v approaches n is taken.
Categories: Bessel functions · Special functions
The modified Bessel function of the first kind of order v and argument z.
bessel_i is defined as
inf
==== - v - 2 k v + 2 k
\ 2 z
> -------------------
/ k! gamma(v + k + 1)
====
k = 0
although the infinite series is not used for computations.
Categories: Bessel functions · Special functions
The modified Bessel function of the second kind of order v and argument z.
bessel_k is defined as
%pi csc(%pi v) (bessel_i(-v, z) - bessel_i(v, z))
-------------------------------------------------
2
when v is not an integer. If v is an integer n, then the limit as v approaches n is taken.
Categories: Bessel functions · Special functions
The Hankel function of the first kind of order v and argument z
(A&S 9.1.3). hankel_1 is defined as
bessel_j(v,z) + %i * bessel_y(v,z)
Maxima evaluates hankel_1 numerically for a complex order v and
complex argument z in float precision. The numerical evaluation in
bigfloat precision is not supported.
When besselexpand is true, hankel_1 is expanded in terms
of elementary functions when the order v is half of an odd integer.
See besselexpand.
Maxima knows the derivative of hankel_1 wrt the argument z.
Examples:
Numerical evaluation:
(%i1) hankel_1(1,0.5); (%o1) 0.24226845767487 - 1.471472392670243 %i (%i2) hankel_1(1,0.5+%i); (%o2) - 0.25582879948621 %i - 0.23957560188301
Expansion of hankel_1 when besselexpand is true:
(%i1) hankel_1(1/2,z),besselexpand:true;
sqrt(2) sin(z) - sqrt(2) %i cos(z)
(%o1) ----------------------------------
sqrt(%pi) sqrt(z)
Derivative of hankel_1 wrt the argument z. The derivative wrt the
order v is not supported. Maxima returns a noun form:
(%i1) diff(hankel_1(v,z),z);
hankel_1(v - 1, z) - hankel_1(v + 1, z)
(%o1) ---------------------------------------
2
(%i2) diff(hankel_1(v,z),v);
d
(%o2) -- (hankel_1(v, z))
dv
Categories: Bessel functions · Special functions
The Hankel function of the second kind of order v and argument z
(A&S 9.1.4). hankel_2 is defined as
bessel_j(v,z) - %i * bessel_y(v,z)
Maxima evaluates hankel_2 numerically for a complex order v and
complex argument z in float precision. The numerical evaluation in
bigfloat precision is not supported.
When besselexpand is true, hankel_2 is expanded in terms
of elementary functions when the order v is half of an odd integer.
See besselexpand.
Maxima knows the derivative of hankel_2 wrt the argument z.
For examples see hankel_1.
Categories: Bessel functions · Special functions
Default value: false
Controls expansion of the Bessel functions when the order is half of
an odd integer. In this case, the Bessel functions can be expanded
in terms of other elementary functions. When besselexpand is true,
the Bessel function is expanded.
(%i1) besselexpand: false$
(%i2) bessel_j (3/2, z);
3
(%o2) bessel_j(-, z)
2
(%i3) besselexpand: true$
(%i4) bessel_j (3/2, z);
sin(z) cos(z)
sqrt(2) sqrt(z) (------ - ------)
2 z
z
(%o4) ---------------------------------
sqrt(%pi)
The scaled modified Bessel function of the first kind of order
v and argument z. That is, scaled_bessel_i(v,z) =
exp(-abs(z))*bessel_i(v, z). This function is particularly useful
for calculating bessel_i for large z, which is large.
However, maxima does not otherwise know much about this function. For
symbolic work, it is probably preferable to work with the expression
exp(-abs(z))*bessel_i(v, z).
Categories: Bessel functions
Identical to scaled_bessel_i(0,z).
Categories: Bessel functions · Special functions
Identical to scaled_bessel_i(1,z).
Lommel's little s[u,v](z) function. Probably Gradshteyn & Ryzhik 8.570.1.
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The Airy functions Ai(x) and Bi(x) are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 10.4.
y = Ai(x) and y = Bi(x) are two linearly independent solutions
of the Airy differential equation diff (y(x), x, 2) - x y(x) = 0.
If the argument x is a real or complex floating point
number, the numerical value of the function is returned.
The Airy function Ai(x). (A&S 10.4.2)
The derivative diff (airy_ai(x), x) is airy_dai(x).
See also airy_bi, airy_dai, airy_dbi.
Categories: Airy functions · Special functions
The derivative of the Airy function Ai airy_ai(x).
See airy_ai.
Categories: Airy functions · Special functions
The Airy function Bi(x). (A&S 10.4.3)
The derivative diff (airy_bi(x), x) is airy_dbi(x).
See airy_ai, airy_dbi.
Categories: Airy functions · Special functions
The derivative of the Airy Bi function airy_bi(x).
See airy_ai and airy_bi.
Categories: Airy functions · Special functions
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The gamma function and the related beta, psi and incomplete gamma functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 6.
Bigfloat version of the factorial (shifted gamma) function. The second argument is how many digits to retain and return, it's a good idea to request a couple of extra.
Categories: Gamma and factorial functions · Numerical evaluation
bfpsi is the polygamma function of real argument z and integer
order n. bfpsi0 is the digamma function.
bfpsi0 (z, fpprec) is equivalent to
bfpsi (0, z, fpprec).
These functions return bigfloat values. fpprec is the bigfloat precision of the return value.
Categories: Gamma and factorial functions · Numerical evaluation
Complex bigfloat factorial.
load ("bffac") loads this function.
Categories: Gamma and factorial functions · Complex variables · Numerical evaluation
The basic definition of the gamma function (A&S 6.1.1) is
inf
/
[ z - 1 - t
gamma(z) = I t %e dt
]
/
0
Maxima simplifies gamma for positive integer and positive and negative
rational numbers. For half integral values the result is a rational number times
sqrt(%pi). The simplification for integer values is controlled by
factlim. For integers greater than factlim the numerical result of
the factorial function, which is used to calculate gamma, will overflow.
The simplification for rational numbers is controlled by gammalim to
avoid internal overflow. See factlim and gammalim.
For negative integers gamma is not defined.
Maxima can evalute gamma numerically for real and complex values in float
and bigfloat precision.
gamma has mirror symmetry.
When gamma_expand is true, Maxima expands gamma for
arguments z+n and z-n where n is an integer.
Maxima knows the derivate of gamma.
Examples:
Simplification for integer, half integral, and rational numbers:
(%i1) map('gamma,[1,2,3,4,5,6,7,8,9]);
(%o1) [1, 1, 2, 6, 24, 120, 720, 5040, 40320]
(%i2) map('gamma,[1/2,3/2,5/2,7/2]);
sqrt(%pi) 3 sqrt(%pi) 15 sqrt(%pi)
(%o2) [sqrt(%pi), ---------, -----------, ------------]
2 4 8
(%i3) map('gamma,[2/3,5/3,7/3]);
2 1
2 gamma(-) 4 gamma(-)
2 3 3
(%o3) [gamma(-), ----------, ----------]
3 3 9
Numerical evaluation for real and complex values:
(%i4) map('gamma,[2.5,2.5b0]);
(%o4) [1.329340388179137, 1.3293403881791370205b0]
(%i5) map('gamma,[1.0+%i,1.0b0+%i]);
(%o5) [0.498015668118356 - .1549498283018107 %i,
4.9801566811835604272b-1 - 1.5494982830181068513b-1 %i]
gamma has mirror symmetry:
(%i6) declare(z,complex)$ (%i7) conjugate(gamma(z)); (%o7) gamma(conjugate(z))
Maxima expands gamma(z+n) and gamma(z-n), when gamma_expand
is true:
(%i8) gamma_expand:true$
(%i9) [gamma(z+1),gamma(z-1),gamma(z+2)/gamma(z+1)];
gamma(z)
(%o9) [z gamma(z), --------, z + 1]
z - 1
The deriviative of gamma:
(%i10) diff(gamma(z),z);
(%o10) psi (z) gamma(z)
0
See also makegamma.
The Euler-Mascheroni constant is %gamma.
Categories: Gamma and factorial functions · Special functions
The natural logarithm of the gamma function.
Categories: Gamma and factorial functions · Special functions
The lower incomplete gamma function (A&S 6.5.2):
z
/
[ a - 1 - t
gamma_greek(a, z) = I t %e dt
]
/
0
See also gamma_incomplete (upper incomplete gamma function).
Categories: Gamma and factorial functions · Special functions
The incomplete upper gamma function (A&S 6.5.3):
inf
/
[ a - 1 - t
gamma_incomplete(a, z) = I t %e dt
]
/
z
See also gamma_expand for controlling how
gamma_incomplete is expressed in terms of elementary functions
and erfc.
Also see the related functions gamma_incomplete_regularized and
gamma_incomplete_generalized.
Categories: Gamma and factorial functions · Special functions
The regularized incomplete upper gamma function (A&S 6.5.1):
gamma_incomplete_regularized(a, z) =
gamma_incomplete(a, z)
----------------------
gamma(a)
See also gamma_expand for controlling how
gamma_incomplete is expressed in terms of elementary functions
and erfc.
Also see gamma_incomplete.
Categories: Gamma and factorial functions · Special functions
The generalized incomplete gamma function.
gamma_incomplete_generalized(a, z1, z2) =
z2
/
[ a - 1 - t
I t %e dt
]
/
z1
Also see gamma_incomplete and gamma_incomplete_regularized.
Categories: Gamma and factorial functions · Special functions
Default value: false
gamma_expand controls expansion of gamma_incomplete.
When gamma_expand is true, gamma_incomplete(v,z)
is expanded in terms of
z, exp(z), and erfc(z) when possible.
(%i1) gamma_incomplete(2,z);
(%o1) gamma_incomplete(2, z)
(%i2) gamma_expand:true;
(%o2) true
(%i3) gamma_incomplete(2,z);
- z
(%o3) (z + 1) %e
(%i4) gamma_incomplete(3/2,z);
- z sqrt(%pi) erfc(sqrt(z))
(%o4) sqrt(z) %e + -----------------------
2
Default value: 10000
gammalim controls simplification of the gamma
function for integral and rational number arguments. If the absolute
value of the argument is not greater than gammalim, then
simplification will occur. Note that the factlim switch controls
simplification of the result of gamma of an integer argument as well.
Transforms instances of binomial, factorial, and beta functions in expr into gamma functions.
See also makefact.
Categories: Gamma and factorial functions
The beta function is defined as gamma(a) gamma(b)/gamma(a+b)
(A&S 6.2.1).
Maxima simplifies the beta function for positive integers and rational
numbers, which sum to an integer. When beta_args_sum_to_integer is
true, Maxima simplifies also general expressions which sum to an integer.
For a or b equal to zero the beta function is not defined.
In general the beta function is not defined for negative integers as an argument. The exception is for a=-n, n a positive integer and b a positive integer with b<=n, it is possible to define an analytic continuation. Maxima gives for this case a result.
When beta_expand is true, expressions like beta(a+n,b) and
beta(a-n,b) or beta(a,b+n) and beta(a,b-n) with n
an integer are simplified.
Maxima can evaluate the beta function for real and complex values in float and
bigfloat precision. For numerical evaluation Maxima uses log_gamma:
- log_gamma(b + a) + log_gamma(b) + log_gamma(a)
%e
Maxima knows that the beta function is symmetric and has mirror symmetry.
Maxima knows the derivatives of the beta function with respect to a or b.
To express the beta function as a ratio of gamma functions see makegamma.
Examples:
Simplification, when one of the arguments is an integer:
(%i1) [beta(2,3),beta(2,1/3),beta(2,a)];
1 9 1
(%o1) [--, -, ---------]
12 4 a (a + 1)
Simplification for two rational numbers as arguments which sum to an integer:
(%i2) [beta(1/2,5/2),beta(1/3,2/3),beta(1/4,3/4)];
3 %pi 2 %pi
(%o2) [-----, -------, sqrt(2) %pi]
8 sqrt(3)
When setting beta_args_sum_to_integer to true more general
expression are simplified, when the sum of the arguments is an integer:
(%i3) beta_args_sum_to_integer:true$
(%i4) beta(a+1,-a+2);
%pi (a - 1) a
(%o4) ------------------
2 sin(%pi (2 - a))
The possible results, when one of the arguments is a negative integer:
(%i5) [beta(-3,1),beta(-3,2),beta(-3,3)];
1 1 1
(%o5) [- -, -, - -]
3 6 3
beta(a+n,b) or beta(a-n) with n an integer simplifies when
beta_expand is true:
(%i6) beta_expand:true$
(%i7) [beta(a+1,b),beta(a-1,b),beta(a+1,b)/beta(a,b+1)];
a beta(a, b) beta(a, b) (b + a - 1) a
(%o7) [------------, ----------------------, -]
b + a a - 1 b
Beta is not defined, when one of the arguments is zero:
(%i7) beta(0,b); beta: expected nonzero arguments; found 0, b -- an error. To debug this try debugmode(true);
Numercial evaluation for real and complex arguments in float or bigfloat precision:
(%i8) beta(2.5,2.3); (%o8) .08694748611299981 (%i9) beta(2.5,1.4+%i); (%o9) 0.0640144950796695 - .1502078053286415 %i (%i10) beta(2.5b0,2.3b0); (%o10) 8.694748611299969b-2 (%i11) beta(2.5b0,1.4b0+%i); (%o11) 6.401449507966944b-2 - 1.502078053286415b-1 %i
Beta is symmetric and has mirror symmetry:
(%i14) beta(a,b)-beta(b,a); (%o14) 0 (%i15) declare(a,complex,b,complex)$ (%i16) conjugate(beta(a,b)); (%o16) beta(conjugate(a), conjugate(b))
The derivative of the beta function wrt a:
(%i17) diff(beta(a,b),a);
(%o17) - beta(a, b) (psi (b + a) - psi (a))
0 0
Categories: Gamma and factorial functions
The basic definition of the incomplete beta function (A&S 6.6.1) is
z
/
[ b - 1 a - 1
I (1 - t) t dt
]
/
0
This definition is possible for realpart(a)>0 and realpart(b)>0 and abs(z)<1. For other values the incomplete beta function can be defined through a generalized hypergeometric function:
gamma(a) hypergeometric_generalized([a, 1 - b], [a + 1], z) z
(See functions.wolfram.com for a complete definition of the incomplete beta function.)
For negative integers a = -n and positive integers b=m with m<=n the incomplete beta function is defined through
m - 1 k
==== (1 - m) z
n - 1 \ k
z > -----------
/ k! (n - k)
====
k = 0
Maxima uses this definition to simplify beta_incomplete for a a
negative integer.
For a a positive integer, beta_incomplete simplifies for any
argument b and z and for b a positive integer for any
argument a and z, with the exception of a a negative integer.
For z=0 and realpart(a)>0, beta_incomplete has the
specific value zero. For z=1 and realpart(b)>0,
beta_incomplete simplifies to the beta function beta(a,b).
Maxima evaluates beta_incomplete numerically for real and complex values
in float or bigfloat precision. For the numerical evaluation an expansion of the
incomplete beta function in continued fractions is used.
When the option variable beta_expand is true, Maxima expands
expressions like beta_incomplete(a+n,b,z) and
beta_incomplete(a-n,b,z) where n is a positive integer.
Maxima knows the derivatives of beta_incomplete with respect to the
variables a, b and z and the integral with respect to the
variable z.
Examples:
Simplification for a a positive integer:
(%i1) beta_incomplete(2,b,z);
b
1 - (1 - z) (b z + 1)
(%o1) ----------------------
b (b + 1)
Simplification for b a positive integer:
(%i2) beta_incomplete(a,2,z);
a
(a (1 - z) + 1) z
(%o2) ------------------
a (a + 1)
Simplification for a and b a positive integer:
(%i3) beta_incomplete(3,2,z);
3
(3 (1 - z) + 1) z
(%o3) ------------------
12
a is a negative integer and b<=(-a), Maxima simplifies:
(%i4) beta_incomplete(-3,1,z);
1
(%o4) - ----
3
3 z
For the specific values z=0 and z=1, Maxima simplifies:
(%i5) assume(a>0,b>0)$ (%i6) beta_incomplete(a,b,0); (%o6) 0 (%i7) beta_incomplete(a,b,1); (%o7) beta(a, b)
Numerical evaluation in float or bigfloat precision:
(%i8) beta_incomplete(0.25,0.50,0.9); (%o8) 4.594959440269333 (%i9) fpprec:25$ (%i10) beta_incomplete(0.25,0.50,0.9b0); (%o10) 4.594959440269324086971203b0
For abs(z)>1 beta_incomplete returns a complex result:
(%i11) beta_incomplete(0.25,0.50,1.7); (%o11) 5.244115108584249 - 1.45518047787844 %i
Results for more general complex arguments:
(%i14) beta_incomplete(0.25+%i,1.0+%i,1.7+%i); (%o14) 2.726960675662536 - .3831175704269199 %i (%i15) beta_incomplete(1/2,5/4*%i,2.8+%i); (%o15) 13.04649635168716 %i - 5.802067956270001 (%i16)
Expansion, when beta_expand is true:
(%i23) beta_incomplete(a+1,b,z),beta_expand:true;
b a
a beta_incomplete(a, b, z) (1 - z) z
(%o23) -------------------------- - -----------
b + a b + a
(%i24) beta_incomplete(a-1,b,z),beta_expand:true;
b a - 1
beta_incomplete(a, b, z) (- b - a + 1) (1 - z) z
(%o24) -------------------------------------- - ---------------
1 - a 1 - a
Derivative and integral for beta_incomplete:
(%i34) diff(beta_incomplete(a, b, z), z);
b - 1 a - 1
(%o34) (1 - z) z
(%i35) integrate(beta_incomplete(a, b, z), z);
b a
(1 - z) z
(%o35) ----------- + beta_incomplete(a, b, z) z
b + a
a beta_incomplete(a, b, z)
- --------------------------
b + a
(%i36) factor(diff(%, z));
(%o36) beta_incomplete(a, b, z)
Categories: Gamma and factorial functions
The regularized incomplete beta function (A&S 6.6.2), defined as
beta_incomplete_regularized(a, b, z) =
beta_incomplete(a, b, z)
------------------------
beta(a, b)
As for beta_incomplete this definition is not complete. See
functions.wolfram.com for a complete definition of
beta_incomplete_regularized.
beta_incomplete_regularized simplifies a or b a positive
integer.
For z=0 and realpart(a)>0, beta_incomplete_regularized has
the specific value 0. For z=1 and realpart(b)>0,
beta_incomplete_regularized simplifies to 1.
Maxima can evaluate beta_incomplete_regularized for real and complex
arguments in float and bigfloat precision.
When beta_expand is true, Maxima expands
beta_incomplete_regularized for arguments a+n or a-n,
where n is an integer.
Maxima knows the derivatives of beta_incomplete_regularized with respect
to the variables a, b, and z and the integral with respect to
the variable z.
Examples:
Simplification for a or b a positive integer:
(%i1) beta_incomplete_regularized(2,b,z);
b
(%o1) 1 - (1 - z) (b z + 1)
(%i2) beta_incomplete_regularized(a,2,z);
a
(%o2) (a (1 - z) + 1) z
(%i3) beta_incomplete_regularized(3,2,z);
3
(%o3) (3 (1 - z) + 1) z
For the specific values z=0 and z=1, Maxima simplifies:
(%i4) assume(a>0,b>0)$ (%i5) beta_incomplete_regularized(a,b,0); (%o5) 0 (%i6) beta_incomplete_regularized(a,b,1); (%o6) 1
Numerical evaluation for real and complex arguments in float and bigfloat precision:
(%i7) beta_incomplete_regularized(0.12,0.43,0.9); (%o7) .9114011367359802 (%i8) fpprec:32$ (%i9) beta_incomplete_regularized(0.12,0.43,0.9b0); (%o9) 9.1140113673598075519946998779975b-1 (%i10) beta_incomplete_regularized(1+%i,3/3,1.5*%i); (%o10) .2865367499935403 %i - 0.122995963334684 (%i11) fpprec:20$ (%i12) beta_incomplete_regularized(1+%i,3/3,1.5b0*%i); (%o12) 2.8653674999354036142b-1 %i - 1.2299596333468400163b-1
Expansion, when beta_expand is true:
(%i13) beta_incomplete_regularized(a+1,b,z);
b a
(1 - z) z
(%o13) beta_incomplete_regularized(a, b, z) - ------------
a beta(a, b)
(%i14) beta_incomplete_regularized(a-1,b,z);
(%o14) beta_incomplete_regularized(a, b, z)
b a - 1
(1 - z) z
- ----------------------
beta(a, b) (b + a - 1)
The derivative and the integral wrt z:
(%i15) diff(beta_incomplete_regularized(a,b,z),z);
b - 1 a - 1
(1 - z) z
(%o15) -------------------
beta(a, b)
(%i16) integrate(beta_incomplete_regularized(a,b,z),z);
(%o16) beta_incomplete_regularized(a, b, z) z
b a
(1 - z) z
a (beta_incomplete_regularized(a, b, z) - ------------)
a beta(a, b)
- -------------------------------------------------------
b + a
Categories: Gamma and factorial functions
The basic definition of the generalized incomplete beta function is
z2
/
[ b - 1 a - 1
I (1 - t) t dt
]
/
z1
Maxima simplifies beta_incomplete_regularized for a and b
a positive integer.
For realpart(a)>0 and z1=0 or z2=0, Maxima simplifies
beta_incomplete_generalized to beta_incomplete. For
realpart(b)>0 and z1=1 or z2=1, Maxima simplifies to an
expression with beta and beta_incomplete.
Maxima evaluates beta_incomplete_regularized for real and complex values
in float and bigfloat precision.
When beta_expand is true, Maxima expands
beta_incomplete_generalized for a+n and a-n, n a
positive integer.
Maxima knows the derivative of beta_incomplete_generalized with respect
to the variables a, b, z1, and z2 and the integrals with
respect to the variables z1 and z2.
Examples:
Maxima simplifies beta_incomplete_generalized for a and b a
positive integer:
(%i1) beta_incomplete_generalized(2,b,z1,z2);
b b
(1 - z1) (b z1 + 1) - (1 - z2) (b z2 + 1)
(%o1) -------------------------------------------
b (b + 1)
(%i2) beta_incomplete_generalized(a,2,z1,z2);
a a
(a (1 - z2) + 1) z2 - (a (1 - z1) + 1) z1
(%o2) -------------------------------------------
a (a + 1)
(%i3) beta_incomplete_generalized(3,2,z1,z2);
2 2 2 2
(1 - z1) (3 z1 + 2 z1 + 1) - (1 - z2) (3 z2 + 2 z2 + 1)
(%o3) -----------------------------------------------------------
12
Simplification for specific values z1=0, z2=0, z1=1, or z2=1:
(%i4) assume(a > 0, b > 0)$ (%i5) beta_incomplete_generalized(a,b,z1,0); (%o5) - beta_incomplete(a, b, z1) (%i6) beta_incomplete_generalized(a,b,0,z2); (%o6) - beta_incomplete(a, b, z2) (%i7) beta_incomplete_generalized(a,b,z1,1); (%o7) beta(a, b) - beta_incomplete(a, b, z1) (%i8) beta_incomplete_generalized(a,b,1,z2); (%o8) beta_incomplete(a, b, z2) - beta(a, b)
Numerical evaluation for real arguments in float or bigfloat precision:
(%i9) beta_incomplete_generalized(1/2,3/2,0.25,0.31); (%o9) .09638178086368676 (%i10) fpprec:32$ (%i10) beta_incomplete_generalized(1/2,3/2,0.25,0.31b0); (%o10) 9.6381780863686935309170054689964b-2
Numerical evaluation for complex arguments in float or bigfloat precision:
(%i11) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31); (%o11) - .09625463003205376 %i - .003323847735353769 (%i12) fpprec:20$ (%i13) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31b0); (%o13) - 9.6254630032054178691b-2 %i - 3.3238477353543591914b-3
Expansion for a+n or a-n, n a positive integer, when
beta_expand is true:
(%i14) beta_expand:true$
(%i15) beta_incomplete_generalized(a+1,b,z1,z2);
b a b a
(1 - z1) z1 - (1 - z2) z2
(%o15) -----------------------------
b + a
a beta_incomplete_generalized(a, b, z1, z2)
+ -------------------------------------------
b + a
(%i16) beta_incomplete_generalized(a-1,b,z1,z2);
beta_incomplete_generalized(a, b, z1, z2) (- b - a + 1)
(%o16) -------------------------------------------------------
1 - a
b a - 1 b a - 1
(1 - z2) z2 - (1 - z1) z1
- -------------------------------------
1 - a
Derivative wrt the variable z1 and integrals wrt z1 and z2:
(%i17) diff(beta_incomplete_generalized(a,b,z1,z2),z1);
b - 1 a - 1
(%o17) - (1 - z1) z1
(%i18) integrate(beta_incomplete_generalized(a,b,z1,z2),z1);
(%o18) beta_incomplete_generalized(a, b, z1, z2) z1
+ beta_incomplete(a + 1, b, z1)
(%i19) integrate(beta_incomplete_generalized(a,b,z1,z2),z2);
(%o19) beta_incomplete_generalized(a, b, z1, z2) z2
- beta_incomplete(a + 1, b, z2)
Categories: Gamma and factorial functions
Default value: false
When beta_expand is true, beta(a,b) and related
functions are expanded for arguments like a+n or a-n,
where n is an integer.
Default value: false
When beta_args_sum_to_integer is true, Maxima simplifies
beta(a,b), when the arguments a and b sum to an integer.
The derivative of log (gamma (x)) of order n+1.
Thus, psi[0](x) is the first derivative,
psi[1](x) is the second derivative, etc.
Maxima does not know how, in general, to compute a numerical value of
psi, but it can compute some exact values for rational args.
Several variables control what range of rational args psi will
return an exact value, if possible. See maxpsiposint,
maxpsinegint, maxpsifracnum, and maxpsifracdenom.
That is, x must lie between maxpsinegint and
maxpsiposint. If the absolute value of the fractional part of
x is rational and has a numerator less than maxpsifracnum
and has a denominator less than maxpsifracdenom, psi
will return an exact value.
The function bfpsi in the bffac package can compute
numerical values.
Categories: Gamma and factorial functions
Default value: 20
maxpsiposint is the largest positive value for which
psi[n](x) will try to compute an exact value.
Categories: Gamma and factorial functions
Default value: -10
maxpsinegint is the most negative value for which
psi[n](x) will try to compute an exact value. That is if
x is less than maxnegint, psi[n](x) will not
return simplified answer, even if it could.
Categories: Gamma and factorial functions
Default value: 6
Let x be a rational number less than one of the form p/q.
If p is greater than maxpsifracnum, then
psi[n](x) will not try to return a simplified
value.
Categories: Gamma and factorial functions
Default value: 6
Let x be a rational number less than one of the form p/q.
If q is greater than maxpsifracdenom, then
psi[n](x) will not try to return a simplified
value.
Categories: Gamma and factorial functions
Transforms instances of binomial, gamma, and beta functions in expr into factorials.
See also makegamma.
Categories: Gamma and factorial functions
Returns the numerical factor multiplying the expression expr, which should be a single term.
content returns the greatest common divisor (gcd) of all terms in a sum.
(%i1) gamma (7/2);
15 sqrt(%pi)
(%o1) ------------
8
(%i2) numfactor (%);
15
(%o2) --
8
Categories: Expressions
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The Exponential Integral and related funtions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 5
The Exponential Integral E1(z) (A&S 5.1.1) defined as
Categories: Exponential Integrals · Special functions
The Exponential Integral Ei(z) (A&S 5.1.2)
Categories: Exponential Integrals · Special functions
The Exponential Integral Li(z) (A&S 5.1.3)
Categories: Exponential Integrals · Special functions
The Exponential Integral En(z) (A&S 5.1.4) defined as
Categories: Exponential Integrals · Special functions
The Exponential Integral Si(z) (A&S 5.2.1) defined as
Categories: Exponential Integrals · Special functions
The Exponential Integral Ci(z) (A&S 5.2.2) defined as
Categories: Exponential Integrals · Special functions
The Exponential Integral Shi(z) (A&S 5.2.3) defined as
Categories: Exponential Integrals · Special functions
The Exponential Integral Chi(z) (A&S 5.2.4) defined as
Categories: Exponential Integrals · Special functions
Default value: false
Change the representation of one of the exponential integrals, expintegral_e(m, z), expintegral_e1, or expintegral_ei to an equivalent form if possible.
Possible values for expintrep are false, gamma_incomplete, expintegral_e1, expintegral_ei, expintegral_li, expintegral_trig, or expintegral_hyp.
false means that the representation is not changed. Other values indicate the representation is to be changed to use the function specified where expintegral_trig means expintegral_si, expintegral_ci, and expintegral_hyp means expintegral_shi or expintegral_chi.
Categories: Exponential Integrals
Default value: false
Expand the Exponential Integral E[n](z) for half integral values in terms of Erfc or Erf and for positive integers in terms of Ei
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The Error function and related funtions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 7
The Error Function erf(z) (A&S 7.1.1)
See also flag erfflag.
The Complementary Error Function erfc(z) (A&S 7.1.2)
erfc(z) = 1-erf(z)
The Imaginary Error Function.
erfi(z) = -%i*erf(%i*z)
Generalized Error function Erf(z1,z2)
The Fresnel Integral C(z) = integrate(cos((%pi/2)*t^2),t,0,z). (A&S 7.3.1)
The simplification fresnel_c(-x) = -fresnel_c(x) is applied when
flag trigsign is true.
The simplification fresnel_c(%i*x) = %i*fresnel_c(x) is applied when
flag %iargs is true.
See flags erf_representation and hypergeometric_representation.
The Fresnel Integral S(z) = integrate(sin((%pi/2)*t^2),t,0,z). (A&S 7.3.2)
The simplification fresnel_s(-x) = -fresnel_s(x) is applied when
flag trigsign is true.
The simplification fresnel_s(%i*x) = -%i*fresnel_s(x) is applied when
flag %iargs is true.
See flags erf_representation and hypergeometric_representation.
Default value: false
When T erfc, erfi, erf_generalized, fresnel_s and fresnel_c are transformed to erf.
Default value: false
Enables transformation to a Hypergeometric representation for fresnel_s and fresnel_c
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The Struve functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 12.
The Struve Function H of order v and argument z. (A&S 12.1.1)
Categories: Special functions
The Modified Struve Function L of order v and argument z. (A&S 12.2.1)
Categories: Special functions
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The Hypergeometric Functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapters 13 and 15.
Maxima has very limited knowledge of these functions. They
can be returned from function hgfred.
Whittaker M function
M[k,u](z) = exp(-z/2)*z^(1/2+u)*M(1/2+u-k,1+2*u,z).
(A&S 13.1.32)
Whittaker W function. (A&S 13.1.33)
The pFq(a1,a2,..ap;b1,b2,..bq;z) hypergeometric function,
where a a list of length p and
b a list of length q.
The hypergeometric function. Unlike Maxima's %f hypergeometric
function, the function hypergeometric is a simplifying
function; also, hypergeometric supports complex double and
big floating point evaluation. For the Gauss hypergeometric function,
that is p = 2 and q = 1, floating point evaluation
outside the unit circle is supported, but in general, it is not
supported.
When the option variable expand_hypergeometric is true (default
is false) and one of the arguments a1 through ap is a
negative integer (a polynomial case), hypergeometric returns an
expanded polynomial.
Examples:
(%i1) hypergeometric([],[],x); (%o1) %e^x
Polynomial cases automatically expand when expand_hypergeometric is true:
(%i2) hypergeometric([-3],[7],x); (%o2) hypergeometric([-3],[7],x) (%i3) hypergeometric([-3],[7],x), expand_hypergeometric : true; (%o3) -x^3/504+3*x^2/56-3*x/7+1
Both double float and big float evaluation is supported:
(%i4) hypergeometric([5.1],[7.1 + %i],0.42);
(%o4) 1.346250786375334 - 0.0559061414208204 %i
(%i5) hypergeometric([5,6],[8], 5.7 - %i);
(%o5) .007375824009774946 - .001049813688578674 %i
(%i6) hypergeometric([5,6],[8], 5.7b0 - %i), fpprec : 30;
(%o6) 7.37582400977494674506442010824b-3
- 1.04981368857867315858055393376b-3 %i
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The Parabolic Cylinder Functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 19.
Maxima has very limited knowledge of these functions. They
can be returned from function hgfred.
The parabolic cylinder function parabolic_cylinder_d(v,z). (A&S 19.3.1)
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Compute the Laplace transform of expr with respect to the variable t. The integrand expr may contain special functions.
The following special functions are handled by specint: incomplete gamma
function, error functions (but not the error function erfi, it is easy to
transform erfi e.g. to the error function erf), exponential
integrals, bessel functions (including products of bessel functions), hankel
functions, hermite and the laguerre polynomials.
Furthermore, specint can handle the hypergeometric function
%f[p,q]([],[],z), the whittaker function of the first kind
%m[u,k](z) and of the second kind %w[u,k](z).
The result may be in terms of special functions and can include unsimplified hypergeometric functions.
When laplace fails to find a Laplace transform, specint is called.
Because laplace knows more general rules for Laplace transforms, it is
preferable to use laplace and not specint.
demo(hypgeo) displays several examples of Laplace transforms computed by
specint.
Examples:
(%i1) assume (p > 0, a > 0)$
(%i2) specint (t^(1/2) * exp(-a*t/4) * exp(-p*t), t);
sqrt(%pi)
(%o2) ------------
a 3/2
2 (p + -)
4
(%i3) specint (t^(1/2) * bessel_j(1, 2 * a^(1/2) * t^(1/2))
* exp(-p*t), t);
- a/p
sqrt(a) %e
(%o3) ---------------
2
p
Examples for exponential integrals:
(%i4) assume(s>0,a>0,s-a>0)$
(%i5) ratsimp(specint(%e^(a*t)
*(log(a)+expintegral_e1(a*t))*%e^(-s*t),t));
log(s)
(%o5) ------
s - a
(%i6) logarc:true$
(%i7) gamma_expand:true$
radcan(specint((cos(t)*expintegral_si(t)
-sin(t)*expintegral_ci(t))*%e^(-s*t),t));
log(s)
(%o8) ------
2
s + 1
ratsimp(specint((2*t*log(a)+2/a*sin(a*t)
-2*t*expintegral_ci(a*t))*%e^(-s*t),t));
2 2
log(s + a )
(%o9) ------------
2
s
Results when using the expansion of gamma_incomplete and when changing
the representation to expintegral_e1:
(%i10) assume(s>0)$
(%i11) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
1
gamma_incomplete(-, k s)
2
(%o11) ------------------------
sqrt(%pi) sqrt(s)
(%i12) gamma_expand:true$
(%i13) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
erfc(sqrt(k) sqrt(s))
(%o13) ---------------------
sqrt(s)
(%i14) expintrep:expintegral_e1$
(%i15) ratsimp(specint(1/(t+a)^2*%e^(-s*t),t));
a s
a s %e expintegral_e1(a s) - 1
(%o15) - ---------------------------------
a
Categories: Laplace transform
hypergeometric_simp simplifies hypergeometric functions
by applying hgfred
to the arguments of any hypergeometric functions in the expression e.
Only instances of hypergeometric are affected;
any %f, %w, and %m in the expression e are not affected.
Any unsimplified hypergeometric functions are returned unchanged
(instead of changing to %f as hgfred would).
load(hypergeometric); loads this function.
See also hgfred.
Examples:
(%i1) load (hypergeometric) $
(%i2) foo : [hypergeometric([1,1], [2], z), hypergeometric([1/2], [1], z)];
(%o2) [hypergeometric([1, 1], [2], z),
1
hypergeometric([-], [1], z)]
2
(%i3) hypergeometric_simp (foo);
log(1 - z) z z/2
(%o3) [- ----------, bessel_i(0, -) %e ]
z 2
(%i4) bar : hypergeometric([n], [m], z + 1);
(%o4) hypergeometric([n], [m], z + 1)
(%i5) hypergeometric_simp (bar);
(%o5) hypergeometric([n], [m], z + 1)
Categories: Hypergeometric functions · Simplification functions · Special functions
Simplify the generalized hypergeometric function in terms of other, simpler, forms. a is a list of numerator parameters and b is a list of the denominator parameters.
If hgfred cannot simplify the hypergeometric function, it returns
an expression of the form %f[p,q]([a], [b], x) where p is
the number of elements in a, and q is the number of elements
in b. This is the usual pFq generalized hypergeometric
function.
(%i1) assume(not(equal(z,0)));
(%o1) [notequal(z, 0)]
(%i2) hgfred([v+1/2],[2*v+1],2*%i*z);
v/2 %i z
4 bessel_j(v, z) gamma(v + 1) %e
(%o2) ---------------------------------------
v
z
(%i3) hgfred([1,1],[2],z);
log(1 - z)
(%o3) - ----------
z
(%i4) hgfred([a,a+1/2],[3/2],z^2);
1 - 2 a 1 - 2 a
(z + 1) - (1 - z)
(%o4) -------------------------------
2 (1 - 2 a) z
It can be beneficial to load orthopoly too as the following example shows. Note that L is the generalized Laguerre polynomial.
(%i5) load(orthopoly)$
(%i6) hgfred([-2],[a],z);
(a - 1)
2 L (z)
2
(%o6) -------------
a (a + 1)
(%i7) ev(%);
2
z 2 z
(%o7) --------- - --- + 1
a (a + 1) a
The principal branch of Lambert's W function W(z), the solution of
z = W(z) * exp(W(z)). (DLMF 4.13)
The k-th branch of Lambert's W function W(z), the solution of
z = W(z) * exp(W(z)). (DLMF 4.13)
The principal branch, denoted Wp(z) in DLMF, is lambert_w(z) = generalized_lambert_w(0,z).
The other branch with real values, denoted Wm(z) in DLMF, is generalized_lambert_w(-1,z).
The Plasma Dispersion Function
nzeta(z) = %i*sqrt(%pi)*exp(-z^2)*(1-erf(-%i*z))
Returns realpart(nzeta(z)).
Returns imagpart(nzeta(z)).
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| 16.1 Introduction to Elliptic Functions and Integrals | ||
| 16.2 Functions and Variables for Elliptic Functions | ||
| 16.3 Functions and Variables for Elliptic Integrals |
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Maxima includes support for Jacobian elliptic functions and for complete and incomplete elliptic integrals. This includes symbolic manipulation of these functions and numerical evaluation as well. Definitions of these functions and many of their properties can by found in Abramowitz and Stegun, Chapter 16-17. As much as possible, we use the definitions and relationships given there.
In particular, all elliptic functions and integrals use the parameter m instead of the modulus k or the modular angle \alpha. This is one area where we differ from Abramowitz and Stegun who use the modular angle for the elliptic functions. The following relationships are true:
The elliptic functions and integrals are primarily intended to support symbolic computation. Therefore, most of derivatives of the functions and integrals are known. However, if floating-point values are given, a floating-point result is returned.
Support for most of the other properties of elliptic functions and integrals other than derivatives has not yet been written.
Some examples of elliptic functions:
(%i1) jacobi_sn (u, m);
(%o1) jacobi_sn(u, m)
(%i2) jacobi_sn (u, 1);
(%o2) tanh(u)
(%i3) jacobi_sn (u, 0);
(%o3) sin(u)
(%i4) diff (jacobi_sn (u, m), u);
(%o4) jacobi_cn(u, m) jacobi_dn(u, m)
(%i5) diff (jacobi_sn (u, m), m);
(%o5) jacobi_cn(u, m) jacobi_dn(u, m)
elliptic_e(asin(jacobi_sn(u, m)), m)
(u - ------------------------------------)/(2 m)
1 - m
2
jacobi_cn (u, m) jacobi_sn(u, m)
+ --------------------------------
2 (1 - m)
Some examples of elliptic integrals:
(%i1) elliptic_f (phi, m);
(%o1) elliptic_f(phi, m)
(%i2) elliptic_f (phi, 0);
(%o2) phi
(%i3) elliptic_f (phi, 1);
phi %pi
(%o3) log(tan(--- + ---))
2 4
(%i4) elliptic_e (phi, 1);
(%o4) sin(phi)
(%i5) elliptic_e (phi, 0);
(%o5) phi
(%i6) elliptic_kc (1/2);
1
(%o6) elliptic_kc(-)
2
(%i7) makegamma (%);
2 1
gamma (-)
4
(%o7) -----------
4 sqrt(%pi)
(%i8) diff (elliptic_f (phi, m), phi);
1
(%o8) ---------------------
2
sqrt(1 - m sin (phi))
(%i9) diff (elliptic_f (phi, m), m);
elliptic_e(phi, m) - (1 - m) elliptic_f(phi, m)
(%o9) (-----------------------------------------------
m
cos(phi) sin(phi)
- ---------------------)/(2 (1 - m))
2
sqrt(1 - m sin (phi))
Support for elliptic functions and integrals was written by Raymond Toy. It is placed under the terms of the General Public License (GPL) that governs the distribution of Maxima.
Categories: Elliptic functions
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The Jacobian elliptic function sn(u,m).
Categories: Elliptic functions
The Jacobian elliptic function cn(u,m).
Categories: Elliptic functions
The Jacobian elliptic function dn(u,m).
Categories: Elliptic functions
The Jacobian elliptic function ns(u,m) = 1/sn(u,m).
Categories: Elliptic functions
The Jacobian elliptic function sc(u,m) = sn(u,m)/cn(u,m).
Categories: Elliptic functions
The Jacobian elliptic function sd(u,m) = sn(u,m)/dn(u,m).
Categories: Elliptic functions
The Jacobian elliptic function nc(u,m) = 1/cn(u,m).
Categories: Elliptic functions
The Jacobian elliptic function cs(u,m) = cn(u,m)/sn(u,m).
Categories: Elliptic functions
The Jacobian elliptic function cd(u,m) = cn(u,m)/dn(u,m).
Categories: Elliptic functions
The Jacobian elliptic function nd(u,m) = 1/dn(u,m).
Categories: Elliptic functions
The Jacobian elliptic function ds(u,m) = dn(u,m)/sn(u,m).
Categories: Elliptic functions
The Jacobian elliptic function dc(u,m) = dn(u,m)/cn(u,m).
Categories: Elliptic functions
The inverse of the Jacobian elliptic function sn(u,m).
Categories: Elliptic functions
The inverse of the Jacobian elliptic function cn(u,m).
Categories: Elliptic functions
The inverse of the Jacobian elliptic function dn(u,m).
Categories: Elliptic functions
The inverse of the Jacobian elliptic function ns(u,m).
Categories: Elliptic functions
The inverse of the Jacobian elliptic function sc(u,m).
Categories: Elliptic functions
The inverse of the Jacobian elliptic function sd(u,m).
Categories: Elliptic functions
The inverse of the Jacobian elliptic function nc(u,m).
Categories: Elliptic functions
The inverse of the Jacobian elliptic function cs(u,m).
Categories: Elliptic functions
The inverse of the Jacobian elliptic function cd(u,m).
Categories: Elliptic functions
The inverse of the Jacobian elliptic function nd(u,m).
Categories: Elliptic functions
The inverse of the Jacobian elliptic function ds(u,m).
Categories: Elliptic functions
The inverse of the Jacobian elliptic function dc(u,m).
Categories: Elliptic functions
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The incomplete elliptic integral of the first kind, defined as
integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)
See also elliptic_e and elliptic_kc.
Categories: Elliptic integrals
The incomplete elliptic integral of the second kind, defined as
elliptic_e(phi, m) = integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)
See also elliptic_f and elliptic_ec.
Categories: Elliptic integrals
The incomplete elliptic integral of the second kind, defined as
integrate(dn(v,m)^2,v,0,u) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t, 0, tau)
where tau = sn(u,m).
This is related to elliptic_e by
elliptic_eu(u, m) = elliptic_e(asin(sn(u,m)),m)
See also elliptic_e.
The incomplete elliptic integral of the third kind, defined as
integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, 0, phi)
Only the derivative with respect to phi is known by Maxima.
Categories: Elliptic integrals
The complete elliptic integral of the first kind, defined as
integrate(1/sqrt(1 - m*sin(x)^2), x, 0, %pi/2)
For certain values of m, the value of the integral is known in
terms of Gamma functions. Use makegamma to evaluate them.
Categories: Elliptic integrals
The complete elliptic integral of the second kind, defined as
integrate(sqrt(1 - m*sin(x)^2), x, 0, %pi/2)
For certain values of m, the value of the integral is known in
terms of Gamma functions. Use makegamma to evaluate them.
Categories: Elliptic integrals
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| 17.1 Functions and Variables for Limits |
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Default value: 4
lhospitallim is the maximum number of times L'Hospital's
rule is used in limit. This prevents infinite looping in cases like
limit (cot(x)/csc(x), x, 0).
Categories: Limits
Computes the limit of expr as the real variable x approaches the
value val from the direction dir. dir may have the value
plus for a limit from above, minus for a limit from below, or
may be omitted (implying a two-sided limit is to be computed).
limit uses the following special symbols: inf (positive infinity)
and minf (negative infinity). On output it may also use und
(undefined), ind (indefinite but bounded) and infinity (complex
infinity).
infinity (complex infinity) is returned when the limit of
the absolute value of the expression is positive infinity, but
the limit of the expression itself is not positive infinity or
negative infinity. This includes cases where the limit of the
complex argument is a constant, as in limit(log(x), x, minf),
cases where the complex argument oscillates, as in
limit((-2)^x, x, inf), and cases where the complex
argument is different for either side of a two-sided limit, as in
limit(1/x, x, 0) and limit(log(x), x, 0).
lhospitallim is the maximum number of times L'Hospital's rule
is used in limit. This prevents infinite looping in cases like
limit (cot(x)/csc(x), x, 0).
tlimswitch when true will allow the limit command to use
Taylor series expansion when necessary.
limsubst prevents limit from attempting substitutions on
unknown forms. This is to avoid bugs like limit (f(n)/f(n+1), n, inf)
giving 1. Setting limsubst to true will allow such
substitutions.
limit with one argument is often called upon to simplify constant
expressions, for example, limit (inf-1).
example (limit) displays some examples.
For the method see Wang, P., "Evaluation of Definite Integrals by Symbolic Manipulation", Ph.D. thesis, MAC TR-92, October 1971.
Categories: Limits
Default value: false
prevents limit from attempting substitutions on unknown forms. This is
to avoid bugs like limit (f(n)/f(n+1), n, inf) giving 1. Setting
limsubst to true will allow such substitutions.
Categories: Limits
Take the limit of the Taylor series expansion of expr in x
at val from direction dir.
Categories: Limits
Default value: true
When tlimswitch is true, the limit command will use a
Taylor series expansion if the limit of the input expression cannot be computed
directly. This allows evaluation of limits such as
limit(x/(x-1)-1/log(x),x,1,plus). When tlimswitch is false
and the limit of input expression cannot be computed directly, limit will
return an unevaluated limit expression.
Categories: Limits
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| 18.1 Functions and Variables for Differentiation |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Returns a two-element list, such that an antiderivative of expr with respect to x can be constructed from the list. The expression expr may contain an unknown function u and its derivatives.
Let L, a list of two elements, be the return value of antid.
Then L[1] + 'integrate (L[2], x)
is an antiderivative of expr with respect to x.
When antid succeeds entirely,
the second element of the return value is zero.
Otherwise, the second element is nonzero,
and the first element is nonzero or zero.
If antid cannot make any progress,
the first element is zero and the second nonzero.
load ("antid") loads this function. The antid package also
defines the functions nonzeroandfreeof and linear.
antid is related to antidiff as follows.
Let L, a list of two elements, be the return value of antid.
Then the return value of antidiff is equal to
L[1] + 'integrate (L[2], x) where x is the
variable of integration.
Examples:
(%i1) load ("antid")$
(%i2) expr: exp (z(x)) * diff (z(x), x) * y(x);
z(x) d
(%o2) y(x) %e (-- (z(x)))
dx
(%i3) a1: antid (expr, x, z(x));
z(x) z(x) d
(%o3) [y(x) %e , - %e (-- (y(x)))]
dx
(%i4) a2: antidiff (expr, x, z(x));
/
z(x) [ z(x) d
(%o4) y(x) %e - I %e (-- (y(x))) dx
] dx
/
(%i5) a2 - (first (a1) + 'integrate (second (a1), x));
(%o5) 0
(%i6) antid (expr, x, y(x));
z(x) d
(%o6) [0, y(x) %e (-- (z(x)))]
dx
(%i7) antidiff (expr, x, y(x));
/
[ z(x) d
(%o7) I y(x) %e (-- (z(x))) dx
] dx
/
Categories: Integral calculus
Returns an antiderivative of expr with respect to x. The expression expr may contain an unknown function u and its derivatives.
When antidiff succeeds entirely, the resulting expression is free of
integral signs (that is, free of the integrate noun).
Otherwise, antidiff returns an expression
which is partly or entirely within an integral sign.
If antidiff cannot make any progress,
the return value is entirely within an integral sign.
load ("antid") loads this function.
The antid package also defines the functions nonzeroandfreeof and
linear.
antidiff is related to antid as follows.
Let L, a list of two elements, be the return value of antid.
Then the return value of antidiff is equal to
L[1] + 'integrate (L[2], x) where x is the
variable of integration.
Examples:
(%i1) load ("antid")$
(%i2) expr: exp (z(x)) * diff (z(x), x) * y(x);
z(x) d
(%o2) y(x) %e (-- (z(x)))
dx
(%i3) a1: antid (expr, x, z(x));
z(x) z(x) d
(%o3) [y(x) %e , - %e (-- (y(x)))]
dx
(%i4) a2: antidiff (expr, x, z(x));
/
z(x) [ z(x) d
(%o4) y(x) %e - I %e (-- (y(x))) dx
] dx
/
(%i5) a2 - (first (a1) + 'integrate (second (a1), x));
(%o5) 0
(%i6) antid (expr, x, y(x));
z(x) d
(%o6) [0, y(x) %e (-- (z(x)))]
dx
(%i7) antidiff (expr, x, y(x));
/
[ z(x) d
(%o7) I y(x) %e (-- (z(x))) dx
] dx
/
Categories: Integral calculus
Evaluates the expression expr with the variables assuming the values as
specified for them in the list of equations [eqn_1, ...,
eqn_n] or the single equation eqn.
If a subexpression depends on any of the variables for which a value is
specified but there is no atvalue specified and it can't be otherwise
evaluated, then a noun form of the at is returned which displays in a
two-dimensional form.
at carries out multiple substitutions in parallel.
See also atvalue. For other functions which carry out substitutions,
see also subst and ev.
Examples:
(%i1) atvalue (f(x,y), [x = 0, y = 1], a^2);
2
(%o1) a
(%i2) atvalue ('diff (f(x,y), x), x = 0, 1 + y);
(%o2) @2 + 1
(%i3) printprops (all, atvalue);
!
d !
--- (f(@1, @2))! = @2 + 1
d@1 !
!@1 = 0
2
f(0, 1) = a
(%o3) done
(%i4) diff (4*f(x, y)^2 - u(x, y)^2, x);
d d
(%o4) 8 f(x, y) (-- (f(x, y))) - 2 u(x, y) (-- (u(x, y)))
dx dx
(%i5) at (%, [x = 0, y = 1]);
!
2 d !
(%o5) 16 a - 2 u(0, 1) (-- (u(x, y))! )
dx !
!x = 0, y = 1
Categories: Evaluation · Differential equations
atomgrad is the atomic gradient property of an expression.
This property is assigned by gradef.
Categories: Differential calculus
Assigns the value c to expr at the point x = a.
Typically boundary values are established by this mechanism.
expr is a function evaluation, f(x_1, ..., x_m),
or a derivative, diff (f(x_1, ..., x_m), x_1,
n_1, ..., x_n, n_m)
in which the function arguments explicitly appear.
n_i is the order of differentiation with respect to x_i.
The point at which the atvalue is established is given by the list of equations
[x_1 = a_1, ..., x_m = a_m].
If there is a single variable x_1,
the sole equation may be given without enclosing it in a list.
printprops ([f_1, f_2, ...], atvalue) displays the atvalues
of the functions f_1, f_2, ... as specified by calls to
atvalue. printprops (f, atvalue) displays the atvalues of
one function f. printprops (all, atvalue) displays the atvalues
of all functions for which atvalues are defined.
The symbols @1, @2, … represent the
variables x_1, x_2, … when atvalues are displayed.
atvalue evaluates its arguments.
atvalue returns c, the atvalue.
Examples:
(%i1) atvalue (f(x,y), [x = 0, y = 1], a^2);
2
(%o1) a
(%i2) atvalue ('diff (f(x,y), x), x = 0, 1 + y);
(%o2) @2 + 1
(%i3) printprops (all, atvalue);
!
d !
--- (f(@1, @2))! = @2 + 1
d@1 !
!@1 = 0
2
f(0, 1) = a
(%o3) done
(%i4) diff (4*f(x,y)^2 - u(x,y)^2, x);
d d
(%o4) 8 f(x, y) (-- (f(x, y))) - 2 u(x, y) (-- (u(x, y)))
dx dx
(%i5) at (%, [x = 0, y = 1]);
!
2 d !
(%o5) 16 a - 2 u(0, 1) (-- (u(x, y))! )
dx !
!x = 0, y = 1
Categories: Differential equations · Declarations and inferences
The exterior calculus of differential forms is a basic tool
of differential geometry developed by Elie Cartan and has important
applications in the theory of partial differential equations.
The cartan package
implements the functions ext_diff and lie_diff,
along with the operators ~ (wedge product) and | (contraction
of a form with a vector.)
Type demo (tensor) to see a brief
description of these commands along with examples.
cartan was implemented by F.B. Estabrook and H.D. Wahlquist.
Categories: Differential geometry
del (x) represents the differential of the variable x.
diff returns an expression containing del
if an independent variable is not specified.
In this case, the return value is the so-called "total differential".
Examples:
(%i1) diff (log (x));
del(x)
(%o1) ------
x
(%i2) diff (exp (x*y));
x y x y
(%o2) x %e del(y) + y %e del(x)
(%i3) diff (x*y*z);
(%o3) x y del(z) + x z del(y) + y z del(x)
Categories: Differential calculus
The Dirac Delta function.
Currently only laplace knows about the delta function.
Example:
(%i1) laplace (delta (t - a) * sin(b*t), t, s);
Is a positive, negative, or zero?
p;
- a s
(%o1) sin(a b) %e
Categories: Mathematical functions · Laplace transform
The variable dependencies is the list of atoms which have functional
dependencies, assigned by depends, the function dependencies, or gradef.
The dependencies list is cumulative:
each call to depends, dependencies, or gradef appends additional items.
The default value of dependencies is [].
The function dependencies(f_1, …, f_n) appends f_1, …, f_n,
to the dependencies list,
where f_1, …, f_n are expressions of the form f(x_1, …, x_m),
and x_1, …, x_m are any number of arguments.
dependencies(f(x_1, …, x_m)) is equivalent to depends(f, [x_1, …, x_m]).
(%i1) dependencies;
(%o1) []
(%i2) depends (foo, [bar, baz]);
(%o2) [foo(bar, baz)]
(%i3) depends ([g, h], [a, b, c]);
(%o3) [g(a, b, c), h(a, b, c)]
(%i4) dependencies;
(%o4) [foo(bar, baz), g(a, b, c), h(a, b, c)]
(%i5) dependencies (quux (x, y), mumble (u));
(%o5) [quux(x, y), mumble(u)]
(%i6) dependencies;
(%o6) [foo(bar, baz), g(a, b, c), h(a, b, c), quux(x, y),
mumble(u)]
(%i7) remove (quux, dependency);
(%o7) done
(%i8) dependencies;
(%o8) [foo(bar, baz), g(a, b, c), h(a, b, c), mumble(u)]
Categories: Declarations and inferences · Global variables
Declares functional dependencies among variables for the purpose of computing
derivatives. In the absence of declared dependence, diff (f, x) yields
zero. If depends (f, x) is declared, diff (f, x) yields a
symbolic derivative (that is, a diff noun).
Each argument f_1, x_1, etc., can be the name of a variable or array, or a list of names. Every element of f_i (perhaps just a single element) is declared to depend on every element of x_i (perhaps just a single element). If some f_i is the name of an array or contains the name of an array, all elements of the array depend on x_i.
diff recognizes indirect dependencies established by depends
and applies the chain rule in these cases.
remove (f, dependency) removes all dependencies declared for
f.
depends returns a list of the dependencies established.
The dependencies are appended to the global variable dependencies.
depends evaluates its arguments.
diff is the only Maxima command which recognizes dependencies established
by depends. Other functions (integrate, laplace, etc.)
only recognize dependencies explicitly represented by their arguments.
For example, integrate does not recognize the dependence of f on
x unless explicitly represented as integrate (f(x), x).
depends(f, [x_1, …, x_n]) is equivalent to dependencies(f(x_1, …, x_n)).
(%i1) depends ([f, g], x);
(%o1) [f(x), g(x)]
(%i2) depends ([r, s], [u, v, w]);
(%o2) [r(u, v, w), s(u, v, w)]
(%i3) depends (u, t);
(%o3) [u(t)]
(%i4) dependencies;
(%o4) [f(x), g(x), r(u, v, w), s(u, v, w), u(t)]
(%i5) diff (r.s, u);
dr ds
(%o5) -- . s + r . --
du du
(%i6) diff (r.s, t);
dr du ds du
(%o6) -- -- . s + r . -- --
du dt du dt
(%i7) remove (r, dependency);
(%o7) done
(%i8) diff (r.s, t);
ds du
(%o8) r . -- --
du dt
Categories: Differential calculus · Declarations and inferences
Default value: false
When derivabbrev is true,
symbolic derivatives (that is, diff nouns) are displayed as subscripts.
Otherwise, derivatives are displayed in the Leibniz notation dy/dx.
Categories: Differential calculus · Global flags
Returns the highest degree of the derivative of the dependent variable y with respect to the independent variable x occurring in expr.
Example:
(%i1) 'diff (y, x, 2) + 'diff (y, z, 3) + 'diff (y, x) * x^2;
3 2
d y d y 2 dy
(%o1) --- + --- + x --
3 2 dx
dz dx
(%i2) derivdegree (%, y, x);
(%o2) 2
Categories: Differential calculus · Expressions
Causes only differentiations with respect to
the indicated variables, within the ev command.
Categories: Differential calculus · Evaluation
Default value: false
When derivsubst is true, a non-syntactic substitution such as
subst (x, 'diff (y, t), 'diff (y, t, 2)) yields 'diff (x, t).
Categories: Differential calculus · Expressions
Returns the derivative or differential of expr with respect to some or all variables in expr.
diff (expr, x, n) returns the n'th derivative of
expr with respect to x.
diff (expr, x_1, n_1, ..., x_m, n_m)
returns the mixed partial derivative of expr with respect to x_1,
…, x_m. It is equivalent to diff (... (diff (expr,
x_m, n_m) ...), x_1, n_1).
diff (expr, x)
returns the first derivative of expr with respect to
the variable x.
diff (expr) returns the total differential of expr, that is,
the sum of the derivatives of expr with respect to each its variables
times the differential del of each variable.
No further simplification of del is offered.
The noun form of diff is required in some contexts,
such as stating a differential equation.
In these cases, diff may be quoted (as 'diff) to yield the noun
form instead of carrying out the differentiation.
When derivabbrev is true, derivatives are displayed as subscripts.
Otherwise, derivatives are displayed in the Leibniz notation, dy/dx.
Examples:
(%i1) diff (exp (f(x)), x, 2);
2
f(x) d f(x) d 2
(%o1) %e (--- (f(x))) + %e (-- (f(x)))
2 dx
dx
(%i2) derivabbrev: true$
(%i3) 'integrate (f(x, y), y, g(x), h(x));
h(x)
/
[
(%o3) I f(x, y) dy
]
/
g(x)
(%i4) diff (%, x);
h(x)
/
[
(%o4) I f(x, y) dy + f(x, h(x)) h(x) - f(x, g(x)) g(x)
] x x x
/
g(x)
For the tensor package, the following modifications have been incorporated:
(1) The derivatives of any indexed objects in expr will have the variables x_i appended as additional arguments. Then all the derivative indices will be sorted.
(2) The x_i may be integers from 1 up to the value of the variable
dimension [default value: 4]. This will cause the differentiation to be
carried out with respect to the x_i'th member of the list
coordinates which should be set to a list of the names of the
coordinates, e.g., [x, y, z, t]. If coordinates is bound to an
atomic variable, then that variable subscripted by x_i will be used for
the variable of differentiation. This permits an array of coordinate names or
subscripted names like X[1], X[2], … to be used. If
coordinates has not been assigned a value, then the variables will be
treated as in (1) above.
Categories: Differential calculus
When diff is present as an evflag in call to ev,
all differentiations indicated in expr are carried out.
Applies the scalar d'Alembertian to the scalar function f.
load ("ctensor") loads this function.
Categories: Differential calculus · Package ctensor
Expands differential operator nouns into expressions in terms of partial
derivatives. express recognizes the operators grad, div,
curl, laplacian. express also expands the cross product
~.
Symbolic derivatives (that is, diff nouns)
in the return value of express may be evaluated by including diff
in the ev function call or command line.
In this context, diff acts as an evfun.
load ("vect") loads this function.
Examples:
(%i1) load ("vect")$
(%i2) grad (x^2 + y^2 + z^2);
2 2 2
(%o2) grad (z + y + x )
(%i3) express (%);
d 2 2 2 d 2 2 2 d 2 2 2
(%o3) [-- (z + y + x ), -- (z + y + x ), -- (z + y + x )]
dx dy dz
(%i4) ev (%, diff);
(%o4) [2 x, 2 y, 2 z]
(%i5) div ([x^2, y^2, z^2]);
2 2 2
(%o5) div [x , y , z ]
(%i6) express (%);
d 2 d 2 d 2
(%o6) -- (z ) + -- (y ) + -- (x )
dz dy dx
(%i7) ev (%, diff);
(%o7) 2 z + 2 y + 2 x
(%i8) curl ([x^2, y^2, z^2]);
2 2 2
(%o8) curl [x , y , z ]
(%i9) express (%);
d 2 d 2 d 2 d 2 d 2 d 2
(%o9) [-- (z ) - -- (y ), -- (x ) - -- (z ), -- (y ) - -- (x )]
dy dz dz dx dx dy
(%i10) ev (%, diff);
(%o10) [0, 0, 0]
(%i11) laplacian (x^2 * y^2 * z^2);
2 2 2
(%o11) laplacian (x y z )
(%i12) express (%);
2 2 2
d 2 2 2 d 2 2 2 d 2 2 2
(%o12) --- (x y z ) + --- (x y z ) + --- (x y z )
2 2 2
dz dy dx
(%i13) ev (%, diff);
2 2 2 2 2 2
(%o13) 2 y z + 2 x z + 2 x y
(%i14) [a, b, c] ~ [x, y, z];
(%o14) [a, b, c] ~ [x, y, z]
(%i15) express (%);
(%o15) [b z - c y, c x - a z, a y - b x]
Categories: Differential calculus · Vectors · Operators
Defines the partial derivatives (i.e., the components of the gradient) of the function f or variable a.
gradef (f(x_1, ..., x_n), g_1, ..., g_m)
defines df/dx_i as g_i, where g_i is an
expression; g_i may be a function call, but not the name of a function.
The number of partial derivatives m may be less than the number of
arguments n, in which case derivatives are defined with respect to
x_1 through x_m only.
gradef (a, x, expr) defines the derivative of variable
a with respect to x as expr. This also establishes the
dependence of a on x (via depends (a, x)).
The first argument f(x_1, ..., x_n) or a is
quoted, but the remaining arguments g_1, ..., g_m are evaluated.
gradef returns the function or variable for which the partial derivatives
are defined.
gradef can redefine the derivatives of Maxima's built-in functions.
For example, gradef (sin(x), sqrt (1 - sin(x)^2)) redefines the
derivative of sin.
gradef cannot define partial derivatives for a subscripted function.
printprops ([f_1, ..., f_n], gradef) displays the partial
derivatives of the functions f_1, ..., f_n, as defined by
gradef.
printprops ([a_n, ..., a_n], atomgrad) displays the partial
derivatives of the variables a_n, ..., a_n, as defined by
gradef.
gradefs is the list of the functions
for which partial derivatives have been defined by gradef.
gradefs does not include any variables
for which partial derivatives have been defined by gradef.
Gradients are needed when, for example, a function is not known explicitly but its first derivatives are and it is desired to obtain higher order derivatives.
Categories: Differential calculus · Declarations and inferences
Default value: []
gradefs is the list of the functions
for which partial derivatives have been defined by gradef.
gradefs does not include any variables
for which partial derivatives have been defined by gradef.
Categories: Differential calculus · Declarations and inferences
Attempts to compute the Laplace transform of expr with respect to the variable t and transform parameter s.
laplace recognizes in expr the functions delta, exp,
log, sin, cos, sinh, cosh, and erf,
as well as derivative, integrate, sum, and ilt. If
laplace fails to find a transform the function specint is called.
specint can find the laplace transform for expressions with special
functions like the bessel functions bessel_j, bessel_i, …
and can handle the unit_step function. See also specint.
If specint cannot find a solution too, a noun laplace is returned.
expr may also be a linear, constant coefficient differential equation in
which case atvalue of the dependent variable is used.
The required atvalue may be supplied either before or after the transform is
computed. Since the initial conditions must be specified at zero, if one has
boundary conditions imposed elsewhere he can impose these on the general
solution and eliminate the constants by solving the general solution
for them and substituting their values back.
laplace recognizes convolution integrals of the form
integrate (f(x) * g(t - x), x, 0, t);
other kinds of convolutions are not recognized.
Functional relations must be explicitly represented in expr;
implicit relations, established by depends, are not recognized.
That is, if f depends on x and y,
f (x, y) must appear in expr.
See also ilt,, the inverse Laplace transform.
Examples:
(%i1) laplace (exp (2*t + a) * sin(t) * t, t, s);
a
%e (2 s - 4)
(%o1) ---------------
2 2
(s - 4 s + 5)
(%i2) laplace ('diff (f (x), x), x, s);
(%o2) s laplace(f(x), x, s) - f(0)
(%i3) diff (diff (delta (t), t), t);
2
d
(%o3) --- (delta(t))
2
dt
(%i4) laplace (%, t, s);
!
d ! 2
(%o4) - -- (delta(t))! + s - delta(0) s
dt !
!t = 0
(%i5) assume(a>0)$
(%i6) laplace(gamma_incomplete(a,t),t,s),gamma_expand:true;
- a - 1
gamma(a) gamma(a) s
(%o6) -------- - -----------------
s 1 a
(- + 1)
s
(%i7) factor(laplace(gamma_incomplete(1/2,t),t,s));
s + 1
sqrt(%pi) (sqrt(s) sqrt(-----) - 1)
s
(%o7) -----------------------------------
3/2 s + 1
s sqrt(-----)
s
(%i8) assume(exp(%pi*s)>1)$
(%i9) laplace(sum((-1)^n*unit_step(t-n*%pi)*sin(t),n,0,inf),t,s),
simpsum;
%i %i
------------------------ - ------------------------
- %pi s - %pi s
(s + %i) (1 - %e ) (s - %i) (1 - %e )
(%o9) ---------------------------------------------------
2
(%i9) factor(%);
%pi s
%e
(%o9) -------------------------------
%pi s
(s - %i) (s + %i) (%e - 1)
Categories: Laplace transform · Differential equations
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 19.1 Introduction to Integration | ||
| 19.2 Functions and Variables for Integration | ||
| 19.3 Introduction to QUADPACK | ||
| 19.4 Functions and Variables for QUADPACK |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Maxima has several routines for handling integration.
The integrate function makes use of most of them. There is also the
antid package, which handles an unspecified function (and its
derivatives, of course). For numerical uses,
there is a set of adaptive integrators from QUADPACK, named quad_qag,
quad_qags, etc., which are described under the heading QUADPACK.
Hypergeometric functions are being worked on,
see specint for details.
Generally speaking, Maxima only handles integrals which are
integrable in terms of the "elementary functions" (rational functions,
trigonometrics, logs, exponentials, radicals, etc.) and a few
extensions (error function, dilogarithm). It does not handle
integrals in terms of unknown functions such as g(x) and h(x).
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Makes the change of variable given by f(x,y) = 0 in all integrals
occurring in expr with integration with respect to x.
The new variable is y.
(%i1) assume(a > 0)$
(%i2) 'integrate (%e**sqrt(a*y), y, 0, 4);
4
/
[ sqrt(a) sqrt(y)
(%o2) I %e dy
]
/
0
(%i3) changevar (%, y-z^2/a, z, y);
0
/
[ abs(z)
2 I z %e dz
]
/
- 2 sqrt(a)
(%o3) - ----------------------------
a
An expression containing a noun form, such as the instances of 'integrate
above, may be evaluated by ev with the nouns flag.
For example, the expression returned by changevar above may be evaluated
by ev (%o3, nouns).
changevar may also be used to changes in the indices of a sum or
product. However, it must be realized that when a change is made in a
sum or product, this change must be a shift, i.e., i = j+ ..., not a
higher degree function. E.g.,
(%i4) sum (a[i]*x^(i-2), i, 0, inf);
inf
====
\ i - 2
(%o4) > a x
/ i
====
i = 0
(%i5) changevar (%, i-2-n, n, i);
inf
====
\ n
(%o5) > a x
/ n + 2
====
n = - 2
Categories: Integral calculus
A double-integral routine which was written in
top-level Maxima and then translated and compiled to machine code.
Use load (dblint) to access this package. It uses the Simpson's rule
method in both the x and y directions to calculate
/b /s(x) | | | | f(x,y) dy dx | | /a /r(x)
The function f must be a translated or compiled function of two variables,
and r and s must each be a translated or compiled function of one
variable, while a and b must be floating point numbers. The routine
has two global variables which determine the number of divisions of the x and y
intervals: dblint_x and dblint_y, both of which are initially 10,
and can be changed independently to other integer values (there are
2*dblint_x+1 points computed in the x direction, and 2*dblint_y+1
in the y direction). The routine subdivides the X axis and then for each value
of X it first computes r(x) and s(x); then the Y axis
between r(x) and s(x) is subdivided and the integral
along the Y axis is performed using Simpson's rule; then the integral along the
X axis is done using Simpson's rule with the function values being the
Y-integrals. This procedure may be numerically unstable for a great variety of
reasons, but is reasonably fast: avoid using it on highly oscillatory functions
and functions with singularities (poles or branch points in the region). The Y
integrals depend on how far apart r(x) and s(x) are,
so if the distance s(x) - r(x) varies rapidly with X, there
may be substantial errors arising from truncation with different step-sizes in
the various Y integrals. One can increase dblint_x and dblint_y
in an effort to improve the coverage of the region, at the expense of
computation time. The function values are not saved, so if the function is very
time-consuming, you will have to wait for re-computation if you change anything
(sorry). It is required that the functions f, r, and s be
either translated or compiled prior to calling dblint. This will result
in orders of magnitude speed improvement over interpreted code in many cases!
demo (dblint) executes a demonstration of dblint applied to an
example problem.
Categories: Integral calculus
Attempts to compute a definite integral. defint is called by
integrate when limits of integration are specified, i.e., when
integrate is called as
integrate (expr, x, a, b).
Thus from the user's point of view, it is sufficient to call integrate.
defint returns a symbolic expression, either the computed integral or the
noun form of the integral. See quad_qag and related functions for
numerical approximation of definite integrals.
Categories: Integral calculus
Default value: true
When erfflag is false, prevents risch from introducing the
erf function in the answer if there were none in the integrand to
begin with.
Categories: Integral calculus
Computes the inverse Laplace transform of expr with
respect to s and parameter t. expr must be a ratio of
polynomials whose denominator has only linear and quadratic factors.
By using the functions laplace and ilt together with the
solve or linsolve functions the user can solve a single
differential or convolution integral equation or a set of them.
(%i1) 'integrate (sinh(a*x)*f(t-x), x, 0, t) + b*f(t) = t**2;
t
/
[ 2
(%o1) I f(t - x) sinh(a x) dx + b f(t) = t
]
/
0
(%i2) laplace (%, t, s);
a laplace(f(t), t, s) 2
(%o2) b laplace(f(t), t, s) + --------------------- = --
2 2 3
s - a s
(%i3) linsolve ([%], ['laplace(f(t), t, s)]);
2 2
2 s - 2 a
(%o3) [laplace(f(t), t, s) = --------------------]
5 2 3
b s + (a - a b) s
(%i4) ilt (rhs (first (%)), s, t);
Is a b (a b - 1) positive, negative, or zero?
pos;
sqrt(a b (a b - 1)) t
2 cosh(---------------------) 2
b a t
(%o4) - ----------------------------- + -------
3 2 2 a b - 1
a b - 2 a b + a
2
+ ------------------
3 2 2
a b - 2 a b + a
Categories: Laplace transform
Default value: true
When true, definite integration tries to find poles in the integrand in
the interval of integration. If there are, then the integral is evaluated
appropriately as a principal value integral. If intanalysis is false,
this check is not performed and integration is done assuming there are no poles.
See also ldefint.
Examples:
Maxima can solve the following integrals, when intanalysis is set to
false:
(%i1) integrate(1/(sqrt(x)+1),x,0,1);
1
/
[ 1
(%o1) I ----------- dx
] sqrt(x) + 1
/
0
(%i2) integrate(1/(sqrt(x)+1),x,0,1),intanalysis:false;
(%o2) 2 - 2 log(2)
(%i3) integrate(cos(a)/sqrt((tan(a))^2 +1),a,-%pi/2,%pi/2);
The number 1 isn't in the domain of atanh
-- an error. To debug this try: debugmode(true);
(%i4) intanalysis:false$
(%i5) integrate(cos(a)/sqrt((tan(a))^2+1),a,-%pi/2,%pi/2);
%pi
(%o5) ---
2
Categories: Integral calculus
Attempts to symbolically compute the integral of expr with respect to
x. integrate (expr, x) is an indefinite integral,
while integrate (expr, x, a, b) is a definite
integral, with limits of integration a and b. The limits should
not contain x, although integrate does not enforce this
restriction. a need not be less than b.
If b is equal to a, integrate returns zero.
See quad_qag and related functions for numerical approximation of
definite integrals. See residue for computation of residues
(complex integration). See antid for an alternative means of computing
indefinite integrals.
The integral (an expression free of integrate) is returned if
integrate succeeds. Otherwise the return value is
the noun form of the integral (the quoted operator 'integrate)
or an expression containing one or more noun forms.
The noun form of integrate is displayed with an integral sign.
In some circumstances it is useful to construct a noun form by hand, by quoting
integrate with a single quote, e.g.,
'integrate (expr, x). For example, the integral may depend
on some parameters which are not yet computed.
The noun may be applied to its arguments by ev (i, nouns)
where i is the noun form of interest.
integrate handles definite integrals separately from indefinite, and
employs a range of heuristics to handle each case. Special cases of definite
integrals include limits of integration equal to zero or infinity (inf or
minf), trigonometric functions with limits of integration equal to zero
and %pi or 2 %pi, rational functions, integrals related to the
definitions of the beta and psi functions, and some logarithmic
and trigonometric integrals. Processing rational functions may include
computation of residues. If an applicable special case is not found, an attempt
will be made to compute the indefinite integral and evaluate it at the limits of
integration. This may include taking a limit as a limit of integration goes to
infinity or negative infinity; see also ldefint.
Special cases of indefinite integrals include trigonometric functions,
exponential and logarithmic functions,
and rational functions.
integrate may also make use of a short table of elementary integrals.
integrate may carry out a change of variable
if the integrand has the form f(g(x)) * diff(g(x), x).
integrate attempts to find a subexpression g(x) such that
the derivative of g(x) divides the integrand.
This search may make use of derivatives defined by the gradef function.
See also changevar and antid.
If none of the preceding heuristics find the indefinite integral, the Risch
algorithm is executed. The flag risch may be set as an evflag,
in a call to ev or on the command line, e.g.,
ev (integrate (expr, x), risch) or
integrate (expr, x), risch. If risch is present,
integrate calls the risch function without attempting heuristics
first. See also risch.
integrate works only with functional relations represented explicitly
with the f(x) notation. integrate does not respect implicit
dependencies established by the depends function.
integrate may need to know some property of a parameter in the integrand.
integrate will first consult the assume database,
and, if the variable of interest is not there,
integrate will ask the user.
Depending on the question,
suitable responses are yes; or no;,
or pos;, zero;, or neg;.
integrate is not, by default, declared to be linear. See declare
and linear.
integrate attempts integration by parts only in a few special cases.
Examples:
(%i1) integrate (sin(x)^3, x);
3
cos (x)
(%o1) ------- - cos(x)
3
(%i2) integrate (x/ sqrt (b^2 - x^2), x);
2 2
(%o2) - sqrt(b - x )
(%i3) integrate (cos(x)^2 * exp(x), x, 0, %pi);
%pi
3 %e 3
(%o3) ------- - -
5 5
(%i4) integrate (x^2 * exp(-x^2), x, minf, inf);
sqrt(%pi)
(%o4) ---------
2
assume and interactive query.
(%i1) assume (a > 1)$
(%i2) integrate (x**a/(x+1)**(5/2), x, 0, inf);
2 a + 2
Is ------- an integer?
5
no;
Is 2 a - 3 positive, negative, or zero?
neg;
3
(%o2) beta(a + 1, - - a)
2
gradef, and one using the
derivation diff(r(x)) of an unspecified function r(x).
(%i3) gradef (q(x), sin(x**2));
(%o3) q(x)
(%i4) diff (log (q (r (x))), x);
d 2
(-- (r(x))) sin(r (x))
dx
(%o4) ----------------------
q(r(x))
(%i5) integrate (%, x);
(%o5) log(q(r(x)))
'integrate noun form. In this example, Maxima
can extract one factor of the denominator of a rational function, but cannot
factor the remainder or otherwise find its integral. grind shows the
noun form 'integrate in the result. See also
integrate_use_rootsof for more on integrals of rational functions.
(%i1) expand ((x-4) * (x^3+2*x+1));
4 3 2
(%o1) x - 4 x + 2 x - 7 x - 4
(%i2) integrate (1/%, x);
/ 2
[ x + 4 x + 18
I ------------- dx
] 3
log(x - 4) / x + 2 x + 1
(%o2) ---------- - ------------------
73 73
(%i3) grind (%);
log(x-4)/73-('integrate((x^2+4*x+18)/(x^3+2*x+1),x))/73$
f_1 in this
example contains the noun form of integrate. The quote-quote operator
'' causes the integral to be evaluated, and the result becomes the
body of f_2.
(%i1) f_1 (a) := integrate (x^3, x, 1, a);
3
(%o1) f_1(a) := integrate(x , x, 1, a)
(%i2) ev (f_1 (7), nouns);
(%o2) 600
(%i3) /* Note parentheses around integrate(...) here */
f_2 (a) := ''(integrate (x^3, x, 1, a));
4
a 1
(%o3) f_2(a) := -- - -
4 4
(%i4) f_2 (7);
(%o4) 600
Categories: Integral calculus
Default value: %c
When a constant of integration is introduced by indefinite integration of an
equation, the name of the constant is constructed by concatenating
integration_constant and integration_constant_counter.
integration_constant may be assigned any symbol.
Examples:
(%i1) integrate (x^2 = 1, x);
3
x
(%o1) -- = x + %c1
3
(%i2) integration_constant : 'k;
(%o2) k
(%i3) integrate (x^2 = 1, x);
3
x
(%o3) -- = x + k2
3
Categories: Integral calculus
Default value: 0
When a constant of integration is introduced by indefinite integration of an
equation, the name of the constant is constructed by concatenating
integration_constant and integration_constant_counter.
integration_constant_counter is incremented before constructing the next
integration constant.
Examples:
(%i1) integrate (x^2 = 1, x);
3
x
(%o1) -- = x + %c1
3
(%i2) integrate (x^2 = 1, x);
3
x
(%o2) -- = x + %c2
3
(%i3) integrate (x^2 = 1, x);
3
x
(%o3) -- = x + %c3
3
(%i4) reset (integration_constant_counter);
(%o4) [integration_constant_counter]
(%i5) integrate (x^2 = 1, x);
3
x
(%o5) -- = x + %c1
3
Categories: Integral calculus
Default value: false
When integrate_use_rootsof is true and the denominator of
a rational function cannot be factored, integrate returns the integral
in a form which is a sum over the roots (not yet known) of the denominator.
For example, with integrate_use_rootsof set to false,
integrate returns an unsolved integral of a rational function in noun
form:
(%i1) integrate_use_rootsof: false$
(%i2) integrate (1/(1+x+x^5), x);
/ 2
[ x - 4 x + 5
I ------------ dx 2 x + 1
] 3 2 2 5 atan(-------)
/ x - x + 1 log(x + x + 1) sqrt(3)
(%o2) ----------------- - --------------- + ---------------
7 14 7 sqrt(3)
Now we set the flag to be true and the unsolved part of the integral will be expressed as a summation over the roots of the denominator of the rational function:
(%i3) integrate_use_rootsof: true$
(%i4) integrate (1/(1+x+x^5), x);
==== 2
\ (%r4 - 4 %r4 + 5) log(x - %r4)
> -------------------------------
/ 2
==== 3 %r4 - 2 %r4
3 2
%r4 in rootsof(%r4 - %r4 + 1, %r4)
(%o4) ----------------------------------------------------------
7
2 x + 1
2 5 atan(-------)
log(x + x + 1) sqrt(3)
- --------------- + ---------------
14 7 sqrt(3)
Alternatively the user may compute the roots of the denominator separately,
and then express the integrand in terms of these roots, e.g.,
1/((x - a)*(x - b)*(x - c)) or 1/((x^2 - (a+b)*x + a*b)*(x - c))
if the denominator is a cubic polynomial.
Sometimes this will help Maxima obtain a more useful result.
Categories: Integral calculus
Attempts to compute the definite integral of expr by using limit
to evaluate the indefinite integral of expr with respect to x
at the upper limit b and at the lower limit a.
If it fails to compute the definite integral,
ldefint returns an expression containing limits as noun forms.
ldefint is not called from integrate, so executing
ldefint (expr, x, a, b) may yield a different
result than integrate (expr, x, a, b).
ldefint always uses the same method to evaluate the definite integral,
while integrate may employ various heuristics and may recognize some
special cases.
Categories: Integral calculus
The calculation makes use of the global variable potentialzeroloc[0]
which must be nonlist or of the form
[indeterminatej=expressionj, indeterminatek=expressionk, ...]
the former being equivalent to the nonlist expression for all right-hand
sides in the latter. The indicated right-hand sides are used as the
lower limit of integration. The success of the integrations may
depend upon their values and order. potentialzeroloc is initially set
to 0.
Computes the residue in the complex plane of the expression expr when the
variable z assumes the value z_0. The residue is the coefficient of
(z - z_0)^(-1) in the Laurent series for expr.
(%i1) residue (s/(s**2+a**2), s, a*%i);
1
(%o1) -
2
(%i2) residue (sin(a*x)/x**4, x, 0);
3
a
(%o2) - --
6
Categories: Integral calculus · Complex variables
Integrates expr with respect to x using the
transcendental case of the Risch algorithm. (The algebraic case of
the Risch algorithm has not been implemented.) This currently
handles the cases of nested exponentials and logarithms which the main
part of integrate can't do. integrate will automatically apply
risch if given these cases.
erfflag, if false, prevents risch from introducing the
erf function in the answer if there were none in the integrand to begin
with.
(%i1) risch (x^2*erf(x), x);
2
3 2 - x
%pi x erf(x) + (sqrt(%pi) x + sqrt(%pi)) %e
(%o1) -------------------------------------------------
3 %pi
(%i2) diff(%, x), ratsimp;
2
(%o2) x erf(x)
Categories: Integral calculus
Equivalent to ldefint with tlimswitch set to true.
Categories: Integral calculus
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QUADPACK is a collection of functions for the numerical computation of one-dimensional definite integrals. It originated from a joint project of R. Piessens (1), E. de Doncker (2), C. Ueberhuber (3), and D. Kahaner (4).
The QUADPACK library included in Maxima is an automatic translation (via the
program f2cl) of the Fortran source code of QUADPACK as it appears in
the SLATEC Common Mathematical Library, Version 4.1 (5).
The SLATEC library is dated July 1993, but the QUADPACK functions
were written some years before.
There is another version of QUADPACK at Netlib (6);
it is not clear how that version differs from the SLATEC version.
The QUADPACK functions included in Maxima are all automatic, in the sense that these functions attempt to compute a result to a specified accuracy, requiring an unspecified number of function evaluations. Maxima's Lisp translation of QUADPACK also includes some non-automatic functions, but they are not exposed at the Maxima level.
Further information about QUADPACK can be found in the QUADPACK book (7).
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quad_qagIntegration of a general function over a finite interval.
quad_qag implements a simple globally adaptive integrator using the
strategy of Aind (Piessens, 1973).
The caller may choose among 6 pairs of Gauss-Kronrod quadrature
formulae for the rule evaluation component.
The high-degree rules are suitable for strongly oscillating integrands.
quad_qagsIntegration of a general function over a finite interval.
quad_qags implements globally adaptive interval subdivision with
extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).
quad_qagiIntegration of a general function over an infinite or semi-infinite interval.
The interval is mapped onto a finite interval and
then the same strategy as in quad_qags is applied.
quad_qawoIntegration of cos(omega x) f(x) or sin(omega x) f(x) over a
finite interval, where omega is a constant.
The rule evaluation component is based on the modified Clenshaw-Curtis
technique. quad_qawo applies adaptive subdivision with extrapolation,
similar to quad_qags.
quad_qawfCalculates a Fourier cosine or Fourier sine transform on a semi-infinite
interval. The same approach as in quad_qawo is applied on successive
finite intervals, and convergence acceleration by means of the Epsilon algorithm
(Wynn, 1956) is applied to the series of the integral contributions.
quad_qawsIntegration of w(x) f(x) over a finite interval [a, b], where w is a function of the form (x - a)^alpha (b - x)^beta v(x) and v(x) is 1 or log(x - a) or log(b - x) or log(x - a) log(b - x), and alpha > -1 and beta > -1.
A globally adaptive subdivision strategy is applied, with modified Clenshaw-Curtis integration on the subintervals which contain a or b.
quad_qawcComputes the Cauchy principal value of f(x)/(x - c) over a finite interval (a, b) and specified c. The strategy is globally adaptive, and modified Clenshaw-Curtis integration is used on the subranges which contain the point x = c.
quad_qagpBasically the same as quad_qags but points of singularity or
discontinuity of the integrand must be supplied. This makes it easier
for the integrator to produce a good solution.
Categories: Integral calculus · Numerical methods · Share packages · Package quadpack
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Integration of a general function over a finite interval. quad_qag
implements a simple globally adaptive integrator using the strategy of Aind
(Piessens, 1973). The caller may choose among 6 pairs of Gauss-Kronrod
quadrature formulae for the rule evaluation component. The high-degree rules
are suitable for strongly oscillating integrands.
quad_qag computes the integral
integrate (f(x), x, a, b)
The function to be integrated is f(x), with dependent variable x, and the function is to be integrated between the limits a and b. key is the integrator to be used and should be an integer between 1 and 6, inclusive. The value of key selects the order of the Gauss-Kronrod integration rule. High-order rules are suitable for strongly oscillating integrands.
The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.
The numerical integration is done adaptively by subdividing the integration region into sub-intervals until the desired accuracy is achieved.
The keyword arguments are optional and may be specified in any order.
They all take the form key=val. The keyword arguments are:
epsrelDesired relative error of approximation. Default is 1d-8.
epsabsDesired absolute error of approximation. Default is 0.
limitSize of internal work array. limit is the maximum number of subintervals to use. Default is 200.
quad_qag returns a list of four elements:
The error code (fourth element of the return value) can have the values:
0if no problems were encountered;
1if too many sub-intervals were done;
2if excessive roundoff error is detected;
3if extremely bad integrand behavior occurs;
6if the input is invalid.
Examples:
(%i1) quad_qag (x^(1/2)*log(1/x), x, 0, 1, 3, 'epsrel=5d-8);
(%o1) [.4444444444492108, 3.1700968502883E-9, 961, 0]
(%i2) integrate (x^(1/2)*log(1/x), x, 0, 1);
4
(%o2) -
9
Categories: Numerical methods · Package quadpack
Integration of a general function over a finite interval.
quad_qags implements globally adaptive interval subdivision with
extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).
quad_qags computes the integral
integrate (f(x), x, a, b)
The function to be integrated is f(x), with dependent variable x, and the function is to be integrated between the limits a and b.
The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.
The keyword arguments are optional and may be specified in any order.
They all take the form key=val. The keyword arguments are:
epsrelDesired relative error of approximation. Default is 1d-8.
epsabsDesired absolute error of approximation. Default is 0.
limitSize of internal work array. limit is the maximum number of subintervals to use. Default is 200.
quad_qags returns a list of four elements:
The error code (fourth element of the return value) can have the values:
0no problems were encountered;
1too many sub-intervals were done;
2excessive roundoff error is detected;
3extremely bad integrand behavior occurs;
4failed to converge
5integral is probably divergent or slowly convergent
6if the input is invalid.
Examples:
(%i1) quad_qags (x^(1/2)*log(1/x), x, 0, 1, 'epsrel=1d-10); (%o1) [.4444444444444448, 1.11022302462516E-15, 315, 0]
Note that quad_qags is more accurate and efficient than quad_qag for this integrand.
Categories: Numerical methods · Package quadpack
Integration of a general function over an infinite or semi-infinite interval.
The interval is mapped onto a finite interval and
then the same strategy as in quad_qags is applied.
quad_qagi evaluates one of the following integrals
integrate (f(x), x, a, inf)
integrate (f(x), x, minf, a)
integrate (f(x), x, minf, inf)
using the Quadpack QAGI routine. The function to be integrated is f(x), with dependent variable x, and the function is to be integrated over an infinite range.
The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.
One of the limits of integration must be infinity. If not, then
quad_qagi will just return the noun form.
The keyword arguments are optional and may be specified in any order.
They all take the form key=val. The keyword arguments are:
epsrelDesired relative error of approximation. Default is 1d-8.
epsabsDesired absolute error of approximation. Default is 0.
limitSize of internal work array. limit is the maximum number of subintervals to use. Default is 200.
quad_qagi returns a list of four elements:
The error code (fourth element of the return value) can have the values:
0no problems were encountered;
1too many sub-intervals were done;
2excessive roundoff error is detected;
3extremely bad integrand behavior occurs;
4failed to converge
5integral is probably divergent or slowly convergent
6if the input is invalid.
Examples:
(%i1) quad_qagi (x^2*exp(-4*x), x, 0, inf, 'epsrel=1d-8);
(%o1) [0.03125, 2.95916102995002E-11, 105, 0]
(%i2) integrate (x^2*exp(-4*x), x, 0, inf);
1
(%o2) --
32
Categories: Numerical methods · Package quadpack
Computes the Cauchy principal value of f(x)/(x - c) over a finite interval. The strategy is globally adaptive, and modified Clenshaw-Curtis integration is used on the subranges which contain the point x = c.
quad_qawc computes the Cauchy principal value of
integrate (f(x)/(x - c), x, a, b)
using the Quadpack QAWC routine. The function to be integrated is
f(x)/(x - c), with dependent variable x, and the
function is to be integrated over the interval a to b.
The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.
The keyword arguments are optional and may be specified in any order.
They all take the form key=val. The keyword arguments are:
epsrelDesired relative error of approximation. Default is 1d-8.
epsabsDesired absolute error of approximation. Default is 0.
limitSize of internal work array. limit is the maximum number of subintervals to use. Default is 200.
quad_qawc returns a list of four elements:
The error code (fourth element of the return value) can have the values:
0no problems were encountered;
1too many sub-intervals were done;
2excessive roundoff error is detected;
3extremely bad integrand behavior occurs;
6if the input is invalid.
Examples:
(%i1) quad_qawc (2^(-5)*((x-1)^2+4^(-5))^(-1), x, 2, 0, 5,
'epsrel=1d-7);
(%o1) [- 3.130120337415925, 1.306830140249558E-8, 495, 0]
(%i2) integrate (2^(-alpha)*(((x-1)^2 + 4^(-alpha))*(x-2))^(-1),
x, 0, 5);
Principal Value
alpha
alpha 9 4 9
4 log(------------- + -------------)
alpha alpha
64 4 + 4 64 4 + 4
(%o2) (-----------------------------------------
alpha
2 4 + 2
3 alpha 3 alpha
------- -------
2 alpha/2 2 alpha/2
2 4 atan(4 4 ) 2 4 atan(4 ) alpha
- --------------------------- - -------------------------)/2
alpha alpha
2 4 + 2 2 4 + 2
(%i3) ev (%, alpha=5, numer);
(%o3) - 3.130120337415917
Categories: Numerical methods · Package quadpack
Calculates a Fourier cosine or Fourier sine transform on a semi-infinite
interval using the Quadpack QAWF function. The same approach as in
quad_qawo is applied on successive finite intervals, and convergence
acceleration by means of the Epsilon algorithm (Wynn, 1956) is applied to the
series of the integral contributions.
quad_qawf computes the integral
integrate (f(x)*w(x), x, a, inf)
The weight function w is selected by trig:
cosw(x) = cos (omega x)
sinw(x) = sin (omega x)
The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.
The keyword arguments are optional and may be specified in any order.
They all take the form key=val. The keyword arguments are:
epsabsDesired absolute error of approximation. Default is 1d-10.
limitSize of internal work array. (limit - limlst)/2 is the maximum number of subintervals to use. Default is 200.
maxp1Maximum number of Chebyshev moments. Must be greater than 0. Default is 100.
limlstUpper bound on the number of cycles. Must be greater than or equal to 3. Default is 10.
quad_qawf returns a list of four elements:
The error code (fourth element of the return value) can have the values:
0no problems were encountered;
1too many sub-intervals were done;
2excessive roundoff error is detected;
3extremely bad integrand behavior occurs;
6if the input is invalid.
Examples:
(%i1) quad_qawf (exp(-x^2), x, 0, 1, 'cos, 'epsabs=1d-9);
(%o1) [.6901942235215714, 2.84846300257552E-11, 215, 0]
(%i2) integrate (exp(-x^2)*cos(x), x, 0, inf);
- 1/4
%e sqrt(%pi)
(%o2) -----------------
2
(%i3) ev (%, numer);
(%o3) .6901942235215714
Categories: Numerical methods · Package quadpack
Integration of
cos (omega x) f(x)
or
sin (omega x) f(x)
over a finite interval,
where
omega
is a constant. The rule evaluation component is based on the modified
Clenshaw-Curtis technique. quad_qawo applies adaptive subdivision with
extrapolation, similar to quad_qags.
quad_qawo computes the integral using the Quadpack QAWO
routine:
integrate (f(x)*w(x), x, a, b)
The weight function w is selected by trig:
cosw(x) = cos (omega x)
sinw(x) = sin (omega x)
The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.
The keyword arguments are optional and may be specified in any order.
They all take the form key=val. The keyword arguments are:
epsrelDesired relative error of approximation. Default is 1d-8.
epsabsDesired absolute error of approximation. Default is 0.
limitSize of internal work array. limit/2 is the maximum number of subintervals to use. Default is 200.
maxp1Maximum number of Chebyshev moments. Must be greater than 0. Default is 100.
limlstUpper bound on the number of cycles. Must be greater than or equal to 3. Default is 10.
quad_qawo returns a list of four elements:
The error code (fourth element of the return value) can have the values:
0no problems were encountered;
1too many sub-intervals were done;
2excessive roundoff error is detected;
3extremely bad integrand behavior occurs;
6if the input is invalid.
Examples:
(%i1) quad_qawo (x^(-1/2)*exp(-2^(-2)*x), x, 1d-8, 20*2^2, 1, cos);
(%o1) [1.376043389877692, 4.72710759424899E-11, 765, 0]
(%i2) rectform (integrate (x^(-1/2)*exp(-2^(-alpha)*x) * cos(x),
x, 0, inf));
alpha/2 - 1/2 2 alpha
sqrt(%pi) 2 sqrt(sqrt(2 + 1) + 1)
(%o2) -----------------------------------------------------
2 alpha
sqrt(2 + 1)
(%i3) ev (%, alpha=2, numer);
(%o3) 1.376043390090716
Categories: Numerical methods · Package quadpack
Integration of w(x) f(x) over a finite interval, where w(x) is a certain algebraic or logarithmic function. A globally adaptive subdivision strategy is applied, with modified Clenshaw-Curtis integration on the subintervals which contain the endpoints of the interval of integration.
quad_qaws computes the integral using the Quadpack QAWS routine:
integrate (f(x)*w(x), x, a, b)
The weight function w is selected by wfun:
1w(x) = (x - a)^alpha (b - x)^beta
2w(x) = (x - a)^alpha (b - x)^beta log(x - a)
3w(x) = (x - a)^alpha (b - x)^beta log(b - x)
4w(x) = (x - a)^alpha (b - x)^beta log(x - a) log(b - x)
The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.
The keyword arguments are optional and may be specified in any order.
They all take the form key=val. The keyword arguments are:
epsrelDesired relative error of approximation. Default is 1d-8.
epsabsDesired absolute error of approximation. Default is 0.
limitSize of internal work array. limitis the maximum number of subintervals to use. Default is 200.
quad_qaws returns a list of four elements:
The error code (fourth element of the return value) can have the values:
0no problems were encountered;
1too many sub-intervals were done;
2excessive roundoff error is detected;
3extremely bad integrand behavior occurs;
6if the input is invalid.
Examples:
(%i1) quad_qaws (1/(x+1+2^(-4)), x, -1, 1, -0.5, -0.5, 1,
'epsabs=1d-9);
(%o1) [8.750097361672832, 1.24321522715422E-10, 170, 0]
(%i2) integrate ((1-x*x)^(-1/2)/(x+1+2^(-alpha)), x, -1, 1);
alpha
Is 4 2 - 1 positive, negative, or zero?
pos;
alpha alpha
2 %pi 2 sqrt(2 2 + 1)
(%o2) -------------------------------
alpha
4 2 + 2
(%i3) ev (%, alpha=4, numer);
(%o3) 8.750097361672829
Categories: Numerical methods · Package quadpack
Integration of a general function over a finite interval.
quad_qagp implements globally adaptive interval subdivision with
extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).
quad_qagp computes the integral
integrate (f(x), x, a, b)
The function to be integrated is f(x), with dependent variable x, and the function is to be integrated between the limits a and b.
The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.
To help the integrator, the user must supply a list of points where the integrand is singular or discontinous.
The keyword arguments are optional and may be specified in any order.
They all take the form key=val. The keyword arguments are:
epsrelDesired relative error of approximation. Default is 1d-8.
epsabsDesired absolute error of approximation. Default is 0.
limitSize of internal work array. limit is the maximum number of subintervals to use. Default is 200.
quad_qagp returns a list of four elements:
The error code (fourth element of the return value) can have the values:
0no problems were encountered;
1too many sub-intervals were done;
2excessive roundoff error is detected;
3extremely bad integrand behavior occurs;
4failed to converge
5integral is probably divergent or slowly convergent
6if the input is invalid.
Examples:
(%i1) quad_qagp(x^3*log(abs((x^2-1)*(x^2-2))),x,0,3,[1,sqrt(2)]); (%o1) [52.74074838347143, 2.6247632689546663e-7, 1029, 0] (%i2) quad_qags(x^3*log(abs((x^2-1)*(x^2-2))), x, 0, 3); (%o2) [52.74074847951494, 4.088443219529836e-7, 1869, 0]
The integrand has singularities at 1 and sqrt(2) so we supply
these points to quad_qagp. We also note that quad_qagp is
more accurate and more efficient that quad_qags.
Categories: Numerical methods · Package quadpack
Control error handling for quadpack. The parameter should be one of the following symbols:
current_errorThe current error number
controlControls if messages are printed or not. If it is set to zero or less, messages are suppressed.
max_messageThe maximum number of times any message is to be printed.
If value is not given, then the current value of the parameter is returned. If value is given, the value of parameter is set to the given value.
Categories: Numerical methods · Package quadpack
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| 20.1 Functions and Variables for Equations |
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Default value: []
%rnum_list is the list of variables introduced in solutions by
solve and algsys. %r variables are added to
%rnum_list in the order they are created. This is convenient for doing
substitutions into the solution later on.
It's recommended to use this list rather than doing concat ('%r, j).
(%i1) solve ([x + y = 3], [x,y]);
(%o1) [[x = 3 - %r1, y = %r1]]
(%i2) %rnum_list;
(%o2) [%r1]
(%i3) sol : solve ([x + 2*y + 3*z = 4], [x,y,z]);
(%o3) [[x = - 2 %r3 - 3 %r2 + 4, y = %r3, z = %r2]]
(%i4) %rnum_list;
(%o4) [%r2, %r3]
(%i5) for i : 1 thru length (%rnum_list) do
sol : subst (t[i], %rnum_list[i], sol)$
(%i6) sol;
(%o6) [[x = - 2 t - 3 t + 4, y = t , z = t ]]
2 1 2 1
Categories: Algebraic equations
Default value: 10^8
algepsilon is used by algsys.
Categories: Algebraic equations
Default value: false
algexact affects the behavior of algsys as follows:
If algexact is true, algsys always calls solve and
then uses realroots on solve's failures.
If algexact is false, solve is called only if the
eliminant was not univariate, or if it was a quadratic or biquadratic.
Thus algexact: true does not guarantee only exact solutions, just that
algsys will first try as hard as it can to give exact solutions, and
only yield approximations when all else fails.
Categories: Algebraic equations
Solves the simultaneous polynomials expr_1, …, expr_m or
polynomial equations eqn_1, …, eqn_m for the variables
x_1, …, x_n. An expression expr is equivalent to an
equation expr = 0. There may be more equations than variables or
vice versa.
algsys returns a list of solutions, with each solution given as a list
of equations stating values of the variables x_1, …, x_n
which satisfy the system of equations. If algsys cannot find a solution,
an empty list [] is returned.
The symbols %r1, %r2, …, are introduced as needed to
represent arbitrary parameters in the solution; these variables are also
appended to the list %rnum_list.
The method is as follows:
solve is called to find an exact
solution.
In some cases, solve is not be able to find a solution, or if it does
the solution may be a very large expression.
If the equation is univariate and is either linear, quadratic, or biquadratic,
then again solve is called if no approximations have been introduced.
If approximations have been introduced or the equation is not univariate and
neither linear, quadratic, or biquadratic, then if the switch
realonly is true, the function realroots is called to find
the real-valued solutions. If realonly is false, then
allroots is called which looks for real and complex-valued solutions.
If algsys produces a solution which has fewer significant digits than
required, the user can change the value of algepsilon to a higher value.
If algexact is set to true, solve will always be called.
When algsys encounters a multivariate equation which contains floating
point approximations (usually due to its failing to find exact solutions at an
earlier stage), then it does not attempt to apply exact methods to such
equations and instead prints the message:
"algsys cannot solve - system too complicated."
Interactions with radcan can produce large or complicated expressions.
In that case, it may be possible to isolate parts of the result with
pickapart or reveal.
Occasionally, radcan may introduce an imaginary unit %i into a
solution which is actually real-valued.
Examples:
(%i1) e1: 2*x*(1 - a1) - 2*(x - 1)*a2;
(%o1) 2 (1 - a1) x - 2 a2 (x - 1)
(%i2) e2: a2 - a1;
(%o2) a2 - a1
(%i3) e3: a1*(-y - x^2 + 1);
2
(%o3) a1 (- y - x + 1)
(%i4) e4: a2*(y - (x - 1)^2);
2
(%o4) a2 (y - (x - 1) )
(%i5) algsys ([e1, e2, e3, e4], [x, y, a1, a2]);
(%o5) [[x = 0, y = %r1, a1 = 0, a2 = 0],
[x = 1, y = 0, a1 = 1, a2 = 1]]
(%i6) e1: x^2 - y^2;
2 2
(%o6) x - y
(%i7) e2: -1 - y + 2*y^2 - x + x^2;
2 2
(%o7) 2 y - y + x - x - 1
(%i8) algsys ([e1, e2], [x, y]);
1 1
(%o8) [[x = - -------, y = -------],
sqrt(3) sqrt(3)
1 1 1 1
[x = -------, y = - -------], [x = - -, y = - -], [x = 1, y = 1]]
sqrt(3) sqrt(3) 3 3
Categories: Algebraic equations
Computes numerical approximations of the real and complex roots of the polynomial expr or polynomial equation eqn of one variable.
The flag polyfactor when true causes allroots to factor
the polynomial over the real numbers if the polynomial is real, or over the
complex numbers, if the polynomial is complex.
allroots may give inaccurate results in case of multiple roots.
If the polynomial is real, allroots (%i*p) may yield
more accurate approximations than allroots (p), as allroots
invokes a different algorithm in that case.
allroots rejects non-polynomials. It requires that the numerator
after rat'ing should be a polynomial, and it requires that the
denominator be at most a complex number. As a result of this allroots
will always return an equivalent (but factored) expression, if
polyfactor is true.
For complex polynomials an algorithm by Jenkins and Traub is used (Algorithm 419, Comm. ACM, vol. 15, (1972), p. 97). For real polynomials the algorithm used is due to Jenkins (Algorithm 493, ACM TOMS, vol. 1, (1975), p.178).
Examples:
(%i1) eqn: (1 + 2*x)^3 = 13.5*(1 + x^5);
3 5
(%o1) (2 x + 1) = 13.5 (x + 1)
(%i2) soln: allroots (eqn);
(%o2) [x = .8296749902129361, x = - 1.015755543828121,
x = .9659625152196369 %i - .4069597231924075,
x = - .9659625152196369 %i - .4069597231924075, x = 1.0]
(%i3) for e in soln
do (e2: subst (e, eqn), disp (expand (lhs(e2) - rhs(e2))));
- 3.5527136788005E-15
- 5.32907051820075E-15
4.44089209850063E-15 %i - 4.88498130835069E-15
- 4.44089209850063E-15 %i - 4.88498130835069E-15
3.5527136788005E-15
(%o3) done
(%i4) polyfactor: true$
(%i5) allroots (eqn);
(%o5) - 13.5 (x - 1.0) (x - .8296749902129361)
2
(x + 1.015755543828121) (x + .8139194463848151 x
+ 1.098699797110288)
Categories: Polynomials · Numerical methods
Computes numerical approximations of the real and complex roots of the polynomial expr or polynomial equation eqn of one variable.
In all respects, bfallroots is identical to allroots except
that bfallroots computes the roots using bigfloats. See
allroots for more information.
Categories: Polynomials · Numerical methods
Default value: true
When backsubst is false, prevents back substitution in
linsolve after the equations have been triangularized. This may
be helpful in very big problems where back substitution would cause
the generation of extremely large expressions.
(%i1) eq1 : x + y + z = 6$ (%i2) eq2 : x - y + z = 2$ (%i3) eq3 : x + y - z = 0$ (%i4) backsubst : false$ (%i5) linsolve ([eq1, eq2, eq3], [x,y,z]); (%o5) [x = z - y, y = 2, z = 3] (%i6) backsubst : true$ (%i7) linsolve ([eq1, eq2, eq3], [x,y,z]); (%o7) [x = 1, y = 2, z = 3]
Categories: Algebraic equations
Default value: true
When breakup is true, solve expresses solutions of cubic
and quartic equations in terms of common subexpressions, which are assigned to
intermediate expression labels (%t1, %t2, etc.).
Otherwise, common subexpressions are not identified.
breakup: true has an effect only when programmode is false.
Examples:
(%i1) programmode: false$
(%i2) breakup: true$
(%i3) solve (x^3 + x^2 - 1);
sqrt(23) 25 1/3
(%t3) (--------- + --)
6 sqrt(3) 54
Solution:
sqrt(3) %i 1
---------- - -
sqrt(3) %i 1 2 2 1
(%t4) x = (- ---------- - -) %t3 + -------------- - -
2 2 9 %t3 3
sqrt(3) %i 1
- ---------- - -
sqrt(3) %i 1 2 2 1
(%t5) x = (---------- - -) %t3 + ---------------- - -
2 2 9 %t3 3
1 1
(%t6) x = %t3 + ----- - -
9 %t3 3
(%o6) [%t4, %t5, %t6]
(%i6) breakup: false$
(%i7) solve (x^3 + x^2 - 1);
Solution:
sqrt(3) %i 1
---------- - -
2 2 sqrt(23) 25 1/3
(%t7) x = --------------------- + (--------- + --)
sqrt(23) 25 1/3 6 sqrt(3) 54
9 (--------- + --)
6 sqrt(3) 54
sqrt(3) %i 1 1
(- ---------- - -) - -
2 2 3
sqrt(23) 25 1/3 sqrt(3) %i 1
(%t8) x = (--------- + --) (---------- - -)
6 sqrt(3) 54 2 2
sqrt(3) %i 1
- ---------- - -
2 2 1
+ --------------------- - -
sqrt(23) 25 1/3 3
9 (--------- + --)
6 sqrt(3) 54
sqrt(23) 25 1/3 1 1
(%t9) x = (--------- + --) + --------------------- - -
6 sqrt(3) 54 sqrt(23) 25 1/3 3
9 (--------- + --)
6 sqrt(3) 54
(%o9) [%t7, %t8, %t9]
Categories: Algebraic equations
dimen is a package for dimensional analysis.
load ("dimen") loads this package.
demo ("dimen") displays a short demonstration.
Categories: Share packages
Default value: true
If set to false within a block will inhibit the display of output
generated by the solve functions called from within the block.
Termination of the block with a dollar sign, $, sets dispflag to
false.
Categories: Algebraic equations · Display flags and variables
Returns [g(t) = ...] or [], depending on whether
or not there exists a rational function g(t) satisfying
eqn, which must be a first order, linear polynomial in (for this case)
g(t) and g(t+1)
(%i1) eqn: (n + 1)*f(n) - (n + 3)*f(n + 1)/(n + 1) =
(n - 1)/(n + 2);
(n + 3) f(n + 1) n - 1
(%o1) (n + 1) f(n) - ---------------- = -----
n + 1 n + 2
(%i2) funcsolve (eqn, f(n));
Dependent equations eliminated: (4 3)
n
(%o2) f(n) = ---------------
(n + 1) (n + 2)
Warning: this is a very rudimentary implementation - many safety checks and obvious generalizations are missing.
Categories: Algebraic equations
Default value: false
When globalsolve is true, solved-for variables are assigned the
solution values found by linsolve, and by solve when solving two
or more linear equations.
When globalsolve is false, solutions found by linsolve and
by solve when solving two or more linear equations are expressed as
equations, and the solved-for variables are not assigned.
When solving anything other than two or more linear equations, solve
ignores globalsolve. Other functions which solve equations (e.g.,
algsys) always ignore globalsolve.
Examples:
(%i1) globalsolve: true$
(%i2) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);
Solution
17
(%t2) x : --
7
1
(%t3) y : - -
7
(%o3) [[%t2, %t3]]
(%i3) x;
17
(%o3) --
7
(%i4) y;
1
(%o4) - -
7
(%i5) globalsolve: false$
(%i6) kill (x, y)$
(%i7) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);
Solution
17
(%t7) x = --
7
1
(%t8) y = - -
7
(%o8) [[%t7, %t8]]
(%i8) x;
(%o8) x
(%i9) y;
(%o9) y
Categories: Linear equations
inteqn is a package for solving integral equations.
load ("inteqn") loads this package.
ie is the integral equation; unk is the unknown function; tech
is the technique to be tried from those given above (tech = first
means: try the first technique which finds a solution; tech = all
means: try all applicable techniques); n is the maximum number of terms
to take for taylor, neumann, firstkindseries, or
fredseries (it is also the maximum depth of recursion for the
differentiation method); guess is the initial guess for neumann or
firstkindseries.
Default values for the 2nd thru 5th parameters are:
unk: p(x), where p is the first function
encountered in an integrand which is unknown to Maxima and x is the
variable which occurs as an argument to the first occurrence of p found
outside of an integral in the case of secondkind equations, or is the
only other variable besides the variable of integration in firstkind
equations. If the attempt to search for x fails, the user will be asked
to supply the independent variable.
tech: first
n: 1
guess: none which will cause neumann and firstkindseries
to use f(x) as an initial guess.
Categories: Integral equations
Default value: true
ieqnprint governs the behavior of the result returned by the
ieqn command. When ieqnprint is false, the lists returned
by the ieqn function are of the form
[solution, technique used, nterms, flag]
where flag is absent if the solution is exact.
Otherwise, it is the word approximate or incomplete corresponding
to an inexact or non-closed form solution, respectively. If a series method was
used, nterms gives the number of terms taken (which could be less than
the n given to ieqn if an error prevented generation of further terms).
Categories: Integral equations
Returns the left-hand side (that is, the first argument) of the expression
expr, when the operator of expr is one of the relational operators
< <= = # equal notequal >= >,
one of the assignment operators := ::= : ::, or a user-defined binary
infix operator, as declared by infix.
When expr is an atom or its operator is something other than the ones
listed above, lhs returns expr.
See also rhs.
Examples:
(%i1) e: aa + bb = cc;
(%o1) bb + aa = cc
(%i2) lhs (e);
(%o2) bb + aa
(%i3) rhs (e);
(%o3) cc
(%i4) [lhs (aa < bb), lhs (aa <= bb), lhs (aa >= bb),
lhs (aa > bb)];
(%o4) [aa, aa, aa, aa]
(%i5) [lhs (aa = bb), lhs (aa # bb), lhs (equal (aa, bb)),
lhs (notequal (aa, bb))];
(%o5) [aa, aa, aa, aa]
(%i6) e1: '(foo(x) := 2*x);
(%o6) foo(x) := 2 x
(%i7) e2: '(bar(y) ::= 3*y);
(%o7) bar(y) ::= 3 y
(%i8) e3: '(x : y);
(%o8) x : y
(%i9) e4: '(x :: y);
(%o9) x :: y
(%i10) [lhs (e1), lhs (e2), lhs (e3), lhs (e4)];
(%o10) [foo(x), bar(y), x, x]
(%i11) infix ("][");
(%o11) ][
(%i12) lhs (aa ][ bb);
(%o12) aa
Categories: Expressions
Solves the list of simultaneous linear equations for the list of variables. The expressions must each be polynomials in the variables and may be equations.
When globalsolve is true, each solved-for variable is bound to
its value in the solution of the equations.
When backsubst is false, linsolve does not carry out back
substitution after the equations have been triangularized. This may be
necessary in very big problems where back substitution would cause the
generation of extremely large expressions.
When linsolve_params is true, linsolve also generates the
%r symbols used to represent arbitrary parameters described in the manual
under algsys. Otherwise, linsolve solves an under-determined
system of equations with some variables expressed in terms of others.
When programmode is false, linsolve displays the solution
with intermediate expression (%t) labels, and returns the list of labels.
(%i1) e1: x + z = y;
(%o1) z + x = y
(%i2) e2: 2*a*x - y = 2*a^2;
2
(%o2) 2 a x - y = 2 a
(%i3) e3: y - 2*z = 2;
(%o3) y - 2 z = 2
(%i4) [globalsolve: false, programmode: true];
(%o4) [false, true]
(%i5) linsolve ([e1, e2, e3], [x, y, z]);
(%o5) [x = a + 1, y = 2 a, z = a - 1]
(%i6) [globalsolve: false, programmode: false];
(%o6) [false, false]
(%i7) linsolve ([e1, e2, e3], [x, y, z]);
Solution
(%t7) z = a - 1
(%t8) y = 2 a
(%t9) x = a + 1
(%o9) [%t7, %t8, %t9]
(%i9) ''%;
(%o9) [z = a - 1, y = 2 a, x = a + 1]
(%i10) [globalsolve: true, programmode: false];
(%o10) [true, false]
(%i11) linsolve ([e1, e2, e3], [x, y, z]);
Solution
(%t11) z : a - 1
(%t12) y : 2 a
(%t13) x : a + 1
(%o13) [%t11, %t12, %t13]
(%i13) ''%;
(%o13) [z : a - 1, y : 2 a, x : a + 1]
(%i14) [x, y, z];
(%o14) [a + 1, 2 a, a - 1]
(%i15) [globalsolve: true, programmode: true];
(%o15) [true, true]
(%i16) linsolve ([e1, e2, e3], '[x, y, z]);
(%o16) [x : a + 1, y : 2 a, z : a - 1]
(%i17) [x, y, z];
(%o17) [a + 1, 2 a, a - 1]
Categories: Linear equations
Default value: true
When linsolvewarn is true, linsolve prints a message
"Dependent equations eliminated".
Categories: Linear equations
Default value: true
When linsolve_params is true, linsolve also generates
the %r symbols used to represent arbitrary parameters described in
the manual under algsys. Otherwise, linsolve solves an
under-determined system of equations with some variables expressed in terms of
others.
Categories: Linear equations
Default value: not_set_yet
multiplicities is set to a list of the multiplicities of the individual
solutions returned by solve or realroots.
Categories: Algebraic equations · Polynomials
Returns the number of real roots of the real univariate polynomial p in
the half-open interval (low, high]. The endpoints of the
interval may be minf or inf.
nroots uses the method of Sturm sequences.
(%i1) p: x^10 - 2*x^4 + 1/2$ (%i2) nroots (p, -6, 9.1); (%o2) 4
Categories: Polynomials · Numerical methods
where p is a polynomial with integer coefficients and n is a
positive integer returns q, a polynomial over the integers, such that
q^n = p or prints an error message indicating that p is not a
perfect nth power. This routine is much faster than factor or even
sqfr.
Categories: Polynomials
Default value: false
The option variable polyfactor when true causes
allroots and bfallroots to factor the polynomial over the real
numbers if the polynomial is real, or over the complex numbers, if the
polynomial is complex.
See allroots for an example.
Categories: Polynomials · Numerical methods
Default value: true
When programmode is true, solve,
realroots, allroots, and linsolve return solutions
as elements in a list.
(Except when backsubst is set to false, in which case
programmode: false is assumed.)
When programmode is false, solve, etc. create intermediate
expression labels %t1, %t2, etc., and assign the solutions to them.
Categories: Algebraic equations · Polynomials
Default value: false
When realonly is true, algsys returns only those solutions
which are free of %i.
Categories: Algebraic equations
Computes rational approximations of the real roots of the polynomial expr
or polynomial equation eqn of one variable, to within a tolerance of
bound. Coefficients of expr or eqn must be literal numbers;
symbol constants such as %pi are rejected.
realroots assigns the multiplicities of the roots it finds
to the global variable multiplicities.
realroots constructs a Sturm sequence to bracket each root, and then
applies bisection to refine the approximations. All coefficients are converted
to rational equivalents before searching for roots, and computations are carried
out by exact rational arithmetic. Even if some coefficients are floating-point
numbers, the results are rational (unless coerced to floats by the
float or numer flags).
When bound is less than 1, all integer roots are found exactly.
When bound is unspecified, it is assumed equal to the global variable
rootsepsilon.
When the global variable programmode is true, realroots
returns a list of the form [x = x_1, x = x_2, ...].
When programmode is false, realroots creates intermediate
expression labels %t1, %t2, …,
assigns the results to them, and returns the list of labels.
Examples:
(%i1) realroots (-1 - x + x^5, 5e-6);
612003
(%o1) [x = ------]
524288
(%i2) ev (%[1], float);
(%o2) x = 1.167303085327148
(%i3) ev (-1 - x + x^5, %);
(%o3) - 7.396496210176905E-6
(%i1) realroots (expand ((1 - x)^5 * (2 - x)^3 * (3 - x)), 1e-20); (%o1) [x = 1, x = 2, x = 3] (%i2) multiplicities; (%o2) [5, 3, 1]
Categories: Polynomials · Numerical methods
Returns the right-hand side (that is, the second argument) of the expression
expr, when the operator of expr is one of the relational operators
< <= = # equal notequal >= >,
one of the assignment operators := ::= : ::, or a user-defined binary
infix operator, as declared by infix.
When expr is an atom or its operator is something other than the ones
listed above, rhs returns 0.
See also lhs.
Examples:
(%i1) e: aa + bb = cc;
(%o1) bb + aa = cc
(%i2) lhs (e);
(%o2) bb + aa
(%i3) rhs (e);
(%o3) cc
(%i4) [rhs (aa < bb), rhs (aa <= bb), rhs (aa >= bb),
rhs (aa > bb)];
(%o4) [bb, bb, bb, bb]
(%i5) [rhs (aa = bb), rhs (aa # bb), rhs (equal (aa, bb)),
rhs (notequal (aa, bb))];
(%o5) [bb, bb, bb, bb]
(%i6) e1: '(foo(x) := 2*x);
(%o6) foo(x) := 2 x
(%i7) e2: '(bar(y) ::= 3*y);
(%o7) bar(y) ::= 3 y
(%i8) e3: '(x : y);
(%o8) x : y
(%i9) e4: '(x :: y);
(%o9) x :: y
(%i10) [rhs (e1), rhs (e2), rhs (e3), rhs (e4)];
(%o10) [2 x, 3 y, y, y]
(%i11) infix ("][");
(%o11) ][
(%i12) rhs (aa ][ bb);
(%o12) bb
Categories: Expressions
Default value: true
rootsconmode governs the behavior of the rootscontract command.
See rootscontract for details.
Categories: Expressions · Simplification flags and variables
Converts products of roots into roots of products. For example,
rootscontract (sqrt(x)*y^(3/2)) yields sqrt(x*y^3).
When radexpand is true and domain is real,
rootscontract converts abs into sqrt, e.g.,
rootscontract (abs(x)*sqrt(y)) yields sqrt(x^2*y).
There is an option rootsconmode affecting rootscontract as
follows:
Problem Value of Result of applying
rootsconmode rootscontract
x^(1/2)*y^(3/2) false (x*y^3)^(1/2)
x^(1/2)*y^(1/4) false x^(1/2)*y^(1/4)
x^(1/2)*y^(1/4) true (x*y^(1/2))^(1/2)
x^(1/2)*y^(1/3) true x^(1/2)*y^(1/3)
x^(1/2)*y^(1/4) all (x^2*y)^(1/4)
x^(1/2)*y^(1/3) all (x^3*y^2)^(1/6)
When rootsconmode is false, rootscontract contracts only
with respect to rational number exponents whose denominators are the same. The
key to the rootsconmode: true examples is simply that 2 divides into 4
but not into 3. rootsconmode: all involves taking the least common
multiple of the denominators of the exponents.
rootscontract uses ratsimp in a manner similar to
logcontract.
Examples:
(%i1) rootsconmode: false$
(%i2) rootscontract (x^(1/2)*y^(3/2));
3
(%o2) sqrt(x y )
(%i3) rootscontract (x^(1/2)*y^(1/4));
1/4
(%o3) sqrt(x) y
(%i4) rootsconmode: true$
(%i5) rootscontract (x^(1/2)*y^(1/4));
(%o5) sqrt(x sqrt(y))
(%i6) rootscontract (x^(1/2)*y^(1/3));
1/3
(%o6) sqrt(x) y
(%i7) rootsconmode: all$
(%i8) rootscontract (x^(1/2)*y^(1/4));
2 1/4
(%o8) (x y)
(%i9) rootscontract (x^(1/2)*y^(1/3));
3 2 1/6
(%o9) (x y )
(%i10) rootsconmode: false$
(%i11) rootscontract (sqrt(sqrt(x) + sqrt(1 + x))
*sqrt(sqrt(1 + x) - sqrt(x)));
(%o11) 1
(%i12) rootsconmode: true$
(%i13) rootscontract (sqrt(5+sqrt(5)) - 5^(1/4)*sqrt(1+sqrt(5)));
(%o13) 0
Categories: Simplification functions
Default value: 1.0e-7
rootsepsilon is the tolerance which establishes the confidence interval
for the roots found by the realroots function.
Categories: Polynomials · Numerical methods
Solves the algebraic equation expr for the variable x and returns a
list of solution equations in x. If expr is not an equation, the
equation expr = 0 is assumed in its place.
x may be a function (e.g. f(x)), or other non-atomic expression
except a sum or product. x may be omitted if expr contains only one
variable. expr may be a rational expression, and may contain
trigonometric functions, exponentials, etc.
The following method is used:
Let E be the expression and X be the variable. If E is linear
in X then it is trivially solved for X. Otherwise if E is of
the form A*X^N + B then the result is (-B/A)^1/N) times the
N'th roots of unity.
If E is not linear in X then the gcd of the exponents of X in
E (say N) is divided into the exponents and the multiplicity of the
roots is multiplied by N. Then solve is called again on the
result. If E factors then solve is called on each of the factors.
Finally solve will use the quadratic, cubic, or quartic formulas where
necessary.
In the case where E is a polynomial in some function of the variable to be
solved for, say F(X), then it is first solved for F(X) (call the
result C), then the equation F(X)=C can be solved for X
provided the inverse of the function F is known.
breakup if false will cause solve to express the solutions
of cubic or quartic equations as single expressions rather than as made
up of several common subexpressions which is the default.
multiplicities - will be set to a list of the multiplicities of the
individual solutions returned by solve, realroots, or
allroots. Try apropos (solve) for the switches which affect
solve. describe may then by used on the individual switch names
if their purpose is not clear.
solve ([eqn_1, ..., eqn_n], [x_1, ..., x_n])
solves a system of simultaneous (linear or non-linear) polynomial equations by
calling linsolve or algsys and returns a list of the solution
lists in the variables. In the case of linsolve this list would contain
a single list of solutions. It takes two lists as arguments. The first list
represents the equations to be solved; the second list is a
list of the unknowns to be determined. If the total number of
variables in the equations is equal to the number of equations, the
second argument-list may be omitted.
When programmode is false, solve displays solutions with
intermediate expression (%t) labels, and returns the list of labels.
When globalsolve is true and the problem is to solve two or more
linear equations, each solved-for variable is bound to its value in the solution
of the equations.
Examples:
(%i1) solve (asin (cos (3*x))*(f(x) - 1), x);
solve: using arc-trig functions to get a solution.
Some solutions will be lost.
%pi
(%o1) [x = ---, f(x) = 1]
6
(%i2) ev (solve (5^f(x) = 125, f(x)), solveradcan);
log(125)
(%o2) [f(x) = --------]
log(5)
(%i3) [4*x^2 - y^2 = 12, x*y - x = 2];
2 2
(%o3) [4 x - y = 12, x y - x = 2]
(%i4) solve (%, [x, y]);
(%o4) [[x = 2, y = 2], [x = .5202594388652008 %i
- .1331240357358706, y = .07678378523787788
- 3.608003221870287 %i], [x = - .5202594388652008 %i
- .1331240357358706, y = 3.608003221870287 %i
+ .07678378523787788], [x = - 1.733751846381093,
y = - .1535675710019696]]
(%i5) solve (1 + a*x + x^3, x);
3
sqrt(3) %i 1 sqrt(4 a + 27) 1 1/3
(%o5) [x = (- ---------- - -) (--------------- - -)
2 2 6 sqrt(3) 2
sqrt(3) %i 1
(---------- - -) a
2 2
- --------------------------, x =
3
sqrt(4 a + 27) 1 1/3
3 (--------------- - -)
6 sqrt(3) 2
3
sqrt(3) %i 1 sqrt(4 a + 27) 1 1/3
(---------- - -) (--------------- - -)
2 2 6 sqrt(3) 2
sqrt(3) %i 1
(- ---------- - -) a
2 2
- --------------------------, x =
3
sqrt(4 a + 27) 1 1/3
3 (--------------- - -)
6 sqrt(3) 2
3
sqrt(4 a + 27) 1 1/3 a
(--------------- - -) - --------------------------]
6 sqrt(3) 2 3
sqrt(4 a + 27) 1 1/3
3 (--------------- - -)
6 sqrt(3) 2
(%i6) solve (x^3 - 1);
sqrt(3) %i - 1 sqrt(3) %i + 1
(%o6) [x = --------------, x = - --------------, x = 1]
2 2
(%i7) solve (x^6 - 1);
sqrt(3) %i + 1 sqrt(3) %i - 1
(%o7) [x = --------------, x = --------------, x = - 1,
2 2
sqrt(3) %i + 1 sqrt(3) %i - 1
x = - --------------, x = - --------------, x = 1]
2 2
(%i8) ev (x^6 - 1, %[1]);
6
(sqrt(3) %i + 1)
(%o8) ----------------- - 1
64
(%i9) expand (%);
(%o9) 0
(%i10) x^2 - 1;
2
(%o10) x - 1
(%i11) solve (%, x);
(%o11) [x = - 1, x = 1]
(%i12) ev (%th(2), %[1]);
(%o12) 0
The symbols %r are used to denote arbitrary constants in a solution.
(%i1) solve([x+y=1,2*x+2*y=2],[x,y]); solve: dependent equations eliminated: (2) (%o1) [[x = 1 - %r1, y = %r1]]
See algsys and %rnum_list for more information.
Categories: Algebraic equations
Default value: true
When solvedecomposes is true, solve calls
polydecomp if asked to solve polynomials.
Categories: Algebraic equations
Default value: false
When solveexplicit is true, inhibits solve from returning
implicit solutions, that is, solutions of the form F(x) = 0 where
F is some function.
Categories: Algebraic equations
Default value: true
When solvefactors is false, solve does not try to factor
the expression. The false setting may be desired in some cases where
factoring is not necessary.
Categories: Algebraic equations
Default value: true
When solvenullwarn is true, solve prints a warning message
if called with either a null equation list or a null variable list. For
example, solve ([], []) would print two warning messages and return
[].
Categories: Algebraic equations
Default value: false
When solveradcan is true, solve calls radcan
which makes solve slower but will allow certain problems containing
exponentials and logarithms to be solved.
Categories: Algebraic equations
Default value: true
When solvetrigwarn is true, solve may print a message
saying that it is using inverse trigonometric functions to solve the equation,
and thereby losing solutions.
Categories: Algebraic equations
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| 21.1 Introduction to Differential Equations | ||
| 21.2 Functions and Variables for Differential Equations |
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This section describes the functions available in Maxima to obtain
analytic solutions for some specific types of first and second-order
equations. To obtain a numerical solution for a system of differential
equations, see the additional package dynamics. For graphical
representations in phase space, see the additional package
plotdf.
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Solves a boundary value problem for a second order differential equation.
Here: solution is a general solution to the equation, as found by
ode2; xval1 specifies the value of the independent variable
in a first point, in the form x = x1, and yval1
gives the value of the dependent variable in that point, in the form
y = y1. The expressions xval2 and yval2
give the values for these variables at a second point, using the same
form.
See ode2 for an example of its usage.
Categories: Differential equations
The function desolve solves systems of linear ordinary
differential equations using Laplace transform. Here the eqn's
are differential equations in the dependent variables x_1, ...,
x_n. The functional dependence of x_1, ..., x_n on an
independent variable, for instance x, must be explicitly indicated
in the variables and its derivatives. For example, this would not be the
correct way to define two equations:
eqn_1: 'diff(f,x,2) = sin(x) + 'diff(g,x); eqn_2: 'diff(f,x) + x^2 - f = 2*'diff(g,x,2);
The correct way would be:
eqn_1: 'diff(f(x),x,2) = sin(x) + 'diff(g(x),x); eqn_2: 'diff(f(x),x) + x^2 - f(x) = 2*'diff(g(x),x,2);
The call to the function desolve would then be
desolve([eqn_1, eqn_2], [f(x),g(x)]);
If initial conditions at x=0 are known, they can be supplied before
calling desolve by using atvalue.
(%i1) 'diff(f(x),x)='diff(g(x),x)+sin(x);
d d
(%o1) -- (f(x)) = -- (g(x)) + sin(x)
dx dx
(%i2) 'diff(g(x),x,2)='diff(f(x),x)-cos(x);
2
d d
(%o2) --- (g(x)) = -- (f(x)) - cos(x)
2 dx
dx
(%i3) atvalue('diff(g(x),x),x=0,a);
(%o3) a
(%i4) atvalue(f(x),x=0,1);
(%o4) 1
(%i5) desolve([%o1,%o2],[f(x),g(x)]);
x
(%o5) [f(x) = a %e - a + 1, g(x) =
x
cos(x) + a %e - a + g(0) - 1]
(%i6) [%o1,%o2],%o5,diff;
x x x x
(%o6) [a %e = a %e , a %e - cos(x) = a %e - cos(x)]
If desolve cannot obtain a solution, it returns false.
Categories: Differential equations · Laplace transform
Solves initial value problems for first order differential equations.
Here solution is a general solution to the equation, as found by
ode2, xval gives an initial value for the independent
variable in the form x = x0, and yval gives the
initial value for the dependent variable in the form y =
y0.
See ode2 for an example of its usage.
Categories: Differential equations
Solves initial value problems for second-order differential equations.
Here solution is a general solution to the equation, as found by
ode2, xval gives the initial value for the independent
variable in the form x = x0, yval gives the
initial value of the dependent variable in the form y =
y0, and dval gives the initial value for the first
derivative of the dependent variable with respect to independent
variable, in the form diff(y,x) = dy0
(diff does not have to be quoted).
See ode2 for an example of its usage.
Categories: Differential equations
The function ode2 solves an ordinary differential equation (ODE)
of first or second order. It takes three arguments: an ODE given by
eqn, the dependent variable dvar, and the independent
variable ivar. When successful, it returns either an explicit or
implicit solution for the dependent variable. %c is used to
represent the integration constant in the case of first-order equations,
and %k1 and %k2 the constants for second-order
equations. The dependence of the dependent variable on the independent
variable does not have to be written explicitly, as in the case of
desolve, but the independent variable must always be given as the
third argument.
If ode2 cannot obtain a solution for whatever reason, it returns
false, after perhaps printing out an error message. The methods
implemented for first order equations in the order in which they are
tested are: linear, separable, exact - perhaps requiring an integrating
factor, homogeneous, Bernoulli's equation, and a generalized homogeneous
method. The types of second-order equations which can be solved are:
constant coefficients, exact, linear homogeneous with non-constant
coefficients which can be transformed to constant coefficients, the
Euler or equi-dimensional equation, equations solvable by the method of
variation of parameters, and equations which are free of either the
independent or of the dependent variable so that they can be reduced to
two first order linear equations to be solved sequentially.
In the course of solving ODE's, several variables are set purely for
informational purposes: method denotes the method of solution
used (e.g., linear), intfactor denotes any integrating
factor used, odeindex denotes the index for Bernoulli's method or
for the generalized homogeneous method, and yp denotes the
particular solution for the variation of parameters technique.
In order to solve initial value problems (IVP) functions ic1 and
ic2 are available for first and second order equations, and to
solve second-order boundary value problems (BVP) the function bc2
can be used.
Example:
(%i1) x^2*'diff(y,x) + 3*y*x = sin(x)/x;
2 dy sin(x)
(%o1) x -- + 3 x y = ------
dx x
(%i2) ode2(%,y,x);
%c - cos(x)
(%o2) y = -----------
3
x
(%i3) ic1(%o2,x=%pi,y=0);
cos(x) + 1
(%o3) y = - ----------
3
x
(%i4) 'diff(y,x,2) + y*'diff(y,x)^3 = 0;
2
d y dy 3
(%o4) --- + y (--) = 0
2 dx
dx
(%i5) ode2(%,y,x);
3
y + 6 %k1 y
(%o5) ------------ = x + %k2
6
(%i6) ratsimp(ic2(%o5,x=0,y=0,'diff(y,x)=2));
3
2 y - 3 y
(%o6) - ---------- = x
6
(%i7) bc2(%o5,x=0,y=1,x=1,y=3);
3
y - 10 y 3
(%o7) --------- = x - -
6 2
Categories: Differential equations
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The fft package comprises functions for the numerical (not symbolic)
computation of the fast Fourier transform.
Categories: Fourier transform · Numerical methods · Share packages · Package fft
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Translates complex values of the form r %e^(%i t) to the form
a + b %i, where r is the magnitude and t is the phase.
r and t are 1-dimensional arrays of the same size.
The array size need not be a power of 2.
The original values of the input arrays are
replaced by the real and imaginary parts, a and b, on return.
The outputs are calculated as
a = r cos(t) b = r sin(t)
polartorect is the inverse function of recttopolar.
load(fft) loads this function. See also fft.
Categories: Package fft · Complex variables
Translates complex values of the form a + b %i to the form
r %e^(%i t), where a is the real part and b is the imaginary
part. a and b are 1-dimensional arrays of the same size.
The array size need not be a power of 2.
The original values of the input arrays are
replaced by the magnitude and angle, r and t, on return.
The outputs are calculated as
r = sqrt(a^2 + b^2) t = atan2(b, a)
The computed angle is in the range -%pi to %pi.
recttopolar is the inverse function of polartorect.
load(fft) loads this function. See also fft.
Categories: Package fft · Complex variables
Computes the inverse complex fast Fourier transform.
y is a list or array (named or unnamed) which contains the data to
transform. The number of elements must be a power of 2.
The elements must be literal numbers (integers, rationals, floats, or bigfloats)
or symbolic constants,
or expressions a + b*%i where a and b are literal numbers
or symbolic constants.
inverse_fft returns a new object of the same type as y,
which is not modified.
Results are always computed as floats
or expressions a + b*%i where a and b are floats.
If bigfloat precision is needed the function bf_inverse_fft can
be used instead as a drop-in replacement of inverse_fft that is
slower, but supports bfloats.
The inverse discrete Fourier transform is defined as follows.
Let x be the output of the inverse transform.
Then for j from 0 through n - 1,
x[j] = sum(y[k] exp(-2 %i %pi j k / n), k, 0, n - 1)
As there are various sign and normalization conventions possible, this definition of the transform may differ from that used by other mathematical software.
load(fft) loads this function.
See also fft (forward transform), recttopolar, and
polartorect.
Examples:
Real data.
(%i1) load (fft) $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
(%i4) L1 : inverse_fft (L);
(%o4) [0.0, 14.49 %i - .8284, 0.0, 2.485 %i + 4.828, 0.0,
4.828 - 2.485 %i, 0.0, - 14.49 %i - .8284]
(%i5) L2 : fft (L1);
(%o5) [1.0, 2.0 - 2.168L-19 %i, 3.0 - 7.525L-20 %i,
4.0 - 4.256L-19 %i, - 1.0, 2.168L-19 %i - 2.0,
7.525L-20 %i - 3.0, 4.256L-19 %i - 4.0]
(%i6) lmax (abs (L2 - L));
(%o6) 3.545L-16
Complex data.
(%i1) load (fft) $ (%i2) fpprintprec : 4 $ (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $ (%i4) L1 : inverse_fft (L); (%o4) [4.0, 2.711L-19 %i + 4.0, 2.0 %i - 2.0, - 2.828 %i - 2.828, 0.0, 5.421L-20 %i + 4.0, - 2.0 %i - 2.0, 2.828 %i + 2.828] (%i5) L2 : fft (L1); (%o5) [4.066E-20 %i + 1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i, 1.55L-19 %i - 1.0, - 4.066E-20 %i - 1.0, 1.0 - 1.0 %i, 1.0 %i + 1.0, 1.0 - 7.368L-20 %i] (%i6) lmax (abs (L2 - L)); (%o6) 6.841L-17
Categories: Package fft
Computes the complex fast Fourier transform.
x is a list or array (named or unnamed) which contains the data to
transform. The number of elements must be a power of 2.
The elements must be literal numbers (integers, rationals, floats, or bigfloats)
or symbolic constants,
or expressions a + b*%i where a and b are literal numbers
or symbolic constants.
fft returns a new object of the same type as x,
which is not modified.
Results are always computed as floats
or expressions a + b*%i where a and b are floats.
If bigfloat precision is needed the function bf_fft can be used
instead as a drop-in replacement of fft that is slower, but
supports bfloats. In addition if it is known that the input consists
of only real values (no imaginary parts), real_fft can be used
which is potentially faster.
The discrete Fourier transform is defined as follows.
Let y be the output of the transform.
Then for k from 0 through n - 1,
y[k] = (1/n) sum(x[j] exp(+2 %i %pi j k / n), j, 0, n - 1)
As there are various sign and normalization conventions possible, this definition of the transform may differ from that used by other mathematical software.
When the data x are real,
real coefficients a and b can be computed such that
x[j] = sum(a[k]*cos(2*%pi*j*k/n)+b[k]*sin(2*%pi*j*k/n), k, 0, n/2)
with
a[0] = realpart (y[0]) b[0] = 0
and, for k from 1 through n/2 - 1,
a[k] = realpart (y[k] + y[n - k]) b[k] = imagpart (y[n - k] - y[k])
and
a[n/2] = realpart (y[n/2]) b[n/2] = 0
load(fft) loads this function.
See also inverse_fft (inverse transform), recttopolar, and
polartorect.. See real_fft for FFTs of a real-valued
input, and bf_fft and bf_real_fft for operations on
bigfloat values.
Examples:
Real data.
(%i1) load (fft) $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
(%i4) L1 : fft (L);
(%o4) [0.0, - 1.811 %i - .1036, 0.0, .6036 - .3107 %i, 0.0,
.3107 %i + .6036, 0.0, 1.811 %i - .1036]
(%i5) L2 : inverse_fft (L1);
(%o5) [1.0, 2.168L-19 %i + 2.0, 7.525L-20 %i + 3.0,
4.256L-19 %i + 4.0, - 1.0, - 2.168L-19 %i - 2.0,
- 7.525L-20 %i - 3.0, - 4.256L-19 %i - 4.0]
(%i6) lmax (abs (L2 - L));
(%o6) 3.545L-16
Complex data.
(%i1) load (fft) $ (%i2) fpprintprec : 4 $ (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $ (%i4) L1 : fft (L); (%o4) [0.5, .3536 %i + .3536, - 0.25 %i - 0.25, 0.5 - 6.776L-21 %i, 0.0, - .3536 %i - .3536, 0.25 %i - 0.25, 0.5 - 3.388L-20 %i] (%i5) L2 : inverse_fft (L1); (%o5) [1.0 - 4.066E-20 %i, 1.0 %i + 1.0, 1.0 - 1.0 %i, - 1.008L-19 %i - 1.0, 4.066E-20 %i - 1.0, 1.0 - 1.0 %i, 1.0 %i + 1.0, 1.947L-20 %i + 1.0] (%i6) lmax (abs (L2 - L)); (%o6) 6.83L-17
Computation of sine and cosine coefficients.
(%i1) load (fft) $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, 5, 6, 7, 8] $
(%i4) n : length (L) $
(%i5) x : make_array (any, n) $
(%i6) fillarray (x, L) $
(%i7) y : fft (x) $
(%i8) a : make_array (any, n/2 + 1) $
(%i9) b : make_array (any, n/2 + 1) $
(%i10) a[0] : realpart (y[0]) $
(%i11) b[0] : 0 $
(%i12) for k : 1 thru n/2 - 1 do
(a[k] : realpart (y[k] + y[n - k]),
b[k] : imagpart (y[n - k] - y[k]));
(%o12) done
(%i13) a[n/2] : y[n/2] $
(%i14) b[n/2] : 0 $
(%i15) listarray (a);
(%o15) [4.5, - 1.0, - 1.0, - 1.0, - 0.5]
(%i16) listarray (b);
(%o16) [0, - 2.414, - 1.0, - .4142, 0]
(%i17) f(j) := sum (a[k]*cos(2*%pi*j*k/n) + b[k]*sin(2*%pi*j*k/n),
k, 0, n/2) $
(%i18) makelist (float (f (j)), j, 0, n - 1);
(%o18) [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]
Categories: Package fft
Computes the fast Fourier transform of a real-valued sequence
x. This is equivalent to performing fft(x), except that
only the first N/2+1 results are returned, where N is
the length of x. N must be power of two.
No check is made that x contains only real values.
The symmetry properites of the Fourier transform of real sequences to
reduce he complexity. In particular the first and last output values
of real_fft are purely real. For larger sequencyes, real_fft
may be computed more quickly than fft.
Since the output length is short, the normal inverse_fft cannot
be directly used. Use inverse_real_fft to compute the inverse.
Computes the inverse Fourier transfrom of y, which must have a
length of N/2+1 where N is a power of two. That is, the
input x is expected to be the output of real_fft.
No check is made to ensure that the input has the correct format. (The first and last elements must be purely real.)
Categories: Package fft
Computes the inverse complex fast Fourier transform. This is the
bigfloat version of inverse_fft that converts the input to
bigfloats and returns a bigfloat result.
Computes the forward complex fast Fourier transform. This is the
bigfloat version of fft that converts the input to
bigfloats and returns a bigfloat result.
Computes the forward fast Fourier transform of a real-valued input
returning a bigfloat result. This is the bigfloat version of
real_fft.
Categories: Package fft
Computes the inverse fast Fourier transfrom with a real-valued
bigfloat output. This is the bigfloat version of inverse_real_fft.
Categories: Package fft
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Returns a rearranged representation of expr as in Horner's rule, using
x as the main variable if it is specified. x may be omitted in
which case the main variable of the canonical rational expression form of
expr is used.
horner sometimes improves stability if expr is
to be numerically evaluated. It is also useful if Maxima is used to
generate programs to be run in Fortran. See also stringout.
(%i1) expr: 1e-155*x^2 - 5.5*x + 5.2e155;
2
(%o1) 1.e-155 x - 5.5 x + 5.2e+155
(%i2) expr2: horner (%, x), keepfloat: true;
(%o2) 1.0 ((1.e-155 x - 5.5) x + 5.2e+155)
(%i3) ev (expr, x=1e155);
Maxima encountered a Lisp error:
arithmetic error FLOATING-POINT-OVERFLOW signalled
Automatically continuing.
To enable the Lisp debugger set *debugger-hook* to nil.
(%i4) ev (expr2, x=1e155);
(%o4) 7.00000000000001e+154
Categories: Numerical methods
Finds a root of the expression expr or the function f over the
closed interval [a, b]. The expression expr may be an
equation, in which case find_root seeks a root of
lhs(expr) - rhs(expr).
Given that Maxima can evaluate expr or f over
[a, b] and that expr or f is continuous,
find_root is guaranteed to find the root,
or one of the roots if there is more than one.
find_root initially applies binary search.
If the function in question appears to be smooth enough,
find_root applies linear interpolation instead.
bf_find_root is a bigfloat version of find_root. The
function is computed using bigfloat arithmetic and a bigfloat result
is returned. Otherwise, bf_find_root is identical to
find_root, and the following description is equally applicable
to bf_find_root.
The accuracy of find_root is governed by abserr and
relerr, which are optional keyword arguments to
find_root. These keyword arguments take the form
key=val. The keyword arguments are
abserrDesired absolute error of function value at root. Default is
find_root_abs.
relerrDesired relative error of root. Default is find_root_rel.
find_root stops when the function in question evaluates to
something less than or equal to abserr, or if successive
approximants x_0, x_1 differ by no more than relerr
* max(abs(x_0), abs(x_1)). The default values of
find_root_abs and find_root_rel are both zero.
find_root expects the function in question to have a different sign at
the endpoints of the search interval.
When the function evaluates to a number at both endpoints
and these numbers have the same sign,
the behavior of find_root is governed by find_root_error.
When find_root_error is true,
find_root prints an error message.
Otherwise find_root returns the value of find_root_error.
The default value of find_root_error is true.
If f evaluates to something other than a number at any step in the search
algorithm, find_root returns a partially-evaluated find_root
expression.
The order of a and b is ignored; the region in which a root is sought is [min(a, b), max(a, b)].
Examples:
(%i1) f(x) := sin(x) - x/2;
x
(%o1) f(x) := sin(x) - -
2
(%i2) find_root (sin(x) - x/2, x, 0.1, %pi);
(%o2) 1.895494267033981
(%i3) find_root (sin(x) = x/2, x, 0.1, %pi);
(%o3) 1.895494267033981
(%i4) find_root (f(x), x, 0.1, %pi);
(%o4) 1.895494267033981
(%i5) find_root (f, 0.1, %pi);
(%o5) 1.895494267033981
(%i6) find_root (exp(x) = y, x, 0, 100);
x
(%o6) find_root(%e = y, x, 0.0, 100.0)
(%i7) find_root (exp(x) = y, x, 0, 100), y = 10;
(%o7) 2.302585092994046
(%i8) log (10.0);
(%o8) 2.302585092994046
(%i9) fpprec:32;
(%o9) 32
(%i10) bf_find_root (exp(x) = y, x, 0, 100), y = 10;
(%o10) 2.3025850929940456840179914546844b0
(%i11) log(10b0);
(%o11) 2.3025850929940456840179914546844b0
Categories: Algebraic equations · Numerical methods
Returns an approximate solution of expr = 0 by Newton's method,
considering expr to be a function of one variable, x.
The search begins with x = x_0
and proceeds until abs(expr) < eps
(with expr evaluated at the current value of x).
newton allows undefined variables to appear in expr,
so long as the termination test abs(expr) < eps evaluates
to true or false.
Thus it is not necessary that expr evaluate to a number.
load(newton1) loads this function.
See also realroots, allroots, find_root and
mnewton
Examples:
(%i1) load (newton1);
(%o1) /maxima/share/numeric/newton1.mac
(%i2) newton (cos (u), u, 1, 1/100);
(%o2) 1.570675277161251
(%i3) ev (cos (u), u = %);
(%o3) 1.2104963335033529e-4
(%i4) assume (a > 0);
(%o4) [a > 0]
(%i5) newton (x^2 - a^2, x, a/2, a^2/100);
(%o5) 1.00030487804878 a
(%i6) ev (x^2 - a^2, x = %);
2
(%o6) 6.098490481853958e-4 a
Categories: Algebraic equations · Numerical methods
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The Ordinary Differential Equations (ODE) solved by the functions in this section should have the form,
dy
-- = F(x,y)
dx
which is a first-order ODE. Higher order differential equations of order n must be written as a system of n first-order equations of that kind. For instance, a second-order ODE should be written as a system of two equations
dx dy
-- = G(x,y,t) -- = F(x,y,t)
dt dt
The first argument in the functions will be a list with the expressions on the right-side of the ODE's. The variables whose derivatives are represented by those expressions should be given in a second list. In the case above those variables are x and y. The independent variable, t in the examples above, might be given in a separated option. If the expressions given do not depend on that independent variable, the system is called autonomous.
Categories: Differential equations · Numerical methods · Plotting
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The function plotdf creates a two-dimensional plot of the direction
field (also called slope field) for a first-order Ordinary Differential
Equation (ODE) or a system of two autonomous first-order ODE's.
Plotdf requires Xmaxima, even if its run from a Maxima session in a console, since the plot will be created by the Tk scripts in Xmaxima. If Xmaxima is not installed plotdf will not work.
dydx, dxdt and dydt are expressions that depend on
x and y. dvdu, dudt and dvdt are
expressions that depend on u and v. In addition to those two
variables, the expressions can also depend on a set of parameters, with
numerical values given with the parameters option (the option
syntax is given below), or with a range of allowed values specified by a
sliders option.
Several other options can be given within the command, or selected in
the menu. Integral curves can be obtained by clicking on the plot, or
with the option trajectory_at. The direction of the integration
can be controlled with the direction option, which can have
values of forward, backward or both. The number of
integration steps is given by nsteps; at each integration
step the time increment will be adjusted automatically to produce
displacements much smaller than the size of the plot window. The
numerical method used is 4th order Runge-Kutta with variable time steps.
Plot window menu:
The menu bar of the plot window has the following seven icons:
An X. Can be used to close the plot window.
A wrench and a screwdriver. Opens the configuration menu with several fields that show the ODE(s) in use and various other settings. If a pair of coordinates are entered in the field Trajectory at and the enter key is pressed, a new integral curve will be shown, in addition to the ones already shown.
Two arrows following a circle. Replots the direction field with the new settings defined in the configuration menu and replots only the last integral curve that was previously plotted.
Hard disk drive with an arrow. Used to save a copy of the plot, in Postscript format, in the file specified in a field of the box that appears when that icon is clicked.
Magnifying glass with a plus sign. Zooms in the plot.
Magnifying glass with a minus sign. Zooms out the plot. The plot can be displaced by holding down the right mouse button while the mouse is moved.
Icon of a plot. Opens another window with a plot of the two variables in terms of time, for the last integral curve that was plotted.
Plot options:
Options can also be given within the plotdf itself, each one being
a list of two or more elements. The first element in each option is the name
of the option, and the remainder is the value or values assigned to the
option.
The options which are recognized by plotdf are the following:
forward, to make the independent variable increase
nsteps times, with increments tstep, backward, to
make the independent variable decrease, or both that will lead to
an integral curve that extends nsteps forward, and nsteps
backward. The keywords right and left can be used as
synonyms for forward and backward.
The default value is both.
versus_t is given any value
different from 0, the second plot window will be displayed. The second
plot window includes another menu, similar to the menu of the main plot
window.
The default value is 0.
name=value.
name=min:max
Examples:
(%i1) plotdf(exp(-x)+y,[trajectory_at,2,-0.1])$
(%i1) plotdf(x-y^2,[xfun,"sqrt(x);-sqrt(x)"],
[trajectory_at,-1,3], [direction,forward],
[y,-5,5], [x,-4,16])$
The graph also shows the function y = sqrt(x).
(%i1) plotdf([v,-k*z/m], [z,v], [parameters,"m=2,k=2"],
[sliders,"m=1:5"], [trajectory_at,6,0])$
(%i1) plotdf([y,-(k*x + c*y + b*x^3)/m],
[parameters,"k=-1,m=1.0,c=0,b=1"],
[sliders,"k=-2:2,m=-1:1"],[tstep,0.1])$
(%i1) plotdf([w,-g*sin(a)/l - b*w/m/l], [a,w],
[parameters,"g=9.8,l=0.5,m=0.3,b=0.05"],
[trajectory_at,1.05,-9],[tstep,0.01],
[a,-10,2], [w,-14,14], [direction,forward],
[nsteps,300], [sliders,"m=0.1:1"], [versus_t,1])$
Categories: Differential equations · Plotting · Numerical methods
Plots equipotential curves for exp, which should be an expression depending on two variables. The curves are obtained by integrating the differential equation that define the orthogonal trajectories to the solutions of the autonomous system obtained from the gradient of the expression given. The plot can also show the integral curves for that gradient system (option fieldlines).
This program also requires Xmaxima, even if its run from a Maxima session in a console, since the plot will be created by the Tk scripts in Xmaxima. By default, the plot region will be empty until the user clicks in a point (or gives its coordinate with in the set-up menu or via the trajectory_at option).
Most options accepted by plotdf can also be used for ploteq and the plot interface is the same that was described in plotdf.
Example:
(%i1) V: 900/((x+1)^2+y^2)^(1/2)-900/((x-1)^2+y^2)^(1/2)$ (%i2) ploteq(V,[x,-2,2],[y,-2,2],[fieldlines,"blue"])$
Clicking on a point will plot the equipotential curve that passes by that point (in red) and the orthogonal trajectory (in blue).
Categories: Differential equations · Plotting · Numerical methods
The first form solves numerically one first-order ordinary differential equation, and the second form solves a system of m of those equations, using the 4th order Runge-Kutta method. var represents the dependent variable. ODE must be an expression that depends only on the independent and dependent variables and defines the derivative of the dependent variable with respect to the independent variable.
The independent variable is specified with domain, which must be a
list of four elements as, for instance:
[t, 0, 10, 0.1]
the first element of the list identifies the independent variable, the second and third elements are the initial and final values for that variable, and the last element sets the increments that should be used within that interval.
If m equations are going to be solved, there should be m
dependent variables v1, v2, ..., vm. The initial values
for those variables will be init1, init2, ..., initm.
There will still be just one independent variable defined by domain,
as in the previous case. ODE1, ..., ODEm are the expressions
that define the derivatives of each dependent variable in
terms of the independent variable. The only variables that may appear in
those expressions are the independent variable and any of the dependent
variables. It is important to give the derivatives ODE1, ...,
ODEm in the list in exactly the same order used for the dependent
variables; for instance, the third element in the list will be interpreted
as the derivative of the third dependent variable.
The program will try to integrate the equations from the initial value of the independent variable until its last value, using constant increments. If at some step one of the dependent variables takes an absolute value too large, the integration will be interrupted at that point. The result will be a list with as many elements as the number of iterations made. Each element in the results list is itself another list with m+1 elements: the value of the independent variable, followed by the values of the dependent variables corresponding to that point.
To solve numerically the differential equation
dx/dt = t - x^2
With initial value x(t=0) = 1, in the interval of t from 0 to 8 and with increments of 0.1 for t, use:
(%i1) results: rk(t-x^2,x,1,[t,0,8,0.1])$ (%i2) plot2d ([discrete, results])$
the results will be saved in the list results and the plot will show the solution obtained, with t on the horizontal axis and x on the vertical axis.
To solve numerically the system:
dx/dt = 4-x^2-4*y^2 dy/dt = y^2-x^2+1
for t between 0 and 4, and with values of -1.25 and 0.75 for x and y at t=0:
(%i1) sol: rk([4-x^2-4*y^2,y^2-x^2+1],[x,y],[-1.25,0.75],[t,0,4,0.02])$ (%i2) plot2d ([discrete,makelist([p[1],p[3]],p,sol)], [xlabel,"t"],[ylabel,"y"])$
The plot will show the solution for variable y as a function of t.
Categories: Differential equations · Numerical methods
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| 23.1 Introduction to Matrices and Linear Algebra | ||
| 23.2 Functions and Variables for Matrices and Linear Algebra |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 23.1.1 Dot | ||
| 23.1.2 Vectors | ||
| 23.1.3 eigen |
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The operator . represents noncommutative multiplication and scalar
product. When the operands are 1-column or 1-row matrices a and
b, the expression a.b is equivalent to
sum (a[i]*b[i], i, 1, length(a)). If a and b are not
complex, this is the scalar product, also called the inner product or dot
product, of a and b. The scalar product is defined as
conjugate(a).b when a and b are complex;
innerproduct in the eigen package provides the complex scalar
product.
When the operands are more general matrices,
the product is the matrix product a and b.
The number of rows of b must equal the number of columns of a,
and the result has number of rows equal to the number of rows of a
and number of columns equal to the number of columns of b.
To distinguish . as an arithmetic operator from the decimal point in a
floating point number, it may be necessary to leave spaces on either side.
For example, 5.e3 is 5000.0 but 5 . e3 is 5
times e3.
There are several flags which govern the simplification of expressions
involving ., namely dot0nscsimp, dot0simp,
dot1simp, dotassoc, dotconstrules,
dotdistrib, dotexptsimp, dotident, and
dotscrules.
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vect is a package of functions for vector analysis. load ("vect")
loads this package, and demo ("vect") displays a demonstration.
The vector analysis package can combine and simplify symbolic expressions including dot products and cross products, together with the gradient, divergence, curl, and Laplacian operators. The distribution of these operators over sums or products is governed by several flags, as are various other expansions, including expansion into components in any specific orthogonal coordinate systems. There are also functions for deriving the scalar or vector potential of a field.
The vect package contains these functions:
vectorsimp, scalefactors, express,
potential, and vectorpotential.
By default the vect package does not declare the dot operator to be a
commutative operator. To get a commutative dot operator ., the command
declare(".", commutative) must be executed.
Categories: Vectors · Share packages · Package vect
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The package eigen contains several functions devoted to the
symbolic computation of eigenvalues and eigenvectors.
Maxima loads the package automatically if one of the functions
eigenvalues or eigenvectors is invoked.
The package may be loaded explicitly as load ("eigen").
demo ("eigen") displays a demonstration of the capabilities
of this package.
batch ("eigen") executes the same demonstration,
but without the user prompt between successive computations.
The functions in the eigen package are:
innerproduct, unitvector, columnvector,
gramschmidt, eigenvalues,
eigenvectors, uniteigenvectors, and
similaritytransform.
Categories: Vectors · Matrices · Share packages · Package eigen
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Appends the column(s) given by the one or more lists (or matrices) onto the matrix M.
Categories: Matrices
Appends the row(s) given by the one or more lists (or matrices) onto the matrix M.
Categories: Matrices
Returns the adjoint of the matrix M. The adjoint matrix is the transpose of the matrix of cofactors of M.
Categories: Matrices
Returns the augmented coefficient matrix for the variables x_1, …, x_n of the system of linear equations eqn_1, …, eqn_m. This is the coefficient matrix with a column adjoined for the constant terms in each equation (i.e., those terms not dependent upon x_1, …, x_n).
(%i1) m: [2*x - (a - 1)*y = 5*b, c + b*y + a*x = 0]$
(%i2) augcoefmatrix (m, [x, y]);
[ 2 1 - a - 5 b ]
(%o2) [ ]
[ a b c ]
Categories: Linear equations · Matrices
Returns a n by m Cauchy matrix with the elements a[i,j]
= 1/(x_i+y_i). The second argument of cauchy_matrix is
optional. For this case the elements of the Cauchy matrix are
a[i,j] = 1/(x_i+x_j).
Remark: In the literature the Cauchy matrix can be found defined in two forms. A second definition is a[i,j] = 1/(x_i-y_i).
Examples:
(%i1) cauchy_matrix([x1,x2],[y1,y2]);
[ 1 1 ]
[ ------- ------- ]
[ y1 + x1 y2 + x1 ]
(%o1) [ ]
[ 1 1 ]
[ ------- ------- ]
[ y1 + x2 y2 + x2 ]
(%i2) cauchy_matrix([x1,x2]);
[ 1 1 ]
[ ---- ------- ]
[ 2 x1 x2 + x1 ]
(%o2) [ ]
[ 1 1 ]
[ ------- ---- ]
[ x2 + x1 2 x2 ]
Categories: Matrices
Returns the characteristic polynomial for the matrix M
with respect to variable x. That is,
determinant (M - diagmatrix (length (M), x)).
(%i1) a: matrix ([3, 1], [2, 4]);
[ 3 1 ]
(%o1) [ ]
[ 2 4 ]
(%i2) expand (charpoly (a, lambda));
2
(%o2) lambda - 7 lambda + 10
(%i3) (programmode: true, solve (%));
(%o3) [lambda = 5, lambda = 2]
(%i4) matrix ([x1], [x2]);
[ x1 ]
(%o4) [ ]
[ x2 ]
(%i5) ev (a . % - lambda*%, %th(2)[1]);
[ x2 - 2 x1 ]
(%o5) [ ]
[ 2 x1 - x2 ]
(%i6) %[1, 1] = 0;
(%o6) x2 - 2 x1 = 0
(%i7) x2^2 + x1^2 = 1;
2 2
(%o7) x2 + x1 = 1
(%i8) solve ([%th(2), %], [x1, x2]);
1 2
(%o8) [[x1 = - -------, x2 = - -------],
sqrt(5) sqrt(5)
1 2
[x1 = -------, x2 = -------]]
sqrt(5) sqrt(5)
Categories: Matrices
Returns the coefficient matrix for the variables x_1, …, x_n of the system of linear equations eqn_1, …, eqn_m.
(%i1) coefmatrix([2*x-(a-1)*y+5*b = 0, b*y+a*x = 3], [x,y]);
[ 2 1 - a ]
(%o1) [ ]
[ a b ]
Categories: Linear equations · Matrices
Returns the i'th column of the matrix M. The return value is a matrix.
Categories: Matrices
Returns a matrix of one column and length (L) rows,
containing the elements of the list L.
covect is a synonym for columnvector.
load ("eigen") loads this function.
This is useful if you want to use parts of the outputs of the functions in this package in matrix calculations.
Example:
(%i1) load ("eigen")$
Warning - you are redefining the Macsyma function eigenvalues
Warning - you are redefining the Macsyma function eigenvectors
(%i2) columnvector ([aa, bb, cc, dd]);
[ aa ]
[ ]
[ bb ]
(%o2) [ ]
[ cc ]
[ ]
[ dd ]
Categories: Matrices
Returns a copy of the matrix M. This is the only way to make a copy aside from copying M element by element.
Note that an assignment of one matrix to another, as in m2: m1, does not
copy m1. An assignment m2 [i,j]: x or setelmx(x, i, j, m2)
also modifies m1 [i,j]. Creating a copy with copymatrix and then
using assignment creates a separate, modified copy.
Categories: Matrices
Computes the determinant of M by a method similar to Gaussian elimination.
The form of the result depends upon the setting of the switch ratmx.
There is a special routine for computing sparse determinants which is called
when the switches ratmx and sparse are both true.
Categories: Matrices
Default value: false
When detout is true, the determinant of a
matrix whose inverse is computed is factored out of the inverse.
For this switch to have an effect doallmxops and doscmxops should
be false (see their descriptions). Alternatively this switch can be
given to ev which causes the other two to be set correctly.
Example:
(%i1) m: matrix ([a, b], [c, d]);
[ a b ]
(%o1) [ ]
[ c d ]
(%i2) detout: true$
(%i3) doallmxops: false$
(%i4) doscmxops: false$
(%i5) invert (m);
[ d - b ]
[ ]
[ - c a ]
(%o5) ------------
a d - b c
Categories: Matrices · Evaluation flags
Returns a diagonal matrix of size n by n with the diagonal elements
all equal to x. diagmatrix (n, 1) returns an identity matrix
(same as ident (n)).
n must evaluate to an integer, otherwise diagmatrix complains with
an error message.
x can be any kind of expression, including another matrix. If x is a matrix, it is not copied; all diagonal elements refer to the same instance, x.
Categories: Matrices
Default value: true
When doallmxops is true,
all operations relating to matrices are carried out.
When it is false then the setting of the
individual dot switches govern which operations are performed.
Categories: Matrices
Default value: true
When domxexpt is true,
a matrix exponential, exp (M) where M is a matrix, is
interpreted as a matrix with element [i,j] equal to exp (m[i,j]).
Otherwise exp (M) evaluates to exp (ev(M)).
domxexpt affects all expressions of the form
base^power where base is an expression assumed scalar
or constant, and power is a list or matrix.
Example:
(%i1) m: matrix ([1, %i], [a+b, %pi]);
[ 1 %i ]
(%o1) [ ]
[ b + a %pi ]
(%i2) domxexpt: false$
(%i3) (1 - c)^m;
[ 1 %i ]
[ ]
[ b + a %pi ]
(%o3) (1 - c)
(%i4) domxexpt: true$
(%i5) (1 - c)^m;
[ %i ]
[ 1 - c (1 - c) ]
(%o5) [ ]
[ b + a %pi ]
[ (1 - c) (1 - c) ]
Categories: Matrices
Default value: true
When domxmxops is true, all matrix-matrix or
matrix-list operations are carried out (but not scalar-matrix
operations); if this switch is false such operations are not carried out.
Categories: Matrices
Default value: false
When domxnctimes is true, non-commutative products of
matrices are carried out.
Categories: Matrices
Default value: []
dontfactor may be set to a list of variables with respect to which
factoring is not to occur. (The list is initially empty.) Factoring also will
not take place with respect to any variables which are less important, according
the variable ordering assumed for canonical rational expression (CRE) form, than
those on the dontfactor list.
Categories: Expressions
Default value: false
When doscmxops is true, scalar-matrix operations are
carried out.
Categories: Matrices
Default value: false
When doscmxplus is true, scalar-matrix operations yield
a matrix result. This switch is not subsumed under doallmxops.
Categories: Matrices
Default value: true
When dot0nscsimp is true, a non-commutative product of zero
and a nonscalar term is simplified to a commutative product.
Categories: Simplification flags and variables
Default value: true
When dot0simp is true,
a non-commutative product of zero and
a scalar term is simplified to a commutative product.
Categories: Simplification flags and variables
Default value: true
When dot1simp is true,
a non-commutative product of one and
another term is simplified to a commutative product.
Categories: Simplification flags and variables
Default value: true
When dotassoc is true, an expression (A.B).C simplifies to
A.(B.C).
Categories: Simplification flags and variables
Default value: true
When dotconstrules is true, a non-commutative product of a
constant and another term is simplified to a commutative product.
Turning on this flag effectively turns on dot0simp,
dot0nscsimp, and dot1simp as well.
Categories: Simplification flags and variables
Default value: false
When dotdistrib is true, an expression A.(B + C) simplifies
to A.B + A.C.
Categories: Simplification flags and variables
Default value: true
When dotexptsimp is true, an expression A.A simplifies to
A^^2.
Categories: Simplification flags and variables
Default value: 1
dotident is the value returned by X^^0.
Categories: Simplification flags and variables
Default value: false
When dotscrules is true, an expression A.SC or SC.A
simplifies to SC*A and A.(SC*B) simplifies to SC*(A.B).
Categories: Simplification flags and variables
Returns the echelon form of the matrix M, as produced by Gaussian elimination. The echelon form is computed from M by elementary row operations such that the first non-zero element in each row in the resulting matrix is one and the column elements under the first one in each row are all zero.
triangularize also carries out Gaussian elimination, but it does not
normalize the leading non-zero element in each row.
lu_factor and cholesky are other functions which yield
triangularized matrices.
(%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]);
[ 3 7 aa bb ]
[ ]
(%o1) [ - 1 8 5 2 ]
[ ]
[ 9 2 11 4 ]
(%i2) echelon (M);
[ 1 - 8 - 5 - 2 ]
[ ]
[ 28 11 ]
[ 0 1 -- -- ]
(%o2) [ 37 37 ]
[ ]
[ 37 bb - 119 ]
[ 0 0 1 ----------- ]
[ 37 aa - 313 ]
Categories: Linear equations · Matrices
Returns a list of two lists containing the eigenvalues of the matrix M. The first sublist of the return value is the list of eigenvalues of the matrix, and the second sublist is the list of the multiplicities of the eigenvalues in the corresponding order.
eivals is a synonym for eigenvalues.
eigenvalues calls the function solve to find the roots of the
characteristic polynomial of the matrix. Sometimes solve may not be able
to find the roots of the polynomial; in that case some other functions in this
package (except innerproduct, unitvector,
columnvector and gramschmidt) will not work.
Sometimes solve may find only a subset of the roots of the polynomial.
This may happen when the factoring of the polynomial contains polynomials
of degree 5 or more. In such cases a warning message is displayed and the
only the roots found and their corresponding multiplicities are returned.
In some cases the eigenvalues found by solve may be complicated
expressions. (This may happen when solve returns a not-so-obviously real
expression for an eigenvalue which is known to be real.) It may be possible to
simplify the eigenvalues using some other functions.
The package eigen.mac is loaded automatically when
eigenvalues or eigenvectors is referenced.
If eigen.mac is not already loaded,
load ("eigen") loads it.
After loading, all functions and variables in the package are available.
Categories: Package eigen
Computes eigenvectors of the matrix M. The return value is a list of two elements. The first is a list of the eigenvalues of M and a list of the multiplicities of the eigenvalues. The second is a list of lists of eigenvectors. There is one list of eigenvectors for each eigenvalue. There may be one or more eigenvectors in each list.
eivects is a synonym for eigenvectors.
The package eigen.mac is loaded automatically when
eigenvalues or eigenvectors is referenced.
If eigen.mac is not already loaded,
load ("eigen") loads it.
After loading, all functions and variables in the package are available.
Note that eigenvectors internally calls eigenvalues to
obtain eigenvalues. So, when eigenvalues returns a subset of
all the eigenvalues, the eigenvectors returns the corresponding
subset of the all the eigenvectors, with the same warning displayed as
eigenvalues.
The flags that affect this function are:
nondiagonalizable is set to true or false depending on
whether the matrix is nondiagonalizable or diagonalizable after
eigenvectors returns.
hermitianmatrix when true, causes the degenerate
eigenvectors of the Hermitian matrix to be orthogonalized using the
Gram-Schmidt algorithm.
knowneigvals when true causes the eigen package to assume
the eigenvalues of the matrix are known to the user and stored under the global
name listeigvals. listeigvals should be set to a list similar
to the output eigenvalues.
The function algsys is used here to solve for the eigenvectors.
Sometimes if the eigenvalues are messy, algsys may not be able to find a
solution. In some cases, it may be possible to simplify the eigenvalues by
first finding them using eigenvalues command and then using other
functions to reduce them to something simpler. Following simplification,
eigenvectors can be called again with the knowneigvals flag set
to true.
See also eigenvalues.
Examples:
A matrix which has just one eigenvector per eigenvalue.
(%i1) M1 : matrix ([11, -1], [1, 7]);
[ 11 - 1 ]
(%o1) [ ]
[ 1 7 ]
(%i2) [vals, vecs] : eigenvectors (M1);
(%o2) [[[9 - sqrt(3), sqrt(3) + 9], [1, 1]],
[[[1, sqrt(3) + 2]], [[1, 2 - sqrt(3)]]]]
(%i3) for i thru length (vals[1]) do disp (val[i] = vals[1][i],
mult[i] = vals[2][i], vec[i] = vecs[i]);
val = 9 - sqrt(3)
1
mult = 1
1
vec = [[1, sqrt(3) + 2]]
1
val = sqrt(3) + 9
2
mult = 1
2
vec = [[1, 2 - sqrt(3)]]
2
(%o3) done
A matrix which has two eigenvectors for one eigenvalue (namely 2).
(%i1) M1 : matrix ([0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 2, 0],
[0, 0, 0, 2]);
[ 0 1 0 0 ]
[ ]
[ 0 0 0 0 ]
(%o1) [ ]
[ 0 0 2 0 ]
[ ]
[ 0 0 0 2 ]
(%i2) [vals, vecs] : eigenvectors (M1);
(%o2) [[[0, 2], [2, 2]], [[[1, 0, 0, 0]],
[[0, 0, 1, 0], [0, 0, 0, 1]]]]
(%i3) for i thru length (vals[1]) do disp (val[i] = vals[1][i],
mult[i] = vals[2][i], vec[i] = vecs[i]);
val = 0
1
mult = 2
1
vec = [[1, 0, 0, 0]]
1
val = 2
2
mult = 2
2
vec = [[0, 0, 1, 0], [0, 0, 0, 1]]
2
(%o3) done
Categories: Package eigen
Returns an m by n matrix, all elements of which
are zero except for the [i, j] element which is x.
Categories: Matrices
Returns an m by n matrix, reading the elements interactively.
If n is equal to m, Maxima prompts for the type of the matrix
(diagonal, symmetric, antisymmetric, or general) and for each element.
Each response is terminated by a semicolon ; or dollar sign $.
If n is not equal to m, Maxima prompts for each element.
The elements may be any expressions, which are evaluated.
entermatrix evaluates its arguments.
(%i1) n: 3$
(%i2) m: entermatrix (n, n)$
Is the matrix 1. Diagonal 2. Symmetric 3. Antisymmetric
4. General
Answer 1, 2, 3 or 4 :
1$
Row 1 Column 1:
(a+b)^n$
Row 2 Column 2:
(a+b)^(n+1)$
Row 3 Column 3:
(a+b)^(n+2)$
Matrix entered.
(%i3) m;
[ 3 ]
[ (b + a) 0 0 ]
[ ]
(%o3) [ 4 ]
[ 0 (b + a) 0 ]
[ ]
[ 5 ]
[ 0 0 (b + a) ]
Categories: Console interaction · Matrices
Returns a matrix generated from a, taking element
a[i_1, j_1] as the upper-left element and
a[i_2, j_2] as the lower-right element of the matrix.
Here a is a declared array (created by array but not by
make_array) or an undeclared array, or an array function, or a lambda
expression of two arguments. (An array function is created like other functions
with := or define, but arguments are enclosed in square
brackets instead of parentheses.)
If j_1 is omitted, it is assumed equal to i_1. If both j_1 and i_1 are omitted, both are assumed equal to 1.
If a selected element i,j of the array is undefined,
the matrix will contain a symbolic element a[i,j].
Examples:
(%i1) h [i, j] := 1 / (i + j - 1);
1
(%o1) h := ---------
i, j i + j - 1
(%i2) genmatrix (h, 3, 3);
[ 1 1 ]
[ 1 - - ]
[ 2 3 ]
[ ]
[ 1 1 1 ]
(%o2) [ - - - ]
[ 2 3 4 ]
[ ]
[ 1 1 1 ]
[ - - - ]
[ 3 4 5 ]
(%i3) array (a, fixnum, 2, 2);
(%o3) a
(%i4) a [1, 1] : %e;
(%o4) %e
(%i5) a [2, 2] : %pi;
(%o5) %pi
(%i6) genmatrix (a, 2, 2);
[ %e 0 ]
(%o6) [ ]
[ 0 %pi ]
(%i7) genmatrix (lambda ([i, j], j - i), 3, 3);
[ 0 1 2 ]
[ ]
(%o7) [ - 1 0 1 ]
[ ]
[ - 2 - 1 0 ]
(%i8) genmatrix (B, 2, 2);
[ B B ]
[ 1, 1 1, 2 ]
(%o8) [ ]
[ B B ]
[ 2, 1 2, 2 ]
Categories: Matrices
Carries out the Gram-Schmidt orthogonalization algorithm on x, which is
either a matrix or a list of lists. x is not modified by
gramschmidt. The inner product employed by gramschmidt is
F, if present, otherwise the inner product is the function
innerproduct.
If x is a matrix, the algorithm is applied to the rows of x. If x is a list of lists, the algorithm is applied to the sublists, which must have equal numbers of elements. In either case, the return value is a list of lists, the sublists of which are orthogonal and span the same space as x. If the dimension of the span of x is less than the number of rows or sublists, some sublists of the return value are zero.
factor is called at each stage of the algorithm to simplify intermediate
results. As a consequence, the return value may contain factored integers.
load(eigen) loads this function.
Example:
Gram-Schmidt algorithm using default inner product function.
(%i1) load (eigen)$
(%i2) x: matrix ([1, 2, 3], [9, 18, 30], [12, 48, 60]);
[ 1 2 3 ]
[ ]
(%o2) [ 9 18 30 ]
[ ]
[ 12 48 60 ]
(%i3) y: gramschmidt (x);
2 2 4 3
3 3 3 5 2 3 2 3
(%o3) [[1, 2, 3], [- ---, - --, ---], [- ----, ----, 0]]
2 7 7 2 7 5 5
(%i4) map (innerproduct, [y[1], y[2], y[3]], [y[2], y[3], y[1]]);
(%o4) [0, 0, 0]
Gram-Schmidt algorithm using a specified inner product function.
(%i1) load (eigen)$
(%i2) ip (f, g) := integrate (f * g, u, a, b);
(%o2) ip(f, g) := integrate(f g, u, a, b)
(%i3) y : gramschmidt([1, sin(u), cos(u)], ip), a= -%pi/2, b=%pi/2;
%pi cos(u) - 2
(%o3) [1, sin(u), --------------]
%pi
(%i4) map (ip, [y[1], y[2], y[3]], [y[2], y[3], y[1]]),
a= -%pi/2, b=%pi/2;
(%o4) [0, 0, 0]
Categories: Package eigen
Returns an n by n identity matrix.
Categories: Matrices
Returns the inner product (also called the scalar product or dot product) of
x and y, which are lists of equal length, or both 1-column or 1-row
matrices of equal length. The return value is conjugate (x) . y,
where . is the noncommutative multiplication operator.
load ("eigen") loads this function.
inprod is a synonym for innerproduct.
Categories: Package eigen
Returns the inverse of the matrix M. The inverse is computed by the adjoint method.
invert_by_adjoint honors the ratmx and detout flags,
the same as invert.
Returns the inverse of the matrix M. The inverse is computed via the LU decomposition.
When ratmx is true,
elements of M are converted to canonical rational expressions (CRE),
and the elements of the return value are also CRE.
When ratmx is false,
elements of M are not converted to a common representation.
In particular, float and bigfloat elements are not converted to rationals.
When detout is true, the determinant is factored out of the inverse.
The global flags doallmxops and doscmxops must be false
to prevent the determinant from being absorbed into the inverse.
xthru can multiply the determinant into the inverse.
invert does not apply any simplifications to the elements of the inverse
apart from the default arithmetic simplifications.
ratsimp and expand can apply additional simplifications.
In particular, when M has polynomial elements,
expand(invert(M)) might be preferable.
invert(M) is equivalent to M^^-1.
Categories: Matrices
Returns a list containing the elements of the matrix M.
Example:
(%i1) list_matrix_entries(matrix([a,b],[c,d])); (%o1) [a, b, c, d]
Categories: Matrices
Default value: [
lmxchar is the character displayed as the left delimiter of a matrix.
See also rmxchar.
Example:
(%i1) lmxchar: "|"$
(%i2) matrix ([a, b, c], [d, e, f], [g, h, i]);
| a b c ]
| ]
(%o2) | d e f ]
| ]
| g h i ]
Categories: Display flags and variables · Matrices
Returns a rectangular matrix which has the rows row_1, …, row_n. Each row is a list of expressions. All rows must be the same length.
The operations + (addition), - (subtraction), *
(multiplication), and / (division), are carried out element by element
when the operands are two matrices, a scalar and a matrix, or a matrix and a
scalar. The operation ^ (exponentiation, equivalently **)
is carried out element by element if the operands are a scalar and a matrix or
a matrix and a scalar, but not if the operands are two matrices.
All operations are normally carried out in full,
including . (noncommutative multiplication).
Matrix multiplication is represented by the noncommutative multiplication
operator .. The corresponding noncommutative exponentiation operator
is ^^. For a matrix A, A.A = A^^2
and A^^-1 is the inverse of A, if it exists.
A^^-1 is equivalent to invert(A).
There are switches for controlling simplification of expressions involving dot
and matrix-list operations. These are
doallmxops, domxexpt, domxmxops,
doscmxops, and doscmxplus.
There are additional options which are related to matrices. These are:
lmxchar, rmxchar, ratmx,
listarith, detout, scalarmatrix and
sparse.
There are a number of functions which take matrices as arguments or yield
matrices as return values.
See eigenvalues, eigenvectors, determinant,
charpoly, genmatrix, addcol,
addrow, copymatrix, transpose,
echelon, and rank.
Examples:
(%i1) x: matrix ([17, 3], [-8, 11]);
[ 17 3 ]
(%o1) [ ]
[ - 8 11 ]
(%i2) y: matrix ([%pi, %e], [a, b]);
[ %pi %e ]
(%o2) [ ]
[ a b ]
(%i3) x + y;
[ %pi + 17 %e + 3 ]
(%o3) [ ]
[ a - 8 b + 11 ]
(%i4) x - y;
[ 17 - %pi 3 - %e ]
(%o4) [ ]
[ - a - 8 11 - b ]
(%i5) x * y;
[ 17 %pi 3 %e ]
(%o5) [ ]
[ - 8 a 11 b ]
(%i6) x / y;
[ 17 - 1 ]
[ --- 3 %e ]
[ %pi ]
(%o6) [ ]
[ 8 11 ]
[ - - -- ]
[ a b ]
(%i7) x ^ 3;
[ 4913 27 ]
(%o7) [ ]
[ - 512 1331 ]
(%i8) exp(y);
[ %pi %e ]
[ %e %e ]
(%o8) [ ]
[ a b ]
[ %e %e ]
matrixexp.
(%i9) x ^ y;
[ %pi %e ]
[ ]
[ a b ]
[ 17 3 ]
(%o9) [ ]
[ - 8 11 ]
(%i10) x . y;
[ 3 a + 17 %pi 3 b + 17 %e ]
(%o10) [ ]
[ 11 a - 8 %pi 11 b - 8 %e ]
(%i11) y . x;
[ 17 %pi - 8 %e 3 %pi + 11 %e ]
(%o11) [ ]
[ 17 a - 8 b 11 b + 3 a ]
b^^m is the same as b^m.
(%i12) x ^^ 3;
[ 3833 1719 ]
(%o12) [ ]
[ - 4584 395 ]
(%i13) %e ^^ y;
[ %pi %e ]
[ %e %e ]
(%o13) [ ]
[ a b ]
[ %e %e ]
(%i14) x ^^ -1;
[ 11 3 ]
[ --- - --- ]
[ 211 211 ]
(%o14) [ ]
[ 8 17 ]
[ --- --- ]
[ 211 211 ]
(%i15) x . (x ^^ -1);
[ 1 0 ]
(%o15) [ ]
[ 0 1 ]
Categories: Matrices
Calculates the matrix exponential
e^(M*V)
. Instead of the vector V a number n can be specified as the second
argument. If this argument is omitted matrixexp replaces it by 1.
The matrix exponential of a matrix M can be expressed as a power series: e^M=sum(M^k/k!,0,inf)
Categories: Matrices
Returns a matrix with element i,j equal to f(M[i,j]).
See also map, fullmap, fullmapl, and
apply.
Categories: Matrices
Returns true if expr is a matrix, otherwise false.
Categories: Predicate functions · Matrices
Default value: +
matrix_element_add is the operation
invoked in place of addition in a matrix multiplication.
matrix_element_add can be assigned any n-ary operator
(that is, a function which handles any number of arguments).
The assigned value may be the name of an operator enclosed in quote marks,
the name of a function,
or a lambda expression.
See also matrix_element_mult and matrix_element_transpose.
Example:
(%i1) matrix_element_add: "*"$
(%i2) matrix_element_mult: "^"$
(%i3) aa: matrix ([a, b, c], [d, e, f]);
[ a b c ]
(%o3) [ ]
[ d e f ]
(%i4) bb: matrix ([u, v, w], [x, y, z]);
[ u v w ]
(%o4) [ ]
[ x y z ]
(%i5) aa . transpose (bb);
[ u v w x y z ]
[ a b c a b c ]
(%o5) [ ]
[ u v w x y z ]
[ d e f d e f ]
Categories: Matrices
Default value: *
matrix_element_mult is the operation
invoked in place of multiplication in a matrix multiplication.
matrix_element_mult can be assigned any binary operator.
The assigned value may be the name of an operator enclosed in quote marks,
the name of a function,
or a lambda expression.
The dot operator . is a useful choice in some contexts.
See also matrix_element_add and matrix_element_transpose.
Example:
(%i1) matrix_element_add: lambda ([[x]], sqrt (apply ("+", x)))$
(%i2) matrix_element_mult: lambda ([x, y], (x - y)^2)$
(%i3) [a, b, c] . [x, y, z];
2 2 2
(%o3) sqrt((c - z) + (b - y) + (a - x) )
(%i4) aa: matrix ([a, b, c], [d, e, f]);
[ a b c ]
(%o4) [ ]
[ d e f ]
(%i5) bb: matrix ([u, v, w], [x, y, z]);
[ u v w ]
(%o5) [ ]
[ x y z ]
(%i6) aa . transpose (bb);
[ 2 2 2 ]
[ sqrt((c - w) + (b - v) + (a - u) ) ]
(%o6) Col 1 = [ ]
[ 2 2 2 ]
[ sqrt((f - w) + (e - v) + (d - u) ) ]
[ 2 2 2 ]
[ sqrt((c - z) + (b - y) + (a - x) ) ]
Col 2 = [ ]
[ 2 2 2 ]
[ sqrt((f - z) + (e - y) + (d - x) ) ]
Categories: Matrices
Default value: false
matrix_element_transpose is the operation
applied to each element of a matrix when it is transposed.
matrix_element_mult can be assigned any unary operator.
The assigned value may be the name of an operator enclosed in quote marks,
the name of a function, or a lambda expression.
When matrix_element_transpose equals transpose,
the transpose function is applied to every element.
When matrix_element_transpose equals nonscalars,
the transpose function is applied to every nonscalar element.
If some element is an atom, the nonscalars option applies
transpose only if the atom is declared nonscalar,
while the transpose option always applies transpose.
The default value, false, means no operation is applied.
See also matrix_element_add and matrix_element_mult.
Examples:
(%i1) declare (a, nonscalar)$
(%i2) transpose ([a, b]);
[ transpose(a) ]
(%o2) [ ]
[ b ]
(%i3) matrix_element_transpose: nonscalars$
(%i4) transpose ([a, b]);
[ transpose(a) ]
(%o4) [ ]
[ b ]
(%i5) matrix_element_transpose: transpose$
(%i6) transpose ([a, b]);
[ transpose(a) ]
(%o6) [ ]
[ transpose(b) ]
(%i7) matrix_element_transpose: lambda ([x], realpart(x)
- %i*imagpart(x))$
(%i8) m: matrix ([1 + 5*%i, 3 - 2*%i], [7*%i, 11]);
[ 5 %i + 1 3 - 2 %i ]
(%o8) [ ]
[ 7 %i 11 ]
(%i9) transpose (m);
[ 1 - 5 %i - 7 %i ]
(%o9) [ ]
[ 2 %i + 3 11 ]
Categories: Matrices
Returns the trace (that is, the sum of the elements on the main diagonal) of the square matrix M.
mattrace is called by ncharpoly, an alternative to Maxima's
charpoly.
load ("nchrpl") loads this function.
Categories: Matrices · Package nchrpl
Returns the i, j minor of the matrix M. That is, M with row i and column j removed.
Categories: Matrices
Returns the characteristic polynomial of the matrix M
with respect to x. This is an alternative to Maxima's charpoly.
ncharpoly works by computing traces of powers of the given matrix,
which are known to be equal to sums of powers of the roots of the
characteristic polynomial. From these quantities the symmetric
functions of the roots can be calculated, which are nothing more than
the coefficients of the characteristic polynomial. charpoly works by
forming the determinant of x * ident [n] - a. Thus
ncharpoly wins, for example, in the case of large dense matrices filled
with integers, since it avoids polynomial arithmetic altogether.
load ("nchrpl") loads this file.
Categories: Matrices · Package nchrpl
Computes the determinant of the matrix M by the Johnson-Gentleman tree
minor algorithm. newdet returns the result in CRE form.
Categories: Matrices
Computes the permanent of the matrix M by the Johnson-Gentleman tree
minor algorithm. A permanent is like a determinant but with no sign changes.
permanent returns the result in CRE form.
See also newdet.
Categories: Matrices
Computes the rank of the matrix M. That is, the order of the largest non-singular subdeterminant of M.
rank may return the wrong answer if it cannot determine that a matrix element that is equivalent to zero is indeed so.
Categories: Matrices
Default value: false
When ratmx is false, determinant and matrix
addition, subtraction, and multiplication are performed in the
representation of the matrix elements and cause the result of
matrix inversion to be left in general representation.
When ratmx is true,
the 4 operations mentioned above are performed in CRE form and the
result of matrix inverse is in CRE form. Note that this may
cause the elements to be expanded (depending on the setting of ratfac)
which might not always be desired.
Categories: Matrices · Rational expressions
Returns the i'th row of the matrix M. The return value is a matrix.
Categories: Matrices
Default value: ]
rmxchar is the character drawn on the right-hand side of a matrix.
See also lmxchar.
Categories: Display flags and variables
Default value: true
When scalarmatrixp is true, then whenever a 1 x 1 matrix
is produced as a result of computing the dot product of matrices it
is simplified to a scalar, namely the sole element of the matrix.
When scalarmatrixp is all,
then all 1 x 1 matrices are simplified to scalars.
When scalarmatrixp is false, 1 x 1 matrices are not simplified
to scalars.
Categories: Matrices · Simplification flags and variables
Here the argument coordinatetransform evaluates to the form
[[expression1, expression2, ...], indeterminate1, indeterminat2, ...],
where the variables indeterminate1, indeterminate2, etc. are the
curvilinear coordinate variables and where a set of rectangular Cartesian
components is given in terms of the curvilinear coordinates by
[expression1, expression2, ...]. coordinates is set to the vector
[indeterminate1, indeterminate2,...], and dimension is set to the
length of this vector. SF[1], SF[2], …, SF[DIMENSION] are set to the
coordinate scale factors, and sfprod is set to the product of these scale
factors. Initially, coordinates is [X, Y, Z], dimension
is 3, and SF[1]=SF[2]=SF[3]=SFPROD=1, corresponding to 3-dimensional rectangular
Cartesian coordinates. To expand an expression into physical components in the
current coordinate system, there is a function with usage of the form
Categories: Package vect
Assigns x to the (i, j)'th element of the matrix M, and returns the altered matrix.
M [i, j]: x has the same effect,
but returns x instead of M.
Categories: Matrices
similaritytransform computes a similarity transform of the matrix
M. It returns a list which is the output of the uniteigenvectors
command. In addition if the flag nondiagonalizable is false two
global matrices leftmatrix and rightmatrix are computed. These
matrices have the property that leftmatrix . M . rightmatrix is a
diagonal matrix with the eigenvalues of M on the diagonal. If
nondiagonalizable is true the left and right matrices are not
computed.
If the flag hermitianmatrix is true then leftmatrix is the
complex conjugate of the transpose of rightmatrix. Otherwise
leftmatrix is the inverse of rightmatrix.
rightmatrix is the matrix the columns of which are the unit
eigenvectors of M. The other flags (see eigenvalues and
eigenvectors) have the same effects since
similaritytransform calls the other functions in the package in order
to be able to form rightmatrix.
load ("eigen") loads this function.
simtran is a synonym for similaritytransform.
Categories: Package eigen
Default value: false
When sparse is true, and if ratmx is true, then
determinant will use special routines for computing sparse determinants.
Categories: Matrices
Returns a new matrix composed of the matrix M with rows i_1, …, i_m deleted, and columns j_1, …, j_n deleted.
Categories: Matrices
Returns the transpose of M.
If M is a matrix, the return value is another matrix N
such that N[i,j] = M[j,i].
If M is a list, the return value is a matrix N
of length (m) rows and 1 column, such that N[i,1] = M[i].
Otherwise M is a symbol,
and the return value is a noun expression 'transpose (M).
Categories: Matrices
Returns the upper triangular form of the matrix M,
as produced by Gaussian elimination.
The return value is the same as echelon,
except that the leading nonzero coefficient in each row is not normalized to 1.
lu_factor and cholesky are other functions which yield
triangularized matrices.
(%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]);
[ 3 7 aa bb ]
[ ]
(%o1) [ - 1 8 5 2 ]
[ ]
[ 9 2 11 4 ]
(%i2) triangularize (M);
[ - 1 8 5 2 ]
[ ]
(%o2) [ 0 - 74 - 56 - 22 ]
[ ]
[ 0 0 626 - 74 aa 238 - 74 bb ]
Categories: Linear equations · Matrices
Computes unit eigenvectors of the matrix M.
The return value is a list of lists, the first sublist of which is the
output of the eigenvalues command, and the other sublists of which are
the unit eigenvectors of the matrix corresponding to those eigenvalues
respectively.
The flags mentioned in the description of the
eigenvectors command have the same effects in this one as well.
When knowneigvects is true, the eigen package assumes
that the eigenvectors of the matrix are known to the user and are
stored under the global name listeigvects. listeigvects should
be set to a list similar to the output of the eigenvectors command.
If knowneigvects is set to true and the list of eigenvectors is
given the setting of the flag nondiagonalizable may not be correct. If
that is the case please set it to the correct value. The author assumes that
the user knows what he is doing and will not try to diagonalize a matrix the
eigenvectors of which do not span the vector space of the appropriate dimension.
load ("eigen") loads this function.
ueivects is a synonym for uniteigenvectors.
Categories: Package eigen
Returns x/norm(x); this is a unit vector in the same direction as x.
load ("eigen") loads this function.
uvect is a synonym for unitvector.
Categories: Package eigen
Returns the vector potential of a given curl vector, in the current coordinate
system. potentialzeroloc has a similar role as for potential, but
the order of the left-hand sides of the equations must be a cyclic permutation
of the coordinate variables.
Categories: Package vect
Applies simplifications and expansions according to the following global flags:
expandall, expanddot, expanddotplus, expandcross, expandcrossplus,
expandcrosscross, expandgrad, expandgradplus, expandgradprod,
expanddiv, expanddivplus, expanddivprod, expandcurl, expandcurlplus,
expandcurlcurl, expandlaplacian, expandlaplacianplus,
and expandlaplacianprod.
All these flags have default value false. The plus suffix refers
to employing additivity or distributivity. The prod suffix refers to the
expansion for an operand that is any kind of product.
expandcrosscrossSimplifies p ~ (q ~ r) to (p . r)*q - (p . q)*r.
expandcurlcurlSimplifies curl curl p to grad div p + div grad p.
expandlaplaciantodivgradSimplifies laplacian p to div grad p.
expandcrossEnables expandcrossplus and expandcrosscross.
expandplusEnables expanddotplus, expandcrossplus, expandgradplus,
expanddivplus, expandcurlplus, and expandlaplacianplus.
expandprodEnables expandgradprod, expanddivprod, and expandlaplacianprod.
These flags have all been declared evflag.
Categories: Package vect · Simplification functions
Default value: false
When vect_cross is true, it allows DIFF(X~Y,T) to work where
~ is defined in SHARE;VECT (where VECT_CROSS is set to true, anyway.)
Categories: Package vect · Differential calculus
Returns an m by n matrix, all elements of which are zero.
Categories: Matrices
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| 24.1 Introduction to Affine | ||
| 24.2 Functions and Variables for Affine |
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affine is a package to work with groups of polynomials.
Categories: Polynomials · Groebner bases · Share packages · Package affine
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Solves the simultaneous linear equations expr_1, ..., expr_m
for the variables x_1, ..., x_n.
Each expr_i may be an equation or a general expression;
if given as a general expression, it is treated as an equation of the form expr_i = 0.
The return value is a list of equations of the form
[x_1 = a_1, ..., x_n = a_n]
where a_1, ..., a_n are all free of x_1, ..., x_n.
fast_linsolve is faster than linsolve for system of equations which
are sparse.
load(affine) loads this function.
Categories: Package affine
Returns a Groebner basis for the equations expr_1, ..., expr_m.
The function polysimp can then
be used to simplify other functions relative to the equations.
grobner_basis ([3*x^2+1, y*x])$ polysimp (y^2*x + x^3*9 + 2) ==> -3*x + 2
polysimp(f) yields 0 if and only if f is in the ideal generated by
expr_1, ..., expr_m, that is,
if and only if f is a polynomial combination of the elements of
expr_1, ..., expr_m.
load(affine) loads this function.
Categories: Package affine
The eqns are polynomial equations in non commutative variables.
The value of current_variables is the
list of variables used for computing degrees. The equations must be
homogeneous, in order for the procedure to terminate.
If you have checked overlapping simplifications in dot_simplifications
above the degree of f, then the following is true:
dotsimp (f) yields 0 if and only if f is in the
ideal generated by the equations, i.e.,
if and only if f is a polynomial combination
of the elements of the equations.
The degree is that returned by nc_degree. This in turn is influenced by
the weights of individual variables.
load(affine) loads this function.
Categories: Package affine
Assigns weights w_1, ..., w_n to x_1, ..., x_n, respectively.
These are the weights used in computing nc_degree.
load(affine) loads this function.
Categories: Package affine
Returns the degree of a noncommutative polynomial p. See declare_weights.
load(affine) loads this function.
Categories: Package affine
Returns 0 if and only if f is in the ideal generated by the equations, i.e., if and only if f is a polynomial combination of the elements of the equations.
load(affine) loads this function.
Categories: Package affine
If set_up_dot_simplifications has been previously done, finds the central polynomials
in the variables x_1, ..., x_n in the given degree, n.
For example:
set_up_dot_simplifications ([y.x + x.y], 3); fast_central_elements ([x, y], 2); [y.y, x.x];
load(affine) loads this function.
Categories: Package affine
Checks the overlaps thru degree n,
making sure that you have sufficient simplification rules in each
degree, for dotsimp to work correctly. This process can be speeded
up if you know before hand what the dimension of the space of monomials is.
If it is of finite global dimension, then hilbert should be used. If you
don't know the monomial dimensions, do not specify a rank_function.
An optional third argument reset, false says don't bother to query
about resetting things.
load(affine) loads this function.
Categories: Package affine
Returns the list of independent monomials relative to the current dot simplifications of degree n in the variables x_1, ..., x_n.
load(affine) loads this function.
Categories: Package affine
Compute the Hilbert series through degree n for the current algebra.
load(affine) loads this function.
Categories: Package affine
Makes a list of the coefficients of the noncommutative polynomials p_1, ..., p_n
of the noncommutative monomials m_1, ..., m_n.
The coefficients should be scalars. Use list_nc_monomials to build the list of
monomials.
load(affine) loads this function.
Categories: Package affine
Returns a list of the non commutative monomials occurring in a polynomial p or a list of polynomials p_1, ..., p_n.
load(affine) loads this function.
Categories: Package affine
Default value: false
When all_dotsimp_denoms is a list,
the denominators encountered by dotsimp are appended to the list.
all_dotsimp_denoms may be initialized to an empty list []
before calling dotsimp.
By default, denominators are not collected by dotsimp.
Categories: Package affine
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| 25.1 Introduction to itensor | ||
| 25.2 Functions and Variables for itensor |
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Maxima implements symbolic tensor manipulation of two distinct types:
component tensor manipulation (ctensor package) and indicial tensor
manipulation (itensor package).
Nota bene: Please see the note on 'new tensor notation' below.
Component tensor manipulation means that geometrical tensor
objects are represented as arrays or matrices. Tensor operations such
as contraction or covariant differentiation are carried out by
actually summing over repeated (dummy) indices with do statements.
That is, one explicitly performs operations on the appropriate tensor
components stored in an array or matrix.
Indicial tensor manipulation is implemented by representing tensors as functions of their covariant, contravariant and derivative indices. Tensor operations such as contraction or covariant differentiation are performed by manipulating the indices themselves rather than the components to which they correspond.
These two approaches to the treatment of differential, algebraic and analytic processes in the context of Riemannian geometry have various advantages and disadvantages which reveal themselves only through the particular nature and difficulty of the user's problem. However, one should keep in mind the following characteristics of the two implementations:
The representation of tensors and tensor operations explicitly in
terms of their components makes ctensor easy to use. Specification of
the metric and the computation of the induced tensors and invariants
is straightforward. Although all of Maxima's powerful simplification
capacity is at hand, a complex metric with intricate functional and
coordinate dependencies can easily lead to expressions whose size is
excessive and whose structure is hidden. In addition, many calculations
involve intermediate expressions which swell causing programs to
terminate before completion. Through experience, a user can avoid
many of these difficulties.
Because of the special way in which tensors and tensor operations
are represented in terms of symbolic operations on their indices,
expressions which in the component representation would be
unmanageable can sometimes be greatly simplified by using the special
routines for symmetrical objects in itensor. In this way the structure
of a large expression may be more transparent. On the other hand, because
of the special indicial representation in itensor, in some cases the
user may find difficulty with the specification of the metric, function
definition, and the evaluation of differentiated "indexed" objects.
The itensor package can carry out differentiation with respect to an indexed
variable, which allows one to use the package when dealing with Lagrangian
and Hamiltonian formalisms. As it is possible to differentiate a field
Lagrangian with respect to an (indexed) field variable, one can use Maxima
to derive the corresponding Euler-Lagrange equations in indicial form. These
equations can be translated into component tensor (ctensor) programs using
the ic_convert function, allowing us to solve the field equations in a
particular coordinate representation, or to recast the equations of motion
in Hamiltonian form. See einhil.dem and bradic.dem for two comprehensive
examples. The first, einhil.dem, uses the Einstein-Hilbert action to derive
the Einstein field tensor in the homogeneous and isotropic case (Friedmann
equations) and the spherically symmetric, static case (Schwarzschild
solution.) The second, bradic.dem, demonstrates how to compute the Friedmann
equations from the action of Brans-Dicke gravity theory, and also derives
the Hamiltonian associated with the theory's scalar field.
Categories: Tensors · Share packages · Package itensor
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Earlier versions of the itensor package in Maxima used a notation that sometimes
led to incorrect index ordering. Consider the following, for instance:
(%i2) imetric(g);
(%o2) done
(%i3) ishow(g([],[j,k])*g([],[i,l])*a([i,j],[]))$
i l j k
(%t3) g g a
i j
(%i4) ishow(contract(%))$
k l
(%t4) a
This result is incorrect unless a happens to be a symmetric tensor.
The reason why this happens is that although itensor correctly maintains
the order within the set of covariant and contravariant indices, once an
index is raised or lowered, its position relative to the other set of
indices is lost.
To avoid this problem, a new notation has been developed that remains fully
compatible with the existing notation and can be used interchangeably. In
this notation, contravariant indices are inserted in the appropriate
positions in the covariant index list, but with a minus sign prepended.
Functions like contract_Itensor and ishow are now aware of this
new index notation and can process tensors appropriately.
In this new notation, the previous example yields a correct result:
(%i5) ishow(g([-j,-k],[])*g([-i,-l],[])*a([i,j],[]))$
i l j k
(%t5) g a g
i j
(%i6) ishow(contract(%))$
l k
(%t6) a
Presently, the only code that makes use of this notation is the lc2kdt
function. Through this notation, it achieves consistent results as it
applies the metric tensor to resolve Levi-Civita symbols without resorting
to numeric indices.
Since this code is brand new, it probably contains bugs. While it has been tested to make sure that it doesn't break anything using the "old" tensor notation, there is a considerable chance that "new" tensors will fail to interoperate with certain functions or features. These bugs will be fixed as they are encountered... until then, caveat emptor!
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The indicial tensor manipulation package may be loaded by
load(itensor). Demos are also available: try demo(tensor).
In itensor a tensor is represented as an "indexed object" . This is a
function of 3 groups of indices which represent the covariant,
contravariant and derivative indices. The covariant indices are
specified by a list as the first argument to the indexed object, and
the contravariant indices by a list as the second argument. If the
indexed object lacks either of these groups of indices then the empty
list [] is given as the corresponding argument. Thus, g([a,b],[c])
represents an indexed object called g which has two covariant indices
(a,b), one contravariant index (c) and no derivative indices.
The derivative indices, if they are present, are appended as
additional arguments to the symbolic function representing the tensor.
They can be explicitly specified by the user or be created in the
process of differentiation with respect to some coordinate variable.
Since ordinary differentiation is commutative, the derivative indices
are sorted alphanumerically, unless iframe_flag is set to true,
indicating that a frame metric is being used. This canonical ordering makes it
possible for Maxima to recognize that, for example, t([a],[b],i,j) is
the same as t([a],[b],j,i). Differentiation of an indexed object with
respect to some coordinate whose index does not appear as an argument
to the indexed object would normally yield zero. This is because
Maxima would not know that the tensor represented by the indexed
object might depend implicitly on the corresponding coordinate. By
modifying the existing Maxima function diff in itensor, Maxima now
assumes that all indexed objects depend on any variable of
differentiation unless otherwise stated. This makes it possible for
the summation convention to be extended to derivative indices. It
should be noted that itensor does not possess the capabilities of
raising derivative indices, and so they are always treated as
covariant.
The following functions are available in the tensor package for
manipulating indexed objects. At present, with respect to the
simplification routines, it is assumed that indexed objects do not
by default possess symmetry properties. This can be overridden by
setting the variable allsym[false] to true, which will
result in treating all indexed objects completely symmetric in their
lists of covariant indices and symmetric in their lists of
contravariant indices.
The itensor package generally treats tensors as opaque objects. Tensorial
equations are manipulated based on algebraic rules, specifically symmetry
and contraction rules. In addition, the itensor package understands
covariant differentiation, curvature, and torsion. Calculations can be
performed relative to a metric of moving frame, depending on the setting
of the iframe_flag variable.
A sample session below demonstrates how to load the itensor package,
specify the name of the metric, and perform some simple calculations.
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) imetric(g);
(%o2) done
(%i3) components(g([i,j],[]),p([i,j],[])*e([],[]))$
(%i4) ishow(g([k,l],[]))$
(%t4) e p
k l
(%i5) ishow(diff(v([i],[]),t))$
(%t5) 0
(%i6) depends(v,t);
(%o6) [v(t)]
(%i7) ishow(diff(v([i],[]),t))$
d
(%t7) -- (v )
dt i
(%i8) ishow(idiff(v([i],[]),j))$
(%t8) v
i,j
(%i9) ishow(extdiff(v([i],[]),j))$
(%t9) v - v
j,i i,j
-----------
2
(%i10) ishow(liediff(v,w([i],[])))$
%3 %3
(%t10) v w + v w
i,%3 ,i %3
(%i11) ishow(covdiff(v([i],[]),j))$
%4
(%t11) v - v ichr2
i,j %4 i j
(%i12) ishow(ev(%,ichr2))$
%4 %5
(%t12) v - (g v (e p + e p - e p - e p
i,j %4 j %5,i ,i j %5 i j,%5 ,%5 i j
+ e p + e p ))/2
i %5,j ,j i %5
(%i13) iframe_flag:true;
(%o13) true
(%i14) ishow(covdiff(v([i],[]),j))$
%6
(%t14) v - v icc2
i,j %6 i j
(%i15) ishow(ev(%,icc2))$
%6
(%t15) v - v ifc2
i,j %6 i j
(%i16) ishow(radcan(ev(%,ifc2,ifc1)))$
%6 %7 %6 %7
(%t16) - (ifg v ifb + ifg v ifb - 2 v
%6 j %7 i %6 i j %7 i,j
%6 %7
- ifg v ifb )/2
%6 %7 i j
(%i17) ishow(canform(s([i,j],[])-s([j,i])))$
(%t17) s - s
i j j i
(%i18) decsym(s,2,0,[sym(all)],[]);
(%o18) done
(%i19) ishow(canform(s([i,j],[])-s([j,i])))$
(%t19) 0
(%i20) ishow(canform(a([i,j],[])+a([j,i])))$
(%t20) a + a
j i i j
(%i21) decsym(a,2,0,[anti(all)],[]);
(%o21) done
(%i22) ishow(canform(a([i,j],[])+a([j,i])))$
(%t22) 0
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Displays the contraction properties of its arguments as were given to
defcon. dispcon (all) displays all the contraction properties
which were defined.
Categories: Display functions
is a function which, by prompting, allows one to create an indexed
object called name with any number of tensorial and derivative
indices. Either a single index or a list of indices (which may be
null) is acceptable input (see the example under covdiff).
Categories: Package itensor
will change the name of all indexed objects called old to new
in expr. old may be either a symbol or a list of the form
[name, m, n] in which case only those indexed objects called
name with m covariant and n contravariant indices will be
renamed to new.
Categories: Package itensor
Lists all tensors in a tensorial expression, complete with their indices. E.g.,
(%i6) ishow(a([i,j],[k])*b([u],[],v)+c([x,y],[])*d([],[])*e)$
k
(%t6) d e c + a b
x y i j u,v
(%i7) ishow(listoftens(%))$
k
(%t7) [a , b , c , d]
i j u,v x y
Categories: Package itensor
displays expr with the indexed objects in it shown having their covariant indices as subscripts and contravariant indices as superscripts. The derivative indices are displayed as subscripts, separated from the covariant indices by a comma (see the examples throughout this document).
Categories: Package itensor
Returns a list of two elements. The first is a list of the free indices in expr (those that occur only once). The second is the list of the dummy indices in expr (those that occur exactly twice) as the following example demonstrates.
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) ishow(a([i,j],[k,l],m,n)*b([k,o],[j,m,p],q,r))$
k l j m p
(%t2) a b
i j,m n k o,q r
(%i3) indices(%);
(%o3) [[l, p, i, n, o, q, r], [k, j, m]]
A tensor product containing the same index more than twice is syntactically
illegal. indices attempts to deal with these expressions in a
reasonable manner; however, when it is called to operate upon such an
illegal expression, its behavior should be considered undefined.
Categories: Package itensor
Returns an expression equivalent to expr but with the dummy indices
in each term chosen from the set [%1, %2,...], if the optional second
argument is omitted. Otherwise, the dummy indices are indexed
beginning at the value of count. Each dummy index in a product
will be different. For a sum, rename will operate upon each term in
the sum resetting the counter with each term. In this way rename can
serve as a tensorial simplifier. In addition, the indices will be
sorted alphanumerically (if allsym is true) with respect to
covariant or contravariant indices depending upon the value of flipflag.
If flipflag is false then the indices will be renamed according
to the order of the contravariant indices. If flipflag is true
the renaming will occur according to the order of the covariant
indices. It often happens that the combined effect of the two renamings will
reduce an expression more than either one by itself.
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) allsym:true;
(%o2) true
(%i3) g([],[%4,%5])*g([],[%6,%7])*ichr2([%1,%4],[%3])*
ichr2([%2,%3],[u])*ichr2([%5,%6],[%1])*ichr2([%7,r],[%2])-
g([],[%4,%5])*g([],[%6,%7])*ichr2([%1,%2],[u])*
ichr2([%3,%5],[%1])*ichr2([%4,%6],[%3])*ichr2([%7,r],[%2]),noeval$
(%i4) expr:ishow(%)$
%4 %5 %6 %7 %3 u %1 %2
(%t4) g g ichr2 ichr2 ichr2 ichr2
%1 %4 %2 %3 %5 %6 %7 r
%4 %5 %6 %7 u %1 %3 %2
- g g ichr2 ichr2 ichr2 ichr2
%1 %2 %3 %5 %4 %6 %7 r
(%i5) flipflag:true;
(%o5) true
(%i6) ishow(rename(expr))$
%2 %5 %6 %7 %4 u %1 %3
(%t6) g g ichr2 ichr2 ichr2 ichr2
%1 %2 %3 %4 %5 %6 %7 r
%4 %5 %6 %7 u %1 %3 %2
- g g ichr2 ichr2 ichr2 ichr2
%1 %2 %3 %4 %5 %6 %7 r
(%i7) flipflag:false;
(%o7) false
(%i8) rename(%th(2));
(%o8) 0
(%i9) ishow(rename(expr))$
%1 %2 %3 %4 %5 %6 %7 u
(%t9) g g ichr2 ichr2 ichr2 ichr2
%1 %6 %2 %3 %4 r %5 %7
%1 %2 %3 %4 %6 %5 %7 u
- g g ichr2 ichr2 ichr2 ichr2
%1 %3 %2 %6 %4 r %5 %7
Categories: Package itensor
Displays expr with the indexed objects in it shown
having covariant indices as subscripts, contravariant indices as
superscripts. The derivative indices are displayed as subscripts,
separated from the covariant indices by a comma.
Categories: Package itensor · Display functions
Default value: false
If false then the indices will be
renamed according to the order of the contravariant indices,
otherwise according to the order of the covariant indices.
If flipflag is false then rename forms a list
of the contravariant indices as they are encountered from left to right
(if true then of the covariant indices). The first dummy
index in the list is renamed to %1, the next to %2, etc.
Then sorting occurs after the rename-ing (see the example
under rename).
Categories: Package itensor
gives tensor_1 the property that the
contraction of a product of tensor_1 and tensor_2 results in tensor_3
with the appropriate indices. If only one argument, tensor_1, is
given, then the contraction of the product of tensor_1 with any indexed
object having the appropriate indices (say my_tensor) will yield an
indexed object with that name, i.e. my_tensor, and with a new set of
indices reflecting the contractions performed.
For example, if imetric:g, then defcon(g) will implement the
raising and lowering of indices through contraction with the metric
tensor.
More than one defcon can be given for the same indexed object; the
latest one given which applies in a particular contraction will be
used.
contractions is a list of those indexed objects which have been given
contraction properties with defcon.
Categories: Package itensor
Removes all the contraction properties
from the (tensor_1, ..., tensor_n). remcon(all) removes all contraction
properties from all indexed objects.
Categories: Package itensor
Carries out the tensorial contractions in expr which may be any
combination of sums and products. This function uses the information
given to the defcon function. For best results, expr
should be fully expanded. ratexpand is the fastest way to expand
products and powers of sums if there are no variables in the denominators
of the terms. The gcd switch should be false if GCD
cancellations are unnecessary.
Categories: Package itensor
Must be executed before assigning components to a tensor for which
a built in value already exists as with ichr1, ichr2,
icurvature. See the example under icurvature.
Categories: Package itensor
permits one to assign an indicial value to an expression
expr giving the values of the components of tensor. These
are automatically substituted for the tensor whenever it occurs with
all of its indices. The tensor must be of the form t([...],[...])
where either list may be empty. expr can be any indexed expression
involving other objects with the same free indices as tensor. When
used to assign values to the metric tensor wherein the components
contain dummy indices one must be careful to define these indices to
avoid the generation of multiple dummy indices. Removal of this
assignment is given to the function remcomps.
It is important to keep in mind that components cares only about
the valence of a tensor, not about any particular index ordering. Thus
assigning components to, say, x([i,-j],[]), x([-j,i],[]), or
x([i],[j]) all produce the same result, namely components being
assigned to a tensor named x with valence (1,1).
Components can be assigned to an indexed expression in four ways, two
of which involve the use of the components command:
1) As an indexed expression. For instance:
(%i2) components(g([],[i,j]),e([],[i])*p([],[j]))$
(%i3) ishow(g([],[i,j]))$
i j
(%t3) e p
2) As a matrix:
(%i5) lg:-ident(4)$lg[1,1]:1$lg;
[ 1 0 0 0 ]
[ ]
[ 0 - 1 0 0 ]
(%o5) [ ]
[ 0 0 - 1 0 ]
[ ]
[ 0 0 0 - 1 ]
(%i6) components(g([i,j],[]),lg);
(%o6) done
(%i7) ishow(g([i,j],[]))$
(%t7) g
i j
(%i8) g([1,1],[]);
(%o8) 1
(%i9) g([4,4],[]);
(%o9) - 1
3) As a function. You can use a Maxima function to specify the
components of a tensor based on its indices. For instance, the following
code assigns kdelta to h if h has the same number
of covariant and contravariant indices and no derivative indices, and
g otherwise:
(%i4) h(l1,l2,[l3]):=if length(l1)=length(l2) and length(l3)=0
then kdelta(l1,l2) else apply(g,append([l1,l2], l3))$
(%i5) ishow(h([i],[j]))$
j
(%t5) kdelta
i
(%i6) ishow(h([i,j],[k],l))$
k
(%t6) g
i j,l
4) Using Maxima's pattern matching capabilities, specifically the
defrule and applyb1 commands:
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) matchdeclare(l1,listp);
(%o2) done
(%i3) defrule(r1,m(l1,[]),(i1:idummy(),
g([l1[1],l1[2]],[])*q([i1],[])*e([],[i1])))$
(%i4) defrule(r2,m([],l1),(i1:idummy(),
w([],[l1[1],l1[2]])*e([i1],[])*q([],[i1])))$
(%i5) ishow(m([i,n],[])*m([],[i,m]))$
i m
(%t5) m m
i n
(%i6) ishow(rename(applyb1(%,r1,r2)))$
%1 %2 %3 m
(%t6) e q w q e g
%1 %2 %3 n
Categories: Package itensor
Unbinds all values from tensor which were assigned with the
components function.
Categories: Package itensor
Shows component assignments of a tensor, as made using the components
command. This function can be particularly useful when a matrix is assigned
to an indicial tensor using components, as demonstrated by the
following example:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) load(itensor);
(%o2) /share/tensor/itensor.lisp
(%i3) lg:matrix([sqrt(r/(r-2*m)),0,0,0],[0,r,0,0],
[0,0,sin(theta)*r,0],[0,0,0,sqrt((r-2*m)/r)]);
[ r ]
[ sqrt(-------) 0 0 0 ]
[ r - 2 m ]
[ ]
[ 0 r 0 0 ]
(%o3) [ ]
[ 0 0 r sin(theta) 0 ]
[ ]
[ r - 2 m ]
[ 0 0 0 sqrt(-------) ]
[ r ]
(%i4) components(g([i,j],[]),lg);
(%o4) done
(%i5) showcomps(g([i,j],[]));
[ r ]
[ sqrt(-------) 0 0 0 ]
[ r - 2 m ]
[ ]
[ 0 r 0 0 ]
(%t5) g = [ ]
i j [ 0 0 r sin(theta) 0 ]
[ ]
[ r - 2 m ]
[ 0 0 0 sqrt(-------) ]
[ r ]
(%o5) false
The showcomps command can also display components of a tensor of
rank higher than 2.
Categories: Package itensor
Increments icounter and returns as its value an index of the form
%n where n is a positive integer. This guarantees that dummy indices
which are needed in forming expressions will not conflict with indices
already in use (see the example under indices).
Categories: Package itensor
Default value: %
Is the prefix for dummy indices (see the example under indices).
Categories: Package itensor
Default value: 1
Determines the numerical suffix to be used in
generating the next dummy index in the tensor package. The prefix is
determined by the option idummy (default: %).
Categories: Package itensor
is the generalized Kronecker delta function defined in
the itensor package with L1 the list of covariant indices and L2
the list of contravariant indices. kdelta([i],[j]) returns the ordinary
Kronecker delta. The command ev(expr,kdelta) causes the evaluation of
an expression containing kdelta([],[]) to the dimension of the
manifold.
In what amounts to an abuse of this notation, itensor also allows
kdelta to have 2 covariant and no contravariant, or 2 contravariant
and no covariant indices, in effect providing a co(ntra)variant "unit matrix"
capability. This is strictly considered a programming aid and not meant to
imply that kdelta([i,j],[]) is a valid tensorial object.
Categories: Package itensor
Symmetrized Kronecker delta, used in some calculations. For instance:
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) kdelta([1,2],[2,1]);
(%o2) - 1
(%i3) kdels([1,2],[2,1]);
(%o3) 1
(%i4) ishow(kdelta([a,b],[c,d]))$
c d d c
(%t4) kdelta kdelta - kdelta kdelta
a b a b
(%i4) ishow(kdels([a,b],[c,d]))$
c d d c
(%t4) kdelta kdelta + kdelta kdelta
a b a b
Categories: Package itensor
is the permutation (or Levi-Civita) tensor which yields 1 if the list L consists of an even permutation of integers, -1 if it consists of an odd permutation, and 0 if some indices in L are repeated.
Categories: Package itensor
Simplifies expressions containing the Levi-Civita symbol, converting these
to Kronecker-delta expressions when possible. The main difference between
this function and simply evaluating the Levi-Civita symbol is that direct
evaluation often results in Kronecker expressions containing numerical
indices. This is often undesirable as it prevents further simplification.
The lc2kdt function avoids this problem, yielding expressions that
are more easily simplified with rename or contract.
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) expr:ishow('levi_civita([],[i,j])
*'levi_civita([k,l],[])*a([j],[k]))$
i j k
(%t2) levi_civita a levi_civita
j k l
(%i3) ishow(ev(expr,levi_civita))$
i j k 1 2
(%t3) kdelta a kdelta
1 2 j k l
(%i4) ishow(ev(%,kdelta))$
i j j i k
(%t4) (kdelta kdelta - kdelta kdelta ) a
1 2 1 2 j
1 2 2 1
(kdelta kdelta - kdelta kdelta )
k l k l
(%i5) ishow(lc2kdt(expr))$
k i j k j i
(%t5) a kdelta kdelta - a kdelta kdelta
j k l j k l
(%i6) ishow(contract(expand(%)))$
i i
(%t6) a - a kdelta
l l
The lc2kdt function sometimes makes use of the metric tensor.
If the metric tensor was not defined previously with imetric,
this results in an error.
(%i7) expr:ishow('levi_civita([],[i,j])
*'levi_civita([],[k,l])*a([j,k],[]))$
i j k l
(%t7) levi_civita levi_civita a
j k
(%i8) ishow(lc2kdt(expr))$
Maxima encountered a Lisp error:
Error in $IMETRIC [or a callee]:
$IMETRIC [or a callee] requires less than two arguments.
Automatically continuing.
To reenable the Lisp debugger set *debugger-hook* to nil.
(%i9) imetric(g);
(%o9) done
(%i10) ishow(lc2kdt(expr))$
%3 i k %4 j l %3 i l %4 j
(%t10) (g kdelta g kdelta - g kdelta g
%3 %4 %3
k
kdelta ) a
%4 j k
(%i11) ishow(contract(expand(%)))$
l i l i j
(%t11) a - g a
j
Categories: Package itensor
Simplification rule used for expressions containing the unevaluated Levi-Civita
symbol (levi_civita). Along with lc_u, it can be used to simplify
many expressions more efficiently than the evaluation of levi_civita.
For example:
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) el1:ishow('levi_civita([i,j,k],[])*a([],[i])*a([],[j]))$
i j
(%t2) a a levi_civita
i j k
(%i3) el2:ishow('levi_civita([],[i,j,k])*a([i])*a([j]))$
i j k
(%t3) levi_civita a a
i j
(%i4) canform(contract(expand(applyb1(el1,lc_l,lc_u))));
(%t4) 0
(%i5) canform(contract(expand(applyb1(el2,lc_l,lc_u))));
(%t5) 0
Categories: Package itensor
Simplification rule used for expressions containing the unevaluated Levi-Civita
symbol (levi_civita). Along with lc_u, it can be used to simplify
many expressions more efficiently than the evaluation of levi_civita.
For details, see lc_l.
Categories: Package itensor
Simplifies expr by renaming (see rename)
and permuting dummy indices. rename is restricted to sums of tensor
products in which no derivatives are present. As such it is limited
and should only be used if canform is not capable of carrying out the
required simplification.
The canten function returns a mathematically correct result only
if its argument is an expression that is fully symmetric in its indices.
For this reason, canten returns an error if allsym is not
set to true.
Categories: Package itensor
Similar to canten but also performs index contraction.
Categories: Package itensor
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Default value: false
If true then all indexed objects
are assumed symmetric in all of their covariant and contravariant
indices. If false then no symmetries of any kind are assumed
in these indices. Derivative indices are always taken to be symmetric
unless iframe_flag is set to true.
Categories: Package itensor
Declares symmetry properties for tensor of m covariant and
n contravariant indices. The cov_i and contr_i are
pseudofunctions expressing symmetry relations among the covariant and
contravariant indices respectively. These are of the form
symoper(index_1, index_2,...) where symoper is one of
sym, anti or cyc and the index_i are integers
indicating the position of the index in the tensor. This will
declare tensor to be symmetric, antisymmetric or cyclic respectively
in the index_i. symoper(all) is also an allowable form which
indicates all indices obey the symmetry condition. For example, given an
object b with 5 covariant indices,
decsym(b,5,3,[sym(1,2),anti(3,4)],[cyc(all)]) declares b
symmetric in its first and second and antisymmetric in its third and
fourth covariant indices, and cyclic in all of its contravariant indices.
Either list of symmetry declarations may be null. The function which
performs the simplifications is canform as the example below
illustrates.
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) expr:contract( expand( a([i1, j1, k1], [])
*kdels([i, j, k], [i1, j1, k1])))$
(%i3) ishow(expr)$
(%t3) a + a + a + a + a + a
k j i k i j j k i j i k i k j i j k
(%i4) decsym(a,3,0,[sym(all)],[]);
(%o4) done
(%i5) ishow(canform(expr))$
(%t5) 6 a
i j k
(%i6) remsym(a,3,0);
(%o6) done
(%i7) decsym(a,3,0,[anti(all)],[]);
(%o7) done
(%i8) ishow(canform(expr))$
(%t8) 0
(%i9) remsym(a,3,0);
(%o9) done
(%i10) decsym(a,3,0,[cyc(all)],[]);
(%o10) done
(%i11) ishow(canform(expr))$
(%t11) 3 a + 3 a
i k j i j k
(%i12) dispsym(a,3,0);
(%o12) [[cyc, [[1, 2, 3]], []]]
Categories: Package itensor
Removes all symmetry properties from tensor which has m covariant indices and n contravariant indices.
Categories: Package itensor
Simplifies expr by renaming dummy
indices and reordering all indices as dictated by symmetry conditions
imposed on them. If allsym is true then all indices are assumed
symmetric, otherwise symmetry information provided by decsym
declarations will be used. The dummy indices are renamed in the same
manner as in the rename function. When canform is applied to a large
expression the calculation may take a considerable amount of time.
This time can be shortened by calling rename on the expression first.
Also see the example under decsym. Note: canform may not be able to
reduce an expression completely to its simplest form although it will
always return a mathematically correct result.
The optional second parameter rename, if set to false, suppresses renaming.
Categories: Package itensor
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is the usual Maxima differentiation function which has been expanded
in its abilities for itensor. It takes the derivative of expr with
respect to v_1 n_1 times, with respect to v_2 n_2
times, etc. For the tensor package, the function has been modified so
that the v_i may be integers from 1 up to the value of the variable
dim. This will cause the differentiation to be carried out with
respect to the v_ith member of the list vect_coords. If
vect_coords is bound to an atomic variable, then that variable
subscripted by v_i will be used for the variable of
differentiation. This permits an array of coordinate names or
subscripted names like x[1], x[2], ... to be used.
A further extension adds the ability to diff to compute derivatives
with respect to an indexed variable. In particular, the tensor package knows
how to differentiate expressions containing combinations of the metric tensor
and its derivatives with respect to the metric tensor and its first and
second derivatives. This capability is particularly useful when considering
Lagrangian formulations of a gravitational theory, allowing one to derive
the Einstein tensor and field equations from the action principle.
Categories: Package itensor
Indicial differentiation. Unlike diff, which differentiates
with respect to an independent variable, idiff) can be used
to differentiate with respect to a coordinate. For an indexed object,
this amounts to appending the v_i as derivative indices.
Subsequently, derivative indices will be sorted, unless iframe_flag
is set to true.
idiff can also differentiate the determinant of the metric
tensor. Thus, if imetric has been bound to G then
idiff(determinant(g),k) will return
2 * determinant(g) * ichr2([%i,k],[%i]) where the dummy index %i
is chosen appropriately.
Categories: Package itensor
Computes the Lie-derivative of the tensorial expression ten with respect to the vector field v. ten should be any indexed tensor expression; v should be the name (without indices) of a vector field. For example:
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) ishow(liediff(v,a([i,j],[])*b([],[k],l)))$
k %2 %2 %2
(%t2) b (v a + v a + v a )
,l i j,%2 ,j i %2 ,i %2 j
%1 k %1 k %1 k
+ (v b - b v + v b ) a
,%1 l ,l ,%1 ,l ,%1 i j
Categories: Package itensor
Evaluates all occurrences of the idiff command in the tensorial
expression ten.
Categories: Package itensor
Returns an expression equivalent to expr but with all derivatives
of indexed objects replaced by the noun form of the idiff function. Its
arguments would yield that indexed object if the differentiation were
carried out. This is useful when it is desired to replace a
differentiated indexed object with some function definition resulting
in expr and then carry out the differentiation by saying
ev(expr, idiff).
Categories: Package itensor
Equivalent to the execution of undiff, followed by ev and
rediff.
The point of this operation is to easily evalute expressions that cannot be directly evaluated in derivative form. For instance, the following causes an error:
(%i1) load(itensor); (%o1) /share/tensor/itensor.lisp (%i2) icurvature([i,j,k],[l],m); Maxima encountered a Lisp error: Error in $ICURVATURE [or a callee]: $ICURVATURE [or a callee] requires less than three arguments. Automatically continuing. To reenable the Lisp debugger set *debugger-hook* to nil.
However, if icurvature is entered in noun form, it can be evaluated
using evundiff:
(%i3) ishow('icurvature([i,j,k],[l],m))$
l
(%t3) icurvature
i j k,m
(%i4) ishow(evundiff(%))$
l l %1 l %1
(%t4) - ichr2 - ichr2 ichr2 - ichr2 ichr2
i k,j m %1 j i k,m %1 j,m i k
l l %1 l %1
+ ichr2 + ichr2 ichr2 + ichr2 ichr2
i j,k m %1 k i j,m %1 k,m i j
Note: In earlier versions of Maxima, derivative forms of the
Christoffel-symbols also could not be evaluated. This has been fixed now,
so evundiff is no longer necessary for expressions like this:
(%i5) imetric(g);
(%o5) done
(%i6) ishow(ichr2([i,j],[k],l))$
k %3
g (g - g + g )
j %3,i l i j,%3 l i %3,j l
(%t6) -----------------------------------------
2
k %3
g (g - g + g )
,l j %3,i i j,%3 i %3,j
+ -----------------------------------
2
Categories: Package itensor
Set to zero, in expr, all occurrences of the tensor_i that have no derivative indices.
Categories: Package itensor
Set to zero, in expr, all occurrences of the tensor_i that have derivative indices.
Categories: Package itensor
Set to zero, in expr, all occurrences of the differentiated object tensor that have n or more derivative indices as the following example demonstrates.
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) ishow(a([i],[J,r],k,r)+a([i],[j,r,s],k,r,s))$
J r j r s
(%t2) a + a
i,k r i,k r s
(%i3) ishow(flushnd(%,a,3))$
J r
(%t3) a
i,k r
Categories: Package itensor
Gives tensor_i the coordinate differentiation property that the
derivative of contravariant vector whose name is one of the
tensor_i yields a Kronecker delta. For example, if coord(x) has
been done then idiff(x([],[i]),j) gives kdelta([i],[j]).
coord is a list of all indexed objects having this property.
Categories: Package itensor
Removes the coordinate differentiation property from the tensor_i
that was established by the function coord. remcoord(all)
removes this property from all indexed objects.
Categories: Package itensor
Display expr in the same manner as show; however,
any tensor d'Alembertian occurring in expr will be indicated using the
symbol []. For example, []p([m],[n]) represents
g([],[i,j])*p([m],[n],i,j).
Categories: Package itensor
Simplifies expressions containing ordinary derivatives of
both covariant and contravariant forms of the metric tensor (the
current restriction). For example, conmetderiv can relate the
derivative of the contravariant metric tensor with the Christoffel
symbols as seen from the following:
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) ishow(g([],[a,b],c))$
a b
(%t2) g
,c
(%i3) ishow(conmetderiv(%,g))$
%1 b a %1 a b
(%t3) - g ichr2 - g ichr2
%1 c %1 c
Categories: Package itensor
Simplifies expressions containing products of the derivatives of the
metric tensor. Specifically, simpmetderiv recognizes two identities:
ab ab ab a g g + g g = (g g ) = (kdelta ) = 0 ,d bc bc,d bc ,d c ,d
hence
ab ab g g = - g g ,d bc bc,d
and
ab ab g g = g g ,j ab,i ,i ab,j
which follows from the symmetries of the Christoffel symbols.
The simpmetderiv function takes one optional parameter which, when
present, causes the function to stop after the first successful
substitution in a product expression. The simpmetderiv function
also makes use of the global variable flipflag which determines
how to apply a "canonical" ordering to the product indices.
Put together, these capabilities can be used to achieve powerful
simplifications that are difficult or impossible to accomplish otherwise.
This is demonstrated through the following example that explicitly uses the
partial simplification features of simpmetderiv to obtain a
contractible expression:
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) imetric(g);
(%o2) done
(%i3) ishow(g([],[a,b])*g([],[b,c])*g([a,b],[],d)*g([b,c],[],e))$
a b b c
(%t3) g g g g
a b,d b c,e
(%i4) ishow(canform(%))$
errexp1 has improper indices
-- an error. Quitting. To debug this try debugmode(true);
(%i5) ishow(simpmetderiv(%))$
a b b c
(%t5) g g g g
a b,d b c,e
(%i6) flipflag:not flipflag;
(%o6) true
(%i7) ishow(simpmetderiv(%th(2)))$
a b b c
(%t7) g g g g
,d ,e a b b c
(%i8) flipflag:not flipflag;
(%o8) false
(%i9) ishow(simpmetderiv(%th(2),stop))$
a b b c
(%t9) - g g g g
,e a b,d b c
(%i10) ishow(contract(%))$
b c
(%t10) - g g
,e c b,d
See also weyl.dem for an example that uses simpmetderiv
and conmetderiv together to simplify contractions of the Weyl tensor.
Categories: Package itensor
Set to zero, in expr, all occurrences of tensor that have
exactly one derivative index.
Categories: Package itensor
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Specifies the metric by assigning the variable imetric:g in
addition, the contraction properties of the metric g are set up by
executing the commands defcon(g), defcon(g, g, kdelta).
The variable imetric (unbound by default), is bound to the metric, assigned by
the imetric(g) command.
Categories: Package itensor
Sets the dimensions of the metric. Also initializes the antisymmetry properties of the Levi-Civita symbols for the given dimension.
Categories: Package itensor
Yields the Christoffel symbol of the first kind via the definition
(g + g - g )/2 .
ik,j jk,i ij,k
To evaluate the Christoffel symbols for a particular metric, the
variable imetric must be assigned a name as in the example under chr2.
Categories: Package itensor
Yields the Christoffel symbol of the second kind defined by the relation
ks
ichr2([i,j],[k]) = g (g + g - g )/2
is,j js,i ij,s
Categories: Package itensor
Yields the Riemann
curvature tensor in terms of the Christoffel symbols of the second
kind (ichr2). The following notation is used:
h h h %1 h
icurvature = - ichr2 - ichr2 ichr2 + ichr2
i j k i k,j %1 j i k i j,k
h %1
+ ichr2 ichr2
%1 k i j
Categories: Package itensor
Yields the covariant derivative of expr with
respect to the variables v_i in terms of the Christoffel symbols of the
second kind (ichr2). In order to evaluate these, one should use
ev(expr,ichr2).
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) entertensor()$
Enter tensor name: a;
Enter a list of the covariant indices: [i,j];
Enter a list of the contravariant indices: [k];
Enter a list of the derivative indices: [];
k
(%t2) a
i j
(%i3) ishow(covdiff(%,s))$
k %1 k %1 k
(%t3) - a ichr2 - a ichr2 + a
i %1 j s %1 j i s i j,s
k %1
+ ichr2 a
%1 s i j
(%i4) imetric:g;
(%o4) g
(%i5) ishow(ev(%th(2),ichr2))$
%1 %4 k
g a (g - g + g )
i %1 s %4,j j s,%4 j %4,s
(%t5) - ------------------------------------------
2
%1 %3 k
g a (g - g + g )
%1 j s %3,i i s,%3 i %3,s
- ------------------------------------------
2
k %2 %1
g a (g - g + g )
i j s %2,%1 %1 s,%2 %1 %2,s k
+ ------------------------------------------- + a
2 i j,s
(%i6)
Categories: Package itensor
Imposes the Lorentz condition by substituting 0 for all indexed objects in expr that have a derivative index identical to a contravariant index.
Categories: Package itensor
Causes undifferentiated Christoffel symbols and
first derivatives of the metric tensor vanish in expr. The name
in the igeodesic_coords function refers to the metric name
(if it appears in expr) while the connection coefficients must be
called with the names ichr1 and/or ichr2. The following example
demonstrates the verification of the cyclic identity satisfied by the Riemann
curvature tensor using the igeodesic_coords function.
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) ishow(icurvature([r,s,t],[u]))$
u u %1 u
(%t2) - ichr2 - ichr2 ichr2 + ichr2
r t,s %1 s r t r s,t
u %1
+ ichr2 ichr2
%1 t r s
(%i3) ishow(igeodesic_coords(%,ichr2))$
u u
(%t3) ichr2 - ichr2
r s,t r t,s
(%i4) ishow(igeodesic_coords(icurvature([r,s,t],[u]),ichr2)+
igeodesic_coords(icurvature([s,t,r],[u]),ichr2)+
igeodesic_coords(icurvature([t,r,s],[u]),ichr2))$
u u u u
(%t4) - ichr2 + ichr2 + ichr2 - ichr2
t s,r t r,s s t,r s r,t
u u
- ichr2 + ichr2
r t,s r s,t
(%i5) canform(%);
(%o5) 0
Categories: Package itensor
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Maxima now has the ability to perform calculations using moving frames. These can be orthonormal frames (tetrads, vielbeins) or an arbitrary frame.
To use frames, you must first set iframe_flag to true. This
causes the Christoffel-symbols, ichr1 and ichr2, to be replaced
by the more general frame connection coefficients icc1 and icc2
in calculations. Speficially, the behavior of covdiff and
icurvature is changed.
The frame is defined by two tensors: the inverse frame field (ifri,
the dual basis tetrad),
and the frame metric ifg. The frame metric is the identity matrix for
orthonormal frames, or the Lorentz metric for orthonormal frames in Minkowski
spacetime. The inverse frame field defines the frame base (unit vectors).
Contraction properties are defined for the frame field and the frame metric.
When iframe_flag is true, many itensor expressions use the frame
metric ifg instead of the metric defined by imetric for
raising and lowerind indices.
IMPORTANT: Setting the variable iframe_flag to true does NOT
undefine the contraction properties of a metric defined by a call to
defcon or imetric. If a frame field is used, it is best to
define the metric by assigning its name to the variable imetric
and NOT invoke the imetric function.
Maxima uses these two tensors to define the frame coefficients (ifc1
and ifc2) which form part of the connection coefficients (icc1
and icc2), as the following example demonstrates:
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) iframe_flag:true;
(%o2) true
(%i3) ishow(covdiff(v([],[i]),j))$
i i %1
(%t3) v + icc2 v
,j %1 j
(%i4) ishow(ev(%,icc2))$
%1 i i
(%t4) v ifc2 + v
%1 j ,j
(%i5) ishow(ev(%,ifc2))$
%1 i %2 i
(%t5) v ifg ifc1 + v
%1 j %2 ,j
(%i6) ishow(ev(%,ifc1))$
%1 i %2
v ifg (ifb - ifb + ifb )
j %2 %1 %2 %1 j %1 j %2 i
(%t6) -------------------------------------------------- + v
2 ,j
(%i7) ishow(ifb([a,b,c]))$
%3 %4
(%t7) (ifri - ifri ) ifr ifr
a %3,%4 a %4,%3 b c
An alternate method is used to compute the frame bracket (ifb) if
the iframe_bracket_form flag is set to false:
(%i8) block([iframe_bracket_form:false],ishow(ifb([a,b,c])))$
%6 %5 %5 %6
(%t8) ifri (ifr ifr - ifr ifr )
a %5 b c,%6 b,%6 c
Since in this version of Maxima, contraction identities for ifr and
ifri are always defined, as is the frame bracket (ifb), this
function does nothing.
Categories: Package itensor
The frame bracket. The contribution of the frame metric to the connection coefficients is expressed using the frame bracket:
- ifb + ifb + ifb
c a b b c a a b c
ifc1 = --------------------------------
abc 2
The frame bracket itself is defined in terms of the frame field and frame
metric. Two alternate methods of computation are used depending on the
value of frame_bracket_form. If true (the default) or if the
itorsion_flag is true:
d e f ifb = ifr ifr (ifri - ifri - ifri itr ) abc b c a d,e a e,d a f d e
Otherwise:
e d d e ifb = (ifr ifr - ifr ifr ) ifri abc b c,e b,e c a d
Categories: Package itensor
Connection coefficients of the first kind. In itensor, defined as
icc1 = ichr1 - ikt1 - inmc1
abc abc abc abc
In this expression, if iframe_flag is true, the Christoffel-symbol
ichr1 is replaced with the frame connection coefficient ifc1.
If itorsion_flag is false, ikt1
will be omitted. It is also omitted if a frame base is used, as the
torsion is already calculated as part of the frame bracket.
Lastly, of inonmet_flag is false,
inmc1 will not be present.
Categories: Package itensor
Connection coefficients of the second kind. In itensor, defined as
c c c c
icc2 = ichr2 - ikt2 - inmc2
ab ab ab ab
In this expression, if iframe_flag is true, the Christoffel-symbol
ichr2 is replaced with the frame connection coefficient ifc2.
If itorsion_flag is false, ikt2
will be omitted. It is also omitted if a frame base is used, as the
torsion is already calculated as part of the frame bracket.
Lastly, of inonmet_flag is false,
inmc2 will not be present.
Categories: Package itensor
Frame coefficient of the first kind (also known as Ricci-rotation coefficients.) This tensor represents the contribution of the frame metric to the connection coefficient of the first kind. Defined as:
- ifb + ifb + ifb
c a b b c a a b c
ifc1 = --------------------------------
abc 2
Categories: Package itensor
Frame coefficient of the second kind. This tensor represents the contribution
of the frame metric to the connection coefficient of the second kind. Defined
as a permutation of the frame bracket (ifb) with the appropriate
indices raised and lowered as necessary:
c cd
ifc2 = ifg ifc1
ab abd
Categories: Package itensor
The frame field. Contracts with the inverse frame field (ifri) to
form the frame metric (ifg).
Categories: Package itensor
The inverse frame field. Specifies the frame base (dual basis vectors). Along with the frame metric, it forms the basis of all calculations based on frames.
Categories: Package itensor
The frame metric. Defaults to kdelta, but can be changed using
components.
Categories: Package itensor
The inverse frame metric. Contracts with the frame metric (ifg)
to kdelta.
Categories: Package itensor
Default value: true
Specifies how the frame bracket (ifb) is computed.
Categories: Package itensor
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Maxima can now take into account torsion and nonmetricity. When the flag
itorsion_flag is set to true, the contribution of torsion
is added to the connection coefficients. Similarly, when the flag
inonmet_flag is true, nonmetricity components are included.
The nonmetricity vector. Conformal nonmetricity is defined through the
covariant derivative of the metric tensor. Normally zero, the metric
tensor's covariant derivative will evaluate to the following when
inonmet_flag is set to true:
g =- g inm ij;k ij k
Categories: Package itensor
Covariant permutation of the nonmetricity vector components. Defined as
g inm - inm g - g inm
ab c a bc ac b
inmc1 = ------------------------------
abc 2
(Substitute ifg in place of g if a frame metric is used.)
Categories: Package itensor
Contravariant permutation of the nonmetricity vector components. Used
in the connection coefficients if inonmet_flag is true. Defined
as:
c c cd
-inm kdelta - kdelta inm + g inm g
c a b a b d ab
inmc2 = -------------------------------------------
ab 2
(Substitute ifg in place of g if a frame metric is used.)
Categories: Package itensor
Covariant permutation of the torsion tensor (also known as contorsion). Defined as:
d d d
-g itr - g itr - itr g
ad cb bd ca ab cd
ikt1 = ----------------------------------
abc 2
(Substitute ifg in place of g if a frame metric is used.)
Categories: Package itensor
Contravariant permutation of the torsion tensor (also known as contorsion). Defined as:
c cd
ikt2 = g ikt1
ab abd
(Substitute ifg in place of g if a frame metric is used.)
Categories: Package itensor
The torsion tensor. For a metric with torsion, repeated covariant differentiation on a scalar function will not commute, as demonstrated by the following example:
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) imetric:g;
(%o2) g
(%i3) covdiff( covdiff( f( [], []), i), j)
- covdiff( covdiff( f( [], []), j), i)$
(%i4) ishow(%)$
%4 %2
(%t4) f ichr2 - f ichr2
,%4 j i ,%2 i j
(%i5) canform(%);
(%o5) 0
(%i6) itorsion_flag:true;
(%o6) true
(%i7) covdiff( covdiff( f( [], []), i), j)
- covdiff( covdiff( f( [], []), j), i)$
(%i8) ishow(%)$
%8 %6
(%t8) f icc2 - f icc2 - f + f
,%8 j i ,%6 i j ,j i ,i j
(%i9) ishow(canform(%))$
%1 %1
(%t9) f icc2 - f icc2
,%1 j i ,%1 i j
(%i10) ishow(canform(ev(%,icc2)))$
%1 %1
(%t10) f ikt2 - f ikt2
,%1 i j ,%1 j i
(%i11) ishow(canform(ev(%,ikt2)))$
%2 %1 %2 %1
(%t11) f g ikt1 - f g ikt1
,%2 i j %1 ,%2 j i %1
(%i12) ishow(factor(canform(rename(expand(ev(%,ikt1))))))$
%3 %2 %1 %1
f g g (itr - itr )
,%3 %2 %1 j i i j
(%t12) ------------------------------------
2
(%i13) decsym(itr,2,1,[anti(all)],[]);
(%o13) done
(%i14) defcon(g,g,kdelta);
(%o14) done
(%i15) subst(g,nounify(g),%th(3))$
(%i16) ishow(canform(contract(%)))$
%1
(%t16) - f itr
,%1 i j
Categories: Package itensor
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The itensor package can perform operations on totally antisymmetric
covariant tensor fields. A totally antisymmetric tensor field of rank
(0,L) corresponds with a differential L-form. On these objects, a
multiplication operation known as the exterior product, or wedge product,
is defined.
Unfortunately, not all authors agree on the definition of the wedge product. Some authors prefer a definition that corresponds with the notion of antisymmetrization: in these works, the wedge product of two vector fields, for instance, would be defined as
a a - a a
i j j i
a /\ a = -----------
i j 2
More generally, the product of a p-form and a q-form would be defined as
1 k1..kp l1..lq A /\ B = ------ D A B i1..ip j1..jq (p+q)! i1..ip j1..jq k1..kp l1..lq
where D stands for the Kronecker-delta.
Other authors, however, prefer a "geometric" definition that corresponds with the notion of the volume element:
a /\ a = a a - a a i j i j j i
and, in the general case
1 k1..kp l1..lq A /\ B = ----- D A B i1..ip j1..jq p! q! i1..ip j1..jq k1..kp l1..lq
Since itensor is a tensor algebra package, the first of these two
definitions appears to be the more natural one. Many applications, however,
utilize the second definition. To resolve this dilemma, a flag has been
implemented that controls the behavior of the wedge product: if
igeowedge_flag is false (the default), the first, "tensorial"
definition is used, otherwise the second, "geometric" definition will
be applied.
The wedge product operator is denoted by the tilde ~. This is
a binary operator. Its arguments should be expressions involving scalars,
covariant tensors of rank one, or covariant tensors of rank l that
have been declared antisymmetric in all covariant indices.
The behavior of the wedge product operator is controlled by the
igeowedge_flag flag, as in the following example:
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) ishow(a([i])~b([j]))$
a b - b a
i j i j
(%t2) -------------
2
(%i3) decsym(a,2,0,[anti(all)],[]);
(%o3) done
(%i4) ishow(a([i,j])~b([k]))$
a b + b a - a b
i j k i j k i k j
(%t4) ---------------------------
3
(%i5) igeowedge_flag:true;
(%o5) true
(%i6) ishow(a([i])~b([j]))$
(%t6) a b - b a
i j i j
(%i7) ishow(a([i,j])~b([k]))$
(%t7) a b + b a - a b
i j k i j k i k j
Categories: Package itensor · Operators
The vertical bar | denotes the "contraction with a vector" binary
operation. When a totally antisymmetric covariant tensor is contracted
with a contravariant vector, the result is the same regardless which index
was used for the contraction. Thus, it is possible to define the
contraction operation in an index-free manner.
In the itensor package, contraction with a vector is always carried out
with respect to the first index in the literal sorting order. This ensures
better simplification of expressions involving the | operator. For instance:
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) decsym(a,2,0,[anti(all)],[]);
(%o2) done
(%i3) ishow(a([i,j],[])|v)$
%1
(%t3) v a
%1 j
(%i4) ishow(a([j,i],[])|v)$
%1
(%t4) - v a
%1 j
Note that it is essential that the tensors used with the | operator be
declared totally antisymmetric in their covariant indices. Otherwise,
the results will be incorrect.
Categories: Package itensor · Operators
Computes the exterior derivative of expr with respect to the index
i. The exterior derivative is formally defined as the wedge
product of the partial derivative operator and a differential form. As
such, this operation is also controlled by the setting of igeowedge_flag.
For instance:
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) ishow(extdiff(v([i]),j))$
v - v
j,i i,j
(%t2) -----------
2
(%i3) decsym(a,2,0,[anti(all)],[]);
(%o3) done
(%i4) ishow(extdiff(a([i,j]),k))$
a - a + a
j k,i i k,j i j,k
(%t4) ------------------------
3
(%i5) igeowedge_flag:true;
(%o5) true
(%i6) ishow(extdiff(v([i]),j))$
(%t6) v - v
j,i i,j
(%i7) ishow(extdiff(a([i,j]),k))$
(%t7) - (a - a + a )
k j,i k i,j j i,k
Categories: Package itensor
Compute the Hodge-dual of expr. For instance:
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) imetric(g);
(%o2) done
(%i3) idim(4);
(%o3) done
(%i4) icounter:100;
(%o4) 100
(%i5) decsym(A,3,0,[anti(all)],[])$
(%i6) ishow(A([i,j,k],[]))$
(%t6) A
i j k
(%i7) ishow(canform(hodge(%)))$
%1 %2 %3 %4
levi_civita g A
%1 %102 %2 %3 %4
(%t7) -----------------------------------------
6
(%i8) ishow(canform(hodge(%)))$
%1 %2 %3 %8 %4 %5 %6 %7
(%t8) levi_civita levi_civita g
%1 %106
g g g A /6
%2 %107 %3 %108 %4 %8 %5 %6 %7
(%i9) lc2kdt(%)$
(%i10) %,kdelta$
(%i11) ishow(canform(contract(expand(%))))$
(%t11) - A
%106 %107 %108
Categories: Package itensor
Default value: false
Controls the behavior of the wedge product and exterior derivative. When
set to false (the default), the notion of differential forms will
correspond with that of a totally antisymmetric covariant tensor field.
When set to true, differential forms will agree with the notion
of the volume element.
Categories: Package itensor
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The itensor package provides limited support for exporting tensor
expressions to TeX. Since itensor expressions appear as function calls,
the regular Maxima tex command will not produce the expected
output. You can try instead the tentex command, which attempts
to translate tensor expressions into appropriately indexed TeX objects.
To use the tentex function, you must first load tentex,
as in the following example:
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) load(tentex);
(%o2) /share/tensor/tentex.lisp
(%i3) idummyx:m;
(%o3) m
(%i4) ishow(icurvature([j,k,l],[i]))$
m1 i m1 i i
(%t4) ichr2 ichr2 - ichr2 ichr2 - ichr2
j k m1 l j l m1 k j l,k
i
+ ichr2
j k,l
(%i5) tentex(%)$
$$\Gamma_{j\,k}^{m_1}\,\Gamma_{l\,m_1}^{i}-\Gamma_{j\,l}^{m_1}\,
\Gamma_{k\,m_1}^{i}-\Gamma_{j\,l,k}^{i}+\Gamma_{j\,k,l}^{i}$$
Note the use of the idummyx assignment, to avoid the appearance
of the percent sign in the TeX expression, which may lead to compile errors.
NB: This version of the tentex function is somewhat experimental.
Categories: Package itensor · TeX output
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The itensor package has the ability to generate Maxima code that can
then be executed in the context of the ctensor package. The function that performs
this task is ic_convert.
Converts the itensor equation eqn to a ctensor assignment statement.
Implied sums over dummy indices are made explicit while indexed
objects are transformed into arrays (the array subscripts are in the
order of covariant followed by contravariant indices of the indexed
objects). The derivative of an indexed object will be replaced by the
noun form of diff taken with respect to ct_coords subscripted
by the derivative index. The Christoffel symbols ichr1 and ichr2
will be translated to lcs and mcs, respectively and if
metricconvert is true then all occurrences of the metric
with two covariant (contravariant) indices will be renamed to lg
(ug). In addition, do loops will be introduced summing over
all free indices so that the
transformed assignment statement can be evaluated by just doing
ev. The following examples demonstrate the features of this
function.
(%i1) load(itensor);
(%o1) /share/tensor/itensor.lisp
(%i2) eqn:ishow(t([i,j],[k])=f([],[])*g([l,m],[])*a([],[m],j)
*b([i],[l,k]))$
k m l k
(%t2) t = f a b g
i j ,j i l m
(%i3) ic_convert(eqn);
(%o3) for i thru dim do (for j thru dim do (
for k thru dim do
t : f sum(sum(diff(a , ct_coords ) b
i, j, k m j i, l, k
g , l, 1, dim), m, 1, dim)))
l, m
(%i4) imetric(g);
(%o4) done
(%i5) metricconvert:true;
(%o5) true
(%i6) ic_convert(eqn);
(%o6) for i thru dim do (for j thru dim do (
for k thru dim do
t : f sum(sum(diff(a , ct_coords ) b
i, j, k m j i, l, k
lg , l, 1, dim), m, 1, dim)))
l, m
Categories: Package itensor · Package ctensor
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The following Maxima words are used by the itensor package internally and
should not be redefined:
Keyword Comments ------------------------------------------ indices2() Internal version of indices() conti Lists contravariant indices covi Lists covariant indices of a indexed object deri Lists derivative indices of an indexed object name Returns the name of an indexed object concan irpmon lc0 _lc2kdt0 _lcprod _extlc
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| 26.1 Introduction to ctensor | ||
| 26.2 Functions and Variables for ctensor |
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ctensor is a component tensor manipulation package. To use the ctensor
package, type load(ctensor).
To begin an interactive session with ctensor, type csetup(). You are
first asked to specify the dimension of the manifold. If the dimension
is 2, 3 or 4 then the list of coordinates defaults to [x,y], [x,y,z]
or [x,y,z,t] respectively.
These names may be changed by assigning a new list of coordinates to
the variable ct_coords (described below) and the user is queried about
this. Care must be taken to avoid the coordinate names conflicting
with other object definitions.
Next, the user enters the metric either directly or from a file by
specifying its ordinal position.
The metric is stored in the matrix
lg. Finally, the metric inverse is computed and stored in the matrix
ug. One has the option of carrying out all calculations in a power
series.
A sample protocol is begun below for the static, spherically symmetric
metric (standard coordinates) which will be applied to the problem of
deriving Einstein's vacuum equations (which lead to the Schwarzschild
solution) as an example. Many of the functions in ctensor will be
displayed for the standard metric as examples.
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) csetup();
Enter the dimension of the coordinate system:
4;
Do you wish to change the coordinate names?
n;
Do you want to
1. Enter a new metric?
2. Enter a metric from a file?
3. Approximate a metric with a Taylor series?
1;
Is the matrix 1. Diagonal 2. Symmetric 3. Antisymmetric 4. General
Answer 1, 2, 3 or 4
1;
Row 1 Column 1:
a;
Row 2 Column 2:
x^2;
Row 3 Column 3:
x^2*sin(y)^2;
Row 4 Column 4:
-d;
Matrix entered.
Enter functional dependencies with the DEPENDS function or 'N' if none
depends([a,d],x);
Do you wish to see the metric?
y;
[ a 0 0 0 ]
[ ]
[ 2 ]
[ 0 x 0 0 ]
[ ]
[ 2 2 ]
[ 0 0 x sin (y) 0 ]
[ ]
[ 0 0 0 - d ]
(%o2) done
(%i3) christof(mcs);
a
x
(%t3) mcs = ---
1, 1, 1 2 a
1
(%t4) mcs = -
1, 2, 2 x
1
(%t5) mcs = -
1, 3, 3 x
d
x
(%t6) mcs = ---
1, 4, 4 2 d
x
(%t7) mcs = - -
2, 2, 1 a
cos(y)
(%t8) mcs = ------
2, 3, 3 sin(y)
2
x sin (y)
(%t9) mcs = - ---------
3, 3, 1 a
(%t10) mcs = - cos(y) sin(y)
3, 3, 2
d
x
(%t11) mcs = ---
4, 4, 1 2 a
(%o11) done
Categories: Tensors · Share packages · Package ctensor
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A function in the ctensor (component tensor) package
which initializes the package and allows the user to enter a metric
interactively. See ctensor for more details.
Categories: Package ctensor
A function in the ctensor (component tensor) package
that computes the metric inverse and sets up the package for
further calculations.
If cframe_flag is false, the function computes the inverse metric
ug from the (user-defined) matrix lg. The metric determinant is
also computed and stored in the variable gdet. Furthermore, the
package determines if the metric is diagonal and sets the value
of diagmetric accordingly. If the optional argument dis
is present and not equal to false, the user is prompted to see
the metric inverse.
If cframe_flag is true, the function expects that the values of
fri (the inverse frame matrix) and lfg (the frame metric) are
defined. From these, the frame matrix fr and the inverse frame
metric ufg are computed.
Categories: Package ctensor
Sets up a predefined coordinate system and metric. The argument coordinate_system can be one of the following symbols:
SYMBOL Dim Coordinates Description/comments
------------------------------------------------------------------
cartesian2d 2 [x,y] Cartesian 2D coordinate
system
polar 2 [r,phi] Polar coordinate system
elliptic 2 [u,v] Elliptic coord. system
confocalelliptic 2 [u,v] Confocal elliptic
coordinates
bipolar 2 [u,v] Bipolar coord. system
parabolic 2 [u,v] Parabolic coord. system
cartesian3d 3 [x,y,z] Cartesian 3D coordinate
system
polarcylindrical 3 [r,theta,z] Polar 2D with
cylindrical z
ellipticcylindrical 3 [u,v,z] Elliptic 2D with
cylindrical z
confocalellipsoidal 3 [u,v,w] Confocal ellipsoidal
bipolarcylindrical 3 [u,v,z] Bipolar 2D with
cylindrical z
paraboliccylindrical 3 [u,v,z] Parabolic 2D with
cylindrical z
paraboloidal 3 [u,v,phi] Paraboloidal coords.
conical 3 [u,v,w] Conical coordinates
toroidal 3 [phi,u,v] Toroidal coordinates
spherical 3 [r,theta,phi] Spherical coord. system
oblatespheroidal 3 [u,v,phi] Oblate spheroidal
coordinates
oblatespheroidalsqrt 3 [u,v,phi]
prolatespheroidal 3 [u,v,phi] Prolate spheroidal
coordinates
prolatespheroidalsqrt 3 [u,v,phi]
ellipsoidal 3 [r,theta,phi] Ellipsoidal coordinates
cartesian4d 4 [x,y,z,t] Cartesian 4D coordinate
system
spherical4d 4 [r,theta,eta,phi] Spherical 4D coordinate
system
exteriorschwarzschild 4 [t,r,theta,phi] Schwarzschild metric
interiorschwarzschild 4 [t,z,u,v] Interior Schwarzschild
metric
kerr_newman 4 [t,r,theta,phi] Charged axially
symmetric metric
coordinate_system can also be a list of transformation functions,
followed by a list containing the coordinate variables. For instance,
you can specify a spherical metric as follows:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) ct_coordsys([r*cos(theta)*cos(phi),r*cos(theta)*sin(phi),
r*sin(theta),[r,theta,phi]]);
(%o2) done
(%i3) lg:trigsimp(lg);
[ 1 0 0 ]
[ ]
[ 2 ]
(%o3) [ 0 r 0 ]
[ ]
[ 2 2 ]
[ 0 0 r cos (theta) ]
(%i4) ct_coords;
(%o4) [r, theta, phi]
(%i5) dim;
(%o5) 3
Transformation functions can also be used when cframe_flag is true:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) cframe_flag:true;
(%o2) true
(%i3) ct_coordsys([r*cos(theta)*cos(phi),r*cos(theta)*sin(phi),
r*sin(theta),[r,theta,phi]]);
(%o3) done
(%i4) fri;
(%o4)
[cos(phi)cos(theta) -cos(phi) r sin(theta) -sin(phi) r cos(theta)]
[ ]
[sin(phi)cos(theta) -sin(phi) r sin(theta) cos(phi) r cos(theta)]
[ ]
[ sin(theta) r cos(theta) 0 ]
(%i5) cmetric();
(%o5) false
(%i6) lg:trigsimp(lg);
[ 1 0 0 ]
[ ]
[ 2 ]
(%o6) [ 0 r 0 ]
[ ]
[ 2 2 ]
[ 0 0 r cos (theta) ]
The optional argument extra_arg can be any one of the following:
cylindrical tells ct_coordsys to attach an additional cylindrical coordinate.
minkowski tells ct_coordsys to attach an additional coordinate with negative metric signature.
all tells ct_coordsys to call cmetric and christof(false) after setting up the metric.
If the global variable verbose is set to true, ct_coordsys displays the values of dim, ct_coords, and either lg or lfg and fri, depending on the value of cframe_flag.
Categories: Package ctensor
Initializes the ctensor package.
The init_ctensor function reinitializes the ctensor package. It removes all arrays and matrices used by ctensor, resets all flags, resets dim to 4, and resets the frame metric to the Lorentz-frame.
Categories: Package ctensor
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The main purpose of the ctensor package is to compute the tensors
of curved space(time), most notably the tensors used in general
relativity.
When a metric base is used, ctensor can compute the following tensors:
lg -- ug
\ \
lcs -- mcs -- ric -- uric
\ \ \
\ tracer - ein -- lein
\
riem -- lriem -- weyl
\
uriem
ctensor can also work using moving frames. When cframe_flag is
set to true, the following tensors can be calculated:
lfg -- ufg
\
fri -- fr -- lcs -- mcs -- lriem -- ric -- uric
\ | \ \ \
lg -- ug | weyl tracer - ein -- lein
|\
| riem
|
\uriem
A function in the ctensor (component tensor)
package. It computes the Christoffel symbols of both
kinds. The argument dis determines which results are to be immediately
displayed. The Christoffel symbols of the first and second kinds are
stored in the arrays lcs[i,j,k] and mcs[i,j,k] respectively and
defined to be symmetric in the first two indices. If the argument to
christof is lcs or mcs then the unique non-zero values of lcs[i,j,k]
or mcs[i,j,k], respectively, will be displayed. If the argument is all
then the unique non-zero values of lcs[i,j,k] and mcs[i,j,k] will be
displayed. If the argument is false then the display of the elements
will not occur. The array elements mcs[i,j,k] are defined in such a
manner that the final index is contravariant.
Categories: Package ctensor
A function in the ctensor (component tensor)
package. ricci computes the covariant (symmetric)
components ric[i,j] of the Ricci tensor. If the argument dis is true,
then the non-zero components are displayed.
Categories: Package ctensor
This function first computes the
covariant components ric[i,j] of the Ricci tensor.
Then the mixed Ricci tensor is computed using the
contravariant metric tensor. If the value of the argument dis
is true, then these mixed components, uric[i,j] (the
index i is covariant and the index j is contravariant), will be displayed
directly. Otherwise, ricci(false) will simply compute the entries
of the array uric[i,j] without displaying the results.
Categories: Package ctensor
Returns the scalar curvature (obtained by contracting the Ricci tensor) of the Riemannian manifold with the given metric.
Categories: Package ctensor
A function in the ctensor (component tensor)
package. einstein computes the mixed Einstein tensor
after the Christoffel symbols and Ricci tensor have been obtained
(with the functions christof and ricci). If the argument dis is
true, then the non-zero values of the mixed Einstein tensor ein[i,j]
will be displayed where j is the contravariant index.
The variable rateinstein will cause the rational simplification on
these components. If ratfac is true then the components will
also be factored.
Categories: Package ctensor
Covariant Einstein-tensor. leinstein stores the values of the covariant Einstein tensor in the array lein. The covariant Einstein-tensor is computed from the mixed Einstein tensor ein by multiplying it with the metric tensor. If the argument dis is true, then the non-zero values of the covariant Einstein tensor are displayed.
Categories: Package ctensor
A function in the ctensor (component tensor)
package. riemann computes the Riemann curvature tensor
from the given metric and the corresponding Christoffel symbols. The following
index conventions are used:
l _l _l _l _m _l _m
R[i,j,k,l] = R = | - | + | | - | |
ijk ij,k ik,j mk ij mj ik
This notation is consistent with the notation used by the itensor
package and its icurvature function.
If the optional argument dis is true,
the unique non-zero components riem[i,j,k,l] will be displayed.
As with the Einstein tensor, various switches set by the user
control the simplification of the components of the Riemann tensor.
If ratriemann is true, then
rational simplification will be done. If ratfac
is true then
each of the components will also be factored.
If the variable cframe_flag is false, the Riemann tensor is
computed directly from the Christoffel-symbols. If cframe_flag is
true, the covariant Riemann-tensor is computed first from the
frame field coefficients.
Categories: Package ctensor
Covariant Riemann-tensor (lriem[]).
Computes the covariant Riemann-tensor as the array lriem. If the
argument dis is true, unique non-zero values are displayed.
If the variable cframe_flag is true, the covariant Riemann
tensor is computed directly from the frame field coefficients. Otherwise,
the (3,1) Riemann tensor is computed first.
For information on index ordering, see riemann.
Categories: Package ctensor
Computes the contravariant components of the Riemann
curvature tensor as array elements uriem[i,j,k,l]. These are displayed
if dis is true.
Categories: Package ctensor
Forms the Kretchmann-invariant (kinvariant) obtained by
contracting the tensors
lriem[i,j,k,l]*uriem[i,j,k,l].
This object is not automatically simplified since it can be very large.
Categories: Package ctensor
Computes the Weyl conformal tensor. If the argument dis is
true, the non-zero components weyl[i,j,k,l] will be displayed to the
user. Otherwise, these components will simply be computed and stored.
If the switch ratweyl is set to true, then the components will be
rationally simplified; if ratfac is true then the results will be
factored as well.
Categories: Package ctensor
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The ctensor package has the ability to truncate results by assuming
that they are Taylor-series approximations. This behavior is controlled by
the ctayswitch variable; when set to true, ctensor makes use
internally of the function ctaylor when simplifying results.
The ctaylor function is invoked by the following ctensor functions:
Function Comments
---------------------------------
christof() For mcs only
ricci()
uricci()
einstein()
riemann()
weyl()
checkdiv()
The ctaylor function truncates its argument by converting
it to a Taylor-series using taylor, and then calling
ratdisrep. This has the combined effect of dropping terms
higher order in the expansion variable ctayvar. The order
of terms that should be dropped is defined by ctaypov; the
point around which the series expansion is carried out is specified
in ctaypt.
As an example, consider a simple metric that is a perturbation of the Minkowski metric. Without further restrictions, even a diagonal metric produces expressions for the Einstein tensor that are far too complex:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) ratfac:true;
(%o2) true
(%i3) derivabbrev:true;
(%o3) true
(%i4) ct_coords:[t,r,theta,phi];
(%o4) [t, r, theta, phi]
(%i5) lg:matrix([-1,0,0,0],[0,1,0,0],[0,0,r^2,0],
[0,0,0,r^2*sin(theta)^2]);
[ - 1 0 0 0 ]
[ ]
[ 0 1 0 0 ]
[ ]
(%o5) [ 2 ]
[ 0 0 r 0 ]
[ ]
[ 2 2 ]
[ 0 0 0 r sin (theta) ]
(%i6) h:matrix([h11,0,0,0],[0,h22,0,0],[0,0,h33,0],[0,0,0,h44]);
[ h11 0 0 0 ]
[ ]
[ 0 h22 0 0 ]
(%o6) [ ]
[ 0 0 h33 0 ]
[ ]
[ 0 0 0 h44 ]
(%i7) depends(l,r);
(%o7) [l(r)]
(%i8) lg:lg+l*h;
[ h11 l - 1 0 0 0 ]
[ ]
[ 0 h22 l + 1 0 0 ]
[ ]
(%o8) [ 2 ]
[ 0 0 r + h33 l 0 ]
[ ]
[ 2 2 ]
[ 0 0 0 r sin (theta) + h44 l ]
(%i9) cmetric(false);
(%o9) done
(%i10) einstein(false);
(%o10) done
(%i11) ntermst(ein);
[[1, 1], 62]
[[1, 2], 0]
[[1, 3], 0]
[[1, 4], 0]
[[2, 1], 0]
[[2, 2], 24]
[[2, 3], 0]
[[2, 4], 0]
[[3, 1], 0]
[[3, 2], 0]
[[3, 3], 46]
[[3, 4], 0]
[[4, 1], 0]
[[4, 2], 0]
[[4, 3], 0]
[[4, 4], 46]
(%o12) done
However, if we recompute this example as an approximation that is
linear in the variable l, we get much simpler expressions:
(%i14) ctayswitch:true;
(%o14) true
(%i15) ctayvar:l;
(%o15) l
(%i16) ctaypov:1;
(%o16) 1
(%i17) ctaypt:0;
(%o17) 0
(%i18) christof(false);
(%o18) done
(%i19) ricci(false);
(%o19) done
(%i20) einstein(false);
(%o20) done
(%i21) ntermst(ein);
[[1, 1], 6]
[[1, 2], 0]
[[1, 3], 0]
[[1, 4], 0]
[[2, 1], 0]
[[2, 2], 13]
[[2, 3], 2]
[[2, 4], 0]
[[3, 1], 0]
[[3, 2], 2]
[[3, 3], 9]
[[3, 4], 0]
[[4, 1], 0]
[[4, 2], 0]
[[4, 3], 0]
[[4, 4], 9]
(%o21) done
(%i22) ratsimp(ein[1,1]);
2 2 4 2 2
(%o22) - (((h11 h22 - h11 ) (l ) r - 2 h33 l r ) sin (theta)
r r r
2 2 4 2
- 2 h44 l r - h33 h44 (l ) )/(4 r sin (theta))
r r r
This capability can be useful, for instance, when working in the weak field limit far from a gravitational source.
Categories: Package ctensor
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When the variable cframe_flag is set to true, the ctensor package
performs its calculations using a moving frame.
The frame bracket (fb[]).
Computes the frame bracket according to the following definition:
c c c d e ifb = ( ifri - ifri ) ifr ifr ab d,e e,d a b
Categories: Package ctensor
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A new feature (as of November, 2004) of ctensor is its ability to
compute the Petrov classification of a 4-dimensional spacetime metric.
For a demonstration of this capability, see the file
share/tensor/petrov.dem.
Computes a Newman-Penrose null tetrad (np) and its raised-index
counterpart (npi). See petrov for an example.
The null tetrad is constructed on the assumption that a four-dimensional orthonormal frame metric with metric signature (-,+,+,+) is being used. The components of the null tetrad are related to the inverse frame matrix as follows:
np = (fri + fri ) / sqrt(2) 1 1 2 np = (fri - fri ) / sqrt(2) 2 1 2 np = (fri + %i fri ) / sqrt(2) 3 3 4 np = (fri - %i fri ) / sqrt(2) 4 3 4
Categories: Package ctensor
Computes the five Newman-Penrose coefficients psi[0]...psi[4].
If dis is set to true, the coefficients are displayed.
See petrov for an example.
These coefficients are computed from the Weyl-tensor in a coordinate base.
If a frame base is used, the Weyl-tensor is first converted to a coordinate
base, which can be a computationally expensive procedure. For this reason,
in some cases it may be more advantageous to use a coordinate base in the
first place before the Weyl tensor is computed. Note however, that
constructing a Newman-Penrose null tetrad requires a frame base. Therefore,
a meaningful computation sequence may begin with a frame base, which
is then used to compute lg (computed automatically by cmetric)
and then ug. See petrov for an example. At this point, you can switch back to a coordinate base
by setting cframe_flag to false before beginning to compute the
Christoffel symbols. Changing to a frame base at a later stage could yield
inconsistent results, as you may end up with a mixed bag of tensors, some
computed in a frame base, some in a coordinate base, with no means to
distinguish between the two.
Categories: Package ctensor
Computes the Petrov classification of the metric characterized by psi[0]...psi[4].
For example, the following demonstrates how to obtain the Petrov-classification of the Kerr metric:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) (cframe_flag:true,gcd:spmod,ctrgsimp:true,ratfac:true);
(%o2) true
(%i3) ct_coordsys(exteriorschwarzschild,all);
(%o3) done
(%i4) ug:invert(lg)$
(%i5) weyl(false);
(%o5) done
(%i6) nptetrad(true);
(%t6) np =
[ sqrt(r - 2 m) sqrt(r) ]
[--------------- --------------------- 0 0 ]
[sqrt(2) sqrt(r) sqrt(2) sqrt(r - 2 m) ]
[ ]
[ sqrt(r - 2 m) sqrt(r) ]
[--------------- - --------------------- 0 0 ]
[sqrt(2) sqrt(r) sqrt(2) sqrt(r - 2 m) ]
[ ]
[ r %i r sin(theta) ]
[ 0 0 ------- --------------- ]
[ sqrt(2) sqrt(2) ]
[ ]
[ r %i r sin(theta)]
[ 0 0 ------- - ---------------]
[ sqrt(2) sqrt(2) ]
sqrt(r) sqrt(r - 2 m)
(%t7) npi = matrix([- ---------------------,---------------, 0, 0],
sqrt(2) sqrt(r - 2 m) sqrt(2) sqrt(r)
sqrt(r) sqrt(r - 2 m)
[- ---------------------, - ---------------, 0, 0],
sqrt(2) sqrt(r - 2 m) sqrt(2) sqrt(r)
1 %i
[0, 0, ---------, --------------------],
sqrt(2) r sqrt(2) r sin(theta)
1 %i
[0, 0, ---------, - --------------------])
sqrt(2) r sqrt(2) r sin(theta)
(%o7) done
(%i7) psi(true);
(%t8) psi = 0
0
(%t9) psi = 0
1
m
(%t10) psi = --
2 3
r
(%t11) psi = 0
3
(%t12) psi = 0
4
(%o12) done
(%i12) petrov();
(%o12) D
The Petrov classification function is based on the algorithm published in "Classifying geometries in general relativity: III Classification in practice" by Pollney, Skea, and d'Inverno, Class. Quant. Grav. 17 2885-2902 (2000). Except for some simple test cases, the implementation is untested as of December 19, 2004, and is likely to contain errors.
Categories: Package ctensor
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ctensor has the ability to compute and include torsion and nonmetricity
coefficients in the connection coefficients.
The torsion coefficients are calculated from a user-supplied tensor
tr, which should be a rank (2,1) tensor. From this, the torsion
coefficients kt are computed according to the following formulae:
m m m
- g tr - g tr - tr g
im kj jm ki ij km
kt = -------------------------------
ijk 2
k km
kt = g kt
ij ijm
Note that only the mixed-index tensor is calculated and stored in the
array kt.
The nonmetricity coefficients are calculated from the user-supplied
nonmetricity vector nm. From this, the nonmetricity coefficients
nmc are computed as follows:
k k km
-nm D - D nm + g nm g
k i j i j m ij
nmc = ------------------------------
ij 2
where D stands for the Kronecker-delta.
When ctorsion_flag is set to true, the values of kt
are subtracted from the mixed-indexed connection coefficients computed by
christof and stored in mcs. Similarly, if cnonmet_flag
is set to true, the values of nmc are subtracted from the
mixed-indexed connection coefficients.
If necessary, christof calls the functions contortion and
nonmetricity in order to compute kt and nm.
Computes the (2,1) contortion coefficients from the torsion tensor tr.
Categories: Package ctensor
Computes the (2,1) nonmetricity coefficients from the nonmetricity vector nm.
Categories: Package ctensor
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A function in the ctensor (component tensor)
package which will perform a coordinate transformation
upon an arbitrary square symmetric matrix M. The user must input the
functions which define the transformation. (Formerly called transform.)
Categories: Package ctensor
returns a list of the unique differential equations (expressions)
corresponding to the elements of the n dimensional square
array A. Presently, n may be 2 or 3. deindex is a global list
containing the indices of A corresponding to these unique
differential equations. For the Einstein tensor (ein), which
is a two dimensional array, if computed for the metric in the example
below, findde gives the following independent differential equations:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) derivabbrev:true;
(%o2) true
(%i3) dim:4;
(%o3) 4
(%i4) lg:matrix([a, 0, 0, 0], [ 0, x^2, 0, 0],
[0, 0, x^2*sin(y)^2, 0], [0,0,0,-d]);
[ a 0 0 0 ]
[ ]
[ 2 ]
[ 0 x 0 0 ]
(%o4) [ ]
[ 2 2 ]
[ 0 0 x sin (y) 0 ]
[ ]
[ 0 0 0 - d ]
(%i5) depends([a,d],x);
(%o5) [a(x), d(x)]
(%i6) ct_coords:[x,y,z,t];
(%o6) [x, y, z, t]
(%i7) cmetric();
(%o7) done
(%i8) einstein(false);
(%o8) done
(%i9) findde(ein,2);
2
(%o9) [d x - a d + d, 2 a d d x - a (d ) x - a d d x
x x x x x x
2 2
+ 2 a d d - 2 a d , a x + a - a]
x x x
(%i10) deindex;
(%o10) [[1, 1], [2, 2], [4, 4]]
Categories: Package ctensor
Computes the covariant gradient of a scalar function allowing the
user to choose the corresponding vector name as the example under
contragrad illustrates.
Categories: Package ctensor
Computes the contravariant gradient of a scalar function allowing the user to choose the corresponding vector name as the example below for the Schwarzschild metric illustrates:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) derivabbrev:true;
(%o2) true
(%i3) ct_coordsys(exteriorschwarzschild,all);
(%o3) done
(%i4) depends(f,r);
(%o4) [f(r)]
(%i5) cograd(f,g1);
(%o5) done
(%i6) listarray(g1);
(%o6) [0, f , 0, 0]
r
(%i7) contragrad(f,g2);
(%o7) done
(%i8) listarray(g2);
f r - 2 f m
r r
(%o8) [0, -------------, 0, 0]
r
Categories: Package ctensor
computes the tensor d'Alembertian of the scalar function once dependencies have been declared upon the function. For example:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) derivabbrev:true;
(%o2) true
(%i3) ct_coordsys(exteriorschwarzschild,all);
(%o3) done
(%i4) depends(p,r);
(%o4) [p(r)]
(%i5) factor(dscalar(p));
2
p r - 2 m p r + 2 p r - 2 m p
r r r r r r
(%o5) --------------------------------------
2
r
Categories: Package ctensor
computes the covariant divergence of the mixed second rank tensor
(whose first index must be covariant) by printing the
corresponding n components of the vector field (the divergence) where
n = dim. If the argument to the function is g then the
divergence of the Einstein tensor will be formed and must be zero.
In addition, the divergence (vector) is given the array name div.
Categories: Package ctensor
A function in the ctensor (component tensor)
package. cgeodesic computes the geodesic equations of
motion for a given metric. They are stored in the array geod[i]. If
the argument dis is true then these equations are displayed.
Categories: Package ctensor
generates the covariant components of the vacuum field equations of
the Brans- Dicke gravitational theory. The scalar field is specified
by the argument f, which should be a (quoted) function name
with functional dependencies, e.g., 'p(x).
The components of the second rank covariant field tensor are
represented by the array bd.
Categories: Package ctensor
generates the mixed Euler- Lagrange tensor (field equations) for the
invariant density of R^2. The field equations are the components of an
array named inv1.
Categories: Package ctensor
*** NOT YET IMPLEMENTED ***
generates the mixed Euler- Lagrange tensor (field equations) for the
invariant density of ric[i,j]*uriem[i,j]. The field equations are the
components of an array named inv2.
Categories: Package ctensor
*** NOT YET IMPLEMENTED ***
generates the field equations of Rosen's bimetric theory. The field
equations are the components of an array named rosen.
Categories: Package ctensor
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Returns true if the first n rows and n columns of M
form a diagonal matrix or (2D) array.
Categories: Package ctensor · Predicate functions
Returns true if M is a n by n symmetric matrix or two-dimensional array,
otherwise false.
If n is less than the size of M,
symmetricp considers only the n by n submatrix (respectively, subarray)
comprising rows 1 through n and columns 1 through n.
Categories: Package ctensor · Predicate functions
gives the user a quick picture of the "size" of the doubly subscripted tensor (array) f. It prints two element lists where the second element corresponds to NTERMS of the components specified by the first elements. In this way, it is possible to quickly find the non-zero expressions and attempt simplification.
Categories: Package ctensor
displays all the elements of the tensor ten, as represented by
a multidimensional array. Tensors of rank 0 and 1, as well as other types
of variables, are displayed as with ldisplay. Tensors of rank 2 are
displayed as 2-dimensional matrices, while tensors of higher rank are displayed
as a list of 2-dimensional matrices. For instance, the Riemann-tensor of
the Schwarzschild metric can be viewed as:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) ratfac:true;
(%o2) true
(%i3) ct_coordsys(exteriorschwarzschild,all);
(%o3) done
(%i4) riemann(false);
(%o4) done
(%i5) cdisplay(riem);
[ 0 0 0 0 ]
[ ]
[ 2 ]
[ 3 m (r - 2 m) m 2 m ]
[ 0 - ------------- + -- - ---- 0 0 ]
[ 4 3 4 ]
[ r r r ]
[ ]
riem = [ m (r - 2 m) ]
1, 1 [ 0 0 ----------- 0 ]
[ 4 ]
[ r ]
[ ]
[ m (r - 2 m) ]
[ 0 0 0 ----------- ]
[ 4 ]
[ r ]
[ 2 m (r - 2 m) ]
[ 0 ------------- 0 0 ]
[ 4 ]
[ r ]
riem = [ ]
1, 2 [ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ m (r - 2 m) ]
[ 0 0 - ----------- 0 ]
[ 4 ]
[ r ]
riem = [ ]
1, 3 [ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ m (r - 2 m) ]
[ 0 0 0 - ----------- ]
[ 4 ]
[ r ]
riem = [ ]
1, 4 [ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
[ ]
[ 2 m ]
[ - ------------ 0 0 0 ]
riem = [ 2 ]
2, 1 [ r (r - 2 m) ]
[ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ 2 m ]
[ ------------ 0 0 0 ]
[ 2 ]
[ r (r - 2 m) ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ m ]
2, 2 [ 0 0 - ------------ 0 ]
[ 2 ]
[ r (r - 2 m) ]
[ ]
[ m ]
[ 0 0 0 - ------------ ]
[ 2 ]
[ r (r - 2 m) ]
[ 0 0 0 0 ]
[ ]
[ m ]
[ 0 0 ------------ 0 ]
riem = [ 2 ]
2, 3 [ r (r - 2 m) ]
[ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
[ ]
[ m ]
[ 0 0 0 ------------ ]
riem = [ 2 ]
2, 4 [ r (r - 2 m) ]
[ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ m ]
3, 1 [ - 0 0 0 ]
[ r ]
[ ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ m ]
3, 2 [ 0 - 0 0 ]
[ r ]
[ ]
[ 0 0 0 0 ]
[ m ]
[ - - 0 0 0 ]
[ r ]
[ ]
[ m ]
[ 0 - - 0 0 ]
riem = [ r ]
3, 3 [ ]
[ 0 0 0 0 ]
[ ]
[ 2 m - r ]
[ 0 0 0 ------- + 1 ]
[ r ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ 2 m ]
3, 4 [ 0 0 0 - --- ]
[ r ]
[ ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ 0 0 0 0 ]
4, 1 [ ]
[ 2 ]
[ m sin (theta) ]
[ ------------- 0 0 0 ]
[ r ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ 0 0 0 0 ]
4, 2 [ ]
[ 2 ]
[ m sin (theta) ]
[ 0 ------------- 0 0 ]
[ r ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ 0 0 0 0 ]
4, 3 [ ]
[ 2 ]
[ 2 m sin (theta) ]
[ 0 0 - --------------- 0 ]
[ r ]
[ 2 ]
[ m sin (theta) ]
[ - ------------- 0 0 0 ]
[ r ]
[ ]
[ 2 ]
[ m sin (theta) ]
riem = [ 0 - ------------- 0 0 ]
4, 4 [ r ]
[ ]
[ 2 ]
[ 2 m sin (theta) ]
[ 0 0 --------------- 0 ]
[ r ]
[ ]
[ 0 0 0 0 ]
(%o5) done
Categories: Package ctensor
Returns a new list consisting of L with the n'th element deleted.
Categories: Package ctensor
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ctensor Default value: 4
An option in the ctensor (component tensor)
package. dim is the dimension of the manifold with the
default 4. The command dim: n will reset the dimension to any other
value n.
Categories: Package ctensor
Default value: false
An option in the ctensor (component tensor)
package. If diagmetric is true special routines compute
all geometrical objects (which contain the metric tensor explicitly)
by taking into consideration the diagonality of the metric. Reduced
run times will, of course, result. Note: this option is set
automatically by csetup if a diagonal metric is specified.
Categories: Package ctensor
Causes trigonometric simplifications to be used when tensors are computed. Presently,
ctrgsimp affects only computations involving a moving frame.
Categories: Package ctensor · Simplification flags and variables
Causes computations to be performed relative to a moving frame as opposed to
a holonomic metric. The frame is defined by the inverse frame array fri
and the frame metric lfg. For computations using a Cartesian frame,
lfg should be the unit matrix of the appropriate dimension; for
computations in a Lorentz frame, lfg should have the appropriate
signature.
Categories: Package ctensor
Causes the contortion tensor to be included in the computation of the
connection coefficients. The contortion tensor itself is computed by
contortion from the user-supplied tensor tr.
Categories: Package ctensor
Causes the nonmetricity coefficients to be included in the computation of
the connection coefficients. The nonmetricity coefficients are computed
from the user-supplied nonmetricity vector nm by the function
nonmetricity.
Categories: Package ctensor
If set to true, causes some ctensor computations to be carried out using
Taylor-series expansions. Presently, christof, ricci,
uricci, einstein, and weyl take into account this
setting.
Categories: Package ctensor
Variable used for Taylor-series expansion if ctayswitch is set to
true.
Categories: Package ctensor
Maximum power used in Taylor-series expansion when ctayswitch is
set to true.
Categories: Package ctensor
Point around which Taylor-series expansion is carried out when
ctayswitch is set to true.
Categories: Package ctensor
The determinant of the metric tensor lg. Computed by cmetric when
cframe_flag is set to false.
Categories: Package ctensor
Causes rational simplification to be applied by christof.
Categories: Package ctensor
Default value: true
If true rational simplification will be
performed on the non-zero components of Einstein tensors; if
ratfac is true then the components will also be factored.
Categories: Package ctensor
Default value: true
One of the switches which controls
simplification of Riemann tensors; if true, then rational
simplification will be done; if ratfac is true then each of the
components will also be factored.
Categories: Package ctensor
Default value: true
If true, this switch causes the weyl function
to apply rational simplification to the values of the Weyl tensor. If
ratfac is true, then the components will also be factored.
Categories: Package ctensor
The covariant frame metric. By default, it is initialized to the
4-dimensional Lorentz frame with signature (+,+,+,-). Used when cframe_flag is true.
Categories: Package ctensor
The inverse frame metric. Computed from lfg when cmetric is called while cframe_flag is set to true.
Categories: Package ctensor
The (3,1) Riemann tensor. Computed when the function riemann is invoked. For information about index ordering, see the description of riemann.
If cframe_flag is true, riem is computed from the covariant Riemann-tensor lriem.
Categories: Package ctensor
The covariant Riemann tensor. Computed by lriemann.
Categories: Package ctensor
The contravariant Riemann tensor. Computed by uriemann.
Categories: Package ctensor
The mixed Ricci-tensor. Computed by ricci.
Categories: Package ctensor
The contravariant Ricci-tensor. Computed by uricci.
Categories: Package ctensor
The metric tensor. This tensor must be specified (as a dim by dim matrix)
before other computations can be performed.
Categories: Package ctensor
The inverse of the metric tensor. Computed by cmetric.
Categories: Package ctensor
The Weyl tensor. Computed by weyl.
Categories: Package ctensor
Frame bracket coefficients, as computed by frame_bracket.
Categories: Package ctensor
The Kretchmann invariant. Computed by rinvariant.
Categories: Package ctensor
A Newman-Penrose null tetrad. Computed by nptetrad.
Categories: Package ctensor
The raised-index Newman-Penrose null tetrad. Computed by nptetrad.
Defined as ug.np. The product np.transpose(npi) is constant:
(%i39) trigsimp(np.transpose(npi));
[ 0 - 1 0 0 ]
[ ]
[ - 1 0 0 0 ]
(%o39) [ ]
[ 0 0 0 1 ]
[ ]
[ 0 0 1 0 ]
Categories: Package ctensor
User-supplied rank-3 tensor representing torsion. Used by contortion.
Categories: Package ctensor
The contortion tensor, computed from tr by contortion.
Categories: Package ctensor
User-supplied nonmetricity vector. Used by nonmetricity.
Categories: Package ctensor
The nonmetricity coefficients, computed from nm by nonmetricity.
Categories: Package ctensor
Variable indicating if the tensor package has been initialized. Set and used by
csetup, reset by init_ctensor.
Categories: Package ctensor
Default value: []
An option in the ctensor (component tensor)
package. ct_coords contains a list of coordinates.
While normally defined when the function csetup is called,
one may redefine the coordinates with the assignment
ct_coords: [j1, j2, ..., jn] where the j's are the new coordinate names.
See also csetup.
Categories: Package ctensor
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The following names are used internally by the ctensor package and
should not be redefined:
Name Description --------------------------------------------------------------------- _lg() Evaluates to lfg if frame metric used, lg otherwise _ug() Evaluates to ufg if frame metric used, ug otherwise cleanup() Removes items drom the deindex list contract4() Used by psi() filemet() Used by csetup() when reading the metric from a file findde1() Used by findde() findde2() Used by findde() findde3() Used by findde() kdelt() Kronecker-delta (not generalized) newmet() Used by csetup() for setting up a metric interactively setflags() Used by init_ctensor() readvalue() resimp() sermet() Used by csetup() for entering a metric as Taylor-series txyzsum() tmetric() Frame metric, used by cmetric() when cframe_flag:true triemann() Riemann-tensor in frame base, used when cframe_flag:true tricci() Ricci-tensor in frame base, used when cframe_flag:true trrc() Ricci rotation coefficients, used by christof() yesp()
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In November, 2004, the ctensor package was extensively rewritten.
Many functions and variables have been renamed in order to make the
package compatible with the commercial version of Macsyma.
New Name Old Name Description --------------------------------------------------------------------- ctaylor() DLGTAYLOR() Taylor-series expansion of an expression lgeod[] EM Geodesic equations ein[] G[] Mixed Einstein-tensor ric[] LR[] Mixed Ricci-tensor ricci() LRICCICOM() Compute the mixed Ricci-tensor ctaypov MINP Maximum power in Taylor-series expansion cgeodesic() MOTION Compute geodesic equations ct_coords OMEGA Metric coordinates ctayvar PARAM Taylor-series expansion variable lriem[] R[] Covariant Riemann-tensor uriemann() RAISERIEMANN() Compute the contravariant Riemann-tensor ratriemann RATRIEMAN Rational simplif. of the Riemann-tensor uric[] RICCI[] Contravariant Ricci-tensor uricci() RICCICOM() Compute the contravariant Ricci-tensor cmetric() SETMETRIC() Set up the metric ctaypt TAYPT Point for Taylor-series expansion ctayswitch TAYSWITCH Taylor-series setting switch csetup() TSETUP() Start interactive setup session ctransform() TTRANSFORM() Interactive coordinate transformation uriem[] UR[] Contravariant Riemann-tensor weyl[] W[] (3,1) Weyl-tensor
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| 27.1 Introduction to atensor | ||
| 27.2 Functions and Variables for atensor |
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atensor is an algebraic tensor manipulation package. To use atensor,
type load(atensor), followed by a call to the init_atensor
function.
The essence of atensor is a set of simplification rules for the
noncommutative (dot) product operator ("."). atensor recognizes
several algebra types; the corresponding simplification rules are put
into effect when the init_atensor function is called.
The capabilities of atensor can be demonstrated by defining the
algebra of quaternions as a Clifford-algebra Cl(0,2) with two basis
vectors. The three quaternionic imaginary units are then the two
basis vectors and their product, i.e.:
i = v j = v k = v . v
1 2 1 2
Although the atensor package has a built-in definition for the
quaternion algebra, it is not used in this example, in which we
endeavour to build the quaternion multiplication table as a matrix:
(%i1) load(atensor);
(%o1) /share/tensor/atensor.mac
(%i2) init_atensor(clifford,0,0,2);
(%o2) done
(%i3) atensimp(v[1].v[1]);
(%o3) - 1
(%i4) atensimp((v[1].v[2]).(v[1].v[2]));
(%o4) - 1
(%i5) q:zeromatrix(4,4);
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
(%o5) [ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
(%i6) q[1,1]:1;
(%o6) 1
(%i7) for i thru adim do q[1,i+1]:q[i+1,1]:v[i];
(%o7) done
(%i8) q[1,4]:q[4,1]:v[1].v[2];
(%o8) v . v
1 2
(%i9) for i from 2 thru 4 do for j from 2 thru 4 do
q[i,j]:atensimp(q[i,1].q[1,j]);
(%o9) done
(%i10) q;
[ 1 v v v . v ]
[ 1 2 1 2 ]
[ ]
[ v - 1 v . v - v ]
[ 1 1 2 2 ]
(%o10) [ ]
[ v - v . v - 1 v ]
[ 2 1 2 1 ]
[ ]
[ v . v v - v - 1 ]
[ 1 2 2 1 ]
atensor recognizes as base vectors indexed symbols, where the symbol
is that stored in asymbol and the index runs between 1 and adim.
For indexed symbols, and indexed symbols only, the bilinear forms
sf, af, and av are evaluated. The evaluation
substitutes the value of aform[i,j] in place of fun(v[i],v[j])
where v represents the value of asymbol and fun is
either af or sf; or, it substitutes v[aform[i,j]]
in place of av(v[i],v[j]).
Needless to say, the functions sf, af and av
can be redefined.
When the atensor package is loaded, the following flags are set:
dotscrules:true; dotdistrib:true; dotexptsimp:false;
If you wish to experiment with a nonassociative algebra, you may also
consider setting dotassoc to false. In this case, however,
atensimp will not always be able to obtain the desired
simplifications.
Categories: Tensors · Share packages · Package atensor
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Initializes the atensor package with the specified algebra type. alg_type
can be one of the following:
universal: The universal algebra has no commutation rules.
grassmann: The Grassman algebra is defined by the commutation
relation u.v+v.u=0.
clifford: The Clifford algebra is defined by the commutation
relation u.v+v.u=-2*sf(u,v) where sf is a symmetric
scalar-valued function. For this algebra, opt_dims can be up
to three nonnegative integers, representing the number of positive,
degenerate, and negative dimensions of the algebra, respectively. If
any opt_dims values are supplied, atensor will configure the
values of adim and aform appropriately. Otherwise,
adim will default to 0 and aform will not be defined.
symmetric: The symmetric algebra is defined by the commutation
relation u.v-v.u=0.
symplectic: The symplectic algebra is defined by the commutation
relation u.v-v.u=2*af(u,v) where af is an antisymmetric
scalar-valued function. For the symplectic algebra, opt_dims can
be up to two nonnegative integers, representing the nondegenerate and
degenerate dimensions, respectively. If any opt_dims values are
supplied, atensor will configure the values of adim and aform
appropriately. Otherwise, adim will default to 0 and aform
will not be defined.
lie_envelop: The algebra of the Lie envelope is defined by the
commutation relation u.v-v.u=2*av(u,v) where av is
an antisymmetric function.
The init_atensor function also recognizes several predefined
algebra types:
complex implements the algebra of complex numbers as the
Clifford algebra Cl(0,1). The call init_atensor(complex) is
equivalent to init_atensor(clifford,0,0,1).
quaternion implements the algebra of quaternions. The call
init_atensor (quaternion) is equivalent to
init_atensor (clifford,0,0,2).
pauli implements the algebra of Pauli-spinors as the Clifford-algebra
Cl(3,0). A call to init_atensor(pauli) is equivalent to
init_atensor(clifford,3).
dirac implements the algebra of Dirac-spinors as the Clifford-algebra
Cl(3,1). A call to init_atensor(dirac) is equivalent to
init_atensor(clifford,3,0,1).
Categories: Package atensor
Simplifies an algebraic tensor expression expr according to the rules
configured by a call to init_atensor. Simplification includes
recursive application of commutation relations and resolving calls
to sf, af, and av where applicable. A
safeguard is used to ensure that the function always terminates, even
for complex expressions.
Categories: Package atensor · Simplification functions
The algebra type. Valid values are universal, grassmann,
clifford, symmetric, symplectic and lie_envelop.
Categories: Package atensor
Default value: 0
The dimensionality of the algebra. atensor uses the value of adim
to determine if an indexed object is a valid base vector. See abasep.
Categories: Package atensor · Global variables
Default value: ident(3)
Default values for the bilinear forms sf, af, and
av. The default is the identity matrix ident(3).
Categories: Package atensor · Global variables
Default value: v
The symbol for base vectors.
Categories: Package atensor · Global variables
A symmetric scalar function that is used in commutation relations.
The default implementation checks if both arguments are base vectors
using abasep and if that is the case, substitutes the
corresponding value from the matrix aform.
Categories: Package atensor
An antisymmetric scalar function that is used in commutation relations.
The default implementation checks if both arguments are base vectors
using abasep and if that is the case, substitutes the
corresponding value from the matrix aform.
Categories: Package atensor
An antisymmetric function that is used in commutation relations.
The default implementation checks if both arguments are base vectors
using abasep and if that is the case, substitutes the
corresponding value from the matrix aform.
For instance:
(%i1) load(atensor);
(%o1) /share/tensor/atensor.mac
(%i2) adim:3;
(%o2) 3
(%i3) aform:matrix([0,3,-2],[-3,0,1],[2,-1,0]);
[ 0 3 - 2 ]
[ ]
(%o3) [ - 3 0 1 ]
[ ]
[ 2 - 1 0 ]
(%i4) asymbol:x;
(%o4) x
(%i5) av(x[1],x[2]);
(%o5) x
3
Categories: Package atensor
Checks if its argument is an atensor base vector. That is, if it is
an indexed symbol, with the symbol being the same as the value of
asymbol, and the index having a numeric value between 1
and adim.
Categories: Package atensor · Predicate functions
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Transforms the expression expr by giving each summation and product a
unique index. This gives changevar greater precision when it is working
with summations or products. The form of the unique index is
jnumber. The quantity number is determined by referring to
gensumnum, which can be changed by the user. For example,
gensumnum:0$ resets it.
Categories: Sums and products
Represents the sum of expr for each element x in L.
A noun form 'lsum is returned if the argument L does not evaluate
to a list.
Examples:
(%i1) lsum (x^i, i, [1, 2, 7]);
7 2
(%o1) x + x + x
(%i2) lsum (i^2, i, rootsof (x^3 - 1, x));
====
\ 2
(%o2) > i
/
====
3
i in rootsof(x - 1, x)
Categories: Sums and products
Moves multiplicative factors outside a summation to inside.
If the index is used in the
outside expression, then the function tries to find a reasonable
index, the same as it does for sumcontract. This is essentially the
reverse idea of the outative property of summations, but note that it
does not remove this property, it only bypasses it.
In some cases, a scanmap (multthru, expr) may be necessary before
the intosum.
Categories: Expressions
Default value: false
When simpproduct is true, the result of a product is simplified.
This simplification may sometimes be able to produce a closed form. If
simpproduct is false or if the quoted form 'product is used, the
value is a product noun form which is a representation of the pi notation used
in mathematics.
Categories: Sums and products · Simplification flags and variables
Represents a product of the values of expr as
the index i varies from i_0 to i_1.
The noun form 'product is displayed as an uppercase letter pi.
product evaluates expr and lower and upper limits i_0 and
i_1, product quotes (does not evaluate) the index i.
If the upper and lower limits differ by an integer, expr is evaluated for each value of the index i, and the result is an explicit product.
Otherwise, the range of the index is indefinite.
Some rules are applied to simplify the product.
When the global variable simpproduct is true, additional rules
are applied. In some cases, simplification yields a result which is not a
product; otherwise, the result is a noun form 'product.
Examples:
(%i1) product (x + i*(i+1)/2, i, 1, 4);
(%o1) (x + 1) (x + 3) (x + 6) (x + 10)
(%i2) product (i^2, i, 1, 7);
(%o2) 25401600
(%i3) product (a[i], i, 1, 7);
(%o3) a a a a a a a
1 2 3 4 5 6 7
(%i4) product (a(i), i, 1, 7);
(%o4) a(1) a(2) a(3) a(4) a(5) a(6) a(7)
(%i5) product (a(i), i, 1, n);
n
/===\
! !
(%o5) ! ! a(i)
! !
i = 1
(%i6) product (k, k, 1, n);
n
/===\
! !
(%o6) ! ! k
! !
k = 1
(%i7) product (k, k, 1, n), simpproduct;
(%o7) n!
(%i8) product (integrate (x^k, x, 0, 1), k, 1, n);
n
/===\
! ! 1
(%o8) ! ! -----
! ! k + 1
k = 1
(%i9) product (if k <= 5 then a^k else b^k, k, 1, 10);
15 40
(%o9) a b
Categories: Sums and products
Default value: false
When simpsum is true, the result of a sum is simplified.
This simplification may sometimes be able to produce a closed form. If
simpsum is false or if the quoted form 'sum is used, the
value is a sum noun form which is a representation of the sigma notation used
in mathematics.
Categories: Sums and products · Simplification flags and variables
Represents a summation of the values of expr as
the index i varies from i_0 to i_1.
The noun form 'sum is displayed as an uppercase letter sigma.
sum evaluates its summand expr and lower and upper limits i_0
and i_1, sum quotes (does not evaluate) the index i.
If the upper and lower limits differ by an integer, the summand expr is evaluated for each value of the summation index i, and the result is an explicit sum.
Otherwise, the range of the index is indefinite.
Some rules are applied to simplify the summation.
When the global variable simpsum is true, additional rules are
applied. In some cases, simplification yields a result which is not a
summation; otherwise, the result is a noun form 'sum.
When the evflag (evaluation flag) cauchysum is true,
a product of summations is expressed as a Cauchy product,
in which the index of the inner summation is a function of the
index of the outer one, rather than varying independently.
The global variable genindex is the alphabetic prefix used to generate
the next index of summation, when an automatically generated index is needed.
gensumnum is the numeric suffix used to generate the next index of
summation, when an automatically generated index is needed.
When gensumnum is false, an automatically-generated index is only
genindex with no numeric suffix.
See also lsum, sumcontract, intosum,
bashindices, niceindices,
nouns, evflag, and zeilberger
Examples:
(%i1) sum (i^2, i, 1, 7);
(%o1) 140
(%i2) sum (a[i], i, 1, 7);
(%o2) a + a + a + a + a + a + a
7 6 5 4 3 2 1
(%i3) sum (a(i), i, 1, 7);
(%o3) a(7) + a(6) + a(5) + a(4) + a(3) + a(2) + a(1)
(%i4) sum (a(i), i, 1, n);
n
====
\
(%o4) > a(i)
/
====
i = 1
(%i5) sum (2^i + i^2, i, 0, n);
n
====
\ i 2
(%o5) > (2 + i )
/
====
i = 0
(%i6) sum (2^i + i^2, i, 0, n), simpsum;
3 2
n + 1 2 n + 3 n + n
(%o6) 2 + --------------- - 1
6
(%i7) sum (1/3^i, i, 1, inf);
inf
====
\ 1
(%o7) > --
/ i
==== 3
i = 1
(%i8) sum (1/3^i, i, 1, inf), simpsum;
1
(%o8) -
2
(%i9) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf);
inf
====
\ 1
(%o9) 30 > --
/ 2
==== i
i = 1
(%i10) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf), simpsum;
2
(%o10) 5 %pi
(%i11) sum (integrate (x^k, x, 0, 1), k, 1, n);
n
====
\ 1
(%o11) > -----
/ k + 1
====
k = 1
(%i12) sum (if k <= 5 then a^k else b^k, k, 1, 10);
10 9 8 7 6 5 4 3 2
(%o12) b + b + b + b + b + a + a + a + a + a
Categories: Sums and products
Combines all sums of an addition that have
upper and lower bounds that differ by constants. The result is an
expression containing one summation for each set of such summations
added to all appropriate extra terms that had to be extracted to form
this sum. sumcontract combines all compatible sums and uses one of
the indices from one of the sums if it can, and then try to form a
reasonable index if it cannot use any supplied.
It may be necessary to do an intosum (expr) before the
sumcontract.
Categories: Sums and products
Default value: false
When sumexpand is true, products of sums and
exponentiated sums simplify to nested sums.
See also cauchysum.
Examples:
(%i1) sumexpand: true$
(%i2) sum (f (i), i, 0, m) * sum (g (j), j, 0, n);
m n
==== ====
\ \
(%o2) > > f(i1) g(i2)
/ /
==== ====
i1 = 0 i2 = 0
(%i3) sum (f (i), i, 0, m)^2;
m m
==== ====
\ \
(%o3) > > f(i3) f(i4)
/ /
==== ====
i3 = 0 i4 = 0
Categories: Sums and products · Simplification flags and variables
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Maxima contains functions taylor and powerseries for finding the
series of differentiable functions. It also has tools such as nusum
capable of finding the closed form of some series. Operations such as addition
and multiplication work as usual on series. This section presents the global
variables which control the expansion.
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Default value: false
When multiplying together sums with inf as their upper limit,
if sumexpand is true and cauchysum is true
then the Cauchy product will be used rather than the usual
product.
In the Cauchy product the index of the inner summation is a
function of the index of the outer one rather than varying
independently.
Example:
(%i1) sumexpand: false$
(%i2) cauchysum: false$
(%i3) s: sum (f(i), i, 0, inf) * sum (g(j), j, 0, inf);
inf inf
==== ====
\ \
(%o3) ( > f(i)) > g(j)
/ /
==== ====
i = 0 j = 0
(%i4) sumexpand: true$
(%i5) cauchysum: true$
(%i6) ''s;
inf i1
==== ====
\ \
(%o6) > > g(i1 - i2) f(i2)
/ /
==== ====
i1 = 0 i2 = 0
Categories: Sums and products
For each function f_i of one variable x_i,
deftaylor defines expr_i as the Taylor series about zero.
expr_i is typically a polynomial in x_i or a summation;
more general expressions are accepted by deftaylor without complaint.
powerseries (f_i(x_i), x_i, 0)
returns the series defined by deftaylor.
deftaylor returns a list of the functions f_1, …, f_n.
deftaylor evaluates its arguments.
Example:
(%i1) deftaylor (f(x), x^2 + sum(x^i/(2^i*i!^2), i, 4, inf));
(%o1) [f]
(%i2) powerseries (f(x), x, 0);
inf
==== i1
\ x 2
(%o2) > -------- + x
/ i1 2
==== 2 i1!
i1 = 4
(%i3) taylor (exp (sqrt (f(x))), x, 0, 4);
2 3 4
x 3073 x 12817 x
(%o3)/T/ 1 + x + -- + ------- + -------- + . . .
2 18432 307200
Categories: Power series
Default value: true
When maxtayorder is true, then during algebraic
manipulation of (truncated) Taylor series, taylor tries to retain
as many terms as are known to be correct.
Categories: Power series
Renames the indices of sums and products in expr. niceindices
attempts to rename each index to the value of niceindicespref[1], unless
that name appears in the summand or multiplicand, in which case
niceindices tries the succeeding elements of niceindicespref in
turn, until an unused variable is found. If the entire list is exhausted,
additional indices are constructed by appending integers to the value of
niceindicespref[1], e.g., i0, i1, i2, …
niceindices returns an expression.
niceindices evaluates its argument.
Example:
(%i1) niceindicespref;
(%o1) [i, j, k, l, m, n]
(%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
inf inf
/===\ ====
! ! \
(%o2) ! ! > f(bar i j + foo)
! ! /
bar = 1 ====
foo = 1
(%i3) niceindices (%);
inf inf
/===\ ====
! ! \
(%o3) ! ! > f(i j l + k)
! ! /
l = 1 ====
k = 1
Categories: Sums and products
Default value: [i, j, k, l, m, n]
niceindicespref is the list from which niceindices
takes the names of indices for sums and products.
The elements of niceindicespref are typically names of variables,
although that is not enforced by niceindices.
Example:
(%i1) niceindicespref: [p, q, r, s, t, u]$
(%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
inf inf
/===\ ====
! ! \
(%o2) ! ! > f(bar i j + foo)
! ! /
bar = 1 ====
foo = 1
(%i3) niceindices (%);
inf inf
/===\ ====
! ! \
(%o3) ! ! > f(i j q + p)
! ! /
q = 1 ====
p = 1
Categories: Sums and products
Carries out indefinite hypergeometric summation of expr with respect to x using a decision procedure due to R.W. Gosper. expr and the result must be expressible as products of integer powers, factorials, binomials, and rational functions.
The terms "definite"
and "indefinite summation" are used analogously to "definite" and
"indefinite integration".
To sum indefinitely means to give a symbolic result
for the sum over intervals of variable length, not just e.g. 0 to
inf. Thus, since there is no formula for the general partial sum of
the binomial series, nusum can't do it.
nusum and unsum know a little about sums and differences of
finite products. See also unsum.
Examples:
(%i1) nusum (n*n!, n, 0, n);
Dependent equations eliminated: (1)
(%o1) (n + 1)! - 1
(%i2) nusum (n^4*4^n/binomial(2*n,n), n, 0, n);
4 3 2 n
2 (n + 1) (63 n + 112 n + 18 n - 22 n + 3) 4 2
(%o2) ------------------------------------------------ - ------
693 binomial(2 n, n) 3 11 7
(%i3) unsum (%, n);
4 n
n 4
(%o3) ----------------
binomial(2 n, n)
(%i4) unsum (prod (i^2, i, 1, n), n);
n - 1
/===\
! ! 2
(%o4) ( ! ! i ) (n - 1) (n + 1)
! !
i = 1
(%i5) nusum (%, n, 1, n);
Dependent equations eliminated: (2 3)
n
/===\
! ! 2
(%o5) ! ! i - 1
! !
i = 1
Categories: Sums and products
Returns a list of all rational functions which have the given Taylor series expansion where the sum of the degrees of the numerator and the denominator is less than or equal to the truncation level of the power series, i.e. are "best" approximants, and which additionally satisfy the specified degree bounds.
taylor_series is a univariate Taylor series. numer_deg_bound and denom_deg_bound are positive integers specifying degree bounds on the numerator and denominator.
taylor_series can also be a Laurent series, and the degree
bounds can be inf which causes all rational functions whose total
degree is less than or equal to the length of the power series to be
returned. Total degree is defined as numer_deg_bound +
denom_deg_bound.
Length of a power series is defined as
"truncation level" + 1 - min(0, "order of series").
(%i1) taylor (1 + x + x^2 + x^3, x, 0, 3);
2 3
(%o1)/T/ 1 + x + x + x + . . .
(%i2) pade (%, 1, 1);
1
(%o2) [- -----]
x - 1
(%i3) t: taylor(-(83787*x^10 - 45552*x^9 - 187296*x^8
+ 387072*x^7 + 86016*x^6 - 1507328*x^5
+ 1966080*x^4 + 4194304*x^3 - 25165824*x^2
+ 67108864*x - 134217728)
/134217728, x, 0, 10);
2 3 4 5 6 7
x 3 x x 15 x 23 x 21 x 189 x
(%o3)/T/ 1 - - + ---- - -- - ----- + ----- - ----- - ------
2 16 32 1024 2048 32768 65536
8 9 10
5853 x 2847 x 83787 x
+ ------- + ------- - --------- + . . .
4194304 8388608 134217728
(%i4) pade (t, 4, 4);
(%o4) []
There is no rational function of degree 4 numerator/denominator, with this power series expansion. You must in general have degree of the numerator and degree of the denominator adding up to at least the degree of the power series, in order to have enough unknown coefficients to solve.
(%i5) pade (t, 5, 5);
5 4 3
(%o5) [- (520256329 x - 96719020632 x - 489651410240 x
2
- 1619100813312 x - 2176885157888 x - 2386516803584)
5 4 3
/(47041365435 x + 381702613848 x + 1360678489152 x
2
+ 2856700692480 x + 3370143559680 x + 2386516803584)]
Categories: Power series
Returns the general form of the power series expansion for expr in the
variable x about the point a (which may be inf for infinity):
inf
====
\ n
> b (x - a)
/ n
====
n = 0
If powerseries is unable to expand expr,
taylor may give the first several terms of the series.
When verbose is true,
powerseries prints progress messages.
(%i1) verbose: true$
(%i2) powerseries (log(sin(x)/x), x, 0);
can't expand
log(sin(x))
so we'll try again after applying the rule:
d
/ -- (sin(x))
[ dx
log(sin(x)) = i ----------- dx
] sin(x)
/
in the first simplification we have returned:
/
[
i cot(x) dx - log(x)
]
/
inf
==== i1 2 i1 2 i1
\ (- 1) 2 bern(2 i1) x
> ------------------------------
/ i1 (2 i1)!
====
i1 = 1
(%o2) -------------------------------------
2
Categories: Power series
Default value: false
When psexpand is true,
an extended rational function expression is displayed fully expanded.
The switch ratexpand has the same effect.
When psexpand is false,
a multivariate expression is displayed just as in the rational function package.
When psexpand is multi,
then terms with the same total degree in the variables are grouped together.
Categories: Display flags and variables
These functions return the reversion of expr, a Taylor series about zero
in the variable x. revert returns a polynomial of degree equal to
the highest power in expr. revert2 returns a polynomial of degree
n, which may be greater than, equal to, or less than the degree of
expr.
load ("revert") loads these functions.
Examples:
(%i1) load ("revert")$
(%i2) t: taylor (exp(x) - 1, x, 0, 6);
2 3 4 5 6
x x x x x
(%o2)/T/ x + -- + -- + -- + --- + --- + . . .
2 6 24 120 720
(%i3) revert (t, x);
6 5 4 3 2
10 x - 12 x + 15 x - 20 x + 30 x - 60 x
(%o3)/R/ - --------------------------------------------
60
(%i4) ratexpand (%);
6 5 4 3 2
x x x x x
(%o4) - -- + -- - -- + -- - -- + x
6 5 4 3 2
(%i5) taylor (log(x+1), x, 0, 6);
2 3 4 5 6
x x x x x
(%o5)/T/ x - -- + -- - -- + -- - -- + . . .
2 3 4 5 6
(%i6) ratsimp (revert (t, x) - taylor (log(x+1), x, 0, 6));
(%o6) 0
(%i7) revert2 (t, x, 4);
4 3 2
x x x
(%o7) - -- + -- - -- + x
4 3 2
Categories: Power series
taylor (expr, x, a, n) expands the expression
expr in a truncated Taylor or Laurent series in the variable x
around the point a,
containing terms through (x - a)^n.
If expr is of the form f(x)/g(x) and
g(x) has no terms up to degree n then taylor
attempts to expand g(x) up to degree 2 n.
If there are still no nonzero terms, taylor doubles the degree of the
expansion of g(x) so long as the degree of the expansion is
less than or equal to n 2^taylordepth.
taylor (expr, [x_1, x_2, ...], a, n)
returns a truncated power series
of degree n in all variables x_1, x_2, …
about the point (a, a, ...).
taylor (expr, [x_1, a_1, n_1], [x_2,
a_2, n_2], ...) returns a truncated power series in the variables
x_1, x_2, … about the point
(a_1, a_2, ...), truncated at n_1, n_2, …
taylor (expr, [x_1, x_2, ...], [a_1,
a_2, ...], [n_1, n_2, ...]) returns a truncated power series
in the variables x_1, x_2, … about the point
(a_1, a_2, ...), truncated at n_1, n_2, …
taylor (expr, [x, a, n, 'asymp]) returns an
expansion of expr in negative powers of x - a.
The highest order term is (x - a)^-n.
When maxtayorder is true, then during algebraic
manipulation of (truncated) Taylor series, taylor tries to retain
as many terms as are known to be correct.
When psexpand is true,
an extended rational function expression is displayed fully expanded.
The switch ratexpand has the same effect.
When psexpand is false,
a multivariate expression is displayed just as in the rational function package.
When psexpand is multi,
then terms with the same total degree in the variables are grouped together.
See also the taylor_logexpand switch for controlling expansion.
Examples:
(%i1) taylor (sqrt (sin(x) + a*x + 1), x, 0, 3);
2 2
(a + 1) x (a + 2 a + 1) x
(%o1)/T/ 1 + --------- - -----------------
2 8
3 2 3
(3 a + 9 a + 9 a - 1) x
+ -------------------------- + . . .
48
(%i2) %^2;
3
x
(%o2)/T/ 1 + (a + 1) x - -- + . . .
6
(%i3) taylor (sqrt (x + 1), x, 0, 5);
2 3 4 5
x x x 5 x 7 x
(%o3)/T/ 1 + - - -- + -- - ---- + ---- + . . .
2 8 16 128 256
(%i4) %^2;
(%o4)/T/ 1 + x + . . .
(%i5) product ((1 + x^i)^2.5, i, 1, inf)/(1 + x^2);
inf
/===\
! ! i 2.5
! ! (x + 1)
! !
i = 1
(%o5) -----------------
2
x + 1
(%i6) ev (taylor(%, x, 0, 3), keepfloat);
2 3
(%o6)/T/ 1 + 2.5 x + 3.375 x + 6.5625 x + . . .
(%i7) taylor (1/log (x + 1), x, 0, 3);
2 3
1 1 x x 19 x
(%o7)/T/ - + - - -- + -- - ----- + . . .
x 2 12 24 720
(%i8) taylor (cos(x) - sec(x), x, 0, 5);
4
2 x
(%o8)/T/ - x - -- + . . .
6
(%i9) taylor ((cos(x) - sec(x))^3, x, 0, 5);
(%o9)/T/ 0 + . . .
(%i10) taylor (1/(cos(x) - sec(x))^3, x, 0, 5);
2 4
1 1 11 347 6767 x 15377 x
(%o10)/T/ - -- + ---- + ------ - ----- - ------- - --------
6 4 2 15120 604800 7983360
x 2 x 120 x
+ . . .
(%i11) taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6);
2 2 4 2 4
k x (3 k - 4 k ) x
(%o11)/T/ 1 - ----- - ----------------
2 24
6 4 2 6
(45 k - 60 k + 16 k ) x
- -------------------------- + . . .
720
(%i12) taylor ((x + 1)^n, x, 0, 4);
2 2 3 2 3
(n - n) x (n - 3 n + 2 n) x
(%o12)/T/ 1 + n x + ----------- + --------------------
2 6
4 3 2 4
(n - 6 n + 11 n - 6 n) x
+ ---------------------------- + . . .
24
(%i13) taylor (sin (y + x), x, 0, 3, y, 0, 3);
3 2
y y
(%o13)/T/ y - -- + . . . + (1 - -- + . . .) x
6 2
3 2
y y 2 1 y 3
+ (- - + -- + . . .) x + (- - + -- + . . .) x + . . .
2 12 6 12
(%i14) taylor (sin (y + x), [x, y], 0, 3);
3 2 2 3
x + 3 y x + 3 y x + y
(%o14)/T/ y + x - ------------------------- + . . .
6
(%i15) taylor (1/sin (y + x), x, 0, 3, y, 0, 3);
1 y 1 1 1 2
(%o15)/T/ - + - + . . . + (- -- + - + . . .) x + (-- + . . .) x
y 6 2 6 3
y y
1 3
+ (- -- + . . .) x + . . .
4
y
(%i16) taylor (1/sin (y + x), [x, y], 0, 3);
3 2 2 3
1 x + y 7 x + 21 y x + 21 y x + 7 y
(%o16)/T/ ----- + ----- + ------------------------------- + . . .
x + y 6 360
Categories: Power series
Default value: 3
If there are still no nonzero terms, taylor doubles the degree of the
expansion of g(x) so long as the degree of the expansion is
less than or equal to n 2^taylordepth.
Categories: Power series
Returns information about the Taylor series expr. The return value is a list of lists. Each list comprises the name of a variable, the point of expansion, and the degree of the expansion.
taylorinfo returns false if expr is not a Taylor series.
Example:
(%i1) taylor ((1 - y^2)/(1 - x), x, 0, 3, [y, a, inf]);
2 2
(%o1)/T/ - (y - a) - 2 a (y - a) + (1 - a )
2 2
+ (1 - a - 2 a (y - a) - (y - a) ) x
2 2 2
+ (1 - a - 2 a (y - a) - (y - a) ) x
2 2 3
+ (1 - a - 2 a (y - a) - (y - a) ) x + . . .
(%i2) taylorinfo(%);
(%o2) [[y, a, inf], [x, 0, 3]]
Categories: Power series
Returns true if expr is a Taylor series,
and false otherwise.
Categories: Predicate functions · Power series
Default value: true
taylor_logexpand controls expansions of logarithms in
taylor series.
When taylor_logexpand is true, all logarithms are expanded fully
so that zero-recognition problems involving logarithmic identities do not
disturb the expansion process. However, this scheme is not always
mathematically correct since it ignores branch information.
When taylor_logexpand is set to false, then the only expansion of
logarithms that occur is that necessary to obtain a formal power series.
Categories: Power series · Exponential and logarithm functions
Default value: true
taylor_order_coefficients controls the ordering of
coefficients in a Taylor series.
When taylor_order_coefficients is true,
coefficients of taylor series are ordered canonically.
Categories: Power series
Simplifies coefficients of the power series expr.
taylor calls this function.
Categories: Power series
Default value: true
When taylor_truncate_polynomials is true,
polynomials are truncated based upon the input truncation levels.
Otherwise,
polynomials input to taylor are considered to have infinite precison.
Categories: Power series
Converts expr from taylor form to canonical rational expression
(CRE) form. The effect is the same as rat (ratdisrep (expr)), but
faster.
Categories: Power series · Rational expressions
Annotates the internal representation of the general expression expr so that it is displayed as if its sums were truncated Taylor series. expr is not otherwise modified.
Example:
(%i1) expr: x^2 + x + 1;
2
(%o1) x + x + 1
(%i2) trunc (expr);
2
(%o2) 1 + x + x + . . .
(%i3) is (expr = trunc (expr));
(%o3) true
Categories: Power series
Returns the first backward difference
f(n) - f(n - 1).
Thus unsum in a sense is the inverse of sum.
See also nusum.
Examples:
(%i1) g(p) := p*4^n/binomial(2*n,n);
n
p 4
(%o1) g(p) := ----------------
binomial(2 n, n)
(%i2) g(n^4);
4 n
n 4
(%o2) ----------------
binomial(2 n, n)
(%i3) nusum (%, n, 0, n);
4 3 2 n
2 (n + 1) (63 n + 112 n + 18 n - 22 n + 3) 4 2
(%o3) ------------------------------------------------ - ------
693 binomial(2 n, n) 3 11 7
(%i4) unsum (%, n);
4 n
n 4
(%o4) ----------------
binomial(2 n, n)
Categories: Sums and products
Default value: false
When verbose is true,
powerseries prints progress messages.
Categories: Power series
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The fourie package comprises functions for the symbolic computation
of Fourier series.
There are functions in the fourie package to calculate Fourier integral
coefficients and some functions for manipulation of expressions.
Categories: Fourier transform · Share packages · Package fourie
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Returns true if equal (x, y) otherwise false
(doesn't give an error message like equal (x, y) would do in this case).
Categories: Package fourie
remfun (f, expr) replaces all occurrences of f
(arg) by arg in expr.
remfun (f, expr, x) replaces all occurrences of
f (arg) by arg in expr only if arg contains
the variable x.
Categories: Package fourie
funp (f, expr)
returns true if expr contains the function f.
funp (f, expr, x)
returns true if expr contains the function f and the variable
x is somewhere in the argument of one of the instances of f.
Categories: Package fourie
absint (f, x, halfplane)
returns the indefinite integral of f with respect to
x in the given halfplane (pos, neg, or both).
f may contain expressions of the form
abs (x), abs (sin (x)), abs (a) * exp (-abs (b) * abs (x)).
absint (f, x) is equivalent to
absint (f, x, pos).
absint (f, x, a, b) returns the definite integral
of f with respect to x from a to b.
f may include absolute values.
Categories: Package fourie · Integral calculus
Returns a list of the Fourier coefficients of f(x) defined
on the interval [-p, p].
Categories: Package fourie
Simplifies sin (n %pi) to 0 if sinnpiflag is true and
cos (n %pi) to (-1)^n if cosnpiflag is true.
Categories: Package fourie · Trigonometric functions · Simplification functions
Default value: true
See foursimp.
Categories: Package fourie
Default value: true
See foursimp.
Categories: Package fourie
Constructs and returns the Fourier series from the list of Fourier coefficients
l up through limit terms (limit may be inf). x
and p have same meaning as in fourier.
Categories: Package fourie
Returns the Fourier cosine coefficients for f(x) defined on
[0, p].
Categories: Package fourie
Returns the Fourier sine coefficients for f(x) defined on
[0, p].
Categories: Package fourie
Returns fourexpand (foursimp (fourier (f, x, p)),
x, p, 'inf).
Categories: Package fourie
Constructs and returns a list of the Fourier integral coefficients of
f(x) defined on [minf, inf].
Categories: Package fourie
Returns the Fourier cosine integral coefficients for f(x)
on [0, inf].
Categories: Package fourie
Returns the Fourier sine integral coefficients for f(x) on
[0, inf].
Categories: Package fourie
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Converts a into a Poisson encoding.
Categories: Poisson series
Converts a from Poisson encoding to general representation. If a is
not in Poisson form, outofpois carries out the conversion,
i.e., the return value is outofpois (intopois (a)).
This function is thus a canonical simplifier
for sums of powers of sine and cosine terms of a particular type.
Categories: Poisson series
Differentiates a with respect to b. b must occur only in the trig arguments or only in the coefficients.
Categories: Poisson series
Functionally identical to intopois (a^b).
b must be a positive integer.
Categories: Poisson series
Integrates in a similarly restricted sense (to poisdiff). Non-periodic
terms in b are dropped if b is in the trig arguments.
Categories: Poisson series
Default value: 5
poislim determines the domain of the coefficients in
the arguments of the trig functions. The initial value of 5
corresponds to the interval [-2^(5-1)+1,2^(5-1)], or [-15,16], but it
can be set to [-2^(n-1)+1, 2^(n-1)].
Categories: Poisson series
will map the functions sinfn on the sine terms and cosfn on the cosine terms of the Poisson series given. sinfn and cosfn are functions of two arguments which are a coefficient and a trigonometric part of a term in series respectively.
Categories: Poisson series
Is functionally identical to intopois (a + b).
Categories: Poisson series
Converts a into a Poisson series for a in general representation.
Categories: Poisson series
The symbol /P/ follows the line label of Poisson series
expressions.
Categories: Poisson series
Substitutes a for b in c. c is a Poisson series.
(1) Where B is a variable u, v, w, x, y,
or z, then a must be an expression linear in those variables (e.g.,
6*u + 4*v).
(2) Where b is other than those variables, then a must also be free of those variables, and furthermore, free of sines or cosines.
poissubst (a, b, c, d, n) is a special type
of substitution which operates on a and b as in type (1) above, but
where d is a Poisson series, expands cos(d) and
sin(d) to order n so as to provide the result of substituting
a + d for b in c. The idea is that d is an
expansion in terms of a small parameter. For example,
poissubst (u, v, cos(v), %e, 3) yields
cos(u)*(1 - %e^2/2) - sin(u)*(%e - %e^3/6).
Categories: Poisson series
Is functionally identical to intopois (a*b).
Categories: Poisson series
is a reserved function name which (if the user has defined
it) gets applied during Poisson multiplication. It is a predicate
function of 6 arguments which are the coefficients of the u, v, ..., z
in a term. Terms for which poistrim is true (for the coefficients of
that term) are eliminated during multiplication.
Categories: Poisson series
Prints a Poisson series in a readable format. In common
with outofpois, it will convert a into a Poisson encoding first, if
necessary.
Categories: Poisson series · Display functions
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| 29.1 Functions and Variables for Number Theory |
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Returns the n'th Bernoulli number for integer n.
Bernoulli numbers equal to zero are suppressed if zerobern is
false.
See also burn.
(%i1) zerobern: true$
(%i2) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
1 1 1 1 1
(%o2) [1, - -, -, 0, - --, 0, --, 0, - --]
2 6 30 42 30
(%i3) zerobern: false$
(%i4) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
1 1 1 1 1 5 691 7
(%o4) [1, - -, -, - --, --, - --, --, - ----, -]
2 6 30 42 30 66 2730 6
Categories: Number theory
Returns the n'th Bernoulli polynomial in the variable x.
Categories: Number theory
Returns the Riemann zeta function for the argument s. The return value is a big float (bfloat); n is the number of digits in the return value.
Categories: Number theory · Numerical evaluation
Returns the Hurwitz zeta function for the arguments s and h. The return value is a big float (bfloat); n is the number of digits in the return value.
The Hurwitz zeta function is defined as
inf
====
\ 1
zeta (s,h) = > --------
/ s
==== (k + h)
k = 0
load ("bffac") loads this function.
Categories: Number theory · Numerical evaluation
Returns a rational number, which is an approximation of the n'th Bernoulli
number for integer n. burn exploits the observation that
(rational) Bernoulli numbers can be approximated by (transcendental) zetas with
tolerable efficiency:
n - 1 1 - 2 n
(- 1) 2 zeta(2 n) (2 n)!
B(2 n) = ------------------------------------
2 n
%pi
burn may be more efficient than bern for large, isolated n
as bern computes all the Bernoulli numbers up to index n before
returning. burn invokes the approximation for even integers n >
255. For odd integers and n <= 255 the function bern is called.
load ("bffac") loads this function. See also bern.
Categories: Number theory
Solves the system of congruences x = r_1 mod m_1, …, x = r_n mod m_n.
The remainders r_n may be arbitrary integers while the moduli m_n have to be
positive and pairwise coprime integers.
(%i1) mods : [1000, 1001, 1003, 1007];
(%o1) [1000, 1001, 1003, 1007]
(%i2) lreduce('gcd, mods);
(%o2) 1
(%i3) x : random(apply("*", mods));
(%o3) 685124877004
(%i4) rems : map(lambda([z], mod(x, z)), mods);
(%o4) [4, 568, 54, 624]
(%i5) chinese(rems, mods);
(%o5) 685124877004
(%i6) chinese([1, 2], [3, n]);
(%o6) chinese([1, 2], [3, n])
(%i7) %, n = 4;
(%o7) 10
Categories: Number theory
Computes a continued fraction approximation.
expr is an expression comprising continued fractions,
square roots of integers, and literal real numbers
(integers, rational numbers, ordinary floats, and bigfloats).
cf computes exact expansions for rational numbers,
but expansions are truncated at ratepsilon for ordinary floats
and 10^(-fpprec) for bigfloats.
Operands in the expression may be combined with arithmetic operators.
Maxima does not know about operations on continued fractions
outside of cf.
cf evaluates its arguments after binding listarith to
false. cf returns a continued fraction, represented as a list.
A continued fraction a + 1/(b + 1/(c + ...)) is represented by the list
[a, b, c, ...]. The list elements a, b, c, …
must evaluate to integers. expr may also contain sqrt (n) where
n is an integer. In this case cf will give as many terms of the
continued fraction as the value of the variable cflength times the
period.
A continued fraction can be evaluated to a number by evaluating the arithmetic
representation returned by cfdisrep. See also cfexpand for
another way to evaluate a continued fraction.
See also cfdisrep, cfexpand, and cflength.
Examples:
(%i1) cf ([5, 3, 1]*[11, 9, 7] + [3, 7]/[4, 3, 2]); (%o1) [59, 17, 2, 1, 1, 1, 27] (%i2) cf ((3/17)*[1, -2, 5]/sqrt(11) + (8/13)); (%o2) [0, 1, 1, 1, 3, 2, 1, 4, 1, 9, 1, 9, 2]
cflength controls how many periods of the continued fraction
are computed for algebraic, irrational numbers.
(%i1) cflength: 1$ (%i2) cf ((1 + sqrt(5))/2); (%o2) [1, 1, 1, 1, 2] (%i3) cflength: 2$ (%i4) cf ((1 + sqrt(5))/2); (%o4) [1, 1, 1, 1, 1, 1, 1, 2] (%i5) cflength: 3$ (%i6) cf ((1 + sqrt(5))/2); (%o6) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]
cfdisrep.
(%i1) cflength: 3$ (%i2) cfdisrep (cf (sqrt (3)))$ (%i3) ev (%, numer); (%o3) 1.731707317073171
cf.
(%i1) cf ([1,1,1,1,1,2] * 3); (%o1) [4, 1, 5, 2] (%i2) cf ([1,1,1,1,1,2]) * 3; (%o2) [3, 3, 3, 3, 3, 6]
Categories: Continued fractions
Constructs and returns an ordinary arithmetic expression
of the form a + 1/(b + 1/(c + ...))
from the list representation of a continued fraction [a, b, c, ...].
(%i1) cf ([1, 2, -3] + [1, -2, 1]);
(%o1) [1, 1, 1, 2]
(%i2) cfdisrep (%);
1
(%o2) 1 + ---------
1
1 + -----
1
1 + -
2
Categories: Continued fractions
Returns a matrix of the numerators and denominators of the last (column 1) and next-to-last (column 2) convergents of the continued fraction x.
(%i1) cf (rat (ev (%pi, numer)));
`rat' replaced 3.141592653589793 by 103993/33102 =3.141592653011902
(%o1) [3, 7, 15, 1, 292]
(%i2) cfexpand (%);
[ 103993 355 ]
(%o2) [ ]
[ 33102 113 ]
(%i3) %[1,1]/%[2,1], numer;
(%o3) 3.141592653011902
Categories: Continued fractions
Default value: 1
cflength controls the number of terms of the continued fraction the
function cf will give, as the value cflength times the period.
Thus the default is to give one period.
(%i1) cflength: 1$ (%i2) cf ((1 + sqrt(5))/2); (%o2) [1, 1, 1, 1, 2] (%i3) cflength: 2$ (%i4) cf ((1 + sqrt(5))/2); (%o4) [1, 1, 1, 1, 1, 1, 1, 2] (%i5) cflength: 3$ (%i6) cf ((1 + sqrt(5))/2); (%o6) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]
Categories: Continued fractions
divsum (n, k) returns the sum of the divisors of n
raised to the k'th power.
divsum (n) returns the sum of the divisors of n.
(%i1) divsum (12); (%o1) 28 (%i2) 1 + 2 + 3 + 4 + 6 + 12; (%o2) 28 (%i3) divsum (12, 2); (%o3) 210 (%i4) 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2; (%o4) 210
Categories: Number theory
Returns the n'th Euler number for nonnegative integer n.
Euler numbers equal to zero are suppressed if zerobern is
false.
For the Euler-Mascheroni constant, see %gamma.
(%i1) zerobern: true$ (%i2) map (euler, [0, 1, 2, 3, 4, 5, 6]); (%o2) [1, 0, - 1, 0, 5, 0, - 61] (%i3) zerobern: false$ (%i4) map (euler, [0, 1, 2, 3, 4, 5, 6]); (%o4) [1, - 1, 5, - 61, 1385, - 50521, 2702765]
Categories: Number theory
Default value: false
Controls the value returned by ifactors. The default false
causes ifactors to provide information about multiplicities of the
computed prime factors. If factors_only is set to true,
ifactors returns nothing more than a list of prime factors.
Example: See ifactors.
Categories: Number theory
Returns the n'th Fibonacci number.
fib(0) is equal to 0 and fib(1) equal to 1, and
fib (-n) equal to (-1)^(n + 1) * fib(n).
After calling fib,
prevfib is equal to fib(n - 1),
the Fibonacci number preceding the last one computed.
(%i1) map (fib, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]); (%o1) [- 3, 2, - 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21]
Categories: Number theory
Expresses Fibonacci numbers in expr in terms of the constant %phi,
which is (1 + sqrt(5))/2, approximately 1.61803399.
Examples:
(%i1) fibtophi (fib (n));
n n
%phi - (1 - %phi)
(%o1) -------------------
2 %phi - 1
(%i2) fib (n-1) + fib (n) - fib (n+1);
(%o2) - fib(n + 1) + fib(n) + fib(n - 1)
(%i3) fibtophi (%);
n + 1 n + 1 n n
%phi - (1 - %phi) %phi - (1 - %phi)
(%o3) - --------------------------- + -------------------
2 %phi - 1 2 %phi - 1
n - 1 n - 1
%phi - (1 - %phi)
+ ---------------------------
2 %phi - 1
(%i4) ratsimp (%);
(%o4) 0
Categories: Number theory
For a positive integer n returns the factorization of n. If
n=p1^e1..pk^nk is the decomposition of n into prime
factors, ifactors returns [[p1, e1], ... , [pk, ek]].
Factorization methods used are trial divisions by primes up to 9973, Pollard's rho and p-1 method and elliptic curves.
If the variable ifactor_verbose is set to true
ifactor produces detailed output about what it is doing including
immediate feedback as soon as a factor has been found.
The value returned by ifactors is controlled by the option variable factors_only.
The default false causes ifactors to provide information about
the multiplicities of the computed prime factors. If factors_only
is set to true, ifactors simply returns the list of
prime factors.
(%i1) ifactors(51575319651600);
(%o1) [[2, 4], [3, 2], [5, 2], [1583, 1], [9050207, 1]]
(%i2) apply("*", map(lambda([u], u[1]^u[2]), %));
(%o2) 51575319651600
(%i3) ifactors(51575319651600), factors_only : true;
(%o3) [2, 3, 5, 1583, 9050207]
Categories: Number theory
Returns a list [a, b, u] where u is the greatest
common divisor of n and k, and u is equal to
a n + b k. The arguments n and k
must be integers.
igcdex implements the Euclidean algorithm. See also gcdex.
The command load(gcdex) loads the function.
Examples:
(%i1) load(gcdex)$ (%i2) igcdex(30,18); (%o2) [- 1, 2, 6] (%i3) igcdex(1526757668, 7835626735736); (%o3) [845922341123, - 164826435, 4] (%i4) igcdex(fib(20), fib(21)); (%o4) [4181, - 2584, 1]
Categories: Number theory
Returns the integer n'th root of the absolute value of x.
(%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$ (%i2) map (lambda ([a], inrt (10^a, 3)), l); (%o2) [2, 4, 10, 21, 46, 100, 215, 464, 1000, 2154, 4641, 10000]
Categories: Number theory
Computes the inverse of n modulo m.
inv_mod (n,m) returns false,
if n is a zero divisor modulo m.
(%i1) inv_mod(3, 41); (%o1) 14 (%i2) ratsimp(3^-1), modulus = 41; (%o2) 14 (%i3) inv_mod(3, 42); (%o3) false
Categories: Number theory
Returns the "integer square root" of the absolute value of x, which is an integer.
Categories: Mathematical functions
Returns the Jacobi symbol of p and q.
(%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$ (%i2) map (lambda ([a], jacobi (a, 9)), l); (%o2) [1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0]
Categories: Number theory
Returns the least common multiple of its arguments. The arguments may be general expressions as well as integers.
load ("functs") loads this function.
Categories: Number theory
Returns the n'th Lucas number.
lucas(0) is equal to 2 and lucas(1) equal to 1, and
lucas(-n) equal to (-1)^(-n) * lucas(n).
(%i1) map (lucas, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]); (%o1) [7, - 4, 3, - 1, 2, 1, 3, 4, 7, 11, 18, 29, 47]
After calling lucas, the global variable
next_lucas is equal to lucas (n + 1),
the Lucas number following the last returned. The example shows
how Fibonacci numbers can be computed via lucas and next_lucas.
(%i1) fib_via_lucas(n) :=
block([lucas : lucas(n)],
signum(n) * (2*next_lucas - lucas)/5 )$
(%i2) map (fib_via_lucas, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
(%o2) [- 3, 2, - 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21]
Categories: Number theory
If x and y are real numbers and y is nonzero, return
x - y * floor(x / y). Further for all real
x, we have mod (x, 0) = x. For a discussion of the
definition mod (x, 0) = x, see Section 3.4, of
"Concrete Mathematics," by Graham, Knuth, and Patashnik. The function
mod (x, 1) is a sawtooth function with period 1 with
mod (1, 1) = 0 and mod (0, 1) = 0.
To find the principal argument (a number in the interval (-%pi, %pi]) of
a complex number, use the function
x |-> %pi - mod (%pi - x, 2*%pi), where x is an
argument.
When x and y are constant expressions (10 * %pi, for
example), mod uses the same big float evaluation scheme that floor
and ceiling uses. Again, it's possible, although unlikely, that
mod could return an erroneous value in such cases.
For nonnumerical arguments x or y, mod knows several
simplification rules:
(%i1) mod (x, 0); (%o1) x (%i2) mod (a*x, a*y); (%o2) a mod(x, y) (%i3) mod (0, x); (%o3) 0
Categories: Mathematical functions
Returns the smallest prime bigger than n.
(%i1) next_prime(27); (%o1) 29
Categories: Number theory
Expands the expression expr in partial fractions
with respect to the main variable var. partfrac does a complete
partial fraction decomposition. The algorithm employed is based on
the fact that the denominators of the partial fraction expansion (the
factors of the original denominator) are relatively prime. The
numerators can be written as linear combinations of denominators, and
the expansion falls out.
(%i1) 1/(1+x)^2 - 2/(1+x) + 2/(2+x);
2 2 1
(%o1) ----- - ----- + --------
x + 2 x + 1 2
(x + 1)
(%i2) ratsimp (%);
x
(%o2) - -------------------
3 2
x + 4 x + 5 x + 2
(%i3) partfrac (%, x);
2 2 1
(%o3) ----- - ----- + --------
x + 2 x + 1 2
(x + 1)
Uses a modular algorithm to compute a^n mod m
where a and n are integers and m is a positive integer.
If n is negative, inv_mod is used to find the modular inverse.
(%i1) power_mod(3, 15, 5); (%o1) 2 (%i2) mod(3^15,5); (%o2) 2 (%i3) power_mod(2, -1, 5); (%o3) 3 (%i4) inv_mod(2,5); (%o4) 3
Categories: Number theory
Primality test. If primep (n) returns false, n is a
composite number and if it returns true, n is a prime number
with very high probability.
For n less than 341550071728321 a deterministic version of
Miller-Rabin's test is used. If primep (n) returns
true, then n is a prime number.
For n bigger than 341550071728321 primep uses
primep_number_of_tests Miller-Rabin's pseudo-primality tests and one
Lucas pseudo-primality test. The probability that a non-prime n will
pass one Miller-Rabin test is less than 1/4. Using the default value 25 for
primep_number_of_tests, the probability of n being
composite is much smaller that 10^-15.
Categories: Predicate functions · Number theory
Default value: 25
Number of Miller-Rabin's tests used in primep.
Categories: Number theory
Returns the list of all primes from start to end.
(%i1) primes(3, 7); (%o1) [3, 5, 7]
Categories: Number theory
Returns the greatest prime smaller than n.
(%i1) prev_prime(27); (%o1) 23
Categories: Number theory
Returns the principal unit of the real quadratic number field
sqrt (n) where n is an integer,
i.e., the element whose norm is unity.
This amounts to solving Pell's equation a^2 - n b^2 = 1.
(%i1) qunit (17); (%o1) sqrt(17) + 4 (%i2) expand (% * (sqrt(17) - 4)); (%o2) 1
Categories: Number theory
Returns the number of integers less than or equal to n which are relatively prime to n.
Categories: Number theory
Default value: true
When zerobern is false, bern excludes the Bernoulli numbers
and euler excludes the Euler numbers which are equal to zero.
See bern and euler.
Categories: Number theory
Returns the Riemann zeta function. If n is a negative integer, 0, or a
positive even integer, the Riemann zeta function simplifies to an exact value.
For a positive even integer the option variable zeta%pi has to be
true in addition (See zeta%pi). For a floating point or bigfloat
number the Riemann zeta function is evaluated numerically. Maxima returns a
noun form zeta (n) for all other arguments, including rational
noninteger, and complex arguments, or for even integers, if zeta%pi has
the value false.
zeta(1) is undefined, but Maxima knows the limit
limit(zeta(x), x, 1) from above and below.
The Riemann zeta function distributes over lists, matrices, and equations.
Examples:
(%i1) zeta([-2, -1, 0, 0.5, 2, 3, 1+%i]);
2
1 1 %pi
(%o1) [0, - --, - -, - 1.460354508809586, ----, zeta(3),
12 2 6
zeta(%i + 1)]
(%i2) limit(zeta(x),x,1,plus);
(%o2) inf
(%i3) limit(zeta(x),x,1,minus);
(%o3) minf
Categories: Number theory
Default value: true
When zeta%pi is true, zeta returns an expression
proportional to %pi^n for even integer n. Otherwise, zeta
returns a noun form zeta (n) for even integer n.
Examples:
(%i1) zeta%pi: true$
(%i2) zeta (4);
4
%pi
(%o2) ----
90
(%i3) zeta%pi: false$
(%i4) zeta (4);
(%o4) zeta(4)
Categories: Number theory
Shows an addition table of all elements in (Z/nZ).
See also zn_mult_table, zn_power_table.
Categories: Number theory
Returns a list containing the characteristic factors of the totient of n.
Using the characteristic factors a multiplication group modulo n can be expressed as a group direct product of cyclic subgroups.
In case the group itself is cyclic the list only contains the totient
and using zn_primroot a generator can be computed.
If the totient splits into more than one characteristic factors
zn_factor_generators finds generators of the corresponding subgroups.
Each of the r factors in the list divides the right following factors.
For the last factor f_r therefore holds a^f_r = 1 (mod n)
for all a coprime to n.
This factor is also known as Carmichael function or Carmichael lambda.
If n > 2, then totient(n)/2^r is the number of quadratic residues,
and each of these has 2^r square roots.
See also totient, zn_primroot, zn_factor_generators.
Examples:
The multiplication group modulo 14 is cyclic and its 6 elements
can be generated by a primitive root.
(%i1) [zn_characteristic_factors(14), phi: totient(14)]; (%o1) [[6], 6] (%i2) [zn_factor_generators(14), g: zn_primroot(14)]; (%o2) [[3], 3] (%i3) M14: makelist(power_mod(g,i,14), i,0,phi-1); (%o3) [1, 3, 9, 13, 11, 5]
The multiplication group modulo 15 is not cyclic and its 8 elements
can be generated by two factor generators.
(%i1) [[f1,f2]: zn_characteristic_factors(15), totient(15)]; (%o1) [[2, 4], 8] (%i2) [[g1,g2]: zn_factor_generators(15), zn_primroot(15)]; (%o2) [[11, 7], false] (%i3) UG1: makelist(power_mod(g1,i,15), i,0,f1-1); (%o3) [1, 11] (%i4) UG2: makelist(power_mod(g2,i,15), i,0,f2-1); (%o4) [1, 7, 4, 13] (%i5) M15: create_list(mod(i*j,15), i,UG1, j,UG2); (%o5) [1, 7, 4, 13, 11, 2, 14, 8]
For the last characteristic factor 4 it holds that a^4 = 1 (mod 15)
for all a in M15.
M15 has two characteristic factors and therefore 8/2^2 quadratic residues,
and each of these has 2^2 square roots.
(%i6) zn_power_table(15);
[ 1 1 1 1 ]
[ ]
[ 2 4 8 1 ]
[ ]
[ 4 1 4 1 ]
[ ]
[ 7 4 13 1 ]
(%o6) [ ]
[ 8 4 2 1 ]
[ ]
[ 11 1 11 1 ]
[ ]
[ 13 4 7 1 ]
[ ]
[ 14 1 14 1 ]
(%i7) map(lambda([i], zn_nth_root(i,2,15)), [1,4]);
(%o7) [[1, 4, 11, 14], [2, 7, 8, 13]]
Categories: Number theory
Returns 1 if n is 1 and otherwise
the greatest characteristic factor of the totient of n.
For remarks and examples see zn_characteristic_factors.
Categories: Number theory
Uses the technique of LU-decomposition to compute the determinant of matrix over (Z/pZ). p must be a prime.
However if the determinant is equal to zero the LU-decomposition might fail.
In that case zn_determinant computes the determinant non-modular
and reduces thereafter.
See also zn_invert_by_lu.
Examples:
(%i1) m : matrix([1,3],[2,4]);
[ 1 3 ]
(%o1) [ ]
[ 2 4 ]
(%i2) zn_determinant(m, 5);
(%o2) 3
(%i3) m : matrix([2,4,1],[3,1,4],[4,3,2]);
[ 2 4 1 ]
[ ]
(%o3) [ 3 1 4 ]
[ ]
[ 4 3 2 ]
(%i4) zn_determinant(m, 5);
(%o4) 0
Categories: Number theory
Returns a list containing factor generators corresponding to the characteristic factors of the totient of n.
For remarks and examples see zn_characteristic_factors.
Categories: Number theory
Uses the technique of LU-decomposition to compute the modular inverse of
matrix over (Z/pZ). p must be a prime and matrix
invertible. zn_invert_by_lu returns false if matrix
is not invertible.
See also zn_determinant.
Example:
(%i1) m : matrix([1,3],[2,4]);
[ 1 3 ]
(%o1) [ ]
[ 2 4 ]
(%i2) zn_determinant(m, 5);
(%o2) 3
(%i3) mi : zn_invert_by_lu(m, 5);
[ 3 4 ]
(%o3) [ ]
[ 1 2 ]
(%i4) matrixmap(lambda([a], mod(a, 5)), m . mi);
[ 1 0 ]
(%o4) [ ]
[ 0 1 ]
Categories: Number theory
Computes the discrete logarithm. Let (Z/nZ)* be a cyclic group, g a
primitive root modulo n and let a be a member of this group.
zn_log (a, g, n) then solves the congruence g^x = a mod n.
The applied algorithm needs a prime factorization of totient(n). This factorization
might be time consuming as well and in some cases it can be useful to factor first
and then to pass the list of factors to zn_log as the fourth argument.
The list must be of the same form as the list returned by ifactors(totient(n))
using the default option factors_only : false.
The algorithm uses a Pohlig-Hellman-reduction and Pollard's Rho-method for
discrete logarithms. The run time of zn_log primarily depends on the
bitlength of the totient's greatest prime factor.
See also zn_primroot, zn_order, ifactors, totient.
Examples:
zn_log (a, g, n) solves the congruence g^x = a mod n.
(%i1) n : 22$ (%i2) g : zn_primroot(n); (%o2) 7 (%i3) ord_7 : zn_order(7, n); (%o3) 10 (%i4) powers_7 : makelist(power_mod(g, x, n), x, 0, ord_7 - 1); (%o4) [1, 7, 5, 13, 3, 21, 15, 17, 9, 19] (%i5) zn_log(21, g, n); (%o5) 5 (%i6) map(lambda([x], zn_log(x, g, n)), powers_7); (%o6) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
The optional fourth argument must be of the same form as the list returned by
ifactors(totient(n)).
The run time primarily depends on the bitlength of the totient's greatest prime factor.
(%i1) (p : 2^127-1, primep(p)); (%o1) true (%i2) ifs : ifactors(p - 1)$ (%i3) g : zn_primroot(p, ifs); (%o3) 43 (%i4) a : power_mod(g, 1234567890, p)$ (%i5) zn_log(a, g, p, ifs); (%o5) 1234567890 (%i6) time(%o5); (%o6) [1.204] (%i7) f_max : last(ifs); (%o7) [77158673929, 1] (%i8) slength( printf(false, "~b", f_max[1]) ); (%o8) 37
Categories: Number theory
Without the optional argument gcd zn_mult_table(n) shows a
multiplication table of all elements in (Z/nZ)* which are all elements
coprime to n.
The optional second argument gcd allows to select a specific
subset of (Z/nZ). If gcd is an integer, a multiplication table of
all residues x with gcd(x,n) = gcd are returned.
Additionally row and column headings are added for better readability.
If necessary, these can be easily removed by submatrix(1, table, 1).
If gcd is set to all, the table is printed for all non-zero
elements in (Z/nZ).
The second example shows an alternative way to create a multiplication table for subgroups.
See also zn_add_table, zn_power_table.
Examples:
The default table shows all elements in (Z/nZ)* and allows to demonstrate and study basic properties of modular multiplication groups. E.g. the principal diagonal contains all quadratic residues, each row and column contains every element, the tables are symmetric, etc..
If gcd is set to all, the table is printed for all non-zero
elements in (Z/nZ).
(%i1) zn_mult_table(8);
[ 1 3 5 7 ]
[ ]
[ 3 1 7 5 ]
(%o1) [ ]
[ 5 7 1 3 ]
[ ]
[ 7 5 3 1 ]
(%i2) zn_mult_table(8, all);
[ 1 2 3 4 5 6 7 ]
[ ]
[ 2 4 6 0 2 4 6 ]
[ ]
[ 3 6 1 4 7 2 5 ]
[ ]
(%o2) [ 4 0 4 0 4 0 4 ]
[ ]
[ 5 2 7 4 1 6 3 ]
[ ]
[ 6 4 2 0 6 4 2 ]
[ ]
[ 7 6 5 4 3 2 1 ]
If gcd is an integer, row and column headings are added for better readability.
If the subset chosen by gcd is a group there is another way to create
a multiplication table. An isomorphic mapping from a group with 1 as
identity builds a table which is easy to read. The mapping is accomplished via CRT.
In the second version of T36_4 the identity, here 28, is placed in
the top left corner, just like in table T9.
(%i1) T36_4: zn_mult_table(36,4);
[ * 4 8 16 20 28 32 ]
[ ]
[ 4 16 32 28 8 4 20 ]
[ ]
[ 8 32 28 20 16 8 4 ]
[ ]
(%o1) [ 16 28 20 4 32 16 8 ]
[ ]
[ 20 8 16 32 4 20 28 ]
[ ]
[ 28 4 8 16 20 28 32 ]
[ ]
[ 32 20 4 8 28 32 16 ]
(%i2) T9: zn_mult_table(36/4);
[ 1 2 4 5 7 8 ]
[ ]
[ 2 4 8 1 5 7 ]
[ ]
[ 4 8 7 2 1 5 ]
(%o2) [ ]
[ 5 1 2 7 8 4 ]
[ ]
[ 7 5 1 8 4 2 ]
[ ]
[ 8 7 5 4 2 1 ]
(%i3) T36_4: matrixmap(lambda([x], chinese([0,x],[4,9])), T9);
[ 28 20 4 32 16 8 ]
[ ]
[ 20 4 8 28 32 16 ]
[ ]
[ 4 8 16 20 28 32 ]
(%o3) [ ]
[ 32 28 20 16 8 4 ]
[ ]
[ 16 32 28 8 4 20 ]
[ ]
[ 8 16 32 4 20 28 ]
Categories: Number theory
Returns a list with all n-th roots of x from the multiplication
subgroup of (Z/mZ) which contains x, or false, if x
is no n-th power modulo m or not contained in any multiplication
subgroup of (Z/mZ).
x is an element of a multiplication subgroup modulo m, if the
greatest common divisor g = gcd(x,m) is coprime to m/g.
zn_nth_root is based on an algorithm by Adleman, Manders and Miller
and on theorems about modulo multiplication groups by Daniel Shanks.
The algorithm needs a prime factorization of the modulus m.
So in case the factorization of m is known, the list of factors
can be passed as the fourth argument. This optional argument
must be of the same form as the list returned by ifactors(m)
using the default option factors_only: false.
Examples:
A power table of the multiplication group modulo 14
followed by a list of lists containing all n-th roots of 1
with n from 1 to 6.
(%i1) zn_power_table(14);
[ 1 1 1 1 1 1 ]
[ ]
[ 3 9 13 11 5 1 ]
[ ]
[ 5 11 13 9 3 1 ]
(%o1) [ ]
[ 9 11 1 9 11 1 ]
[ ]
[ 11 9 1 11 9 1 ]
[ ]
[ 13 1 13 1 13 1 ]
(%i2) makelist(zn_nth_root(1,n,14), n,1,6);
(%o2) [[1], [1, 13], [1, 9, 11], [1, 13], [1], [1, 3, 5, 9, 11, 13]]
In the following example x is not coprime to m, but is a member of a multiplication subgroup of (Z/mZ) and any n-th root is a member of the same subgroup.
The residue class 3 is no member of any multiplication subgroup of (Z/63Z)
and is therefore not returned as a third root of 27.
Here zn_power_table shows all residues x in (Z/63Z)
with gcd(x,63) = 9. This subgroup is isomorphic to (Z/7Z)*
and its identity 36 is computed via CRT.
(%i1) m: 7*9$
(%i2) zn_power_table(m,9);
[ 9 18 36 9 18 36 ]
[ ]
[ 18 9 36 18 9 36 ]
[ ]
[ 27 36 27 36 27 36 ]
(%o2) [ ]
[ 36 36 36 36 36 36 ]
[ ]
[ 45 9 27 18 54 36 ]
[ ]
[ 54 18 27 9 45 36 ]
(%i3) zn_nth_root(27,3,m);
(%o3) [27, 45, 54]
(%i4) id7:1$ id63_9: chinese([id7,0],[7,9]);
(%o5) 36
In the following RSA-like example, where the modulus N is squarefree,
i.e. it splits into
exclusively first power factors, every x from 0 to N-1
is contained in a multiplication subgroup.
The process of decryption needs the e-th root.
e is coprime to totient(N) and therefore the e-th root is unique.
In this case zn_nth_root effectively performs CRT-RSA.
(Please note that flatten removes braces but no solutions.)
(%i1) [p,q,e]: [5,7,17]$ N: p*q$ (%i3) xs: makelist(x,x,0,N-1)$ (%i4) ys: map(lambda([x],power_mod(x,e,N)),xs)$ (%i5) zs: flatten(map(lambda([y], zn_nth_root(y,e,N)), ys))$ (%i6) is(zs = xs); (%o6) true
In the following example the factorization of the modulus is known and passed as the fourth argument.
(%i1) p: 2^107-1$ q: 2^127-1$ N: p*q$ (%i4) ibase: obase: 16$ (%i5) msg: 11223344556677889900aabbccddeeff$ (%i6) enc: power_mod(msg, 10001, N); (%o6) 1a8db7892ae588bdc2be25dd5107a425001fe9c82161abc673241c8b383 (%i7) zn_nth_root(enc, 10001, N, [[p,1],[q,1]]); (%o7) [11223344556677889900aabbccddeeff]
Categories: Number theory
Returns the order of x if it is a unit of the finite group (Z/nZ)*
or returns false. x is a unit modulo n if it is coprime to n.
The applied algorithm needs a prime factorization of totient(n). This factorization
might be time consuming in some cases and it can be useful to factor first
and then to pass the list of factors to zn_log as the third argument.
The list must be of the same form as the list returned by ifactors(totient(n))
using the default option factors_only : false.
See also zn_primroot, ifactors, totient.
Examples:
zn_order computes the order of the unit x in (Z/nZ)*.
(%i1) n : 22$ (%i2) g : zn_primroot(n); (%o2) 7 (%i3) units_22 : sublist(makelist(i,i,1,21), lambda([x], gcd(x, n) = 1)); (%o3) [1, 3, 5, 7, 9, 13, 15, 17, 19, 21] (%i4) (ord_7 : zn_order(7, n)) = totient(n); (%o4) 10 = 10 (%i5) powers_7 : makelist(power_mod(g,i,n), i,0,ord_7 - 1); (%o5) [1, 7, 5, 13, 3, 21, 15, 17, 9, 19] (%i6) map(lambda([x], zn_order(x, n)), powers_7); (%o6) [1, 10, 5, 10, 5, 2, 5, 10, 5, 10] (%i7) map(lambda([x], ord_7/gcd(x, ord_7)), makelist(i, i,0,ord_7 - 1)); (%o7) [1, 10, 5, 10, 5, 2, 5, 10, 5, 10] (%i8) totient(totient(n)); (%o8) 4
The optional third argument must be of the same form as the list returned by
ifactors(totient(n)).
(%i1) (p : 2^142 + 217, primep(p)); (%o1) true (%i2) ifs : ifactors( totient(p) )$ (%i3) g : zn_primroot(p, ifs); (%o3) 3 (%i4) is( (ord_3 : zn_order(g, p, ifs)) = totient(p) ); (%o4) true (%i5) map(lambda([x], ord_3/zn_order(x, p, ifs)), makelist(i,i,2,15)); (%o5) [22, 1, 44, 10, 5, 2, 22, 2, 8, 2, 1, 1, 20, 1]
Categories: Number theory
Without any optional argument zn_power_table(n)
shows a power table of all elements in (Z/nZ)*
which are all residue classes coprime to n.
The exponent loops from 1 to the greatest characteristic factor of
totient(n) (also known as Carmichael function or Carmichael lambda)
and the table ends with a column of ones on the right side.
The optional second argument gcd allows to select powers of a specific
subset of (Z/nZ). If gcd is an integer, powers of all residue
classes x with gcd(x,n) = gcd are returned,
i.e. the default value for gcd is 1.
If gcd is set to all, the table contains powers of all elements
in (Z/nZ).
If the optional third argument max_exp is given, the exponent loops from
1 to max_exp.
See also zn_add_table, zn_mult_table.
Examples:
The default which is gcd = 1 allows to demonstrate and study basic
theorems of e.g. Fermat and Euler.
The argument gcd allows to select subsets of (Z/nZ) and to study
multiplication subgroups and isomorphisms.
E.g. the groups G10 and G10_2 are under multiplication both
isomorphic to G5. 1 is the identity in G5.
So are 1 resp. 6 the identities in G10 resp. G10_2.
There are corresponding mappings for primitive roots, n-th roots, etc..
(%i1) zn_power_table(10);
[ 1 1 1 1 ]
[ ]
[ 3 9 7 1 ]
(%o1) [ ]
[ 7 9 3 1 ]
[ ]
[ 9 1 9 1 ]
(%i2) zn_power_table(10,2);
[ 2 4 8 6 ]
[ ]
[ 4 6 4 6 ]
(%o2) [ ]
[ 6 6 6 6 ]
[ ]
[ 8 4 2 6 ]
(%i3) zn_power_table(10,5);
(%o3) [ 5 5 5 5 ]
(%i4) zn_power_table(10,10);
(%o4) [ 0 0 0 0 ]
(%i5) G5: [1,2,3,4];
(%o6) [1, 2, 3, 4]
(%i6) G10_2: map(lambda([x], chinese([0,x],[2,5])), G5);
(%o6) [6, 2, 8, 4]
(%i7) G10: map(lambda([x], power_mod(3, zn_log(x,2,5), 10)), G5);
(%o7) [1, 3, 7, 9]
If gcd is set to all, the table contains powers of all elements
in (Z/nZ).
The third argument max_exp allows to set the highest exponent. The following table shows a very small example of RSA.
(%i1) N:2*5$ phi:totient(N)$ e:7$ d:inv_mod(e,phi)$
(%i5) zn_power_table(N, all, e*d);
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]
[ ]
[ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
[ ]
[ 2 4 8 6 2 4 8 6 2 4 8 6 2 4 8 6 2 4 8 6 2 ]
[ ]
[ 3 9 7 1 3 9 7 1 3 9 7 1 3 9 7 1 3 9 7 1 3 ]
[ ]
[ 4 6 4 6 4 6 4 6 4 6 4 6 4 6 4 6 4 6 4 6 4 ]
(%o5) [ ]
[ 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 ]
[ ]
[ 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 ]
[ ]
[ 7 9 3 1 7 9 3 1 7 9 3 1 7 9 3 1 7 9 3 1 7 ]
[ ]
[ 8 4 2 6 8 4 2 6 8 4 2 6 8 4 2 6 8 4 2 6 8 ]
[ ]
[ 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 ]
Categories: Number theory
If the multiplicative group (Z/nZ)* is cyclic, zn_primroot computes the
smallest primitive root modulo n. (Z/nZ)* is cyclic if n is equal to
2, 4, p^k or 2*p^k, where p is prime and
greater than 2 and k is a natural number. zn_primroot
performs an according pretest if the option variable zn_primroot_pretest
(default: false) is set to true. In any case the computation is limited
by the upper bound zn_primroot_limit.
If (Z/nZ)* is not cyclic or if there is no primitive root up to
zn_primroot_limit, zn_primroot returns false.
The applied algorithm needs a prime factorization of totient(n). This factorization
might be time consuming in some cases and it can be useful to factor first
and then to pass the list of factors to zn_log as an additional argument.
The list must be of the same form as the list returned by ifactors(totient(n))
using the default option factors_only : false.
See also zn_primroot_p, zn_order, ifactors, totient.
Examples:
zn_primroot computes the smallest primitive root modulo n or returns
false.
(%i1) n : 14$ (%i2) g : zn_primroot(n); (%o2) 3 (%i3) zn_order(g, n) = totient(n); (%o3) 6 = 6 (%i4) n : 15$ (%i5) zn_primroot(n); (%o5) false
The optional second argument must be of the same form as the list returned by
ifactors(totient(n)).
(%i1) (p : 2^142 + 217, primep(p));
(%o1) true
(%i2) ifs : ifactors( totient(p) )$
(%i3) g : zn_primroot(p, ifs);
(%o3) 3
(%i4) [time(%o2), time(%o3)];
(%o4) [[15.556972], [0.004]]
(%i5) is(zn_order(g, p, ifs) = p - 1);
(%o5) true
(%i6) n : 2^142 + 216$
(%i7) ifs : ifactors(totient(n))$
(%i8) zn_primroot(n, ifs),
zn_primroot_limit : 200, zn_primroot_verbose : true;
`zn_primroot' stopped at zn_primroot_limit = 200
(%o8) false
Categories: Number theory
Default value: 1000
If zn_primroot cannot find a primitve root, it stops at this upper bound.
If the option variable zn_primroot_verbose (default: false) is
set to true, a message will be printed when zn_primroot_limit is reached.
Categories: Number theory
Checks whether x is a primitive root in the multiplicative group (Z/nZ)*.
The applied algorithm needs a prime factorization of totient(n). This factorization
might be time consuming and in case zn_primroot_p will be consecutively
applied to a list of candidates it can be useful to factor first and then to
pass the list of factors to zn_log as a third argument.
The list must be of the same form as the list returned by ifactors(totient(n))
using the default option factors_only : false.
See also zn_primroot, zn_order, ifactors, totient.
Examples:
zn_primroot_p as a predicate function.
(%i1) n : 14$ (%i2) units_14 : sublist(makelist(i,i,1,13), lambda([i], gcd(i, n) = 1)); (%o2) [1, 3, 5, 9, 11, 13] (%i3) zn_primroot_p(13, n); (%o3) false (%i4) sublist(units_14, lambda([x], zn_primroot_p(x, n))); (%o4) [3, 5] (%i5) map(lambda([x], zn_order(x, n)), units_14); (%o5) [1, 6, 6, 3, 3, 2]
The optional third argument must be of the same form as the list returned by
ifactors(totient(n)).
(%i1) (p : 2^142 + 217, primep(p)); (%o1) true (%i2) ifs : ifactors( totient(p) )$ (%i3) sublist(makelist(i,i,1,50), lambda([x], zn_primroot_p(x, p, ifs))); (%o3) [3, 12, 13, 15, 21, 24, 26, 27, 29, 33, 38, 42, 48] (%i4) [time(%o2), time(%o3)]; (%o4) [[7.748484], [0.036002]]
Categories: Predicate functions · Number theory
Default value: false
The multiplicative group (Z/nZ)* is cyclic if n is equal to
2, 4, p^k or 2*p^k, where p is prime and
greater than 2 and k is a natural number.
zn_primroot_pretest controls whether zn_primroot will check
if one of these cases occur before it computes the smallest primitive root.
Only if zn_primroot_pretest is set to true this pretest will be
performed.
Categories: Number theory
Default value: false
Controls whether zn_primroot prints a message when reaching
zn_primroot_limit.
Categories: Number theory
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| 30.1 Introduction to Symmetries | ||
| 30.2 Functions and Variables for Symmetries |
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sym is a package for working with symmetric groups of polynomials.
It was written for Macsyma-Symbolics by Annick Valibouze (http://www-calfor.lip6.fr/~avb/). The algorithms are described in the following papers:
Categories: Group theory · Polynomials · Share packages · Package sym
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implements passing from the complete symmetric functions given in the list L to the elementary symmetric functions from 0 to n. If the list L contains fewer than n+1 elements, it will be completed with formal values of the type h1, h2, etc. If the first element of the list L exists, it specifies the size of the alphabet, otherwise the size is set to n.
(%i1) comp2pui (3, [4, g]);
2 2
(%o1) [4, g, 2 h2 - g , 3 h3 - g h2 + g (g - 2 h2)]
Categories: Package sym
goes from the elementary symmetric functions to the complete functions.
Similar to comp2ele and comp2pui.
Other functions for changing bases: comp2ele.
Categories: Package sym
Goes from the elementary symmetric functions to the compete functions.
Similar to comp2ele and comp2pui.
Other functions for changing bases: comp2ele.
Categories: Package sym
decomposes the symmetric polynomial sym, in the variables
contained in the list lvar, in terms of the elementary symmetric
functions given in the list ele. If the first element of
ele is given, it will be the size of the alphabet, otherwise the
size will be the degree of the polynomial sym. If values are
missing in the list ele, formal values of the type e1,
e2, etc. will be added. The polynomial sym may be given in
three different forms: contracted (elem should then be 1, its
default value), partitioned (elem should be 3), or extended
(i.e. the entire polynomial, and elem should then be 2). The
function pui is used in the same way.
On an alphabet of size 3 with e1, the first elementary symmetric function, with value 7, the symmetric polynomial in 3 variables whose contracted form (which here depends on only two of its variables) is x^4-2*x*y decomposes as follows in elementary symmetric functions:
(%i1) elem ([3, 7], x^4 - 2*x*y, [x, y]);
(%o1) 7 (e3 - 7 e2 + 7 (49 - e2)) + 21 e3
+ (- 2 (49 - e2) - 2) e2
(%i2) ratsimp (%);
2
(%o2) 28 e3 + 2 e2 - 198 e2 + 2401
Other functions for changing bases: comp2ele.
Categories: Package sym
The list L represents the Schur function S_L: we have L = [i_1, i_2, ..., i_q], with i_1 <= i_2 <= ... <= i_q. The Schur function S_[i_1, i_2, ..., i_q] is the minor of the infinite matrix h_[i-j], i <= 1, j <= 1, consisting of the q first rows and the columns 1 + i_1, 2 + i_2, ..., q + i_q.
This Schur function can be written in terms of monomials by using
treinat and kostka. The form returned is a symmetric
polynomial in a contracted representation in the variables
x_1,x_2,...
(%i1) mon2schur ([1, 1, 1]);
(%o1) x1 x2 x3
(%i2) mon2schur ([3]);
2 3
(%o2) x1 x2 x3 + x1 x2 + x1
(%i3) mon2schur ([1, 2]);
2
(%o3) 2 x1 x2 x3 + x1 x2
which means that for 3 variables this gives:
2 x1 x2 x3 + x1^2 x2 + x2^2 x1 + x1^2 x3 + x3^2 x1
+ x2^2 x3 + x3^2 x2
Other functions for changing bases: comp2ele.
Categories: Package sym
decomposes a multi-symmetric polynomial in the multi-contracted form multi_pc in the groups of variables contained in the list of lists l_var in terms of the elementary symmetric functions contained in l_elem.
(%i1) multi_elem ([[2, e1, e2], [2, f1, f2]], a*x + a^2 + x^3,
[[x, y], [a, b]]);
3
(%o1) - 2 f2 + f1 (f1 + e1) - 3 e1 e2 + e1
(%i2) ratsimp (%);
2 3
(%o2) - 2 f2 + f1 + e1 f1 - 3 e1 e2 + e1
Other functions for changing bases: comp2ele.
Categories: Package sym
is to the function pui what the function multi_elem is to
the function elem.
(%i1) multi_pui ([[2, p1, p2], [2, t1, t2]], a*x + a^2 + x^3,
[[x, y], [a, b]]);
3
3 p1 p2 p1
(%o1) t2 + p1 t1 + ------- - ---
2 2
Categories: Package sym
decomposes the symmetric polynomial sym, in the variables in the
list lvar, in terms of the power functions in the list L.
If the first element of L is given, it will be the size of the
alphabet, otherwise the size will be the degree of the polynomial
sym. If values are missing in the list L, formal values of
the type p1, p2 , etc. will be added. The polynomial
sym may be given in three different forms: contracted (elem
should then be 1, its default value), partitioned (elem should be
3), or extended (i.e. the entire polynomial, and elem should then
be 2). The function pui is used in the same way.
(%i1) pui;
(%o1) 1
(%i2) pui ([3, a, b], u*x*y*z, [x, y, z]);
2
a (a - b) u (a b - p3) u
(%o2) ------------ - ------------
6 3
(%i3) ratsimp (%);
3
(2 p3 - 3 a b + a ) u
(%o3) ---------------------
6
Other functions for changing bases: comp2ele.
Categories: Package sym
renders the list of the first n complete functions (with the
length first) in terms of the power functions given in the list
lpui. If the list lpui is empty, the cardinal is n,
otherwise it is its first element (as in comp2ele and
comp2pui).
(%i1) pui2comp (2, []);
2
p2 + p1
(%o1) [2, p1, --------]
2
(%i2) pui2comp (3, [2, a1]);
2
a1 (p2 + a1 )
2 p3 + ------------- + a1 p2
p2 + a1 2
(%o2) [2, a1, --------, --------------------------]
2 3
(%i3) ratsimp (%);
2 3
p2 + a1 2 p3 + 3 a1 p2 + a1
(%o3) [2, a1, --------, --------------------]
2 6
Other functions for changing bases: comp2ele.
Categories: Package sym
effects the passage from power functions to the elementary symmetric functions.
If the flag pui2ele is girard, it will return the list of
elementary symmetric functions from 1 to n, and if the flag is
close, it will return the n-th elementary symmetric function.
Other functions for changing bases: comp2ele.
Categories: Package sym
lpui is a list whose first element is an integer m.
puireduc gives the first n power functions in terms of the
first m.
(%i1) puireduc (3, [2]);
2
p1 (p1 - p2)
(%o1) [2, p1, p2, p1 p2 - -------------]
2
(%i2) ratsimp (%);
3
3 p1 p2 - p1
(%o2) [2, p1, p2, -------------]
2
Categories: Package sym
P is a polynomial in the variables of the list l_var. Each
of these variables represents a complete symmetric function. In
l_var the i-th complete symmetric function is represented by
the concatenation of the letter h and the integer i:
hi. This function expresses P in terms of Schur
functions.
(%i1) schur2comp (h1*h2 - h3, [h1, h2, h3]);
(%o1) s
1, 2
(%i2) schur2comp (a*h3, [h3]);
(%o2) s a
3
Categories: Package sym
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returns the partitioned polynomial associated to the contracted form pc whose variables are in lvar.
(%i1) pc: 2*a^3*b*x^4*y + x^5;
3 4 5
(%o1) 2 a b x y + x
(%i2) cont2part (pc, [x, y]);
3
(%o2) [[1, 5, 0], [2 a b, 4, 1]]
Categories: Package sym
returns a contracted form (i.e. a monomial orbit under the action of the
symmetric group) of the polynomial psym in the variables contained
in the list lvar. The function explose performs the
inverse operation. The function tcontract tests the symmetry of
the polynomial.
(%i1) psym: explose (2*a^3*b*x^4*y, [x, y, z]);
3 4 3 4 3 4 3 4
(%o1) 2 a b y z + 2 a b x z + 2 a b y z + 2 a b x z
3 4 3 4
+ 2 a b x y + 2 a b x y
(%i2) contract (psym, [x, y, z]);
3 4
(%o2) 2 a b x y
Categories: Package sym
returns the symmetric polynomial associated with the contracted form pc. The list lvar contains the variables.
(%i1) explose (a*x + 1, [x, y, z]); (%o1) a z + a y + a x + 1
Categories: Package sym
goes from the partitioned form to the contracted form of a symmetric polynomial. The contracted form is rendered with the variables in lvar.
(%i1) part2cont ([[2*a^3*b, 4, 1]], [x, y]);
3 4
(%o1) 2 a b x y
Categories: Package sym
psym is a symmetric polynomial in the variables of the list lvar. This function retturns its partitioned representation.
(%i1) partpol (-a*(x + y) + 3*x*y, [x, y]); (%o1) [[3, 1, 1], [- a, 1, 0]]
Categories: Package sym
tests if the polynomial pol is symmetric in the variables of the
list lvar. If so, it returns a contracted representation like the
function contract.
Categories: Package sym
tests if the polynomial pol is symmetric in the variables of the
list lvar. If so, it returns its partitioned representation like
the function partpol.
Categories: Package sym
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calculates the direct image (see M. Giusti, D. Lazard et A. Valibouze, ISSAC 1988, Rome) associated to the function f, in the lists of variables lvar_1, ..., lvar_n, and in the polynomials p_1, ..., p_n in a variable y. The arity of the function f is important for the calulation. Thus, if the expression for f does not depend on some variable, it is useless to include this variable, and not including it will also considerably reduce the amount of computation.
(%i1) direct ([z^2 - e1* z + e2, z^2 - f1* z + f2],
z, b*v + a*u, [[u, v], [a, b]]);
2
(%o1) y - e1 f1 y
2 2 2 2
- 4 e2 f2 - (e1 - 2 e2) (f1 - 2 f2) + e1 f1
+ -----------------------------------------------
2
(%i2) ratsimp (%);
2 2 2
(%o2) y - e1 f1 y + (e1 - 4 e2) f2 + e2 f1
(%i3) ratsimp (direct ([z^3-e1*z^2+e2*z-e3,z^2 - f1* z + f2],
z, b*v + a*u, [[u, v], [a, b]]));
6 5 2 2 2 4
(%o3) y - 2 e1 f1 y + ((2 e1 - 6 e2) f2 + (2 e2 + e1 ) f1 ) y
3 3 3
+ ((9 e3 + 5 e1 e2 - 2 e1 ) f1 f2 + (- 2 e3 - 2 e1 e2) f1 ) y
2 2 4 2
+ ((9 e2 - 6 e1 e2 + e1 ) f2
2 2 2 2 4
+ (- 9 e1 e3 - 6 e2 + 3 e1 e2) f1 f2 + (2 e1 e3 + e2 ) f1 )
2 2 2 3 2
y + (((9 e1 - 27 e2) e3 + 3 e1 e2 - e1 e2) f1 f2
2 2 3 5
+ ((15 e2 - 2 e1 ) e3 - e1 e2 ) f1 f2 - 2 e2 e3 f1 ) y
2 3 3 2 2 3
+ (- 27 e3 + (18 e1 e2 - 4 e1 ) e3 - 4 e2 + e1 e2 ) f2
2 3 3 2 2
+ (27 e3 + (e1 - 9 e1 e2) e3 + e2 ) f1 f2
2 4 2 6
+ (e1 e2 e3 - 9 e3 ) f1 f2 + e3 f1
Finding the polynomial whose roots are the sums a+u where a is a root of z^2 - e_1 z + e_2 and u is a root of z^2 - f_1 z + f_2.
(%i1) ratsimp (direct ([z^2 - e1* z + e2, z^2 - f1* z + f2],
z, a + u, [[u], [a]]));
4 3 2
(%o1) y + (- 2 f1 - 2 e1) y + (2 f2 + f1 + 3 e1 f1 + 2 e2
2 2 2 2
+ e1 ) y + ((- 2 f1 - 2 e1) f2 - e1 f1 + (- 2 e2 - e1 ) f1
2 2 2
- 2 e1 e2) y + f2 + (e1 f1 - 2 e2 + e1 ) f2 + e2 f1 + e1 e2 f1
2
+ e2
direct accepts two flags: elementaires and
puissances (default) which allow decomposing the symmetric
polynomials appearing in the calculation into elementary symmetric
functions, or power functions, respectively.
Functions of sym used in this function:
multi_orbit (so orbit), pui_direct, multi_elem
(so elem), multi_pui (so pui), pui2ele, ele2pui
(if the flag direct is in puissances).
Categories: Package sym
P is a polynomial in the set of variables contained in the lists lvar_1, lvar_2, ..., lvar_p. This function returns the orbit of the polynomial P under the action of the product of the symmetric groups of the sets of variables represented in these p lists.
(%i1) multi_orbit (a*x + b*y, [[x, y], [a, b]]); (%o1) [b y + a x, a y + b x] (%i2) multi_orbit (x + y + 2*a, [[x, y], [a, b, c]]); (%o2) [y + x + 2 c, y + x + 2 b, y + x + 2 a]
Also see: orbit for the action of a single symmetric group.
Categories: Package sym
returns the product of the two symmetric polynomials in n variables by working only modulo the action of the symmetric group of order n. The polynomials are in their partitioned form.
Given the 2 symmetric polynomials in x, y: 3*(x + y)
+ 2*x*y and 5*(x^2 + y^2) whose partitioned forms are [[3,
1], [2, 1, 1]] and [[5, 2]], their product will be
(%i1) multsym ([[3, 1], [2, 1, 1]], [[5, 2]], 2); (%o1) [[10, 3, 1], [15, 3, 0], [15, 2, 1]]
that is 10*(x^3*y + y^3*x) + 15*(x^2*y + y^2*x) + 15*(x^3 + y^3).
Functions for changing the representations of a symmetric polynomial:
contract, cont2part, explose, part2cont,
partpol, tcontract, tpartpol.
Categories: Package sym
computes the orbit of the polynomial P in the variables in the list lvar under the action of the symmetric group of the set of variables in the list lvar.
(%i1) orbit (a*x + b*y, [x, y]);
(%o1) [a y + b x, b y + a x]
(%i2) orbit (2*x + x^2, [x, y]);
2 2
(%o2) [y + 2 y, x + 2 x]
See also multi_orbit for the action of a product of symmetric
groups on a polynomial.
Categories: Package sym
Let f be a polynomial in n blocks of variables lvar_1,
..., lvar_n. Let c_i be the number of variables in
lvar_i, and SC be the product of n symmetric groups of
degree c_1, ..., c_n. This group acts naturally on f.
The list orbite is the orbit, denoted SC(f), of
the function f under the action of SC. (This list may be
obtained by the function multi_orbit.) The di are integers
s.t.
c_1 <= d_1, c_2 <= d_2, ..., c_n <= d_n.
Let SD be the product of the symmetric groups S_[d_1] x
S_[d_2] x ... x S_[d_n].
The function pui_direct returns
the first n power functions of SD(f) deduced
from the power functions of SC(f), where n is
the size of SD(f).
The result is in multi-contracted form w.r.t. SD, i.e. only one element is kept per orbit, under the action of SD.
(%i1) l: [[x, y], [a, b]];
(%o1) [[x, y], [a, b]]
(%i2) pui_direct (multi_orbit (a*x + b*y, l), l, [2, 2]);
2 2
(%o2) [a x, 4 a b x y + a x ]
(%i3) pui_direct (multi_orbit (a*x + b*y, l), l, [3, 2]);
2 2 2 2 3 3
(%o3) [2 a x, 4 a b x y + 2 a x , 3 a b x y + 2 a x ,
2 2 2 2 3 3 4 4
12 a b x y + 4 a b x y + 2 a x ,
3 2 3 2 4 4 5 5
10 a b x y + 5 a b x y + 2 a x ,
3 3 3 3 4 2 4 2 5 5 6 6
40 a b x y + 15 a b x y + 6 a b x y + 2 a x ]
(%i4) pui_direct ([y + x + 2*c, y + x + 2*b, y + x + 2*a],
[[x, y], [a, b, c]], [2, 3]);
2 2
(%o4) [3 x + 2 a, 6 x y + 3 x + 4 a x + 4 a ,
2 3 2 2 3
9 x y + 12 a x y + 3 x + 6 a x + 12 a x + 8 a ]
Categories: Package sym
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written by P. Esperet, calculates the Kostka number of the partition part_1 and part_2.
(%i1) kostka ([3, 3, 3], [2, 2, 2, 1, 1, 1]); (%o1) 6
Categories: Package sym
returns the list of partitions of weight n and length m.
(%i1) lgtreillis (4, 2); (%o1) [[3, 1], [2, 2]]
Also see: ltreillis, treillis and treinat.
Categories: Package sym
returns the list of partitions of weight n and length less than or equal to m.
(%i1) ltreillis (4, 2); (%o1) [[4, 0], [3, 1], [2, 2]]
Also see: lgtreillis, treillis and treinat.
Categories: Package sym
returns all partitions of weight n.
(%i1) treillis (4); (%o1) [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]
See also: lgtreillis, ltreillis and treinat.
Categories: Package sym
retruns the list of partitions inferior to the partition part w.r.t. the natural order.
(%i1) treinat ([5]);
(%o1) [[5]]
(%i2) treinat ([1, 1, 1, 1, 1]);
(%o2) [[5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1],
[1, 1, 1, 1, 1]]
(%i3) treinat ([3, 2]);
(%o3) [[5], [4, 1], [3, 2]]
See also: lgtreillis, ltreillis and treillis.
Categories: Package sym
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returns the polynomial in z s.t. the elementary symmetric
functions of its roots are in the list L = [n,
e_1, ..., e_n], where n is the degree of the
polynomial and e_i the i-th elementary symmetric function.
(%i1) ele2polynome ([2, e1, e2], z);
2
(%o1) z - e1 z + e2
(%i2) polynome2ele (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x);
(%o2) [7, 0, - 14, 0, 56, 0, - 56, - 22]
(%i3) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x);
7 5 3
(%o3) x - 14 x + 56 x - 56 x + 22
The inverse: polynome2ele (P, z).
Also see:
polynome2ele, pui2polynome.
Categories: Package sym
gives the list l = [n, e_1, ..., e_n]
where n is the degree of the polynomial P in the variable
x and e_i is the i-the elementary symmetric function
of the roots of P.
(%i1) polynome2ele (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x);
(%o1) [7, 0, - 14, 0, 56, 0, - 56, - 22]
(%i2) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x);
7 5 3
(%o2) x - 14 x + 56 x - 56 x + 22
The inverse: ele2polynome (l, x)
Categories: Package sym
L is a list containing the elementary symmetric functions
on a set A. prodrac returns the polynomial whose roots
are the k by k products of the elements of A.
Also see somrac.
Categories: Package sym
calculates the polynomial in x whose power functions of the roots are given in the list lpui.
(%i1) pui;
(%o1) 1
(%i2) kill(labels);
(%o0) done
(%i1) polynome2ele (x^3 - 4*x^2 + 5*x - 1, x);
(%o1) [3, 4, 5, 1]
(%i2) ele2pui (3, %);
(%o2) [3, 4, 6, 7]
(%i3) pui2polynome (x, %);
3 2
(%o3) x - 4 x + 5 x - 1
See also:
polynome2ele, ele2polynome.
Categories: Package sym
The list L contains elementary symmetric functions of a polynomial P . The function computes the polynomial whose roots are the k by k distinct sums of the roots of P.
Also see prodrac.
Categories: Package sym
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calculates the resolvent of the polynomial P in x of degree
n >= d by the function f expressed in the variables
x_1, ..., x_d. For efficiency of computation it is
important to not include in the list [x_1, ..., x_d]
variables which do not appear in the transformation function f.
To increase the efficiency of the computation one may set flags in
resolvante so as to use appropriate algorithms:
If the function f is unitary:
(x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1 -
(x1*x3 + x3*x5 + x5*x2 + x2*x4 + x4*x1))^2
general,
the flag of resolvante may be, respectively:
(%i1) resolvante: unitaire$
(%i2) resolvante (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x, x^3 - 1,
[x]);
" resolvante unitaire " [7, 0, 28, 0, 168, 0, 1120, - 154, 7840,
- 2772, 56448, - 33880,
413952, - 352352, 3076668, - 3363360, 23114112, - 30494464,
175230832, - 267412992, 1338886528, - 2292126760]
3 6 3 9 6 3
[x - 1, x - 2 x + 1, x - 3 x + 3 x - 1,
12 9 6 3 15 12 9 6 3
x - 4 x + 6 x - 4 x + 1, x - 5 x + 10 x - 10 x + 5 x
18 15 12 9 6 3
- 1, x - 6 x + 15 x - 20 x + 15 x - 6 x + 1,
21 18 15 12 9 6 3
x - 7 x + 21 x - 35 x + 35 x - 21 x + 7 x - 1]
[- 7, 1127, - 6139, 431767, - 5472047, 201692519, - 3603982011]
7 6 5 4 3 2
(%o2) y + 7 y - 539 y - 1841 y + 51443 y + 315133 y
+ 376999 y + 125253
(%i3) resolvante: lineaire$
(%i4) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);
" resolvante lineaire "
24 20 16 12 8
(%o4) y + 80 y + 7520 y + 1107200 y + 49475840 y
4
+ 344489984 y + 655360000
(%i5) resolvante: general$
(%i6) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);
" resolvante generale "
24 20 16 12 8
(%o6) y + 80 y + 7520 y + 1107200 y + 49475840 y
4
+ 344489984 y + 655360000
(%i7) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3, x4]);
" resolvante generale "
24 20 16 12 8
(%o7) y + 80 y + 7520 y + 1107200 y + 49475840 y
4
+ 344489984 y + 655360000
(%i8) direct ([x^4 - 1], x, x1 + 2*x2 + 3*x3, [[x1, x2, x3]]);
24 20 16 12 8
(%o8) y + 80 y + 7520 y + 1107200 y + 49475840 y
4
+ 344489984 y + 655360000
(%i9) resolvante :lineaire$
(%i10) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);
" resolvante lineaire "
4
(%o10) y - 1
(%i11) resolvante: symetrique$
(%i12) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);
" resolvante symetrique "
4
(%o12) y - 1
(%i13) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);
" resolvante symetrique "
6 2
(%o13) y - 4 y - 1
(%i14) resolvante: alternee$
(%i15) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);
" resolvante alternee "
12 8 6 4 2
(%o15) y + 8 y + 26 y - 112 y + 216 y + 229
(%i16) resolvante: produit$
(%i17) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);
" resolvante produit "
35 33 29 28 27 26
(%o17) y - 7 y - 1029 y + 135 y + 7203 y - 756 y
24 23 22 21 20
+ 1323 y + 352947 y - 46305 y - 2463339 y + 324135 y
19 18 17 15
- 30618 y - 453789 y - 40246444 y + 282225202 y
14 12 11 10
- 44274492 y + 155098503 y + 12252303 y + 2893401 y
9 8 7 6
- 171532242 y + 6751269 y + 2657205 y - 94517766 y
5 3
- 3720087 y + 26040609 y + 14348907
(%i18) resolvante: symetrique$
(%i19) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);
" resolvante symetrique "
35 33 29 28 27 26
(%o19) y - 7 y - 1029 y + 135 y + 7203 y - 756 y
24 23 22 21 20
+ 1323 y + 352947 y - 46305 y - 2463339 y + 324135 y
19 18 17 15
- 30618 y - 453789 y - 40246444 y + 282225202 y
14 12 11 10
- 44274492 y + 155098503 y + 12252303 y + 2893401 y
9 8 7 6
- 171532242 y + 6751269 y + 2657205 y - 94517766 y
5 3
- 3720087 y + 26040609 y + 14348907
(%i20) resolvante: cayley$
(%i21) resolvante (x^5 - 4*x^2 + x + 1, x, a, []);
" resolvante de Cayley "
6 5 4 3 2
(%o21) x - 40 x + 4080 x - 92928 x + 3772160 x + 37880832 x
+ 93392896
For the Cayley resolvent, the 2 last arguments are neutral and the input polynomial must necessarily be of degree 5.
See also:
resolvante_bipartite, resolvante_produit_sym,
resolvante_unitaire, resolvante_alternee1, resolvante_klein,
resolvante_klein3, resolvante_vierer, resolvante_diedrale.
Categories: Package sym
calculates the transformation
P(x) of degree n by the function
product(x_i - x_j, 1 <= i < j <= n - 1).
See also:
resolvante_produit_sym, resolvante_unitaire,
resolvante, resolvante_klein, resolvante_klein3,
resolvante_vierer, resolvante_diedrale, resolvante_bipartite.
Categories: Package sym
calculates the transformation of
P(x) of even degree n by the function
x_1 x_2 ... x_[n/2] + x_[n/2 + 1] ... x_n.
(%i1) resolvante_bipartite (x^6 + 108, x);
10 8 6 4
(%o1) y - 972 y + 314928 y - 34012224 y
See also:
resolvante_produit_sym, resolvante_unitaire,
resolvante, resolvante_klein, resolvante_klein3,
resolvante_vierer, resolvante_diedrale, resolvante_alternee1.
Categories: Package sym
calculates the transformation of P(x) by the function
x_1 x_2 + x_3 x_4.
(%i1) resolvante_diedrale (x^5 - 3*x^4 + 1, x);
15 12 11 10 9 8 7
(%o1) x - 21 x - 81 x - 21 x + 207 x + 1134 x + 2331 x
6 5 4 3 2
- 945 x - 4970 x - 18333 x - 29079 x - 20745 x - 25326 x
- 697
See also:
resolvante_produit_sym, resolvante_unitaire,
resolvante_alternee1, resolvante_klein, resolvante_klein3,
resolvante_vierer, resolvante.
Categories: Package sym
calculates the transformation of P(x) by the function
x_1 x_2 x_4 + x_4.
See also:
resolvante_produit_sym, resolvante_unitaire,
resolvante_alternee1, resolvante, resolvante_klein3,
resolvante_vierer, resolvante_diedrale.
Categories: Package sym
calculates the transformation of P(x) by the function
x_1 x_2 x_4 + x_4.
See also:
resolvante_produit_sym, resolvante_unitaire,
resolvante_alternee1, resolvante_klein, resolvante,
resolvante_vierer, resolvante_diedrale.
Categories: Package sym
calculates the list of all product resolvents of the polynomial
P(x).
(%i1) resolvante_produit_sym (x^5 + 3*x^4 + 2*x - 1, x);
5 4 10 8 7 6 5
(%o1) [y + 3 y + 2 y - 1, y - 2 y - 21 y - 31 y - 14 y
4 3 2 10 8 7 6 5 4
- y + 14 y + 3 y + 1, y + 3 y + 14 y - y - 14 y - 31 y
3 2 5 4
- 21 y - 2 y + 1, y - 2 y - 3 y - 1, y - 1]
(%i2) resolvante: produit$
(%i3) resolvante (x^5 + 3*x^4 + 2*x - 1, x, a*b*c, [a, b, c]);
" resolvante produit "
10 8 7 6 5 4 3 2
(%o3) y + 3 y + 14 y - y - 14 y - 31 y - 21 y - 2 y + 1
See also:
resolvante, resolvante_unitaire,
resolvante_alternee1, resolvante_klein,
resolvante_klein3, resolvante_vierer,
resolvante_diedrale.
Categories: Package sym
computes the resolvent of the polynomial P(x) by the
polynomial Q(x).
See also:
resolvante_produit_sym, resolvante,
resolvante_alternee1, resolvante_klein, resolvante_klein3,
resolvante_vierer, resolvante_diedrale.
Categories: Package sym
computes the transformation of
P(x) by the function x_1 x_2 -
x_3 x_4.
See also:
resolvante_produit_sym, resolvante_unitaire,
resolvante_alternee1, resolvante_klein, resolvante_klein3,
resolvante, resolvante_diedrale.
Categories: Package sym
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where r is the weight of the partition part. This function
returns the associate multinomial coefficient: if the parts of
part are i_1, i_2, ..., i_k, the result is
r!/(i_1! i_2! ... i_k!).
Categories: Package sym
returns the list of permutations of the list L.
Categories: Package sym · Lists
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| 31.1 Functions and Variables for Groups |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Find the order of G/H where G is the Free Group modulo relations, and
H is the subgroup of G generated by subgroup. subgroup is an optional
argument, defaulting to []. In doing this it produces a
multiplication table for the right action of G on G/H, where the
cosets are enumerated [H,Hg2,Hg3,...]. This can be seen internally in
the variable todd_coxeter_state.
Example:
(%i1) symet(n):=create_list(
if (j - i) = 1 then (p(i,j))^^3 else
if (not i = j) then (p(i,j))^^2 else
p(i,i) , j, 1, n-1, i, 1, j);
<3>
(%o1) symet(n) := create_list(if j - i = 1 then p(i, j)
<2>
else (if not i = j then p(i, j) else p(i, i)), j, 1, n - 1,
i, 1, j)
(%i2) p(i,j) := concat(x,i).concat(x,j);
(%o2) p(i, j) := concat(x, i) . concat(x, j)
(%i3) symet(5);
<2> <3> <2> <2> <3>
(%o3) [x1 , (x1 . x2) , x2 , (x1 . x3) , (x2 . x3) ,
<2> <2> <2> <3> <2>
x3 , (x1 . x4) , (x2 . x4) , (x3 . x4) , x4 ]
(%i4) todd_coxeter(%o3);
Rows tried 426
(%o4) 120
(%i5) todd_coxeter(%o3,[x1]);
Rows tried 213
(%o5) 60
(%i6) todd_coxeter(%o3,[x1,x2]);
Rows tried 71
(%o6) 20
Categories: Group theory
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| 32.1 Introduction for Runtime Environment | ||
| 32.2 Interrupts | ||
| 32.3 Functions and Variables for Runtime Environment |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
maxima-init.mac is a file which is loaded automatically when Maxima
starts. You can use maxima-init.mac to customize your Maxima
environment. maxima-init.mac, if it exists, is typically placed in the
directory named by maxima_userdir,
although it can be in any directory searched by the function file_search.
Here is an example maxima-init.mac file:
setup_autoload ("specfun.mac", ultraspherical, assoc_legendre_p);
showtime:all;
In this example, setup_autoload tells Maxima to load the
specified file
(specfun.mac) if any of the functions (ultraspherical,
assoc_legendre_p) are called but not yet defined.
Thus you needn't remember to load the file before calling the functions.
The statement showtime: all tells Maxima to set the showtime
variable. The maxima-init.mac file can contain any other assignments or
other Maxima statements.
Categories: Session management
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The user can stop a time-consuming computation with the ^C (control-C) character. The default action is to stop the computation and print another user prompt. In this case, it is not possible to restart a stopped computation.
If the Lisp variable *debugger-hook* is set to nil, by executing
:lisp (setq *debugger-hook* nil)
then upon receiving ^C, Maxima will enter the Lisp debugger,
and the user may use the debugger to inspect the Lisp environment.
The stopped computation can be restarted by entering
continue in the Lisp debugger.
The means of returning to Maxima from the Lisp debugger
(other than running the computation to completion)
is different for each version of Lisp.
On Unix systems, the character ^Z (control-Z) causes Maxima
to stop altogether, and control is returned to the shell prompt.
The fg command causes Maxima
to resume from the point at which it was stopped.
Categories: Console interaction
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maxima_tempdir names the directory in which Maxima creates some temporary
files. In particular, temporary files for plotting are created in
maxima_tempdir.
The initial value of maxima_tempdir is the user's home directory, if
Maxima can locate it; otherwise Maxima makes a guess about a suitable directory.
maxima_tempdir may be assigned a string which names a directory.
Categories: Global variables
maxima_userdir names a directory which Maxima searches to find Maxima and
Lisp files. (Maxima searches some other directories as well;
file_search_maxima and file_search_lisp are the complete lists.)
The initial value of maxima_userdir is a subdirectory of the user's home
directory, if Maxima can locate it; otherwise Maxima makes a guess about a
suitable directory.
maxima_userdir may be assigned a string which names a directory.
However, assigning to maxima_userdir does not automatically change
file_search_maxima and file_search_lisp;
those variables must be changed separately.
Categories: Global variables
Prints out a description of the state of storage and
stack management in Maxima. room calls the Lisp function of
the same name.
room () prints out a moderate description.
room (true) prints out a verbose description.
room (false) prints out a terse description.
Categories: Debugging
When keyword is the symbol feature, item is put on the list
of system features. After sstatus (keyword, item) is executed,
status (feature, item) returns true. If keyword is the
symbol nofeature, item is deleted from the list of system features.
This can be useful for package writers, to keep track of what features they have
loaded in.
See also status.
Categories: Programming
feature) feature, item)
Returns information about the presence or absence of certain system-dependent features.
status (feature) returns a list of system features. These include Lisp
version, operating system type, etc. The list may vary from one Lisp type to
another.
status (feature, item) returns true if item is on the
list of items returned by status (feature) and false otherwise.
status quotes the argument item. The quote-quote operator
'' defeats quotation. A feature whose name contains a special
character, such as a hyphen, must be given as a string argument. For example,
status (feature, "ansi-cl").
See also sstatus.
The variable features contains a list of features which apply to
mathematical expressions. See features and featurep for more
information.
Categories: Programming
Executes command as a separate process. The command is passed to the
default shell for execution. system is not supported by all operating
systems, but generally exists in Unix and Unix-like environments.
Supposing _hist.out is a list of frequencies which you wish to plot as a
bar graph using xgraph.
(%i1) (with_stdout("_hist.out",
for i:1 thru length(hist) do (
print(i,hist[i]))),
system("xgraph -bar -brw .7 -nl < _hist.out"));
In order to make the plot be done in the background (returning control to Maxima) and remove the temporary file after it is done do:
system("(xgraph -bar -brw .7 -nl < _hist.out; rm -f _hist.out)&")
Returns a list of the times, in seconds, taken to compute the output lines
%o1, %o2, %o3, … The time returned is Maxima's
estimate of the internal computation time, not the elapsed time. time
can only be applied to output line variables; for any other variables,
time returns unknown.
Set showtime: true to make Maxima print out the computation time
and elapsed time with each output line.
Categories: Debugging
timedate() with no argument
returns a string representing the current time and date.
The string has the format YYYY-MM-DD HH:MM:SS[+|-]ZZ:ZZ,
where the fields are year, month, day, hours, minutes, seconds, and time zone
offset in hours and minutes.
timedate(T) returns the time T
as a string with the format YYYY-MM-DD HH:MM:SS[+|-]ZZ:ZZ.
T is interpreted as the number of seconds since midnight, January 1, 1900,
as returned by absolute_real_time.
Example:
timedate with no argument returns a string representing the current time and date.
(%i1) d : timedate ();
(%o1) 2010-06-08 04:08:09+01:00
(%i2) print ("timedate reports current time", d) $
timedate reports current time 2010-06-08 04:08:09+01:00
timedate with an argument returns a string representing the argument.
(%i1) timedate (0); (%o1) 1900-01-01 01:00:00+01:00 (%i2) timedate (absolute_real_time () - 7*24*3600); (%o2) 2010-06-01 04:19:51+01:00
Categories: Time and date functions
Parses a string S representing a date or date and time of day and returns the number of seconds since midnight, January 1, 1900 GMT. If there is a nonzero milliseconds part, the value returned is a rational number, otherwise, it is an integer.
The string S must have one of the following formats:
YYYY-MM-DD[ T]hh:mm:ss[,.]nnn
YYYY-MM-DD[ T]hh:mm:ss
YYYY-MM-DD
where the fields are year, month, day, hours, minutes, seconds, and milliseconds, and square brackets indicate acceptable alternatives.
See also timedate and absolute_real_time.
Examples:
Midnight, January 1, 1900, in the local time zone, in each acceptable format.
(%i1) parse_timedate ("1900-01-01 00:00:00,000");
(%o1) 28800
(%i2) parse_timedate ("1900-01-01 00:00:00.000");
(%o2) 28800
(%i3) parse_timedate ("1900-01-01T00:00:00,000");
(%o3) 28800
(%i4) parse_timedate ("1900-01-01T00:00:00.000");
(%o4) 28800
(%i5) parse_timedate ("1900-01-01 00:00:00");
(%o5) 28800
(%i6) parse_timedate ("1900-01-01T00:00:00");
(%o6) 28800
(%i7) parse_timedate ("1900-01-01");
(%o7) 28800
Categories: Time and date functions
Returns the number of seconds since midnight, January 1, 1900 UTC. The return value is an integer.
See also elapsed_real_time and elapsed_run_time.
Example:
(%i1) absolute_real_time (); (%o1) 3385045277 (%i2) 1900 + absolute_real_time () / (365.25 * 24 * 3600); (%o2) 2007.265612087104
Categories: Time and date functions
Returns the number of seconds (including fractions of a second) since Maxima was most recently started or restarted. The return value is a floating-point number.
See also absolute_real_time and elapsed_run_time.
Example:
(%i1) elapsed_real_time (); (%o1) 2.559324 (%i2) expand ((a + b)^500)$ (%i3) elapsed_real_time (); (%o3) 7.552087
Categories: Time and date functions
Returns an estimate of the number of seconds (including fractions of a second) which Maxima has spent in computations since Maxima was most recently started or restarted. The return value is a floating-point number.
See also absolute_real_time and elapsed_real_time.
Example:
(%i1) elapsed_run_time (); (%o1) 0.04 (%i2) expand ((a + b)^500)$ (%i3) elapsed_run_time (); (%o3) 1.26
Categories: Time and date functions
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| 33.1 Introduction to Miscellaneous Options | ||
| 33.2 Share | ||
| 33.3 Functions and Variables for Miscellaneous Options |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
In this section various options are discussed which have a global effect on the operation of Maxima. Also various lists such as the list of all user defined functions, are discussed.
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
The Maxima "share" directory contains programs and other files
of interest to Maxima users, but not part of the core implementation of Maxima.
These programs are typically loaded via load or setup_autoload.
:lisp *maxima-sharedir* displays the location of the share directory
within the user's file system.
printfile ("share.usg") prints an out-of-date list of share packages.
Users may find it more informative to browse the share directory using a file
system browser.
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When asksign is called,
askexp is the expression asksign is testing.
At one time, it was possible for a user to inspect askexp
by entering a Maxima break with control-A.
Categories: Declarations and inferences
Default value: i
genindex is the alphabetic prefix used to generate the
next variable of summation when necessary.
Categories: Sums and products
Default value: 0
gensumnum is the numeric suffix used to generate the next variable
of summation. If it is set to false then the index will consist only
of genindex with no numeric suffix.
Categories: Sums and products
gensym() creates and returns a fresh symbol.
The name of the new-symbol is the concatenation of a prefix, which defaults to "g", and a suffix, which is the decimal representation of a number that defaults to the value of a Lisp internal counter.
If x is supplied, and is a string, then that string is used as a prefix instead of "g" for this call to gensym only.
If x is supplied, and is a nonnegative integer, then that integer, instead of the value of the internal Lisp integer, is used as the suffix for this call to gensym only.
If and only if no explicit suffix is supplied, the Lisp internal integer is incremented after it is used.
Examples:
(%i1) gensym();
(%o1) g887
(%i2) gensym("new");
(%o2) new888
(%i3) gensym(123);
(%o3) g123
Default value: false
Package designers who use save or translate to create packages
(files) for others to use may want to set packagefile: true to prevent
information from being added to Maxima's information-lists (e.g.
values, functions) except where necessary when the file is
loaded in. In this way, the contents of the package will not get in the user's
way when he adds his own data. Note that this will not solve the problem of
possible name conflicts. Also note that the flag simply affects what is output
to the package file. Setting the flag to true is also useful for
creating Maxima init files.
Categories: Translation flags and variables
Removes the values of user variables name_1, …, name_n (which can be subscripted) from the system.
remvalue (all) removes the values of all variables in values,
the list of all variables given names by the user
(as opposed to those which are automatically assigned by Maxima).
See also values.
Categories: Evaluation
Transforms expr by combining all terms of expr that have
identical denominators or denominators that differ from each other by
numerical factors only. This is slightly different from the behavior
of combine, which collects terms that have identical denominators.
Setting pfeformat: true and using combine yields results similar
to those that can be obtained with rncombine, but rncombine takes
the additional step of cross-multiplying numerical denominator factors.
This results in neater forms, and the possibility of recognizing some
cancellations.
load(rncomb) loads this function.
Categories: Expressions
Specifies that if any of function_1, …, function_n are
referenced and not yet defined, filename is loaded via load.
filename usually contains definitions for the functions specified,
although that is not enforced.
setup_autoload does not work for array functions.
setup_autoload quotes its arguments.
Example:
(%i1) legendre_p (1, %pi);
(%o1) legendre_p(1, %pi)
(%i2) setup_autoload ("specfun.mac", legendre_p, ultraspherical);
(%o2) done
(%i3) ultraspherical (2, 1/2, %pi);
Warning - you are redefining the Macsyma function ultraspherical
Warning - you are redefining the Macsyma function legendre_p
2
3 (%pi - 1)
(%o3) ------------ + 3 (%pi - 1) + 1
2
(%i4) legendre_p (1, %pi);
(%o4) %pi
(%i5) legendre_q (1, %pi);
%pi + 1
%pi log(-------)
1 - %pi
(%o5) ---------------- - 1
2
Categories: Function definition · File input
Prints elements of a list enclosed by curly braces { },
suitable as part of a program in the Tcl/Tk language.
tcl_output (list, i0, skip)
prints list, beginning with element i0 and printing elements
i0 + skip, i0 + 2 skip, etc.
tcl_output (list, i0)
is equivalent to tcl_output (list, i0, 2).
tcl_output ([list_1, ..., list_n], i)
prints the i'th elements of list_1, …, list_n.
Examples:
(%i1) tcl_output ([1, 2, 3, 4, 5, 6], 1, 3)$
{1.000000000 4.000000000
}
(%i2) tcl_output ([1, 2, 3, 4, 5, 6], 2, 3)$
{2.000000000 5.000000000
}
(%i3) tcl_output ([3/7, 5/9, 11/13, 13/17], 1)$
{((RAT SIMP) 3 7) ((RAT SIMP) 11 13)
}
(%i4) tcl_output ([x1, y1, x2, y2, x3, y3], 2)$
{$Y1 $Y2 $Y3
}
(%i5) tcl_output ([[1, 2, 3], [11, 22, 33]], 1)$
{SIMP 1.000000000 11.00000000
}
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| 34.1 Introduction to Rules and Patterns | ||
| 34.2 Functions and Variables for Rules and Patterns |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
This section describes user-defined pattern matching and simplification rules.
There are two groups of functions which implement somewhat different pattern
matching schemes. In one group are tellsimp, tellsimpafter,
defmatch, defrule, apply1, applyb1, and
apply2. In the other group are let and letsimp.
Both schemes define patterns in terms of pattern variables declared by
matchdeclare.
Pattern-matching rules defined by tellsimp and tellsimpafter are
applied automatically by the Maxima simplifier. Rules defined by
defmatch, defrule, and let are applied by an explicit
function call.
There are additional mechanisms for rules applied to polynomials by
tellrat, and for commutative and noncommutative algebra in affine
package.
Categories: Simplification · Rules and patterns
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Repeatedly applies rule_1 to expr until it fails, then repeatedly applies the same rule to all subexpressions of expr, left to right, until rule_1 has failed on all subexpressions. Call the result of transforming expr in this manner expr_2. Then rule_2 is applied in the same fashion starting at the top of expr_2. When rule_n fails on the final subexpression, the result is returned.
maxapplydepth is the depth of the deepest subexpressions processed by
apply1 and apply2.
See also applyb1, apply2 and let.
Categories: Rules and patterns
If rule_1 fails on a given subexpression, then rule_2 is repeatedly applied, etc. Only if all rules fail on a given subexpression is the whole set of rules repeatedly applied to the next subexpression. If one of the rules succeeds, then the same subexpression is reprocessed, starting with the first rule.
maxapplydepth is the depth of the deepest subexpressions processed by
apply1 and apply2.
Categories: Rules and patterns
Repeatedly applies rule_1 to the deepest subexpression of expr until it fails, then repeatedly applies the same rule one level higher (i.e., larger subexpressions), until rule_1 has failed on the top-level expression. Then rule_2 is applied in the same fashion to the result of rule_1. After rule_n has been applied to the top-level expression, the result is returned.
applyb1 is similar to apply1 but works from
the bottom up instead of from the top down.
maxapplyheight is the maximum height which applyb1 reaches
before giving up.
See also apply1, apply2 and let.
Categories: Rules and patterns
Default value: default_let_rule_package
current_let_rule_package is the name of the rule package that is used by
functions in the let package (letsimp, etc.) if no other rule package is specified.
This variable may be assigned the name of any rule package defined
via the let command.
If a call such as letsimp (expr, rule_pkg_name) is made,
the rule package rule_pkg_name is used for that function call only,
and the value of current_let_rule_package is not changed.
Categories: Rules and patterns
Default value: default_let_rule_package
default_let_rule_package is the name of the rule package used when one
is not explicitly set by the user with let or by changing the value of
current_let_rule_package.
Categories: Rules and patterns
Defines a function progname(expr, x_1, ..., x_n)
which tests expr to see if it matches pattern.
pattern is an expression containing the pattern arguments x_1,
…, x_n (if any) and some pattern variables (if any). The pattern
arguments are given explicitly as arguments to defmatch while the pattern
variables are declared by the matchdeclare function. Any variable not
declared as a pattern variable in matchdeclare or as a pattern argument
in defmatch matches only itself.
The first argument to the created function progname is an expression to be matched against the pattern and the other arguments are the actual arguments which correspond to the dummy variables x_1, …, x_n in the pattern.
If the match is successful, progname returns a list of equations whose
left sides are the pattern arguments and pattern variables, and whose right
sides are the subexpressions which the pattern arguments and variables matched.
The pattern variables, but not the pattern arguments, are assigned the
subexpressions they match. If the match fails, progname returns
false.
A literal pattern (that is, a pattern which contains neither pattern arguments
nor pattern variables) returns true if the match succeeds.
See also matchdeclare, defrule, tellsimp and
tellsimpafter.
Examples:
Define a function linearp(expr, x) which
tests expr to see if it is of the form a*x + b
such that a and b do not contain x and a is nonzero.
This match function matches expressions which are linear in any variable,
because the pattern argument x is given to defmatch.
(%i1) matchdeclare (a, lambda ([e], e#0 and freeof(x, e)), b,
freeof(x));
(%o1) done
(%i2) defmatch (linearp, a*x + b, x);
(%o2) linearp
(%i3) linearp (3*z + (y + 1)*z + y^2, z);
2
(%o3) [b = y , a = y + 4, x = z]
(%i4) a;
(%o4) y + 4
(%i5) b;
2
(%o5) y
(%i6) x;
(%o6) x
Define a function linearp(expr) which tests expr
to see if it is of the form a*x + b
such that a and b do not contain x and a is nonzero.
This match function only matches expressions linear in x,
not any other variable, because no pattern argument is given to defmatch.
(%i1) matchdeclare (a, lambda ([e], e#0 and freeof(x, e)), b,
freeof(x));
(%o1) done
(%i2) defmatch (linearp, a*x + b);
(%o2) linearp
(%i3) linearp (3*z + (y + 1)*z + y^2);
(%o3) false
(%i4) linearp (3*x + (y + 1)*x + y^2);
2
(%o4) [b = y , a = y + 4]
Define a function checklimits(expr) which tests expr
to see if it is a definite integral.
(%i1) matchdeclare ([a, f], true);
(%o1) done
(%i2) constinterval (l, h) := constantp (h - l);
(%o2) constinterval(l, h) := constantp(h - l)
(%i3) matchdeclare (b, constinterval (a));
(%o3) done
(%i4) matchdeclare (x, atom);
(%o4) done
(%i5) simp : false;
(%o5) false
(%i6) defmatch (checklimits, 'integrate (f, x, a, b));
(%o6) checklimits
(%i7) simp : true;
(%o7) true
(%i8) 'integrate (sin(t), t, %pi + x, 2*%pi + x);
x + 2 %pi
/
[
(%o8) I sin(t) dt
]
/
x + %pi
(%i9) checklimits (%);
(%o9) [b = x + 2 %pi, a = x + %pi, x = t, f = sin(t)]
Categories: Rules and patterns
Defines and names a replacement rule for the given pattern. If the rule named
rulename is applied to an expression (by apply1, applyb1, or
apply2), every subexpression matching the pattern will be replaced by the
replacement. All variables in the replacement which have been
assigned values by the pattern match are assigned those values in the
replacement which is then simplified.
The rules themselves can be
treated as functions which transform an expression by one
operation of the pattern match and replacement.
If the match fails, the rule function returns false.
Categories: Rules and patterns
Display rules with the names rulename_1, …, rulename_n,
as returned by defrule, tellsimp, or tellsimpafter,
or a pattern defined by defmatch.
Each rule is displayed with an intermediate expression label (%t).
disprule (all) displays all rules.
disprule quotes its arguments.
disprule returns the list of intermediate expression labels corresponding
to the displayed rules.
See also letrules, which displays rules defined by let.
Examples:
(%i1) tellsimpafter (foo (x, y), bar (x) + baz (y));
(%o1) [foorule1, false]
(%i2) tellsimpafter (x + y, special_add (x, y));
(%o2) [+rule1, simplus]
(%i3) defmatch (quux, mumble (x));
(%o3) quux
(%i4) disprule (foorule1, "+rule1", quux);
(%t4) foorule1 : foo(x, y) -> baz(y) + bar(x)
(%t5) +rule1 : y + x -> special_add(x, y)
(%t6) quux : mumble(x) -> []
(%o6) [%t4, %t5, %t6]
(%i6) ''%;
(%o6) [foorule1 : foo(x, y) -> baz(y) + bar(x),
+rule1 : y + x -> special_add(x, y), quux : mumble(x) -> []]
Categories: Rules and patterns · Display functions
Defines a substitution rule for letsimp such that prod is replaced
by repl. prod is a product of positive or negative powers of the
following terms:
letsimp will search for literally unless previous to calling
letsimp the matchdeclare function is used to associate a
predicate with the atom. In this case letsimp will match the atom to
any term of a product satisfying the predicate.
sin(x), n!, f(x,y), etc. As with atoms
above letsimp will look for a literal match unless matchdeclare
is used to associate a predicate with the argument of the kernel.
A term to a positive power will only match a term having at least that
power. A term to a negative power
on the other hand will only match a term with a power at least as
negative. In the case of negative powers in prod the switch
letrat must be set to true.
See also letrat.
If a predicate is included in the let function followed by a list of
arguments, a tentative match (i.e. one that would be accepted if the predicate
were omitted) is accepted only if predname (arg_1', ..., arg_n')
evaluates to true where arg_i' is the value matched to arg_i.
The arg_i may be the name of any atom or the argument of any kernel
appearing in prod.
repl may be any rational expression. If any of the atoms or arguments from prod appear in repl the
appropriate substitutions are made.
The global flag letrat controls the simplification of quotients by
letsimp. When letrat is false, letsimp simplifies
the numerator and denominator of expr separately, and does not simplify
the quotient. Substitutions such as n!/n goes to (n-1)! then
fail. When letrat is true, then the numerator, denominator, and
the quotient are simplified in that order.
These substitution functions allow you to work with several rule packages at
once. Each rule package can contain any number of let rules and is
referenced by a user-defined name. The command let ([prod,
repl, predname, arg_1, ..., arg_n], package_name)
adds the rule predname to the rule package package_name. The
command letsimp (expr, package_name) applies the rules in
package_name. letsimp (expr, package_name1,
package_name2, ...) is equivalent to letsimp (expr,
package_name1) followed by letsimp (%, package_name2),
…
current_let_rule_package is the name of the rule package that is
presently being used. This variable may be assigned the name of any rule
package defined via the let command. Whenever any of the functions
comprising the let package are called with no package name, the package
named by current_let_rule_package is used. If a call such as
letsimp (expr, rule_pkg_name) is made, the rule package
rule_pkg_name is used for that letsimp command only, and
current_let_rule_package is not changed. If not otherwise specified,
current_let_rule_package defaults to default_let_rule_package.
(%i1) matchdeclare ([a, a1, a2], true)$
(%i2) oneless (x, y) := is (x = y-1)$
(%i3) let (a1*a2!, a1!, oneless, a2, a1);
(%o3) a1 a2! --> a1! where oneless(a2, a1)
(%i4) letrat: true$
(%i5) let (a1!/a1, (a1-1)!);
a1!
(%o5) --- --> (a1 - 1)!
a1
(%i6) letsimp (n*m!*(n-1)!/m);
(%o6) (m - 1)! n!
(%i7) let (sin(a)^2, 1 - cos(a)^2);
2 2
(%o7) sin (a) --> 1 - cos (a)
(%i8) letsimp (sin(x)^4);
4 2
(%o8) cos (x) - 2 cos (x) + 1
Categories: Rules and patterns
Default value: false
When letrat is false, letsimp simplifies the
numerator and denominator of a ratio separately,
and does not simplify the quotient.
When letrat is true,
the numerator, denominator, and their quotient are simplified in that order.
(%i1) matchdeclare (n, true)$
(%i2) let (n!/n, (n-1)!);
n!
(%o2) -- --> (n - 1)!
n
(%i3) letrat: false$
(%i4) letsimp (a!/a);
a!
(%o4) --
a
(%i5) letrat: true$
(%i6) letsimp (a!/a);
(%o6) (a - 1)!
Categories: Rules and patterns
Displays the rules in a rule package.
letrules () displays the rules in the current rule package.
letrules (package_name) displays the rules in package_name.
The current rule package is named by current_let_rule_package.
If not otherwise specified, current_let_rule_package
defaults to default_let_rule_package.
See also disprule, which displays rules defined by tellsimp and
tellsimpafter.
Categories: Rules and patterns
Repeatedly applies the substitution rules defined by let
until no further change is made to expr.
letsimp (expr) uses the rules from current_let_rule_package.
letsimp (expr, package_name) uses the rules from
package_name without changing current_let_rule_package.
letsimp (expr, package_name_1, ..., package_name_n)
is equivalent to letsimp (expr, package_name_1),
followed by letsimp (%, package_name_2), and so on.
Categories: Rules and patterns
Default value: [default_let_rule_package]
let_rule_packages is a list of all user-defined let rule packages
plus the default package default_let_rule_package.
Categories: Rules and patterns
Associates a predicate pred_k
with a variable or list of variables a_k
so that a_k matches expressions
for which the predicate returns anything other than false.
A predicate is the name of a function,
or a lambda expression,
or a function call or lambda call missing the last argument,
or true or all.
Any expression matches true or all.
If the predicate is specified as a function call or lambda call,
the expression to be tested is appended to the list of arguments;
the arguments are evaluated at the time the match is evaluated.
Otherwise, the predicate is specified as a function name or lambda expression,
and the expression to be tested is the sole argument.
A predicate function need not be defined when matchdeclare is called;
the predicate is not evaluated until a match is attempted.
A predicate may return a Boolean expression as well as true or
false. Boolean expressions are evaluated by is within the
constructed rule function, so it is not necessary to call is within the
predicate.
If an expression satisfies a match predicate, the match variable is assigned the
expression, except for match variables which are operands of addition +
or multiplication *. Only addition and multiplication are handled
specially; other n-ary operators (both built-in and user-defined) are treated
like ordinary functions.
In the case of addition and multiplication, the match variable may be assigned a single expression which satisfies the match predicate, or a sum or product (respectively) of such expressions. Such multiple-term matching is greedy: predicates are evaluated in the order in which their associated variables appear in the match pattern, and a term which satisfies more than one predicate is taken by the first predicate which it satisfies. Each predicate is tested against all operands of the sum or product before the next predicate is evaluated. In addition, if 0 or 1 (respectively) satisfies a match predicate, and there are no other terms which satisfy the predicate, 0 or 1 is assigned to the match variable associated with the predicate.
The algorithm for processing addition and multiplication patterns makes some match results (for example, a pattern in which a "match anything" variable appears) dependent on the ordering of terms in the match pattern and in the expression to be matched. However, if all match predicates are mutually exclusive, the match result is insensitive to ordering, as one match predicate cannot accept terms matched by another.
Calling matchdeclare with a variable a as an argument changes the
matchdeclare property for a, if one was already declared; only the
most recent matchdeclare is in effect when a rule is defined. Later
changes to the matchdeclare property (via matchdeclare or
remove) do not affect existing rules.
propvars (matchdeclare) returns the list of all variables for which there
is a matchdeclare property. printprops (a, matchdeclare)
returns the predicate for variable a.
printprops (all, matchdeclare) returns the list of predicates for all
matchdeclare variables. remove (a, matchdeclare) removes
the matchdeclare property from a.
The functions defmatch, defrule, tellsimp,
tellsimpafter, and let construct rules which test expressions
against patterns.
matchdeclare quotes its arguments.
matchdeclare always returns done.
Examples:
A predicate is the name of a function,
or a lambda expression,
or a function call or lambda call missing the last argument,
or true or all.
(%i1) matchdeclare (aa, integerp); (%o1) done (%i2) matchdeclare (bb, lambda ([x], x > 0)); (%o2) done (%i3) matchdeclare (cc, freeof (%e, %pi, %i)); (%o3) done (%i4) matchdeclare (dd, lambda ([x, y], gcd (x, y) = 1) (1728)); (%o4) done (%i5) matchdeclare (ee, true); (%o5) done (%i6) matchdeclare (ff, all); (%o6) done
If an expression satisfies a match predicate, the match variable is assigned the expression.
(%i1) matchdeclare (aa, integerp, bb, atom);
(%o1) done
(%i2) defrule (r1, bb^aa, ["integer" = aa, "atom" = bb]);
aa
(%o2) r1 : bb -> [integer = aa, atom = bb]
(%i3) r1 (%pi^8);
(%o3) [integer = 8, atom = %pi]
In the case of addition and multiplication, the match variable may be assigned a single expression which satisfies the match predicate, or a sum or product (respectively) of such expressions.
(%i1) matchdeclare (aa, atom, bb, lambda ([x], not atom(x)));
(%o1) done
(%i2) defrule (r1, aa + bb, ["all atoms" = aa, "all nonatoms" =
bb]);
bb + aa partitions `sum'
(%o2) r1 : bb + aa -> [all atoms = aa, all nonatoms = bb]
(%i3) r1 (8 + a*b + sin(x));
(%o3) [all atoms = 8, all nonatoms = sin(x) + a b]
(%i4) defrule (r2, aa * bb, ["all atoms" = aa, "all nonatoms" =
bb]);
bb aa partitions `product'
(%o4) r2 : aa bb -> [all atoms = aa, all nonatoms = bb]
(%i5) r2 (8 * (a + b) * sin(x));
(%o5) [all atoms = 8, all nonatoms = (b + a) sin(x)]
When matching arguments of + and *,
if all match predicates are mutually exclusive,
the match result is insensitive to ordering,
as one match predicate cannot accept terms matched by another.
(%i1) matchdeclare (aa, atom, bb, lambda ([x], not atom(x)));
(%o1) done
(%i2) defrule (r1, aa + bb, ["all atoms" = aa, "all nonatoms" =
bb]);
bb + aa partitions `sum'
(%o2) r1 : bb + aa -> [all atoms = aa, all nonatoms = bb]
(%i3) r1 (8 + a*b + %pi + sin(x) - c + 2^n);
n
(%o3) [all atoms = %pi + 8, all nonatoms = sin(x) + 2 - c + a b]
(%i4) defrule (r2, aa * bb, ["all atoms" = aa, "all nonatoms" =
bb]);
bb aa partitions `product'
(%o4) r2 : aa bb -> [all atoms = aa, all nonatoms = bb]
(%i5) r2 (8 * (a + b) * %pi * sin(x) / c * 2^n);
n
(b + a) 2 sin(x)
(%o5) [all atoms = 8 %pi, all nonatoms = -----------------]
c
The functions propvars and printprops return information about
match variables.
(%i1) matchdeclare ([aa, bb, cc], atom, [dd, ee], integerp);
(%o1) done
(%i2) matchdeclare (ff, floatnump, gg, lambda ([x], x > 100));
(%o2) done
(%i3) propvars (matchdeclare);
(%o3) [aa, bb, cc, dd, ee, ff, gg]
(%i4) printprops (ee, matchdeclare);
(%o4) [integerp(ee)]
(%i5) printprops (gg, matchdeclare);
(%o5) [lambda([x], x > 100, gg)]
(%i6) printprops (all, matchdeclare);
(%o6) [lambda([x], x > 100, gg), floatnump(ff), integerp(ee),
integerp(dd), atom(cc), atom(bb), atom(aa)]
Categories: Rules and patterns · Declarations and inferences
Default value: 10000
maxapplydepth is the maximum depth to which apply1
and apply2 will delve.
Categories: Function application
Default value: 10000
maxapplyheight is the maximum height to which applyb1
will reach before giving up.
Categories: Function application
Deletes the substitution rule, prod -> repl, most
recently defined by the let function. If name is supplied the rule is
deleted from the rule package name.
remlet() and remlet(all) delete all substitution rules from the
current rule package. If the name of a rule package is supplied, e.g.
remlet (all, name), the rule package name is also deleted.
If a substitution is to be changed using the same
product, remlet need not be called, just redefine the substitution
using the same product (literally) with the let function and the new
replacement and/or predicate name. Should remlet (prod) now be
called the original substitution rule is revived.
See also remrule, which removes a rule defined by tellsimp or
tellsimpafter.
Categories: Rules and patterns
Removes rules defined by tellsimp or tellsimpafter.
remrule (op, rulename)
removes the rule with the name rulename from the operator op.
When op is a built-in or user-defined operator
(as defined by infix, prefix, etc.),
op and rulename must be enclosed in double quote marks.
remrule (op, all) removes all rules for the operator op.
See also remlet, which removes a rule defined by let.
Examples:
(%i1) tellsimp (foo (aa, bb), bb - aa);
(%o1) [foorule1, false]
(%i2) tellsimpafter (aa + bb, special_add (aa, bb));
(%o2) [+rule1, simplus]
(%i3) infix ("@@");
(%o3) @@
(%i4) tellsimp (aa @@ bb, bb/aa);
(%o4) [@@rule1, false]
(%i5) tellsimpafter (quux (%pi, %e), %pi - %e);
(%o5) [quuxrule1, false]
(%i6) tellsimpafter (quux (%e, %pi), %pi + %e);
(%o6) [quuxrule2, quuxrule1, false]
(%i7) [foo (aa, bb), aa + bb, aa @@ bb, quux (%pi, %e),
quux (%e, %pi)];
bb
(%o7) [bb - aa, special_add(aa, bb), --, %pi - %e, %pi + %e]
aa
(%i8) remrule (foo, foorule1);
(%o8) foo
(%i9) remrule ("+", ?\+rule1);
(%o9) +
(%i10) remrule ("@@", ?\@\@rule1);
(%o10) @@
(%i11) remrule (quux, all);
(%o11) quux
(%i12) [foo (aa, bb), aa + bb, aa @@ bb, quux (%pi, %e),
quux (%e, %pi)];
(%o12) [foo(aa, bb), bb + aa, aa @@ bb, quux(%pi, %e),
quux(%e, %pi)]
Categories: Rules and patterns
is similar to tellsimpafter but places
new information before old so that it is applied before the built-in
simplification rules.
tellsimp is used when it is important to modify
the expression before the simplifier works on it, for instance if the
simplifier "knows" something about the expression, but what it returns
is not to your liking.
If the simplifier "knows" something about the
main operator of the expression, but is simply not doing enough for
you, you probably want to use tellsimpafter.
The pattern may not be a sum, product, single variable, or number.
The system variable rules is the list of rules defined by
defrule, defmatch, tellsimp, and tellsimpafter.
Examples:
(%i1) matchdeclare (x, freeof (%i));
(%o1) done
(%i2) %iargs: false$
(%i3) tellsimp (sin(%i*x), %i*sinh(x));
(%o3) [sinrule1, simp-%sin]
(%i4) trigexpand (sin (%i*y + x));
(%o4) sin(x) cos(%i y) + %i cos(x) sinh(y)
(%i5) %iargs:true$
(%i6) errcatch(0^0);
0
0 has been generated
(%o6) []
(%i7) ev (tellsimp (0^0, 1), simp: false);
(%o7) [^rule1, simpexpt]
(%i8) 0^0;
(%o8) 1
(%i9) remrule ("^", %th(2)[1]);
(%o9) ^
(%i10) tellsimp (sin(x)^2, 1 - cos(x)^2);
(%o10) [^rule2, simpexpt]
(%i11) (1 + sin(x))^2;
2
(%o11) (sin(x) + 1)
(%i12) expand (%);
2
(%o12) 2 sin(x) - cos (x) + 2
(%i13) sin(x)^2;
2
(%o13) 1 - cos (x)
(%i14) kill (rules);
(%o14) done
(%i15) matchdeclare (a, true);
(%o15) done
(%i16) tellsimp (sin(a)^2, 1 - cos(a)^2);
(%o16) [^rule3, simpexpt]
(%i17) sin(y)^2;
2
(%o17) 1 - cos (y)
Categories: Rules and patterns
Defines a simplification rule which the Maxima simplifier applies after built-in
simplification rules. pattern is an expression, comprising pattern
variables (declared by matchdeclare) and other atoms and operators,
considered literals for the purpose of pattern matching. replacement is
substituted for an actual expression which matches pattern; pattern
variables in replacement are assigned the values matched in the actual
expression.
pattern may be any nonatomic expression in which the main operator is not
a pattern variable; the simplification rule is associated with the main
operator. The names of functions (with one exception, described below), lists,
and arrays may appear in pattern as the main operator only as literals
(not pattern variables); this rules out expressions such as aa(x) and
bb[y] as patterns, if aa and bb are pattern variables.
Names of functions, lists, and arrays which are pattern variables may appear as
operators other than the main operator in pattern.
There is one exception to the above rule concerning names of functions.
The name of a subscripted function in an expression such as aa[x](y)
may be a pattern variable, because the main operator is not aa but rather
the Lisp atom mqapply. This is a consequence of the representation of
expressions involving subscripted functions.
Simplification rules are applied after evaluation
(if not suppressed through quotation or the flag noeval).
Rules established by tellsimpafter are applied in the order they were
defined, and after any built-in rules.
Rules are applied bottom-up, that is,
applied first to subexpressions before application to the whole expression.
It may be necessary to repeatedly simplify a result (for example, via the
quote-quote operator '' or the flag infeval)
to ensure that all rules are applied.
Pattern variables are treated as local variables in simplification rules.
Once a rule is defined, the value of a pattern variable
does not affect the rule, and is not affected by the rule.
An assignment to a pattern variable which results from a successful rule match
does not affect the current assignment (or lack of it) of the pattern variable.
However, as with all atoms in Maxima, the properties of pattern variables (as
declared by put and related functions) are global.
The rule constructed by tellsimpafter is named after the main operator of
pattern. Rules for built-in operators, and user-defined operators defined
by infix, prefix, postfix, matchfix, and
nofix, have names which are Lisp identifiers.
Rules for other functions have names which are Maxima identifiers.
The treatment of noun and verb forms is slightly confused. If a rule is defined for a noun (or verb) form and a rule for the corresponding verb (or noun) form already exists, the newly-defined rule applies to both forms (noun and verb). If a rule for the corresponding verb (or noun) form does not exist, the newly-defined rule applies only to the noun (or verb) form.
The rule constructed by tellsimpafter is an ordinary Lisp function.
If the name of the rule is $foorule1,
the construct :lisp (trace $foorule1) traces the function,
and :lisp (symbol-function '$foorule1) displays its definition.
tellsimpafter quotes its arguments.
tellsimpafter returns the list of rules for the main operator of
pattern, including the newly established rule.
See also matchdeclare, defmatch, defrule, tellsimp,
let, kill, remrule and clear_rules.
Examples:
pattern may be any nonatomic expression in which the main operator is not a pattern variable.
(%i1) matchdeclare (aa, atom, [ll, mm], listp, xx, true)$
(%i2) tellsimpafter (sin (ll), map (sin, ll));
(%o2) [sinrule1, simp-%sin]
(%i3) sin ([1/6, 1/4, 1/3, 1/2, 1]*%pi);
1 sqrt(2) sqrt(3)
(%o3) [-, -------, -------, 1, 0]
2 2 2
(%i4) tellsimpafter (ll^mm, map ("^", ll, mm));
(%o4) [^rule1, simpexpt]
(%i5) [a, b, c]^[1, 2, 3];
2 3
(%o5) [a, b , c ]
(%i6) tellsimpafter (foo (aa (xx)), aa (foo (xx)));
(%o6) [foorule1, false]
(%i7) foo (bar (u - v));
(%o7) bar(foo(u - v))
Rules are applied in the order they were defined. If two rules can match an expression, the rule which was defined first is applied.
(%i1) matchdeclare (aa, integerp); (%o1) done (%i2) tellsimpafter (foo (aa), bar_1 (aa)); (%o2) [foorule1, false] (%i3) tellsimpafter (foo (aa), bar_2 (aa)); (%o3) [foorule2, foorule1, false] (%i4) foo (42); (%o4) bar_1(42)
Pattern variables are treated as local variables in simplification rules.
(Compare to defmatch, which treats pattern variables as global
variables.)
(%i1) matchdeclare (aa, integerp, bb, atom);
(%o1) done
(%i2) tellsimpafter (foo(aa, bb), bar('aa=aa, 'bb=bb));
(%o2) [foorule1, false]
(%i3) bb: 12345;
(%o3) 12345
(%i4) foo (42, %e);
(%o4) bar(aa = 42, bb = %e)
(%i5) bb;
(%o5) 12345
As with all atoms, properties of pattern variables are global even though values
are local. In this example, an assignment property is declared via
define_variable. This is a property of the atom bb throughout
Maxima.
(%i1) matchdeclare (aa, integerp, bb, atom);
(%o1) done
(%i2) tellsimpafter (foo(aa, bb), bar('aa=aa, 'bb=bb));
(%o2) [foorule1, false]
(%i3) foo (42, %e);
(%o3) bar(aa = 42, bb = %e)
(%i4) define_variable (bb, true, boolean);
(%o4) true
(%i5) foo (42, %e);
Error: bb was declared mode boolean, has value: %e
-- an error. Quitting. To debug this try debugmode(true);
Rules are named after main operators. Names of rules for built-in and user-defined operators are Lisp identifiers, while names for other functions are Maxima identifiers.
(%i1) tellsimpafter (foo (%pi + %e), 3*%pi);
(%o1) [foorule1, false]
(%i2) tellsimpafter (foo (%pi * %e), 17*%e);
(%o2) [foorule2, foorule1, false]
(%i3) tellsimpafter (foo (%i ^ %e), -42*%i);
(%o3) [foorule3, foorule2, foorule1, false]
(%i4) tellsimpafter (foo (9) + foo (13), quux (22));
(%o4) [+rule1, simplus]
(%i5) tellsimpafter (foo (9) * foo (13), blurf (22));
(%o5) [*rule1, simptimes]
(%i6) tellsimpafter (foo (9) ^ foo (13), mumble (22));
(%o6) [^rule1, simpexpt]
(%i7) rules;
(%o7) [foorule1, foorule2, foorule3, +rule1, *rule1, ^rule1]
(%i8) foorule_name: first (%o1);
(%o8) foorule1
(%i9) plusrule_name: first (%o4);
(%o9) +rule1
(%i10) remrule (foo, foorule1);
(%o10) foo
(%i11) remrule ("^", ?\^rule1);
(%o11) ^
(%i12) rules;
(%o12) [foorule2, foorule3, +rule1, *rule1]
A worked example: anticommutative multiplication.
(%i1) gt (i, j) := integerp(j) and i < j;
(%o1) gt(i, j) := integerp(j) and i < j
(%i2) matchdeclare (i, integerp, j, gt(i));
(%o2) done
(%i3) tellsimpafter (s[i]^^2, 1);
(%o3) [^^rule1, simpncexpt]
(%i4) tellsimpafter (s[i] . s[j], -s[j] . s[i]);
(%o4) [.rule1, simpnct]
(%i5) s[1] . (s[1] + s[2]);
(%o5) s . (s + s )
1 2 1
(%i6) expand (%);
(%o6) 1 - s . s
2 1
(%i7) factor (expand (sum (s[i], i, 0, 9)^^5));
(%o7) 100 (s + s + s + s + s + s + s + s + s + s )
9 8 7 6 5 4 3 2 1 0
Categories: Rules and patterns
Executes kill (rules) and then resets the next rule number to 1
for addition +, multiplication *, and exponentiation ^.
Categories: Rules and patterns
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| 35.1 Introduction to Sets | ||
| 35.2 Functions and Variables for Sets |
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Maxima provides set functions, such as intersection and union, for finite sets that are defined by explicit enumeration. Maxima treats lists and sets as distinct objects. This feature makes it possible to work with sets that have members that are either lists or sets.
In addition to functions for finite sets, Maxima provides some functions related to combinatorics; these include the Stirling numbers of the first and second kind, the Bell numbers, multinomial coefficients, partitions of nonnegative integers, and a few others. Maxima also defines a Kronecker delta function.
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To construct a set with members a_1, ..., a_n, write
set(a_1, ..., a_n) or {a_1, ..., a_n};
to construct the empty set, write set() or {}.
In input, set(...) and { ... } are equivalent.
Sets are always displayed with curly braces.
If a member is listed more than once, simplification eliminates the redundant member.
(%i1) set();
(%o1) {}
(%i2) set(a, b, a);
(%o2) {a, b}
(%i3) set(a, set(b));
(%o3) {a, {b}}
(%i4) set(a, [b]);
(%o4) {a, [b]}
(%i5) {};
(%o5) {}
(%i6) {a, b, a};
(%o6) {a, b}
(%i7) {a, {b}};
(%o7) {a, {b}}
(%i8) {a, [b]};
(%o8) {a, [b]}
Two would-be elements x and y are redundant
(i.e., considered the same for the purpose of set construction)
if and only if is(x = y) yields true.
Note that is(equal(x, y)) can yield true
while is(x = y) yields false;
in that case the elements x and y are considered distinct.
(%i1) x: a/c + b/c;
b a
(%o1) - + -
c c
(%i2) y: a/c + b/c;
b a
(%o2) - + -
c c
(%i3) z: (a + b)/c;
b + a
(%o3) -----
c
(%i4) is (x = y);
(%o4) true
(%i5) is (y = z);
(%o5) false
(%i6) is (equal (y, z));
(%o6) true
(%i7) y - z;
b + a b a
(%o7) - ----- + - + -
c c c
(%i8) ratsimp (%);
(%o8) 0
(%i9) {x, y, z};
b + a b a
(%o9) {-----, - + -}
c c c
To construct a set from the elements of a list, use setify.
(%i1) setify ([b, a]);
(%o1) {a, b}
Set members x and y are equal provided is(x = y)
evaluates to true. Thus rat(x) and x are equal as set
members; consequently,
(%i1) {x, rat(x)};
(%o1) {x}
Further, since is((x - 1)*(x + 1) = x^2 - 1) evaluates to false,
(x - 1)*(x + 1) and x^2 - 1 are distinct set members; thus
(%i1) {(x - 1)*(x + 1), x^2 - 1};
2
(%o1) {(x - 1) (x + 1), x - 1}
To reduce this set to a singleton set, apply rat to each set member:
(%i1) {(x - 1)*(x + 1), x^2 - 1};
2
(%o1) {(x - 1) (x + 1), x - 1}
(%i2) map (rat, %);
2
(%o2)/R/ {x - 1}
To remove redundancies from other sets, you may need to use other
simplification functions. Here is an example that uses trigsimp:
(%i1) {1, cos(x)^2 + sin(x)^2};
2 2
(%o1) {1, sin (x) + cos (x)}
(%i2) map (trigsimp, %);
(%o2) {1}
A set is simplified when its members are non-redundant and
sorted. The current version of the set functions uses the Maxima function
orderlessp to order sets; however, future versions of
the set functions might use a different ordering function.
Some operations on sets, such as substitution, automatically force a re-simplification; for example,
(%i1) s: {a, b, c}$
(%i2) subst (c=a, s);
(%o2) {a, b}
(%i3) subst ([a=x, b=x, c=x], s);
(%o3) {x}
(%i4) map (lambda ([x], x^2), set (-1, 0, 1));
(%o4) {0, 1}
Maxima treats lists and sets as distinct objects;
functions such as union and intersection complain
if any argument is not a set. If you need to apply a set
function to a list, use the setify function to convert it
to a set. Thus
(%i1) union ([1, 2], {a, b});
Function union expects a set, instead found [1,2]
-- an error. Quitting. To debug this try debugmode(true);
(%i2) union (setify ([1, 2]), {a, b});
(%o2) {1, 2, a, b}
To extract all set elements of a set s that satisfy a predicate
f, use subset(s, f). (A predicate is a
boolean-valued function.) For example, to find the equations
in a given set that do not depend on a variable z, use
(%i1) subset ({x + y + z, x - y + 4, x + y - 5},
lambda ([e], freeof (z, e)));
(%o1) {- y + x + 4, y + x - 5}
The section Functions and Variables for Sets has a complete list of the set functions in Maxima.
Categories: Sets
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There two ways to to iterate over set members. One way is the use
map; for example:
(%i1) map (f, {a, b, c});
(%o1) {f(a), f(b), f(c)}
The other way is to use for x in s do
(%i1) s: {a, b, c};
(%o1) {a, b, c}
(%i2) for si in s do print (concat (si, 1));
a1
b1
c1
(%o2) done
The Maxima functions first and rest work
correctly on sets. Applied to a set, first returns the first
displayed element of a set; which element that is may be
implementation-dependent. If s is a set, then
rest(s) is equivalent to disjoin(first(s), s).
Currently, there are other Maxima functions that work correctly
on sets.
In future versions of the set functions,
first and rest may function differently or not at all.
Maxima's orderless and ordergreat mechanisms are
incompatible with the set functions. If you need to use either orderless
or ordergreat, call those functions before constructing any sets,
and do not call unorder.
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Stavros Macrakis of Cambridge, Massachusetts and Barton Willis of the University of Nebraska at Kearney (UNK) wrote the Maxima set functions and their documentation.
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Returns the union of the set a with {x}.
adjoin complains if a is not a literal set.
adjoin(x, a) and union(set(x), a)
are equivalent;
however, adjoin may be somewhat faster than union.
See also disjoin.
Examples:
(%i1) adjoin (c, {a, b});
(%o1) {a, b, c}
(%i2) adjoin (a, {a, b});
(%o2) {a, b}
Categories: Sets
Represents the n-th Bell number.
belln(n) is the number of partitions of a set with n members.
For nonnegative integers n,
belln(n) simplifies to the n-th Bell number.
belln does not simplify for any other arguments.
belln distributes over equations, lists, matrices, and sets.
Examples:
belln applied to nonnegative integers.
(%i1) makelist (belln (i), i, 0, 6);
(%o1) [1, 1, 2, 5, 15, 52, 203]
(%i2) is (cardinality (set_partitions ({})) = belln (0));
(%o2) true
(%i3) is (cardinality (set_partitions ({1, 2, 3, 4, 5, 6})) =
belln (6));
(%o3) true
belln applied to arguments which are not nonnegative integers.
(%i1) [belln (x), belln (sqrt(3)), belln (-9)]; (%o1) [belln(x), belln(sqrt(3)), belln(- 9)]
Categories: Sets
Returns the number of distinct elements of the set a.
cardinality ignores redundant elements
even when simplification is disabled.
Examples:
(%i1) cardinality ({});
(%o1) 0
(%i2) cardinality ({a, a, b, c});
(%o2) 3
(%i3) simp : false;
(%o3) false
(%i4) cardinality ({a, a, b, c});
(%o4) 3
Categories: Sets
Returns a set of lists of the form [x_1, ..., x_n], where
x_1, ..., x_n are elements of the sets b_1, ... , b_n,
respectively.
cartesian_product complains if any argument is not a literal set.
Examples:
(%i1) cartesian_product ({0, 1});
(%o1) {[0], [1]}
(%i2) cartesian_product ({0, 1}, {0, 1});
(%o2) {[0, 0], [0, 1], [1, 0], [1, 1]}
(%i3) cartesian_product ({x}, {y}, {z});
(%o3) {[x, y, z]}
(%i4) cartesian_product ({x}, {-1, 0, 1});
(%o4) {[x, - 1], [x, 0], [x, 1]}
Categories: Sets
Returns the set a without the member x. If x is not a member of a, return a unchanged.
disjoin complains if a is not a literal set.
disjoin(x, a), delete(x, a), and
setdifference(a, set(x)) are all equivalent.
Of these, disjoin is generally faster than the others.
Examples:
(%i1) disjoin (a, {a, b, c, d});
(%o1) {b, c, d}
(%i2) disjoin (a + b, {5, z, a + b, %pi});
(%o2) {5, %pi, z}
(%i3) disjoin (a - b, {5, z, a + b, %pi});
(%o3) {5, %pi, b + a, z}
Categories: Sets
Returns true if and only if the sets a and b are disjoint.
disjointp complains if either a or b is not a literal set.
Examples:
(%i1) disjointp ({a, b, c}, {1, 2, 3});
(%o1) true
(%i2) disjointp ({a, b, 3}, {1, 2, 3});
(%o2) false
Categories: Sets · Predicate functions
Represents the set of divisors of n.
divisors(n) simplifies to a set of integers
when n is a nonzero integer.
The set of divisors includes the members 1 and n.
The divisors of a negative integer are the divisors of its absolute value.
divisors distributes over equations, lists, matrices, and sets.
Examples:
We can verify that 28 is a perfect number: the sum of its divisors (except for itself) is 28.
(%i1) s: divisors(28);
(%o1) {1, 2, 4, 7, 14, 28}
(%i2) lreduce ("+", args(s)) - 28;
(%o2) 28
divisors is a simplifying function.
Substituting 8 for a in divisors(a)
yields the divisors without reevaluating divisors(8).
(%i1) divisors (a);
(%o1) divisors(a)
(%i2) subst (8, a, %);
(%o2) {1, 2, 4, 8}
divisors distributes over equations, lists, matrices, and sets.
(%i1) divisors (a = b);
(%o1) divisors(a) = divisors(b)
(%i2) divisors ([a, b, c]);
(%o2) [divisors(a), divisors(b), divisors(c)]
(%i3) divisors (matrix ([a, b], [c, d]));
[ divisors(a) divisors(b) ]
(%o3) [ ]
[ divisors(c) divisors(d) ]
(%i4) divisors ({a, b, c});
(%o4) {divisors(a), divisors(b), divisors(c)}
Categories: Integers
Returns true if and only if x is a member of the
set a.
elementp complains if a is not a literal set.
Examples:
(%i1) elementp (sin(1), {sin(1), sin(2), sin(3)});
(%o1) true
(%i2) elementp (sin(1), {cos(1), cos(2), cos(3)});
(%o2) false
Categories: Sets · Predicate functions
Return true if and only if a is the empty set or
the empty list.
Examples:
(%i1) map (emptyp, [{}, []]);
(%o1) [true, true]
(%i2) map (emptyp, [a + b, {{}}, %pi]);
(%o2) [false, false, false]
Categories: Sets · Predicate functions
Returns a set of the equivalence classes of the set s with respect to the equivalence relation F.
F is a function of two variables defined on the Cartesian product of s with s.
The return value of F is either true or false,
or an expression expr such that is(expr) is either true or false.
When F is not an equivalence relation,
equiv_classes accepts it without complaint,
but the result is generally incorrect in that case.
Examples:
The equivalence relation is a lambda expression which returns true or false.
(%i1) equiv_classes ({1, 1.0, 2, 2.0, 3, 3.0},
lambda ([x, y], is (equal (x, y))));
(%o1) {{1, 1.0}, {2, 2.0}, {3, 3.0}}
The equivalence relation is the name of a relational function
which is evaluates to true or false.
(%i1) equiv_classes ({1, 1.0, 2, 2.0, 3, 3.0}, equal);
(%o1) {{1, 1.0}, {2, 2.0}, {3, 3.0}}
The equivalence classes are numbers which differ by a multiple of 3.
(%i1) equiv_classes ({1, 2, 3, 4, 5, 6, 7},
lambda ([x, y], remainder (x - y, 3) = 0));
(%o1) {{1, 4, 7}, {2, 5}, {3, 6}}
Categories: Sets
Returns true if the predicate f is true for all given arguments.
Given one set as the second argument,
every(f, s) returns true
if is(f(a_i)) returns true for all a_i in s.
every may or may not evaluate f for all a_i in s.
Since sets are unordered,
every may evaluate f(a_i) in any order.
Given one or more lists as arguments,
every(f, L_1, ..., L_n) returns true
if is(f(x_1, ..., x_n)) returns true
for all x_1, ..., x_n in L_1, ..., L_n, respectively.
every may or may not evaluate
f for every combination x_1, ..., x_n.
every evaluates lists in the order of increasing index.
Given an empty set {} or empty lists [] as arguments,
every returns true.
When the global flag maperror is true, all lists
L_1, ..., L_n must have equal lengths.
When maperror is false, list arguments are
effectively truncated to the length of the shortest list.
Return values of the predicate f which evaluate (via is)
to something other than true or false
are governed by the global flag prederror.
When prederror is true,
such values are treated as false,
and the return value from every is false.
When prederror is false,
such values are treated as unknown,
and the return value from every is unknown.
Examples:
every applied to a single set.
The predicate is a function of one argument.
(%i1) every (integerp, {1, 2, 3, 4, 5, 6});
(%o1) true
(%i2) every (atom, {1, 2, sin(3), 4, 5 + y, 6});
(%o2) false
every applied to two lists.
The predicate is a function of two arguments.
(%i1) every ("=", [a, b, c], [a, b, c]);
(%o1) true
(%i2) every ("#", [a, b, c], [a, b, c]);
(%o2) false
Return values of the predicate f which evaluate
to something other than true or false
are governed by the global flag prederror.
(%i1) prederror : false;
(%o1) false
(%i2) map (lambda ([a, b], is (a < b)), [x, y, z],
[x^2, y^2, z^2]);
(%o2) [unknown, unknown, unknown]
(%i3) every ("<", [x, y, z], [x^2, y^2, z^2]);
(%o3) unknown
(%i4) prederror : true;
(%o4) true
(%i5) every ("<", [x, y, z], [x^2, y^2, z^2]);
(%o5) false
Categories: Sets
Returns the subset of s for which the function f takes on maximum or minimum values.
extremal_subset(s, f, max) returns the subset of the set or
list s for which the real-valued function f takes on its maximum value.
extremal_subset(s, f, min) returns the subset of the set or
list s for which the real-valued function f takes on its minimum value.
Examples:
(%i1) extremal_subset ({-2, -1, 0, 1, 2}, abs, max);
(%o1) {- 2, 2}
(%i2) extremal_subset ({sqrt(2), 1.57, %pi/2}, sin, min);
(%o2) {sqrt(2)}
Categories: Sets
Collects arguments of subexpressions which have the same operator as expr and constructs an expression from these collected arguments.
Subexpressions in which the operator is different from the main operator of expr
are copied without modification,
even if they, in turn, contain some subexpressions in which the operator is the same as for expr.
It may be possible for flatten to construct expressions in which the number
of arguments differs from the declared arguments for an operator;
this may provoke an error message from the simplifier or evaluator.
flatten does not try to detect such situations.
Expressions with special representations, for example, canonical rational expressions (CRE),
cannot be flattened; in such cases, flatten returns its argument unchanged.
Examples:
Applied to a list, flatten gathers all list elements that are lists.
(%i1) flatten ([a, b, [c, [d, e], f], [[g, h]], i, j]); (%o1) [a, b, c, d, e, f, g, h, i, j]
Applied to a set, flatten gathers all members of set elements that are sets.
(%i1) flatten ({a, {b}, {{c}}});
(%o1) {a, b, c}
(%i2) flatten ({a, {[a], {a}}});
(%o2) {a, [a]}
flatten is similar to the effect of declaring the main operator n-ary.
However, flatten has no effect on subexpressions which have an operator
different from the main operator, while an n-ary declaration affects those.
(%i1) expr: flatten (f (g (f (f (x))))); (%o1) f(g(f(f(x)))) (%i2) declare (f, nary); (%o2) done (%i3) ev (expr); (%o3) f(g(f(x)))
flatten treats subscripted functions the same as any other operator.
(%i1) flatten (f[5] (f[5] (x, y), z));
(%o1) f (x, y, z)
5
It may be possible for flatten to construct expressions in which the number
of arguments differs from the declared arguments for an operator;
(%i1) 'mod (5, 'mod (7, 4)); (%o1) mod(5, mod(7, 4)) (%i2) flatten (%); (%o2) mod(5, 7, 4) (%i3) ''%, nouns; Wrong number of arguments to mod -- an error. Quitting. To debug this try debugmode(true);
Replaces every set operator in a by a list operator,
and returns the result.
full_listify replaces set operators in nested subexpressions,
even if the main operator is not set.
listify replaces only the main operator.
Examples:
(%i1) full_listify ({a, b, {c, {d, e, f}, g}});
(%o1) [a, b, [c, [d, e, f], g]]
(%i2) full_listify (F (G ({a, b, H({c, d, e})})));
(%o2) F(G([a, b, H([c, d, e])]))
Categories: Sets
When a is a list, replaces the list operator with a set operator,
and applies fullsetify to each member which is a set.
When a is not a list, it is returned unchanged.
setify replaces only the main operator.
Examples:
In line (%o2), the argument of f isn't converted to a set
because the main operator of f([b]) isn't a list.
(%i1) fullsetify ([a, [a]]);
(%o1) {a, {a}}
(%i2) fullsetify ([a, f([b])]);
(%o2) {a, f([b])}
Categories: Lists
Returns x for any argument x.
Examples:
identity may be used as a predicate when the arguments
are already Boolean values.
(%i1) every (identity, [true, true]); (%o1) true
Returns integer partitions of n, that is, lists of integers which sum to n.
integer_partitions(n) returns the set of
all partitions of the integer n.
Each partition is a list sorted from greatest to least.
integer_partitions(n, len)
returns all partitions that have length len or less; in this
case, zeros are appended to each partition with fewer than len
terms to make each partition have exactly len terms.
Each partition is a list sorted from greatest to least.
A list [a_1, ..., a_m] is a partition of a nonnegative integer n when (1) each a_i is a nonzero integer, and (2) a_1 + ... + a_m = n. Thus 0 has no partitions.
Examples:
(%i1) integer_partitions (3);
(%o1) {[1, 1, 1], [2, 1], [3]}
(%i2) s: integer_partitions (25)$
(%i3) cardinality (s);
(%o3) 1958
(%i4) map (lambda ([x], apply ("+", x)), s);
(%o4) {25}
(%i5) integer_partitions (5, 3);
(%o5) {[2, 2, 1], [3, 1, 1], [3, 2, 0], [4, 1, 0], [5, 0, 0]}
(%i6) integer_partitions (5, 2);
(%o6) {[3, 2], [4, 1], [5, 0]}
To find all partitions that satisfy a condition, use the function subset;
here is an example that finds all partitions of 10 that consist of prime numbers.
(%i1) s: integer_partitions (10)$
(%i2) cardinality (s);
(%o2) 42
(%i3) xprimep(x) := integerp(x) and (x > 1) and primep(x)$
(%i4) subset (s, lambda ([x], every (xprimep, x)));
(%o4) {[2, 2, 2, 2, 2], [3, 3, 2, 2], [5, 3, 2], [5, 5], [7, 3]}
Categories: Integers
intersect is the same as intersection, which see.
Categories: Sets
Returns a set containing the elements that are common to the sets a_1 through a_n.
intersection complains if any argument is not a literal set.
Examples:
(%i1) S_1 : {a, b, c, d};
(%o1) {a, b, c, d}
(%i2) S_2 : {d, e, f, g};
(%o2) {d, e, f, g}
(%i3) S_3 : {c, d, e, f};
(%o3) {c, d, e, f}
(%i4) S_4 : {u, v, w};
(%o4) {u, v, w}
(%i5) intersection (S_1, S_2);
(%o5) {d}
(%i6) intersection (S_2, S_3);
(%o6) {d, e, f}
(%i7) intersection (S_1, S_2, S_3);
(%o7) {d}
(%i8) intersection (S_1, S_2, S_3, S_4);
(%o8) {}
Categories: Sets
Represents the Kronecker delta function.
kron_delta simplifies to 1 when xi and yj are equal
for all pairs of arguments, and it simplifies to 0 when xi and
yj are not equal for some pair of arguments. Equality is
determined using is(equal(xi,xj)) and inequality by
is(notequal(xi,xj)). For exactly one argument, kron_delta
signals an error.
Examples:
(%i1) kron_delta(a,a); (%o1) 1 (%i2) kron_delta(a,b,a,b); (%o2) kron_delta(a, b) (%i3) kron_delta(a,a,b,a+1); (%o3) 0 (%i4) assume(equal(x,y)); (%o4) [equal(x, y)] (%i5) kron_delta(x,y); (%o5) 1
Returns a list containing the members of a when a is a set.
Otherwise, listify returns a.
full_listify replaces all set operators in a by list operators.
Examples:
(%i1) listify ({a, b, c, d});
(%o1) [a, b, c, d]
(%i2) listify (F ({a, b, c, d}));
(%o2) F({a, b, c, d})
Categories: Sets
Extends the binary function F to an n-ary function by composition, where s is a list.
lreduce(F, s) returns F(... F(F(s_1, s_2), s_3), ... s_n).
When the optional argument s_0 is present,
the result is equivalent to lreduce(F, cons(s_0, s)).
The function F is first applied to the leftmost list elements, thus the name "lreduce".
See also rreduce, xreduce, and tree_reduce.
Examples:
lreduce without the optional argument.
(%i1) lreduce (f, [1, 2, 3]); (%o1) f(f(1, 2), 3) (%i2) lreduce (f, [1, 2, 3, 4]); (%o2) f(f(f(1, 2), 3), 4)
lreduce with the optional argument.
(%i1) lreduce (f, [1, 2, 3], 4); (%o1) f(f(f(4, 1), 2), 3)
lreduce applied to built-in binary operators.
/ is the division operator.
(%i1) lreduce ("^", args ({a, b, c, d}));
b c d
(%o1) ((a ) )
(%i2) lreduce ("/", args ({a, b, c, d}));
a
(%o2) -----
b c d
Categories: Lists
Returns a set with members generated from the expression expr, where x is a list of variables in expr, and s is a set or list of lists. To generate each set member, expr is evaluated with the variables x bound in parallel to a member of s.
Each member of s must have the same length as x. The list of variables x must be a list of symbols, without subscripts. Even if there is only one symbol, x must be a list of one element, and each member of s must be a list of one element.
See also makelist.
Examples:
(%i1) makeset (i/j, [i, j], [[1, a], [2, b], [3, c], [4, d]]);
1 2 3 4
(%o1) {-, -, -, -}
a b c d
(%i2) S : {x, y, z}$
(%i3) S3 : cartesian_product (S, S, S);
(%o3) {[x, x, x], [x, x, y], [x, x, z], [x, y, x], [x, y, y],
[x, y, z], [x, z, x], [x, z, y], [x, z, z], [y, x, x],
[y, x, y], [y, x, z], [y, y, x], [y, y, y], [y, y, z],
[y, z, x], [y, z, y], [y, z, z], [z, x, x], [z, x, y],
[z, x, z], [z, y, x], [z, y, y], [z, y, z], [z, z, x],
[z, z, y], [z, z, z]}
(%i4) makeset (i + j + k, [i, j, k], S3);
(%o4) {3 x, 3 y, y + 2 x, 2 y + x, 3 z, z + 2 x, z + y + x,
z + 2 y, 2 z + x, 2 z + y}
(%i5) makeset (sin(x), [x], {[1], [2], [3]});
(%o5) {sin(1), sin(2), sin(3)}
Categories: Sets
Represents the Moebius function.
When n is product of k distinct primes,
moebius(n) simplifies to (-1)^k;
when n = 1, it simplifies to 1;
and it simplifies to 0 for all other positive integers.
moebius distributes over equations, lists, matrices, and sets.
Examples:
(%i1) moebius (1);
(%o1) 1
(%i2) moebius (2 * 3 * 5);
(%o2) - 1
(%i3) moebius (11 * 17 * 29 * 31);
(%o3) 1
(%i4) moebius (2^32);
(%o4) 0
(%i5) moebius (n);
(%o5) moebius(n)
(%i6) moebius (n = 12);
(%o6) moebius(n) = 0
(%i7) moebius ([11, 11 * 13, 11 * 13 * 15]);
(%o7) [- 1, 1, 1]
(%i8) moebius (matrix ([11, 12], [13, 14]));
[ - 1 0 ]
(%o8) [ ]
[ - 1 1 ]
(%i9) moebius ({21, 22, 23, 24});
(%o9) {- 1, 0, 1}
Categories: Integers
Returns the multinomial coefficient.
When each a_k is a nonnegative integer, the multinomial coefficient
gives the number of ways of placing a_1 + ... + a_n
distinct objects into n boxes with a_k elements in the
k'th box. In general, multinomial_coeff (a_1, ..., a_n)
evaluates to (a_1 + ... + a_n)!/(a_1! ... a_n!).
multinomial_coeff() (with no arguments) evaluates to 1.
minfactorial may be able to simplify the value returned by multinomial_coeff.
Examples:
(%i1) multinomial_coeff (1, 2, x);
(x + 3)!
(%o1) --------
2 x!
(%i2) minfactorial (%);
(x + 1) (x + 2) (x + 3)
(%o2) -----------------------
2
(%i3) multinomial_coeff (-6, 2);
(- 4)!
(%o3) --------
2 (- 6)!
(%i4) minfactorial (%);
(%o4) 10
Categories: Integers
Returns the number of distinct integer partitions of n
when n is a nonnegative integer.
Otherwise, num_distinct_partitions returns a noun expression.
num_distinct_partitions(n, list) returns a
list of the number of distinct partitions of 1, 2, 3, ..., n.
A distinct partition of n is a list of distinct positive integers k_1, ..., k_m such that n = k_1 + ... + k_m.
Examples:
(%i1) num_distinct_partitions (12); (%o1) 15 (%i2) num_distinct_partitions (12, list); (%o2) [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15] (%i3) num_distinct_partitions (n); (%o3) num_distinct_partitions(n)
Categories: Integers
Returns the number of integer partitions of n
when n is a nonnegative integer.
Otherwise, num_partitions returns a noun expression.
num_partitions(n, list) returns a
list of the number of integer partitions of 1, 2, 3, ..., n.
For a nonnegative integer n, num_partitions(n) is equal to
cardinality(integer_partitions(n)); however, num_partitions
does not actually construct the set of partitions, so it is much faster.
Examples:
(%i1) num_partitions (5) = cardinality (integer_partitions (5)); (%o1) 7 = 7 (%i2) num_partitions (8, list); (%o2) [1, 1, 2, 3, 5, 7, 11, 15, 22] (%i3) num_partitions (n); (%o3) num_partitions(n)
Categories: Integers
Partitions the set a according to the predicate f.
partition_set returns a list of two sets.
The first set comprises the elements of a for which f evaluates to false,
and the second comprises any other elements of a.
partition_set does not apply is to the return value of f.
partition_set complains if a is not a literal set.
See also subset.
Examples:
(%i1) partition_set ({2, 7, 1, 8, 2, 8}, evenp);
(%o1) [{1, 7}, {2, 8}]
(%i2) partition_set ({x, rat(y), rat(y) + z, 1},
lambda ([x], ratp(x)));
(%o2)/R/ [{1, x}, {y, y + z}]
Categories: Sets
Returns a set of all distinct permutations of the members of the list or set a. Each permutation is a list, not a set.
When a is a list, duplicate members of a are included in the permutations.
permutations complains if a is not a literal list or set.
See also random_permutation.
Examples:
(%i1) permutations ([a, a]);
(%o1) {[a, a]}
(%i2) permutations ([a, a, b]);
(%o2) {[a, a, b], [a, b, a], [b, a, a]}
Returns the set of all subsets of a, or a subset of that set.
powerset(a) returns the set of all subsets of the set a.
powerset(a) has 2^cardinality(a) members.
powerset(a, n) returns the set of all subsets of a that have
cardinality n.
powerset complains if a is not a literal set,
or if n is not a nonnegative integer.
Examples:
(%i1) powerset ({a, b, c});
(%o1) {{}, {a}, {a, b}, {a, b, c}, {a, c}, {b}, {b, c}, {c}}
(%i2) powerset ({w, x, y, z}, 4);
(%o2) {{w, x, y, z}}
(%i3) powerset ({w, x, y, z}, 3);
(%o3) {{w, x, y}, {w, x, z}, {w, y, z}, {x, y, z}}
(%i4) powerset ({w, x, y, z}, 2);
(%o4) {{w, x}, {w, y}, {w, z}, {x, y}, {x, z}, {y, z}}
(%i5) powerset ({w, x, y, z}, 1);
(%o5) {{w}, {x}, {y}, {z}}
(%i6) powerset ({w, x, y, z}, 0);
(%o6) {{}}
Categories: Sets
Returns a random permutation of the set or list a, as constructed by the Knuth shuffle algorithm.
The return value is a new list, which is distinct from the argument even if all elements happen to be the same. However, the elements of the argument are not copied.
Examples:
(%i1) random_permutation ([a, b, c, 1, 2, 3]);
(%o1) [c, 1, 2, 3, a, b]
(%i2) random_permutation ([a, b, c, 1, 2, 3]);
(%o2) [b, 3, 1, c, a, 2]
(%i3) random_permutation ({x + 1, y + 2, z + 3});
(%o3) [y + 2, z + 3, x + 1]
(%i4) random_permutation ({x + 1, y + 2, z + 3});
(%o4) [x + 1, y + 2, z + 3]
Extends the binary function F to an n-ary function by composition, where s is a list.
rreduce(F, s) returns F(s_1, ... F(s_{n - 2}, F(s_{n - 1}, s_n))).
When the optional argument s_{n + 1} is present,
the result is equivalent to rreduce(F, endcons(s_{n + 1}, s)).
The function F is first applied to the rightmost list elements, thus the name "rreduce".
See also lreduce, tree_reduce, and xreduce.
Examples:
rreduce without the optional argument.
(%i1) rreduce (f, [1, 2, 3]); (%o1) f(1, f(2, 3)) (%i2) rreduce (f, [1, 2, 3, 4]); (%o2) f(1, f(2, f(3, 4)))
rreduce with the optional argument.
(%i1) rreduce (f, [1, 2, 3], 4); (%o1) f(1, f(2, f(3, 4)))
rreduce applied to built-in binary operators.
/ is the division operator.
(%i1) rreduce ("^", args ({a, b, c, d}));
d
c
b
(%o1) a
(%i2) rreduce ("/", args ({a, b, c, d}));
a c
(%o2) ---
b d
Categories: Lists
Returns a set containing the elements in the set a that are not in the set b.
setdifference complains if either a or b is not a literal set.
Examples:
(%i1) S_1 : {a, b, c, x, y, z};
(%o1) {a, b, c, x, y, z}
(%i2) S_2 : {aa, bb, c, x, y, zz};
(%o2) {aa, bb, c, x, y, zz}
(%i3) setdifference (S_1, S_2);
(%o3) {a, b, z}
(%i4) setdifference (S_2, S_1);
(%o4) {aa, bb, zz}
(%i5) setdifference (S_1, S_1);
(%o5) {}
(%i6) setdifference (S_1, {});
(%o6) {a, b, c, x, y, z}
(%i7) setdifference ({}, S_1);
(%o7) {}
Categories: Sets
Returns true if sets a and b have the same number of elements
and is(x = y) is true
for x in the elements of a
and y in the elements of b,
considered in the order determined by listify.
Otherwise, setequalp returns false.
Examples:
(%i1) setequalp ({1, 2, 3}, {1, 2, 3});
(%o1) true
(%i2) setequalp ({a, b, c}, {1, 2, 3});
(%o2) false
(%i3) setequalp ({x^2 - y^2}, {(x + y) * (x - y)});
(%o3) false
Categories: Sets · Predicate functions
Constructs a set from the elements of the list a. Duplicate
elements of the list a are deleted and the elements
are sorted according to the predicate orderlessp.
setify complains if a is not a literal list.
Examples:
(%i1) setify ([1, 2, 3, a, b, c]);
(%o1) {1, 2, 3, a, b, c}
(%i2) setify ([a, b, c, a, b, c]);
(%o2) {a, b, c}
(%i3) setify ([7, 13, 11, 1, 3, 9, 5]);
(%o3) {1, 3, 5, 7, 9, 11, 13}
Categories: Lists
Returns true if and only if a is a Maxima set.
setp returns true for unsimplified sets (that is, sets with redundant members)
as well as simplified sets.
setp is equivalent to the Maxima function
setp(a) := not atom(a) and op(a) = 'set.
Examples:
(%i1) simp : false;
(%o1) false
(%i2) {a, a, a};
(%o2) {a, a, a}
(%i3) setp (%);
(%o3) true
Categories: Sets · Predicate functions
Returns the set of all partitions of a, or a subset of that set.
set_partitions(a, n) returns a set of all
decompositions of a into n nonempty disjoint subsets.
set_partitions(a) returns the set of all partitions.
stirling2 returns the cardinality of the set of partitions of a set.
A set of sets P is a partition of a set S when
Examples:
The empty set is a partition of itself, the conditions 1 and 2 being vacuously true.
(%i1) set_partitions ({});
(%o1) {{}}
The cardinality of the set of partitions of a set can be found using stirling2.
(%i1) s: {0, 1, 2, 3, 4, 5}$
(%i2) p: set_partitions (s, 3)$
(%i3) cardinality(p) = stirling2 (6, 3);
(%o3) 90 = 90
Each member of p should have n = 3 members; let's check.
(%i1) s: {0, 1, 2, 3, 4, 5}$
(%i2) p: set_partitions (s, 3)$
(%i3) map (cardinality, p);
(%o3) {3}
Finally, for each member of p, the union of its members should
equal s; again let's check.
(%i1) s: {0, 1, 2, 3, 4, 5}$
(%i2) p: set_partitions (s, 3)$
(%i3) map (lambda ([x], apply (union, listify (x))), p);
(%o3) {{0, 1, 2, 3, 4, 5}}
Categories: Sets
Returns true if the predicate f is true for one or more given arguments.
Given one set as the second argument,
some(f, s) returns true
if is(f(a_i)) returns true for one or more a_i in s.
some may or may not evaluate f for all a_i in s.
Since sets are unordered,
some may evaluate f(a_i) in any order.
Given one or more lists as arguments,
some(f, L_1, ..., L_n) returns true
if is(f(x_1, ..., x_n)) returns true
for one or more x_1, ..., x_n in L_1, ..., L_n, respectively.
some may or may not evaluate
f for some combinations x_1, ..., x_n.
some evaluates lists in the order of increasing index.
Given an empty set {} or empty lists [] as arguments,
some returns false.
When the global flag maperror is true, all lists
L_1, ..., L_n must have equal lengths.
When maperror is false, list arguments are
effectively truncated to the length of the shortest list.
Return values of the predicate f which evaluate (via is)
to something other than true or false
are governed by the global flag prederror.
When prederror is true,
such values are treated as false.
When prederror is false,
such values are treated as unknown.
Examples:
some applied to a single set.
The predicate is a function of one argument.
(%i1) some (integerp, {1, 2, 3, 4, 5, 6});
(%o1) true
(%i2) some (atom, {1, 2, sin(3), 4, 5 + y, 6});
(%o2) true
some applied to two lists.
The predicate is a function of two arguments.
(%i1) some ("=", [a, b, c], [a, b, c]);
(%o1) true
(%i2) some ("#", [a, b, c], [a, b, c]);
(%o2) false
Return values of the predicate f which evaluate
to something other than true or false
are governed by the global flag prederror.
(%i1) prederror : false;
(%o1) false
(%i2) map (lambda ([a, b], is (a < b)), [x, y, z],
[x^2, y^2, z^2]);
(%o2) [unknown, unknown, unknown]
(%i3) some ("<", [x, y, z], [x^2, y^2, z^2]);
(%o3) unknown
(%i4) some ("<", [x, y, z], [x^2, y^2, z + 1]);
(%o4) true
(%i5) prederror : true;
(%o5) true
(%i6) some ("<", [x, y, z], [x^2, y^2, z^2]);
(%o6) false
(%i7) some ("<", [x, y, z], [x^2, y^2, z + 1]);
(%o7) true
Represents the Stirling number of the first kind.
When n and m are nonnegative
integers, the magnitude of stirling1 (n, m) is the number of
permutations of a set with n members that have m cycles.
stirling1 is a simplifying function.
Maxima knows the following identities:
These identities are applied when the arguments are literal integers
or symbols declared as integers, and the first argument is nonnegative.
stirling1 does not simplify for non-integer arguments.
Examples:
(%i1) declare (n, integer)$ (%i2) assume (n >= 0)$ (%i3) stirling1 (n, n); (%o3) 1
Categories: Integers
Represents the Stirling number of the second kind.
When n and m are nonnegative
integers, stirling2 (n, m) is the number of ways a set with
cardinality n can be partitioned into m disjoint subsets.
stirling2 is a simplifying function.
Maxima knows the following identities.
These identities are applied when the arguments are literal integers
or symbols declared as integers, and the first argument is nonnegative.
stirling2 does not simplify for non-integer arguments.
Examples:
(%i1) declare (n, integer)$ (%i2) assume (n >= 0)$ (%i3) stirling2 (n, n); (%o3) 1
stirling2 does not simplify for non-integer arguments.
(%i1) stirling2 (%pi, %pi); (%o1) stirling2(%pi, %pi)
Categories: Integers
Returns the subset of the set a that satisfies the predicate f.
subset returns a set which comprises the elements of a
for which f returns anything other than false.
subset does not apply is to the return value of f.
subset complains if a is not a literal set.
See also partition_set.
Examples:
(%i1) subset ({1, 2, x, x + y, z, x + y + z}, atom);
(%o1) {1, 2, x, z}
(%i2) subset ({1, 2, 7, 8, 9, 14}, evenp);
(%o2) {2, 8, 14}
Categories: Sets
Returns true if and only if the set a is a subset of b.
subsetp complains if either a or b is not a literal set.
Examples:
(%i1) subsetp ({1, 2, 3}, {a, 1, b, 2, c, 3});
(%o1) true
(%i2) subsetp ({a, 1, b, 2, c, 3}, {1, 2, 3});
(%o2) false
Categories: Sets · Predicate functions
Returns the symmetric difference of sets a_1, …, a_n.
Given two arguments, symmdifference (a, b) is the same as
union (setdifference (a, b), setdifference (b,
a)).
symmdifference complains if any argument is not a literal set.
Examples:
(%i1) S_1 : {a, b, c};
(%o1) {a, b, c}
(%i2) S_2 : {1, b, c};
(%o2) {1, b, c}
(%i3) S_3 : {a, b, z};
(%o3) {a, b, z}
(%i4) symmdifference ();
(%o4) {}
(%i5) symmdifference (S_1);
(%o5) {a, b, c}
(%i6) symmdifference (S_1, S_2);
(%o6) {1, a}
(%i7) symmdifference (S_1, S_2, S_3);
(%o7) {1, b, z}
(%i8) symmdifference ({}, S_1, S_2, S_3);
(%o8) {1,b, z}
Categories: Sets
Extends the binary function F to an n-ary function by composition, where s is a set or list.
tree_reduce is equivalent to the following:
Apply F to successive pairs of elements
to form a new list [F(s_1, s_2), F(s_3, s_4), ...],
carrying the final element unchanged if there are an odd number of elements.
Then repeat until the list is reduced to a single element, which is the return value.
When the optional argument s_0 is present,
the result is equivalent tree_reduce(F, cons(s_0, s)).
For addition of floating point numbers,
tree_reduce may return a sum that has a smaller rounding error
than either rreduce or lreduce.
The elements of s and the partial results may be arranged in a minimum-depth binary tree, thus the name "tree_reduce".
Examples:
tree_reduce applied to a list with an even number of elements.
(%i1) tree_reduce (f, [a, b, c, d]); (%o1) f(f(a, b), f(c, d))
tree_reduce applied to a list with an odd number of elements.
(%i1) tree_reduce (f, [a, b, c, d, e]); (%o1) f(f(f(a, b), f(c, d)), e)
Returns the union of the sets a_1 through a_n.
union() (with no arguments) returns the empty set.
union complains if any argument is not a literal set.
Examples:
(%i1) S_1 : {a, b, c + d, %e};
(%o1) {%e, a, b, d + c}
(%i2) S_2 : {%pi, %i, %e, c + d};
(%o2) {%e, %i, %pi, d + c}
(%i3) S_3 : {17, 29, 1729, %pi, %i};
(%o3) {17, 29, 1729, %i, %pi}
(%i4) union ();
(%o4) {}
(%i5) union (S_1);
(%o5) {%e, a, b, d + c}
(%i6) union (S_1, S_2);
(%o6) {%e, %i, %pi, a, b, d + c}
(%i7) union (S_1, S_2, S_3);
(%o7) {17, 29, 1729, %e, %i, %pi, a, b, d + c}
(%i8) union ({}, S_1, S_2, S_3);
(%o8) {17, 29, 1729, %e, %i, %pi, a, b, d + c}
Categories: Sets
Extends the function F to an n-ary function by composition,
or, if F is already n-ary, applies F to s.
When F is not n-ary, xreduce is the same as lreduce.
The argument s is a list.
Functions known to be n-ary include
addition +, multiplication *, and, or, max,
min, and append.
Functions may also be declared n-ary by declare(F, nary).
For these functions,
xreduce is expected to be faster than either rreduce or lreduce.
When the optional argument s_0 is present,
the result is equivalent to xreduce(s, cons(s_0, s)).
Floating point addition is not exactly associative; be that as it may,
xreduce applies Maxima's n-ary addition when s contains floating point numbers.
Examples:
xreduce applied to a function known to be n-ary.
F is called once, with all arguments.
(%i1) declare (F, nary);
(%o1) done
(%i2) F ([L]) := L;
(%o2) F([L]) := L
(%i3) xreduce (F, [a, b, c, d, e]);
(%o3) [[[[[("[", simp), a], b], c], d], e]
xreduce applied to a function not known to be n-ary.
G is called several times, with two arguments each time.
(%i1) G ([L]) := L;
(%o1) G([L]) := L
(%i2) xreduce (G, [a, b, c, d, e]);
(%o2) [[[[[("[", simp), a], b], c], d], e]
(%i3) lreduce (G, [a, b, c, d, e]);
(%o3) [[[[a, b], c], d], e]
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| 36.1 Introduction to Function Definition | ||
| 36.2 Function | ||
| 36.3 Macros | ||
| 36.4 Functions and Variables for Function Definition |
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Categories: Function definition · Programming
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To define a function in Maxima you use the := operator.
E.g.
f(x) := sin(x)
defines a function f.
Anonymous functions may also be created using lambda.
For example
lambda ([i, j], ...)
can be used instead of f
where
f(i,j) := block ([], ...); map (lambda ([i], i+1), l)
would return a list with 1 added to each term.
You may also define a function with a variable number of arguments, by having a final argument which is assigned to a list of the extra arguments:
(%i1) f ([u]) := u; (%o1) f([u]) := u (%i2) f (1, 2, 3, 4); (%o2) [1, 2, 3, 4] (%i3) f (a, b, [u]) := [a, b, u]; (%o3) f(a, b, [u]) := [a, b, u] (%i4) f (1, 2, 3, 4, 5, 6); (%o4) [1, 2, [3, 4, 5, 6]]
The right hand side of a function is an expression. Thus if you want a sequence of expressions, you do
f(x) := (expr1, expr2, ...., exprn);
and the value of exprn is what is returned by the function.
If you wish to make a return from some expression inside the
function then you must use block and return.
block ([], expr1, ..., if (a > 10) then return(a), ..., exprn)
is itself an expression, and so could take the place of the right hand side of a function definition. Here it may happen that the return happens earlier than the last expression.
The first [] in the block, may contain a list of variables and
variable assignments, such as [a: 3, b, c: []], which would cause the
three variables a,b,and c to not refer to their
global values, but rather have these special values for as long as the
code executes inside the block, or inside functions called from
inside the block. This is called dynamic binding, since the
variables last from the start of the block to the time it exits. Once
you return from the block, or throw out of it, the old values (if
any) of the variables will be restored. It is certainly a good idea
to protect your variables in this way. Note that the assignments
in the block variables, are done in parallel. This means, that if
you had used c: a in the above, the value of c would
have been the value of a at the time you just entered the block,
but before a was bound. Thus doing something like
block ([a: a], expr1, ... a: a+3, ..., exprn)
will protect the external value of a from being altered, but
would let you access what that value was. Thus the right hand
side of the assignments, is evaluated in the entering context, before
any binding occurs.
Using just block ([x], ...) would cause the x to have itself
as value, just as if it would have if you entered a fresh Maxima
session.
The actual arguments to a function are treated in exactly same way as the variables in a block. Thus in
f(x) := (expr1, ..., exprn);
and
f(1);
we would have a similar context for evaluation of the expressions as if we had done
block ([x: 1], expr1, ..., exprn)
Inside functions, when the right hand side of a definition,
may be computed at runtime, it is useful to use define and
possibly buildq.
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An array function stores the function value the first time it is called with a given argument, and returns the stored value, without recomputing it, when that same argument is given. Such a function is often called a memoizing function.
Array function names are appended to the global list arrays
(not the global list functions). arrayinfo returns the list of
arguments for which there are stored values, and listarray returns the
stored values. dispfun and fundef return the array function
definition.
arraymake constructs an array function call,
analogous to funmake for ordinary functions.
arrayapply applies an array function to its arguments,
analogous to apply for ordinary functions.
There is nothing exactly analogous to map for array functions,
although map(lambda([x], a[x]), L) or
makelist(a[x], x, L), where L is a list,
are not too far off the mark.
remarray removes an array function definition (including any stored
function values), analogous to remfunction for ordinary functions.
kill(a[x]) removes the value of the array function a
stored for the argument x;
the next time a is called with argument x,
the function value is recomputed.
However, there is no way to remove all of the stored values at once,
except for kill(a) or remarray(a),
which also remove the function definition.
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Substitutes variables named by the list L into the expression expr,
in parallel, without evaluating expr. The resulting expression is
simplified, but not evaluated, after buildq carries out the substitution.
The elements of L are symbols or assignment expressions
symbol: value, evaluated in parallel. That is, the binding
of a variable on the right-hand side of an assignment is the binding of that
variable in the context from which buildq was called, not the binding of
that variable in the variable list L. If some variable in L is not
given an explicit assignment, its binding in buildq is the same as in
the context from which buildq was called.
Then the variables named by L are substituted into expr in parallel. That is, the substitution for every variable is determined before any substitution is made, so the substitution for one variable has no effect on any other.
If any variable x appears as splice (x) in expr,
then x must be bound to a list,
and the list is spliced (interpolated) into expr instead of substituted.
Any variables in expr not appearing in L are carried into the result
verbatim, even if they have bindings in the context from which buildq
was called.
Examples
a is explicitly bound to x, while b has the same binding
(namely 29) as in the calling context, and c is carried through verbatim.
The resulting expression is not evaluated until the explicit evaluation
''%.
(%i1) (a: 17, b: 29, c: 1729)$ (%i2) buildq ([a: x, b], a + b + c); (%o2) x + c + 29 (%i3) ''%; (%o3) x + 1758
e is bound to a list, which appears as such in the arguments of
foo, and interpolated into the arguments of bar.
(%i1) buildq ([e: [a, b, c]], foo (x, e, y)); (%o1) foo(x, [a, b, c], y) (%i2) buildq ([e: [a, b, c]], bar (x, splice (e), y)); (%o2) bar(x, a, b, c, y)
The result is simplified after substitution. If simplification were applied before substitution, these two results would be the same.
(%i1) buildq ([e: [a, b, c]], splice (e) + splice (e)); (%o1) 2 c + 2 b + 2 a (%i2) buildq ([e: [a, b, c]], 2 * splice (e)); (%o2) 2 a b c
The variables in L are bound in parallel; if bound sequentially,
the first result would be foo (b, b).
Substitutions are carried out in parallel;
compare the second result with the result of subst,
which carries out substitutions sequentially.
(%i1) buildq ([a: b, b: a], foo (a, b));
(%o1) foo(b, a)
(%i2) buildq ([u: v, v: w, w: x, x: y, y: z, z: u],
bar (u, v, w, x, y, z));
(%o2) bar(v, w, x, y, z, u)
(%i3) subst ([u=v, v=w, w=x, x=y, y=z, z=u],
bar (u, v, w, x, y, z));
(%o3) bar(u, u, u, u, u, u)
Construct a list of equations with some variables or expressions on the
left-hand side and their values on the right-hand side. macroexpand
shows the expression returned by show_values.
(%i1) show_values ([L]) ::= buildq ([L], map ("=", 'L, L));
(%o1) show_values([L]) ::= buildq([L], map("=", 'L, L))
(%i2) (a: 17, b: 29, c: 1729)$
(%i3) show_values (a, b, c - a - b);
(%o3) [a = 17, b = 29, c - b - a = 1683]
(%i4) macroexpand (show_values (a, b, c - a - b));
(%o4) map(=, '([a, b, c - b - a]), [a, b, c - b - a])
Given a function of several arguments, create another function for which some of the arguments are fixed.
(%i1) curry (f, [a]) :=
buildq ([f, a], lambda ([[x]], apply (f, append (a, x))))$
(%i2) by3 : curry ("*", 3);
(%o2) lambda([[x]], apply(*, append([3], x)))
(%i3) by3 (a + b);
(%o3) 3 (b + a)
Categories: Function definition
Returns the macro expansion of expr without evaluating it,
when expr is a macro function call.
Otherwise, macroexpand returns expr.
If the expansion of expr yields another macro function call, that macro function call is also expanded.
macroexpand quotes its argument.
However, if the expansion of a macro function call has side effects,
those side effects are executed.
See also ::=, macros, and macroexpand1..
Examples
(%i1) g (x) ::= x / 99;
x
(%o1) g(x) ::= --
99
(%i2) h (x) ::= buildq ([x], g (x - a));
(%o2) h(x) ::= buildq([x], g(x - a))
(%i3) a: 1234;
(%o3) 1234
(%i4) macroexpand (h (y));
y - a
(%o4) -----
99
(%i5) h (y);
y - 1234
(%o5) --------
99
Categories: Function application
Returns the macro expansion of expr without evaluating it,
when expr is a macro function call.
Otherwise, macroexpand1 returns expr.
macroexpand1 quotes its argument.
However, if the expansion of a macro function call has side effects,
those side effects are executed.
If the expansion of expr yields another macro function call, that macro function call is not expanded.
See also ::=, macros, and macroexpand.
Examples
(%i1) g (x) ::= x / 99;
x
(%o1) g(x) ::= --
99
(%i2) h (x) ::= buildq ([x], g (x - a));
(%o2) h(x) ::= buildq([x], g(x - a))
(%i3) a: 1234;
(%o3) 1234
(%i4) macroexpand1 (h (y));
(%o4) g(y - a)
(%i5) h (y);
y - 1234
(%o5) --------
99
Categories: Function application
Default value: []
macros is the list of user-defined macro functions.
The macro function definition operator ::= puts a new macro function
onto this list, and kill, remove, and remfunction remove
macro functions from the list.
See also infolists.
Categories: Function definition · Global variables
Splices (interpolates) the list named by the atom a into an expression,
but only if splice appears within buildq;
otherwise, splice is treated as an undefined function.
If appearing within buildq as a alone (without splice),
a is substituted (not interpolated) as a list into the result.
The argument of splice can only be an atom;
it cannot be a literal list or an expression which yields a list.
Typically splice supplies the arguments for a function or operator.
For a function f, the expression f (splice (a)) within
buildq expands to f (a[1], a[2], a[3], ...).
For an operator o, the expression "o" (splice (a)) within
buildq expands to "o" (a[1], a[2], a[3], ...),
where o may be any type of operator (typically one which takes multiple
arguments). Note that the operator must be enclosed in double quotes ".
Examples
(%i1) buildq ([x: [1, %pi, z - y]], foo (splice (x)) / length (x));
foo(1, %pi, z - y)
(%o1) -----------------------
length([1, %pi, z - y])
(%i2) buildq ([x: [1, %pi]], "/" (splice (x)));
1
(%o2) ---
%pi
(%i3) matchfix ("<>", "<>");
(%o3) <>
(%i4) buildq ([x: [1, %pi, z - y]], "<>" (splice (x)));
(%o4) <>1, %pi, z - y<>
Categories: Function definition
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Constructs and evaluates an expression F(arg_1, ...,
arg_n).
apply does not attempt to distinguish array functions from ordinary
functions; when F is the name of an array function, apply evaluates
F(...) (that is, a function call with parentheses instead of square
brackets). arrayapply evaluates a function call with square brackets in
this case.
Examples:
apply evaluates its arguments.
In this example, min is applied to the value of L.
(%i1) L : [1, 5, -10.2, 4, 3]; (%o1) [1, 5, - 10.2, 4, 3] (%i2) apply (min, L); (%o2) - 10.2
apply evaluates arguments, even if the function F quotes them.
(%i1) F (x) := x / 1729;
x
(%o1) F(x) := ----
1729
(%i2) fname : F;
(%o2) F
(%i3) dispfun (F);
x
(%t3) F(x) := ----
1729
(%o3) [%t3]
(%i4) dispfun (fname);
fundef: no such function: fname
-- an error. To debug this try: debugmode(true);
(%i5) apply (dispfun, [fname]);
x
(%t5) F(x) := ----
1729
(%o5) [%t5]
apply evaluates the function name F.
Single quote ' defeats evaluation.
demoivre is the name of a global variable and also a function.
(%i1) demoivre;
(%o1) false
(%i2) demoivre (exp (%i * x));
(%o2) %i sin(x) + cos(x)
(%i3) apply (demoivre, [exp (%i * x)]);
apply: found false where a function was expected.
-- an error. To debug this try: debugmode(true);
(%i4) apply ('demoivre, [exp (%i * x)]);
(%o4) %i sin(x) + cos(x)
How to convert a nested list into a matrix:
(%i1) a:[[1,2],[3,4]];
(%o1) [[1, 2], [3, 4]]
(%i2) apply(matrix,a);
[ 1 2 ]
(%o2) [ ]
[ 3 4 ]
Categories: Function application
The function block allows to make the variables v_1, …,
v_m to be local for a sequence of commands. If these variables
are already bound block saves the current values of the
variables v_1, …, v_m (if any) upon entry to the
block, then unbinds the variables so that they evaluate to themselves;
The local variables may be bound to arbitrary values within the block
but when the block is exited the saved values are restored, and the
values assigned within the block are lost.
If there is no need to define local variables then the list at the
beginning of the block command may be omitted.
In this case if neither return nor go are used
block behaves similar to the following construct:
( expr_1, expr_2,... , expr_n );
expr_1, …, expr_n will be evaluated in sequence and
the value of the last expression will be returned. The sequence can be
modified by the go, throw, and return functions. The last
expression is expr_n unless return or an expression containing
throw is evaluated.
The declaration local(v_1, ..., v_m) within block
saves the properties associated with the symbols v_1, …, v_m,
removes any properties before evaluating other expressions, and restores any
saved properties on exit from the block. Some declarations are implemented as
properties of a symbol, including :=, array, dependencies,
atvalue, matchdeclare, atomgrad, constant,
nonscalar, assume, and some others. The effect of local
is to make such declarations effective only within the block; otherwise
declarations within a block are actually global declarations.
block may appear within another block.
Local variables are established each time a new block is evaluated.
Local variables appear to be global to any enclosed blocks.
If a variable is non-local in a block,
its value is the value most recently assigned by an enclosing block, if any,
otherwise, it is the value of the variable in the global environment.
This policy may coincide with the usual understanding of "dynamic scope".
The value of the block is the value of the last statement or the
value of the argument to the function return which may be used to exit
explicitly from the block. The function go may be used to transfer
control to the statement of the block that is tagged with the argument
to go. To tag a statement, precede it by an atomic argument as
another statement in the block. For example:
block ([x], x:1, loop, x: x+1, ..., go(loop), ...). The argument to
go must be the name of a tag appearing within the block. One cannot use
go to transfer to a tag in a block other than the one containing the
go.
Blocks typically appear on the right side of a function definition but can be used in other places as well.
Categories: Expressions · Programming
Evaluates and prints expr_1, …, expr_n and then
causes a Maxima break at which point the user can examine and change
his environment. Upon typing exit; the computation resumes.
Categories: Debugging
Evaluates expr_1, …, expr_n one by one; if any
leads to the evaluation of an expression of the
form throw (arg), then the value of the catch is the value of
throw (arg), and no further expressions are evaluated.
This "non-local return" thus goes through any depth of
nesting to the nearest enclosing catch. If there is no catch
enclosing a throw, an error message is printed.
If the evaluation of the arguments does not lead to the evaluation of any
throw then the value of catch is the value of expr_n.
(%i1) lambda ([x], if x < 0 then throw(x) else f(x))$
(%i2) g(l) := catch (map (''%, l))$
(%i3) g ([1, 2, 3, 7]);
(%o3) [f(1), f(2), f(3), f(7)]
(%i4) g ([1, 2, -3, 7]);
(%o4) - 3
The function g returns a list of f of each element of l if
l consists only of non-negative numbers; otherwise, g "catches"
the first negative element of l and "throws" it up.
Categories: Programming
Translates Maxima functions into Lisp and writes the translated code into the file filename.
compfile(filename, f_1, ..., f_n) translates the
specified functions. compfile (filename, functions) and
compfile (filename, all) translate all user-defined functions.
The Lisp translations are not evaluated, nor is the output file processed by
the Lisp compiler.
translate creates and evaluates Lisp translations. compile_file
translates Maxima into Lisp, and then executes the Lisp compiler.
See also translate, translate_file, and compile_file.
Categories: Translation and compilation
Translates Maxima functions f_1, …, f_n into Lisp, evaluates
the Lisp translations, and calls the Lisp function COMPILE on each
translated function. compile returns a list of the names of the
compiled functions.
compile (all) or compile (functions) compiles all user-defined
functions.
compile quotes its arguments;
the quote-quote operator '' defeats quotation.
Compiling a function to native code can mean a big increase in speed and might cause the memory footprint to reduce drastically. Code tends to be especially effective when the flexibility it needs to provide is limited. If compilation doesn't provide the speed that is needed a few ways to limit the code's functionality are the following:
mode_declare or a statement like the following one:
put(x_1, bigfloat, numerical_type)
'
tells the compiler that the text is meant as an option.
Categories: Translation and compilation
Defines a function named f with arguments x_1, …, x_n
and function body expr. define always evaluates its second
argument (unless explicitly quoted). The function so defined may be an ordinary
Maxima function (with arguments enclosed in parentheses) or an array function
(with arguments enclosed in square brackets).
When the last or only function argument x_n is a list of one element,
the function defined by define accepts a variable number of arguments.
Actual arguments are assigned one-to-one to formal arguments x_1, …,
x_(n - 1), and any further actual arguments, if present, are assigned to
x_n as a list.
When the first argument of define is an expression of the form
f(x_1, ..., x_n) or f[x_1, ...,
x_n], the function arguments are evaluated but f is not evaluated,
even if there is already a function or variable by that name.
When the first argument is an expression with operator funmake,
arraymake, or ev, the first argument is evaluated;
this allows for the function name to be computed, as well as the body.
All function definitions appear in the same namespace; defining a function
f within another function g does not automatically limit the scope
of f to g. However, local(f) makes the definition of
function f effective only within the block or other compound expression
in which local appears.
If some formal argument x_k is a quoted symbol (after evaluation), the
function defined by define does not evaluate the corresponding actual
argument. Otherwise all actual arguments are evaluated.
Examples:
define always evaluates its second argument (unless explicitly quoted).
(%i1) expr : cos(y) - sin(x); (%o1) cos(y) - sin(x) (%i2) define (F1 (x, y), expr); (%o2) F1(x, y) := cos(y) - sin(x) (%i3) F1 (a, b); (%o3) cos(b) - sin(a) (%i4) F2 (x, y) := expr; (%o4) F2(x, y) := expr (%i5) F2 (a, b); (%o5) cos(y) - sin(x)
The function defined by define may be an ordinary Maxima function or an
array function.
(%i1) define (G1 (x, y), x.y - y.x);
(%o1) G1(x, y) := x . y - y . x
(%i2) define (G2 [x, y], x.y - y.x);
(%o2) G2 := x . y - y . x
x, y
When the last or only function argument x_n is a list of one element,
the function defined by define accepts a variable number of arguments.
(%i1) define (H ([L]), '(apply ("+", L)));
(%o1) H([L]) := apply("+", L)
(%i2) H (a, b, c);
(%o2) c + b + a
When the first argument is an expression with operator funmake,
arraymake, or ev, the first argument is evaluated.
(%i1) [F : I, u : x];
(%o1) [I, x]
(%i2) funmake (F, [u]);
(%o2) I(x)
(%i3) define (funmake (F, [u]), cos(u) + 1);
(%o3) I(x) := cos(x) + 1
(%i4) define (arraymake (F, [u]), cos(u) + 1);
(%o4) I := cos(x) + 1
x
(%i5) define (foo (x, y), bar (y, x));
(%o5) foo(x, y) := bar(y, x)
(%i6) define (ev (foo (x, y)), sin(x) - cos(y));
(%o6) bar(y, x) := sin(x) - cos(y)
Categories: Function definition
Introduces a global variable into the Maxima environment.
define_variable is useful in user-written packages, which are often
translated or compiled.
define_variable carries out the following steps:
mode_declare (name, mode) declares the mode of name to
the translator. See mode_declare for a list of the possible modes.
The value_check property can be assigned to any variable which has been
defined via define_variable with a mode other than any.
The value_check property is a lambda expression or the name of a function
of one variable, which is called when an attempt is made to assign a value to
the variable. The argument of the value_check function is the would-be
assigned value.
define_variable evaluates default_value, and quotes name
and mode. define_variable returns the current value of
name, which is default_value if name was unbound before,
and otherwise it is the previous value of name.
Examples:
foo is a Boolean variable, with the initial value true.
(%i1) define_variable (foo, true, boolean);
(%o1) true
(%i2) foo;
(%o2) true
(%i3) foo: false;
(%o3) false
(%i4) foo: %pi;
translator: foo was declared with mode boolean
, but it has value: %pi
-- an error. To debug this try: debugmode(true);
(%i5) foo;
(%o5) false
bar is an integer variable, which must be prime.
(%i1) define_variable (bar, 2, integer);
(%o1) 2
(%i2) qput (bar, prime_test, value_check);
(%o2) prime_test
(%i3) prime_test (y) := if not primep(y) then
error (y, "is not prime.");
(%o3) prime_test(y) := if not primep(y)
then error(y, "is not prime.")
(%i4) bar: 1439;
(%o4) 1439
(%i5) bar: 1440;
1440 is not prime.
-- an error. To debug this try: debugmode(true);
(%i6) bar;
(%o6) 1439
baz_quux is a variable which cannot be assigned a value.
The mode any_check is like any, but any_check enables the
value_check mechanism, and any does not.
(%i1) define_variable (baz_quux, 'baz_quux, any_check);
(%o1) baz_quux
(%i2) F: lambda ([y], if y # 'baz_quux then
error ("Cannot assign to `baz_quux'."));
(%o2) lambda([y], if y # 'baz_quux
then error(Cannot assign to `baz_quux'.))
(%i3) qput (baz_quux, ''F, value_check);
(%o3) lambda([y], if y # 'baz_quux
then error(Cannot assign to `baz_quux'.))
(%i4) baz_quux: 'baz_quux;
(%o4) baz_quux
(%i5) baz_quux: sqrt(2);
Cannot assign to `baz_quux'.
-- an error. To debug this try: debugmode(true);
(%i6) baz_quux;
(%o6) baz_quux
Categories: Translation and compilation
Displays the definition of the user-defined functions f_1, …,
f_n. Each argument may be the name of a macro (defined with ::=),
an ordinary function (defined with := or define), an array
function (defined with := or define, but enclosing arguments in
square brackets [ ]), a subscripted function (defined with := or
define, but enclosing some arguments in square brackets and others in
parentheses ( )), one of a family of subscripted functions selected by a
particular subscript value, or a subscripted function defined with a constant
subscript.
dispfun (all) displays all user-defined functions as
given by the functions, arrays, and macros lists,
omitting subscripted functions defined with constant subscripts.
dispfun creates an intermediate expression label
(%t1, %t2, etc.)
for each displayed function, and assigns the function definition to the label.
In contrast, fundef returns the function definition.
dispfun quotes its arguments; the quote-quote operator ''
defeats quotation. dispfun returns the list of intermediate expression
labels corresponding to the displayed functions.
Examples:
(%i1) m(x, y) ::= x^(-y);
- y
(%o1) m(x, y) ::= x
(%i2) f(x, y) := x^(-y);
- y
(%o2) f(x, y) := x
(%i3) g[x, y] := x^(-y);
- y
(%o3) g := x
x, y
(%i4) h[x](y) := x^(-y);
- y
(%o4) h (y) := x
x
(%i5) i[8](y) := 8^(-y);
- y
(%o5) i (y) := 8
8
(%i6) dispfun (m, f, g, h, h[5], h[10], i[8]);
- y
(%t6) m(x, y) ::= x
- y
(%t7) f(x, y) := x
- y
(%t8) g := x
x, y
- y
(%t9) h (y) := x
x
1
(%t10) h (y) := --
5 y
5
1
(%t11) h (y) := ---
10 y
10
- y
(%t12) i (y) := 8
8
(%o12) [%t6, %t7, %t8, %t9, %t10, %t11, %t12]
(%i13) ''%;
- y - y - y
(%o13) [m(x, y) ::= x , f(x, y) := x , g := x ,
x, y
- y 1 1 - y
h (y) := x , h (y) := --, h (y) := ---, i (y) := 8 ]
x 5 y 10 y 8
5 10
Categories: Function definition · Display functions
Similar to map, but fullmap keeps mapping down all subexpressions
until the main operators are no longer the same.
fullmap is used by the Maxima simplifier for certain matrix
manipulations; thus, Maxima sometimes generates an error message concerning
fullmap even though fullmap was not explicitly called by the user.
Examples:
(%i1) a + b * c; (%o1) b c + a (%i2) fullmap (g, %); (%o2) g(b) g(c) + g(a) (%i3) map (g, %th(2)); (%o3) g(b c) + g(a)
Categories: Function application · Expressions
Similar to fullmap, but fullmapl only maps onto lists and
matrices.
Example:
(%i1) fullmapl ("+", [3, [4, 5]], [[a, 1], [0, -1.5]]);
(%o1) [[a + 3, 4], [4, 3.5]]
Categories: Function application · Expressions
Default value: []
functions is the list of ordinary Maxima functions
in the current session.
An ordinary function is a function constructed by
define or := and called with parentheses ().
A function may be defined at the Maxima prompt
or in a Maxima file loaded by load or batch.
Array functions (called with square brackets, e.g., F[x]) and subscripted
functions (called with square brackets and parentheses, e.g., F[x](y))
are listed by the global variable arrays, and not by functions.
Lisp functions are not kept on any list.
Examples:
(%i1) F_1 (x) := x - 100;
(%o1) F_1(x) := x - 100
(%i2) F_2 (x, y) := x / y;
x
(%o2) F_2(x, y) := -
y
(%i3) define (F_3 (x), sqrt (x));
(%o3) F_3(x) := sqrt(x)
(%i4) G_1 [x] := x - 100;
(%o4) G_1 := x - 100
x
(%i5) G_2 [x, y] := x / y;
x
(%o5) G_2 := -
x, y y
(%i6) define (G_3 [x], sqrt (x));
(%o6) G_3 := sqrt(x)
x
(%i7) H_1 [x] (y) := x^y;
y
(%o7) H_1 (y) := x
x
(%i8) functions;
(%o8) [F_1(x), F_2(x, y), F_3(x)]
(%i9) arrays;
(%o9) [G_1, G_2, G_3, H_1]
Categories: Function definition · Global variables
Returns the definition of the function f.
The argument may be
::=),
:= or define),
:= or define, but enclosing arguments in square brackets [ ]),
:= or define,
but enclosing some arguments in square brackets and others in parentheses
( )),
fundef quotes its argument;
the quote-quote operator '' defeats quotation.
fundef (f) returns the definition of f.
In contrast, dispfun (f) creates an intermediate expression label
and assigns the definition to the label.
Categories: Function definition
Returns an expression F(arg_1, ..., arg_n).
The return value is simplified, but not evaluated,
so the function F is not called, even if it exists.
funmake does not attempt to distinguish array functions from ordinary
functions; when F is the name of an array function,
funmake returns F(...)
(that is, a function call with parentheses instead of square brackets).
arraymake returns a function call with square brackets in this case.
funmake evaluates its arguments.
Examples:
funmake applied to an ordinary Maxima function.
(%i1) F (x, y) := y^2 - x^2;
2 2
(%o1) F(x, y) := y - x
(%i2) funmake (F, [a + 1, b + 1]);
(%o2) F(a + 1, b + 1)
(%i3) ''%;
2 2
(%o3) (b + 1) - (a + 1)
funmake applied to a macro.
(%i1) G (x) ::= (x - 1)/2;
x - 1
(%o1) G(x) ::= -----
2
(%i2) funmake (G, [u]);
(%o2) G(u)
(%i3) ''%;
u - 1
(%o3) -----
2
funmake applied to a subscripted function.
(%i1) H [a] (x) := (x - 1)^a;
a
(%o1) H (x) := (x - 1)
a
(%i2) funmake (H [n], [%e]);
n
(%o2) lambda([x], (x - 1) )(%e)
(%i3) ''%;
n
(%o3) (%e - 1)
(%i4) funmake ('(H [n]), [%e]);
(%o4) H (%e)
n
(%i5) ''%;
n
(%o5) (%e - 1)
funmake applied to a symbol which is not a defined function of any kind.
(%i1) funmake (A, [u]); (%o1) A(u) (%i2) ''%; (%o2) A(u)
funmake evaluates its arguments, but not the return value.
(%i1) det(a,b,c) := b^2 -4*a*c;
2
(%o1) det(a, b, c) := b - 4 a c
(%i2) (x : 8, y : 10, z : 12);
(%o2) 12
(%i3) f : det;
(%o3) det
(%i4) funmake (f, [x, y, z]);
(%o4) det(8, 10, 12)
(%i5) ''%;
(%o5) - 284
Maxima simplifies funmake's return value.
(%i1) funmake (sin, [%pi / 2]); (%o1) 1
Categories: Function application · Expressions
Defines and returns a lambda expression (that is, an anonymous function). The function may have required arguments x_1, …, x_m and/or optional arguments L, which appear within the function body as a list. The return value of the function is expr_n. A lambda expression can be assigned to a variable and evaluated like an ordinary function. A lambda expression may appear in some contexts in which a function name is expected.
When the function is evaluated, unbound local variables x_1, …,
x_m are created. lambda may appear within block or another
lambda; local variables are established each time another block or
lambda is evaluated. Local variables appear to be global to any enclosed
block or lambda. If a variable is not local, its value is the
value most recently assigned in an enclosing block or lambda, if
any, otherwise, it is the value of the variable in the global environment.
This policy may coincide with the usual understanding of "dynamic scope".
After local variables are established, expr_1 through expr_n are
evaluated in turn. The special variable %%, representing the value of
the preceding expression, is recognized. throw and catch may also
appear in the list of expressions.
return cannot appear in a lambda expression unless enclosed by
block, in which case return defines the return value of the block
and not of the lambda expression, unless the block happens to be expr_n.
Likewise, go cannot appear in a lambda expression unless enclosed by
block.
lambda quotes its arguments;
the quote-quote operator '' defeats quotation.
Examples:
(%i1) f: lambda ([x], x^2);
2
(%o1) lambda([x], x )
(%i2) f(a);
2
(%o2) a
(%i1) lambda ([x], x^2) (a);
2
(%o1) a
(%i2) apply (lambda ([x], x^2), [a]);
2
(%o2) a
(%i3) map (lambda ([x], x^2), [a, b, c, d, e]);
2 2 2 2 2
(%o3) [a , b , c , d , e ]
''.
(%i1) a: %pi$
(%i2) b: %e$
(%i3) g: lambda ([a], a*b);
(%o3) lambda([a], a b)
(%i4) b: %gamma$
(%i5) g(1/2);
%gamma
(%o5) ------
2
(%i6) g2: lambda ([a], a*''b);
(%o6) lambda([a], a %gamma)
(%i7) b: %e$
(%i8) g2(1/2);
%gamma
(%o8) ------
2
(%i1) h: lambda ([a, b], h2: lambda ([a], a*b), h2(1/2));
1
(%o1) lambda([a, b], h2 : lambda([a], a b), h2(-))
2
(%i2) h(%pi, %gamma);
%gamma
(%o2) ------
2
lambda quotes its arguments, lambda expression i below does
not define a "multiply by a" function. Such a function can be defined
via buildq, as in lambda expression i2 below.
(%i1) i: lambda ([a], lambda ([x], a*x));
(%o1) lambda([a], lambda([x], a x))
(%i2) i(1/2);
(%o2) lambda([x], a x)
(%i3) i2: lambda([a], buildq([a: a], lambda([x], a*x)));
(%o3) lambda([a], buildq([a : a], lambda([x], a x)))
(%i4) i2(1/2);
1
(%o4) lambda([x], (-) x)
2
(%i5) i2(1/2)(%pi);
%pi
(%o5) ---
2
[L] as the sole or final argument.
The arguments appear within the function body as a list.
(%i1) f : lambda ([aa, bb, [cc]], aa * cc + bb);
(%o1) lambda([aa, bb, [cc]], aa cc + bb)
(%i2) f (foo, %i, 17, 29, 256);
(%o2) [17 foo + %i, 29 foo + %i, 256 foo + %i]
(%i3) g : lambda ([[aa]], apply ("+", aa));
(%o3) lambda([[aa]], apply(+, aa))
(%i4) g (17, 29, x, y, z, %e);
(%o4) z + y + x + %e + 46
Categories: Function definition
Saves the properties associated with the symbols v_1, …, v_n,
removes any properties before evaluating other expressions,
and restores any saved properties on exit
from the block or other compound expression in which local appears.
Some declarations are implemented as properties of a symbol, including
:=, array, dependencies, atvalue,
matchdeclare, atomgrad, constant, nonscalar,
assume, and some others. The effect of local is to make such
declarations effective only within the block or other compound expression in
which local appears; otherwise such declarations are global declarations.
local can only appear in block
or in the body of a function definition or lambda expression,
and only one occurrence is permitted in each.
local quotes its arguments.
local returns done.
Example:
A local function definition.
(%i1) foo (x) := 1 - x; (%o1) foo(x) := 1 - x (%i2) foo (100); (%o2) - 99 (%i3) block (local (foo), foo (x) := 2 * x, foo (100)); (%o3) 200 (%i4) foo (100); (%o4) - 99
Categories: Function definition · Programming
Default value: false
macroexpansion controls whether the expansion (that is, the return value)
of a macro function is substituted for the macro function call.
A substitution may speed up subsequent expression evaluations,
at the cost of storing the expansion.
falseThe expansion of a macro function is not substituted for the macro function call.
expandThe first time a macro function call is evaluated,
the expansion is stored.
The expansion is not recomputed on subsequent calls;
any side effects (such as print or assignment to global variables) happen
only when the macro function call is first evaluated.
Expansion in an expression does not affect other expressions
which have the same macro function call.
displaceThe first time a macro function call is evaluated, the expansion is substituted for the call, thus modifying the expression from which the macro function was called. The expansion is not recomputed on subsequent calls; any side effects happen only when the macro function call is first evaluated. Expansion in an expression does not affect other expressions which have the same macro function call.
Examples
When macroexpansion is false,
a macro function is called every time the calling expression is evaluated,
and the calling expression is not modified.
(%i1) f (x) := h (x) / g (x);
h(x)
(%o1) f(x) := ----
g(x)
(%i2) g (x) ::= block (print ("x + 99 is equal to", x),
return (x + 99));
(%o2) g(x) ::= block(print("x + 99 is equal to", x),
return(x + 99))
(%i3) h (x) ::= block (print ("x - 99 is equal to", x),
return (x - 99));
(%o3) h(x) ::= block(print("x - 99 is equal to", x),
return(x - 99))
(%i4) macroexpansion: false;
(%o4) false
(%i5) f (a * b);
x - 99 is equal to x
x + 99 is equal to x
a b - 99
(%o5) --------
a b + 99
(%i6) dispfun (f);
h(x)
(%t6) f(x) := ----
g(x)
(%o6) [%t6]
(%i7) f (a * b);
x - 99 is equal to x
x + 99 is equal to x
a b - 99
(%o7) --------
a b + 99
When macroexpansion is expand,
a macro function is called once,
and the calling expression is not modified.
(%i1) f (x) := h (x) / g (x);
h(x)
(%o1) f(x) := ----
g(x)
(%i2) g (x) ::= block (print ("x + 99 is equal to", x),
return (x + 99));
(%o2) g(x) ::= block(print("x + 99 is equal to", x),
return(x + 99))
(%i3) h (x) ::= block (print ("x - 99 is equal to", x),
return (x - 99));
(%o3) h(x) ::= block(print("x - 99 is equal to", x),
return(x - 99))
(%i4) macroexpansion: expand;
(%o4) expand
(%i5) f (a * b);
x - 99 is equal to x
x + 99 is equal to x
a b - 99
(%o5) --------
a b + 99
(%i6) dispfun (f);
mmacroexpanded(x - 99, h(x))
(%t6) f(x) := ----------------------------
mmacroexpanded(x + 99, g(x))
(%o6) [%t6]
(%i7) f (a * b);
a b - 99
(%o7) --------
a b + 99
When macroexpansion is displace,
a macro function is called once,
and the calling expression is modified.
(%i1) f (x) := h (x) / g (x);
h(x)
(%o1) f(x) := ----
g(x)
(%i2) g (x) ::= block (print ("x + 99 is equal to", x),
return (x + 99));
(%o2) g(x) ::= block(print("x + 99 is equal to", x),
return(x + 99))
(%i3) h (x) ::= block (print ("x - 99 is equal to", x),
return (x - 99));
(%o3) h(x) ::= block(print("x - 99 is equal to", x),
return(x - 99))
(%i4) macroexpansion: displace;
(%o4) displace
(%i5) f (a * b);
x - 99 is equal to x
x + 99 is equal to x
a b - 99
(%o5) --------
a b + 99
(%i6) dispfun (f);
x - 99
(%t6) f(x) := ------
x + 99
(%o6) [%t6]
(%i7) f (a * b);
a b - 99
(%o7) --------
a b + 99
Categories: Function application · Global flags
Default value: true
When mode_checkp is true, mode_declare checks the modes
of bound variables.
Categories: Translation flags and variables
Default value: false
When mode_check_errorp is true, mode_declare calls
error.
Categories: Translation flags and variables
Default value: true
When mode_check_warnp is true, mode errors are
described.
Categories: Translation flags and variables
mode_declare is used to declare the modes of variables and functions for
subsequent translation or compilation of functions. mode_declare is
typically placed at the beginning of a function definition, at the beginning of
a Maxima script, or executed at the interactive prompt.
The arguments of mode_declare are pairs consisting of a variable and a
mode which is one of boolean, fixnum, number,
rational, or float. Each variable may also be a list of variables
all of which are declared to have the same mode.
If a variable is an array, and if every element of the array which is
referenced has a value then array (yi, complete, dim1, dim2, ...)
rather than
array(yi, dim1, dim2, ...)
should be used when first declaring the bounds of the array.
If all the elements of the array are of mode fixnum (float), use
fixnum (float) instead of complete.
Also if every element of the array is of the same mode, say m, then
mode_declare (completearray (yi), m))
should be used for efficient translation.
Numeric code using arrays might run faster by declaring the expected size of the array, as in:
mode_declare (completearray (a [10, 10]), float)
for a floating point number array which is 10 x 10.
One may declare the mode of the result of a function by using
function (f_1, f_2, ...) as an argument; here f_1, f_2,
… are the names of functions. For example the expression,
mode_declare ([function (f_1, f_2, ...)], fixnum)
declares that the values returned by f_1, f_2, … are
single-word integers.
modedeclare is a synonym for mode_declare.
Categories: Translation and compilation
A special form used with mode_declare and macros to declare, e.g.,
a list of lists of flonums, or other compound data object. The first argument
to mode_identity is a primitive value mode name as given to
mode_declare (i.e., one of float, fixnum, number,
list, or any), and the second argument is an expression which is
evaluated and returned as the value of mode_identity. However, if the
return value is not allowed by the mode declared in the first
argument, an error or warning is signalled. The important thing is
that the mode of the expression as determined by the Maxima to Lisp
translator, will be that given as the first argument, independent of
anything that goes on in the second argument.
E.g., x: 3.3; mode_identity (fixnum, x); yields an error.
mode_identity (flonum, x) returns 3.3 .
This has a number of uses, e.g., if you knew that first (l) returned a
number then you might write mode_identity (number, first (l)). However,
a more efficient way to do it would be to define a new primitive,
firstnumb (x) ::= buildq ([x], mode_identity (number, first(x)));
and use firstnumb
every time you take the first of a list of numbers.
Categories: Translation and compilation
Unbinds the function definitions of the symbols f_1, …, f_n.
The arguments may be the names of ordinary functions (created by := or
define) or macro functions (created by ::=).
remfunction (all) unbinds all function definitions.
remfunction quotes its arguments.
remfunction returns a list of the symbols for which the function
definition was unbound. false is returned in place of any symbol for
which there is no function definition.
remfunction does not apply to array functions or subscripted functions.
remarray applies to those types of functions.
Categories: Function definition
Default value: true
When savedef is true, the Maxima version of a user function is
preserved when the function is translated. This permits the definition to be
displayed by dispfun and allows the function to be edited.
When savedef is false, the names of translated functions are
removed from the functions list.
Categories: Translation flags and variables
Default value: true
When transcompile is true, translate and
translate_file generate declarations to make the translated code more
suitable for compilation.
compfile sets transcompile: true for the duration.
Categories: Translation flags and variables
Translates the user-defined functions f_1, …, f_n from the Maxima language into Lisp and evaluates the Lisp translations. Typically the translated functions run faster than the originals.
translate (all) or translate (functions) translates all
user-defined functions.
Functions to be translated should include a call to mode_declare at the
beginning when possible in order to produce more efficient code. For example:
f (x_1, x_2, ...) := block ([v_1, v_2, ...],
mode_declare (v_1, mode_1, v_2, mode_2, ...), ...)
where the x_1, x_2, … are the parameters to the function and the v_1, v_2, … are the local variables.
The names of translated functions are removed from the functions list
if savedef is false (see below) and are added to the props
lists.
Functions should not be translated unless they are fully debugged.
Expressions are assumed simplified; if they are not, correct but non-optimal
code gets generated. Thus, the user should not set the simp switch to
false which inhibits simplification of the expressions to be translated.
The switch translate, if true, causes automatic
translation of a user's function to Lisp.
Note that translated
functions may not run identically to the way they did before
translation as certain incompatibilities may exist between the Lisp
and Maxima versions. Principally, the rat function with more than
one argument and the ratvars function should not be used if any
variables are mode_declare'd canonical rational expressions (CRE).
Also the prederror: false setting
will not translate.
savedef - if true will cause the Maxima version of a user
function to remain when the function is translate'd. This permits the
definition to be displayed by dispfun and allows the function to be
edited.
transrun - if false will cause the interpreted version of all
functions to be run (provided they are still around) rather than the
translated version.
The result returned by translate is a list of the names of the
functions translated.
Categories: Translation and compilation
Translates a file of Maxima code into a file of Lisp code.
translate_file returns a list of three filenames:
the name of the Maxima file, the name of the Lisp file, and the name of file
containing additional information about the translation.
translate_file evaluates its arguments.
translate_file ("foo.mac"); load("foo.LISP") is the same as the command
batch ("foo.mac") except for certain restrictions, the use of
'' and %, for example.
translate_file (maxima_filename) translates a Maxima file
maxima_filename into a similarly-named Lisp file.
For example, foo.mac is translated into foo.LISP.
The Maxima filename may include a directory name or names,
in which case the Lisp output file is written
to the same directory from which the Maxima input comes.
translate_file (maxima_filename, lisp_filename) translates
a Maxima file maxima_filename into a Lisp file lisp_filename.
translate_file ignores the filename extension, if any, of
lisp_filename; the filename extension of the Lisp output file is always
LISP. The Lisp filename may include a directory name or names,
in which case the Lisp output file is written to the specified directory.
translate_file also writes a file of translator warning
messages of various degrees of severity.
The filename extension of this file is UNLISP.
This file may contain valuable information, though possibly obscure,
for tracking down bugs in translated code.
The UNLISP file is always written
to the same directory from which the Maxima input comes.
translate_file emits Lisp code which causes
some declarations and definitions to take effect as soon
as the Lisp code is compiled.
See compile_file for more on this topic.
See also
tr_array_as_ref
tr_bound_function_applyp,
tr_exponent
tr_file_tty_messagesp,
tr_float_can_branch_complex,
tr_function_call_default,
tr_numer,
tr_optimize_max_loop,
tr_semicompile,
tr_state_vars,
tr_warnings_get,
tr_warn_bad_function_calls
tr_warn_fexpr,
tr_warn_meval,
tr_warn_mode,
tr_warn_undeclared,
and tr_warn_undefined_variable.
Categories: Translation and compilation
Default value: true
When transrun is false will cause the interpreted
version of all functions to be run (provided they are still around)
rather than the translated version.
Categories: Translation flags and variables
Default value: true
If translate_fast_arrays is false, array references in Lisp code
emitted by translate_file are affected by tr_array_as_ref.
When tr_array_as_ref is true,
array names are evaluated,
otherwise array names appear as literal symbols in translated code.
tr_array_as_ref has no effect if translate_fast_arrays is
true.
Categories: Translation flags and variables
Default value: true
When tr_bound_function_applyp is true, Maxima gives a warning if a
bound variable (such as a function argument) is found being used as a function.
tr_bound_function_applyp does not affect the code generated in such
cases.
For example, an expression such as g (f, x) := f (x+1) will trigger
the warning message.
Categories: Translation flags and variables
Default value: false
When tr_file_tty_messagesp is true, messages generated by
translate_file during translation of a file are displayed on the console
and inserted into the UNLISP file. When false, messages about
translation of the file are only inserted into the UNLISP file.
Categories: Translation flags and variables
Default value: true
Tells the Maxima-to-Lisp translator to assume that the functions
acos, asin, asec, and acsc can return complex results.
The ostensible effect of tr_float_can_branch_complex is the following.
However, it appears that this flag has no effect on the translator output.
When it is true then acos(x) is of mode any
even if x is of mode float (as set by mode_declare).
When false then acos(x) is of mode
float if and only if x is of mode float.
Categories: Translation flags and variables
Default value: general
false means give up and call meval, expr means assume Lisp
fixed arg function. general, the default gives code good for
mexprs and mlexprs but not macros. general assures
variable bindings are correct in compiled code. In general mode, when
translating F(X), if F is a bound variable, then it assumes that
apply (f, [x]) is meant, and translates a such, with appropriate warning.
There is no need to turn this off. With the default settings, no warning
messages implies full compatibility of translated and compiled code with the
Maxima interpreter.
Categories: Translation flags and variables
Default value: false
When tr_numer is true, numer properties are used for
atoms which have them, e.g. %pi.
Categories: Translation flags and variables
Default value: 100
tr_optimize_max_loop is the maximum number of times the
macro-expansion and optimization pass of the translator will loop in
considering a form. This is to catch macro expansion errors, and
non-terminating optimization properties.
Categories: Translation flags and variables
Default value: false
When tr_semicompile is true, translate_file and
compfile output forms which will be macroexpanded but not compiled into
machine code by the Lisp compiler.
Categories: Translation flags and variables
Default value:
[transcompile, tr_semicompile, tr_warn_undeclared, tr_warn_meval, tr_warn_fexpr, tr_warn_mode, tr_warn_undefined_variable, tr_function_call_default, tr_array_as_ref,tr_numer]
The list of the switches that affect the form of the translated output. This information is useful to system people when trying to debug the translator. By comparing the translated product to what should have been produced for a given state, it is possible to track down bugs.
Categories: Translation flags and variables
Prints a list of warnings which have been given by the translator during the current translation.
Categories: Translation and compilation
Default value: true
- Gives a warning when when function calls are being made which may not be correct due to improper declarations that were made at translate time.
Categories: Translation flags and variables
Default value: compfile
- Gives a warning if any FEXPRs are encountered. FEXPRs should not normally be output in translated code, all legitimate special program forms are translated.
Categories: Translation flags and variables
Default value: compfile
- Gives a warning if the function meval gets called. If meval is
called that indicates problems in the translation.
Categories: Translation flags and variables
Default value: all
- Gives a warning when variables are assigned values inappropriate for their mode.
Categories: Translation flags and variables
Default value: compile
- Determines when to send warnings about undeclared variables to the TTY.
Categories: Translation flags and variables
Default value: all
- Gives a warning when undefined global variables are seen.
Categories: Translation flags and variables
Translates the Maxima file filename into Lisp, executes the Lisp compiler, and, if the translation and compilation succeed, loads the compiled code into Maxima.
compile_file returns a list of the names of four files: the original
Maxima file, the Lisp translation, notes on translation, and the compiled code.
If the compilation fails, the fourth item is false.
Some declarations and definitions take effect as soon
as the Lisp code is compiled (without loading the compiled code).
These include functions defined with the := operator,
macros define with the ::= operator,
alias, declare,
define_variable, mode_declare,
and
infix, matchfix,
nofix, postfix, prefix,
and compfile.
Assignments and function calls are not evaluated until the compiled code is
loaded. In particular, within the Maxima file, assignments to the translation
flags (tr_numer, etc.) have no effect on the translation.
filename may not contain :lisp statements.
compile_file evaluates its arguments.
Categories: Translation and compilation
When translating a file of Maxima code
to Lisp, it is important for the translator to know which functions it
sees in the file are to be called as translated or compiled functions,
and which ones are just Maxima functions or undefined. Putting this
declaration at the top of the file, lets it know that although a symbol
does which does not yet have a Lisp function value, will have one at
call time. (MFUNCTION-CALL fn arg1 arg2 ...) is generated when
the translator does not know fn is going to be a Lisp function.
Categories: Translation and compilation
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| 37.1 Lisp and Maxima | ||
| 37.2 Garbage Collection | ||
| 37.3 Introduction to Program Flow | ||
| 37.4 Functions and Variables for Program Flow |
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Maxima is written in Lisp, and it is easy to access Lisp functions and variables
from Maxima and vice versa. Lisp and Maxima symbols are distinguished by a
naming convention. A Lisp symbol which begins with a dollar sign $
corresponds to a Maxima symbol without the dollar sign.
A Maxima symbol which begins with a question mark ? corresponds to a Lisp
symbol without the question mark. For example, the Maxima symbol foo
corresponds to the Lisp symbol $FOO, while the Maxima symbol ?foo
corresponds to the Lisp symbol FOO. Note that ?foo is written
without a space between ? and foo; otherwise it might be mistaken
for describe ("foo").
Hyphen -, asterisk *, or other special characters in Lisp symbols
must be escaped by backslash \ where they appear in Maxima code. For
example, the Lisp identifier *foo-bar* is written ?\*foo\-bar\*
in Maxima.
Lisp code may be executed from within a Maxima session. A single line of Lisp
(containing one or more forms) may be executed by the special command
:lisp. For example,
(%i1) :lisp (foo $x $y)
calls the Lisp function foo with Maxima variables x and y
as arguments. The :lisp construct can appear at the interactive prompt
or in a file processed by batch or demo, but not in a file
processed by load, batchload,
translate_file, or compile_file.
The function to_lisp opens an interactive Lisp session.
Entering (to-maxima) closes the Lisp session and returns to Maxima.
Lisp functions and variables which are to be visible in Maxima as functions and
variables with ordinary names (no special punctuation) must have Lisp names
beginning with the dollar sign $.
Maxima is case-sensitive, distinguishing between lowercase and uppercase letters in identifiers. There are some rules governing the translation of names between Lisp and Maxima.
$foo, $FOO, and $Foo
all correspond to Maxima foo. But this is because $foo,
$FOO and $Foo are converted by the Lisp reader by default to the
Lisp symbol $FOO.
|$FOO| and |$foo| correspond to Maxima foo and FOO,
respectively.
|$Foo| corresponds to Maxima Foo.
The #$ Lisp macro allows the use of Maxima expressions in Lisp code.
#$expr$ expands to a Lisp expression equivalent to the Maxima
expression expr.
(msetq $foo #$[x, y]$)
This has the same effect as entering
(%i1) foo: [x, y];
The Lisp function displa prints an expression in Maxima format.
(%i1) :lisp #$[x, y, z]$ ((MLIST SIMP) $X $Y $Z) (%i1) :lisp (displa '((MLIST SIMP) $X $Y $Z)) [x, y, z] NIL
Functions defined in Maxima are not ordinary Lisp functions. The Lisp function
mfuncall calls a Maxima function. For example:
(%i1) foo(x,y) := x*y$ (%i2) :lisp (mfuncall '$foo 'a 'b) ((MTIMES SIMP) A B)
Some Lisp functions are shadowed in the Maxima package, namely the following.
complement continue // float functionp array exp listen signum atan asin acos asinh acosh atanh tanh cosh sinh tan break gcd
Categories: Programming
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Symbolic computation tends to create a good deal of garbage (temporary or intermediate results that are eventually not used), and effective handling of this can be crucial to successful completion of some programs.
Under GCL, on UNIX systems where the mprotect system call is available (including SUN OS 4.0 and some variants of BSD) a stratified garbage collection is available. This limits the collection to pages which have been recently written to. See the GCL documentation under ALLOCATE and GBC. At the Lisp level doing (setq si::*notify-gbc* t) will help you determine which areas might need more space.
For other Lisps that run Maxima, we refer the reader to the documentation for that Lisp on how to control GC.
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Maxima provides a do loop for iteration, as well as more primitive
constructs such as go.
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Prints the call stack, that is, the list of functions which called the currently active function.
backtrace() prints the entire call stack.
backtrace (n) prints the n most recent
functions, including the currently active function.
backtrace can be called from a script, a function, or the interactive
prompt (not only in a debugging context).
Examples:
backtrace() prints the entire call stack.
(%i1) h(x) := g(x/7)$
(%i2) g(x) := f(x-11)$
(%i3) f(x) := e(x^2)$
(%i4) e(x) := (backtrace(), 2*x + 13)$
(%i5) h(10);
#0: e(x=4489/49)
#1: f(x=-67/7)
#2: g(x=10/7)
#3: h(x=10)
9615
(%o5) ----
49
backtrace (n) prints the n most recent
functions, including the currently active function.
(%i1) h(x) := (backtrace(1), g(x/7))$
(%i2) g(x) := (backtrace(1), f(x-11))$
(%i3) f(x) := (backtrace(1), e(x^2))$
(%i4) e(x) := (backtrace(1), 2*x + 13)$
(%i5) h(10);
#0: h(x=10)
#0: g(x=10/7)
#0: f(x=-67/7)
#0: e(x=4489/49)
9615
(%o5) ----
49
Categories: Debugging
The do statement is used for performing iteration. Due to its
great generality the do statement will be described in two parts.
First the usual form will be given which is analogous to that used in
several other programming languages (Fortran, Algol, PL/I, etc.); then
the other features will be mentioned.
There are three variants of this form that differ only in their terminating conditions. They are:
for variable: initial_value step increment
thru limit do body
for variable: initial_value step increment
while condition do body
for variable: initial_value step increment
unless condition do body
(Alternatively, the step may be given after the termination condition
or limit.)
initial_value, increment, limit, and body can be any
expressions. If the increment is 1 then "step 1" may be omitted.
The execution of the do statement proceeds by first assigning
the initial_value to the variable (henceforth called the
control-variable). Then: (1) If the control-variable has exceeded the
limit of a thru specification, or if the condition of the
unless is true, or if the condition of the while
is false then the do terminates. (2) The body is
evaluated. (3) The increment is added to the control-variable. The
process from (1) to (3) is performed repeatedly until the termination
condition is satisfied. One may also give several termination
conditions in which case the do terminates when any of them is
satisfied.
In general the thru test is satisfied when the control-variable
is greater than the limit if the increment was
non-negative, or when the control-variable is less than the
limit if the increment was negative. The
increment and limit may be non-numeric expressions as
long as this inequality can be determined. However, unless the
increment is syntactically negative (e.g. is a negative number)
at the time the do statement is input, Maxima assumes it will
be positive when the do is executed. If it is not positive,
then the do may not terminate properly.
Note that the limit, increment, and termination condition are
evaluated each time through the loop. Thus if any of these involve
much computation, and yield a result that does not change during all
the executions of the body, then it is more efficient to set a
variable to their value prior to the do and use this variable in the
do form.
The value normally returned by a do statement is the atom
done. However, the function return may be used inside
the body to exit the do prematurely and give it any
desired value. Note however that a return within a do
that occurs in a block will exit only the do and not the
block. Note also that the go function may not be used
to exit from a do into a surrounding block.
The control-variable is always local to the do and thus any
variable may be used without affecting the value of a variable with
the same name outside of the do. The control-variable is unbound
after the do terminates.
(%i1) for a:-3 thru 26 step 7 do display(a)$
a = - 3
a = 4
a = 11
a = 18
a = 25
(%i1) s: 0$ (%i2) for i: 1 while i <= 10 do s: s+i; (%o2) done (%i3) s; (%o3) 55
Note that the condition while i <= 10
is equivalent to unless i > 10 and also thru 10.
(%i1) series: 1$
(%i2) term: exp (sin (x))$
(%i3) for p: 1 unless p > 7 do
(term: diff (term, x)/p,
series: series + subst (x=0, term)*x^p)$
(%i4) series;
7 6 5 4 2
x x x x x
(%o4) -- - --- - -- - -- + -- + x + 1
90 240 15 8 2
which gives 8 terms of the Taylor series for e^sin(x).
(%i1) poly: 0$
(%i2) for i: 1 thru 5 do
for j: i step -1 thru 1 do
poly: poly + i*x^j$
(%i3) poly;
5 4 3 2
(%o3) 5 x + 9 x + 12 x + 14 x + 15 x
(%i4) guess: -3.0$
(%i5) for i: 1 thru 10 do
(guess: subst (guess, x, 0.5*(x + 10/x)),
if abs (guess^2 - 10) < 0.00005 then return (guess));
(%o5) - 3.162280701754386
This example computes the negative square root of 10 using the
Newton- Raphson iteration a maximum of 10 times. Had the convergence
criterion not been met the value returned would have been done.
Instead of always adding a quantity to the control-variable one
may sometimes wish to change it in some other way for each iteration.
In this case one may use next expression instead of
step increment. This will cause the control-variable to be set to
the result of evaluating expression each time through the loop.
(%i6) for count: 2 next 3*count thru 20 do display (count)$
count = 2
count = 6
count = 18
As an alternative to for variable: value ...do...
the syntax for variable from value ...do... may be
used. This permits the from value to be placed after the
step or next value or after the termination condition.
If from value is omitted then 1 is used as the initial
value.
Sometimes one may be interested in performing an iteration where the control-variable is never actually used. It is thus permissible to give only the termination conditions omitting the initialization and updating information as in the following example to compute the square-root of 5 using a poor initial guess.
(%i1) x: 1000$ (%i2) thru 20 do x: 0.5*(x + 5.0/x)$ (%i3) x; (%o3) 2.23606797749979 (%i4) sqrt(5), numer; (%o4) 2.23606797749979
If it is desired one may even omit the termination conditions entirely
and just give do body which will continue to evaluate the
body indefinitely. In this case the function return
should be used to terminate execution of the do.
(%i1) newton (f, x):= ([y, df, dfx], df: diff (f ('x), 'x),
do (y: ev(df), x: x - f(x)/y,
if abs (f (x)) < 5e-6 then return (x)))$
(%i2) sqr (x) := x^2 - 5.0$
(%i3) newton (sqr, 1000);
(%o3) 2.236068027062195
(Note that return, when executed, causes the current value of x to
be returned as the value of the do. The block is exited and this
value of the do is returned as the value of the block because the
do is the last statement in the block.)
One other form of the do is available in Maxima. The syntax is:
for variable in list end_tests do body
The elements of list are any expressions which will successively
be assigned to the variable on each iteration of the
body. The optional termination tests end_tests can be
used to terminate execution of the do; otherwise it will
terminate when the list is exhausted or when a return is
executed in the body. (In fact, list may be any
non-atomic expression, and successive parts are taken.)
(%i1) for f in [log, rho, atan] do ldisp(f(1))$
(%t1) 0
(%t2) rho(1)
%pi
(%t3) ---
4
(%i4) ev(%t3,numer);
(%o4) 0.78539816
Categories: Programming
Evaluates expr_1, …, expr_n one by one and
returns [expr_n] (a list) if no error occurs. If an
error occurs in the evaluation of any argument, errcatch
prevents the error from propagating and
returns the empty list [] without evaluating any more arguments.
errcatch
is useful in batch files where one suspects an error might occur which
would terminate the batch if the error weren't caught.
Categories: Programming
Evaluates and prints expr_1, …, expr_n,
and then causes an error return to top level Maxima
or to the nearest enclosing errcatch.
The variable error is set to a list describing the error.
The first element of error is a format string, which merges all the
strings among the arguments expr_1, …, expr_n,
and the remaining elements are the values of any non-string arguments.
errormsg() formats and prints error.
This is effectively reprinting the most recent error message.
Categories: Programming
Default value: 10
error_size modifies error messages according to the size of expressions
which appear in them. If the size of an expression (as determined by the Lisp
function ERROR-SIZE) is greater than error_size, the expression is
replaced in the message by a symbol, and the symbol is assigned the expression.
The symbols are taken from the list error_syms.
Otherwise, the expression is smaller than error_size, and the expression
is displayed in the message.
See also error and error_syms.
Example:
The size of U, as determined by ERROR-SIZE, is 24.
(%i1) U: (C^D^E + B + A)/(cos(X-1) + 1)$
(%i2) error_size: 20$
(%i3) error ("Example expression is", U);
Example expression is errexp1
-- an error. Quitting. To debug this try debugmode(true);
(%i4) errexp1;
E
D
C + B + A
(%o4) --------------
cos(X - 1) + 1
(%i5) error_size: 30$
(%i6) error ("Example expression is", U);
E
D
C + B + A
Example expression is --------------
cos(X - 1) + 1
-- an error. Quitting. To debug this try debugmode(true);
Categories: Debugging · Display flags and variables
Default value: [errexp1, errexp2, errexp3]
In error messages, expressions larger than error_size are replaced by
symbols, and the symbols are set to the expressions. The symbols are taken from
the list error_syms. The first too-large expression is replaced by
error_syms[1], the second by error_syms[2], and so on.
If there are more too-large expressions than there are elements of
error_syms, symbols are constructed automatically, with the n-th
symbol equivalent to concat ('errexp, n).
See also error and error_size.
Categories: Debugging · Display flags and variables
Reprints the most recent error message.
The variable error holds the message,
and errormsg formats and prints it.
Categories: Programming
Default value: true
When false the output of error messages is suppressed.
The option variable errormsg can not be set in a block to a local
value. The global value of errormsg is always present.
(%i1) errormsg; (%o1) true (%i2) sin(a,b); sin: wrong number of arguments. -- an error. To debug this try: debugmode(true); (%i3) errormsg:false; (%o3) false (%i4) sin(a,b); -- an error. To debug this try: debugmode(true);
The option variable errormsg can not be set in a block to a local value.
(%i1) f(bool):=block([errormsg:bool], print ("value of errormsg is",errormsg))$
(%i2) errormsg:true;
(%o2) true
(%i3) f(false);
value of errormsg is true
(%o3) true
(%i4) errormsg:false;
(%o4) false
(%i5) f(true);
value of errormsg is false
(%o5) false
Categories: Programming
Used in iterations. See do for a description of
Maxima's iteration facilities.
Categories: Programming
is used within a block to transfer control to the statement
of the block which is tagged with the argument to go. To tag a
statement, precede it by an atomic argument as another statement in
the block. For example:
block ([x], x:1, loop, x+1, ..., go(loop), ...)
The argument to go must be the name of a tag appearing in the same
block. One cannot use go to transfer to tag in a block
other than the one containing the go.
Categories: Programming
Represents conditional evaluation. Various forms of if expressions are
recognized.
if cond_1 then expr_1 else expr_0
evaluates to expr_1 if cond_1 evaluates to true,
otherwise the expression evaluates to expr_0.
The command if cond_1 then expr_1 elseif cond_2 then
expr_2 elseif ... else expr_0 evaluates to expr_k if
cond_k is true and all preceding conditions are false. If
none of the conditions are true, the expression evaluates to
expr_0.
A trailing else false is assumed if else is missing. That is,
the command if cond_1 then expr_1 is equivalent to
if cond_1 then expr_1 else false, and the command
if cond_1 then expr_1 elseif ... elseif cond_n then
expr_n is equivalent to if cond_1 then expr_1 elseif
... elseif cond_n then expr_n else false.
The alternatives expr_0, …, expr_n may be any Maxima
expressions, including nested if expressions. The alternatives are
neither simplified nor evaluated unless the corresponding condition is
true.
The conditions cond_1, …, cond_n are expressions which
potentially or actually evaluate to true or false.
When a condition does not actually evaluate to true or false,
the behavior of if is governed by the global flag prederror.
When prederror is true, it is an error if any evaluated condition
does not evaluate to true or false. Otherwise, conditions which
do not evaluate to true or false are accepted, and the result is
a conditional expression.
Among other elements, conditions may comprise relational and logical operators as follows.
Operation Symbol Type less than < relational infix less than <= or equal to relational infix equality (syntactic) = relational infix negation of = # relational infix equality (value) equal relational function negation of equal notequal relational function greater than >= or equal to relational infix greater than > relational infix and and logical infix or or logical infix not not logical prefix
Categories: Programming · Predicate functions
Returns an expression whose leading operator is the same as that of the
expressions expr_1, …, expr_n but whose subparts are the
results of applying f to the corresponding subparts of the expressions.
f is either the name of a function of n arguments or is a
lambda form of n arguments.
maperror - if false will cause all of the mapping
functions to (1) stop when they finish going down the shortest
expr_i if not all of the expr_i are of the same length and
(2) apply f to [expr_1, expr_2, …] if the
expr_i are not all the same type of object. If maperror
is true then an error message will be given in the above two
instances.
One of the uses of this function is to map a function (e.g.
partfrac) onto each term of a very large expression where it ordinarily
wouldn't be possible to use the function on the entire expression due to an
exhaustion of list storage space in the course of the computation.
(%i1) map(f,x+a*y+b*z);
(%o1) f(b z) + f(a y) + f(x)
(%i2) map(lambda([u],partfrac(u,x)),x+1/(x^3+4*x^2+5*x+2));
1 1 1
(%o2) ----- - ----- + -------- + x
x + 2 x + 1 2
(x + 1)
(%i3) map(ratsimp, x/(x^2+x)+(y^2+y)/y);
1
(%o3) y + ----- + 1
x + 1
(%i4) map("=",[a,b],[-0.5,3]);
(%o4) [a = - 0.5, b = 3]
Categories: Function application
Returns true if and only if expr is treated by the mapping
routines as an atom. "Mapatoms" are atoms, numbers
(including rational numbers), and subscripted variables.
Categories: Predicate functions
Default value: true
When maperror is false, causes all of the mapping functions,
for example
map (f, expr_1, expr_2, ...)
to (1) stop when they finish going down the shortest expr_i if not all of the expr_i are of the same length and (2) apply f to [expr_1, expr_2, …] if the expr_i are not all the same type of object.
If maperror is true then an error message
is displayed in the above two instances.
Categories: Function application
Default value: true
When mapprint is true, various information messages from
map, maplist, and fullmap are produced in certain
situations. These include situations where map would use
apply, or map is truncating on the shortest list.
If mapprint is false, these messages are suppressed.
Categories: Function application
Returns a list of the applications of f to the parts of the expressions expr_1, …, expr_n. f is the name of a function, or a lambda expression.
maplist differs from map(f, expr_1, ..., expr_n)
which returns an expression with the same main operator as expr_i has
(except for simplifications and the case where map does an apply).
Categories: Function application
Default value: false
When prederror is true, an error message is displayed whenever the
predicate of an if statement or an is function fails to evaluate
to either true or false.
If false, unknown is returned
instead in this case. The prederror: false mode is not supported in
translated code;
however, maybe is supported in translated code.
Categories: Programming · Predicate functions
May be used to exit explicitly from the current block, while,
for or do loop bringing its argument. It therefore can be compared
with the return statement found in other programming languages but it yields
one difference: In maxima only returns from the current block, not from the entire
function it was called in. In this aspect it more closely resembles the break
statement from C.
(%i1) for i:1 thru 10 do o:i;
(%o1) done
(%i2) for i:1 thru 10 do if i=3 then return(i);
(%o2) 3
(%i3) for i:1 thru 10 do
(
block([i],
i:3,
return(i)
),
return(8)
);
(%o3) 8
(%i4) block([i],
i:4,
block([o],
o:5,
return(o)
),
return(i),
return(10)
);
(%o4) 4
See also for, while, do and block.
Categories: Programming
Recursively applies f to expr, in a top down manner. This is most useful when complete factorization is desired, for example:
(%i1) exp:(a^2+2*a+1)*y + x^2$
(%i2) scanmap(factor,exp);
2 2
(%o2) (a + 1) y + x
Note the way in which scanmap applies the given function
factor to the constituent subexpressions of expr; if
another form of expr is presented to scanmap then the
result may be different. Thus, %o2 is not recovered when
scanmap is applied to the expanded form of exp:
(%i3) scanmap(factor,expand(exp));
2 2
(%o3) a y + 2 a y + y + x
Here is another example of the way in which scanmap recursively
applies a given function to all subexpressions, including exponents:
(%i4) expr : u*v^(a*x+b) + c$
(%i5) scanmap('f, expr);
f(f(f(a) f(x)) + f(b))
(%o5) f(f(f(u) f(f(v) )) + f(c))
scanmap (f, expr, bottomup) applies f to expr in a
bottom-up manner. E.g., for undefined f,
scanmap(f,a*x+b) ->
f(a*x+b) -> f(f(a*x)+f(b)) -> f(f(f(a)*f(x))+f(b))
scanmap(f,a*x+b,bottomup) -> f(a)*f(x)+f(b)
-> f(f(a)*f(x))+f(b) ->
f(f(f(a)*f(x))+f(b))
In this case, you get the same answer both ways.
Categories: Function application
Evaluates expr and throws the value back to the most recent
catch. throw is used with catch as a nonlocal return
mechanism.
Categories: Programming
See do.
Categories: Programming
Applies the function f to each one of the elements of the outer product a_1 cross a_2 … cross a_n.
f is the name of a function of n arguments or a lambda expression of n arguments. Each argument a_k may be a list or nested list, or a matrix, or any other kind of expression.
The outermap return value is a nested structure. Let x be the
return value. Then x has the same structure as the first list, nested
list, or matrix argument, x[i_1]...[i_m] has the same structure as
the second list, nested list, or matrix argument,
x[i_1]...[i_m][j_1]...[j_n] has the same structure as the third
list, nested list, or matrix argument, and so on, where m, n,
… are the numbers of indices required to access the elements of each
argument (one for a list, two for a matrix, one or more for a nested list).
Arguments which are not lists or matrices have no effect on the structure of
the return value.
Note that the effect of outermap is different from that of applying
f to each one of the elements of the outer product returned by
cartesian_product. outermap preserves the structure of the
arguments in the return value, while cartesian_product does not.
outermap evaluates its arguments.
See also map, maplist, and apply.
Examples:
Elementary examples of outermap.
To show the argument combinations more clearly, F is left undefined.
(%i1) outermap (F, [a, b, c], [1, 2, 3]);
(%o1) [[F(a, 1), F(a, 2), F(a, 3)], [F(b, 1), F(b, 2), F(b, 3)],
[F(c, 1), F(c, 2), F(c, 3)]]
(%i2) outermap (F, matrix ([a, b], [c, d]), matrix ([1, 2], [3, 4]));
[ [ F(a, 1) F(a, 2) ] [ F(b, 1) F(b, 2) ] ]
[ [ ] [ ] ]
[ [ F(a, 3) F(a, 4) ] [ F(b, 3) F(b, 4) ] ]
(%o2) [ ]
[ [ F(c, 1) F(c, 2) ] [ F(d, 1) F(d, 2) ] ]
[ [ ] [ ] ]
[ [ F(c, 3) F(c, 4) ] [ F(d, 3) F(d, 4) ] ]
(%i3) outermap (F, [a, b], x, matrix ([1, 2], [3, 4]));
[ F(a, x, 1) F(a, x, 2) ] [ F(b, x, 1) F(b, x, 2) ]
(%o3) [[ ], [ ]]
[ F(a, x, 3) F(a, x, 4) ] [ F(b, x, 3) F(b, x, 4) ]
(%i4) outermap (F, [a, b], matrix ([1, 2]), matrix ([x], [y]));
[ [ F(a, 1, x) ] [ F(a, 2, x) ] ]
(%o4) [[ [ ] [ ] ],
[ [ F(a, 1, y) ] [ F(a, 2, y) ] ]
[ [ F(b, 1, x) ] [ F(b, 2, x) ] ]
[ [ ] [ ] ]]
[ [ F(b, 1, y) ] [ F(b, 2, y) ] ]
(%i5) outermap ("+", [a, b, c], [1, 2, 3]);
(%o5) [[a + 1, a + 2, a + 3], [b + 1, b + 2, b + 3],
[c + 1, c + 2, c + 3]]
A closer examination of the outermap return value. The first, second,
and third arguments are a matrix, a list, and a matrix, respectively.
The return value is a matrix.
Each element of that matrix is a list,
and each element of each list is a matrix.
(%i1) arg_1 : matrix ([a, b], [c, d]);
[ a b ]
(%o1) [ ]
[ c d ]
(%i2) arg_2 : [11, 22];
(%o2) [11, 22]
(%i3) arg_3 : matrix ([xx, yy]);
(%o3) [ xx yy ]
(%i4) xx_0 : outermap (lambda ([x, y, z], x / y + z), arg_1,
arg_2, arg_3);
[ [ a a ] [ a a ] ]
[ [[ xx + -- yy + -- ], [ xx + -- yy + -- ]] ]
[ [ 11 11 ] [ 22 22 ] ]
(%o4) Col 1 = [ ]
[ [ c c ] [ c c ] ]
[ [[ xx + -- yy + -- ], [ xx + -- yy + -- ]] ]
[ [ 11 11 ] [ 22 22 ] ]
[ [ b b ] [ b b ] ]
[ [[ xx + -- yy + -- ], [ xx + -- yy + -- ]] ]
[ [ 11 11 ] [ 22 22 ] ]
Col 2 = [ ]
[ [ d d ] [ d d ] ]
[ [[ xx + -- yy + -- ], [ xx + -- yy + -- ]] ]
[ [ 11 11 ] [ 22 22 ] ]
(%i5) xx_1 : xx_0 [1][1];
[ a a ] [ a a ]
(%o5) [[ xx + -- yy + -- ], [ xx + -- yy + -- ]]
[ 11 11 ] [ 22 22 ]
(%i6) xx_2 : xx_0 [1][1] [1];
[ a a ]
(%o6) [ xx + -- yy + -- ]
[ 11 11 ]
(%i7) xx_3 : xx_0 [1][1] [1] [1][1];
a
(%o7) xx + --
11
(%i8) [op (arg_1), op (arg_2), op (arg_3)];
(%o8) [matrix, [, matrix]
(%i9) [op (xx_0), op (xx_1), op (xx_2)];
(%o9) [matrix, [, matrix]
outermap preserves the structure of the arguments in the return value,
while cartesian_product does not.
(%i1) outermap (F, [a, b, c], [1, 2, 3]);
(%o1) [[F(a, 1), F(a, 2), F(a, 3)], [F(b, 1), F(b, 2), F(b, 3)],
[F(c, 1), F(c, 2), F(c, 3)]]
(%i2) setify (flatten (%));
(%o2) {F(a, 1), F(a, 2), F(a, 3), F(b, 1), F(b, 2), F(b, 3),
F(c, 1), F(c, 2), F(c, 3)}
(%i3) map (lambda ([L], apply (F, L)),
cartesian_product ({a, b, c}, {1, 2, 3}));
(%o3) {F(a, 1), F(a, 2), F(a, 3), F(b, 1), F(b, 2), F(b, 3),
F(c, 1), F(c, 2), F(c, 3)}
(%i4) is (equal (%, %th (2)));
(%o4) true
Categories: Function application
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 38.1 Source Level Debugging | ||
| 38.2 Keyword Commands | ||
| 38.3 Functions and Variables for Debugging |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Maxima has a built-in source level debugger. The user can set a breakpoint at a function, and then step line by line from there. The call stack may be examined, together with the variables bound at that level.
The command :help or :h shows the list of debugger commands.
(In general,
commands may be abbreviated if the abbreviation is unique. If not
unique, the alternatives will be listed.)
Within the debugger, the user can also use any ordinary Maxima
functions to examine, define, and manipulate variables and expressions.
A breakpoint is set by the :br command at the Maxima prompt.
Within the debugger,
the user can advance one line at a time using the :n ("next") command.
The :bt ("backtrace") command shows a list of stack frames.
The :r ("resume") command exits the debugger and continues with
execution. These commands are demonstrated in the example below.
(%i1) load ("/tmp/foobar.mac");
(%o1) /tmp/foobar.mac
(%i2) :br foo
Turning on debugging debugmode(true)
Bkpt 0 for foo (in /tmp/foobar.mac line 1)
(%i2) bar (2,3);
Bkpt 0:(foobar.mac 1)
/tmp/foobar.mac:1::
(dbm:1) :bt <-- :bt typed here gives a backtrace
#0: foo(y=5)(foobar.mac line 1)
#1: bar(x=2,y=3)(foobar.mac line 9)
(dbm:1) :n <-- Here type :n to advance line
(foobar.mac 2)
/tmp/foobar.mac:2::
(dbm:1) :n <-- Here type :n to advance line
(foobar.mac 3)
/tmp/foobar.mac:3::
(dbm:1) u; <-- Investigate value of u
28
(dbm:1) u: 33; <-- Change u to be 33
33
(dbm:1) :r <-- Type :r to resume the computation
(%o2) 1094
The file /tmp/foobar.mac is the following:
foo(y) := block ([u:y^2], u: u+3, u: u^2, u); bar(x,y) := ( x: x+2, y: y+2, x: foo(y), x+y);
USE OF THE DEBUGGER THROUGH EMACS
If the user is running the code under GNU emacs in a shell window (dbl shell), or is running the graphical interface version, Xmaxima, then if he stops at a break point, he will see his current position in the source file which will be displayed in the other half of the window, either highlighted in red, or with a little arrow pointing at the right line. He can advance single lines at a time by typing M-n (Alt-n).
Under Emacs you should run in a dbl shell, which requires the
dbl.el file in the elisp directory.
Make sure you install the elisp files or add the Maxima elisp directory to
your path:
e.g., add the following to your `.emacs' file or the `site-init.el'
(setq load-path (cons "/usr/share/maxima/5.9.1/emacs" load-path)) (autoload 'dbl "dbl")
then in emacs
M-x dbl
should start a shell window in which you can run programs, for example Maxima, gcl, gdb etc. This shell window also knows about source level debugging, and display of source code in the other window.
The user may set a break point at a certain line of the
file by typing C-x space. This figures out which function
the cursor is in, and then it sees which line of that function
the cursor is on. If the cursor is on, say, line 2 of foo, then it will
insert in the other window the command, ":br foo 2", to
break foo at its second line. To have this enabled, the user must have
maxima-mode.el turned on in the window in which the file foobar.mac is
visiting. There are additional commands available in that file window, such as
evaluating the function into the Maxima, by typing Alt-Control-x.
Categories: Debugging
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Keyword commands are special keywords which are not interpreted as Maxima
expressions. A keyword command can be entered at the Maxima prompt or the
debugger prompt, although not at the break prompt.
Keyword commands start with a colon, ':'.
For example, to evaluate a Lisp form you
may type :lisp followed by the form to be evaluated.
(%i1) :lisp (+ 2 3) 5
The number of arguments taken depends on the particular command. Also,
you need not type the whole command, just enough to be unique among
the break keywords. Thus :br would suffice for :break.
The keyword commands are listed below.
:break F nSet a breakpoint in function F at line offset n
from the beginning of the function.
If F is given as a string, then it is assumed to be
a file, and n is the offset from the beginning of the file.
The offset is optional. If not given, it is assumed to be zero
(first line of the function or file).
:btPrint a backtrace of the stack frames
:continueContinue the computation
:deleteDelete the specified breakpoints, or all if none are specified
:disableDisable the specified breakpoints, or all if none are specified
:enableEnable the specified breakpoints, or all if none are specified
:frame nPrint stack frame n, or the current frame if none is specified
:helpPrint help on a debugger command, or all commands if none is specified
:infoPrint information about item
:lisp some-formEvaluate some-form as a Lisp form
:lisp-quiet some-formEvaluate Lisp form some-form without any output
:nextLike :step, except :next steps over function calls
:quitQuit the current debugger level without completing the computation
:resumeContinue the computation
:stepContinue the computation until it reaches a new source line
:topReturn to the Maxima prompt (from any debugger level) without completing the computation
Categories: Debugging
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Default value: false
When a Maxima error occurs, Maxima will start the debugger if debugmode
is true. The user may enter commands to examine the call stack, set
breakpoints, step through Maxima code, and so on. See debugging for a
list of debugger commands.
Enabling debugmode will not catch Lisp errors.
Categories: Debugging · Global flags
Default value: false
When refcheck is true, Maxima prints a message
each time a bound variable is used for the first time in a
computation.
Categories: Evaluation · Console interaction · Global flags
Default value: false
If setcheck is set to a list of variables (which can
be subscripted),
Maxima prints a message whenever the variables, or
subscripted occurrences of them, are bound with the
ordinary assignment operator :, the :: assignment
operator, or function argument binding,
but not the function assignment := nor the macro assignment
::= operators.
The message comprises the name of the variable and the
value it is bound to.
setcheck may be set to all or true thereby
including all variables.
Each new assignment of setcheck establishes a new list of variables to
check, and any variables previously assigned to setcheck are forgotten.
The names assigned to setcheck must be quoted if they would otherwise
evaluate to something other than themselves.
For example, if x, y, and z are already bound, then enter
setcheck: ['x, 'y, 'z]$
to put them on the list of variables to check.
No printout is generated when a
variable on the setcheck list is assigned to itself, e.g., X: 'X.
Categories: Console interaction · Global flags
Default value: false
When setcheckbreak is true,
Maxima will present a break prompt
whenever a variable on the setcheck list is assigned a new value.
The break occurs before the assignment is carried out.
At this point, setval holds the value to which the variable is
about to be assigned.
Hence, one may assign a different value by assigning to setval.
Categories: Console interaction · Global flags
Holds the value to which a variable is about to be set when
a setcheckbreak occurs.
Hence, one may assign a different value by assigning to setval.
See also setcheck and setcheckbreak.
Categories: Console interaction · Global variables
Given functions f_1, …, f_n, timer puts each one on the
list of functions for which timing statistics are collected.
timer(f)$ timer(g)$ puts f and then g onto the list;
the list accumulates from one call to the next.
timer(all) puts all user-defined functions (as named by the global
variable functions) on the list of timed functions.
With no arguments, timer returns the list of timed functions.
Maxima records how much time is spent executing each function
on the list of timed functions.
timer_info returns the timing statistics, including the
average time elapsed per function call, the number of calls, and the
total time elapsed.
untimer removes functions from the list of timed functions.
timer quotes its arguments.
f(x) := x^2$ g:f$ timer(g)$ does not put f on the timer list.
If trace(f) is in effect, then timer(f) has no effect;
trace and timer cannot both be in effect at the same time.
See also timer_devalue.
Categories: Debugging
Given functions f_1, …, f_n,
untimer removes each function from the timer list.
With no arguments, untimer removes all functions currently on the timer
list.
After untimer (f) is executed, timer_info (f) still returns
previously collected timing statistics,
although timer_info() (with no arguments) does not
return information about any function not currently on the timer list.
timer (f) resets all timing statistics to zero
and puts f on the timer list again.
Categories: Debugging
Default value: false
When timer_devalue is true, Maxima subtracts from each timed
function the time spent in other timed functions. Otherwise, the time reported
for each function includes the time spent in other functions.
Note that time spent in untimed functions is not subtracted from the
total time.
See also timer and timer_info.
Categories: Debugging · Global flags
Given functions f_1, ..., f_n, timer_info returns a matrix
containing timing information for each function.
With no arguments, timer_info returns timing information for
all functions currently on the timer list.
The matrix returned by timer_info contains the function name,
time per function call, number of function calls, total time,
and gctime, which meant "garbage collection time" in the original Macsyma
but is now always zero.
The data from which timer_info constructs its return value
can also be obtained by the get function:
get(f, 'calls); get(f, 'runtime); get(f, 'gctime);
See also timer.
Categories: Debugging
Given functions f_1, …, f_n, trace instructs Maxima to
print out debugging information whenever those functions are called.
trace(f)$ trace(g)$ puts f and then g onto the list of
functions to be traced; the list accumulates from one call to the next.
trace(all) puts all user-defined functions (as named by the global
variable functions) on the list of functions to be traced.
With no arguments,
trace returns a list of all the functions currently being traced.
The untrace function disables tracing.
See also trace_options.
trace quotes its arguments. Thus,
f(x) := x^2$ g:f$ trace(g)$ does not put f on the trace list.
When a function is redefined, it is removed from the timer list.
Thus after timer(f)$ f(x) := x^2$,
function f is no longer on the timer list.
If timer (f) is in effect, then trace (f) has no effect;
trace and timer can't both be in effect for the same function.
Categories: Debugging
Sets the trace options for function f.
Any previous options are superseded.
trace_options (f, ...) has no effect unless trace (f)
is also called (either before or after trace_options).
trace_options (f) resets all options to their default values.
The option keywords are:
noprint
Do not print a message at function entry and exit.
break
Put a breakpoint before the function is entered,
and after the function is exited. See break.
lisp_print
Display arguments and return values as Lisp objects.
info
Print -> true at function entry and exit.
errorcatch
Catch errors, giving the option to signal an error,
retry the function call, or specify a return value.
Trace options are specified in two forms. The presence of the option
keyword alone puts the option into effect unconditionally.
(Note that option foo is not put into effect by specifying
foo: true or a similar form; note also that keywords need not
be quoted.) Specifying the option keyword with a predicate
function makes the option conditional on the predicate.
The argument list to the predicate function is always
[level, direction, function, item] where level is the recursion
level for the function, direction is either enter or exit,
function is the name of the function, and item is the argument
list (on entering) or the return value (on exiting).
Here is an example of unconditional trace options:
(%i1) ff(n) := if equal(n, 0) then 1 else n * ff(n - 1)$ (%i2) trace (ff)$ (%i3) trace_options (ff, lisp_print, break)$ (%i4) ff(3);
Here is the same function, with the break option conditional
on a predicate:
(%i5) trace_options (ff, break(pp))$
(%i6) pp (level, direction, function, item) := block (print (item),
return (function = 'ff and level = 3 and direction = exit))$
(%i7) ff(6);
Categories: Debugging
Given functions f_1, …, f_n,
untrace disables tracing enabled by the trace function.
With no arguments, untrace disables tracing for all functions.
untrace returns a list of the functions for which
it disabled tracing.
Categories: Debugging
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| 39.1 Introduction to alt-display | ||
| 39.2 Functions and Variables for alt-display |
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The alt-display package provides a means to change the way that Maxima displays its output. The *alt-display1d* and *alt-display2d* Lisp hooks were introduced to Maxima in 2002, but were not easily accessible from the Maxima REPL until the introduction of this package.
The package provides a general purpose function to define alternative display functions, and a separate function to set the display function. The package also provides customized display functions to produce output in TeX, Texinfo, XML and all three output formats within Texinfo.
Here is a sample session:
(%i1) load("alt-display.mac")$
(%i2) set_alt_display(2,tex_display)$
(%i3) x/(x^2+y^2) = 1;
\mbox{\tt\red({\it \%o_3}) \black}$${{x}\over{y^2+x^2}}=1$$
(%i4) set_alt_display(2,mathml_display)$
(%i5) x/(x^2+y^2) = 1;
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi>
<mfenced separators=""><msub><mi>%o</mi> <mn>5</mn></msub>
<mo>,</mo><mfrac><mrow><mi>x</mi> </mrow> <mrow><msup><mrow>
<mi>y</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup><mrow>
<mi>x</mi> </mrow> <mn>2</mn> </msup> </mrow></mfrac> <mo>=</mo>
<mn>1</mn> </mfenced> </math>
(%i6) set_alt_display(2,multi_display_for_texinfo)$
(%i7) x/(x^2+y^2) = 1;
@iftex
@tex
\mbox{\tt\red({\it \%o_7}) \black}$${{x}\over{y^2+x^2}}=1$$
@end tex
@end iftex
@ifhtml
@html
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi>
<mfenced separators=""><msub><mi>%o</mi> <mn>7</mn></msub>
<mo>,</mo><mfrac><mrow><mi>x</mi> </mrow> <mrow><msup><mrow>
<mi>y</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup><mrow>
<mi>x</mi> </mrow> <mn>2</mn> </msup> </mrow></mfrac> <mo>=</mo>
<mn>1</mn> </mfenced> </math>
@end html
@end ifhtml
@ifinfo
@example
(%o7) x/(y^2+x^2) = 1
@end example
@end ifinfo
If the alternative display function causes an error, the error is
trapped and the display function is reset to the default display. In the
following example, the error function is set to display the
output. This throws an error, which is handled by resetting the
2d-display to the default.
(%i8) set_alt_display(2,?error)$
(%i9) x;
Error in *alt-display2d*.
Messge: Condition designator ((MLABEL) $%O9 $X) is not of type (OR SYMBOL STRING
FUNCTION).
*alt-display2d* reset to nil.
-- an error. To debug this try: debugmode(true);
(%i10) x;
(%o10) x
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This function is similar to define: it evaluates its arguments
and expands into a function definition. The function is a
function of a single input input. For convenience, a substitution
is applied to expr after evaluation, to provide easy access to
Lisp variable names.
Set a time-stamp on each prompt:
(%i1) load("alt-display.mac")$
(%i2) display2d:false$
(%i3) define_alt_display(time_stamp(x),
block([alt_display1d:false,alt_display2d:false],
prompt_prefix:printf(false,"~a~%",timedate()),
displa(x)));
(%o3) time_stamp(x):=block([simp:false],
block([?\*alt\-display1d\*:false,?\*alt\-display2d\*:false],
?\*prompt\-prefix\*:printf(false,"~a~%",timedate()),
?displa(x)))
(%i4) set_alt_display(1,time_stamp);
(%o4) done
2014-01-07 13:41:50-05:00
(%i5)
The input line %i3 defines time_stamp using
define_alt_display. The output line %o3 shows that the
Maxima variable names alt_display1d, alt_display2d and
prompt_prefix have been replaced by their Lisp translations, as
has displa been replaced by ?displa (the display
function).
The display variables alt_display1d and alt_display2d are
both bound to false in the body of time_stamp to prevent
an infinite recursion in displa.
Categories: Package alt-display
This is an alias for the default 1-d display function. It may be used as an alternative 1-d or 2-d display function.
(%i1) load("alt-display.mac")$
(%i2) set_alt_display(2,info_display);
(%o2) done
(%i3) x/y;
(%o3) x/y
Categories: Package alt-display
Produces MathML output.
(%i1) load("alt-display.mac")$
(%i2) set_alt_display(2,mathml_display);
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi>
<mfenced separators=""><msub><mi>%o</mi> <mn>2</mn></msub>
<mo>,</mo><mi>done</mi> </mfenced> </math>
Categories: Package alt-display
Produces TeX output.
(%i2) set_alt_display(2,tex_display);
\mbox{\tt\red({\it \%o_2}) \black}$$\mathbf{done}$$
(%i3) x/(x^2+y^2);
\mbox{\tt\red({\it \%o_3}) \black}$${{x}\over{y^2+x^2}}$$
Categories: Package alt-display
Produces Texinfo output using all three display functions.
(%i2) set_alt_display(2,multi_display_for_texinfo)$
(%i3) x/(x^2+y^2);
@iftex
@tex
\mbox{\tt\red({\it \%o_3}) \black}$${{x}\over{y^2+x^2}}$$
@end tex
@end iftex
@ifhtml
@html
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi>
<mfenced separators=""><msub><mi>%o</mi> <mn>3</mn></msub>
<mo>,</mo><mfrac><mrow><mi>x</mi> </mrow> <mrow><msup><mrow>
<mi>y</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup><mrow>
<mi>x</mi> </mrow> <mn>2</mn> </msup> </mrow></mfrac> </mfenced> </math>
@end html
@end ifhtml
@ifinfo
@example
(%o3) x/(y^2+x^2)
@end example
@end ifinfo
Categories: Package alt-display
Resets the prompt prefix and suffix to the empty string, and sets both 1-d and 2-d display functions to the default.
Categories: Package alt-display
The input num is the display to set; it may be either 1 or 2. The
second input display-function is the display function to use. The
display function may be either a Maxima function or a lambda
expression.
Here is an example where the display function is a lambda
expression; it just displays the result as TeX.
(%i1) load("alt-display.mac")$
(%i2) set_alt_display(2, lambda([form], tex(?caddr(form))))$
(%i3) integrate(exp(-t^2),t,0,inf);
$${{\sqrt{\pi}}\over{2}}$$
A user-defined display function should take care that it prints its output. A display function that returns a string will appear to display nothing, nor cause any errors.
Categories: Package alt-display
Set the prompt prefix or suffix to expr. The input fix must
evaluate to one of prefix, suffix, general,
prolog or epilog. The input expr must evaluate to
either a string or false; if false, the fix is reset
to the default value.
(%i1) load("alt-display.mac")$
(%i2) set_prompt('prefix,printf(false,"It is now: ~a~%",timedate()))$
It is now: 2014-01-07 15:23:23-05:00
(%i3)
The following example shows the effect of each option, except
prolog. Note that the epilog prompt is printed as Maxima
closes down. The general is printed between the end of input and
the output, unless the input line ends in $.
Here is an example to show where the prompt strings are placed.
(%i1) load("alt-display.mac")$
(%i2) set_prompt(prefix,"<<prefix>> ",suffix,"<<suffix>> ",general,
printf(false,"<<general>>~%"),epilog,printf(false,"<<epilog>>~%"));
(%o2) done
<<prefix>> (%i3) <<suffix>> x/y;
<<general>>
x
(%o3) -
y
<<prefix>> (%i4) <<suffix>> quit();
<<general>>
<<epilog>>
Here is an example that shows how to colorize the input and output when Maxima is running in a terminal or terminal emulator like Emacs(8).
Each prompt string starts with the ASCII escape character (27) followed by an open square bracket (91); each string ends with a lower-case m (109). The webpages http://misc.flogisoft.com/bash/tip_colors_and_formatting and http://www.tldp.org/HOWTO/Bash-Prompt-HOWTO/x329.html provide information on how to use control strings to set the terminal colors.
Categories: Package alt-display
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| 40.1 Introduction to asympa | ||
| 40.2 Functions and variables for asympa |
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asympa is a package for asymptotic analysis. The package contains
simplification functions for asymptotic analysis, including the "big O"
and "little o" functions that are widely used in complexity analysis and
numerical analysis.
load ("asympa") loads this package.
Categories: Package asympa
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| 41.1 Functions and Variables for augmented_lagrangian |
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Returns an approximate minimum of the expression FOM with respect to the variables xx, holding the constraints C equal to zero. yy is a list of initial guesses for xx. The method employed is the augmented Lagrangian method (see Refs [1] and [2]).
grad, if present, is the gradient of FOM with respect to xx, represented as a list of expressions, one for each variable in xx. If not present, the gradient is constructed automatically.
FOM and each element of grad, if present, must be ordinary expressions, not names of functions or lambda expressions.
optional_args represents additional arguments,
specified as symbol = value.
The optional arguments recognized are:
niterNumber of iterations of the augmented Lagrangian algorithm
lbfgs_toleranceTolerance supplied to LBFGS
iprintIPRINT parameter (a list of two integers which controls verbosity) supplied to LBFGS
%lambdaInitial value of %lambda to be used for calculating the augmented Lagrangian
This implementation minimizes the augmented Lagrangian by applying the limited-memory BFGS (LBFGS) algorithm, which is a quasi-Newton algorithm.
load(augmented_lagrangian) loads this function.
See also lbfgs
References:
[1] http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/constrained/nonlinearcon/auglag.html
[2] http://www.cs.ubc.ca/spider/ascher/542/chap10.pdf
Examples:
(%i1) load (lbfgs);
(%o1) /maxima/share/lbfgs/lbfgs.mac
(%i2) load (augmented_lagrangian);
(%o2)
/maxima/share/contrib/augmented_lagrangian.mac
(%i3) FOM: x^2 + 2*y^2;
2 2
(%o3) 2 y + x
(%i4) xx: [x, y];
(%o4) [x, y]
(%i5) C: [x + y - 1];
(%o5) [y + x - 1]
(%i6) yy: [1, 1];
(%o6) [1, 1]
(%i7) augmented_lagrangian_method(FOM, xx, C, yy, iprint=[-1,0]);
(%o7) [[x = 0.66665984108002, y = 0.33334027245545],
%lambda = [- 1.333337940892525]]
Same example as before, but this time the gradient is supplied as an argument.
(%i1) load (lbfgs)$
(%i2) load (augmented_lagrangian)$
(%i3) FOM: x^2 + 2*y^2;
2 2
(%o3) 2 y + x
(%i4) FOM: x^2 + 2*y^2;
2 2
(%o4) 2 y + x
(%i5) xx: [x, y];
(%o5) [x, y]
(%i6) grad : [2*x, 4*y];
(%o6) [2 x, 4 y]
(%i7) C: [x + y - 1];
(%o7) [y + x - 1]
(%i8) yy: [1, 1];
(%o8) [1, 1]
(%i9) augmented_lagrangian_method ([FOM, grad], xx, C, yy,
iprint = [-1, 0]);
(%o9) [[x = 0.666659841080025, y = .3333402724554462],
%lambda = [- 1.333337940892543]]
Categories: Numerical methods · Optimization · Share packages · Package augmented_lagrangian
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| 42.1 Functions and Variables for Bernstein |
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Provided k is not a negative integer, the Bernstein polynomials
are defined by bernstein_poly(k,n,x) = binomial(n,k) x^k
(1-x)^(n-k); for a negative integer k, the Bernstein polynomial
bernstein_poly(k,n,x) vanishes. When either k or n are
non integers, the option variable bernstein_explicit
controls the expansion of the Bernstein polynomials into its explicit form;
example:
(%i1) load(bernstein)$
(%i2) bernstein_poly(k,n,x);
(%o2) bernstein_poly(k, n, x)
(%i3) bernstein_poly(k,n,x), bernstein_explicit : true;
n - k k
(%o3) binomial(n, k) (1 - x) x
The Bernstein polynomials have both a gradef property and an integrate property:
(%i4) diff(bernstein_poly(k,n,x),x);
(%o4) (bernstein_poly(k - 1, n - 1, x)
- bernstein_poly(k, n - 1, x)) n
(%i5) integrate(bernstein_poly(k,n,x),x);
(%o5)
k + 1
hypergeometric([k + 1, k - n], [k + 2], x) binomial(n, k) x
----------------------------------------------------------------
k + 1
For numeric inputs, both real and complex, the Bernstein polynomials evaluate to a numeric result:
(%i6) bernstein_poly(5,9, 1/2 + %i);
39375 %i 39375
(%o6) -------- + -----
128 256
(%i7) bernstein_poly(5,9, 0.5b0 + %i);
(%o7) 3.076171875b2 %i + 1.5380859375b2
To use bernstein_poly, first load("bernstein").
Default value: false
When either k or n are non integers, the option variable
bernstein_explicit controls the expansion of bernstein(k,n,x)
into its explicit form; example:
(%i1) bernstein_poly(k,n,x);
(%o1) bernstein_poly(k, n, x)
(%i2) bernstein_poly(k,n,x), bernstein_explicit : true;
n - k k
(%o2) binomial(n, k) (1 - x) x
When both k and n are explicitly integers, bernstein(k,n,x)
always expands to its explicit form.
The multibernstein polynomial multibernstein_poly ([k1, k2, ...,
kp], [n1, n2, ..., np], [x1, x2, ..., xp]) is the product of
bernstein polynomials bernstein_poly(k1, n1, x1)
bernstein_poly(k2, n2, x2) ... bernstein_poly(kp, np, xp).
To use multibernstein_poly, first load("bernstein").
Return the n-th order uniform Bernstein polynomial approximation for the
function (x1, x2, ..., xn) |--> f.
Examples
(%i1) bernstein_approx(f(x),[x], 2);
2 1 2
(%o1) f(1) x + 2 f(-) (1 - x) x + f(0) (1 - x)
2
(%i2) bernstein_approx(f(x,y),[x,y], 2);
2 2 1 2
(%o2) f(1, 1) x y + 2 f(-, 1) (1 - x) x y
2
2 2 1 2
+ f(0, 1) (1 - x) y + 2 f(1, -) x (1 - y) y
2
1 1 1 2
+ 4 f(-, -) (1 - x) x (1 - y) y + 2 f(0, -) (1 - x) (1 - y) y
2 2 2
2 2 1 2
+ f(1, 0) x (1 - y) + 2 f(-, 0) (1 - x) x (1 - y)
2
2 2
+ f(0, 0) (1 - x) (1 - y)
To use bernstein_approx, first load("bernstein").
Express the polynomial e exactly as a linear combination of multi-variable
Bernstein polynomials.
(%i1) bernstein_expand(x*y+1,[x,y]); (%o1) 2 x y + (1 - x) y + x (1 - y) + (1 - x) (1 - y) (%i2) expand(%); (%o2) x y + 1
Maxima signals an error when the first argument isn't a polynomial.
To use bernstein_expand, first load("bernstein").
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The package bitwise provides functions that allow to manipulate
bits of integer constants. As always maxima attempts to simplify the result
of the operation if the actual value of a constant isn't known considering
attributes that might be known for the variables, see the declare
mechanism.
| 43.1 Functions and Variables for bitwise |
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Inverts all bits of a signed integer. The result of this action reads
-int - 1.
(%i1) load("bitwise")$
(%i2) bit_not(i);
(%o2) bit_not(i)
(%i3) bit_not(bit_not(i));
(%o3) i
(%i4) bit_not(3);
(%o4) - 4
(%i5) bit_not(100);
(%o5) - 101
(%i6) bit_not(-101);
(%o6) 100
Categories: Number theory Binary operations Package bitwise
This function calculates a bitwise and of two or more signed integers.
(%i1) load("bitwise")$
(%i2) bit_and(i,i);
(%o2) i
(%i3) bit_and(i,i,i);
(%o3) i
(%i4) bit_and(1,3);
(%o4) 1
(%i5) bit_and(-7,7);
(%o5) 1
If it is known if one of the parameters to bit_and is even this information
is taken into consideration by the function.
(%i1) load("bitwise")$
(%i2) declare(e,even,o,odd);
(%o2) done
(%i3) bit_and(1,e);
(%o3) 0
(%i4) bit_and(1,o);
(%o4) 1
Categories: Number theory Binary operations Package bitwise
This function calculates a bitwise or of two or more signed integers.
(%i1) load("bitwise")$
(%i2) bit_or(i,i);
(%o2) i
(%i3) bit_or(i,i,i);
(%o3) i
(%i4) bit_or(1,3);
(%o4) 3
(%i5) bit_or(-7,7);
(%o5) - 1
If it is known if one of the parameters to bit_or is even this information
is taken into consideration by the function.
(%i1) load("bitwise")$
(%i2) declare(e,even,o,odd);
(%o2) done
(%i3) bit_or(1,e);
(%o3) e + 1
(%i4) bit_or(1,o);
(%o4) o
Categories: Number theory Binary operations Package bitwise
This function calculates a bitwise or of two or more signed integers.
(%i1) load("bitwise")$
(%i2) bit_xor(i,i);
(%o2) 0
(%i3) bit_xor(i,i,i);
(%o3) i
(%i4) bit_xor(1,3);
(%o4) 2
(%i5) bit_xor(-7,7);
(%o5) - 2
If it is known if one of the parameters to bit_xor is even this information
is taken into consideration by the function.
(%i1) load("bitwise")$
(%i2) declare(e,even,o,odd);
(%o2) done
(%i3) bit_xor(1,e);
(%o3) e + 1
(%i4) bit_xor(1,o);
(%o4) o - 1
Categories: Number theory Binary operations Package bitwise
This function shifts all bits of the signed integer int to the left by
nBits bits. The width of the integer is extended by nBits for
this process. The result of bit_lsh therefore is int * 2.
(%i1) load("bitwise")$
(%i2) bit_lsh(0,1);
(%o2) 0
(%i3) bit_lsh(1,0);
(%o3) 1
(%i4) bit_lsh(1,1);
(%o4) 2
(%i5) bit_lsh(1,i);
(%o5) bit_lsh(1, i)
(%i6) bit_lsh(-3,1);
(%o6) - 6
(%i7) bit_lsh(-2,1);
(%o7) - 4
Categories: Number theory Binary operations Package bitwise
This function shifts all bits of the signed integer int to the right by
nBits bits. The width of the integer is reduced by nBits for
this process.
(%i1) load("bitwise")$
(%i2) bit_rsh(0,1);
(%o2) 0
(%i3) bit_rsh(2,0);
(%o3) 2
(%i4) bit_rsh(2,1);
(%o4) 1
(%i5) bit_rsh(2,2);
(%o5) 0
(%i6) bit_rsh(-3,1);
(%o6) - 2
(%i7) bit_rsh(-2,1);
(%o7) - 1
(%i8) bit_rsh(-2,2);
(%o8) - 1
Categories: Number theory Binary operations Package bitwise
determines how many bits a variable needs to be long in order to store the
number int. This function only operates on positive numbers.
(%i1) load("bitwise")$
(%i2) bit_length(0);
(%o2) 0
(%i3) bit_length(1);
(%o3) 1
(%i4) bit_length(7);
(%o4) 3
(%i5) bit_length(8);
(%o5) 4
Categories: Number theory Binary operations Package bitwise
determines if bits nBit is set in the signed integer int.
(%i1) load("bitwise")$
(%i2) bit_onep(85,0);
(%o2) true
(%i3) bit_onep(85,1);
(%o3) false
(%i4) bit_onep(85,2);
(%o4) true
(%i5) bit_onep(85,3);
(%o5) false
(%i6) bit_onep(85,100);
(%o6) false
(%i7) bit_onep(i,100);
(%o7) bit_onep(i, 100)
For signed numbers the sign bit is interpreted to be more than nBit to the
left of the leftmost bit of int that reads 1.
(%i1) load("bitwise")$
(%i2) bit_onep(-2,0);
(%o2) false
(%i3) bit_onep(-2,1);
(%o3) true
(%i4) bit_onep(-2,2);
(%o4) true
(%i5) bit_onep(-2,3);
(%o5) true
(%i6) bit_onep(-2,4);
(%o6) true
If it is known if the number to be tested is even this information is taken into consideration by the function.
(%i1) load("bitwise")$
(%i2) declare(e,even,o,odd);
(%o2) done
(%i3) bit_onep(e,0);
(%o3) false
(%i4) bit_onep(o,0);
(%o4) true
Categories: Number theory Binary operations Package bitwise
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| 44.1 Functions and Variables for bode |
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Function to draw Bode gain plots.
Examples (1 through 7 from
http://www.swarthmore.edu/NatSci/echeeve1/Ref/Bode/BodeHow.html,
8 from Ron Crummett):
(%i1) load("bode")$
(%i2) H1 (s) := 100 * (1 + s) / ((s + 10) * (s + 100))$
(%i3) bode_gain (H1 (s), [w, 1/1000, 1000])$
(%i4) H2 (s) := 1 / (1 + s/omega0)$
(%i5) bode_gain (H2 (s), [w, 1/1000, 1000]), omega0 = 10$
(%i6) H3 (s) := 1 / (1 + s/omega0)^2$
(%i7) bode_gain (H3 (s), [w, 1/1000, 1000]), omega0 = 10$
(%i8) H4 (s) := 1 + s/omega0$
(%i9) bode_gain (H4 (s), [w, 1/1000, 1000]), omega0 = 10$
(%i10) H5 (s) := 1/s$
(%i11) bode_gain (H5 (s), [w, 1/1000, 1000])$
(%i12) H6 (s) := 1/((s/omega0)^2 + 2 * zeta * (s/omega0) + 1)$
(%i13) bode_gain (H6 (s), [w, 1/1000, 1000]),
omega0 = 10, zeta = 1/10$
(%i14) H7 (s) := (s/omega0)^2 + 2 * zeta * (s/omega0) + 1$
(%i15) bode_gain (H7 (s), [w, 1/1000, 1000]),
omega0 = 10, zeta = 1/10$
(%i16) H8 (s) := 0.5 / (0.0001 * s^3 + 0.002 * s^2 + 0.01 * s)$
(%i17) bode_gain (H8 (s), [w, 1/1000, 1000])$
To use this function write first load("bode"). See also bode_phase.
Categories: Plotting · Package bode
Function to draw Bode phase plots.
Examples (1 through 7 from
http://www.swarthmore.edu/NatSci/echeeve1/Ref/Bode/BodeHow.html,
8 from Ron Crummett):
(%i1) load("bode")$
(%i2) H1 (s) := 100 * (1 + s) / ((s + 10) * (s + 100))$
(%i3) bode_phase (H1 (s), [w, 1/1000, 1000])$
(%i4) H2 (s) := 1 / (1 + s/omega0)$
(%i5) bode_phase (H2 (s), [w, 1/1000, 1000]), omega0 = 10$
(%i6) H3 (s) := 1 / (1 + s/omega0)^2$
(%i7) bode_phase (H3 (s), [w, 1/1000, 1000]), omega0 = 10$
(%i8) H4 (s) := 1 + s/omega0$
(%i9) bode_phase (H4 (s), [w, 1/1000, 1000]), omega0 = 10$
(%i10) H5 (s) := 1/s$
(%i11) bode_phase (H5 (s), [w, 1/1000, 1000])$
(%i12) H6 (s) := 1/((s/omega0)^2 + 2 * zeta * (s/omega0) + 1)$
(%i13) bode_phase (H6 (s), [w, 1/1000, 1000]),
omega0 = 10, zeta = 1/10$
(%i14) H7 (s) := (s/omega0)^2 + 2 * zeta * (s/omega0) + 1$
(%i15) bode_phase (H7 (s), [w, 1/1000, 1000]),
omega0 = 10, zeta = 1/10$
(%i16) H8 (s) := 0.5 / (0.0001 * s^3 + 0.002 * s^2 + 0.01 * s)$
(%i17) bode_phase (H8 (s), [w, 1/1000, 1000])$
(%i18) block ([bode_phase_unwrap : false],
bode_phase (H8 (s), [w, 1/1000, 1000]));
(%i19) block ([bode_phase_unwrap : true],
bode_phase (H8 (s), [w, 1/1000, 1000]));
To use this function write first load("bode"). See also bode_gain.
Categories: Plotting · Package bode
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| 45.1 Introduction to celine |
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Maxima implementation of Sister Celine's method. Barton Willis wrote this code. It is released under the Creative Commons CC0 license.
Celine's method is described in Sections 4.1-4.4 of the book "A=B", by Marko Petkovsek, Herbert S. Wilf, and Doron Zeilberger. This book is available at http://www.math.rutgers.edu/~zeilberg/AeqB.pdf
Let f = F(n,k). The function celine returns a set of recursion relations for F of the form
p_0(n) * fff(n,k) + p_1(n) * fff(n+1,k) + ... + p_p(n) * fff(n+p,k+q),
where p_0 through p_p are polynomials. If Maxima is unable to determine that sum(sum(a(i,j) * F(n+i,k+j),i,0,p),j,0,q) / F(n,k) is a rational function of n and k, celine returns the empty set. When f involves parameters (variables other than n or k), celine might make assumptions about these parameters. Using 'put' with a key of 'proviso,' Maxima saves these assumptions on the input label.
To use this function, first load the package integer_sequence, opsubst, and to_poly_solve.
Examples:
(%i1) load("integer_sequence")$
(%i2) load("opsubst")$
(%i3) load("to_poly_solve")$
(%i4) load("celine")$
(%i5) celine(n!,n,k,1,0);
(%o5) {fff(n + 1, k) - n fff(n, k) - fff(n, k)}
Verification that this result is correct:
(%i1) load("integer_sequence")$
(%i2) load("opsubst")$
(%i3) load("to_poly_solve")$
(%i4) load("celine")$
(%i5) g1:{fff(n+1,k)-n*fff(n,k)-fff(n,k)};
(%o5) {fff(n + 1, k) - n fff(n, k) - fff(n, k)}
(%i6) ratsimp(minfactorial(first(g1))),fff(n,k) := n!;
(%o6) 0
An example with parameters including the test that the result of the example is correct:
(%i1) load("integer_sequence")$
(%i2) load("opsubst")$
(%i3) load("to_poly_solve")$
(%i4) load("celine")$
(%i5) e : pochhammer(a,k) * pochhammer(-k,n) / (pochhammer(b,k));
(a) (- k)
k n
(%o5) -----------
(b)
k
(%i6) recur : celine(e,n,k,2,1);
(%o6) {fff(n + 2, k + 1) - fff(n + 2, k) - b fff(n + 1, k + 1)
+ n ((- fff(n + 1, k + 1)) + 2 fff(n + 1, k) - a fff(n, k)
- fff(n, k)) + a (fff(n + 1, k) - fff(n, k)) + 2 fff(n + 1, k)
2
- n fff(n, k)}
(%i7) /* Test this result for correctness */
(%i8) first(%), fff(n,k) := ''(e)$
(%i9) makefact(makegamma(%))$
(%o9) 0
(%i10) minfactorial(factor(minfactorial(factor(%))));
The proviso data suggests that setting a = b may result in a lower order recursion which is shown by the following example:
(%i1) load("integer_sequence")$
(%i2) load("opsubst")$
(%i3) load("to_poly_solve")$
(%i4) load("celine")$
(%i5) e : pochhammer(a,k) * pochhammer(-k,n) / (pochhammer(b,k));
(a) (- k)
k n
(%o5) -----------
(b)
k
(%i6) recur : celine(e,n,k,2,1);
(%o6) {fff(n + 2, k + 1) - fff(n + 2, k) - b fff(n + 1, k + 1)
+ n ((- fff(n + 1, k + 1)) + 2 fff(n + 1, k) - a fff(n, k)
- fff(n, k)) + a (fff(n + 1, k) - fff(n, k)) + 2 fff(n + 1, k)
2
- n fff(n, k)}
(%i7) get('%,'proviso);
(%o7) false
(%i8) celine(subst(b=a,e),n,k,1,1);
(%o8) {fff(n + 1, k + 1) - fff(n + 1, k) + n fff(n, k)
+ fff(n, k)}
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| 46.1 Functions and Variables for clebsch_gordan |
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Compute the Clebsch-Gordan coefficient <j1, j2, m1, m2 | j, m>.
Categories: clebsch_gordan
Compute Racah's V coefficient (computed in terms of a related Clebsch-Gordan coefficient).
Categories: clebsch_gordan
Compute Racah's W coefficient (computed in terms of a Wigner 6j symbol)
Categories: clebsch_gordan
Compute Wigner's 3j symbol (computed in terms of a related Clebsch-Gordan coefficient).
Categories: clebsch_gordan
Compute Wigner's 6j symbol.
Categories: clebsch_gordan
Compute Wigner's 9j symbol.
Categories: clebsch_gordan
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| 47.1 Introduction to cobyla | ||
| 47.2 Functions and Variables for cobyla | ||
| 47.3 Examples for cobyla |
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fmin_cobyla is a Common Lisp translation (via f2cl) of the
Fortran constrained optimization routine COBYLA by Powell[1][2][3].
COBYLA minimizes an objective function F(X) subject to M inequality constraints of the form g(X) >= 0 on X, where X is a vector of variables that has N components.
Equality constraints g(X)=0 can often be implemented by a pair of inequality constraints g(X)>=0 and -g(X)>= 0. Maxima's interface to COBYLA allows equality constraints and internally converts the equality constraints to a pair of inequality constraints.
The algorithm employs linear approximations to the objective and constraint functions, the approximations being formed by linear interpolation at N+1 points in the space of the variables. The interpolation points are regarded as vertices of a simplex. The parameter RHO controls the size of the simplex and it is reduced automatically from RHOBEG to RHOEND. For each RHO the subroutine tries to achieve a good vector of variables for the current size, and then RHO is reduced until the value RHOEND is reached. Therefore RHOBEG and RHOEND should be set to reasonable initial changes to and the required accuracy in the variables respectively, but this accuracy should be viewed as a subject for experimentation because it is not guaranteed. The routine treats each constraint individually when calculating a change to the variables, rather than lumping the constraints together into a single penalty function. The name of the subroutine is derived from the phrase Constrained Optimization BY Linear Approximations.
References:
[1] Fortran Code is from http://plato.asu.edu/sub/nlores.html#general
[2] M. J. D. Powell, "A direct search optimization method that models the objective and constraint functions by linear interpolation," in Advances in Optimization and Numerical Analysis, eds. S. Gomez and J.-P. Hennart (Kluwer Academic: Dordrecht, 1994), p. 51-67.
[3] M. J. D. Powell, "Direct search algorithms for optimization calculations," Acta Numerica 7, 287-336 (1998). Also available as University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Numerical Analysis Group, Report NA1998/04 from http://www.damtp.cam.ac.uk/user/na/reports.html
Categories: Numerical methods · Optimization · Share packages · Package cobyla
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Returns an approximate minimum of the expression F with respect to the variables X, subject to an optional set of constraints. Y is a list of initial guesses for X.
F must be an ordinary expressions, not names of functions or lambda expressions.
optional_args represents additional arguments,
specified as symbol = value.
The optional arguments recognized are:
constraintsList of inequality and equality constraints that must be satisfied by
X. The inequality constraints must be actual inequalities of
the form g(X) >= h(X) or g(X) <=
h(X). The equality constraints must be of the form
g(X) = h(X).
rhobegInitial value of the internal RHO variable which controls the size of simplex. (Defaults to 1.0)
rhoend The desired final value rho parameter. It is approximately the accuracy in the variables. (Defaults to 1d-6.)
iprintVerbose output level. (Defaults to 0)
maxfunThe maximum number of function evaluations. (Defaults to 1000).
On return, a vector is given:
var = value for each of the
variables listed in X.
load(fmin_cobyla) loads this function.
This function is identical to fmin_cobyla, except that bigfloat
operations are used, and the default value for rhoend is
10^(fpprec/2).
See fmin_cobyla for more information.
load(bf_fmin_cobyla) loads this function.
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Minimize x1*x2 with 1-x1^2-x2^2 >= 0. The theoretical solution is x1 = 1/sqrt(2), x2 = -1/sqrt(2).
(%i1) load(fmin_cobyla)$
(%i2) fmin_cobyla(x1*x2, [x1, x2], [1,1],
constraints = [x1^2+x2^2<=1], iprint=1);
Normal return from subroutine COBYLA
NFVALS = 66 F =-5.000000E-01 MAXCV = 1.999845E-12
X = 7.071058E-01 -7.071077E-01
(%o2) [[x1 = 0.70710584934848, x2 = - 0.7071077130248],
- 0.49999999999926, [[-1.999955756559757e-12],[]], 66]
There are additional examples in the share/cobyla/ex directory.
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| 48.1 Introduction to contrib_ode | ||
| 48.2 Functions and Variables for contrib_ode | ||
| 48.3 Possible improvements to contrib_ode | ||
| 48.4 Test cases for contrib_ode | ||
| 48.5 References for contrib_ode |
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Maxima's ordinary differential equation (ODE) solver ode2 solves
elementary linear ODEs of first and second order. The function
contrib_ode extends ode2 with additional methods for linear
and non-linear first order ODEs and linear homogeneous second order ODEs.
The code is still under development and the calling sequence may change
in future releases. Once the code has stabilized it may be
moved from the contrib directory and integrated into Maxima.
This package must be loaded with the command load('contrib_ode)
before use.
The calling convention for contrib_ode is identical to ode2.
It takes
three arguments: an ODE (only the left hand side need be given if the
right hand side is 0), the dependent variable, and the independent
variable. When successful, it returns a list of solutions.
The form of the solution differs from ode2.
As non-linear equations can have multiple solutions,
contrib_ode returns a list of solutions. Each solution can
have a number of forms:
%t, or
%u.
%c is used to represent the constant of integration for first order equations.
%k1 and %k2 are the constants for second order equations.
If contrib_ode
cannot obtain a solution for whatever reason, it returns false, after
perhaps printing out an error message.
It is necessary to return a list of solutions, as even first order non-linear ODEs can have multiple solutions. For example:
(%i1) load('contrib_ode)$
(%i2) eqn:x*'diff(y,x)^2-(1+x*y)*'diff(y,x)+y=0;
dy 2 dy
(%o2) x (--) - (1 + x y) -- + y = 0
dx dx
(%i3) contrib_ode(eqn,y,x);
dy 2 dy
(%t3) x (--) - (1 + x y) -- + y = 0
dx dx
first order equation not linear in y'
x
(%o3) [y = log(x) + %c, y = %c %e ]
(%i4) method;
(%o4) factor
Nonlinear ODEs can have singular solutions without constants of integration, as in the second solution of the following example:
(%i1) load('contrib_ode)$
(%i2) eqn:'diff(y,x)^2+x*'diff(y,x)-y=0;
dy 2 dy
(%o2) (--) + x -- - y = 0
dx dx
(%i3) contrib_ode(eqn,y,x);
dy 2 dy
(%t3) (--) + x -- - y = 0
dx dx
first order equation not linear in y'
2
2 x
(%o3) [y = %c x + %c , y = - --]
4
(%i4) method;
(%o4) clairault
The following ODE has two parametric solutions in terms of the dummy
variable %t. In this case the parametric solutions can be manipulated
to give explicit solutions.
(%i1) load('contrib_ode)$
(%i2) eqn:'diff(y,x)=(x+y)^2;
dy 2
(%o2) -- = (x + y)
dx
(%i3) contrib_ode(eqn,y,x);
(%o3) [[x = %c - atan(sqrt(%t)), y = (- x) - sqrt(%t)],
[x = atan(sqrt(%t)) + %c, y = sqrt(%t) - x]]
(%i4) method;
(%o4) lagrange
The following example (Kamke 1.112) demonstrates an implicit solution.
(%i1) load('contrib_ode)$
(%i2) assume(x>0,y>0);
(%o2) [x > 0, y > 0]
(%i3) eqn:x*'diff(y,x)-x*sqrt(y^2+x^2)-y;
dy 2 2
(%o3) x -- - x sqrt(y + x ) - y
dx
(%i4) contrib_ode(eqn,y,x);
y
(%o4) [x - asinh(-) = %c]
x
(%i5) method;
(%o5) lie
The following Riccati equation is transformed into a linear
second order ODE in the variable %u. Maxima is unable to
solve the new ODE, so it is returned unevaluated.
(%i1) load('contrib_ode)$
(%i2) eqn:x^2*'diff(y,x)=a+b*x^n+c*x^2*y^2;
2 dy 2 2 n
(%o2) x -- = c x y + b x + a
dx
(%i3) contrib_ode(eqn,y,x);
d%u
--- 2
dx 2 a n - 2 d %u
(%o3) [[y = - ----, %u c (-- + b x ) + ---- c = 0]]
%u c 2 2
x dx
(%i4) method;
(%o4) riccati
For first order ODEs contrib_ode calls ode2. It then tries the
following methods: factorization, Clairault, Lagrange, Riccati,
Abel and Lie symmetry methods. The Lie method is not attempted
on Abel equations if the Abel method fails, but it is tried
if the Riccati method returns an unsolved second order ODE.
For second order ODEs contrib_ode calls ode2 then odelin.
Extensive debugging traces and messages are displayed if the command
put('contrib_ode,true,'verbose) is executed.
Categories: Differential equations · Share packages · Package contrib_ode
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Returns a list of solutions of the ODE eqn with independent variable x and dependent variable y.
Categories: Package contrib_ode
odelin solves linear homogeneous ODEs of first and
second order with
independent variable x and dependent variable y.
It returns a fundamental solution set of the ODE.
For second order ODEs, odelin uses a method, due to Bronstein
and Lafaille, that searches for solutions in terms of given
special functions.
(%i1) load('contrib_ode)$
(%i2) odelin(x*(x+1)*'diff(y,x,2)+(x+5)*'diff(y,x,1)+(-4)*y,y,x);
gauss_a(- 6, - 2, - 3, - x) gauss_b(- 6, - 2, - 3, - x)
(%o2) {---------------------------, ---------------------------}
4 4
x x
Categories: Package contrib_ode
Returns the value of ODE eqn after substituting a possible solution soln. The value is equivalent to zero if soln is a solution of eqn.
(%i1) load('contrib_ode)$
(%i2) eqn:'diff(y,x,2)+(a*x+b)*y;
2
d y
(%o2) --- + (b + a x) y
2
dx
(%i3) ans:[y = bessel_y(1/3,2*(a*x+b)^(3/2)/(3*a))*%k2*sqrt(a*x+b)
+bessel_j(1/3,2*(a*x+b)^(3/2)/(3*a))*%k1*sqrt(a*x+b)];
3/2
1 2 (b + a x)
(%o3) [y = bessel_y(-, --------------) %k2 sqrt(a x + b)
3 3 a
3/2
1 2 (b + a x)
+ bessel_j(-, --------------) %k1 sqrt(a x + b)]
3 3 a
(%i4) ode_check(eqn,ans[1]);
(%o4) 0
Categories: Package contrib_ode
The variable method is set to the successful solution
method.
Categories: Package contrib_ode
%c is the integration constant for first order ODEs.
Categories: Package contrib_ode
%k1 is the first integration constant for second order ODEs.
Categories: Package contrib_ode
%k2 is the second integration constant for second order ODEs.
Categories: Package contrib_ode
gauss_a(a,b,c,x) and gauss_b(a,b,c,x) are 2F1
geometric functions. They represent any two independent
solutions of the hypergeometric differential equation
x(1-x) diff(y,x,2) + [c-(a+b+1)x] diff(y,x) - aby = 0 (A&S 15.5.1).
The only use of these functions is in solutions of ODEs returned by
odelin and contrib_ode. The definition and use of these
functions may change in future releases of Maxima.
See also gauss_b, dgauss_a and gauss_b.
Categories: Package contrib_ode
See gauss_a.
Categories: Package contrib_ode
The derivative with respect to x of gauss_a(a, b, c, x).
Categories: Package contrib_ode
The derivative with respect to x of gauss_b(a, b, c, x).
Categories: Package contrib_ode
Kummer's M function, as defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 13.1.2.
The only use of this function is in solutions of ODEs returned by
odelin and contrib_ode. The definition and use of this
function may change in future releases of Maxima.
See also kummer_u, dkummer_m, and dkummer_u.
Categories: Package contrib_ode
Kummer's U function, as defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 13.1.3.
See kummer_m.
Categories: Package contrib_ode
The derivative with respect to x of kummer_m(a, b, x).
Categories: Package contrib_ode
The derivative with respect to x of kummer_u(a, b, x).
Categories: Package contrib_ode
Simplifies expressions containing Bessel functions bessel_j, bessel_y, bessel_i, bessel_k, hankel_1, hankel_2, strauve_h and strauve_l. Recurrence relations (given in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 9.1.27) are used to replace functions of highest order n by functions of order n-1 and n-2.
This process repeated until all the orders differ by less than 2.
(%i1) load('contrib_ode)$
(%i2) bessel_simplify(4*bessel_j(n,x^2)*(x^2-n^2/x^2)
+x*((bessel_j(n-2,x^2)-bessel_j(n,x^2))*x
-(bessel_j(n,x^2)-bessel_j(n+2,x^2))*x)
-2*bessel_j(n+1,x^2)+2*bessel_j(n-1,x^2));
(%o2) 0
(%i3) bessel_simplify(-2*bessel_j(1,z)*z^3-10*bessel_j(2,z)*z^2
+15*%pi*bessel_j(1,z)*struve_h(3,z)*z-15*%pi*struve_h(1,z)*bessel_j(3,z)*z
-15*%pi*bessel_j(0,z)*struve_h(2,z)*z+15*%pi*struve_h(0,z)*bessel_j(2,z)*z
-30*%pi*bessel_j(1,z)*struve_h(2,z)+30*%pi*struve_h(1,z)*bessel_j(2,z));
(%o3) 0
Categories: Package contrib_ode Bessel functions Special functions
Simplify expressions containing exponential integral expintegral_e using the recurrence (A&S 5.1.14).
expintegral_e(n+1,z) = (1/n) * (exp(-z)-z*expintegral_e(n,z)) n = 1,2,3 ....
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These routines are work in progress. I still need to:
ode1_factor to work for multiple roots.
ode1_factor to attempt to solve higher
order factors. At present it only attemps to solve linear factors.
ode1_lagrange to prefer real roots over
complex roots.
ode1_lie. There are quite a
few problems with it: some parts are unimplemented; some test cases
seem to run forever; other test cases crash; yet others return very
complex "solutions". I wonder if it really ready for release yet.
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The routines have been tested on a approximately one thousand test cases from Murphy, Kamke, Zwillinger and elsewhere. These are included in the tests subdirectory.
ode1_clairault finds all known solutions,
including singular solutions, of the Clairault equations in Murphy and
Kamke.
ode1_lie are overly complex and
impossible to check.
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| 49.1 Introduction to descriptive | ||
| 49.2 Functions and Variables for data manipulation | ||
| 49.3 Functions and Variables for descriptive statistics | ||
| 49.4 Functions and Variables for statistical graphs |
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Package descriptive contains a set of functions for
making descriptive statistical computations and graphing.
Together with the source code there are three data sets in
your Maxima tree: pidigits.data, wind.data and biomed.data.
Any statistics manual can be used as a reference to the functions in package descriptive.
For comments, bugs or suggestions, please contact me at 'mario AT edu DOT xunta DOT es'.
Here is a simple example on how the descriptive functions in descriptive do they work, depending on the nature of their arguments, lists or matrices,
(%i1) load (descriptive)$
(%i2) /* univariate sample */ mean ([a, b, c]);
c + b + a
(%o2) ---------
3
(%i3) matrix ([a, b], [c, d], [e, f]);
[ a b ]
[ ]
(%o3) [ c d ]
[ ]
[ e f ]
(%i4) /* multivariate sample */ mean (%);
e + c + a f + d + b
(%o4) [---------, ---------]
3 3
Note that in multivariate samples the mean is calculated for each column.
In case of several samples with possible different sizes, the Maxima function map can be used to get the desired results for each sample,
(%i1) load (descriptive)$
(%i2) map (mean, [[a, b, c], [d, e]]);
c + b + a e + d
(%o2) [---------, -----]
3 2
In this case, two samples of sizes 3 and 2 were stored into a list.
Univariate samples must be stored in lists like
(%i1) s1 : [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5]; (%o1) [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5]
and multivariate samples in matrices as in
(%i1) s2 : matrix ([13.17, 9.29], [14.71, 16.88], [18.50, 16.88],
[10.58, 6.63], [13.33, 13.25], [13.21, 8.12]);
[ 13.17 9.29 ]
[ ]
[ 14.71 16.88 ]
[ ]
[ 18.5 16.88 ]
(%o1) [ ]
[ 10.58 6.63 ]
[ ]
[ 13.33 13.25 ]
[ ]
[ 13.21 8.12 ]
In this case, the number of columns equals the random variable dimension and the number of rows is the sample size.
Data can be introduced by hand, but big samples are usually stored in plain text files. For example, file pidigits.data contains the first 100 digits of number %pi:
3
1
4
1
5
9
2
6
5
3 ...
In order to load these digits in Maxima,
(%i1) s1 : read_list (file_search ("pidigits.data"))$
(%i2) length (s1);
(%o2) 100
On the other hand, file wind.data contains daily average
wind speeds at 5 meteorological stations in the Republic of Ireland
(This is part of a data set taken at 12 meteorological stations. The
original file is freely downloadable from the StatLib Data Repository
and its analysis is discused in Haslett, J., Raftery, A. E. (1989) Space-time Modelling with Long-memory Dependence: Assessing Ireland's Wind Power Resource, with Discussion. Applied Statistics 38, 1-50). This loads the data:
(%i1) s2 : read_matrix (file_search ("wind.data"))$
(%i2) length (s2);
(%o2) 100
(%i3) s2 [%]; /* last record */
(%o3) [3.58, 6.0, 4.58, 7.62, 11.25]
Some samples contain non numeric data. As an example, file biomed.data
(which is part of another bigger one downloaded from the StatLib Data
Repository) contains four blood measures taken from two groups of
patients, A and B, of different ages,
(%i1) s3 : read_matrix (file_search ("biomed.data"))$
(%i2) length (s3);
(%o2) 100
(%i3) s3 [1]; /* first record */
(%o3) [A, 30, 167.0, 89.0, 25.6, 364]
The first individual belongs to group A, is 30 years old and his/her blood measures were 167.0, 89.0, 25.6 and 364.
One must take care when working with categorical data. In the next example, symbol a is assigned a value in some previous moment and then a sample with categorical value a is taken,
(%i1) a : 1$
(%i2) matrix ([a, 3], [b, 5]);
[ 1 3 ]
(%o2) [ ]
[ b 5 ]
Categories: Descriptive statistics · Share packages · Package descriptive
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Builds a sample from a table of absolute frequencies. The input table can be a matrix or a list of lists, all of them of equal size. The number of columns or the length of the lists must be greater than 1. The last element of each row or list is interpreted as the absolute frequency. The output is always a sample in matrix form.
Examples:
Univariate frequency table.
(%i1) load (descriptive)$
(%i2) sam1: build_sample([[6,1], [j,2], [2,1]]);
[ 6 ]
[ ]
[ j ]
(%o2) [ ]
[ j ]
[ ]
[ 2 ]
(%i3) mean(sam1);
2 j + 8
(%o3) [-------]
4
(%i4) barsplot(sam1) $
Multivariate frequency table.
(%i1) load (descriptive)$
(%i2) sam2: build_sample([[6,3,1], [5,6,2], [u,2,1],[6,8,2]]) ;
[ 6 3 ]
[ ]
[ 5 6 ]
[ ]
[ 5 6 ]
(%o2) [ ]
[ u 2 ]
[ ]
[ 6 8 ]
[ ]
[ 6 8 ]
(%i3) cov(sam2);
[ 2 2 ]
[ u + 158 (u + 28) 2 u + 174 11 (u + 28) ]
[ -------- - --------- --------- - ----------- ]
(%o3) [ 6 36 6 12 ]
[ ]
[ 2 u + 174 11 (u + 28) 21 ]
[ --------- - ----------- -- ]
[ 6 12 4 ]
(%i4) barsplot(sam2, grouping=stacked) $
Categories: Package descriptive
The argument of continuous_freq must be a list of numbers.
Divides the range in intervals and counts how many values
are inside them. The second argument is optional and either
equals the number of classes we want, 10 by default, or
equals a list containing the class limits and the number of
classes we want, or a list containing only the limits.
Argument list must be a list of (2 or 3) real numbers.
If sample values are all equal, this function returns only
one class of amplitude 2.
Examples:
Optional argument indicates the number of classes we want.
The first list in the output contains the interval limits, and
the second the corresponding counts: there are 16 digits inside
the interval [0, 1.8], 24 digits in (1.8, 3.6], and so on.
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) continuous_freq (s1, 5);
(%o3) [[0, 1.8, 3.6, 5.4, 7.2, 9.0], [16, 24, 18, 17, 25]]
Optional argument indicates we want 7 classes with limits -2 and 12:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) continuous_freq (s1, [-2,12,7]);
(%o3) [[- 2, 0, 2, 4, 6, 8, 10, 12], [8, 20, 22, 17, 20, 13, 0]]
Optional argument indicates we want the default number of classes with limits -2 and 12:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) continuous_freq (s1, [-2,12]);
3 4 11 18 32 39 46 53
(%o3) [[- 2, - -, -, --, --, 5, --, --, --, --, 12],
5 5 5 5 5 5 5 5
[0, 8, 20, 12, 18, 9, 8, 25, 0, 0]]
Categories: Package descriptive
Counts absolute frequencies in discrete samples, both numeric and categorical. Its unique argument is a list,
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) discrete_freq (s1);
(%o3) [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9],
[8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]
The first list gives the sample values and the second their absolute frequencies. Commands ? col and ? transpose should help you to understand the last input.
Categories: Package descriptive
Subtracts to each element of the list the sample mean and divides
the result by the standard deviation. When the input is a matrix,
standardize subtracts to each row the multivariate mean, and then
divides each component by the corresponding standard deviation.
Categories: Package descriptive
This is a sort of variant of the Maxima submatrix function.
The first argument is the data matrix, the second is a predicate function
and optional additional arguments are the numbers of the columns to be taken.
Its behaviour is better understood with examples.
These are multivariate records in which the wind speed
in the first meteorological station were greater than 18.
See that in the lambda expression the i-th component is
referred to as v[i].
(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) subsample (s2, lambda([v], v[1] > 18));
[ 19.38 15.37 15.12 23.09 25.25 ]
[ ]
[ 18.29 18.66 19.08 26.08 27.63 ]
(%o3) [ ]
[ 20.25 21.46 19.95 27.71 23.38 ]
[ ]
[ 18.79 18.96 14.46 26.38 21.84 ]
In the following example, we request only the first, second and fifth components of those records with wind speeds greater or equal than 16 in station number 1 and less than 25 knots in station number 4. The sample contains only data from stations 1, 2 and 5. In this case, the predicate function is defined as an ordinary Maxima function.
(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) g(x):= x[1] >= 16 and x[4] < 25$
(%i4) subsample (s2, g, 1, 2, 5);
[ 19.38 15.37 25.25 ]
[ ]
[ 17.33 14.67 19.58 ]
(%o4) [ ]
[ 16.92 13.21 21.21 ]
[ ]
[ 17.25 18.46 23.87 ]
Here is an example with the categorical variables of biomed.data.
We want the records corresponding to those patients in group B
who are older than 38 years.
(%i1) load (descriptive)$
(%i2) s3 : read_matrix (file_search ("biomed.data"))$
(%i3) h(u):= u[1] = B and u[2] > 38 $
(%i4) subsample (s3, h);
[ B 39 28.0 102.3 17.1 146 ]
[ ]
[ B 39 21.0 92.4 10.3 197 ]
[ ]
[ B 39 23.0 111.5 10.0 133 ]
[ ]
[ B 39 26.0 92.6 12.3 196 ]
(%o4) [ ]
[ B 39 25.0 98.7 10.0 174 ]
[ ]
[ B 39 21.0 93.2 5.9 181 ]
[ ]
[ B 39 18.0 95.0 11.3 66 ]
[ ]
[ B 39 39.0 88.5 7.6 168 ]
Probably, the statistical analysis will involve only the blood measures,
(%i1) load (descriptive)$
(%i2) s3 : read_matrix (file_search ("biomed.data"))$
(%i3) subsample (s3, lambda([v], v[1] = B and v[2] > 38),
3, 4, 5, 6);
[ 28.0 102.3 17.1 146 ]
[ ]
[ 21.0 92.4 10.3 197 ]
[ ]
[ 23.0 111.5 10.0 133 ]
[ ]
[ 26.0 92.6 12.3 196 ]
(%o3) [ ]
[ 25.0 98.7 10.0 174 ]
[ ]
[ 21.0 93.2 5.9 181 ]
[ ]
[ 18.0 95.0 11.3 66 ]
[ ]
[ 39.0 88.5 7.6 168 ]
This is the multivariate mean of s3,
(%i1) load (descriptive)$
(%i2) s3 : read_matrix (file_search ("biomed.data"))$
(%i3) mean (s3);
65 B + 35 A 317 6 NA + 8144.999999999999
(%o3) [-----------, ---, 87.178, ------------------------,
100 10 100
3 NA + 19587
18.123, ------------]
100
Here, the first component is meaningless, since A and B
are categorical, the second component is the mean age of individuals in
rational form, and the fourth and last values exhibit some strange
behaviour. This is because symbol NA is used here to indicate non available data, and the two means are nonsense. A possible solution would be to take out from the matrix those rows with NA symbols, although this deserves some loss of information.
(%i1) load (descriptive)$
(%i2) s3 : read_matrix (file_search ("biomed.data"))$
(%i3) g(v):= v[4] # NA and v[6] # NA $
(%i4) mean (subsample (s3, g, 3, 4, 5, 6));
(%o4) [79.4923076923077, 86.2032967032967, 16.93186813186813,
2514
----]
13
Categories: Package descriptive
Transforms the sample matrix, where each column is called according to varlist, following expressions in exprlist.
Examples:
The second argument assigns names to the three columns. With these names, a list of expressions define the transformation of the sample.
(%i1) load (descriptive)$
(%i2) data: matrix([3,2,7],[3,7,2],[8,2,4],[5,2,4]) $
(%i3) transform_sample(data, [a,b,c], [c, a*b, log(a)]);
[ 7 6 log(3) ]
[ ]
[ 2 21 log(3) ]
(%o3) [ ]
[ 4 16 log(8) ]
[ ]
[ 4 10 log(5) ]
Add a constant column and remove the third variable.
(%i1) load (descriptive)$
(%i2) data: matrix([3,2,7],[3,7,2],[8,2,4],[5,2,4]) $
(%i3) transform_sample(data, [a,b,c], [makelist(1,k,length(data)),a,b]);
[ 1 3 2 ]
[ ]
[ 1 3 7 ]
(%o3) [ ]
[ 1 8 2 ]
[ ]
[ 1 5 2 ]
Categories: Package descriptive
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This is the sample mean, defined as
n
====
_ 1 \
x = - > x
n / i
====
i = 1
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) mean (s1);
471
(%o3) ---
100
(%i4) %, numer;
(%o4) 4.71
(%i5) s2 : read_matrix (file_search ("wind.data"))$
(%i6) mean (s2);
(%o6) [9.9485, 10.1607, 10.8685, 15.7166, 14.8441]
Categories: Package descriptive
This is the sample variance, defined as
n
====
2 1 \ _ 2
s = - > (x - x)
n / i
====
i = 1
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) var (s1), numer;
(%o3) 8.425899999999999
See also function var1.
Categories: Package descriptive
This is the sample variance, defined as
n
====
1 \ _ 2
--- > (x - x)
n-1 / i
====
i = 1
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) var1 (s1), numer;
(%o3) 8.5110101010101
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) var1 (s2);
(%o5) [17.39586540404041, 15.13912778787879, 15.63204924242424,
32.50152569696971, 24.66977392929294]
See also function var.
Categories: Package descriptive
This is the square root of the function var, the variance with denominator n.
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) std (s1), numer;
(%o3) 2.902740084816414
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) std (s2);
(%o5) [4.149928523480858, 3.871399812729241, 3.933920277534866,
5.672434260526957, 4.941970881136392]
See also functions var and std1.
Categories: Package descriptive
This is the square root of the function var1, the variance with denominator n-1.
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) std1 (s1), numer;
(%o3) 2.917363553109228
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) std1 (s2);
(%o5) [4.170835096721089, 3.89090320978032, 3.953738641137555,
5.701010936401517, 4.966867617451963]
See also functions var1 and std.
Categories: Package descriptive
The non central moment of order k, defined as
n
====
1 \ k
- > x
n / i
====
i = 1
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) noncentral_moment (s1, 1), numer; /* the mean */
(%o3) 4.71
(%i5) s2 : read_matrix (file_search ("wind.data"))$
(%i6) noncentral_moment (s2, 5);
(%o6) [319793.8724761505, 320532.1923892463,
391249.5621381556, 2502278.205988911, 1691881.797742255]
See also function central_moment.
Categories: Package descriptive
The central moment of order k, defined as
n
====
1 \ _ k
- > (x - x)
n / i
====
i = 1
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) central_moment (s1, 2), numer; /* the variance */
(%o3) 8.425899999999999
(%i5) s2 : read_matrix (file_search ("wind.data"))$
(%i6) central_moment (s2, 3);
(%o6) [11.29584771375004, 16.97988248298583, 5.626661952750102,
37.5986572057918, 25.85981904394192]
See also functions central_moment and mean.
Categories: Package descriptive
The variation coefficient is the quotient between the sample standard deviation (std) and the mean,
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) cv (s1), numer;
(%o3) .6193977819764815
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) cv (s2);
(%o5) [.4192426091090204, .3829365309260502, 0.363779605385983,
.3627381836021478, .3346021393989506]
See also functions std and mean.
Categories: Package descriptive
This is the minimum value of the sample list.
When the argument is a matrix, smin returns
a list containing the minimum values of the columns,
which are associated to statistical variables.
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) smin (s1);
(%o3) 0
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) smin (s2);
(%o5) [0.58, 0.5, 2.67, 5.25, 5.17]
See also function smax.
Categories: Package descriptive
This is the maximum value of the sample list.
When the argument is a matrix, smax returns
a list containing the maximum values of the columns,
which are associated to statistical variables.
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) smax (s1);
(%o3) 9
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) smax (s2);
(%o5) [20.25, 21.46, 20.04, 29.63, 27.63]
See also function smin.
Categories: Package descriptive
The range is the difference between the extreme values.
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) range (s1);
(%o3) 9
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) range (s2);
(%o5) [19.67, 20.96, 17.37, 24.38, 22.46]
Categories: Package descriptive
This is the p-quantile, with p a number in [0, 1], of the sample list. Although there are several definitions for the sample quantile (Hyndman, R. J., Fan, Y. (1996) Sample quantiles in statistical packages. American Statistician, 50, 361-365), the one based on linear interpolation is implemented in package descriptive
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) /* 1st and 3rd quartiles */
[quantile (s1, 1/4), quantile (s1, 3/4)], numer;
(%o3) [2.0, 7.25]
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) quantile (s2, 1/4);
(%o5) [7.2575, 7.477500000000001, 7.82, 11.28, 11.48]
Categories: Package descriptive
Once the sample is ordered, if the sample size is odd the median is the central value, otherwise it is the mean of the two central values.
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) median (s1);
9
(%o3) -
2
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) median (s2);
(%o5) [10.06, 9.855, 10.73, 15.48, 14.105]
The median is the 1/2-quantile.
See also function quantile.
Categories: Package descriptive
The interquartilic range is the difference between the third and first quartiles, quantile(list,3/4) - quantile(list,1/4),
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) qrange (s1);
21
(%o3) --
4
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) qrange (s2);
(%o5) [5.385, 5.572499999999998, 6.022500000000001,
8.729999999999999, 6.649999999999999]
See also function quantile.
Categories: Package descriptive
The mean deviation, defined as
n
====
1 \ _
- > |x - x|
n / i
====
i = 1
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) mean_deviation (s1);
51
(%o3) --
20
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) mean_deviation (s2);
(%o5) [3.287959999999999, 3.075342, 3.23907, 4.715664000000001,
4.028546000000002]
See also function mean.
Categories: Package descriptive
The median deviation, defined as
n
====
1 \
- > |x - med|
n / i
====
i = 1
where med is the median of list.
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) median_deviation (s1);
5
(%o3) -
2
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) median_deviation (s2);
(%o5) [2.75, 2.755, 3.08, 4.315, 3.31]
See also function mean.
Categories: Package descriptive
The harmonic mean, defined as
n
--------
n
====
\ 1
> --
/ x
==== i
i = 1
Example:
(%i1) load (descriptive)$
(%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
(%i3) harmonic_mean (y), numer;
(%o3) 3.901858027632205
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) harmonic_mean (s2);
(%o5) [6.948015590052786, 7.391967752360356, 9.055658197151745,
13.44199028193692, 13.01439145898509]
See also functions mean and geometric_mean.
Categories: Package descriptive
The geometric mean, defined as
/ n \ 1/n
| /===\ |
| ! ! |
| ! ! x |
| ! ! i|
| i = 1 |
\ /
Example:
(%i1) load (descriptive)$
(%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
(%i3) geometric_mean (y), numer;
(%o3) 4.454845412337012
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) geometric_mean (s2);
(%o5) [8.82476274347979, 9.22652604739361, 10.0442675714889,
14.61274126349021, 13.96184163444275]
See also functions mean and harmonic_mean.
Categories: Package descriptive
The kurtosis coefficient, defined as
n
====
1 \ _ 4
---- > (x - x) - 3
4 / i
n s ====
i = 1
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) kurtosis (s1), numer;
(%o3) - 1.273247946514421
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) kurtosis (s2);
(%o5) [- .2715445622195385, 0.119998784429451,
- .4275233490482861, - .6405361979019522, - .4952382132352935]
See also functions mean, var and skewness.
Categories: Package descriptive
The skewness coefficient, defined as
n
====
1 \ _ 3
---- > (x - x)
3 / i
n s ====
i = 1
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) skewness (s1), numer;
(%o3) .009196180476450424
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) skewness (s2);
(%o5) [.1580509020000978, .2926379232061854, .09242174416107717,
.2059984348148687, .2142520248890831]
See also functions mean,, var and kurtosis.
Categories: Package descriptive
Pearson's skewness coefficient, defined as
_
3 (x - med)
-----------
s
where med is the median of list.
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) pearson_skewness (s1), numer;
(%o3) .2159484029093895
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) pearson_skewness (s2);
(%o5) [- .08019976629211892, .2357036272952649,
.1050904062491204, .1245042340592368, .4464181795804519]
See also functions mean, var and median.
Categories: Package descriptive
The quartile skewness coefficient, defined as
c - 2 c + c
3/4 1/2 1/4
--------------------
c - c
3/4 1/4
where c_p is the p-quantile of sample list.
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) quartile_skewness (s1), numer;
(%o3) .04761904761904762
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) quartile_skewness (s2);
(%o5) [- 0.0408542246982353, .1467025572005382,
0.0336239103362392, .03780068728522298, .2105263157894735]
See also function quantile.
Categories: Package descriptive
The covariance matrix of the multivariate sample, defined as
n
====
1 \ _ _
S = - > (X - X) (X - X)'
n / j j
====
j = 1
where X_j is the j-th row of the sample matrix.
Example:
(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) fpprintprec : 7$ /* change precision for pretty output */
(%i4) cov (s2);
[ 17.22191 13.61811 14.37217 19.39624 15.42162 ]
[ ]
[ 13.61811 14.98774 13.30448 15.15834 14.9711 ]
[ ]
(%o4) [ 14.37217 13.30448 15.47573 17.32544 16.18171 ]
[ ]
[ 19.39624 15.15834 17.32544 32.17651 20.44685 ]
[ ]
[ 15.42162 14.9711 16.18171 20.44685 24.42308 ]
See also function cov1.
Categories: Package descriptive
The covariance matrix of the multivariate sample, defined as
n
====
1 \ _ _
S = --- > (X - X) (X - X)'
1 n-1 / j j
====
j = 1
where X_j is the j-th row of the sample matrix.
Example:
(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) fpprintprec : 7$ /* change precision for pretty output */
(%i4) cov1 (s2);
[ 17.39587 13.75567 14.51734 19.59216 15.5774 ]
[ ]
[ 13.75567 15.13913 13.43887 15.31145 15.12232 ]
[ ]
(%o4) [ 14.51734 13.43887 15.63205 17.50044 16.34516 ]
[ ]
[ 19.59216 15.31145 17.50044 32.50153 20.65338 ]
[ ]
[ 15.5774 15.12232 16.34516 20.65338 24.66977 ]
See also function cov.
Categories: Package descriptive
Function global_variances returns a list of global variance measures:
trace(S_1),
trace(S_1)/p,
determinant(S_1),
sqrt(determinant(S_1)),
determinant(S_1)^(1/p), (defined in: Peña, D. (2002) Análisis de datos multivariantes; McGraw-Hill, Madrid.)
determinant(S_1)^(1/(2*p)).
where p is the dimension of the multivariate random variable and S_1 the covariance matrix returned by cov1.
Option:
'data, default 'true, indicates whether the input matrix contains the sample data,
in which case the covariance matrix cov1 must be calculated, or not, and then the covariance
matrix (symmetric) must be given, instead of the data.
Example:
(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) global_variances (s2);
(%o3) [105.338342060606, 21.06766841212119, 12874.34690469686,
113.4651792608501, 6.636590811800795, 2.576158149609762]
Calculate the global_variances from the covariance matrix.
(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) s : cov1 (s2)$
(%i4) global_variances (s, data=false);
(%o4) [105.338342060606, 21.06766841212119, 12874.34690469686,
113.4651792608501, 6.636590811800795, 2.576158149609762]
Categories: Package descriptive
The correlation matrix of the multivariate sample.
Option:
'data, default 'true, indicates whether the input matrix contains the sample data,
in which case the covariance matrix cov1 must be calculated, or not, and then the covariance
matrix (symmetric) must be given, instead of the data.
Example:
(%i1) load (descriptive)$
(%i2) fpprintprec : 7 $
(%i3) s2 : read_matrix (file_search ("wind.data"))$
(%i4) cor (s2);
[ 1.0 .8476339 .8803515 .8239624 .7519506 ]
[ ]
[ .8476339 1.0 .8735834 .6902622 0.782502 ]
[ ]
(%o4) [ .8803515 .8735834 1.0 .7764065 .8323358 ]
[ ]
[ .8239624 .6902622 .7764065 1.0 .7293848 ]
[ ]
[ .7519506 0.782502 .8323358 .7293848 1.0 ]
Calculate de correlation matrix from the covariance matrix.
(%i1) load (descriptive)$
(%i2) fpprintprec : 7 $
(%i3) s2 : read_matrix (file_search ("wind.data"))$
(%i4) s : cov1 (s2)$
(%i5) cor (s, data=false); /* this is faster */
[ 1.0 .8476339 .8803515 .8239624 .7519506 ]
[ ]
[ .8476339 1.0 .8735834 .6902622 0.782502 ]
[ ]
(%o5) [ .8803515 .8735834 1.0 .7764065 .8323358 ]
[ ]
[ .8239624 .6902622 .7764065 1.0 .7293848 ]
[ ]
[ .7519506 0.782502 .8323358 .7293848 1.0 ]
Categories: Package descriptive
Function list_correlations returns a list of correlation measures:
-1 ij
S = (s )
1 i,j = 1,2,...,p
2 1
R = 1 - -------
i ii
s s
ii
being an indicator of the goodness of fit of the linear multivariate regression model on X_i when the rest of variables are used as regressors.
ij
s
r = - ------------
ij.rest / ii jj\ 1/2
|s s |
\ /
Option:
'data, default 'true, indicates whether the input matrix contains the sample data,
in which case the covariance matrix cov1 must be calculated, or not, and then the covariance
matrix (symmetric) must be given, instead of the data.
Example:
(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) z : list_correlations (s2)$
(%i4) fpprintprec : 5$ /* for pretty output */
(%i5) z[1]; /* precision matrix */
[ .38486 - .13856 - .15626 - .10239 .031179 ]
[ ]
[ - .13856 .34107 - .15233 .038447 - .052842 ]
[ ]
(%o5) [ - .15626 - .15233 .47296 - .024816 - .10054 ]
[ ]
[ - .10239 .038447 - .024816 .10937 - .034033 ]
[ ]
[ .031179 - .052842 - .10054 - .034033 .14834 ]
(%i6) z[2]; /* multiple correlation vector */
(%o6) [.85063, .80634, .86474, .71867, .72675]
(%i7) z[3]; /* partial correlation matrix */
[ - 1.0 .38244 .36627 .49908 - .13049 ]
[ ]
[ .38244 - 1.0 .37927 - .19907 .23492 ]
[ ]
(%o7) [ .36627 .37927 - 1.0 .10911 .37956 ]
[ ]
[ .49908 - .19907 .10911 - 1.0 .26719 ]
[ ]
[ - .13049 .23492 .37956 .26719 - 1.0 ]
Categories: Package descriptive
Calculates the principal componentes of a multivariate sample. Principal components are used in multivariate statistical analysis to reduce the dimensionality of the sample.
Option:
'data, default 'true, indicates whether the input matrix contains the sample data,
in which case the covariance matrix cov1 must be calculated, or not, and then the covariance
matrix (symmetric) must be given, instead of the data.
The output of function principal_components is a list with the following results:
Examples:
In this sample, the first component explains 83.13 per cent of total variance.
(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) fpprintprec:4 $
(%i4) res: principal_components(s2);
0 errors, 0 warnings
(%o4) [[87.57, 8.753, 5.515, 1.889, 1.613],
[83.13, 8.31, 5.235, 1.793, 1.531],
[ .4149 .03379 - .4757 - 0.581 - .5126 ]
[ ]
[ 0.369 - .3657 - .4298 .7237 - .1469 ]
[ ]
[ .3959 - .2178 - .2181 - .2749 .8201 ]]
[ ]
[ .5548 .7744 .1857 .2319 .06498 ]
[ ]
[ .4765 - .4669 0.712 - .09605 - .1969 ]
(%i5) /* accumulated percentages */
block([ap: copy(res[2])],
for k:2 thru length(ap) do ap[k]: ap[k]+ap[k-1],
ap);
(%o5) [83.13, 91.44, 96.68, 98.47, 100.0]
(%i6) /* sample dimension */
p: length(first(res));
(%o6) 5
(%i7) /* plot percentages to select number of
principal components for further work */
draw2d(
fill_density = 0.2,
apply(bars, makelist([k, res[2][k], 1/2], k, p)),
points_joined = true,
point_type = filled_circle,
point_size = 3,
points(makelist([k, res[2][k]], k, p)),
xlabel = "Variances",
ylabel = "Percentages",
xtics = setify(makelist([concat("PC",k),k], k, p))) $
In case de covariance matrix is known, it can be passed to the function,
but option data=false must be used.
(%i1) load (descriptive)$
(%i2) S: matrix([1,-2,0],[-2,5,0],[0,0,2]);
[ 1 - 2 0 ]
[ ]
(%o2) [ - 2 5 0 ]
[ ]
[ 0 0 2 ]
(%i3) fpprintprec:4 $
(%i4) /* the argumment is a covariance matrix */
res: principal_components(S, data=false);
0 errors, 0 warnings
[ - .3827 0.0 .9239 ]
[ ]
(%o4) [[5.828, 2.0, .1716], [72.86, 25.0, 2.145], [ .9239 0.0 .3827 ]]
[ ]
[ 0.0 1.0 0.0 ]
(%i5) /* transformation to get the principal components
from original records */
matrix([a1,b2,c3],[a2,b2,c2]).last(res);
[ .9239 b2 - .3827 a1 1.0 c3 .3827 b2 + .9239 a1 ]
(%o5) [ ]
[ .9239 b2 - .3827 a2 1.0 c2 .3827 b2 + .9239 a2 ]
Categories: Package descriptive
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Plots bars diagrams for discrete statistical variables, both for one or multiple samples.
data can be a list of outcomes representing one sample, or a matrix of m rows and n columns, representing n samples of size m each.
Available options are:
3/4): relative width of rectangles. This
value must be in the range [0,1].
clustered): indicates how multiple samples are
shown. Valid values are: clustered and stacked.
1): a positive integer number representing
the gap between two consecutive groups of bars.
[]): a list of colors for multiple samples.
When there are more samples than specified colors, the extra necessary colors
are chosen at random. See color to learn more about them.
absolute): indicates the scale of the
ordinates. Possible values are: absolute, relative,
and percent.
orderlessp): possible values are orderlessp or ordergreatp,
indicating how statistical outcomes should be ordered on the x-axis.
[]): a list with the strings to be used in the legend.
When the list length is other than 0 or the number of samples, an error message is returned.
0): indicates where the plot begins to be plotted on the
x axis.
draw options, except xtics, which is
internally assigned by barsplot.
If you want to set your own values for this option or want to build
complex scenes, make use of barsplot_description. See example below.
key, color_draw,
fill_color, fill_density and line_width.
See also
barsplot.
There is also a function wxbarsplot for creating embedded
histograms in interfaces wxMaxima and iMaxima. barsplot in a
multiplot context.
Examples:
Univariate sample in matrix form. Absolute frequencies.
(%i1) load (descriptive)$
(%i2) m : read_matrix (file_search ("biomed.data"))$
(%i3) barsplot(
col(m,2),
title = "Ages",
xlabel = "years",
box_width = 1/2,
fill_density = 3/4)$
Two samples of different sizes, with relative frequencies and user declared colors.
(%i1) load (descriptive)$
(%i2) l1:makelist(random(10),k,1,50)$
(%i3) l2:makelist(random(10),k,1,100)$
(%i4) barsplot(
l1,l2,
box_width = 1,
fill_density = 1,
bars_colors = [black, grey],
frequency = relative,
sample_keys = ["A", "B"])$
Four non numeric samples of equal size.
(%i1) load (descriptive)$
(%i2) barsplot(
makelist([Yes, No, Maybe][random(3)+1],k,1,50),
makelist([Yes, No, Maybe][random(3)+1],k,1,50),
makelist([Yes, No, Maybe][random(3)+1],k,1,50),
makelist([Yes, No, Maybe][random(3)+1],k,1,50),
title = "Asking for something to four groups",
ylabel = "# of individuals",
groups_gap = 3,
fill_density = 0.5,
ordering = ordergreatp)$
Stacked bars.
(%i1) load (descriptive)$
(%i2) barsplot(
makelist([Yes, No, Maybe][random(3)+1],k,1,50),
makelist([Yes, No, Maybe][random(3)+1],k,1,50),
makelist([Yes, No, Maybe][random(3)+1],k,1,50),
makelist([Yes, No, Maybe][random(3)+1],k,1,50),
title = "Asking for something to four groups",
ylabel = "# of individuals",
grouping = stacked,
fill_density = 0.5,
ordering = ordergreatp)$
For bars diagrams related options, see barsplot of package draw
See also functions histogram and piechart.
Categories: Package descriptive · Plotting
Function barsplot_description creates a graphic object
suitable for creating complex scenes, together with other
graphic objects.
Example: barsplot in a multiplot context.
(%i1) load (descriptive)$
(%i2) l1:makelist(random(10),k,1,50)$
(%i3) l2:makelist(random(10),k,1,100)$
(%i4) bp1 :
barsplot_description(
l1,
box_width = 1,
fill_density = 0.5,
bars_colors = [blue],
frequency = relative)$
(%i5) bp2 :
barsplot_description(
l2,
box_width = 1,
fill_density = 0.5,
bars_colors = [red],
frequency = relative)$
(%i6) draw(gr2d(bp1), gr2d(bp2))$
Categories: Package descriptive · Plotting
This function plots box-and-whishker diagrams. Argument data can be a list,
which is not of great interest, since these diagrams are mainly used for
comparing different samples, or a matrix, so it is possible to compare
two or more components of a multivariate statistical variable.
But it is also allowed data to be a list of samples with
possible different sample sizes, in fact this is the only function
in package descriptive that admits this type of data structure.
Available options are:
3/4): relative width of boxes.
This value must be in the range [0,1].
vertical): possible values: vertical
and horizontal.
draw options, except points_joined, point_size, point_type,
xtics, ytics, xrange, and yrange, which are
internally assigned by boxplot.
If you want to set your own values for this options or want to build
complex scenes, make use of boxplot_description.
draw options: key, color,
and line_width.
There is also a function wxboxplot for creating embedded
histograms in interfaces wxMaxima and iMaxima.
Examples:
Box-and-whishker diagram from a multivariate sample.
(%i1) load (descriptive)$
(%i2) s2 : read_matrix(file_search("wind.data"))$
(%i3) boxplot(s2,
box_width = 0.2,
title = "Windspeed in knots",
xlabel = "Stations",
color = red,
line_width = 2)$
Box-and-whishker diagram from three samples of different sizes.
(%i1) load (descriptive)$
(%i2) A :
[[6, 4, 6, 2, 4, 8, 6, 4, 6, 4, 3, 2],
[8, 10, 7, 9, 12, 8, 10],
[16, 13, 17, 12, 11, 18, 13, 18, 14, 12]]$
(%i3) boxplot (A, box_orientation = horizontal)$
Categories: Package descriptive · Plotting
Function boxplot_description creates a graphic object
suitable for creating complex scenes, together with other
graphic objects.
Categories: Package descriptive · Plotting
This function plots an histogram from a continuous sample. Sample data must be stored in a list of numbers or a one dimensional matrix.
Available options are:
10): number of classes of the histogram, or
a list indicating the limits of the classes and the number of them, or
only the limits.
absolute): indicates the scale of the
ordinates. Possible values are: absolute, relative, percent,
and density. With density, the histogram area has a total area of one.
auto): format of the histogram tics. Possible
values are: auto, endpoints, intervals, or a list
of labels.
draw options, except xrange, yrange,
and xtics, which are internally assigned by histogram.
If you want to set your own values for these options, make use of
histogram_description. See examples bellow.
key, color,
fill_color, fill_density and line_width. See also
barsplot.
There is also a function wxhistogram for creating embedded
histograms in interfaces wxMaxima and iMaxima.
Examples:
A simple with eight classes:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) histogram (
s1,
nclasses = 8,
title = "pi digits",
xlabel = "digits",
ylabel = "Absolute frequency",
fill_color = grey,
fill_density = 0.6)$
Setting the limits of the histogram to -2 and 12, with 3 classes. Also, we introduce predefined tics:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) histogram (
s1,
nclasses = [-2,12,3],
htics = ["A", "B", "C"],
terminal = png,
fill_color = "#23afa0",
fill_density = 0.6)$
Categories: Package descriptive · Plotting
Function histogram_description creates a graphic object
suitable for creating complex scenes, together with other
graphic objects. We make use of histogram_description for setting
the xrange and adding an explicit curve into the scene:
(%i1) load (descriptive)$
(%i2) ( load("distrib"),
m: 14, s: 2,
s2: random_normal(m, s, 1000) ) $
(%i3) draw2d(
grid = true,
xrange = [5, 25],
histogram_description(
s2,
nclasses = 9,
frequency = density,
fill_density = 0.5),
explicit(pdf_normal(x,m,s), x, m - 3*s, m + 3* s))$
Categories: Package descriptive · Plotting
Similar to barsplot, but plots sectors instead of rectangles.
Available options are:
[]): a list of colors for sectors.
When there are more sectors than specified colors, the extra necessary colors
are chosen at random. See color to learn more about them.
[0,0]): diagram's center.
1): diagram's radius.
draw options, except key, which is
internally assigned by piechart.
If you want to set your own values for this option or want to build
complex scenes, make use of piechart_description.
draw options: key, color,
fill_density and line_width. See also
ellipse
There is also a function wxpiechart for
creating embedded histograms in interfaces wxMaxima and iMaxima.
Example:
(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) piechart(
s1,
xrange = [-1.1, 1.3],
yrange = [-1.1, 1.1],
title = "Digit frequencies in pi")$
See also function barsplot.
Categories: Package descriptive · Plotting
Function piechart_description creates a graphic object
suitable for creating complex scenes, together with other
graphic objects.
Categories: Package descriptive · Plotting
Plots scatter diagrams both for univariate (list) and multivariate (matrix) samples.
Available options are the same admitted by histogram.
There is also a function wxscatterplot for
creating embedded histograms in interfaces wxMaxima and iMaxima.
Examples:
Univariate scatter diagram from a simulated Gaussian sample.
(%i1) load (descriptive)$
(%i2) load (distrib)$
(%i3) scatterplot(
random_normal(0,1,200),
xaxis = true,
point_size = 2,
dimensions = [600,150])$
Two dimensional scatter plot.
(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) scatterplot(
submatrix(s2, 1,2,3),
title = "Data from stations #4 and #5",
point_type = diamant,
point_size = 2,
color = blue)$
Three dimensional scatter plot.
(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) scatterplot(submatrix (s2, 1,2), nclasses=4)$
Five dimensional scatter plot, with five classes histograms.
(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) scatterplot(
s2,
nclasses = 5,
frequency = relative,
fill_color = blue,
fill_density = 0.3,
xtics = 5)$
For plotting isolated or line-joined points in two and three dimensions,
see points. See also histogram.
Categories: Package descriptive · Plotting
Function scatterplot_description creates a graphic object
suitable for creating complex scenes, together with other
graphic objects.
Categories: Package descriptive · Plotting
Plots star diagrams for discrete statistical variables, both for one or multiple samples.
data can be a list of outcomes representing one sample, or a matrix of m rows and n columns, representing n samples of size m each.
Available options are:
[]): a list of colors for multiple samples.
When there are more samples than specified colors, the extra necessary colors
are chosen at random. See color to learn more about them.
absolute): indicates the scale of the
radii. Possible values are: absolute and relative.
orderlessp): possible values are orderlessp or ordergreatp,
indicating how statistical outcomes should be ordered.
[]): a list with the strings to be used in the legend.
When the list length is other than 0 or the number of samples, an error message is returned.
[0,0]): diagram's center.
1): diagram's radius.
draw options, except points_joined, point_type,
and key, which are internally assigned by starplot.
If you want to set your own values for this options or want to build
complex scenes, make use of starplot_description.
draw option: line_width.
There is also a function wxstarplot for
creating embedded histograms in interfaces wxMaxima and iMaxima.
Example:
Plot based on absolute frequencies. Location and radius defined by the user.
(%i1) load (descriptive)$
(%i2) l1: makelist(random(10),k,1,50)$
(%i3) l2: makelist(random(10),k,1,200)$
(%i4) starplot(
l1, l2,
stars_colors = [blue,red],
sample_keys = ["1st sample", "2nd sample"],
star_center = [1,2],
star_radius = 4,
proportional_axes = xy,
line_width = 2 ) $
Categories: Package descriptive · Plotting
Function starplot_description creates a graphic object
suitable for creating complex scenes, together with other
graphic objects.
Categories: Package descriptive · Plotting
Plots stem and leaf diagrams.
Unique available option is:
1): indicates the unit of the leaves; must be a
power of 10.
Example:
(%i1) load (descriptive)$
(%i2) load(distrib)$
(%i3) stemplot(
random_normal(15, 6, 100),
leaf_unit = 0.1);
-5|4
0|37
1|7
3|6
4|4
5|4
6|57
7|0149
8|3
9|1334588
10|07888
11|01144467789
12|12566889
13|24778
14|047
15|223458
16|4
17|11557
18|000247
19|4467799
20|00
21|1
22|2335
23|01457
24|12356
25|455
27|79
key: 6|3 = 6.3
(%o3) done
Categories: Package descriptive · Plotting
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| 50.1 Functions and Variables for diag |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Constructs a matrix that is the block sum of the elements of lm. The elements of lm are assumed to be matrices; if an element is scalar, it treated as a 1 by 1 matrix.
The resulting matrix will be square if each of the elements of lm is square.
Example:
(%i1) load("diag")$
(%i2) a1:matrix([1,2,3],[0,4,5],[0,0,6])$
(%i3) a2:matrix([1,1],[1,0])$
(%i4) diag([a1,x,a2]);
[ 1 2 3 0 0 0 ]
[ ]
[ 0 4 5 0 0 0 ]
[ ]
[ 0 0 6 0 0 0 ]
(%o4) [ ]
[ 0 0 0 x 0 0 ]
[ ]
[ 0 0 0 0 1 1 ]
[ ]
[ 0 0 0 0 1 0 ]
(%i5) diag ([matrix([1,2]), 3]);
[ 1 2 0 ]
(%o5) [ ]
[ 0 0 3 ]
To use this function write first load("diag").
Categories: Matrices · Share packages · Package diag
Returns the Jordan cell of order n with eigenvalue lambda.
Example:
(%i1) load("diag")$
(%i2) JF(2,5);
[ 2 1 0 0 0 ]
[ ]
[ 0 2 1 0 0 ]
[ ]
(%o2) [ 0 0 2 1 0 ]
[ ]
[ 0 0 0 2 1 ]
[ ]
[ 0 0 0 0 2 ]
(%i3) JF(3,2);
[ 3 1 ]
(%o3) [ ]
[ 0 3 ]
To use this function write first load("diag").
Categories: Package diag
Returns the Jordan form of matrix mat, encoded as a list in a
particular format. To get the corresponding matrix, call the function
dispJordan using the output of jordan as the argument.
The elements of the returned list are themselves lists. The first element of each is an eigenvalue of mat. The remaining elements are positive integers which are the lengths of the Jordan blocks for this eigenvalue. These integers are listed in decreasing order. Eigenvalues are not repeated.
The functions dispJordan, minimalPoly and
ModeMatrix expect the output of a call to jordan as an
argument. If you construct this argument by hand, rather than by
calling jordan, you must ensure that each eigenvalue only
appears once and that the block sizes are listed in decreasing order,
otherwise the functions might give incorrect answers.
Example:
(%i1) load("diag")$
(%i2) A: matrix([2,0,0,0,0,0,0,0],
[1,2,0,0,0,0,0,0],
[-4,1,2,0,0,0,0,0],
[2,0,0,2,0,0,0,0],
[-7,2,0,0,2,0,0,0],
[9,0,-2,0,1,2,0,0],
[-34,7,1,-2,-1,1,2,0],
[145,-17,-16,3,9,-2,0,3])$
(%i3) jordan (A);
(%o3) [[2, 3, 3, 1], [3, 1]]
(%i4) dispJordan (%);
[ 2 1 0 0 0 0 0 0 ]
[ ]
[ 0 2 1 0 0 0 0 0 ]
[ ]
[ 0 0 2 0 0 0 0 0 ]
[ ]
[ 0 0 0 2 1 0 0 0 ]
(%o4) [ ]
[ 0 0 0 0 2 1 0 0 ]
[ ]
[ 0 0 0 0 0 2 0 0 ]
[ ]
[ 0 0 0 0 0 0 2 0 ]
[ ]
[ 0 0 0 0 0 0 0 3 ]
To use this function write first load("diag"). See also dispJordan and minimalPoly.
Categories: Package diag
Returns a matrix in Jordan canonical form (JCF) corresponding to the
list of eigenvalues and multiplicities given by l. This list
should be in the format given by the jordan function. See
jordan for details of this format.
Example:
(%i1) load("diag")$
(%i2) b1:matrix([0,0,1,1,1],
[0,0,0,1,1],
[0,0,0,0,1],
[0,0,0,0,0],
[0,0,0,0,0])$
(%i3) jordan(b1);
(%o3) [[0, 3, 2]]
(%i4) dispJordan(%);
[ 0 1 0 0 0 ]
[ ]
[ 0 0 1 0 0 ]
[ ]
(%o4) [ 0 0 0 0 0 ]
[ ]
[ 0 0 0 0 1 ]
[ ]
[ 0 0 0 0 0 ]
To use this function write first load("diag"). See also jordan and minimalPoly.
Categories: Package diag
Returns the minimal polynomial of the matrix whose Jordan form is
described by the list l. This list should be in the format given
by the jordan function. See jordan for details of this
format.
Example:
(%i1) load("diag")$
(%i2) a:matrix([2,1,2,0],
[-2,2,1,2],
[-2,-1,-1,1],
[3,1,2,-1])$
(%i3) jordan(a);
(%o3) [[- 1, 1], [1, 3]]
(%i4) minimalPoly(%);
3
(%o4) (x - 1) (x + 1)
To use this function write first load("diag"). See also jordan and dispJordan.
Categories: Package diag
Returns an invertible matrix M such that (M^^-1).A.M is the Jordan form of A.
To calculate this, Maxima must find the Jordan form of A, which
might be quite computationally expensive. If that has already been
calculated by a previous call to jordan, pass it as a second
argument, jordan_info. See jordan for details of the
required format.
Example:
(%i1) load("diag")$
(%i2) A: matrix([2,1,2,0], [-2,2,1,2], [-2,-1,-1,1], [3,1,2,-1])$
(%i3) M: ModeMatrix (A);
[ 1 - 1 1 1 ]
[ ]
[ 1 ]
[ - - - 1 0 0 ]
[ 9 ]
[ ]
(%o3) [ 13 ]
[ - -- 1 - 1 0 ]
[ 9 ]
[ ]
[ 17 ]
[ -- - 1 1 1 ]
[ 9 ]
(%i4) is ((M^^-1) . A . M = dispJordan (jordan (A)));
(%o4) true
Note that, in this example, the Jordan form of A is computed
twice. To avoid this, we could have stored the output of
jordan(A) in a variable and passed that to both
ModeMatrix and dispJordan.
To use this function write first load("diag"). See also
jordan and dispJordan.
Categories: Package diag
Returns f(A), where f is an analytic function and A a matrix. This computation is based on the Taylor expansion of f. It is not efficient for numerical evaluation, but can give symbolic answers for small matrices.
Example 1:
The exponential of a matrix. We only give the first row of the answer, since the output is rather large.
(%i1) load("diag")$
(%i2) A: matrix ([0,1,0], [0,0,1], [-1,-3,-3])$
(%i3) ratsimp (mat_function (exp, t*A)[1]);
2 - t 2 - t
(t + 2 t + 2) %e 2 - t t %e
(%o3) [--------------------, (t + t) %e , --------]
2 2
Example 2:
Comparison with the Taylor series for the exponential and also
comparing exp(%i*A) with sine and cosine.
(%i1) load("diag")$
(%i2) A: matrix ([0,1,1,1],
[0,0,0,1],
[0,0,0,1],
[0,0,0,0])$
(%i3) ratsimp (mat_function (exp, t*A));
[ 2 ]
[ 1 t t t + t ]
[ ]
(%o3) [ 0 1 0 t ]
[ ]
[ 0 0 1 t ]
[ ]
[ 0 0 0 1 ]
(%i4) minimalPoly (jordan (A));
3
(%o4) x
(%i5) ratsimp (ident(4) + t*A + 1/2*(t^2)*A^^2);
[ 2 ]
[ 1 t t t + t ]
[ ]
(%o5) [ 0 1 0 t ]
[ ]
[ 0 0 1 t ]
[ ]
[ 0 0 0 1 ]
(%i6) ratsimp (mat_function (exp, %i*t*A));
[ 2 ]
[ 1 %i t %i t %i t - t ]
[ ]
(%o6) [ 0 1 0 %i t ]
[ ]
[ 0 0 1 %i t ]
[ ]
[ 0 0 0 1 ]
(%i7) ratsimp (mat_function (cos, t*A) + %i*mat_function (sin, t*A));
[ 2 ]
[ 1 %i t %i t %i t - t ]
[ ]
(%o7) [ 0 1 0 %i t ]
[ ]
[ 0 0 1 %i t ]
[ ]
[ 0 0 0 1 ]
Example 3:
Power operations.
(%i1) load("diag")$
(%i2) A: matrix([1,2,0], [0,1,0], [1,0,1])$
(%i3) integer_pow(x) := block ([k], declare (k, integer), x^k)$
(%i4) mat_function (integer_pow, A);
[ 1 2 k 0 ]
[ ]
(%o4) [ 0 1 0 ]
[ ]
[ k (k - 1) k 1 ]
(%i5) A^^20;
[ 1 40 0 ]
[ ]
(%o5) [ 0 1 0 ]
[ ]
[ 20 380 1 ]
To use this function write first load("diag").
Categories: Package diag
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 51.1 Introduction to distrib | ||
| 51.2 Functions and Variables for continuous distributions | ||
| 51.3 Functions and Variables for discrete distributions |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Package distrib contains a set of functions for making probability computations on both discrete and continuous univariate models.
What follows is a short reminder of basic probabilistic related definitions.
Let f(x) be the density function of an absolute continuous random variable X. The distribution function is defined as
x
/
[
F(x) = I f(u) du
]
/
minf
which equals the probability Pr(X <= x).
The mean value is a localization parameter and is defined as
inf
/
[
E[X] = I x f(x) dx
]
/
minf
The variance is a measure of variation,
inf
/
[ 2
V[X] = I f(x) (x - E[X]) dx
]
/
minf
which is a positive real number. The square root of the variance is the standard deviation, D[X]=sqrt(V[X]), and it is another measure of variation.
The skewness coefficient is a measure of non-symmetry,
inf
/
1 [ 3
SK[X] = ----- I f(x) (x - E[X]) dx
3 ]
D[X] /
minf
And the kurtosis coefficient measures the peakedness of the distribution,
inf
/
1 [ 4
KU[X] = ----- I f(x) (x - E[X]) dx - 3
4 ]
D[X] /
minf
If X is gaussian, KU[X]=0. In fact, both skewness and kurtosis are shape parameters used to measure the non-gaussianity of a distribution.
If the random variable X is discrete, the density, or probability, function f(x) takes positive values within certain countable set of numbers x_i, and zero elsewhere. In this case, the distribution function is
====
\
F(x) = > f(x )
/ i
====
x <= x
i
The mean, variance, standard deviation, skewness coefficient and kurtosis coefficient take the form
====
\
E[X] = > x f(x ) ,
/ i i
====
x
i
====
\ 2
V[X] = > f(x ) (x - E[X]) ,
/ i i
====
x
i
D[X] = sqrt(V[X]),
====
1 \ 3
SK[X] = ------- > f(x ) (x - E[X])
D[X]^3 / i i
====
x
i
and
====
1 \ 4
KU[X] = ------- > f(x ) (x - E[X]) - 3 ,
D[X]^4 / i i
====
x
i
respectively.
There is a naming convention in package distrib. Every function name has two parts, the first one makes reference to the function or parameter we want to calculate,
Functions: Density function (pdf_*) Distribution function (cdf_*) Quantile (quantile_*) Mean (mean_*) Variance (var_*) Standard deviation (std_*) Skewness coefficient (skewness_*) Kurtosis coefficient (kurtosis_*) Random variate (random_*)
The second part is an explicit reference to the probabilistic model,
Continuous distributions: Normal (*normal) Student (*student_t) Chi^2 (*chi2) Noncentral Chi^2 (*noncentral_chi2) F (*f) Exponential (*exp) Lognormal (*lognormal) Gamma (*gamma) Beta (*beta) Continuous uniform (*continuous_uniform) Logistic (*logistic) Pareto (*pareto) Weibull (*weibull) Rayleigh (*rayleigh) Laplace (*laplace) Cauchy (*cauchy) Gumbel (*gumbel) Discrete distributions: Binomial (*binomial) Poisson (*poisson) Bernoulli (*bernoulli) Geometric (*geometric) Discrete uniform (*discrete_uniform) hypergeometric (*hypergeometric) Negative binomial (*negative_binomial) Finite discrete (*general_finite_discrete)
For example, pdf_student_t(x,n) is the density function of the Student distribution with n degrees of freedom, std_pareto(a,b) is the standard deviation of the Pareto distribution with parameters a and b and kurtosis_poisson(m) is the kurtosis coefficient of the Poisson distribution with mean m.
In order to make use of package distrib you need first to load it by typing
(%i1) load(distrib)$
For comments, bugs or suggestions, please contact the author at 'mario AT edu DOT xunta DOT es'.
Categories: Statistical functions · Share packages · Package distrib
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Returns the value at x of the density function of a Normal(m,s) random variable, with s>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Normal(m,s) random variable, with s>0. This function is defined in terms of Maxima's built-in error function erf.
(%i1) load (distrib)$
(%i2) assume(s>0)$ cdf_normal(x,m,s);
x - m
erf(---------)
sqrt(2) s 1
(%o3) -------------- + -
2 2
See also erf.
Categories: Package distrib
Returns the q-quantile of a Normal(m,s) random variable, with s>0; in other words, this is the inverse of cdf_normal. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
(%i1) load (distrib)$
(%i2) quantile_normal(95/100,0,1);
9
(%o2) sqrt(2) inverse_erf(--)
10
(%i3) float(%);
(%o3) 1.644853626951472
Categories: Package distrib
Returns the mean of a Normal(m,s) random variable, with s>0, namely m. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Normal(m,s) random variable, with s>0, namely s^2. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Normal(m,s) random variable, with s>0, namely s. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Normal(m,s) random variable, with s>0, which is always equal to 0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Normal(m,s) random variable, with s>0, which is always equal to 0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Normal(m,s) random variate, with s>0. Calling random_normal with a third argument n, a random sample of size n will be simulated.
This is an implementation of the Box-Mueller algorithm, as described in Knuth, D.E. (1981) Seminumerical Algorithms. The Art of Computer Programming. Addison-Wesley.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a Student random variable t(n), with n>0 degrees of freedom. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Student random variable t(n), with n>0 degrees of freedom.
(%i1) load (distrib)$
(%i2) cdf_student_t(1/2, 7/3);
7 1 28
beta_incomplete_regularized(-, -, --)
6 2 31
(%o2) 1 - -------------------------------------
2
(%i3) float(%);
(%o3) .6698450596140415
Categories: Package distrib
Returns the q-quantile of a Student random variable t(n), with n>0; in other words, this is the inverse of cdf_student_t. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Student random variable t(n), with n>0, which is always equal to 0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Student random variable t(n), with n>2.
(%i1) load (distrib)$
(%i2) assume(n>2)$ var_student_t(n);
n
(%o3) -----
n - 2
Categories: Package distrib
Returns the standard deviation of a Student random variable t(n), with n>2. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Student random variable t(n), with n>3, which is always equal to 0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Student random variable t(n), with n>4. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Student random variate t(n), with n>0. Calling random_student_t with a second argument m, a random sample of size m will be simulated.
The implemented algorithm is based on the fact that if Z is a normal random variable N(0,1) and S^2 is a chi square random variable with n degrees of freedom, Chi^2(n), then
Z
X = -------------
/ 2 \ 1/2
| S |
| --- |
\ n /
is a Student random variable with n degrees of freedom, t(n).
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a noncentral Student random variable nc_t(n,ncp), with n>0 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first load(distrib).
Sometimes an extra work is necessary to get the final result.
(%i1) load (distrib)$
(%i2) expand(pdf_noncentral_student_t(3,5,0.1));
.01370030107589574 sqrt(5)
(%o2) --------------------------
sqrt(2) sqrt(14) sqrt(%pi)
1.654562884111515E-4 sqrt(5)
+ ----------------------------
sqrt(%pi)
.02434921505438663 sqrt(5)
+ --------------------------
%pi
(%i3) float(%);
(%o3) .02080593159405669
Categories: Package distrib
Returns the value at x of the distribution function of a noncentral Student random variable nc_t(n,ncp), with n>0 degrees of freedom and noncentrality parameter ncp. This function has no closed form and it is numerically computed if the global variable numer equals true or at least one of the arguments is a float, otherwise it returns a nominal expression.
(%i1) load (distrib)$ (%i2) cdf_noncentral_student_t(-2,5,-5); (%o2) cdf_noncentral_student_t(- 2, 5, - 5) (%i3) cdf_noncentral_student_t(-2.0,5,-5); (%o3) .9952030093319743
Categories: Package distrib
Returns the q-quantile of a noncentral Student random variable nc_t(n,ncp), with n>0 degrees of freedom and noncentrality parameter ncp; in other words, this is the inverse of cdf_noncentral_student_t. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a noncentral Student random variable nc_t(n,ncp), with n>1 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first load(distrib).
(%i1) load (distrib)$
(%i2) (assume(df>1), mean_noncentral_student_t(df,k));
df - 1
gamma(------) sqrt(df) k
2
(%o2) ------------------------
df
sqrt(2) gamma(--)
2
Categories: Package distrib
Returns the variance of a noncentral Student random variable nc_t(n,ncp), with n>2 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a noncentral Student random variable nc_t(n,ncp), with n>2 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a noncentral Student random variable nc_t(n,ncp), with n>3 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a noncentral Student random variable nc_t(n,ncp), with n>4 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a noncentral Student random variate nc_t(n,ncp), with n>0. Calling random_noncentral_student_t with a third argument m, a random sample of size m will be simulated.
The implemented algorithm is based on the fact that if X is a normal random variable N(ncp,1) and S^2 is a chi square random variable with n degrees of freedom, Chi^2(n), then
X
U = -------------
/ 2 \ 1/2
| S |
| --- |
\ n /
is a noncentral Student random variable with n degrees of freedom and noncentrality parameter ncp, nc_t(n,ncp).
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a Chi-square random variable Chi^2(n), with n>0.
The Chi^2(n) random variable is equivalent to the Gamma(n/2,2), therefore when Maxima has not enough information to get the result, a noun form based on the gamma density is returned.
(%i1) load (distrib)$
(%i2) pdf_chi2(x,n);
n
(%o2) pdf_gamma(x, -, 2)
2
(%i3) assume(x>0, n>0)$ pdf_chi2(x,n);
n/2 - 1 - x/2
x %e
(%o4) ----------------
n/2 n
2 gamma(-)
2
Categories: Package distrib
Returns the value at x of the distribution function of a Chi-square random variable Chi^2(n), with n>0.
(%i1) load (distrib)$
(%i2) cdf_chi2(3,4);
3
(%o2) 1 - gamma_incomplete_regularized(2, -)
2
(%i3) float(%);
(%o3) .4421745996289256
Categories: Package distrib
Returns the q-quantile of a Chi-square random variable Chi^2(n), with n>0; in other words, this is the inverse of cdf_chi2. Argument q must be an element of [0,1].
This function has no closed form and it is numerically computed if the global variable numer equals true, otherwise it returns a nominal expression based on the gamma quantile function, since the Chi^2(n) random variable is equivalent to the Gamma(n/2,2).
(%i1) load (distrib)$
(%i2) quantile_chi2(0.99,9);
(%o2) 21.66599433346194
(%i3) quantile_chi2(0.99,n);
n
(%o3) quantile_gamma(0.99, -, 2)
2
Categories: Package distrib
Returns the mean of a Chi-square random variable Chi^2(n), with n>0.
The Chi^2(n) random variable is equivalent to the Gamma(n/2,2), therefore when Maxima has not enough information to get the result, a noun form based on the gamma mean is returned.
(%i1) load (distrib)$
(%i2) mean_chi2(n);
n
(%o2) mean_gamma(-, 2)
2
(%i3) assume(n>0)$ mean_chi2(n);
(%o4) n
Categories: Package distrib
Returns the variance of a Chi-square random variable Chi^2(n), with n>0.
The Chi^2(n) random variable is equivalent to the Gamma(n/2,2), therefore when Maxima has not enough information to get the result, a noun form based on the gamma variance is returned.
(%i1) load (distrib)$
(%i2) var_chi2(n);
n
(%o2) var_gamma(-, 2)
2
(%i3) assume(n>0)$ var_chi2(n);
(%o4) 2 n
Categories: Package distrib
Returns the standard deviation of a Chi-square random variable Chi^2(n), with n>0.
The Chi^2(n) random variable is equivalent to the Gamma(n/2,2), therefore when Maxima has not enough information to get the result, a noun form based on the gamma standard deviation is returned.
(%i1) load (distrib)$
(%i2) std_chi2(n);
n
(%o2) std_gamma(-, 2)
2
(%i3) assume(n>0)$ std_chi2(n);
(%o4) sqrt(2) sqrt(n)
Categories: Package distrib
Returns the skewness coefficient of a Chi-square random variable Chi^2(n), with n>0.
The Chi^2(n) random variable is equivalent to the Gamma(n/2,2), therefore when Maxima has not enough information to get the result, a noun form based on the gamma skewness coefficient is returned.
(%i1) load (distrib)$
(%i2) skewness_chi2(n);
n
(%o2) skewness_gamma(-, 2)
2
(%i3) assume(n>0)$ skewness_chi2(n);
2 sqrt(2)
(%o4) ---------
sqrt(n)
Categories: Package distrib
Returns the kurtosis coefficient of a Chi-square random variable Chi^2(n), with n>0.
The Chi^2(n) random variable is equivalent to the Gamma(n/2,2), therefore when Maxima has not enough information to get the result, a noun form based on the gamma kurtosis coefficient is returned.
(%i1) load (distrib)$
(%i2) kurtosis_chi2(n);
n
(%o2) kurtosis_gamma(-, 2)
2
(%i3) assume(n>0)$ kurtosis_chi2(n);
12
(%o4) --
n
Categories: Package distrib
Returns a Chi-square random variate Chi^2(n), with n>0. Calling random_chi2 with a second argument m, a random sample of size m will be simulated.
The simulation is based on the Ahrens-Cheng algorithm. See random_gamma for details.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the q-quantile of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0; in other words, this is the inverse of cdf_noncentral_chi2. Argument q must be an element of [0,1].
This function has no closed form and it is numerically computed if the global variable numer equals true, otherwise it returns a nominal expression.
Categories: Package distrib
Returns the mean of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Categories: Package distrib
Returns the variance of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Categories: Package distrib
Returns the standard deviation of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Categories: Package distrib
Returns the skewness coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Categories: Package distrib
Returns the kurtosis coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Categories: Package distrib
Returns a noncentral Chi-square random variate nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0. Calling random_noncentral_chi2 with a third argument m, a random sample of size m will be simulated.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a F random variable F(m,n), with m,n>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a F random variable F(m,n), with m,n>0.
(%i1) load (distrib)$
(%i2) cdf_f(2,3,9/4);
9 3 3
(%o2) 1 - beta_incomplete_regularized(-, -, --)
8 2 11
(%i3) float(%);
(%o3) 0.66756728179008
Categories: Package distrib
Returns the q-quantile of a F random variable F(m,n), with m,n>0; in other words, this is the inverse of cdf_f. Argument q must be an element of [0,1].
This function has no closed form and it is numerically computed if the global variable numer equals true, otherwise it returns a nominal expression.
(%i1) load (distrib)$
(%i2) quantile_f(2/5,sqrt(3),5);
2
(%o2) quantile_f(-, sqrt(3), 5)
5
(%i3) %,numer;
(%o3) 0.518947838573693
Categories: Package distrib
Returns the mean of a F random variable F(m,n), with m>0, n>2. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a F random variable F(m,n), with m>0, n>4. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a F random variable F(m,n), with m>0, n>4. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a F random variable F(m,n), with m>0, n>6. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a F random variable F(m,n), with m>0, n>8. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a F random variate F(m,n), with m,n>0. Calling random_f with a third argument k, a random sample of size k will be simulated.
The simulation algorithm is based on the fact that if X is a Chi^2(m) random variable and Y is a Chi^2(n) random variable, then
n X
F = ---
m Y
is a F random variable with m and n degrees of freedom, F(m,n).
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of an Exponential(m) random variable, with m>0.
The Exponential(m) random variable is equivalent to the Weibull(1,1/m), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull density is returned.
(%i1) load (distrib)$
(%i2) pdf_exp(x,m);
1
(%o2) pdf_weibull(x, 1, -)
m
(%i3) assume(x>0,m>0)$ pdf_exp(x,m);
- m x
(%o4) m %e
Categories: Package distrib
Returns the value at x of the distribution function of an Exponential(m) random variable, with m>0.
The Exponential(m) random variable is equivalent to the Weibull(1,1/m), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull distribution is returned.
(%i1) load (distrib)$
(%i2) cdf_exp(x,m);
1
(%o2) cdf_weibull(x, 1, -)
m
(%i3) assume(x>0,m>0)$ cdf_exp(x,m);
- m x
(%o4) 1 - %e
Categories: Package distrib
Returns the q-quantile of an Exponential(m) random variable, with m>0; in other words, this is the inverse of cdf_exp. Argument q must be an element of [0,1].
The Exponential(m) random variable is equivalent to the Weibull(1,1/m), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull quantile is returned.
(%i1) load (distrib)$
(%i2) quantile_exp(0.56,5);
(%o2) .1641961104139661
(%i3) quantile_exp(0.56,m);
1
(%o3) quantile_weibull(0.56, 1, -)
m
Categories: Package distrib
Returns the mean of an Exponential(m) random variable, with m>0.
The Exponential(m) random variable is equivalent to the Weibull(1,1/m), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull mean is returned.
(%i1) load (distrib)$
(%i2) mean_exp(m);
1
(%o2) mean_weibull(1, -)
m
(%i3) assume(m>0)$ mean_exp(m);
1
(%o4) -
m
Categories: Package distrib
Returns the variance of an Exponential(m) random variable, with m>0.
The Exponential(m) random variable is equivalent to the Weibull(1,1/m), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull variance is returned.
(%i1) load (distrib)$
(%i2) var_exp(m);
1
(%o2) var_weibull(1, -)
m
(%i3) assume(m>0)$ var_exp(m);
1
(%o4) --
2
m
Categories: Package distrib
Returns the standard deviation of an Exponential(m) random variable, with m>0.
The Exponential(m) random variable is equivalent to the Weibull(1,1/m), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull standard deviation is returned.
(%i1) load (distrib)$
(%i2) std_exp(m);
1
(%o2) std_weibull(1, -)
m
(%i3) assume(m>0)$ std_exp(m);
1
(%o4) -
m
Categories: Package distrib
Returns the skewness coefficient of an Exponential(m) random variable, with m>0.
The Exponential(m) random variable is equivalent to the Weibull(1,1/m), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull skewness coefficient is returned.
(%i1) load (distrib)$
(%i2) skewness_exp(m);
1
(%o2) skewness_weibull(1, -)
m
(%i3) assume(m>0)$ skewness_exp(m);
(%o4) 2
Categories: Package distrib
Returns the kurtosis coefficient of an Exponential(m) random variable, with m>0.
The Exponential(m) random variable is equivalent to the Weibull(1,1/m), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull kurtosis coefficient is returned.
(%i1) load (distrib)$
(%i2) kurtosis_exp(m);
1
(%o2) kurtosis_weibull(1, -)
m
(%i3) assume(m>0)$ kurtosis_exp(m);
(%o4) 6
Categories: Package distrib
Returns an Exponential(m) random variate, with m>0. Calling random_exp with a second argument k, a random sample of size k will be simulated.
The simulation algorithm is based on the general inverse method.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Lognormal(m,s) random variable, with s>0. This function is defined in terms of Maxima's built-in error function erf.
(%i1) load (distrib)$
(%i2) assume(x>0, s>0)$ cdf_lognormal(x,m,s);
log(x) - m
erf(----------)
sqrt(2) s 1
(%o3) --------------- + -
2 2
See also erf.
Categories: Package distrib
Returns the q-quantile of a Lognormal(m,s) random variable, with s>0; in other words, this is the inverse of cdf_lognormal. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
(%i1) load (distrib)$
(%i2) quantile_lognormal(95/100,0,1);
sqrt(2) inverse_erf(9/10)
(%o2) %e
(%i3) float(%);
(%o3) 5.180251602233015
Categories: Package distrib
Returns the mean of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Lognormal(m,s) random variate, with s>0. Calling random_lognormal with a third argument n, a random sample of size n will be simulated.
Log-normal variates are simulated by means of random normal variates. See random_normal for details.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Gamma(a,b) random variable, with a,b>0.
(%i1) load (distrib)$
(%i2) cdf_gamma(3,5,21);
1
(%o2) 1 - gamma_incomplete_regularized(5, -)
7
(%i3) float(%);
(%o3) 4.402663157376807E-7
Categories: Package distrib
Returns the q-quantile of a Gamma(a,b) random variable, with a,b>0; in other words, this is the inverse of cdf_gamma. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Gamma(a,b) random variate, with a,b>0. Calling random_gamma with a third argument n, a random sample of size n will be simulated.
The implemented algorithm is a combinantion of two procedures, depending on the value of parameter a:
For a>=1, Cheng, R.C.H. and Feast, G.M. (1979). Some simple gamma variate generators. Appl. Stat., 28, 3, 290-295.
For 0<a<1, Ahrens, J.H. and Dieter, U. (1974). Computer methods for sampling from gamma, beta, poisson and binomial cdf_tributions. Computing, 12, 223-246.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Beta(a,b) random variable, with a,b>0.
(%i1) load (distrib)$
(%i2) cdf_beta(1/3,15,2);
11
(%o2) --------
14348907
(%i3) float(%);
(%o3) 7.666089131388195E-7
Categories: Package distrib
Returns the q-quantile of a Beta(a,b) random variable, with a,b>0; in other words, this is the inverse of cdf_beta. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Beta(a,b) random variate, with a,b>0. Calling random_beta with a third argument n, a random sample of size n will be simulated.
The implemented algorithm is defined in Cheng, R.C.H. (1978). Generating Beta Variates with Nonintegral Shape Parameters. Communications of the ACM, 21:317-322
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the q-quantile of a Continuous Uniform(a,b) random variable, with a<b; in other words, this is the inverse of cdf_continuous_uniform. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Continuous Uniform(a,b) random variate, with a<b. Calling random_continuous_uniform with a third argument n, a random sample of size n will be simulated.
This is a direct application of the random built-in Maxima function.
See also random. To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the q-quantile of a Logistic(a,b) random variable , with b>0; in other words, this is the inverse of cdf_logistic. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Logistic(a,b) random variate, with b>0. Calling random_logistic with a third argument n, a random sample of size n will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a Pareto(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Pareto(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the q-quantile of a Pareto(a,b) random variable, with a,b>0; in other words, this is the inverse of cdf_pareto. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Pareto(a,b) random variable, with a>1,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Pareto(a,b) random variable, with a>2,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Pareto(a,b) random variable, with a>2,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Pareto(a,b) random variable, with a>3,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Pareto(a,b) random variable, with a>4,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Pareto(a,b) random variate, with a>0,b>0. Calling random_pareto with a third argument n, a random sample of size n will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the q-quantile of a Weibull(a,b) random variable, with a,b>0; in other words, this is the inverse of cdf_weibull. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Weibull(a,b) random variate, with a,b>0. Calling random_weibull with a third argument n, a random sample of size n will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a Rayleigh(b) random variable, with b>0.
The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull density is returned.
(%i1) load (distrib)$
(%i2) pdf_rayleigh(x,b);
1
(%o2) pdf_weibull(x, 2, -)
b
(%i3) assume(x>0,b>0)$ pdf_rayleigh(x,b);
2 2
2 - b x
(%o4) 2 b x %e
Categories: Package distrib
Returns the value at x of the distribution function of a Rayleigh(b) random variable, with b>0.
The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull distribution is returned.
(%i1) load (distrib)$
(%i2) cdf_rayleigh(x,b);
1
(%o2) cdf_weibull(x, 2, -)
b
(%i3) assume(x>0,b>0)$ cdf_rayleigh(x,b);
2 2
- b x
(%o4) 1 - %e
Categories: Package distrib
Returns the q-quantile of a Rayleigh(b) random variable, with b>0; in other words, this is the inverse of cdf_rayleigh. Argument q must be an element of [0,1].
The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull quantile is returned.
(%i1) load (distrib)$
(%i2) quantile_rayleigh(0.99,b);
1
(%o2) quantile_weibull(0.99, 2, -)
b
(%i3) assume(x>0,b>0)$ quantile_rayleigh(0.99,b);
2.145966026289347
(%o4) -----------------
b
Categories: Package distrib
Returns the mean of a Rayleigh(b) random variable, with b>0.
The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull mean is returned.
(%i1) load (distrib)$
(%i2) mean_rayleigh(b);
1
(%o2) mean_weibull(2, -)
b
(%i3) assume(b>0)$ mean_rayleigh(b);
sqrt(%pi)
(%o4) ---------
2 b
Categories: Package distrib
Returns the variance of a Rayleigh(b) random variable, with b>0.
The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull variance is returned.
(%i1) load (distrib)$
(%i2) var_rayleigh(b);
1
(%o2) var_weibull(2, -)
b
(%i3) assume(b>0)$ var_rayleigh(b);
%pi
1 - ---
4
(%o4) -------
2
b
Categories: Package distrib
Returns the standard deviation of a Rayleigh(b) random variable, with b>0.
The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull standard deviation is returned.
(%i1) load (distrib)$
(%i2) std_rayleigh(b);
1
(%o2) std_weibull(2, -)
b
(%i3) assume(b>0)$ std_rayleigh(b);
%pi
sqrt(1 - ---)
4
(%o4) -------------
b
Categories: Package distrib
Returns the skewness coefficient of a Rayleigh(b) random variable, with b>0.
The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull skewness coefficient is returned.
(%i1) load (distrib)$
(%i2) skewness_rayleigh(b);
1
(%o2) skewness_weibull(2, -)
b
(%i3) assume(b>0)$ skewness_rayleigh(b);
3/2
%pi 3 sqrt(%pi)
------ - -----------
4 4
(%o4) --------------------
%pi 3/2
(1 - ---)
4
Categories: Package distrib
Returns the kurtosis coefficient of a Rayleigh(b) random variable, with b>0.
The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b), therefore when Maxima has not enough information to get the result, a noun form based on the Weibull kurtosis coefficient is returned.
(%i1) load (distrib)$
(%i2) kurtosis_rayleigh(b);
1
(%o2) kurtosis_weibull(2, -)
b
(%i3) assume(b>0)$ kurtosis_rayleigh(b);
2
3 %pi
2 - ------
16
(%o4) ---------- - 3
%pi 2
(1 - ---)
4
Categories: Package distrib
Returns a Rayleigh(b) random variate, with b>0. Calling random_rayleigh with a second argument n, a random sample of size n will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the q-quantile of a Laplace(a,b) random variable, with b>0; in other words, this is the inverse of cdf_laplace. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Laplace(a,b) random variate, with b>0. Calling random_laplace with a third argument n, a random sample of size n will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a Cauchy(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Cauchy(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the q-quantile of a Cauchy(a,b) random variable, with b>0; in other words, this is the inverse of cdf_cauchy. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Cauchy(a,b) random variate, with b>0. Calling random_cauchy with a third argument n, a random sample of size n will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the density function of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the q-quantile of a Gumbel(a,b) random variable, with b>0; in other words, this is the inverse of cdf_gumbel. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Gumbel(a,b) random variable, with b>0.
(%i1) load (distrib)$ (%i2) assume(b>0)$ mean_gumbel(a,b); (%o3) %gamma b + a
where symbol %gamma stands for the Euler-Mascheroni constant. See also %gamma.
Categories: Package distrib
Returns the variance of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Gumbel(a,b) random variable, with b>0.
(%i1) load (distrib)$
(%i2) assume(b>0)$ skewness_gumbel(a,b);
12 sqrt(6) zeta(3)
(%o3) ------------------
3
%pi
(%i4) numer:true$ skewness_gumbel(a,b);
(%o5) 1.139547099404649
where zeta stands for the Riemann's zeta function.
Categories: Package distrib
Returns the kurtosis coefficient of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load(distrib).
Categories: Package distrib · Package distrib
Returns a Gumbel(a,b) random variate, with b>0. Calling random_gumbel with a third argument n, a random sample of size n will be simulated.
The implemented algorithm is based on the general inverse method.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
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Returns the value at x of the probability function of a general finite discrete random variable, with vector probabilities v, such that Pr(X=i) = v_i. Vector v
can be a list of nonnegative expressions, whose components will be
normalized to get a vector of probabilities. To make use of this
function, write first load(distrib).
(%i1) load (distrib)$
(%i2) pdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
4
(%o2) -
7
(%i3) pdf_general_finite_discrete(2, [1, 4, 2]);
4
(%o3) -
7
Categories: Package distrib
Returns the value at x of the distribution function of a general finite discrete random variable, with vector probabilities v.
See pdf_general_finite_discrete for more details.
(%i1) load (distrib)$
(%i2) cdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
5
(%o2) -
7
(%i3) cdf_general_finite_discrete(2, [1, 4, 2]);
5
(%o3) -
7
(%i4) cdf_general_finite_discrete(2+1/2, [1, 4, 2]);
5
(%o4) -
7
Categories: Package distrib
Returns the q-quantile of a general finite discrete random variable, with vector probabilities v.
See pdf_general_finite_discrete for more details.
Categories: Package distrib
Returns the mean of a general finite discrete random variable, with vector probabilities v.
See pdf_general_finite_discrete for more details.
Categories: Package distrib
Returns the variance of a general finite discrete random variable, with vector probabilities v.
See pdf_general_finite_discrete for more details.
Categories: Package distrib
Returns the standard deviation of a general finite discrete random variable, with vector probabilities v.
See pdf_general_finite_discrete for more details.
Categories: Package distrib
Returns the skewness coefficient of a general finite discrete random variable, with vector probabilities v.
See pdf_general_finite_discrete for more details.
Categories: Package distrib
Returns the kurtosis coefficient of a general finite discrete random variable, with vector probabilities v.
See pdf_general_finite_discrete for more details.
Categories: Package distrib
Returns a general finite discrete random variate, with vector probabilities v. Calling random_general_finite_discrete with a second argument m, a random sample of size m will be simulated.
See pdf_general_finite_discrete for more details.
(%i1) load (distrib)$ (%i2) random_general_finite_discrete([1,3,1,5]); (%o2) 4 (%i3) random_general_finite_discrete([1,3,1,5], 10); (%o3) [4, 2, 2, 3, 2, 4, 4, 1, 2, 2]
Categories: Package distrib · Random numbers
Returns the value at x of the probability function of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).
4
(%o6) -
7
Returns the value at x of the distribution function of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer.
(%i1) load (distrib)$
(%i2) cdf_binomial(5,7,1/6);
7775
(%o2) ----
7776
(%i3) float(%);
(%o3) .9998713991769548
Categories: Package distrib
Returns the q-quantile of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer; in other words, this is the inverse of cdf_binomial. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Binomial(n,p) random variable, with 0<p<1 and n a positive integer. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Binomial(n,p) random variate, with 0<p<1 and n a positive integer. Calling random_binomial with a third argument m, a random sample of size m will be simulated.
The implemented algorithm is based on the one described in Kachitvichyanukul, V. and Schmeiser, B.W. (1988) Binomial Random Variate Generation. Communications of the ACM, 31, Feb., 216.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the probability function of a Poisson(m) random variable, with m>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Poisson(m) random variable, with m>0.
(%i1) load (distrib)$ (%i2) cdf_poisson(3,5); (%o2) gamma_incomplete_regularized(4, 5) (%i3) float(%); (%o3) .2650259152973623
Categories: Package distrib
Returns the q-quantile of a Poisson(m) random variable, with m>0; in other words, this is the inverse of cdf_poisson. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Poisson(m) random variable, with m>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Poisson(m) random variable, with m>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Poisson(m) random variable, with m>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Poisson(m) random variable, with m>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Poisson random variable Poi(m), with m>0. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Poisson(m) random variate, with m>0. Calling random_poisson with a second argument n, a random sample of size n will be simulated.
The implemented algorithm is the one described in Ahrens, J.H. and Dieter, U. (1982) Computer Generation of Poisson Deviates From Modified Normal Distributions. ACM Trans. Math. Software, 8, 2, June,163-179.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the probability function of a Bernoulli(p) random variable, with 0<p<1.
The Bernoulli(p) random variable is equivalent to the Binomial(1,p), therefore when Maxima has not enough information to get the result, a noun form based on the binomial probability function is returned.
(%i1) load (distrib)$ (%i2) pdf_bernoulli(1,p); (%o2) pdf_binomial(1, 1, p) (%i3) assume(0<p,p<1)$ pdf_bernoulli(1,p); (%o4) p
Categories: Package distrib
Returns the value at x of the distribution function of a Bernoulli(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the q-quantile of a Bernoulli(p) random variable, with 0<p<1; in other words, this is the inverse of cdf_bernoulli. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Bernoulli(p) random variable, with 0<p<1.
The Bernoulli(p) random variable is equivalent to the Binomial(1,p), therefore when Maxima has not enough information to get the result, a noun form based on the binomial mean is returned.
(%i1) load (distrib)$ (%i2) mean_bernoulli(p); (%o2) mean_binomial(1, p) (%i3) assume(0<p,p<1)$ mean_bernoulli(p); (%o4) p
Categories: Package distrib
Returns the variance of a Bernoulli(p) random variable, with 0<p<1.
The Bernoulli(p) random variable is equivalent to the Binomial(1,p), therefore when Maxima has not enough information to get the result, a noun form based on the binomial variance is returned.
(%i1) load (distrib)$ (%i2) var_bernoulli(p); (%o2) var_binomial(1, p) (%i3) assume(0<p,p<1)$ var_bernoulli(p); (%o4) (1 - p) p
Categories: Package distrib
Returns the standard deviation of a Bernoulli(p) random variable, with 0<p<1.
The Bernoulli(p) random variable is equivalent to the Binomial(1,p), therefore when Maxima has not enough information to get the result, a noun form based on the binomial standard deviation is returned.
(%i1) load (distrib)$ (%i2) std_bernoulli(p); (%o2) std_binomial(1, p) (%i3) assume(0<p,p<1)$ std_bernoulli(p); (%o4) sqrt(1 - p) sqrt(p)
Categories: Package distrib
Returns the skewness coefficient of a Bernoulli(p) random variable, with 0<p<1.
The Bernoulli(p) random variable is equivalent to the Binomial(1,p), therefore when Maxima has not enough information to get the result, a noun form based on the binomial skewness coefficient is returned.
(%i1) load (distrib)$
(%i2) skewness_bernoulli(p);
(%o2) skewness_binomial(1, p)
(%i3) assume(0<p,p<1)$ skewness_bernoulli(p);
1 - 2 p
(%o4) -------------------
sqrt(1 - p) sqrt(p)
Categories: Package distrib
Returns the kurtosis coefficient of a Bernoulli(p) random variable, with 0<p<1.
The Bernoulli(p) random variable is equivalent to the Binomial(1,p), therefore when Maxima has not enough information to get the result, a noun form based on the binomial kurtosis coefficient is returned.
(%i1) load (distrib)$
(%i2) kurtosis_bernoulli(p);
(%o2) kurtosis_binomial(1, p)
(%i3) assume(0<p,p<1)$ kurtosis_bernoulli(p);
1 - 6 (1 - p) p
(%o4) ---------------
(1 - p) p
Categories: Package distrib
Returns a Bernoulli(p) random variate, with 0<p<1. Calling random_bernoulli with a second argument n, a random sample of size n will be simulated.
This is a direct application of the random built-in Maxima function.
See also random. To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the probability function of a Geometric(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Geometric(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the q-quantile of a Geometric(p) random variable, with 0<p<1; in other words, this is the inverse of cdf_geometric. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Geometric(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Geometric(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Geometric(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Geometric(p) random variable, with 0<p<1. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a geometric random variable Geo(p), with 0<p<1. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Geometric(p) random variate, with 0<p<1. Calling random_geometric with a second argument n, a random sample of size n will be simulated.
The algorithm is based on simulation of Bernoulli trials.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the probability function of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the q-quantile of a Discrete Uniform(n) random variable, with n a strictly positive integer; in other words, this is the inverse of cdf_discrete_uniform. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Discrete Uniform(n) random variate, with n a strictly positive integer. Calling random_discrete_uniform with a second argument m, a random sample of size m will be simulated.
This is a direct application of the random built-in Maxima function.
See also random. To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the probability function of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. Being n1 the number of objects of class A, n2 the number of objects of class B, and n the size of the sample without replacement, this function returns the probability of event "exactly x objects are of class A".
To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Hypergeometric(n1,n2,n)
random variable, with n1, n2 and n non negative integers and n<=n1+n2.
See pdf_hypergeometric for a more complete description.
To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the q-quantile of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2; in other words, this is the inverse of cdf_hypergeometric. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a discrete uniform random variable Hyp(n1,n2,n), with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a hypergeometric random variable Hyp(n1,n2,n), with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Hypergeometric(n1,n2,n) random variate, with n1, n2 and n non negative integers and n<=n1+n2. Calling random_hypergeometric with a fourth argument m, a random sample of size m will be simulated.
Algorithm described in Kachitvichyanukul, V., Schmeiser, B.W. (1985) Computer generation of hypergeometric random variates. Journal of Statistical Computation and Simulation 22, 127-145.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
Returns the value at x of the probability function of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive number. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the value at x of the distribution function of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive number.
(%i1) load (distrib)$
(%i2) cdf_negative_binomial(3,4,1/8);
3271
(%o2) ------
524288
(%i3) float(%);
(%o3) .006238937377929687
Categories: Package distrib
Returns the q-quantile of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive number; in other words, this is the inverse of cdf_negative_binomial. Argument q must be an element of [0,1]. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the mean of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive number. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the variance of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive number. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the standard deviation of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive number. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the skewness coefficient of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive number. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns the kurtosis coefficient of a Negative Binomial(n,p) random variable, with 0<p<1 and n a positive number. To make use of this function, write first load(distrib).
Categories: Package distrib
Returns a Negative Binomial(n,p) random variate, with 0<p<1 and n a positive number. Calling random_negative_binomial with a third argument m, a random sample of size m will be simulated.
Algorithm described in Devroye, L. (1986) Non-Uniform Random Variate Generation. Springer Verlag, p. 480.
To make use of this function, write first load(distrib).
Categories: Package distrib · Random numbers
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 52.1 Introduction to draw | ||
| 52.2 Functions and Variables for draw | ||
| 52.3 Functions and Variables for pictures | ||
| 52.4 Functions and Variables for worldmap |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
draw is a Maxima-Gnuplot and a Maxima-vtk interface.
There are three main functions to be used at Maxima level:
draw2d, draw3d and draw.
Follow this links for more elaborated examples of this package:
http://riotorto.users.sourceforge.net/gnuplot
http://riotorto.users.sourceforge.net/vtk
You need Gnuplot 4.2 or newer to run this program.
Categories: Plotting · Share packages · Package draw
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Function gr2d builds an object describing a 2D scene. Arguments are
graphic options, graphic objects, or lists containing both graphic options and objects.
This scene is interpreted sequentially: graphic options affect those graphic objects
placed on its right. Some graphic options affect the global appearance of the scene.
This is the list of graphic objects available for scenes in two dimensions:
bars, ellipse, explicit, image, implicit, label,
parametric, points, polar, polygon, quadrilateral,
rectangle, triangle, vector and geomap
(this one defined in package worldmap).
(%i1) load("draw");
(%o1) /maxima/share/draw/draw.lisp
(%i2) draw(
gr2d(
key="sin (x)",grid=[2,2],
explicit(
sin(x),
x,0,2*%pi
)
),
gr2d(
key="cos (x)",grid=[2,2],
explicit(
cos(x),
x,0,2*%pi
)
)
);
(%o2) [gr2d(explicit), gr2d(explicit)]
Categories: Package draw
Function gr3d builds an object describing a 3d scene. Arguments are
graphic options, graphic objects, or lists containing both graphic options
and objects. This scene is interpreted sequentially: graphic options affect those
graphic objects placed on its right. Some graphic options affect the global
appearance of the scene.
This is the list of graphic objects available for scenes in three
dimensions:
cylindrical, elevation_grid, explicit, implicit,
label, mesh, parametric,
parametric_surface, points, quadrilateral,
spherical, triangle, tube,
vector, and geomap (this one defined in package worldmap).
Categories: Package draw
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Plots a series of scenes; its arguments are gr2d and/or gr3d
objects, together with some options, or lists of scenes and options.
By default, the scenes are put together
in one column.
Function draw accepts the following global options: terminal,
columns, dimensions, file_name and delay.
Functions draw2d and draw3d are short cuts to be used
when only one scene is required, in two or three dimensions, respectively.
Example:
(%i1) scene1: gr2d(title="Ellipse",
nticks=30,
parametric(2*cos(t),5*sin(t),t,0,2*%pi))$
(%i2) scene2: gr2d(title="Triangle",
polygon([4,5,7],[6,4,2]))$
(%i3) draw(scene1, scene2, columns = 2)$
The two draw sentences are equivalent:
(%i1) draw(gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1))); (%o1) [gr3d(explicit)] (%i2) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1)); (%o2) [gr3d(explicit)]
An animated gif file:
(%i1) draw(
delay = 100,
file_name = "zzz",
terminal = 'animated_gif,
gr2d(explicit(x^2,x,-1,1)),
gr2d(explicit(x^3,x,-1,1)),
gr2d(explicit(x^4,x,-1,1)));
End of animation sequence
(%o1) [gr2d(explicit), gr2d(explicit), gr2d(explicit)]
gr2d, gr3d, draw2d and draw3d.
Categories: Package draw · File output
This function is a short cut for
draw(gr2d(options, ..., graphic_object, ...)).
It can be used to plot a unique scene in 2d.
Categories: Package draw · File output
This function is a short cut for
draw(gr3d(options, ..., graphic_object, ...)).
It can be used to plot a unique scene in 3d.
Categories: Package draw · File output
Saves the current plot into a file. Accepted graphics options are:
terminal, dimensions and file_name.
Example:
(%i1) /* screen plot */
draw(gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1)))$
(%i2) /* same plot in eps format */
draw_file(terminal = eps,
dimensions = [5,5]) $
Categories: Package draw · File output
This function enables Maxima to work in one-window multiplot mode with terminal
term; accepted arguments for this function are screen,
wxt, aquaterm and none.
When multiplot mode is enabled, each call to draw sends a new plot to the
same window, without erasing the previous ones. To disable the multiplot mode,
write multiplot_mode(none).
When multiplot mode is enabled, global option terminal is blocked and you
have to disable this working mode before changing to another terminal.
This feature does not work in Windows platforms.
Example:
(%i1) set_draw_defaults(
xrange = [-1,1],
yrange = [-1,1],
grid = true,
title = "Step by step plot" )$
(%i2) multiplot_mode(screen)$
(%i3) draw2d(color=blue, explicit(x^2,x,-1,1))$
(%i4) draw2d(color=red, explicit(x^3,x,-1,1))$
(%i5) draw2d(color=brown, explicit(x^4,x,-1,1))$
(%i6) multiplot_mode(none)$
Categories: Package draw · File output
Sets user graphics options. This function is useful for plotting a sequence of graphics with common graphics options. Calling this function without arguments removes user defaults.
Example:
(%i1) set_draw_defaults(
xrange = [-10,10],
yrange = [-2, 2],
color = blue,
grid = true)$
(%i2) /* plot with user defaults */
draw2d(explicit(((1+x)**2/(1+x*x))-1,x,-10,10))$
(%i3) set_draw_defaults()$
(%i4) /* plot with standard defaults */
draw2d(explicit(((1+x)**2/(1+x*x))-1,x,-10,10))$
Categories: Package draw
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Default value: 10
adapt_depth is the maximum number of splittings used by the adaptive plotting routine.
This option is relevant only for 2d explicit functions.
See also nticks
Categories: Package draw
Default value: false
With option allocation it is possible to place a scene in the
output window at will; this is of interest in multiplots. When false,
the scene is placed automatically, depending on the value assigned to option
columns. In any other case, allocation must be set to a list of
two pairs of numbers; the first corresponds to the position of the lower left
corner of the scene, and the second pair gives the width and height of the plot.
All quantities must be given in relative coordinates, between 0 and 1.
Examples:
In site graphics.
(%i1) draw(
gr2d(
explicit(x^2,x,-1,1)),
gr2d(
allocation = [[1/4, 1/4],[1/2, 1/2]],
explicit(x^3,x,-1,1),
grid = true) ) $
Multiplot with selected dimensions.
(%i1) draw(
terminal = wxt,
gr2d(
grid=[5,5],
allocation = [[0, 0],[1, 1/4]],
explicit(x^2,x,-1,1)),
gr3d(
allocation = [[0, 1/4],[1, 3/4]],
explicit(x^2+y^2,x,-1,1,y,-1,1) ))$
See also option columns.
Categories: Package draw
Default value: true
If axis_3d is true, the x, y and z axis are shown in 3d scenes.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d(axis_3d = false,
explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$
See also axis_bottom, axis_left, axis_top, and axis_right for axis in 2d.
Categories: Package draw
Default value: true
If axis_bottom is true, the bottom axis is shown in 2d scenes.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(axis_bottom = false,
explicit(x^3,x,-1,1))$
See also axis_left, axis_top, axis_right and axis_3d.
Categories: Package draw
Default value: true
If axis_left is true, the left axis is shown in 2d scenes.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(axis_left = false,
explicit(x^3,x,-1,1))$
See also axis_bottom, axis_top, axis_right and axis_3d.
Categories: Package draw
Default value: true
If axis_right is true, the right axis is shown in 2d scenes.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(axis_right = false,
explicit(x^3,x,-1,1))$
See also axis_bottom, axis_left, axis_top and axis_3d.
Categories: Package draw
Default value: true
If axis_top is true, the top axis is shown in 2d scenes.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(axis_top = false,
explicit(x^3,x,-1,1))$
See also axis_bottom, axis_left, axis_right, and axis_3d.
Categories: Package draw
Default value: white
Sets the background color for terminals. Default background color is white.
Since this is a global graphics option, its position in the scene description does not matter.
This option das not work with terminals epslatex and epslatex_standalone.
See also color
Categories: Package draw
Default value: true
If border is true, borders of polygons are painted
according to line_type and line_width.
This option affects the following graphic objects:
Example:
(%i1) draw2d(color = brown,
line_width = 8,
polygon([[3,2],[7,2],[5,5]]),
border = false,
fill_color = blue,
polygon([[5,2],[9,2],[7,5]]) )$
Categories: Package draw
Default value: [false, false]
A list with two possible elements, true and false,
indicating whether the extremes of a graphic object tube remain closed
or open. By default, both extremes are left open.
Setting capping = false is equivalent to capping = [false, false],
and capping = true is equivalent to capping = [true, true].
Example:
(%i1) draw3d(
capping = [false, true],
tube(0, 0, a, 1,
a, 0, 8) )$
Categories: Package draw
Default value: auto
If cbrange is auto, the range for the values which are
colored when enhanced3d is not false is computed
automatically. Values outside of the color range use color of the
nearest extreme.
When enhanced3d or colorbox is false, option cbrange has
no effect.
If the user wants a specific interval for the colored values, it must
be given as a Maxima list, as in cbrange=[-2, 3].
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d (
enhanced3d = true,
color = green,
cbrange = [-3,10],
explicit(x^2+y^2, x,-2,2,y,-2,2)) $
See also enhanced3d, colorbox and cbtics.
Categories: Package draw
Default value: auto
This graphic option controls the way tic marks are drawn on the colorbox
when option enhanced3d is not false.
When enhanced3d or colorbox is false, option cbtics has
no effect.
See xtics for a complete description.
Example :
(%i1) draw3d (
enhanced3d = true,
color = green,
cbtics = {["High",10],["Medium",05],["Low",0]},
cbrange = [0, 10],
explicit(x^2+y^2, x,-2,2,y,-2,2)) $
See also enhanced3d, colorbox and cbrange.
Categories: Package draw
Default value: blue
color specifies the color for plotting lines, points, borders of
polygons and labels.
Colors can be given as names or in hexadecimal rgb code.
Available color names are:
white black gray0 grey0 gray10 grey10 gray20 grey20 gray30 grey30 gray40 grey40 gray50 grey50 gray60 grey60 gray70 grey70 gray80 grey80 gray90 grey90 gray100 grey100 gray grey light_gray light_grey dark_gray dark_grey red light_red dark_red yellow light_yellow dark_yellow green light_green dark_green spring_green forest_green sea_green blue light_blue dark_blue midnight_blue navy medium_blue royalblue skyblue cyan light_cyan dark_cyan magenta light_magenta dark_magenta turquoise light_turquoise dark_turquoise pink light_pink dark_pink coral light_coral orange_red salmon light_salmon dark_salmon aquamarine khaki dark_khaki goldenrod light_goldenrod dark_goldenrod gold beige brown orange dark_orange violet dark_violet plum purple
Cromatic componentes in hexadecimal code are introduced in the form "#rrggbb".
Example:
(%i1) draw2d(explicit(x^2,x,-1,1), /* default is black */
color = red,
explicit(0.5 + x^2,x,-1,1),
color = blue,
explicit(1 + x^2,x,-1,1),
color = light_blue,
explicit(1.5 + x^2,x,-1,1),
color = "#23ab0f",
label(["This is a label",0,1.2]) )$
See also fill_color.
Categories: Package draw
Default value: true
If colorbox is true, a color scale without label is drawn together with
image 2D objects, or coloured 3d objects. If colorbox is false, no
color scale is shown. If colorbox is a string, a color scale with label is drawn.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
Color scale and images.
(%i1) im: apply('matrix,
makelist(makelist(random(200),i,1,30),i,1,30))$
(%i2) draw(
gr2d(image(im,0,0,30,30)),
gr2d(colorbox = false, image(im,0,0,30,30))
)$
(%i1) draw3d(
colorbox = "Magnitude",
enhanced3d = true,
explicit(x^2+y^2,x,-1,1,y,-1,1))$
See also palette_draw.
Categories: Package draw
Default value: 1
columns is the number of columns in multiple plots.
Since this is a global graphics option, its position in the scene description
does not matter. It can be also used as an argument of function draw.
Example:
(%i1) scene1: gr2d(title="Ellipse",
nticks=30,
parametric(2*cos(t),5*sin(t),t,0,2*%pi))$
(%i2) scene2: gr2d(title="Triangle",
polygon([4,5,7],[6,4,2]))$
(%i3) draw(scene1, scene2, columns = 2)$
Categories: Package draw
Default value: none
Option contour enables the user to select where to plot contour lines.
Possible values are:
none:
no contour lines are plotted.
base:
contour lines are projected on the xy plane.
surface:
contour lines are plotted on the surface.
both:
two contour lines are plotted: on the xy plane and on the surface.
map:
contour lines are projected on the xy plane, and the view point is
set just in the vertical.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d(explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
contour_levels = 15,
contour = both,
surface_hide = true) $
(%i1) draw3d(explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
contour_levels = 15,
contour = map
) $
Categories: Package draw
Default value: 5
This graphic option controls the way contours are drawn.
contour_levels can be set to a positive integer number, a list of three
numbers or an arbitrary set of numbers:
contour_levels is bounded to positive integer n,
n contour lines will be drawn at equal intervals. By default, five
equally spaced contours are plotted.
contour_levels is bounded to a list of length three of the
form [lowest,s,highest], contour lines are plotted from lowest
to highest in steps of s.
contour_levels is bounded to a set of numbers of the
form {n1, n2, ...}, contour lines are plotted at values n1,
n2, ...
Since this is a global graphics option, its position in the scene description does not matter.
Examples:
Ten equally spaced contour lines. The actual number of levels can be adjusted to give simple labels.
(%i1) draw3d(color = green,
explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
contour_levels = 10,
contour = both,
surface_hide = true) $
From -8 to 8 in steps of 4.
(%i1) draw3d(color = green,
explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
contour_levels = [-8,4,8],
contour = both,
surface_hide = true) $
Isolines at levels -7, -6, 0.8 and 5.
(%i1) draw3d(color = green,
explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
contour_levels = {-7, -6, 0.8, 5},
contour = both,
surface_hide = true) $
See also contour.
Categories: Package draw
Default value: "data.gnuplot"
This is the name of the file with the numeric data needed by Gnuplot to build the requested plot.
Since this is a global graphics option, its position in the scene description
does not matter. It can be also used as an argument of function draw.
See example in gnuplot_file_name.
Categories: Package draw
Default value: 5
This is the delay in 1/100 seconds of frames in animated gif files.
Since this is a global graphics option, its position in the scene description
does not matter. It can be also used as an argument of function draw.
Example:
(%i1) draw(
delay = 100,
file_name = "zzz",
terminal = 'animated_gif,
gr2d(explicit(x^2,x,-1,1)),
gr2d(explicit(x^3,x,-1,1)),
gr2d(explicit(x^4,x,-1,1)));
End of animation sequence
(%o2) [gr2d(explicit), gr2d(explicit), gr2d(explicit)]
Option delay is only active in animated gif's; it is ignored in
any other case.
See also terminal, and dimensions.
Categories: Package draw
Default value: [600,500]
Dimensions of the output terminal. Its value is a list formed by the width and the height. The meaning of the two numbers depends on the terminal you are working with.
With terminals gif, animated_gif, png, jpg,
svg, screen, wxt, and aquaterm,
the integers represent the number of points in each direction. If they
are not intergers, they are rounded.
With terminals eps, eps_color, pdf, and
pdfcairo, both numbers represent hundredths of cm, which
means that, by default, pictures in these formats are 6 cm in
width and 5 cm in height.
Since this is a global graphics option, its position in the scene description
does not matter. It can be also used as an argument of function draw.
Examples:
Option dimensions applied to file output
and to wxt canvas.
(%i1) draw2d(
dimensions = [300,300],
terminal = 'png,
explicit(x^4,x,-1,1)) $
(%i2) draw2d(
dimensions = [300,300],
terminal = 'wxt,
explicit(x^4,x,-1,1)) $
Option dimensions applied to eps output.
We want an eps file with A4 portrait dimensions.
(%i1) A4portrait: 100*[21, 29.7]$
(%i2) draw3d(
dimensions = A4portrait,
terminal = 'eps,
explicit(x^2-y^2,x,-2,2,y,-2,2)) $
Categories: Package draw
Default value: true
When true, functions to be drawn are considered as complex functions whose
real part value should be plotted; when false, nothing will be plotted when
the function does not give a real value.
This option affects objects explicit and parametric in 2D and 3D, and
parametric_surface.
Example:
Option draw_realpart affects objects explicit and parametric.
(%i1) draw2d(
draw_realpart = false,
explicit(sqrt(x^2 - 4*x) - x, x, -1, 5),
color = red,
draw_realpart = true,
parametric(x,sqrt(x^2 - 4*x) - x + 1, x, -1, 5) );
Categories: Package draw
Default value: none
If enhanced3d is none, surfaces are not colored in 3D plots.
In order to get a colored surface, a list must be assigned to option
enhanced3d, where the first element is an expression and the rest
are the names of the variables or parameters used in that expression. A list such
[f(x,y,z), x, y, z] means that point [x,y,z] of the surface
is assigned number f(x,y,z), which will be colored according to
the actual palette. For those 3D graphic objects defined in terms of
parameters, it is possible to define the color number in terms of
the parameters, as in [f(u), u], as in objects parametric and
tube, or [f(u,v), u, v], as in object parametric_surface.
While all 3D objects admit the model based on absolute coordinates,
[f(x,y,z), x, y, z], only two of them, namely explicit
and elevation_grid, accept also models defined on the [x,y] coordinates,
[f(x,y), x, y]. 3D graphic object implicit accepts only the
[f(x,y,z), x, y, z] model. Object points accepts also the
[f(x,y,z), x, y, z] model, but when points have a chronological nature,
model [f(k), k] is also valid, being k an ordering parameter.
When enhanced3d is assigned something different to none, options
color and surface_hide are ignored.
The names of the variables defined in the lists may be different to those used in the definitions of the graphic objects.
In order to maintain back compatibility, enhanced3d = false is equivalent
to enhanced3d = none, and enhanced3d = true is equivalent to
enhanced3d = [z, x, y, z]. If an expression is given to enhanced3d,
its variables must be the same used in the surface definition. This is not
necessary when using lists.
See option palette to learn how palettes are specified.
Examples:
explicit object with coloring defined by the [f(x,y,z), x, y, z] model.
(%i1) draw3d(
enhanced3d = [x-z/10,x,y,z],
palette = gray,
explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$
explicit object with coloring defined by the [f(x,y), x, y] model.
The names of the variables defined in the lists may be different to those
used in the definitions of the graphic objects; in this case, r corresponds
to x, and s to y.
(%i1) draw3d(
enhanced3d = [sin(r*s),r,s],
explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$
parametric object with coloring defined by the [f(x,y,z), x, y, z] model.
(%i1) draw3d(
nticks = 100,
line_width = 2,
enhanced3d = [if y>= 0 then 1 else 0, x, y, z],
parametric(sin(u)^2,cos(u),u,u,0,4*%pi)) $
parametric object with coloring defined by the [f(u), u] model.
In this case, (u-1)^2 is a shortcut for [(u-1)^2,u].
(%i1) draw3d(
nticks = 60,
line_width = 3,
enhanced3d = (u-1)^2,
parametric(cos(5*u)^2,sin(7*u),u-2,u,0,2))$
elevation_grid object with coloring defined by the [f(x,y), x, y] model.
(%i1) m: apply(
matrix,
makelist(makelist(cos(i^2/80-k/30),k,1,30),i,1,20)) $
(%i2) draw3d(
enhanced3d = [cos(x*y*10),x,y],
elevation_grid(m,-1,-1,2,2),
xlabel = "x",
ylabel = "y");
tube object with coloring defined by the [f(x,y,z), x, y, z] model.
(%i1) draw3d(
enhanced3d = [cos(x-y),x,y,z],
palette = gray,
xu_grid = 50,
tube(cos(a), a, 0, 1, a, 0, 4*%pi) )$
tube object with coloring defined by the [f(u), u] model.
Here, enhanced3d = -a would be the shortcut for enhanced3d = [-foo,foo].
(%i1) draw3d(
capping = [true, false],
palette = [26,15,-2],
enhanced3d = [-foo, foo],
tube(a, a, a^2, 1, a, -2, 2) )$
implicit and points objects with coloring defined by the [f(x,y,z), x, y, z] model.
(%i1) draw3d(
enhanced3d = [x-y,x,y,z],
implicit((x^2+y^2+z^2-1)*(x^2+(y-1.5)^2+z^2-0.5)=0.015,
x,-1,1,y,-1.2,2.3,z,-1,1)) $
(%i2) m: makelist([random(1.0),random(1.0),random(1.0)],k,1,2000)$
(%i3) draw3d(
point_type = filled_circle,
point_size = 2,
enhanced3d = [u+v-w,u,v,w],
points(m) ) $
When points have a chronological nature, model [f(k), k] is also valid,
being k an ordering parameter.
(%i1) m:makelist([random(1.0), random(1.0), random(1.0)],k,1,5)$
(%i2) draw3d(
enhanced3d = [sin(j), j],
point_size = 3,
point_type = filled_circle,
points_joined = true,
points(m)) $
Categories: Package draw
Default value: y
Depending on its value, which can be x, y, or xy,
graphic object errors will draw points with horizontal, vertical,
or both, error bars. When error_type=boxes, boxes will be drawn
instead of crosses.
See also errors.
Default value: "maxima_out"
This is the name of the file where terminals png, jpg, gif,
eps, eps_color, pdf, pdfcairo and svg
will save the graphic.
Since this is a global graphics option, its position in the scene description
does not matter. It can be also used as an argument of function draw.
Example:
(%i1) draw2d(file_name = "myfile",
explicit(x^2,x,-1,1),
terminal = 'png)$
See also terminal, dimensions_draw.
Categories: Package draw
Default value: "red"
fill_color specifies the color for filling polygons and
2d explicit functions.
See color to learn how colors are specified.
Categories: Package draw
Default value: 0
fill_density is a number between 0 and 1 that specifies
the intensity of the fill_color in bars objects.
See bars for examples.
Default value: false
Option filled_func controls how regions limited by functions
should be filled. When filled_func is true, the region
bounded by the function defined with object explicit and the
bottom of the graphic window is filled with fill_color. When
filled_func contains a function expression, then the region bounded
by this function and the function defined with object explicit
will be filled. By default, explicit functions are not filled.
A useful special case is filled_func=0, which generates the region
bond by the horizontal axis and the explicit function.
This option affects only the 2d graphic object explicit.
Example:
Region bounded by an explicit object and the bottom of the
graphic window.
(%i1) draw2d(fill_color = red,
filled_func = true,
explicit(sin(x),x,0,10) )$
Region bounded by an explicit object and the function
defined by option filled_func. Note that the variable in
filled_func must be the same as that used in explicit.
(%i1) draw2d(fill_color = grey,
filled_func = sin(x),
explicit(-sin(x),x,0,%pi));
fill_color and explicit.
Categories: Package draw
Default value: "" (empty string)
This option can be used to set the font face to be used by the terminal. Only one font face and size can be used throughout the plot.
Since this is a global graphics option, its position in the scene description does not matter.
See also font_size.
Gnuplot doesn't handle fonts by itself, it leaves this task to the support libraries of the different terminals, each one with its own philosophy about it. A brief summary follows:
Example:
(%i1) draw2d(font = "Arial",
font_size = 20,
label(["Arial font, size 20",1,1]))$
GDFONTPATH; in this case, it is only necessary to
set option font to the font's name. It is also possible to
give the complete path to the font file.
Examples:
Option font can be given the complete path to the font file:
(%i1) path: "/usr/share/fonts/truetype/freefont/" $
(%i2) file: "FreeSerifBoldItalic.ttf" $
(%i3) draw2d(
font = concat(path, file),
font_size = 20,
color = red,
label(["FreeSerifBoldItalic font, size 20",1,1]),
terminal = png)$
If environment variable GDFONTPATH is set to the
path where font files are allocated, it is possible to
set graphic option font to the name of the font.
(%i1) draw2d(
font = "FreeSerifBoldItalic",
font_size = 20,
color = red,
label(["FreeSerifBoldItalic font, size 20",1,1]),
terminal = png)$
"Times-Roman", "Times-Italic", "Times-Bold",
"Times-BoldItalic","Helvetica", "Helvetica-Oblique", "Helvetica-Bold","Helvetic-BoldOblique", "Courier",
"Courier-Oblique", "Courier-Bold","Courier-BoldOblique".
Example:
(%i1) draw2d(
font = "Courier-Oblique",
font_size = 15,
label(["Courier-Oblique font, size 15",1,1]),
terminal = eps)$
fontconfig utility.
"Times-Roman".
The gnuplot documentation is an important source of information about terminals and fonts.
Categories: Package draw
Default value: 10
This option can be used to set the font size to be used by the terminal.
Only one font face and size can be used throughout the plot. font_size is
active only when option font is not equal to the empty string.
Since this is a global graphics option, its position in the scene description does not matter.
See also font.
Categories: Package draw
Default value: "maxout_xxx.gnuplot" with "xxx"
being a number that is unique to each concurrently-running
maxima process.
This is the name of the file with the necessary commands to be processed by Gnuplot.
Since this is a global graphics option, its position in the scene description
does not matter. It can be also used as an argument of function draw.
Example:
(%i1) draw2d(
file_name = "my_file",
gnuplot_file_name = "my_commands_for_gnuplot",
data_file_name = "my_data_for_gnuplot",
terminal = png,
explicit(x^2,x,-1,1)) $
See also data_file_name.
Categories: Package draw
Default value: false
If grid is not false, a grid will be drawn on the xy plane.
If grid is assigned true, one grid line per tick of each axis is drawn.
If grid is assigned a list nx,ny with [nx,ny] > [0,0]
instead nx lines per tick of the x axis and ny lines per tick of
the y axis are drawn.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(grid = true,
explicit(exp(u),u,-2,2))$
(%i1) draw2d(grid = [2,2],
explicit(sin(x),x,0,2*%pi))$
Categories: Package draw
Default value: 45
head_angle indicates the angle, in degrees, between the arrow heads and
the segment.
This option is relevant only for vector objects.
Example:
(%i1) draw2d(xrange = [0,10],
yrange = [0,9],
head_length = 0.7,
head_angle = 10,
vector([1,1],[0,6]),
head_angle = 20,
vector([2,1],[0,6]),
head_angle = 30,
vector([3,1],[0,6]),
head_angle = 40,
vector([4,1],[0,6]),
head_angle = 60,
vector([5,1],[0,6]),
head_angle = 90,
vector([6,1],[0,6]),
head_angle = 120,
vector([7,1],[0,6]),
head_angle = 160,
vector([8,1],[0,6]),
head_angle = 180,
vector([9,1],[0,6]) )$
See also head_both, head_length, and head_type.
Categories: Package draw
Default value: false
If head_both is true, vectors are plotted with two arrow heads.
If false, only one arrow is plotted.
This option is relevant only for vector objects.
Example:
(%i1) draw2d(xrange = [0,8],
yrange = [0,8],
head_length = 0.7,
vector([1,1],[6,0]),
head_both = true,
vector([1,7],[6,0]) )$
See also head_length, head_angle, and head_type.
Categories: Package draw
Default value: 2
head_length indicates, in x-axis units, the length of arrow heads.
This option is relevant only for vector objects.
Example:
(%i1) draw2d(xrange = [0,12],
yrange = [0,8],
vector([0,1],[5,5]),
head_length = 1,
vector([2,1],[5,5]),
head_length = 0.5,
vector([4,1],[5,5]),
head_length = 0.25,
vector([6,1],[5,5]))$
See also head_both, head_angle, and head_type.
Categories: Package draw
Default value: filled
head_type is used to specify how arrow heads are plotted. Possible
values are: filled (closed and filled arrow heads), empty
(closed but not filled arrow heads), and nofilled (open arrow heads).
This option is relevant only for vector objects.
Example:
(%i1) draw2d(xrange = [0,12],
yrange = [0,10],
head_length = 1,
vector([0,1],[5,5]), /* default type */
head_type = 'empty,
vector([3,1],[5,5]),
head_type = 'nofilled,
vector([6,1],[5,5]))$
See also head_both, head_angle, and head_length.
Categories: Package draw
Default value: false
This option is relevant only when enhanced3d is not false.
When interpolate_color is false, surfaces are colored with
homogeneous quadrangles. When true, color transitions are smoothed
by interpolation.
interpolate_color also accepts a list of two numbers, [m,n].
For positive m and n, each quadrangle or triangle is interpolated
m times and n times in the respective direction. For negative
m and n, the interpolation frequency is chosen so that there will be at least
|m| and |n| points drawn; you can consider this as a special gridding function.
Zeros, i.e. interpolate_color=[0,0], will automatically choose an
optimal number of interpolated surface points.
Also, interpolate_color=true is equivalent to interpolate_color=[0,0].
Examples:
Color interpolation with explicit functions.
(%i1) draw3d(
enhanced3d = sin(x*y),
explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $
(%i2) draw3d(
interpolate_color = true,
enhanced3d = sin(x*y),
explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $
(%i3) draw3d(
interpolate_color = [-10,0],
enhanced3d = sin(x*y),
explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $
Color interpolation with the mesh graphic object.
Interpolating colors in parametric surfaces can give unexpected results.
(%i1) draw3d(
enhanced3d = true,
mesh([[1,1,3], [7,3,1],[12,-2,4],[15,0,5]],
[[2,7,8], [4,3,1],[10,5,8], [12,7,1]],
[[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $
(%i2) draw3d(
enhanced3d = true,
interpolate_color = true,
mesh([[1,1,3], [7,3,1],[12,-2,4],[15,0,5]],
[[2,7,8], [4,3,1],[10,5,8], [12,7,1]],
[[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $
(%i3) draw3d(
enhanced3d = true,
interpolate_color = true,
view=map,
mesh([[1,1,3], [7,3,1],[12,-2,4],[15,0,5]],
[[2,7,8], [4,3,1],[10,5,8], [12,7,1]],
[[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $
See also enhanced3d.
Categories: Package draw
Default value: [50, 50]
ip_grid sets the grid for the first sampling in implicit plots.
This option is relevant only for implicit objects.
Categories: Package draw
Default value: [5, 5]
ip_grid_in sets the grid for the second sampling in implicit plots.
This option is relevant only for implicit objects.
Categories: Package draw
Default value: "" (empty string)
key is the name of a function in the legend. If key is an
empty string, no key is assigned to the function.
This option affects the following graphic objects:
gr2d: points, polygon, rectangle,
ellipse, vector, explicit, implicit,
parametric and polar.
gr3d: points, explicit, parametric
and parametric_surface.
Example:
(%i1) draw2d(key = "Sinus",
explicit(sin(x),x,0,10),
key = "Cosinus",
color = red,
explicit(cos(x),x,0,10) )$
Default value: "" (empty string)
key_pos defines at which position the legend will be drawn. If key is an
empty string, "top_right" is used.
Available position specifiers are: top_left, top_center, top_right,
center_left, center, center_right,
bottom_left, bottom_center, and bottom_right.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(
key_pos = top_left,
key = "x",
explicit(x, x,0,10),
color= red,
key = "x squared",
explicit(x^2,x,0,10))$
(%i3) draw3d(
key_pos = center,
key = "x",
explicit(x+y,x,0,10,y,0,10),
color= red,
key = "x squared",
explicit(x^2+y^2,x,0,10,y,0,10))$
Categories: Package draw
Default value: center
label_alignment is used to specify where to write labels with
respect to the given coordinates. Possible values are: center,
left, and right.
This option is relevant only for label objects.
Example:
(%i1) draw2d(xrange = [0,10],
yrange = [0,10],
points_joined = true,
points([[5,0],[5,10]]),
color = blue,
label(["Centered alignment (default)",5,2]),
label_alignment = 'left,
label(["Left alignment",5,5]),
label_alignment = 'right,
label(["Right alignment",5,8]))$
See also label_orientation, and color
Categories: Package draw
Default value: horizontal
label_orientation is used to specify orientation of labels.
Possible values are: horizontal, and vertical.
This option is relevant only for label objects.
Example:
In this example, a dummy point is added to get an image.
Package draw needs always data to draw an scene.
(%i1) draw2d(xrange = [0,10],
yrange = [0,10],
point_size = 0,
points([[5,5]]),
color = navy,
label(["Horizontal orientation (default)",5,2]),
label_orientation = 'vertical,
color = "#654321",
label(["Vertical orientation",1,5]))$
See also label_alignment and color
Categories: Package draw
Default value: solid
line_type indicates how lines are displayed; possible values are
solid and dots, both available in all terminals, and
dashes, short_dashes, short_long_dashes, short_short_long_dashes,
and dot_dash, which are not available in png, jpg, and gif terminals.
This option affects the following graphic objects:
gr2d: points, polygon, rectangle,
ellipse, vector, explicit, implicit,
parametric and polar.
gr3d: points, explicit, parametric and parametric_surface.
Example:
(%i1) draw2d(line_type = dots,
explicit(1 + x^2,x,-1,1),
line_type = solid, /* default */
explicit(2 + x^2,x,-1,1))$
See also line_width.
Categories: Package draw
Default value: 1
line_width is the width of plotted lines.
Its value must be a positive number.
This option affects the following graphic objects:
gr2d: points, polygon, rectangle,
ellipse, vector, explicit, implicit,
parametric and polar.
gr3d: points and parametric.
Example:
(%i1) draw2d(explicit(x^2,x,-1,1), /* default width */
line_width = 5.5,
explicit(1 + x^2,x,-1,1),
line_width = 10,
explicit(2 + x^2,x,-1,1))$
See also line_type.
Categories: Package draw
Default value: false
If logcb is true, the tics in the colorbox will be drawn in the
logarithmic scale.
When enhanced3d or colorbox is false, option logcb has
no effect.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d (
enhanced3d = true,
color = green,
logcb = true,
logz = true,
palette = [-15,24,-9],
explicit(exp(x^2-y^2), x,-2,2,y,-2,2)) $
See also enhanced3d, colorbox and cbrange.
Categories: Package draw
Default value: false
If logx is true, the x axis will be drawn in the
logarithmic scale.
Since this is a global graphics option, its position in the scene description
does not matter, with the exception that it should be written before any
2D explicit object, so that draw can produce a better plot.
Example:
(%i1) draw2d(logx = true,
explicit(log(x),x,0.01,5))$
See also logy, logx_secondary, logy_secondary, and logz.
Categories: Package draw
Default value: false
If logx_secondary is true, the secondary x axis
will be drawn in the logarithmic scale.
This option is relevant only for 2d scenes.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(
grid = true,
key="x^2, linear scale",
color=red,
explicit(x^2,x,1,100),
xaxis_secondary = true,
xtics_secondary = true,
logx_secondary = true,
key = "x^2, logarithmic x scale",
color = blue,
explicit(x^2,x,1,100) )$
See also logx_draw, logy_draw, logy_secondary, and logz.
Categories: Package draw
Default value: false
If logy is true, the y axis will be drawn in the
logarithmic scale.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(logy = true,
explicit(exp(x),x,0,5))$
See also logx_draw, logx_secondary, logy_secondary, and logz.
Categories: Package draw
Default value: false
If logy_secondary is true, the secondary y axis
will be drawn in the logarithmic scale.
This option is relevant only for 2d scenes.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(
grid = true,
key="x^2, linear scale",
color=red,
explicit(x^2,x,1,100),
yaxis_secondary = true,
ytics_secondary = true,
logy_secondary = true,
key = "x^2, logarithmic y scale",
color = blue,
explicit(x^2,x,1,100) )$
See also logx_draw, logy_draw, logx_secondary, and logz.
Categories: Package draw
Default value: false
If logz is true, the z axis will be drawn in the
logarithmic scale.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d(logz = true,
explicit(exp(u^2+v^2),u,-2,2,v,-2,2))$
See also logx_draw and logy_draw.
Categories: Package draw
Default value: 29
In 2d, nticks gives the initial number of points used by the
adaptive plotting routine for explicit objects. It is also the
number of points that will be shown in parametric and polar curves.
This option affects the following graphic objects:
gr2d: ellipse, explicit, parametric and polar.
gr3d: parametric.
See also adapt_depth
Example:
(%i1) draw2d(transparent = true,
ellipse(0,0,4,2,0,180),
nticks = 5,
ellipse(0,0,4,2,180,180) )$
Categories: Package draw
Default value: color
palette indicates how to map gray levels onto color components.
It works together with option enhanced3d in 3D graphics,
who associates every point of a surfaces to a real number or gray level.
It also works with gray images. With palette, levels are transformed into colors.
There are two ways for defining these transformations.
First, palette can be a vector of length three with components
ranging from -36 to +36; each value is an index for a formula mapping the levels
onto red, green and blue colors, respectively:
0: 0 1: 0.5 2: 1 3: x 4: x^2 5: x^3 6: x^4 7: sqrt(x) 8: sqrt(sqrt(x)) 9: sin(90x) 10: cos(90x) 11: |x-0.5| 12: (2x-1)^2 13: sin(180x) 14: |cos(180x)| 15: sin(360x) 16: cos(360x) 17: |sin(360x)| 18: |cos(360x)| 19: |sin(720x)| 20: |cos(720x)| 21: 3x 22: 3x-1 23: 3x-2 24: |3x-1| 25: |3x-2| 26: (3x-1)/2 27: (3x-2)/2 28: |(3x-1)/2| 29: |(3x-2)/2| 30: x/0.32-0.78125 31: 2*x-0.84 32: 4x;1;-2x+1.84;x/0.08-11.5 33: |2*x - 0.5| 34: 2*x 35: 2*x - 0.5 36: 2*x - 1
negative numbers mean negative colour component.
palette = gray and palette = color are short cuts for
palette = [3,3,3] and palette = [7,5,15], respectively.
Second, palette can be a user defined lookup table. In this case,
the format for building a lookup table of length n is
palette=[color_1, color_2, ..., color_n], where color_i is
a well formed color (see option color) such that color_1 is
assigned to the lowest gray level and color_n to the highest. The rest
of colors are interpolated.
Since this is a global graphics option, its position in the scene description does not matter.
Examples:
It works together with option enhanced3d in 3D graphics.
(%i1) draw3d(
enhanced3d = [z-x+2*y,x,y,z],
palette = [32, -8, 17],
explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$
It also works with gray images.
(%i1) im: apply(
'matrix,
makelist(makelist(random(200),i,1,30),i,1,30))$
(%i2) /* palette = color, default */
draw2d(image(im,0,0,30,30))$
(%i3) draw2d(palette = gray, image(im,0,0,30,30))$
(%i4) draw2d(palette = [15,20,-4],
colorbox=false,
image(im,0,0,30,30))$
palette can be a user defined lookup table.
In this example, low values of x are colored
in red, and higher values in yellow.
(%i1) draw3d(
palette = [red, blue, yellow],
enhanced3d = x,
explicit(x^2+y^2,x,-1,1,y,-1,1)) $
See also colorbox and enhanced3d.
Categories: Package draw
Default value: 1
point_size sets the size for plotted points. It must be a
non negative number.
This option has no effect when graphic option point_type is
set to dot.
This option affects the following graphic objects:
Example:
(%i1) draw2d(points(makelist([random(20),random(50)],k,1,10)),
point_size = 5,
points(makelist(k,k,1,20),makelist(random(30),k,1,20)))$
Categories: Package draw
Default value: 1
point_type indicates how isolated points are displayed; the value of this
option can be any integer index greater or equal than -1, or the name of
a point style: $none (-1), dot (0), plus (1), multiply (2),
asterisk (3), square (4), filled_square (5), circle (6),
filled_circle (7), up_triangle (8), filled_up_triangle (9),
down_triangle (10), filled_down_triangle (11), diamant (12) and
filled_diamant (13).
This option affects the following graphic objects:
Example:
(%i1) draw2d(xrange = [0,10],
yrange = [0,10],
point_size = 3,
point_type = diamant,
points([[1,1],[5,1],[9,1]]),
point_type = filled_down_triangle,
points([[1,2],[5,2],[9,2]]),
point_type = asterisk,
points([[1,3],[5,3],[9,3]]),
point_type = filled_diamant,
points([[1,4],[5,4],[9,4]]),
point_type = 5,
points([[1,5],[5,5],[9,5]]),
point_type = 6,
points([[1,6],[5,6],[9,6]]),
point_type = filled_circle,
points([[1,7],[5,7],[9,7]]),
point_type = 8,
points([[1,8],[5,8],[9,8]]),
point_type = filled_diamant,
points([[1,9],[5,9],[9,9]]) )$
Categories: Package draw
Default value: false
When points_joined is true, points are joined by lines; when false,
isolated points are drawn. A third possible value for this graphic option is
impulses; in such case, vertical segments are drawn from points to the x-axis (2D)
or to the xy-plane (3D).
This option affects the following graphic objects:
Example:
(%i1) draw2d(xrange = [0,10],
yrange = [0,4],
point_size = 3,
point_type = up_triangle,
color = blue,
points([[1,1],[5,1],[9,1]]),
points_joined = true,
point_type = square,
line_type = dots,
points([[1,2],[5,2],[9,2]]),
point_type = circle,
color = red,
line_width = 7,
points([[1,3],[5,3],[9,3]]) )$
Categories: Package draw
Default value: none
When proportional_axes is equal to xy or xyz,
the aspect ratio of the axis units will be set to 1:1 resulting in a 2D or 3D
scene that will be drawn with axes proportional to their relative lengths.
Since this is a global graphics option, its position in the scene description does not matter.
This option works with Gnuplot version 4.2.6 or greater.
Examples:
Single 2D plot.
(%i1) draw2d(
ellipse(0,0,1,1,0,360),
transparent=true,
color = blue,
line_width = 4,
ellipse(0,0,2,1/2,0,360),
proportional_axes = xy) $
Multiplot.
(%i1) draw(
terminal = wxt,
gr2d(proportional_axes = xy,
explicit(x^2,x,0,1)),
gr2d(explicit(x^2,x,0,1),
xrange = [0,1],
yrange = [0,2],
proportional_axes=xy),
gr2d(explicit(x^2,x,0,1)))$
Categories: Package draw
Default value: false
If surface_hide is true, hidden parts are not plotted in 3d surfaces.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw(columns=2,
gr3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3)),
gr3d(surface_hide = true,
explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3)) )$
Categories: Package draw
Default value: screen
Selects the terminal to be used by Gnuplot; possible values are:
screen (default), png, pngcairo, jpg, gif,
eps, eps_color, epslatex, epslatex_standalone,
svg, canvas, dumb, dumb_file, pdf, pdfcairo,
wxt, animated_gif, multipage_pdfcairo, multipage_pdf,
multipage_eps, multipage_eps_color, and aquaterm.
Terminals screen, wxt and aquaterm can be also defined as a list
with two elements: the name of the terminal itself and a non negative integer number.
In this form, multiple windows can be opened at the same time, each with its
corresponding number. This feature does not work in Windows platforms.
Since this is a global graphics option, its position in the scene description
does not matter. It can be also used as an argument of function draw.
N.B. pdfcairo requires Gnuplot 4.3 or newer.
pdf requires Gnuplot to be compiled with the option --enable-pdf and libpdf must
be installed. The pdf library is available from: http://www.pdflib.com/en/download/pdflib-family/pdflib-lite/
Examples:
(%i1) /* screen terminal (default) */
draw2d(explicit(x^2,x,-1,1))$
(%i2) /* png file */
draw2d(terminal = 'png,
explicit(x^2,x,-1,1))$
(%i3) /* jpg file */
draw2d(terminal = 'jpg,
dimensions = [300,300],
explicit(x^2,x,-1,1))$
(%i4) /* eps file */
draw2d(file_name = "myfile",
explicit(x^2,x,-1,1),
terminal = 'eps)$
(%i5) /* pdf file */
draw2d(file_name = "mypdf",
dimensions = 100*[12.0,8.0],
explicit(x^2,x,-1,1),
terminal = 'pdf)$
(%i6) /* wxwidgets window */
draw2d(explicit(x^2,x,-1,1),
terminal = 'wxt)$
Multiple windows.
(%i1) draw2d(explicit(x^5,x,-2,2), terminal=[screen, 3])$ (%i2) draw2d(explicit(x^2,x,-2,2), terminal=[screen, 0])$
An animated gif file.
(%i1) draw(
delay = 100,
file_name = "zzz",
terminal = 'animated_gif,
gr2d(explicit(x^2,x,-1,1)),
gr2d(explicit(x^3,x,-1,1)),
gr2d(explicit(x^4,x,-1,1)));
End of animation sequence
(%o1) [gr2d(explicit), gr2d(explicit), gr2d(explicit)]
Option delay is only active in animated gif's; it is ignored in
any other case.
Multipage output in eps format.
(%i1) draw(
file_name = "parabol",
terminal = multipage_eps,
dimensions = 100*[10,10],
gr2d(explicit(x^2,x,-1,1)),
gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1))) $
See also file_name, dimensions_draw and delay.
Categories: Package draw
Default value: "" (empty string)
Option title, a string, is the main title for the scene.
By default, no title is written.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(explicit(exp(u),u,-2,2),
title = "Exponential function")$
Categories: Package draw
Default value: none
If transform is none, the space is not transformed and
graphic objects are drawn as defined. When a space transformation is
desired, a list must be assigned to option transform. In case of
a 2D scene, the list takes the form [f1(x,y), f2(x,y), x, y].
In case of a 3D scene, the list is of the form
[f1(x,y,z), f2(x,y,z), f3(x,y,z), x, y, z].
The names of the variables defined in the lists may be different to those used in the definitions of the graphic objects.
Examples:
Rotation in 2D.
(%i1) th : %pi / 4$
(%i2) draw2d(
color = "#e245f0",
proportional_axes = 'xy,
line_width = 8,
triangle([3,2],[7,2],[5,5]),
border = false,
fill_color = yellow,
transform = [cos(th)*x - sin(th)*y,
sin(th)*x + cos(th)*y, x, y],
triangle([3,2],[7,2],[5,5]) )$
Translation in 3D.
(%i1) draw3d(
color = "#a02c00",
explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3),
transform = [x+10,y+10,z+10,x,y,z],
color = blue,
explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3) )$
Categories: Package draw
Default value: false
If transparent is false, interior regions of polygons are
filled according to fill_color.
This option affects the following graphic objects:
Example:
(%i1) draw2d(polygon([[3,2],[7,2],[5,5]]),
transparent = true,
color = blue,
polygon([[5,2],[9,2],[7,5]]) )$
Categories: Package draw
Default value: false
If unit_vectors is true, vectors are plotted with module 1.
This is useful for plotting vector fields. If unit_vectors is false,
vectors are plotted with its original length.
This option is relevant only for vector objects.
Example:
(%i1) draw2d(xrange = [-1,6],
yrange = [-1,6],
head_length = 0.1,
vector([0,0],[5,2]),
unit_vectors = true,
color = red,
vector([0,3],[5,2]))$
Categories: Package draw
Default value: "" (empty string)
Expert Gnuplot users can make use of this option to fine tune Gnuplot's
behaviour by writing settings to be sent before the plot or splot
command.
The value of this option must be a string or a list of strings (one per line).
Since this is a global graphics option, its position in the scene description does not matter.
Example:
Tell Gnuplot to draw axes and grid on top of graphics objects,
(%i1) draw2d(
xaxis =true, xaxis_type=solid,
yaxis =true, yaxis_type=solid,
user_preamble="set grid front",
region(x^2+y^2<1 ,x,-1.5,1.5,y,-1.5,1.5))$
Tell gnuplot to draw all contour lines in black
(%i1) wxdraw3d(
contour=both,
surface_hide=true,enhanced3d=true,wired_surface=true,
contour_levels=10,
user_preamble="set for [i=1:8] linetype i dashtype i linecolor 0",
explicit(sin(x)*cos(y),x,1,10,y,1,10)
);
Categories: Package draw
Default value: [60,30]
A pair of angles, measured in degrees, indicating the view direction in a 3D scene. The first angle is the vertical rotation around the x axis, in the range [0, 360]. The second one is the horizontal rotation around the z axis, in the range [0, 360].
If option view is given the value map, the view direction is set
to be perpendicular to the xy-plane.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d(view = [170, 50],
enhanced3d = true,
explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$
(%i2) draw3d(view = map,
enhanced3d = true,
explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$
Categories: Package draw
Default value: false
Indicates whether 3D surfaces in enhanced3d mode show the grid joinning
the points or not.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d(
enhanced3d = [sin(x),x,y],
wired_surface = true,
explicit(x^2+y^2,x,-1,1,y,-1,1)) $
Categories: Package draw
Default value: 10
x_voxel is the number of voxels in the x direction to
be used by the marching cubes algorithm implemented
by the 3d implicit object. It is also used by graphic
object region.
Categories: Package draw
Default value: false
If xaxis is true, the x axis is drawn.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(explicit(x^3,x,-1,1),
xaxis = true,
xaxis_color = blue)$
See also xaxis_width, xaxis_type and xaxis_color.
Categories: Package draw
Default value: "black"
xaxis_color specifies the color for the x axis. See
color to know how colors are defined.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(explicit(x^3,x,-1,1),
xaxis = true,
xaxis_color = red)$
See also xaxis, xaxis_width and xaxis_type.
Categories: Package draw
Default value: false
If xaxis_secondary is true, function values can be plotted with
respect to the second x axis, which will be drawn on top of the scene.
Note that this is a local graphics option which only affects to 2d plots.
Example:
(%i1) draw2d(
key = "Bottom x-axis",
explicit(x+1,x,1,2),
color = red,
key = "Above x-axis",
xtics_secondary = true,
xaxis_secondary = true,
explicit(x^2,x,-1,1)) $
See also xrange_secondary, xtics_secondary, xtics_rotate_secondary,
xtics_axis_secondary and xaxis_secondary.
Default value: dots
xaxis_type indicates how the x axis is displayed;
possible values are solid and dots
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(explicit(x^3,x,-1,1),
xaxis = true,
xaxis_type = solid)$
See also xaxis, xaxis_width and xaxis_color.
Categories: Package draw
Default value: 1
xaxis_width is the width of the x axis.
Its value must be a positive number.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(explicit(x^3,x,-1,1),
xaxis = true,
xaxis_width = 3)$
See also xaxis, xaxis_type and xaxis_color.
Categories: Package draw
Default value: "x"
Option xlabel, a string, is the label for the x axis.
By default, the axis is labeled with string "x".
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(xlabel = "Time",
explicit(exp(u),u,-2,2),
ylabel = "Population")$
See also xlabel_secondary, ylabel, ylabel_secondary and zlabel_draw.
Categories: Package draw
Default value: "" (empty string)
Option xlabel_secondary, a string, is the label for the secondary x axis.
By default, no label is written.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(
xaxis_secondary=true,yaxis_secondary=true,
xtics_secondary=true,ytics_secondary=true,
xlabel_secondary="t[s]",
ylabel_secondary="U[V]",
explicit(sin(t),t,0,10) )$
See also xlabel_draw, ylabel_draw, ylabel_secondary and zlabel_draw.
Categories: Package draw
Default value: auto
If xrange is auto, the range for the x coordinate is
computed automatically.
If the user wants a specific interval for x, it must be given as a
Maxima list, as in xrange=[-2, 3].
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(xrange = [-3,5],
explicit(x^2,x,-1,1))$
Categories: Package draw
Default value: auto
If xrange_secondary is auto, the range for the second x axis is
computed automatically.
If the user wants a specific interval for the second x axis, it must be given as a
Maxima list, as in xrange_secondary=[-2, 3].
Since this is a global graphics option, its position in the scene description does not matter.
See also xrange, yrange, zrange and yrange_secondary.
Categories: Package draw
Default value: true
This graphic option controls the way tic marks are drawn on the x axis.
xtics is bounded to symbol true, tic marks are
drawn automatically.
xtics is bounded to symbol false, tic marks are
not drawn.
xtics is bounded to a positive number, this is the distance
between two consecutive tic marks.
xtics is bounded to a list of length three of the
form [start,incr,end], tic marks are plotted from start
to end at intervals of length incr.
xtics is bounded to a set of numbers of the
form {n1, n2, ...}, tic marks are plotted at values n1,
n2, ...
xtics is bounded to a set of pairs of the
form {["label1", n1], ["label2", n2], ...}, tic marks corresponding to values n1,
n2, ... are labeled with "label1", "label2", ..., respectively.
Since this is a global graphics option, its position in the scene description does not matter.
Examples:
Disable tics.
(%i1) draw2d(xtics = 'false,
explicit(x^3,x,-1,1) )$
Tics every 1/4 units.
(%i1) draw2d(xtics = 1/4,
explicit(x^3,x,-1,1) )$
Tics from -3/4 to 3/4 in steps of 1/8.
(%i1) draw2d(xtics = [-3/4,1/8,3/4],
explicit(x^3,x,-1,1) )$
Tics at points -1/2, -1/4 and 3/4.
(%i1) draw2d(xtics = {-1/2,-1/4,3/4},
explicit(x^3,x,-1,1) )$
Labeled tics.
(%i1) draw2d(xtics = {["High",0.75],["Medium",0],["Low",-0.75]},
explicit(x^3,x,-1,1) )$
Categories: Package draw
Default value: false
If xtics_axis is true, tic marks and their labels are plotted just
along the x axis, if it is false tics are plotted on the border.
Since this is a global graphics option, its position in the scene description does not matter.
Categories: Package draw
Default value: false
If xtics_rotate is true, tic marks on the x axis are rotated
90 degrees.
Since this is a global graphics option, its position in the scene description does not matter.
Categories: Package draw
Default value: false
If xtics_rotate_secondary is true, tic marks on the secondary x axis are rotated
90 degrees.
Since this is a global graphics option, its position in the scene description does not matter.
Categories: Package draw
Default value: auto
This graphic option controls the way tic marks are drawn on the second x axis.
See xtics for a complete description.
Categories: Package draw
Default value: false
If xtics_secondary_axis is true, tic marks and their labels are plotted just
along the secondary x axis, if it is false tics are plotted on the border.
Since this is a global graphics option, its position in the scene description does not matter.
Categories: Package draw
Default value: 30
xu_grid is the number of coordinates of the first variable
(x in explicit and u in parametric 3d surfaces) to
build the grid of sample points.
This option affects the following graphic objects:
gr3d: explicit and parametric_surface.
Example:
(%i1) draw3d(xu_grid = 10,
yv_grid = 50,
explicit(x^2+y^2,x,-3,3,y,-3,3) )$
See also yv_grid.
Categories: Package draw
Default value: "" (empty string)
xy_file is the name of the file where the coordinates will be saved
after clicking with the mouse button and hitting the 'x' key. By default,
no coordinates are saved.
Since this is a global graphics option, its position in the scene description does not matter.
Categories: Package draw
Default value: false
Allocates the xy-plane in 3D scenes. When xyplane is
false, the xy-plane is placed automatically; when it is
a real number, the xy-plane intersects the z-axis at this level.
This option has no effect in 2D scenes.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d(xyplane = %e-2,
explicit(x^2+y^2,x,-1,1,y,-1,1))$
Categories: Package draw
Default value: 10
y_voxel is the number of voxels in the y direction to
be used by the marching cubes algorithm implemented
by the 3d implicit object. It is also used by graphic
object region.
Categories: Package draw
Default value: false
If yaxis is true, the y axis is drawn.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(explicit(x^3,x,-1,1),
yaxis = true,
yaxis_color = blue)$
See also yaxis_width, yaxis_type and yaxis_color.
Categories: Package draw
Default value: "black"
yaxis_color specifies the color for the y axis. See
color to know how colors are defined.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(explicit(x^3,x,-1,1),
yaxis = true,
yaxis_color = red)$
See also yaxis, yaxis_width and yaxis_type.
Categories: Package draw
Default value: false
If yaxis_secondary is true, function values can be plotted with
respect to the second y axis, which will be drawn on the right side of the
scene.
Note that this is a local graphics option which only affects to 2d plots.
Example:
(%i1) draw2d(
explicit(sin(x),x,0,10),
yaxis_secondary = true,
ytics_secondary = true,
color = blue,
explicit(100*sin(x+0.1)+2,x,0,10));
See also yrange_secondary, ytics_secondary, ytics_rotate_secondary
and ytics_axis_secondary
Categories: Package draw
Default value: dots
yaxis_type indicates how the y axis is displayed;
possible values are solid and dots.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(explicit(x^3,x,-1,1),
yaxis = true,
yaxis_type = solid)$
See also yaxis, yaxis_width and yaxis_color.
Categories: Package draw
Default value: 1
yaxis_width is the width of the y axis.
Its value must be a positive number.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(explicit(x^3,x,-1,1),
yaxis = true,
yaxis_width = 3)$
See also yaxis, yaxis_type and yaxis_color.
Categories: Package draw
Default value: "y"
Option ylabel, a string, is the label for the y axis.
By default, the axis is labeled with string "y".
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(xlabel = "Time",
ylabel = "Population",
explicit(exp(u),u,-2,2) )$
See also xlabel_draw, xlabel_secondary, ylabel_secondary, and zlabel_draw.
Categories: Package draw
Default value: "" (empty string)
Option ylabel_secondary, a string, is the label for the secondary y axis.
By default, no label is written.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(
key_pos=bottom_right,
key="current",
xlabel="t[s]",
ylabel="I[A]",ylabel_secondary="P[W]",
explicit(sin(t),t,0,10),
yaxis_secondary=true,
ytics_secondary=true,
color=red,key="Power",
explicit((sin(t))^2,t,0,10)
)$
See also xlabel_draw, xlabel_secondary, ylabel_draw and zlabel_draw.
Categories: Package draw
Default value: auto
If yrange is auto, the range for the y coordinate is
computed automatically.
If the user wants a specific interval for y, it must be given as a
Maxima list, as in yrange=[-2, 3].
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(yrange = [-2,3],
explicit(x^2,x,-1,1),
xrange = [-3,3])$
See also xrange, yrange_secondary and zrange.
Categories: Package draw
Default value: auto
If yrange_secondary is auto, the range for the second y axis is
computed automatically.
If the user wants a specific interval for the second y axis, it must be given as a
Maxima list, as in yrange_secondary=[-2, 3].
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw2d(
explicit(sin(x),x,0,10),
yaxis_secondary = true,
ytics_secondary = true,
yrange = [-3, 3],
yrange_secondary = [-20, 20],
color = blue,
explicit(100*sin(x+0.1)+2,x,0,10)) $
See also xrange, yrange and zrange.
Categories: Package draw
Default value: true
This graphic option controls the way tic marks are drawn on the y axis.
See xtics for a complete description.
Categories: Package draw
Default value: false
If ytics_axis is true, tic marks and their labels are plotted just
along the y axis, if it is false tics are plotted on the border.
Since this is a global graphics option, its position in the scene description does not matter.
Categories: Package draw
Default value: false
If ytics_rotate is true, tic marks on the y axis are rotated
90 degrees.
Since this is a global graphics option, its position in the scene description does not matter.
Categories: Package draw
Default value: false
If ytics_rotate_secondary is true, tic marks on the secondary y axis are rotated
90 degrees.
Since this is a global graphics option, its position in the scene description does not matter.
Categories: Package draw
Default value: auto
This graphic option controls the way tic marks are drawn on the second y axis.
See xtics for a complete description.
Categories: Package draw
Default value: false
If ytics_secondary_axis is true, tic marks and their labels are plotted just
along the secondary y axis, if it is false tics are plotted on the border.
Since this is a global graphics option, its position in the scene description does not matter.
Categories: Package draw
Default value: 30
yv_grid is the number of coordinates of the second variable
(y in explicit and v in parametric 3d surfaces) to
build the grid of sample points.
This option affects the following graphic objects:
gr3d: explicit and parametric_surface.
Example:
(%i1) draw3d(xu_grid = 10,
yv_grid = 50,
explicit(x^2+y^2,x,-3,3,y,-3,3) )$
See also xu_grid.
Categories: Package draw
Default value: 10
z_voxel is the number of voxels in the z direction to
be used by the marching cubes algorithm implemented
by the 3d implicit object.
Categories: Package draw
Default value: false
If zaxis is true, the z axis is drawn in 3D plots.
This option has no effect in 2D scenes.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
zaxis = true,
zaxis_type = solid,
zaxis_color = blue)$
See also zaxis_width, zaxis_type and zaxis_color.
Categories: Package draw
Default value: "black"
zaxis_color specifies the color for the z axis. See
color to know how colors are defined.
This option has no effect in 2D scenes.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
zaxis = true,
zaxis_type = solid,
zaxis_color = red)$
See also zaxis, zaxis_width and zaxis_type.
Categories: Package draw
Default value: dots
zaxis_type indicates how the z axis is displayed;
possible values are solid and dots.
This option has no effect in 2D scenes.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
zaxis = true,
zaxis_type = solid)$
See also zaxis, zaxis_width and zaxis_color.
Categories: Package draw
Default value: 1
zaxis_width is the width of the z axis.
Its value must be a positive number. This option has no effect in 2D scenes.
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
zaxis = true,
zaxis_type = solid,
zaxis_width = 3)$
See also zaxis, zaxis_type and zaxis_color.
Categories: Package draw
Default value: "z"
Option zlabel, a string, is the label for the z axis.
By default, the axis is labeled with string "z".
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d(zlabel = "Z variable",
ylabel = "Y variable",
explicit(sin(x^2+y^2),x,-2,2,y,-2,2),
xlabel = "X variable" )$
See also xlabel_draw, ylabel_draw, and zlabel_rotate.
Categories: Package draw
Default value: "auto"
This graphics option allows to choose if the z axis label of 3d plots is
drawn horizontal (false), vertical (true) or if maxima
automatically chooses an orientation based on the length of the label
(auto).
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d(
explicit(sin(x)*sin(y),x,0,10,y,0,10),
zlabel_rotate=false
)$
See also zlabel_draw.
Categories: Package draw
Default value: auto
If zrange is auto, the range for the z coordinate is
computed automatically.
If the user wants a specific interval for z, it must be given as a
Maxima list, as in zrange=[-2, 3].
Since this is a global graphics option, its position in the scene description does not matter.
Example:
(%i1) draw3d(yrange = [-3,3],
zrange = [-2,5],
explicit(x^2+y^2,x,-1,1,y,-1,1),
xrange = [-3,3])$
Categories: Package draw
Default value: true
This graphic option controls the way tic marks are drawn on the z axis.
See xtics for a complete description.
Categories: Package draw
Default value: false
If ztics_axis is true, tic marks and their labels are plotted just
along the z axis, if it is false tics are plotted on the border.
Since this is a global graphics option, its position in the scene description does not matter.
Categories: Package draw
Default value: false
If ztics_rotate is true, tic marks on the z axis are rotated
90 degrees.
Since this is a global graphics option, its position in the scene description does not matter.
Categories: Package draw
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Draws vertical bars in 2D.
2D
bars ([x1,h1,w1], [x2,h2,w2, ...])
draws bars centered at values x1, x2, ... with heights h1, h2, ...
and widths w1, w2, ...
This object is affected by the following graphic options: key,
fill_color, fill_density and line_width.
Example:
(%i1) draw2d(
key = "Group A",
fill_color = blue,
fill_density = 0.2,
bars([0.8,5,0.4],[1.8,7,0.4],[2.8,-4,0.4]),
key = "Group B",
fill_color = red,
fill_density = 0.6,
line_width = 4,
bars([1.2,4,0.4],[2.2,-2,0.4],[3.2,5,0.4]),
xaxis = true);
Categories: Package draw
Draws 3D functions defined in cylindrical coordinates.
3D
cylindrical(radius, z, minz, maxz, azi,
minazi, maxazi) plots the function radius(z,
azi) defined in cylindrical coordinates, with variable z taking
values from minz to maxz and azimuth azi taking values
from minazi to maxazi.
This object is affected by the following graphic options: xu_grid,
yv_grid, line_type, key, wired_surface, enhanced3d and color
Example:
(%i1) draw3d(cylindrical(1,z,-2,2,az,0,2*%pi))$
Categories: Package draw
Draws matrix mat in 3D space. z values are taken from mat, the abscissas range from x0 to x0 + width and ordinates from y0 to y0 + height. Element a(1,1) is projected on point (x0,y0+height), a(1,n) on (x0+width,y0+height), a(m,1) on (x0,y0), and a(m,n) on (x0+width,y0).
This object is affected by the following graphic options: line_type,,
line_width key, wired_surface, enhanced3d and color
In older versions of Maxima, elevation_grid was called mesh.
See also mesh.
Example:
(%i1) m: apply(
matrix,
makelist(makelist(random(10.0),k,1,30),i,1,20)) $
(%i2) draw3d(
color = blue,
elevation_grid(m,0,0,3,2),
xlabel = "x",
ylabel = "y",
surface_hide = true);
Categories: Package draw
Draws ellipses and circles in 2D.
2D
ellipse (xc, yc, a, b, ang1, ang2)
plots an ellipse centered at [xc, yc] with horizontal and vertical
semi axis a and b, respectively, starting at angle ang1 with an amplitude
equal to angle ang2.
This object is affected by the following graphic options: nticks,
transparent, fill_color, border, line_width,
line_type, key and color
Example:
(%i1) draw2d(transparent = false,
fill_color = red,
color = gray30,
transparent = false,
line_width = 5,
ellipse(0,6,3,2,270,-270),
/* center (x,y), a, b, start & end in degrees */
transparent = true,
color = blue,
line_width = 3,
ellipse(2.5,6,2,3,30,-90),
xrange = [-3,6],
yrange = [2,9] )$
Categories: Package draw
Draws points with error bars, horizontally, vertically or both, depending on the
value of option error_type.
2D
If error_type = x, arguments to errors must be of the form
[x, y, xdelta] or [x, y, xlow, xhigh]. If error_type = y,
arguments must be of the form [x, y, ydelta] or
[x, y, ylow, yhigh]. If error_type = xy or
error_type = boxes, arguments to errors must be of the form
[x, y, xdelta, ydelta] or [x, y, xlow, xhigh, ylow, yhigh].
See also error_type.
This object is affected by the following graphic options: error_type,
points_joined, line_width, key, line_type,
color fill_density, xaxis_secondary and yaxis_secondary.
Option fill_density is only relevant when error_type=boxes.
Examples:
Horizontal error bars.
(%i1) draw2d(
error_type = y,
errors([[1,2,1], [3,5,3], [10,3,1], [17,6,2]]))$
Vertical and horizontal error bars.
(%i1) draw2d(
error_type = xy,
points_joined = true,
color = blue,
errors([[1,2,1,2], [3,5,2,1], [10,3,1,1], [17,6,1/2,2]]));
Categories: Package draw
Draws explicit functions in 2D and 3D.
2D
explicit(fcn,var,minval,maxval) plots explicit function fcn,
with variable var taking values from minval to maxval.
This object is affected by the following graphic options: nticks,
adapt_depth, draw_realpart, line_width, line_type, key,
filled_func, fill_color and color
Example:
(%i1) draw2d(line_width = 3,
color = blue,
explicit(x^2,x,-3,3) )$
(%i2) draw2d(fill_color = brown,
filled_func = true,
explicit(x^2,x,-3,3) )$
3D
explicit(fcn, var1, minval1, maxval1, var2,
minval2, maxval2) plots the explicit function fcn, with
variable var1 taking values from minval1 to maxval1 and
variable var2 taking values from minval2 to maxval2.
This object is affected by the following graphic options: draw_realpart, xu_grid,
yv_grid, line_type, line_width, key, wired_surface,
enhanced3d and color.
Example:
(%i1) draw3d(key = "Gauss",
color = "#a02c00",
explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3),
yv_grid = 10,
color = blue,
key = "Plane",
explicit(x+y,x,-5,5,y,-5,5),
surface_hide = true)$
See also filled_func for filled functions.
Categories: Package draw
Renders images in 2D.
2D
image (im,x0,y0,width,height) plots image im in the rectangular
region from vertex (x0,y0) to (x0+width,y0+height) on the real
plane. Argument im must be a matrix of real numbers, a matrix of
vectors of length three or a picture object.
If im is a matrix of real numbers or a levels picture object,
pixel values are interpreted according to graphic option palette,
which is a vector of length three with components
ranging from -36 to +36; each value is an index for a formula mapping the levels
onto red, green and blue colors, respectively:
0: 0 1: 0.5 2: 1 3: x 4: x^2 5: x^3 6: x^4 7: sqrt(x) 8: sqrt(sqrt(x)) 9: sin(90x) 10: cos(90x) 11: |x-0.5| 12: (2x-1)^2 13: sin(180x) 14: |cos(180x)| 15: sin(360x) 16: cos(360x) 17: |sin(360x)| 18: |cos(360x)| 19: |sin(720x)| 20: |cos(720x)| 21: 3x 22: 3x-1 23: 3x-2 24: |3x-1| 25: |3x-2| 26: (3x-1)/2 27: (3x-2)/2 28: |(3x-1)/2| 29: |(3x-2)/2| 30: x/0.32-0.78125 31: 2*x-0.84 32: 4x;1;-2x+1.84;x/0.08-11.5 33: |2*x - 0.5| 34: 2*x 35: 2*x - 0.5 36: 2*x - 1
negative numbers mean negative colour component.
palette = gray and palette = color are short cuts for
palette = [3,3,3] and palette = [7,5,15], respectively.
If im is a matrix of vectors of length three or an rgb picture object, they are interpreted as red, green and blue color components.
Examples:
If im is a matrix of real numbers, pixel values are interpreted according
to graphic option palette.
(%i1) im: apply(
'matrix,
makelist(makelist(random(200),i,1,30),i,1,30))$
(%i2) /* palette = color, default */
draw2d(image(im,0,0,30,30))$
(%i3) draw2d(palette = gray, image(im,0,0,30,30))$
(%i4) draw2d(palette = [15,20,-4],
colorbox=false,
image(im,0,0,30,30))$
See also colorbox.
If im is a matrix of vectors of length three, they are interpreted as red, green and blue color components.
(%i1) im: apply(
'matrix,
makelist(
makelist([random(300),
random(300),
random(300)],i,1,30),i,1,30))$
(%i2) draw2d(image(im,0,0,30,30))$
Package draw automatically loads package picture. In this
example, a level picture object is built by hand and then rendered.
(%i1) im: make_level_picture([45,87,2,134,204,16],3,2);
(%o1) picture(level, 3, 2, {Array: #(45 87 2 134 204 16)})
(%i2) /* default color palette */
draw2d(image(im,0,0,30,30))$
(%i3) /* gray palette */
draw2d(palette = gray,
image(im,0,0,30,30))$
An xpm file is read and then rendered.
(%i1) load(draw)$
(%i2) im: read_xpm("myfile.xpm")$
(%i3) draw2d(image(im,0,0,10,7))$
See also make_level_picture, make_rgb_picture and read_xpm.
http://www.telefonica.net/web2/biomates/maxima/gpdraw/image
contains more elaborated examples.
Categories: Package draw
Draws implicit functions in 2D and 3D.
2D
implicit(fcn,x,xmin,xmax,y,ymin,ymax)
plots the implicit function defined by fcn, with variable x taking values
from xmin to xmax, and variable y taking values
from ymin to ymax.
This object is affected by the following graphic options: ip_grid,
ip_grid_in, line_width, line_type, key and color.
Example:
(%i1) draw2d(grid = true,
line_type = solid,
key = "y^2=x^3-2*x+1",
implicit(y^2=x^3-2*x+1, x, -4,4, y, -4,4),
line_type = dots,
key = "x^3+y^3 = 3*x*y^2-x-1",
implicit(x^3+y^3 = 3*x*y^2-x-1, x,-4,4, y,-4,4),
title = "Two implicit functions" )$
3D
implicit (fcn,x,xmin,xmax, y,ymin,ymax, z,zmin,zmax)
plots the implicit surface defined by fcn, with variable x taking values
from xmin to xmax, variable y taking values
from ymin to ymax and variable z taking values
from zmin to zmax. This object implements the marching cubes algorithm.
This object is affected by the following graphic options: x_voxel,
y_voxel, z_voxel, line_width, line_type, key,
wired_surface, enhanced3d and color.
Example:
(%i1) draw3d(
color=blue,
implicit((x^2+y^2+z^2-1)*(x^2+(y-1.5)^2+z^2-0.5)=0.015,
x,-1,1,y,-1.2,2.3,z,-1,1),
surface_hide=true);
Categories: Package draw
Writes labels in 2D and 3D.
Colored labels work only with Gnuplot 4.3 and up.
This object is affected by the following graphic options: label_alignment,
label_orientation and color.
2D
label([string,x,y]) writes the string at point
[x,y].
Example:
(%i1) draw2d(yrange = [0.1,1.4],
color = red,
label(["Label in red",0,0.3]),
color = "#0000ff",
label(["Label in blue",0,0.6]),
color = light_blue,
label(["Label in light-blue",0,0.9],
["Another light-blue",0,1.2]) )$
3D
label([string,x,y,z]) writes the string at point
[x,y,z].
Example:
(%i1) draw3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3),
color = red,
label(["UP 1",-2,0,3], ["UP 2",1.5,0,4]),
color = blue,
label(["DOWN 1",2,0,-3]) )$
Categories: Package draw
Draws a quadrangular mesh in 3D.
3D
Argument row_i is a list of n 3D points of the form
[[x_i1,y_i1,z_i1], ...,[x_in,y_in,z_in]], and all rows are
of equal length. All these points define an arbitrary surface in 3D and
in some sense it's a generalization of the elevation_grid object.
This object is affected by the following graphic options: line_type,
line_width, color, key, wired_surface, enhanced3d and transform.
Examples:
A simple example.
(%i1) draw3d(
mesh([[1,1,3], [7,3,1],[12,-2,4],[15,0,5]],
[[2,7,8], [4,3,1],[10,5,8], [12,7,1]],
[[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $
Plotting a triangle in 3D.
(%i1) draw3d(
line_width = 2,
mesh([[1,0,0],[0,1,0]],
[[0,0,1],[0,0,1]])) $
Two quadrilaterals.
(%i1) draw3d(
surface_hide = true,
line_width = 3,
color = red,
mesh([[0,0,0], [0,1,0]],
[[2,0,2], [2,2,2]]),
color = blue,
mesh([[0,0,2], [0,1,2]],
[[2,0,4], [2,2,4]])) $
Categories: Package draw
Draws parametric functions in 2D and 3D.
This object is affected by the following graphic options: nticks,
line_width, line_type, key, color and enhanced3d.
2D
The command parametric(xfun, yfun, par, parmin,
parmax) plots the parametric function [xfun, yfun],
with parameter par taking values from parmin to parmax.
Example:
(%i1) draw2d(explicit(exp(x),x,-1,3),
color = red,
key = "This is the parametric one!!",
parametric(2*cos(rrr),rrr^2,rrr,0,2*%pi))$
3D
parametric(xfun, yfun, zfun, par, parmin,
parmax) plots the parametric curve [xfun, yfun,
zfun], with parameter par taking values from parmin to
parmax.
Example:
(%i1) draw3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3),
color = royalblue,
parametric(cos(5*u)^2,sin(7*u),u-2,u,0,2),
color = turquoise,
line_width = 2,
parametric(t^2,sin(t),2+t,t,0,2),
surface_hide = true,
title = "Surface & curves" )$
Categories: Package draw
Draws parametric surfaces in 3D.
3D
The command parametric_surface(xfun, yfun, zfun,
par1, par1min, par1max, par2, par2min,
par2max) plots the parametric surface [xfun, yfun,
zfun], with parameter par1 taking values from par1min to
par1max and parameter par2 taking values from par2min to
par2max.
This object is affected by the following graphic options: draw_realpart, xu_grid,
yv_grid, line_type, line_width, key, wired_surface, enhanced3d
and color.
Example:
(%i1) draw3d(title = "Sea shell",
xu_grid = 100,
yv_grid = 25,
view = [100,20],
surface_hide = true,
parametric_surface(0.5*u*cos(u)*(cos(v)+1),
0.5*u*sin(u)*(cos(v)+1),
u*sin(v) - ((u+3)/8*%pi)^2 - 20,
u, 0, 13*%pi, v, -%pi, %pi) )$
Categories: Package draw
Draws points in 2D and 3D.
This object is affected by the following graphic options: point_size,
point_type, points_joined, line_width, key,
line_type and color. In 3D mode, it is also affected by enhanced3d
2D
points ([[x1,y1], [x2,y2],...]) or
points ([x1,x2,...], [y1,y2,...])
plots points [x1,y1], [x2,y2], etc. If abscissas
are not given, they are set to consecutive positive integers, so that
points ([y1,y2,...]) draws points [1,y1], [2,y2], etc.
If matrix is a two-column or two-row matrix, points (matrix)
draws the associated points. If matrix is a one-column or one-row matrix,
abscissas are assigned automatically.
If 1d_y_array is a 1D lisp array of numbers, points (1d_y_array) plots them
setting abscissas to consecutive positive integers. points (1d_x_array, 1d_y_array)
plots points with their coordinates taken from the two arrays passed as arguments. If
2d_xy_array is a 2D array with two columns, or with two rows, points (2d_xy_array)
plots the corresponding points on the plane.
Examples:
Two types of arguments for points, a list of pairs and two lists
of separate coordinates.
(%i1) draw2d(
key = "Small points",
points(makelist([random(20),random(50)],k,1,10)),
point_type = circle,
point_size = 3,
points_joined = true,
key = "Great points",
points(makelist(k,k,1,20),makelist(random(30),k,1,20)),
point_type = filled_down_triangle,
key = "Automatic abscissas",
color = red,
points([2,12,8]))$
Drawing impulses.
(%i1) draw2d(
points_joined = impulses,
line_width = 2,
color = red,
points(makelist([random(20),random(50)],k,1,10)))$
Array with ordinates.
(%i1) a: make_array (flonum, 100) $ (%i2) for i:0 thru 99 do a[i]: random(1.0) $ (%i3) draw2d(points(a)) $
Two arrays with separate coordinates.
(%i1) x: make_array (flonum, 100) $
(%i2) y: make_array (fixnum, 100) $
(%i3) for i:0 thru 99 do (
x[i]: float(i/100),
y[i]: random(10) ) $
(%i4) draw2d(points(x, y)) $
A two-column 2D array.
(%i1) xy: make_array(flonum, 100, 2) $
(%i2) for i:0 thru 99 do (
xy[i, 0]: float(i/100),
xy[i, 1]: random(10) ) $
(%i3) draw2d(points(xy)) $
Drawing an array filled with function read_array.
(%i1) a: make_array(flonum,100) $
(%i2) read_array (file_search ("pidigits.data"), a) $
(%i3) draw2d(points(a)) $
3D
points([[x1, y1, z1], [x2, y2, z2],
...]) or points([x1, x2, ...], [y1, y2, ...],
[z1, z2,...]) plots points [x1, y1, z1],
[x2, y2, z2], etc. If matrix is a three-column
or three-row matrix, points (matrix) draws the associated points.
When arguments are lisp arrays, points (1d_x_array, 1d_y_array, 1d_z_array)
takes coordinates from the three 1D arrays. If 2d_xyz_array is a 2D array with three columns,
or with three rows, points (2d_xyz_array) plots the corresponding points.
Examples:
One tridimensional sample,
(%i1) load (numericalio)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) draw3d(title = "Daily average wind speeds",
point_size = 2,
points(args(submatrix (s2, 4, 5))) )$
Two tridimensional samples,
(%i1) load (numericalio)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) draw3d(
title = "Daily average wind speeds. Two data sets",
point_size = 2,
key = "Sample from stations 1, 2 and 3",
points(args(submatrix (s2, 4, 5))),
point_type = 4,
key = "Sample from stations 1, 4 and 5",
points(args(submatrix (s2, 2, 3))) )$
Unidimensional arrays,
(%i1) x: make_array (fixnum, 10) $
(%i2) y: make_array (fixnum, 10) $
(%i3) z: make_array (fixnum, 10) $
(%i4) for i:0 thru 9 do (
x[i]: random(10),
y[i]: random(10),
z[i]: random(10) ) $
(%i5) draw3d(points(x,y,z)) $
Bidimensional colored array,
(%i1) xyz: make_array(fixnum, 10, 3) $
(%i2) for i:0 thru 9 do (
xyz[i, 0]: random(10),
xyz[i, 1]: random(10),
xyz[i, 2]: random(10) ) $
(%i3) draw3d(
enhanced3d = true,
points_joined = true,
points(xyz)) $
Color numbers explicitly specified by the user.
(%i1) pts: makelist([t,t^2,cos(t)], t, 0, 15)$
(%i2) col_num: makelist(k, k, 1, length(pts))$
(%i3) draw3d(
enhanced3d = ['part(col_num,k),k],
point_size = 3,
point_type = filled_circle,
points(pts))$
Categories: Package draw
Draws 2D functions defined in polar coordinates.
2D
polar (radius,ang,minang,maxang) plots function
radius(ang) defined in polar coordinates, with variable
ang taking values from
minang to maxang.
This object is affected by the following graphic options: nticks,
line_width, line_type, key and color.
Example:
(%i1) draw2d(user_preamble = "set grid polar",
nticks = 200,
xrange = [-5,5],
yrange = [-5,5],
color = blue,
line_width = 3,
title = "Hyperbolic Spiral",
polar(10/theta,theta,1,10*%pi) )$
Categories: Package draw
Draws polygons in 2D.
2D
The commands polygon([[x1, y1], [x2, y2], ...])
or polygon([x1, x2, ...], [y1, y2, ...]) plot on
the plane a polygon with vertices [x1, y1], [x2,
y2], etc.
This object is affected by the following graphic options: transparent,
fill_color, border, line_width, key,
line_type and color.
Example:
(%i1) draw2d(color = "#e245f0",
line_width = 8,
polygon([[3,2],[7,2],[5,5]]),
border = false,
fill_color = yellow,
polygon([[5,2],[9,2],[7,5]]) )$
Categories: Package draw
Draws a quadrilateral.
2D
quadrilateral([x1, y1], [x2, y2],
[x3, y3], [x4, y4]) draws a quadrilateral with vertices
[x1, y1], [x2, y2],
[x3, y3], and [x4, y4].
This object is affected by the following graphic options:
transparent, fill_color, border, line_width,
key, xaxis_secondary, yaxis_secondary, line_type,
transform and color.
Example:
(%i1) draw2d(
quadrilateral([1,1],[2,2],[3,-1],[2,-2]))$
3D
quadrilateral([x1, y1, z1], [x2, y2,
z2], [x3, y3, z3], [x4, y4, z4])
draws a quadrilateral with vertices [x1, y1, z1],
[x2, y2, z2], [x3, y3, z3],
and [x4, y4, z4].
This object is affected by the following graphic options: line_type,
line_width, color, key, enhanced3d and
transform.
Categories: Package draw
Draws rectangles in 2D.
2D
rectangle ([x1,y1], [x2,y2]) draws a rectangle with opposite vertices
[x1,y1] and [x2,y2].
This object is affected by the following graphic options: transparent,
fill_color, border, line_width, key,
line_type and color.
Example:
(%i1) draw2d(fill_color = red,
line_width = 6,
line_type = dots,
transparent = false,
fill_color = blue,
rectangle([-2,-2],[8,-1]), /* opposite vertices */
transparent = true,
line_type = solid,
line_width = 1,
rectangle([9,4],[2,-1.5]),
xrange = [-3,10],
yrange = [-3,4.5] )$
Categories: Package draw
Plots a region on the plane defined by inequalities.
2D
expr is an expression formed by inequalities and boolean operators
and, or, and not. The region is bounded by the rectangle
defined by [minval1, maxval1] and [minval2, maxval2].
This object is affected by the following graphic options: fill_color,
key, x_voxel and y_voxel.
Example:
(%i1) draw2d(
x_voxel = 30,
y_voxel = 30,
region(x^2+y^2<1 and x^2+y^2 > 1/2,
x, -1.5, 1.5, y, -1.5, 1.5));
Draws 3D functions defined in spherical coordinates.
3D
spherical(radius, azi, minazi, maxazi, zen,
minzen, maxzen) plots the function radius(azi,
zen) defined in spherical coordinates, with azimuth azi taking
values from minazi to maxazi and zenith zen taking values
from minzen to maxzen.
This object is affected by the following graphic options: xu_grid,
yv_grid, line_type, key, wired_surface, enhanced3d and color.
Example:
(%i1) draw3d(spherical(1,a,0,2*%pi,z,0,%pi))$
Categories: Package draw
Draws a triangle.
2D
triangle ([x1,y1], [x2,y2], [x3,y3]) draws a triangle with vertices [x1,y1], [x2,y2],
and [x3,y3].
This object is affected by the following graphic options:
transparent, fill_color, border, line_width,
key, xaxis_secondary, yaxis_secondary, line_type,
transform and color.
Example:
(%i1) draw2d(
triangle([1,1],[2,2],[3,-1]))$
3D
triangle ([x1,y1,z1], [x2,y2,z2], [x3,y3,z3]) draws a triangle with vertices [x1,y1,z1],
[x2,y2,z2], and [x3,y3,z3].
This object is affected by the following graphic options: line_type,
line_width, color, key, enhanced3d and transform.
Categories: Package draw
Draws a tube in 3D with varying diameter.
3D
[xfun,yfun,zfun]
is the parametric curve with parameter p taking values from pmin
to pmax. Circles of radius rfun are placed with their centers on
the parametric curve and perpendicular to it.
This object is affected by the following graphic options: xu_grid,
yv_grid, line_type, line_width, key, wired_surface, enhanced3d,
color and capping.
Example:
(%i1) draw3d(
enhanced3d = true,
xu_grid = 50,
tube(cos(a), a, 0, cos(a/10)^2,
a, 0, 4*%pi) )$
Categories: Package draw
Draws vectors in 2D and 3D.
This object is affected by the following graphic options: head_both,
head_length, head_angle, head_type, line_width,
line_type, key and color.
2D
vector([x,y], [dx,dy]) plots vector
[dx,dy] with origin in [x,y].
Example:
(%i1) draw2d(xrange = [0,12],
yrange = [0,10],
head_length = 1,
vector([0,1],[5,5]), /* default type */
head_type = 'empty,
vector([3,1],[5,5]),
head_both = true,
head_type = 'nofilled,
line_type = dots,
vector([6,1],[5,5]))$
3D
vector([x,y,z], [dx,dy,dz])
plots vector [dx,dy,dz] with
origin in [x,y,z].
Example:
(%i1) draw3d(color = cyan,
vector([0,0,0],[1,1,1]/sqrt(3)),
vector([0,0,0],[1,-1,0]/sqrt(2)),
vector([0,0,0],[1,1,-2]/sqrt(6)) )$
Categories: Package draw
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Returns pixel from picture. Coordinates x and y range from 0 to
width-1 and height-1, respectively.
Categories: Package draw
Returns a levels picture object. make_level_picture (data)
builds the picture object from matrix data.
make_level_picture (data,width,height)
builds the object from a list of numbers; in this case, both the
width and the height must be given.
The returned picture object contains the following four parts:
level
Example:
Level picture from matrix.
(%i1) make_level_picture(matrix([3,2,5],[7,-9,3000]));
(%o1) picture(level, 3, 2, {Array: #(3 2 5 7 0 255)})
Level picture from numeric list.
(%i1) make_level_picture([-2,0,54,%pi],2,2);
(%o1) picture(level, 2, 2, {Array: #(0 0 54 3)})
Categories: Package draw
Returns an rgb-coloured picture object. All three arguments must be levels picture; with red, green and blue levels.
The returned picture object contains the following four parts:
rgb
Example:
(%i1) red: make_level_picture(matrix([3,2],[7,260]));
(%o1) picture(level, 2, 2, {Array: #(3 2 7 255)})
(%i2) green: make_level_picture(matrix([54,23],[73,-9]));
(%o2) picture(level, 2, 2, {Array: #(54 23 73 0)})
(%i3) blue: make_level_picture(matrix([123,82],[45,32.5698]));
(%o3) picture(level, 2, 2, {Array: #(123 82 45 33)})
(%i4) make_rgb_picture(red,green,blue);
(%o4) picture(rgb, 2, 2,
{Array: #(3 54 123 2 23 82 7 73 45 255 0 33)})
Categories: Package draw
Returns the negative of a (level or rgb) picture.
Categories: Package draw
Returns true in case of equal pictures, and false otherwise.
Categories: Package draw · Predicate functions
Returns true if the argument is a well formed image,
and false otherwise.
Categories: Package draw · Predicate functions
Reads a file in xpm and returns a picture object.
Categories: Package draw
Transforms an rgb picture into a level one by averaging the red, green and blue channels.
Categories: Package draw
If argument color is red, green or blue,
function take_channel returns the corresponding color channel of
picture im.
Example:
(%i1) red: make_level_picture(matrix([3,2],[7,260]));
(%o1) picture(level, 2, 2, {Array: #(3 2 7 255)})
(%i2) green: make_level_picture(matrix([54,23],[73,-9]));
(%o2) picture(level, 2, 2, {Array: #(54 23 73 0)})
(%i3) blue: make_level_picture(matrix([123,82],[45,32.5698]));
(%o3) picture(level, 2, 2, {Array: #(123 82 45 33)})
(%i4) make_rgb_picture(red,green,blue);
(%o4) picture(rgb, 2, 2,
{Array: #(3 54 123 2 23 82 7 73 45 255 0 33)})
(%i5) take_channel(%,'green); /* simple quote!!! */
(%o5) picture(level, 2, 2, {Array: #(54 23 73 0)})
Categories: Package draw
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This package automatically loads package draw.
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Default value: false
boundaries_array is where the graphic object geomap looks
for boundaries coordinates.
Each component of boundaries_array is an array of floating
point quantities, the coordinates of a polygonal segment or map boundary.
See also geomap.
Categories: Package draw
Draws a list of polygonal segments (boundaries), labeled by
its numbers (boundaries_array coordinates). This is of great
help when building new geographical entities.
Example:
Map of Europe labeling borders with their component number in
boundaries_array.
(%i1) load(worldmap)$
(%i2) european_borders:
region_boundaries(-31.81,74.92,49.84,32.06)$
(%i3) numbered_boundaries(european_borders)$
Categories: Package draw
Makes the necessary polygons to draw a colored continent or a list of countries.
Example:
(%i1) load(worldmap)$
(%i2) /* A continent */
make_poly_continent(Africa)$
(%i3) apply(draw2d, %)$
(%i4) /* A list of countries */
make_poly_continent([Germany,Denmark,Poland])$
(%i5) apply(draw2d, %)$
Categories: Package draw
Makes the necessary polygons to draw a colored country. If islands exist, one country can be defined with more than just one polygon.
Example:
(%i1) load(worldmap)$ (%i2) make_poly_country(India)$ (%i3) apply(draw2d, %)$
Categories: Package draw
Returns a polygon object from boundary indices. Argument
nlist is a list of components of boundaries_array.
Example:
Bhutan is defined by boundary numbers 171, 173
and 1143, so that make_polygon([171,173,1143])
appends arrays of coordinates boundaries_array[171],
boundaries_array[173] and boundaries_array[1143] and
returns a polygon object suited to be plotted by
draw. To avoid an error message, arrays must be
compatible in the sense that any two consecutive
arrays have two coordinates in the extremes in common. In this
example, the two first components of boundaries_array[171] are
equal to the last two coordinates of boundaries_array[173], and
the two first of boundaries_array[173] are equal to the two first
of boundaries_array[1143]; in conclussion, boundary numbers
171, 173 and 1143 (in this order) are compatible and the colored
polygon can be drawn.
(%i1) load(worldmap)$
(%i2) Bhutan;
(%o2) [[171, 173, 1143]]
(%i3) boundaries_array[171];
(%o3) {Array:
#(88.750549 27.14727 88.806351 27.25305 88.901367 27.282221
88.917877 27.321039)}
(%i4) boundaries_array[173];
(%o4) {Array:
#(91.659554 27.76511 91.6008 27.66666 91.598022 27.62499
91.631348 27.536381 91.765533 27.45694 91.775253 27.4161
92.007751 27.471939 92.11441 27.28583 92.015259 27.168051
92.015533 27.08083 92.083313 27.02277 92.112183 26.920271
92.069977 26.86194 91.997192 26.85194 91.915253 26.893881
91.916924 26.85416 91.8358 26.863331 91.712479 26.799999
91.542191 26.80444 91.492188 26.87472 91.418854 26.873329
91.371353 26.800831 91.307457 26.778049 90.682457 26.77417
90.392197 26.903601 90.344131 26.894159 90.143044 26.75333
89.98996 26.73583 89.841919 26.70138 89.618301 26.72694
89.636093 26.771111 89.360786 26.859989 89.22081 26.81472
89.110237 26.829161 88.921631 26.98777 88.873016 26.95499
88.867737 27.080549 88.843307 27.108601 88.750549
27.14727)}
(%i5) boundaries_array[1143];
(%o5) {Array:
#(91.659554 27.76511 91.666924 27.88888 91.65831 27.94805
91.338028 28.05249 91.314972 28.096661 91.108856 27.971109
91.015808 27.97777 90.896927 28.05055 90.382462 28.07972
90.396088 28.23555 90.366074 28.257771 89.996353 28.32333
89.83165 28.24888 89.58609 28.139999 89.35997 27.87166
89.225517 27.795 89.125793 27.56749 88.971077 27.47361
88.917877 27.321039)}
(%i6) Bhutan_polygon: make_polygon([171,173,1143])$
(%i7) draw2d(Bhutan_polygon)$
Categories: Package draw
Detects polygonal segments of global variable boundaries_array
fully contained in the rectangle with vertices (x1,y1) -upper left-
and (x2,y2) -bottom right-.
Example:
Returns segment numbers for plotting southern Italy.
(%i1) load(worldmap)$ (%i2) region_boundaries(10.4,41.5,20.7,35.4); (%o2) [1846, 1863, 1864, 1881, 1888, 1894] (%i3) draw2d(geomap(%))$
Categories: Package draw
Detects polygonal segments of global variable boundaries_array
containing at least one vertex in the rectangle defined by vertices (x1,y1)
-upper left- and (x2,y2) -bottom right-.
Example:
(%i1) load(worldmap)$
(%i2) region_boundaries_plus(10.4,41.5,20.7,35.4);
(%o2) [1060, 1062, 1076, 1835, 1839, 1844, 1846, 1858,
1861, 1863, 1864, 1871, 1881, 1888, 1894, 1897]
(%i3) draw2d(geomap(%))$
Categories: Package draw
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Draws cartographic maps in 2D and 3D.
2D
This function works together with global variable boundaries_array.
Argument numlist is a list containing numbers or lists of numbers.
All these numbers must be integers greater or equal than zero,
representing the components of global array boundaries_array.
Each component of boundaries_array is an array of floating
point quantities, the coordinates of a polygonal segment or map boundary.
geomap (numlist) flattens its arguments and draws the
associated boundaries in boundaries_array.
This object is affected by the following graphic options: line_width,
line_type and color.
Examples:
A simple map defined by hand:
(%i1) load(worldmap)$
(%i2) /* Vertices of boundary #0: {(1,1),(2,5),(4,3)} */
( bnd0: make_array(flonum,6),
bnd0[0]:1.0, bnd0[1]:1.0, bnd0[2]:2.0,
bnd0[3]:5.0, bnd0[4]:4.0, bnd0[5]:3.0 )$
(%i3) /* Vertices of boundary #1: {(4,3),(5,4),(6,4),(5,1)} */
( bnd1: make_array(flonum,8),
bnd1[0]:4.0, bnd1[1]:3.0, bnd1[2]:5.0, bnd1[3]:4.0,
bnd1[4]:6.0, bnd1[5]:4.0, bnd1[6]:5.0, bnd1[7]:1.0)$
(%i4) /* Vertices of boundary #2: {(5,1), (3,0), (1,1)} */
( bnd2: make_array(flonum,6),
bnd2[0]:5.0, bnd2[1]:1.0, bnd2[2]:3.0,
bnd2[3]:0.0, bnd2[4]:1.0, bnd2[5]:1.0 )$
(%i5) /* Vertices of boundary #3: {(1,1), (4,3)} */
( bnd3: make_array(flonum,4),
bnd3[0]:1.0, bnd3[1]:1.0, bnd3[2]:4.0, bnd3[3]:3.0)$
(%i6) /* Vertices of boundary #4: {(4,3), (5,1)} */
( bnd4: make_array(flonum,4),
bnd4[0]:4.0, bnd4[1]:3.0, bnd4[2]:5.0, bnd4[3]:1.0)$
(%i7) /* Pack all together in boundaries_array */
( boundaries_array: make_array(any,5),
boundaries_array[0]: bnd0, boundaries_array[1]: bnd1,
boundaries_array[2]: bnd2, boundaries_array[3]: bnd3,
boundaries_array[4]: bnd4 )$
(%i8) draw2d(geomap([0,1,2,3,4]))$
The auxiliary package worldmap sets the global variable
boundaries_array to real world boundaries in
(longitude, latitude) coordinates. These data are in the
public domain and come from
https://web.archive.org/web/20100310124019/http://www-cger.nies.go.jp/grid-e/gridtxt/grid19.html.
Package worldmap defines also boundaries for countries,
continents and coastlines as lists with the necessary components of
boundaries_array (see file share/draw/worldmap.mac
for more information). Package worldmap automatically loads
package worldmap.
(%i1) load(worldmap)$
(%i2) c1: gr2d(geomap([Canada,United_States,
Mexico,Cuba]))$
(%i3) c2: gr2d(geomap(Africa))$
(%i4) c3: gr2d(geomap([Oceania,China,Japan]))$
(%i5) c4: gr2d(geomap([France,Portugal,Spain,
Morocco,Western_Sahara]))$
(%i6) draw(columns = 2,
c1,c2,c3,c4)$
Package worldmap is also useful for plotting
countries as polygons. In this case, graphic object
geomap is no longer necessary and the polygon
object is used instead. Since lists are now used and not
arrays, maps rendering will be slower. See also make_poly_country
and make_poly_continent to understand the following code.
(%i1) load(worldmap)$ (%i2) mymap: append( [color = white], /* borders are white */ [fill_color = red], make_poly_country(Bolivia), [fill_color = cyan], make_poly_country(Paraguay), [fill_color = green], make_poly_country(Colombia), [fill_color = blue], make_poly_country(Chile), [fill_color = "#23ab0f"], make_poly_country(Brazil), [fill_color = goldenrod], make_poly_country(Argentina), [fill_color = "midnight-blue"], make_poly_country(Uruguay))$ (%i3) apply(draw2d, mymap)$
3D
geomap (numlist) projects map boundaries on the sphere of radius 1
centered at (0,0,0). It is possible to change the sphere or the projection type
by using geomap (numlist,3Dprojection).
Available 3D projections:
[spherical_projection,x,y,z,r]: projects map boundaries on the sphere of
radius r centered at (x,y,z).
(%i1) load(worldmap)$
(%i2) draw3d(geomap(Australia), /* default projection */
geomap(Australia,
[spherical_projection,2,2,2,3]))$
[cylindrical_projection,x,y,z,r,rc]: re-projects spherical map boundaries on the cylinder of radius
rc and axis passing through the poles of the globe of radius r centered at (x,y,z).
(%i1) load(worldmap)$
(%i2) draw3d(geomap([America_coastlines,Eurasia_coastlines],
[cylindrical_projection,2,2,2,3,4]))$
[conic_projection,x,y,z,r,alpha]: re-projects spherical map boundaries on the cones of angle alpha,
with axis passing through the poles of the globe of radius r centered at (x,y,z). Both
the northern and southern cones are tangent to sphere.
(%i1) load(worldmap)$
(%i2) draw3d(geomap(World_coastlines,
[conic_projection,0,0,0,1,90]))$
See also http://riotorto.users.sf.net/gnuplot/geomap for more elaborated examples.
Categories: Package draw
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| 53.1 Introduction to drawdf | ||
| 53.2 Functions and Variables for drawdf |
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The function drawdf draws the direction field of a first-order
Ordinary Differential Equation (ODE) or a system of two autonomous
first-order ODE's.
Since this is an additional package, in order to use it you must first
load it with load(drawdf). Drawdf is built upon the draw
package, which requires Gnuplot 4.2.
To plot the direction field of a single ODE, the ODE must be written in the form:
dy
-- = F(x,y)
dx
and the function F should be given as the argument for
drawdf. If the independent and dependent variables are not x,
and y, as in the equation above, then those two variables should
be named explicitly in a list given as an argument to the drawdf command
(see the examples).
To plot the direction field of a set of two autonomous ODE's, they must be written in the form
dx dy
-- = G(x,y) -- = F(x,y)
dt dt
and the argument for drawdf should be a list with the two
functions G and F, in that order; namely, the first
expression in the list will be taken to be the time derivative of the
variable represented on the horizontal axis, and the second expression
will be the time derivative of the variable represented on the vertical
axis. Those two variables do not have to be x and y, but if
they are not, then the second argument given to drawdf must be another
list naming the two variables, first the one on the horizontal axis and
then the one on the vertical axis.
If only one ODE is given, drawdf will implicitly admit
x=t, and G(x,y)=1, transforming the non-autonomous
equation into a system of two autonomous equations.
Categories: Differential equations · Plotting Share packages · Package drawdf Package draw
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Function drawdf draws a 2D direction field with optional
solution curves and other graphics using the draw package.
The first argument specifies the derivative(s), and must be either an expression or a list of two expressions. dydx, dxdt and dydt are expressions that depend on x and y. dvdu, dudt and dvdt are expressions that depend on u and v.
If the independent and dependent variables are not x and
y, then their names must be specified immediately following the
derivative(s), either as a list of two names
[u,v], or as two lists of the form
[u,umin,umax] and
[v,vmin,vmax].
The remaining arguments are graphic options, graphic objects,
or lists containing graphic options and objects, nested to arbitrary
depth. The set of graphic options and objects supported by
drawdf is a superset of those supported by draw2d and
gr2d from the draw package.
The arguments are interpreted sequentially: graphic options affect all following graphic objects. Furthermore, graphic objects are drawn on the canvas in order specified, and may obscure graphics drawn earlier. Some graphic options affect the global appearance of the scene.
The additional graphic objects supported by drawdf include:
solns_at, points_at, saddles_at, soln_at,
point_at, and saddle_at.
The additional graphic options supported by drawdf include:
field_degree, soln_arrows, field_arrows,
field_grid, field_color, show_field,
tstep, nsteps, duration, direction,
field_tstep, field_nsteps, and field_duration.
Commonly used graphic objects inherited from the draw
package include: explicit, implicit, parametric,
polygon, points, vector, label, and all
others supported by draw2d and gr2d.
Commonly used graphic options inherited from the draw
package include:
points_joined, color,
point_type, point_size, line_width,
line_type, key, title, xlabel,
ylabel, user_preamble, terminal,
dimensions, file_name, and all
others supported by draw2d and gr2d.
See also draw2d.
Users of wxMaxima or Imaxima may optionally use wxdrawdf, which
is identical to drawdf except that the graphics are drawn
within the notebook using wxdraw.
To make use of this function, write first load(drawdf).
Examples:
(%i1) load(drawdf)$ (%i2) drawdf(exp(-x)+y)$ /* default vars: x,y */ (%i3) drawdf(exp(-t)+y, [t,y])$ /* default range: [-10,10] */ (%i4) drawdf([y,-9*sin(x)-y/5], [x,1,5], [y,-2,2])$
For backward compatibility, drawdf accepts
most of the parameters supported by plotdf.
(%i5) drawdf(2*cos(t)-1+y, [t,y], [t,-5,10], [y,-4,9],
[trajectory_at,0,0])$
soln_at and solns_at draw solution curves
passing through the specified points, using a slightly
enhanced 4th-order Runge Kutta numerical integrator.
(%i6) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
solns_at([0,0.1],[0,-0.1]),
color=blue, soln_at(0,0))$
field_degree=2 causes the field to be composed of quadratic
splines, based on the first and second derivatives at each grid point.
field_grid=[COLS,ROWS] specifies the number
of columns and rows in the grid.
(%i7) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
field_degree=2, field_grid=[20,15],
solns_at([0,0.1],[0,-0.1]),
color=blue, soln_at(0,0))$
soln_arrows=true adds arrows to the solution curves, and (by
default) removes them from the direction field. It also changes the
default colors to emphasize the solution curves.
(%i8) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
soln_arrows=true,
solns_at([0,0.1],[0,-0.1],[0,0]))$
duration=40 specifies the time duration of numerical
integration (default 10). Integration will also stop automatically if
the solution moves too far away from the plotted region, or if the
derivative becomes complex or infinite. Here we also specify
field_degree=2 to plot quadratic splines. The equations below
model a predator-prey system.
(%i9) drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1],
field_degree=2, duration=40,
soln_arrows=true, point_at(1/2,1/2),
solns_at([0.1,0.2], [0.2,0.1], [1,0.8], [0.8,1],
[0.1,0.1], [0.6,0.05], [0.05,0.4],
[1,0.01], [0.01,0.75]))$
field_degree='solns causes the field to be composed
of many small solution curves computed by 4th-order
Runge Kutta, with better results in this case.
(%i10) drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1],
field_degree='solns, duration=40,
soln_arrows=true, point_at(1/2,1/2),
solns_at([0.1,0.2], [0.2,0.1], [1,0.8],
[0.8,1], [0.1,0.1], [0.6,0.05],
[0.05,0.4], [1,0.01], [0.01,0.75]))$
saddles_at attempts to automatically linearize the equation at
each saddle, and to plot a numerical solution corresponding to each
eigenvector, including the separatrices. tstep=0.05 specifies
the maximum time step for the numerical integrator (the default is
0.1). Note that smaller time steps will sometimes be used in order to
keep the x and y steps small. The equations below model a damped
pendulum.
(%i11) drawdf([y,-9*sin(x)-y/5], tstep=0.05,
soln_arrows=true, point_size=0.5,
points_at([0,0], [2*%pi,0], [-2*%pi,0]),
field_degree='solns,
saddles_at([%pi,0], [-%pi,0]))$
show_field=false suppresses the field entirely.
(%i12) drawdf([y,-9*sin(x)-y/5], tstep=0.05,
show_field=false, soln_arrows=true,
point_size=0.5,
points_at([0,0], [2*%pi,0], [-2*%pi,0]),
saddles_at([3*%pi,0], [-3*%pi,0],
[%pi,0], [-%pi,0]))$
drawdf passes all unrecognized parameters to draw2d or
gr2d, allowing you to combine the full power of the draw
package with drawdf.
(%i13) drawdf(x^2+y^2, [x,-2,2], [y,-2,2], field_color=gray,
key="soln 1", color=black, soln_at(0,0),
key="soln 2", color=red, soln_at(0,1),
key="isocline", color=green, line_width=2,
nticks=100, parametric(cos(t),sin(t),t,0,2*%pi))$
drawdf accepts nested lists of graphic options and objects,
allowing convenient use of makelist and other function calls to
generate graphics.
(%i14) colors : ['red,'blue,'purple,'orange,'green]$
(%i15) drawdf([x-x*y/2, (x*y - 3*y)/4],
[x,2.5,3.5], [y,1.5,2.5],
field_color = gray,
makelist([ key = concat("soln",k),
color = colors[k],
soln_at(3, 2 + k/20) ],
k,1,5))$
Categories: Package drawdf
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| 54.1 The dynamics package | ||
| 54.2 Graphical analysis of discrete dynamical systems | ||
| 54.3 Visualization with VTK |
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Package dynamics includes functions for 3D visualization,
animations, graphical analysis of differential and difference equations
and numerical solution of differential equations. The functions for
differential equations are described in the section on Numerical Methods and the functions to plot the Mandelbrot and Julia
sets are described in the section on Plotting.
All the functions in this package will be loaded automatically the first time they are used.
Categories: Dynamical systems · Share packages · Package dynamics
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Implements the so-called chaos game: the initial point (x0,
y0) is plotted and then one of the m points
[x1, y1]…xm, ym]
will be selected at random. The next point plotted will be on the
segment from the previous point plotted to the point chosen randomly, at a
distance from the random point which will be b times that segment's
length. The procedure is repeated n times. The options are the
same as for plot2d.
Example. A plot of Sierpinsky's triangle:
(%i1) chaosgame([[0, 0], [1, 0], [0.5, sqrt(3)/2]], [0.1, 0.1], 1/2,
30000, [style, dots]);
Categories: Package dynamics · Plotting
Draws n+1 points in a two-dimensional graph, where the horizontal coordinates of the points are the integers 0, 1, 2, ..., n, and the vertical coordinates are the corresponding values y(n) of the sequence defined by the recurrence relation
y(n+1) = F(y(n))
With initial value y(0) equal to y0. F must be an
expression that depends only on one variable (in the example, it
depend on y, but any other variable can be used),
y0 must be a real number and n must be a positive integer.
This function accepts the same options as plot2d.
Example.
(%i1) evolution(cos(y), 2, 11);
Categories: Package dynamics · Plotting
Shows, in a two-dimensional plot, the first n+1 points in the sequence of points defined by the two-dimensional discrete dynamical system with recurrence relations
u(n+1) = F(u(n), v(n)) v(n+1) = G(u(n), v(n))
With initial values u0 and v0. F and G must be
two expressions that depend only on two variables, u and
v, which must be named explicitly in a list. The options are the
same as for plot2d.
Example. Evolution of a two-dimensional discrete dynamical system:
(%i1) f: 0.6*x*(1+2*x)+0.8*y*(x-1)-y^2-0.9$ (%i2) g: 0.1*x*(1-6*x+4*y)+0.1*y*(1+9*y)-0.4$ (%i3) evolution2d([f,g], [x,y], [-0.5,0], 50000, [style,dots]);
And an enlargement of a small region in that fractal:
(%i9) evolution2d([f,g], [x,y], [-0.5,0], 300000, [x,-0.8,-0.6],
[y,-0.4,-0.2], [style, dots]);
Categories: Package dynamics · Plotting
Implements the Iterated Function System method. This method is similar
to the method described in the function chaosgame. but instead of
shrinking the segment from the current point to the randomly chosen
point, the 2 components of that segment will be multiplied by the 2 by 2
matrix Ai that corresponds to the point chosen randomly.
The random choice of one of the m attractive points can be made
with a non-uniform probability distribution defined by the weights
r1,...,rm. Those weights are given in cumulative form; for
instance if there are 3 points with probabilities 0.2, 0.5 and 0.3, the
weights r1, r2 and r3 could be 2, 7 and 10. The
options are the same as for plot2d.
Example. Barnsley's fern, obtained with 4 matrices and 4 points:
(%i1) a1: matrix([0.85,0.04],[-0.04,0.85])$ (%i2) a2: matrix([0.2,-0.26],[0.23,0.22])$ (%i3) a3: matrix([-0.15,0.28],[0.26,0.24])$ (%i4) a4: matrix([0,0],[0,0.16])$ (%i5) p1: [0,1.6]$ (%i6) p2: [0,1.6]$ (%i7) p3: [0,0.44]$ (%i8) p4: [0,0]$ (%i9) w: [85,92,99,100]$ (%i10) ifs(w, [a1,a2,a3,a4], [p1,p2,p3,p4], [5,0], 50000, [style,dots]);
Categories: Package dynamics · Plotting
Draws the orbits diagram for a family of one-dimensional discrete dynamical systems, with one parameter x; that kind of diagram is used to study the bifurcations of a one-dimensional discrete system.
The function F(y) defines a sequence with a starting value of
y0, as in the case of the function evolution, but in this
case that function will also depend on a parameter x that will
take values in the interval from x0 to xf with increments of
xstep. Each value used for the parameter x is shown on the
horizontal axis. The vertical axis will show the n2 values
of the sequence y(n1+1),..., y(n1+n2+1) obtained after letting
the sequence evolve n1 iterations. In addition to the options
accepted by plot2d, it accepts an option pixels that
sets up the maximum number of different points that will be represented
in the vertical direction.
Example. Orbits diagram of the quadratic map, with a parameter a:
(%i1) orbits(x^2+a, 0, 50, 200, [a, -2, 0.25], [style, dots]);
To enlarge the region around the lower bifurcation near x = -1.25 use:
(%i2) orbits(x^2+a, 0, 100, 400, [a,-1,-1.53], [x,-1.6,-0.8],
[nticks, 400], [style,dots]);
Categories: Package dynamics · Plotting
Draws a staircase diagram for the sequence defined by the recurrence relation
y(n+1) = F(y(n))
The interpretation and allowed values of the input parameters is the
same as for the function evolution. A staircase diagram consists
of a plot of the function F(y), together with the line G(y)
= y. A vertical segment is drawn from the point (y0,
y0) on that line until the point where it intersects the function
F. From that point a horizontal segment is drawn until it reaches
the point (y1, y1) on the line, and the procedure is
repeated n times until the point (yn, yn) is
reached. The options are the same as for plot2d.
Example.
(%i1) staircase(cos(y), 1, 11, [y, 0, 1.2]);
Categories: Package dynamics · Plotting
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Function scene creates 3D images and animations using the Visualization ToolKit (VTK) software. In order to use that function, Xmaxima and VTK should be installed in your system (including the TCL bindings of VTK, which in some system might come in a separate package).
Accepts an empty list or a list of several objects
and options. The program launches Xmaxima, which
opens an external window representing the given objects in a
3-dimensional space and applying the options given. Each object must
belong to one of the following 4 classes: sphere, cube, cylinder or cone
(see Scene objects). Objects are identified by
giving their name or by a list in which the first element is the class
name and the following elements are options for that object.
Example. A hexagonal pyramid with a blue background:
(%i1) scene(cone, [background,"#9980e5"])$
By holding down the left button of the mouse while it is moved on the
graphics window, the camera can be rotated showing different views of
the pyramid. The two plot options elevation and
azimuth can also be used to change the initial
orientation of the viewing camera. The camera can be moved by holding
the middle mouse button while moving it and holding the right-side mouse
button while moving it up or down will zoom in or out.
Each object option should be a list starting with the option name,
followed by its value. The list of allowed options can be found in the
Scene object's options section.
Example. This will show a sphere falling to the ground and bouncing off without losing any energy. To start or pause the animation, press the play/pause button.
(%i1) p: makelist ([0,0,2.1- 9.8*t^2/2], t, 0, 0.64, 0.01)$ (%i2) p: append (p, reverse(p))$ (%i3) ball: [sphere, [radius,0.1], [thetaresolution,20], [phiresolution,20], [position,0,0,2.1], [color,red], [animate,position,p]]$ (%i4) ground: [cube, [xlength,2], [ylength,2], [zlength,0.2], [position,0,0,-0.1],[color,violet]]$ (%i5) scene (ball, ground, restart)$
The restart option was used to make the animation restart
automatically every time the last point in the position list is reached.
The accepted values for the colors are the same as for the color
option of plot2d.
Categories: Package dynamics · Plotting
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Default value: 135
The rotation of the camera on the horizontal (x, y) plane. angle must be a real number; an angle of 0 means that the camera points in the direction of the y axis and the x axis will appear on the right.
Categories: Package dynamics · Plotting
Default value: black
The color of the graphics window's background. It accepts color names or
hexadecimal red-green-blue strings (see the color option of plot2d).
Categories: Package dynamics · Plotting
Default value: 30
The vertical rotation of the camera. The angle must be a real number; an angle of 0 means that the camera points on the horizontal, and the default angle of 30 means that the camera is pointing 30 degrees down from the horizontal.
Categories: Package dynamics · Plotting
Default value: 500
The height, in pixels, of the graphics window. pixels must be a positive integer number.
Categories: Package dynamics · Plotting
Default value: false
A true value means that animations will restart automatically when the end of the list is reached. Writing just "restart" is equivalent to [restart, true].
Categories: Package dynamics · Plotting
Default value: 10
The amount of time, in mili-seconds, between iterations among consecutive animation frames. time must be a real number.
Categories: Package dynamics · Plotting
Default value: 500
The width, in pixels, of the graphics window. pixels must be a positive integer number.
Categories: Package dynamics · Plotting
Default value: .scene
name must be a string that can be used as the name of the Tk
window created by Xmaxima for the scene graphics. The default
value .scene implies that a new top level window will be created.
Categories: Package dynamics · Plotting
Default value: Xmaxima: scene
name must be a string that will be written in the title of the
window created by scene.
Categories: Package dynamics · Plotting
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Creates a regular pyramid with height equal to 1 and a hexagonal base
with vertices 0.5 units away from the axis. Options
height and radius can be used
to change those defaults and option resolution
can be used to change the number of edges of the base; higher values
will make it look like a cone. By default, the axis will be along the x
axis, the middle point of the axis will be at the origin and the vertex
on the positive side of the x axis; use options
orientation and center to
change those defaults.
Example. This shows a pyramid that starts rotating around the z axis when the play button is pressed.
(%i1) scene([cone, [orientation,0,30,0], [tstep,100], [animate,orientation,makelist([0,30,i],i,5,360,5)]], restart)$
Categories: Package dynamics · Plotting
A cube with edges of 1 unit and faces parallel to the xy, xz and yz
planes. The lengths of the three edges can be changed with options
xlength, ylength and
zlength, turning it into a rectangular box and
the faces can be rotated with option orientation.
Categories: Package dynamics · Plotting
Creates a regular prism with height equal to 1 and a hexagonal base with
vertices 0.5 units away from the axis. Options
height and radius can be
used to change those defaults and option resolution can be used to change the number of edges of the base;
higher values will make it look like a cylinder. The default height can
be changed with the option height. By default,
the axis will be along the x axis and the middle point of the axis will
be at the origin; use options orientation and
center to change those defaults.
Categories: Package dynamics · Plotting
A sphere with default radius of 0.5 units and center at the origin.
Categories: Package dynamics · Plotting
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property should be one of the following 4 object's properties:
origin, scale,
position or
orientation and positions should be a
list of points. When the play button is pressed, the object property
will be changed sequentially through all the values in the list, at
intervals of time given by the option tstep. The
rewind button can be used to point at the start of the sequence making
the animation restart after the play button is pressed again.
See also track.
Categories: Package dynamics · Plotting
Default value: 1
In a cone or a cylinder, it defines whether the base (or bases) will be shown. A value of 1 for number makes the base visible and a value of 0 makes it invisible.
Categories: Package dynamics · Plotting
Default value: [0, 0, 0]
The coordinates of the object's geometric center, with respect to its
position. point can be a list with 3
real numbers, or 3 real numbers separated by commas. In a cylinder, cone
or cube it will be at half its height and in a sphere at its center.
Categories: Package dynamics · Plotting
Default value: white
The color of the object. It accepts color names or hexadecimal
red-green-blue strings (see the color option of plot2d).
Categories: Package dynamics · Plotting
Default value: 180
In a sphere phi is the angle on the vertical plane that passes through the z axis, measured from the positive part of the z axis. angle must be a number between 0 and 180 that sets the final value of phi at which the surface will end. A value smaller than 180 will eliminate a part of the sphere's surface.
See also startphi and
phiresolution.
Categories: Package dynamics · Plotting
Default value: 360
In a sphere theta is the angle on the horizontal plane (longitude), measured from the positive part of the x axis. angle must be a number between 0 and 360 that sets the final value of theta at which the surface will end. A value smaller than 360 will eliminate a part of the sphere's surface.
See also starttheta and
thetaresolution.
Categories: Package dynamics · Plotting
Default value: 1
value must be a positive number which sets the height of a cone or a cylinder.
Categories: Package dynamics · Plotting
Default value: 1
The width of the lines, when option wireframe is
used. value must be a positive number.
Categories: Package dynamics · Plotting
Default value: 1
value must be a number between 0 and 1. The lower the number, the more transparent the object will become. The default value of 1 means a completely opaque object.
Categories: Package dynamics · Plotting
Default value: [0, 0, 0]
Three angles by which the object will be rotated with respect to the
three axis. angles can be a list with 3 real numbers, or 3 real
numbers separated by commas. Example: [0, 0, 90] rotates
the x axis of the object to the y axis of the reference frame.
Categories: Package dynamics · Plotting
Default value: [0, 0, 0]
The coordinates of the object's origin, with respect to which its other dimensions are defined. point can be a list with 3 real numbers, or 3 real numbers separated by commas.
Categories: Package dynamics · Plotting
Default value:
The number of sub-intervals into which the phi angle interval from
startphi to endphi
will be divided. num must be a positive integer.
Categories: Package dynamics · Plotting
Only the vertices of the triangulation used to render the surface will
be shown. Example: [sphere, [points]]
See also surface and
wireframe.
Categories: Package dynamics · Plotting
Default value: 1
The size of the points, when option points is
used. value must be a positive number.
Categories: Package dynamics · Plotting
Default value: [0, 0, 0]
The coordinates of the object's position. point can be a list with 3 real numbers, or 3 real numbers separated by commas.
Categories: Package dynamics · Plotting
Default value: 0.5
The radius or a sphere or the distance from the axis to the base's vertices in a cylinder or a cone. value must be a positive number.
Categories: Package dynamics · Plotting
Default value: 6
number must be a integer greater than 2 that sets the number of edges in the base of a cone or a cylinder.
Categories: Package dynamics · Plotting
Default value: [1, 1, 1]
Three numbers by which the object will be scaled with respect to the
three axis. factors can be a list with 3 real numbers, or 3 real
numbers separated by commas. Example: [2, 0.5, 1]
enlarges the object to twice its size in the x direction, reduces the
dimensions in the y direction to half and leaves the z dimensions
unchanged.
Categories: Package dynamics · Plotting
Default value: 0
In a sphere phi is the angle on the vertical plane that passes through the z axis, measured from the positive part of the z axis. angle must be a number between 0 and 180 that sets the initial value of phi at which the surface will start. A value bigger than 0 will eliminate a part of the sphere's surface.
See also endphi and
phiresolution.
Categories: Package dynamics · Plotting
Default value: 0
In a sphere theta is the angle on the horizontal plane (longitude), measured from the positive part of the x axis. angle must be a number between 0 and 360 that sets the initial value of theta at which the surface will start. A value bigger than 0 will eliminate a part of the sphere's surface.
See also endtheta and
thetaresolution.
Categories: Package dynamics · Plotting
The surfaces of the object will be rendered and the lines and points of
the triangulation used to build the surface will not be shown. This is
the default behavior, which can be changed using either the option
points or wireframe.
Categories: Package dynamics · Plotting
Default value:
The number of sub-intervals into which the theta angle interval from
starttheta to endtheta
will be divided. num must be a positive integer.
See also starttheta and
endtheta.
Categories: Package dynamics · Plotting
positions should be a list of points. When the play button is
pressed, the object position will be changed sequentially through all
the points in the list, at intervals of time given by the option
tstep, leaving behind a track of the object's
trajectory. The rewind button can be used to point at the start of the
sequence making the animation restart after the play button is pressed
again.
Example. This will show the trajectory of a ball thrown with speed of 5 m/s, at an angle of 45 degrees, when the air resistance can be neglected:
(%i1) p: makelist ([0,4*t,4*t- 9.8*t^2/2], t, 0, 0.82, 0.01)$
(%i2) ball: [sphere, [radius,0.1], [color,red], [track,p]]$
(%i3) ground: [cube, [xlength,2], [ylength,4], [zlength,0.2],
[position,0,1.5,-0.2],[color,green]]$
(%i4) scene (ball, ground)$
See also animation.
Categories: Package dynamics · Plotting
Default value: 1
The height of a cube in the x direction. length must be a positive
number. See also ylength and
zlength.
Categories: Package dynamics · Plotting
Default value: 1
The height of a cube in the y direction. length must be a positive
number. See also xlength and
zlength.
Categories: Package dynamics · Plotting
Default value: 1
The height of a cube in z the direction. length must be a positive
number. See also xlength and
ylength.
Categories: Package dynamics · Plotting
Only the edges of the triangulation used to render the surface will be
shown. Example: [cube, [wireframe]]
Categories: Package dynamics · Plotting
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Engineering-format changes the way maxima outputs floating-point numbers
to the notation engineers are used to: a*10^b with b dividable by
three.
| 55.1 Functions and Variables for engineering-format | ||
| 55.2 Known Bugs |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Default value: true
This variable allows to temporarily switch off engineering-format.
(%i1) load("engineering-format");
(%o1)
/maxima/share/contrib/engineering-format.lisp
(%i2) float(sin(10)/10000);
(%o2) - 54.40211108893698e-6
(%i3) engineering_format_floats:false$
(%i4) float(sin(10)/10000);
(%o4) - 5.440211108893698e-5
See also fpprintprec and float.
Categories: Display functions · Global flags · Share packages
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
The output routine of SBCL 1.3.0 has a bug that sometimes causes the exponent not to be dividable by three. The value of the displayed number is still valid in this case.
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 56.1 Introduction to ezunits | ||
| 56.2 Introduction to physical_constants | ||
| 56.3 Functions and Variables for ezunits |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
ezunits is a package for working with dimensional quantities,
including some functions for dimensional analysis.
ezunits can carry out arithmetic operations on dimensional quantities and unit conversions.
The built-in units include Systeme Internationale (SI) and US customary units,
and other units can be declared.
See also physical_constants, a collection of physical constants.
load(ezunits) loads this package.
demo(ezunits) displays several examples.
The convenience function known_units returns a list of
the built-in and user-declared units,
while display_known_unit_conversions displays
the set of known conversions in an easy-to-read format.
An expression a ` b represents a dimensional quantity,
with a indicating a nondimensional quantity and b indicating the dimensional units.
A symbol can be used as a unit without declaring it as such;
unit symbols need not have any special properties.
The quantity and unit of an expression a ` b can
be extracted by the qty and units functions, respectively.
A symbol may be declared to be a dimensional quantity, with specified quantity or specified units or both.
An expression a ` b `` c converts from unit b to unit c.
ezunits has built-in conversions for SI base units,
SI derived units, and some non-SI units.
Unit conversions not already known to ezunits can be declared.
The unit conversions known to ezunits are specified by the
global variable known_unit_conversions,
which comprises built-in and user-defined conversions.
Conversions for products, quotients, and powers of units are
derived from the set of known unit conversions.
As Maxima generally prefers exact numbers (integers or rationals)
to inexact (float or bigfloat),
so ezunits preserves exact numbers when they appear
in dimensional quantities.
All built-in unit conversions are expressed in terms of exact numbers;
inexact numbers in declared conversions are coerced to exact.
There is no preferred system for display of units;
input units are not converted to other units
unless conversion is explicitly indicated.
ezunits recognizes the prefixes m-, k-, M, and G-
(for milli-, kilo-, mega-, and giga-)
as applied to SI base units and SI derived units,
but such prefixes are applied only when indicated by an explicit conversion.
Arithmetic operations on dimensional quantities are carried out by conventional rules for such operations.
y is nondimensional.
ezunits does not require that units in a sum have the same dimensions;
such terms are not added together, and no error is reported.
ezunits includes functions for elementary dimensional analysis,
namely the fundamental dimensions and fundamental units
of a dimensional quantity,
and computation of dimensionless quantities and natural units.
The functions for dimensional analysis were adapted from similar
functions in another package, written by Barton Willis.
For the purpose of dimensional analysis, a list of fundamental dimensions and an associated list of fundamental units are maintained; by default the fundamental dimensions are length, mass, time, charge, temperature, and quantity, and the fundamental units are the associated SI units, but other fundamental dimensions and units can be declared.
Categories: Physical units · Share packages · Package ezunits
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
physical_constants is a collection of physical constants,
copied from CODATA 2006 recommended values
(http://physics.nist.gov/constants).
load(physical_constants) loads this package,
and loads ezunits also, if it is not already loaded.
A physical constant is represented as a symbol which has a property
which is the constant value.
The constant value is a dimensional quantity, as represented by ezunits.
The function constvalue fetches the constant value;
the constant value is not the ordinary value of the symbol,
so symbols of physical constants persist in evaluated expressions until their
values are fetched by constvalue.
physical_constants includes some auxiliary information,
namely, a description string for each constant,
an estimate of the error of its numerical value,
and a property for TeX display.
To identify physical constants, each symbol has the
physical_constant property;
propvars(physical_constant) therefore shows the list
of all such symbols.
physical_constants comprises the following constants.
%cspeed of light in vacuum
%mu_0magnetic constant
%e_0electric constant
%Z_0characteristic impedance of vacuum
%GNewtonian constant of gravitation
%hPlanck constant
%h_barPlanck constant
%m_PPlanck mass
%T_PPlanck temperature
%l_PPlanck length
%t_PPlanck time
%%eelementary charge
%Phi_0magnetic flux quantum
%G_0conductance quantum
%K_JJosephson constant
%R_Kvon Klitzing constant
%mu_BBohr magneton
%mu_Nnuclear magneton
%alphafine-structure constant
%R_infRydberg constant
%a_0Bohr radius
%E_hHartree energy
%ratio_h_mequantum of circulation
%m_eelectron mass
%N_AAvogadro constant
%m_uatomic mass constant
%FFaraday constant
%Rmolar gas constant
%%kBoltzmann constant
%V_mmolar volume of ideal gas
%n_0Loschmidt constant
%ratio_S0_RSackur-Tetrode constant (absolute entropy constant)
%sigmaStefan-Boltzmann constant
%c_1first radiation constant
%c_1Lfirst radiation constant for spectral radiance
%c_2second radiation constant
%bWien displacement law constant
%b_primeWien displacement law constant
Reference: http://physics.nist.gov/constants
Examples:
The list of all symbols which have the physical_constant property.
(%i1) load (physical_constants)$ (%i2) propvars (physical_constant); (%o2) [%c, %mu_0, %e_0, %Z_0, %G, %h, %h_bar, %m_P, %T_P, %l_P, %t_P, %%e, %Phi_0, %G_0, %K_J, %R_K, %mu_B, %mu_N, %alpha, %R_inf, %a_0, %E_h, %ratio_h_me, %m_e, %N_A, %m_u, %F, %R, %%k, %V_m, %n_0, %ratio_S0_R, %sigma, %c_1, %c_1L, %c_2, %b, %b_prime]
Properties of the physical constant %c.
(%i1) load (physical_constants)$
(%i2) constantp (%c);
(%o2) true
(%i3) get (%c, description);
(%o3) speed of light in vacuum
(%i4) constvalue (%c);
m
(%o4) 299792458 ` -
s
(%i5) get (%c, RSU);
(%o5) 0
(%i6) tex (%c);
$$c$$
(%o6) false
The energy equivalent of 1 pound-mass.
The symbol %c persists until its value is fetched by constvalue.
(%i1) load (physical_constants)$
(%i2) m * %c^2;
2
(%o2) %c m
(%i3) %, m = 1 ` lbm;
2
(%o3) %c ` lbm
(%i4) constvalue (%);
2
lbm m
(%o4) 89875517873681764 ` ------
2
s
(%i5) E : % `` J;
Computing conversions to base units; may take a moment.
366838848464007200
(%o5) ------------------ ` J
9
(%i6) E `` GJ;
458548560580009
(%o6) --------------- ` GJ
11250000
(%i7) float (%);
(%o7) 4.0759872051556356e+7 ` GJ
Categories: Physical units · Share packages · Package physical_constants
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
The dimensional quantity operator.
An expression a ` b represents a dimensional quantity,
with a indicating a nondimensional quantity and b indicating the dimensional units.
A symbol can be used as a unit without declaring it as such;
unit symbols need not have any special properties.
The quantity and unit of an expression a ` b can
be extracted by the qty and units functions, respectively.
Arithmetic operations on dimensional quantities are carried out by conventional rules for such operations.
y is nondimensional.
ezunits does not require that units in a sum have the same dimensions;
such terms are not added together, and no error is reported.
load(ezunits) enables this operator.
Examples:
SI (Systeme Internationale) units.
(%i1) load (ezunits)$ (%i2) foo : 10 ` m; (%o2) 10 ` m (%i3) qty (foo); (%o3) 10 (%i4) units (foo); (%o4) m (%i5) dimensions (foo); (%o5) length
"Customary" units.
(%i1) load (ezunits)$
(%i2) bar : x ` acre;
(%o2) x ` acre
(%i3) dimensions (bar);
2
(%o3) length
(%i4) fundamental_units (bar);
2
(%o4) m
Units ad hoc.
(%i1) load (ezunits)$
(%i2) baz : 3 ` sheep + 8 ` goat + 1 ` horse;
(%o2) 8 ` goat + 3 ` sheep + 1 ` horse
(%i3) subst ([sheep = 3*goat, horse = 10*goat], baz);
(%o3) 27 ` goat
(%i4) baz2 : 1000`gallon/fortnight;
gallon
(%o4) 1000 ` ---------
fortnight
(%i5) subst (fortnight = 14*day, baz2);
500 gallon
(%o5) --- ` ------
7 day
Arithmetic operations on dimensional quantities.
(%i1) load (ezunits)$
(%i2) 100 ` kg + 200 ` kg;
(%o2) 300 ` kg
(%i3) 100 ` m^3 - 100 ` m^3;
3
(%o3) 0 ` m
(%i4) (10 ` kg) * (17 ` m/s^2);
kg m
(%o4) 170 ` ----
2
s
(%i5) (x ` m) / (y ` s);
x m
(%o5) - ` -
y s
(%i6) (a ` m)^2;
2 2
(%o6) a ` m
Categories: Package ezunits
The unit conversion operator.
An expression a ` b `` c converts from unit b to unit c.
ezunits has built-in conversions for SI base units,
SI derived units, and some non-SI units.
Unit conversions not already known to ezunits can be declared.
The unit conversions known to ezunits are specified by the
global variable known_unit_conversions,
which comprises built-in and user-defined conversions.
Conversions for products, quotients, and powers of units are
derived from the set of known unit conversions.
There is no preferred system for display of units;
input units are not converted to other units
unless conversion is explicitly indicated.
ezunits does not attempt to simplify units by prefixes
(milli-, centi-, deci-, etc)
unless such conversion is explicitly indicated.
load(ezunits) enables this operator.
Examples:
The set of known unit conversions.
(%i1) load (ezunits)$
(%i2) display2d : false$
(%i3) known_unit_conversions;
(%o3) {acre = 4840*yard^2,Btu = 1055*J,cfm = feet^3/minute,
cm = m/100,day = 86400*s,feet = 381*m/1250,ft = feet,
g = kg/1000,gallon = 757*l/200,GHz = 1000000000*Hz,
GOhm = 1000000000*Ohm,GPa = 1000000000*Pa,
GWb = 1000000000*Wb,Gg = 1000000*kg,Gm = 1000000000*m,
Gmol = 1000000*mol,Gs = 1000000000*s,ha = hectare,
hectare = 100*m^2,hour = 3600*s,Hz = 1/s,inch = feet/12,
km = 1000*m,kmol = 1000*mol,ks = 1000*s,l = liter,
lbf = pound_force,lbm = pound_mass,liter = m^3/1000,
metric_ton = Mg,mg = kg/1000000,MHz = 1000000*Hz,
microgram = kg/1000000000,micrometer = m/1000000,
micron = micrometer,microsecond = s/1000000,
mile = 5280*feet,minute = 60*s,mm = m/1000,
mmol = mol/1000,month = 2629800*s,MOhm = 1000000*Ohm,
MPa = 1000000*Pa,ms = s/1000,MWb = 1000000*Wb,
Mg = 1000*kg,Mm = 1000000*m,Mmol = 1000000000*mol,
Ms = 1000000*s,ns = s/1000000000,ounce = pound_mass/16,
oz = ounce,Ohm = s*J/C^2,
pound_force = 32*ft*pound_mass/s^2,
pound_mass = 200*kg/441,psi = pound_force/inch^2,
Pa = N/m^2,week = 604800*s,Wb = J/A,yard = 3*feet,
year = 31557600*s,C = s*A,F = C^2/J,GA = 1000000000*A,
GC = 1000000000*C,GF = 1000000000*F,GH = 1000000000*H,
GJ = 1000000000*J,GK = 1000000000*K,GN = 1000000000*N,
GS = 1000000000*S,GT = 1000000000*T,GV = 1000000000*V,
GW = 1000000000*W,H = J/A^2,J = m*N,kA = 1000*A,
kC = 1000*C,kF = 1000*F,kH = 1000*H,kHz = 1000*Hz,
kJ = 1000*J,kK = 1000*K,kN = 1000*N,kOhm = 1000*Ohm,
kPa = 1000*Pa,kS = 1000*S,kT = 1000*T,kV = 1000*V,
kW = 1000*W,kWb = 1000*Wb,mA = A/1000,mC = C/1000,
mF = F/1000,mH = H/1000,mHz = Hz/1000,mJ = J/1000,
mK = K/1000,mN = N/1000,mOhm = Ohm/1000,mPa = Pa/1000,
mS = S/1000,mT = T/1000,mV = V/1000,mW = W/1000,
mWb = Wb/1000,MA = 1000000*A,MC = 1000000*C,
MF = 1000000*F,MH = 1000000*H,MJ = 1000000*J,
MK = 1000000*K,MN = 1000000*N,MS = 1000000*S,
MT = 1000000*T,MV = 1000000*V,MW = 1000000*W,
N = kg*m/s^2,R = 5*K/9,S = 1/Ohm,T = J/(m^2*A),V = J/C,
W = J/s}
Elementary unit conversions.
(%i1) load (ezunits)$
(%i2) 1 ` ft `` m;
Computing conversions to base units; may take a moment.
381
(%o2) ---- ` m
1250
(%i3) %, numer;
(%o3) 0.3048 ` m
(%i4) 1 ` kg `` lbm;
441
(%o4) --- ` lbm
200
(%i5) %, numer;
(%o5) 2.205 ` lbm
(%i6) 1 ` W `` Btu/hour;
720 Btu
(%o6) --- ` ----
211 hour
(%i7) %, numer;
Btu
(%o7) 3.412322274881517 ` ----
hour
(%i8) 100 ` degC `` degF;
(%o8) 212 ` degF
(%i9) -40 ` degF `` degC;
(%o9) (- 40) ` degC
(%i10) 1 ` acre*ft `` m^3;
60228605349 3
(%o10) ----------- ` m
48828125
(%i11) %, numer;
3
(%o11) 1233.48183754752 ` m
Coercing quantities in feet and meters to one or the other.
(%i1) load (ezunits)$
(%i2) 100 ` m + 100 ` ft;
(%o2) 100 ` m + 100 ` ft
(%i3) (100 ` m + 100 ` ft) `` ft;
Computing conversions to base units; may take a moment.
163100
(%o3) ------ ` ft
381
(%i4) %, numer;
(%o4) 428.0839895013123 ` ft
(%i5) (100 ` m + 100 ` ft) `` m;
3262
(%o5) ---- ` m
25
(%i6) %, numer;
(%o6) 130.48 ` m
Dimensional analysis to find fundamental dimensions and fundamental units.
(%i1) load (ezunits)$
(%i2) foo : 1 ` acre * ft;
(%o2) 1 ` acre ft
(%i3) dimensions (foo);
3
(%o3) length
(%i4) fundamental_units (foo);
3
(%o4) m
(%i5) foo `` m^3;
Computing conversions to base units; may take a moment.
60228605349 3
(%o5) ----------- ` m
48828125
(%i6) %, numer;
3
(%o6) 1233.48183754752 ` m
Declared unit conversions.
(%i1) load (ezunits)$
(%i2) declare_unit_conversion (MMBtu = 10^6*Btu, kW = 1000*W);
(%o2) done
(%i3) declare_unit_conversion (kWh = kW*hour, MWh = 1000*kWh,
bell = 1800*s);
(%o3) done
(%i4) 1 ` kW*s `` MWh;
Computing conversions to base units; may take a moment.
1
(%o4) ------- ` MWh
3600000
(%i5) 1 ` kW/m^2 `` MMBtu/bell/ft^2;
1306449 MMBtu
(%o5) ---------- ` --------
8242187500 2
bell ft
Categories: Package ezunits
Shows the value and the units of one of the constants declared by package
physical_constants, which includes a list of physical constants, or
of a new constant declared in package ezunits (see
declare_constvalue).
Note that constant values as recognized by constvalue
are separate from values declared by numerval and
recognized by constantp.
Example:
(%i1) load (physical_constants)$
(%i2) constvalue (%G);
3
m
(%o2) 6.67428 ` -----
2
kg s
(%i3) get ('%G, 'description);
(%o3) Newtonian constant of gravitation
Categories: Package ezunits
Declares the value of a constant to be used in package ezunits. This
function should be loaded with load(ezunits).
Example:
(%i1) load (ezunits)$
(%i2) declare_constvalue (FOO, 100 ` lbm / acre);
lbm
(%o2) 100 ` ----
acre
(%i3) FOO * (50 ` acre);
(%o3) 50 FOO ` acre
(%i4) constvalue (%);
(%o4) 5000 ` lbm
Categories: Package ezunits
Reverts the effect of declare_constvalue. This function should be
loaded with load(ezunits).
Categories: Package ezunits
Returns the units of a dimensional quantity x, or returns 1 if x is nondimensional.
x may be a literal dimensional expression a ` b,
a symbol with declared units via declare_units,
or an expression containing either or both of those.
This function should be loaded with load(ezunits).
Example:
(%i1) load (ezunits)$
(%i2) foo : 100 ` kg;
(%o2) 100 ` kg
(%i3) bar : x ` m/s;
m
(%o3) x ` -
s
(%i4) units (foo);
(%o4) kg
(%i5) units (bar);
m
(%o5) -
s
(%i6) units (foo * bar);
kg m
(%o6) ----
s
(%i7) units (foo / bar);
kg s
(%o7) ----
m
(%i8) units (foo^2);
2
(%o8) kg
Categories: Package ezunits
Declares that units should return units u for a,
where u is an expression. This function should be loaded with
load(ezunits).
Example:
(%i1) load (ezunits)$
(%i2) units (aa);
(%o2) 1
(%i3) declare_units (aa, J);
(%o3) J
(%i4) units (aa);
(%o4) J
(%i5) units (aa^2);
2
(%o5) J
(%i6) foo : 100 ` kg;
(%o6) 100 ` kg
(%i7) units (aa * foo);
(%o7) kg J
Categories: Package ezunits
Returns the nondimensional part of a dimensional quantity x, or returns x if x is nondimensional. x may be a literal dimensional expression a ` b, a symbol with declared quantity, or an expression containing either or both of those.
This function should be loaded with load(ezunits).
Example:
(%i1) load (ezunits)$
(%i2) foo : 100 ` kg;
(%o2) 100 ` kg
(%i3) qty (foo);
(%o3) 100
(%i4) bar : v ` m/s;
m
(%o4) v ` -
s
(%i5) foo * bar;
kg m
(%o5) 100 v ` ----
s
(%i6) qty (foo * bar);
(%o6) 100 v
Categories: Package ezunits
Declares that qty should return x for symbol a, where
x is a nondimensional quantity. This function should be loaded
with load(ezunits).
Example:
(%i1) load (ezunits)$
(%i2) declare_qty (aa, xx);
(%o2) xx
(%i3) qty (aa);
(%o3) xx
(%i4) qty (aa^2);
2
(%o4) xx
(%i5) foo : 100 ` kg;
(%o5) 100 ` kg
(%i6) qty (aa * foo);
(%o6) 100 xx
Categories: Package ezunits
Returns true if x is a literal dimensional expression,
a symbol declared dimensional,
or an expression in which the main operator is declared dimensional.
unitp returns false otherwise.
load(ezunits) loads this function.
Examples:
unitp applied to a literal dimensional expression.
(%i1) load (ezunits)$ (%i2) unitp (100 ` kg); (%o2) true
unitp applied to a symbol declared dimensional.
(%i1) load (ezunits)$ (%i2) unitp (foo); (%o2) false (%i3) declare (foo, dimensional); (%o3) done (%i4) unitp (foo); (%o4) true
unitp applied to an expression in which the main operator is declared dimensional.
(%i1) load (ezunits)$ (%i2) unitp (bar (x, y, z)); (%o2) false (%i3) declare (bar, dimensional); (%o3) done (%i4) unitp (bar (x, y, z)); (%o4) true
Categories: Package ezunits
Appends equations u = v, ... to the list of unit conversions known to the unit conversion operator ``. u and v are both multiplicative terms, in which any variables are units, or both literal dimensional expressions.
At present, it is necessary to express conversions such that the left-hand side of each equation is a simple unit (not a multiplicative expression) or a literal dimensional expression with the quantity equal to 1 and the unit being a simple unit. This limitation might be relaxed in future versions.
known_unit_conversions is the list of known unit conversions.
This function should be loaded with load(ezunits).
Examples:
Unit conversions expressed by equations of multiplicative terms.
(%i1) load (ezunits)$
(%i2) declare_unit_conversion (nautical_mile = 1852 * m,
fortnight = 14 * day);
(%o2) done
(%i3) 100 ` nautical_mile / fortnight `` m/s;
Computing conversions to base units; may take a moment.
463 m
(%o3) ---- ` -
3024 s
Unit conversions expressed by equations of literal dimensional expressions.
(%i1) load (ezunits)$ (%i2) declare_unit_conversion (1 ` fluid_ounce = 2 ` tablespoon); (%o2) done (%i3) declare_unit_conversion (1 ` tablespoon = 3 ` teaspoon); (%o3) done (%i4) 15 ` fluid_ounce `` teaspoon; Computing conversions to base units; may take a moment. (%o4) 90 ` teaspoon
Categories: Package ezunits
Declares a_1, ..., a_n to have dimensions d_1, ..., d_n, respectively.
Each a_k is a symbol or a list of symbols. If it is a list, then every symbol in a_k is declared to have dimension d_k.
load(ezunits) loads these functions.
Examples:
(%i1) load (ezunits) $
(%i2) declare_dimensions ([x, y, z], length, [t, u], time);
(%o2) done
(%i3) dimensions (y^2/u);
2
length
(%o3) -------
time
(%i4) fundamental_units (y^2/u);
0 errors, 0 warnings
2
m
(%o4) --
s
Categories: Package ezunits
Reverts the effect of declare_dimensions. This function should be
loaded with load(ezunits).
Categories: Package ezunits
declare_fundamental_dimensions declares fundamental dimensions.
Symbols d_1, d_2, d_3, ... are appended to the list of
fundamental dimensions, if they are not already on the list.
remove_fundamental_dimensions reverts the effect of declare_fundamental_dimensions.
fundamental_dimensions is the list of fundamental dimensions.
By default, the list comprises several physical dimensions.
load(ezunits) loads these functions.
Examples:
(%i1) load (ezunits) $
(%i2) fundamental_dimensions;
(%o2) [length, mass, time, current, temperature, quantity]
(%i3) declare_fundamental_dimensions (money, cattle, happiness);
(%o3) done
(%i4) fundamental_dimensions;
(%o4) [length, mass, time, current, temperature, quantity,
money, cattle, happiness]
(%i5) remove_fundamental_dimensions (cattle, happiness);
(%o5) done
(%i6) fundamental_dimensions;
(%o6) [length, mass, time, current, temperature, quantity, money]
Categories: Package ezunits
declare_fundamental_units declares u_1, ..., u_n
to have dimensions d_1, ..., d_n, respectively.
All arguments must be symbols.
After calling declare_fundamental_units,
dimensions(u_k) returns d_k for each argument u_1, ..., u_n,
and fundamental_units(d_k) returns u_k for each argument d_1, ..., d_n.
remove_fundamental_units reverts the effect of declare_fundamental_units.
load(ezunits) loads these functions.
Examples:
(%i1) load (ezunits) $
(%i2) declare_fundamental_dimensions (money, cattle, happiness);
(%o2) done
(%i3) declare_fundamental_units (dollar, money, goat, cattle,
smile, happiness);
(%o3) [dollar, goat, smile]
(%i4) dimensions (100 ` dollar/goat/km^2);
money
(%o4) --------------
2
cattle length
(%i5) dimensions (x ` smile/kg);
happiness
(%o5) ---------
mass
(%i6) fundamental_units (money*cattle/happiness);
0 errors, 0 warnings
dollar goat
(%o6) -----------
smile
Categories: Package ezunits
dimensions returns the dimensions of the dimensional quantity x
as an expression comprising products and powers of base dimensions.
dimensions_as_list returns the dimensions of the dimensional quantity x
as a list, in which each element is an integer which indicates the power of the
corresponding base dimension in the dimensions of x.
load(ezunits) loads these functions.
Examples:
(%i1) load (ezunits)$
(%i2) dimensions (1000 ` kg*m^2/s^3);
2
length mass
(%o2) ------------
3
time
(%i3) declare_units (foo, acre*ft/hour);
acre ft
(%o3) -------
hour
(%i4) dimensions (foo);
3
length
(%o4) -------
time
(%i1) load (ezunits)$
(%i2) fundamental_dimensions;
(%o2) [length, mass, time, charge, temperature, quantity]
(%i3) dimensions_as_list (1000 ` kg*m^2/s^3);
(%o3) [2, 1, - 3, 0, 0, 0]
(%i4) declare_units (foo, acre*ft/hour);
acre ft
(%o4) -------
hour
(%i5) dimensions_as_list (foo);
(%o5) [3, 0, - 1, 0, 0, 0]
Categories: Package ezunits
fundamental_units(x) returns the units
associated with the fundamental dimensions of x.
as determined by dimensions(x).
x may be a literal dimensional expression a ` b,
a symbol with declared units via declare_units,
or an expression containing either or both of those.
fundamental_units() returns the list of all known fundamental units,
as declared by declare_fundamental_units.
load(ezunits) loads this function.
Examples:
(%i1) load (ezunits)$
(%i2) fundamental_units ();
(%o2) [m, kg, s, A, K, mol]
(%i3) fundamental_units (100 ` mile/hour);
m
(%o3) -
s
(%i4) declare_units (aa, g/foot^2);
g
(%o4) -----
2
foot
(%i5) fundamental_units (aa);
kg
(%o5) --
2
m
Categories: Package ezunits
Returns a basis for the dimensionless quantities which can be formed from a list L of dimensional quantities.
load(ezunits) loads this function.
Examples:
(%i1) load (ezunits) $
(%i2) dimensionless ([x ` m, y ` m/s, z ` s]);
0 errors, 0 warnings
0 errors, 0 warnings
y z
(%o2) [---]
x
Dimensionless quantities derived from fundamental physical quantities. Note that the first element on the list is proportional to the fine-structure constant.
(%i1) load (ezunits) $
(%i2) load (physical_constants) $
(%i3) dimensionless([%h_bar, %m_e, %m_P, %%e, %c, %e_0]);
0 errors, 0 warnings
0 errors, 0 warnings
2
%%e %m_e
(%o3) [--------------, ----]
%c %e_0 %h_bar %m_P
Categories: Package ezunits
Finds exponents e_1, ..., e_n such that
dimension(expr) = dimension(v_1^e_1 ... v_n^e_n).
load(ezunits) loads this function.
Examples:
Categories: Package ezunits
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| 57.1 Functions and Variables for f90 |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Prints one or more expressions expr_1, …, expr_n as a Fortran 90 program. Output is printed to the standard output.
f90 prints output in the so-called "free form" input format for
Fortran 90: there is no special attention to column positions.
Long lines are split at a fixed width with the ampersand & continuation
character.
load(f90) loads this function. See also the function fortran.
Examples:
(%i1) load (f90)$
(%i2) foo : expand ((xxx + yyy + 7)^4);
4 3 3 2 2 2
(%o2) yyy + 4 xxx yyy + 28 yyy + 6 xxx yyy + 84 xxx yyy
2 3 2
+ 294 yyy + 4 xxx yyy + 84 xxx yyy + 588 xxx yyy + 1372 yyy
4 3 2
+ xxx + 28 xxx + 294 xxx + 1372 xxx + 2401
(%i3) f90 ('foo = foo);
foo = yyy**4+4*xxx*yyy**3+28*yyy**3+6*xxx**2*yyy**2+84*xxx*yyy**2&
+294*yyy**2+4*xxx**3*yyy+84*xxx**2*yyy+588*xxx*yyy+1372*yyy+xxx**&
4+28*xxx**3+294*xxx**2+1372*xxx+2401
(%o3) false
Multiple expressions.
Capture standard output into a file via the with_stdout function.
(%i1) load (f90)$
(%i2) foo : sin (3*x + 1) - cos (7*x - 2);
(%o2) sin(3 x + 1) - cos(7 x - 2)
(%i3) with_stdout ("foo.f90",
f90 (x=0.25, y=0.625, 'foo=foo, 'stop, 'end));
(%o3) false
(%i4) printfile ("foo.f90");
x = 0.25
y = 0.625
foo = sin(3*x+1)-cos(7*x-2)
stop
end
(%o4) foo.f90
Categories: Translation and compilation · Share packages · Package f90
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| 58.1 Introduction to finance | ||
| 58.2 Functions and Variables for finance |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
This is the Finance Package (Ver 0.1).
In all the functions, rate is the compound interest rate, num is the number of periods and must be positive and flow refers to cash flow so if you have an Output the flow is negative and positive for Inputs.
Note that before using the functions defined in this
package, you have to load it writing load(finance)$.
Author: Nicolas Guarin Zapata.
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Calculates the distance between 2 dates, assuming 360 days years, 30 days months.
Example:
(%i1) load(finance)$ (%i2) days360(2008,12,16,2007,3,25); (%o2) - 621
Categories: Package finance
We can calculate the future value of a Present one given a certain interest rate. rate is the interest rate, PV is the present value and num is the number of periods.
Example:
(%i1) load(finance)$ (%i2) fv(0.12,1000,3); (%o2) 1404.928
Categories: Package finance
We can calculate the present value of a Future one given a certain interest rate. rate is the interest rate, FV is the future value and num is the number of periods.
Example:
(%i1) load(finance)$ (%i2) pv(0.12,1000,3); (%o2) 711.7802478134108
Categories: Package finance
Plots the money flow in a time line, the positive values are in blue and upside; the negative ones are in red and downside. The direction of the flow is given by the sign of the value. val is a list of flow values.
Example:
(%i1) load(finance)$ (%i2) graph_flow([-5000,-3000,800,1300,1500,2000])$
Categories: Package finance
We can calculate the annuity knowing the present value (like an amount), it is a constant and periodic payment. rate is the interest rate, PV is the present value and num is the number of periods.
Example:
(%i1) load(finance)$ (%i2) annuity_pv(0.12,5000,10); (%o2) 884.9208207992202
Categories: Package finance
We can calculate the annuity knowing the desired value (future value), it is a constant and periodic payment. rate is the interest rate, FV is the future value and num is the number of periods.
Example:
(%i1) load(finance)$ (%i2) annuity_fv(0.12,65000,10); (%o2) 3703.970670389863
Categories: Package finance
We can calculate the annuity knowing the present value (like an amount), in a growing periodic payment. rate is the interest rate, growing_rate is the growing rate, PV is the present value and num is the number of periods.
Example:
(%i1) load(finance)$ (%i2) geo_annuity_pv(0.14,0.05,5000,10); (%o2) 802.6888176505123
Categories: Package finance
We can calculate the annuity knowing the desired value (future value), in a growing periodic payment. rate is the interest rate, growing_rate is the growing rate, FV is the future value and num is the number of periods.
Example:
(%i1) load(finance)$ (%i2) geo_annuity_fv(0.14,0.05,5000,10); (%o2) 216.5203395312695
Categories: Package finance
Amortization table determined by a specific rate. rate is the interest rate, amount is the amount value, and num is the number of periods.
Example:
(%i1) load(finance)$
(%i2) amortization(0.05,56000,12)$
"n" "Balance" "Interest" "Amortization" "Payment"
0.000 56000.000 0.000 0.000 0.000
1.000 52481.777 2800.000 3518.223 6318.223
2.000 48787.643 2624.089 3694.134 6318.223
3.000 44908.802 2439.382 3878.841 6318.223
4.000 40836.019 2245.440 4072.783 6318.223
5.000 36559.597 2041.801 4276.422 6318.223
6.000 32069.354 1827.980 4490.243 6318.223
7.000 27354.599 1603.468 4714.755 6318.223
8.000 22404.106 1367.730 4950.493 6318.223
9.000 17206.088 1120.205 5198.018 6318.223
10.000 11748.170 860.304 5457.919 6318.223
11.000 6017.355 587.408 5730.814 6318.223
12.000 0.000 300.868 6017.355 6318.223
Categories: Package finance
The amortization table determined by a specific rate and with growing payment
can be claculated by arit_amortization.
Notice that the payment is not constant, it presents
an arithmetic growing, increment is then the difference between two
consecutive rows in the "Payment" column.
rate is the interest rate, increment is the increment, amount
is the amount value, and num is the number of periods.
Example:
(%i1) load(finance)$
(%i2) arit_amortization(0.05,1000,56000,12)$
"n" "Balance" "Interest" "Amortization" "Payment"
0.000 56000.000 0.000 0.000 0.000
1.000 57403.679 2800.000 -1403.679 1396.321
2.000 57877.541 2870.184 -473.863 2396.321
3.000 57375.097 2893.877 502.444 3396.321
4.000 55847.530 2868.755 1527.567 4396.321
5.000 53243.586 2792.377 2603.945 5396.321
6.000 49509.443 2662.179 3734.142 6396.321
7.000 44588.594 2475.472 4920.849 7396.321
8.000 38421.703 2229.430 6166.892 8396.321
9.000 30946.466 1921.085 7475.236 9396.321
10.000 22097.468 1547.323 8848.998 10396.321
11.000 11806.020 1104.873 10291.448 11396.321
12.000 -0.000 590.301 11806.020 12396.321
Categories: Package finance
The amortization table determined by rate, amount,
and number of periods can be found by geo_amortization.
Notice that the payment is not constant, it presents
a geometric growing, growing_rate is then the quotient between two
consecutive rows in the "Payment" column.
rate is the interest rate, amount
is the amount value, and num is the number of periods.
Example:
(%i1) load(finance)$
(%i2) geo_amortization(0.05,0.03,56000,12)$
"n" "Balance" "Interest" "Amortization" "Payment"
0.000 56000.000 0.000 0.000 0.000
1.000 53365.296 2800.000 2634.704 5434.704
2.000 50435.816 2668.265 2929.480 5597.745
3.000 47191.930 2521.791 3243.886 5765.677
4.000 43612.879 2359.596 3579.051 5938.648
5.000 39676.716 2180.644 3936.163 6116.807
6.000 35360.240 1983.836 4316.475 6300.311
7.000 30638.932 1768.012 4721.309 6489.321
8.000 25486.878 1531.947 5152.054 6684.000
9.000 19876.702 1274.344 5610.176 6884.520
10.000 13779.481 993.835 6097.221 7091.056
11.000 7164.668 688.974 6614.813 7303.787
12.000 0.000 358.233 7164.668 7522.901
Categories: Package finance
The table that represents the values in a constant and periodic
saving can be found by saving.
amount represents the desired quantity and num the number
of periods to save.
Example:
(%i1) load(finance)$
(%i2) saving(0.15,12000,15)$
"n" "Balance" "Interest" "Payment"
0.000 0.000 0.000 0.000
1.000 252.205 0.000 252.205
2.000 542.240 37.831 252.205
3.000 875.781 81.336 252.205
4.000 1259.352 131.367 252.205
5.000 1700.460 188.903 252.205
6.000 2207.733 255.069 252.205
7.000 2791.098 331.160 252.205
8.000 3461.967 418.665 252.205
9.000 4233.467 519.295 252.205
10.000 5120.692 635.020 252.205
11.000 6141.000 768.104 252.205
12.000 7314.355 921.150 252.205
13.000 8663.713 1097.153 252.205
14.000 10215.474 1299.557 252.205
15.000 12000.000 1532.321 252.205
Categories: Package finance
Calculates the present value of a value series to evaluate the viability in a project. val is a list of varying cash flows.
Example:
(%i1) load(finance)$ (%i2) npv(0.25,[100,500,323,124,300]); (%o2) 714.4703999999999
Categories: Package finance
IRR (Internal Rate of Return) is the value of rate which makes Net Present Value zero. flowValues is a list of varying cash flows, I0 is the initial investment.
Example:
(%i1) load(finance)$ (%i2) res:irr([-5000,0,800,1300,1500,2000],0)$ (%i3) rhs(res[1][1]); (%o3) .03009250374237132
Categories: Package finance
Calculates the ratio Benefit/Cost. Benefit is the Net Present Value (NPV) of the inputs, and Cost is the Net Present Value (NPV) of the outputs. Notice that if there is not an input or output value in a specific period, the input/output would be a zero for that period. rate is the interest rate, input is a list of input values, and output is a list of output values.
Example:
(%i1) load(finance)$ (%i2) benefit_cost(0.24,[0,300,500,150],[100,320,0,180]); (%o2) 1.427249324905784
Categories: Package finance
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| 59.1 Introduction to fractals | ||
| 59.2 Definitions for IFS fractals | ||
| 59.3 Definitions for complex fractals | ||
| 59.4 Definitions for Koch snowflakes | ||
| 59.5 Definitions for Peano maps |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
This package defines some well known fractals:
- with random IFS (Iterated Function System): the Sierpinsky triangle, a Tree and a Fern
- Complex Fractals: the Mandelbrot and Julia Sets
- the Koch snowflake sets
- Peano maps: the Sierpinski and Hilbert maps
Author: José Ramírez Labrador.
For questions, suggestions and bugs, please feel free to contact me at
pepe DOT ramirez AAATTT uca DOT es
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Some fractals can be generated by iterative applications of contractive affine transformations in a random way; see
Hoggar S. G., "Mathematics for computer graphics", Cambridge University Press 1994.
We define a list with several contractive affine transformations, and we randomly select the transformation in a recursive way. The probability of the choice of a transformation must be related with the contraction ratio.
You can change the transformations and find another fractal
Sierpinski Triangle: 3 contractive maps; .5 contraction constant and translations; all maps have the same contraction ratio. Argument n must be great enougth, 10000 or greater.
Example:
(%i1) load(fractals)$ (%i2) n: 10000$ (%i3) plot2d([discrete,sierpinskiale(n)], [style,dots])$
Categories: Package fractals
3 contractive maps all with the same contraction ratio. Argument n must be great enougth, 10000 or greater.
Example:
(%i1) load(fractals)$ (%i2) n: 10000$ (%i3) plot2d([discrete,treefale(n)], [style,dots])$
Categories: Package fractals
4 contractive maps, the probability to choice a transformation must be related with the contraction ratio. Argument n must be great enougth, 10000 or greater.
Example:
(%i1) load(fractals)$ (%i2) n: 10000$ (%i3) plot2d([discrete,fernfale(n)], [style,dots])$
Categories: Package fractals
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Mandelbrot set.
Example:
This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.
(%i1) load(fractals)$
(%i2) plot3d (mandelbrot_set, [x, -2.5, 1], [y, -1.5, 1.5],
[gnuplot_preamble, "set view map"],
[gnuplot_pm3d, true],
[grid, 150, 150])$
Categories: Package fractals
Julia sets.
This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.
Example:
(%i1) load(fractals)$
(%i2) plot3d (julia_set, [x, -2, 1], [y, -1.5, 1.5],
[gnuplot_preamble, "set view map"],
[gnuplot_pm3d, true],
[grid, 150, 150])$
See also julia_parameter.
Categories: Package fractals
Default value: %i
Complex parameter for Julia fractals.
Its default value is %i; we suggest the values -.745+%i*.113002,
-.39054-%i*.58679, -.15652+%i*1.03225, -.194+%i*.6557 and
.011031-%i*.67037.
Categories: Package fractals
While function julia_set implements the transformation julia_parameter+z^2,
function julia_sin implements julia_parameter*sin(z). See source code
for more details.
This program runs slowly because it calculates a lot of sines.
Example:
This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.
(%i1) load(fractals)$
(%i2) julia_parameter:1+.1*%i$
(%i3) plot3d (julia_sin, [x, -2, 2], [y, -3, 3],
[gnuplot_preamble, "set view map"],
[gnuplot_pm3d, true],
[grid, 150, 150])$
See also julia_parameter.
Categories: Package fractals
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Koch snowflake sets. Function snowmap plots the snow Koch map
over the vertex of an initial closed polygonal, in the complex plane. Here
the orientation of the polygon is important. Argument nn is the number of
recursive applications of Koch transformation; nn must be small (5 or 6).
Examples:
(%i1) load(fractals)$
(%i2) plot2d([discrete,
snowmap([1,exp(%i*%pi*2/3),exp(-%i*%pi*2/3),1],4)])$
(%i3) plot2d([discrete,
snowmap([1,exp(-%i*%pi*2/3),exp(%i*%pi*2/3),1],4)])$
(%i4) plot2d([discrete, snowmap([0,1,1+%i,%i,0],4)])$
(%i5) plot2d([discrete, snowmap([0,%i,1+%i,1,0],4)])$
Categories: Package fractals
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Continuous curves that cover an area. Warning: the number of points exponentially grows with n.
Hilbert map. Argument nn must be small (5, for example). Maxima can crash if nn is 7 or greater.
Example:
(%i1) load(fractals)$ (%i2) plot2d([discrete,hilbertmap(6)])$
Categories: Package fractals
Sierpinski map. Argument nn must be small (5, for example). Maxima can crash if nn is 7 or greater.
Example:
(%i1) load(fractals)$ (%i2) plot2d([discrete,sierpinskimap(6)])$
Categories: Package fractals
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| 60.1 Functions and Variables for ggf |
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Default value: 3
This is an option variable for function ggf.
When computing the continued fraction of the generating function, a partial quotient having a degree (strictly) greater than GGFINFINITY will be discarded and the current convergent will be considered as the exact value of the generating function; most often the degree of all partial quotients will be 0 or 1; if you use a greater value, then you should give enough terms in order to make the computation accurate enough.
See also ggf.
Categories: Package ggf
Default value: 3
This is an option variable for function ggf.
When computing the continued fraction of the generating function, if no good result has been found (see the GGFINFINITY flag) after having computed GGFCFMAX partial quotients, the generating function will be considered as not being a fraction of two polynomials and the function will exit. Put freely a greater value for more complicated generating functions.
See also ggf.
Categories: Package ggf
Compute the generating function (if it is a fraction of two polynomials) of a sequence, its first terms being given. l is a list of numbers.
The solution is returned as a fraction of two polynomials.
If no solution has been found, it returns with done.
This function is controlled by global variables GGFINFINITY and GGFCFMAX. See also GGFINFINITY and GGFCFMAX.
To use this function write first load("ggf").
Categories: Generating functions · Share packages · Package ggf
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| 61.1 Introduction to graphs | ||
| 61.2 Functions and Variables for graphs |
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The graphs package provides graph and digraph data structure for
Maxima. Graphs and digraphs are simple (have no multiple edges nor
loops), although digraphs can have a directed edge from u to
v and a directed edge from v to u.
Internally graphs are represented by adjacency lists and implemented as a lisp structures. Vertices are identified by their ids (an id is an integer). Edges/arcs are represented by lists of length 2. Labels can be assigned to vertices of graphs/digraphs and weights can be assigned to edges/arcs of graphs/digraphs.
There is a draw_graph function for drawing graphs. Graphs are
drawn using a force based vertex positioning
algorithm. draw_graph can also use graphviz programs available
from http://www.graphviz.org. draw_graph is based on the maxima
draw package.
To use the graphs package, first load it with load(graphs).
Categories: Share packages · Package graphs
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Creates a new graph on the set of vertices v_list and with edges e_list.
v_list is a list of vertices ([v1, v2,..., vn]) or a
list of vertices together with vertex labels ([[v1,l1], [v2,l2],..., [vn,ln]]).
n is the number of vertices. Vertices will be identified by integers from 0 to n-1.
e_list is a list of edges ([e1, e2,..., em]) or a list of
edges together with edge-weights ([[e1, w1], ..., [em, wm]]).
If directed is not false, a directed graph will be returned.
Example 1: create a cycle on 3 vertices:
(%i1) load (graphs)$ (%i2) g : create_graph([1,2,3], [[1,2], [2,3], [1,3]])$ (%i3) print_graph(g)$ Graph on 3 vertices with 3 edges. Adjacencies: 3 : 1 2 2 : 3 1 1 : 3 2
Example 2: create a cycle on 3 vertices with edge weights:
(%i1) load (graphs)$
(%i2) g : create_graph([1,2,3], [[[1,2], 1.0], [[2,3], 2.0],
[[1,3], 3.0]])$
(%i3) print_graph(g)$
Graph on 3 vertices with 3 edges.
Adjacencies:
3 : 1 2
2 : 3 1
1 : 3 2
Example 3: create a directed graph:
(%i1) load (graphs)$
(%i2) d : create_graph(
[1,2,3,4],
[
[1,3], [1,4],
[2,3], [2,4]
],
'directed = true)$
(%i3) print_graph(d)$
Digraph on 4 vertices with 4 arcs.
Adjacencies:
4 :
3 :
2 : 4 3
1 : 4 3
Categories: Package graphs
Returns a copy of the graph g.
Categories: Package graphs · Package graphs - constructions
Returns the circulant graph with parameters n and d.
Example:
(%i1) load (graphs)$ (%i2) g : circulant_graph(10, [1,3])$ (%i3) print_graph(g)$ Graph on 10 vertices with 20 edges. Adjacencies: 9 : 2 6 0 8 8 : 1 5 9 7 7 : 0 4 8 6 6 : 9 3 7 5 5 : 8 2 6 4 4 : 7 1 5 3 3 : 6 0 4 2 2 : 9 5 3 1 1 : 8 4 2 0 0 : 7 3 9 1
Categories: Package graphs · Package graphs - constructions
Returns the Clebsch graph.
Categories: Package graphs · Package graphs - constructions
Returns the complement of the graph g.
Categories: Package graphs · Package graphs - constructions
Returns the complete bipartite graph on n+m vertices.
Categories: Package graphs · Package graphs - constructions
Returns the complete graph on n vertices.
Categories: Package graphs · Package graphs - constructions
Returns the directed cycle on n vertices.
Categories: Package graphs · Package graphs - constructions
Returns the cycle on n vertices.
Categories: Package graphs · Package graphs - constructions
Returns the cuboctahedron graph.
Categories: Package graphs · Package graphs - constructions
Returns the n-dimensional cube.
Categories: Package graphs · Package graphs - constructions
Returns the dodecahedron graph.
Categories: Package graphs · Package graphs - constructions
Returns the empty graph on n vertices.
Categories: Package graphs · Package graphs - constructions
Returns the flower graph on 4n vertices.
Example:
(%i1) load (graphs)$ (%i2) f5 : flower_snark(5)$ (%i3) chromatic_index(f5); (%o3) 4
Categories: Package graphs · Package graphs - constructions
Returns the graph represented by its adjacency matrix A.
Categories: Package graphs · Package graphs - constructions
Returns the Frucht graph.
Categories: Package graphs · Package graphs - constructions
Returns the direct product of graphs g1 and g2.
Example:
(%i1) load (graphs)$ (%i2) grid : graph_product(path_graph(3), path_graph(4))$ (%i3) draw_graph(grid)$
Categories: Package graphs · Package graphs - constructions
Returns the union (sum) of graphs g1 and g2.
Categories: Package graphs · Package graphs - constructions
Returns the n x m grid.
Categories: Package graphs · Package graphs - constructions
Returns the great rhombicosidodecahedron graph.
Categories: Package graphs · Package graphs - constructions
Returns the great rhombicuboctahedron graph.
Categories: Package graphs · Package graphs - constructions
Returns the Grotzch graph.
Categories: Package graphs · Package graphs - constructions
Returns the Heawood graph.
Categories: Package graphs · Package graphs - constructions
Returns the icosahedron graph.
Categories: Package graphs · Package graphs - constructions
Returns the icosidodecahedron graph.
Categories: Package graphs · Package graphs - constructions
Returns the graph induced on the subset V of vertices of the graph g.
Example:
(%i1) load (graphs)$ (%i2) p : petersen_graph()$ (%i3) V : [0,1,2,3,4]$ (%i4) g : induced_subgraph(V, p)$ (%i5) print_graph(g)$ Graph on 5 vertices with 5 edges. Adjacencies: 4 : 3 0 3 : 2 4 2 : 1 3 1 : 0 2 0 : 1 4
Categories: Package graphs · Package graphs - constructions
Returns the line graph of the graph g.
Categories: Package graphs · Package graphs - constructions
Creates a graph using a predicate function f.
vrt is a list/set of vertices or an integer. If vrt is an integer, then vertices of the graph will be integers from 1 to vrt.
f is a predicate function. Two vertices a and b will
be connected if f(a,b)=true.
If directed is not false, then the graph will be directed.
Example 1:
(%i1) load(graphs)$
(%i2) g : make_graph(powerset({1,2,3,4,5}, 2), disjointp)$
(%i3) is_isomorphic(g, petersen_graph());
(%o3) true
(%i4) get_vertex_label(1, g);
(%o4) {1, 2}
Example 2:
(%i1) load(graphs)$ (%i2) f(i, j) := is (mod(j, i)=0)$ (%i3) g : make_graph(20, f, directed=true)$ (%i4) out_neighbors(4, g); (%o4) [8, 12, 16, 20] (%i5) in_neighbors(18, g); (%o5) [1, 2, 3, 6, 9]
Categories: Package graphs · Package graphs - constructions
Returns the mycielskian graph of the graph g.
Categories: Package graphs · Package graphs - constructions
Returns the graph with no vertices and no edges.
Categories: Package graphs · Package graphs - constructions
Returns the directed path on n vertices.
Categories: Package graphs · Package graphs - constructions
Returns the path on n vertices.
Categories: Package graphs · Package graphs - constructions
Returns the petersen graph P_{n,d}. The default values for
n and d are n=5 and d=2.
Categories: Package graphs · Package graphs - constructions
Returns a random bipartite graph on a+b vertices. Each edge is
present with probability p.
Categories: Package graphs · Package graphs - constructions
Returns a random directed graph on n vertices. Each arc is present with probability p.
Categories: Package graphs · Package graphs - constructions
Returns a random d-regular graph on n vertices. The default
value for d is d=3.
Categories: Package graphs · Package graphs - constructions
Returns a random graph on n vertices. Each edge is present with probability p.
Categories: Package graphs · Package graphs - constructions
Returns a random graph on n vertices and random m edges.
Categories: Package graphs · Package graphs - constructions
Returns a random network on n vertices. Each arc is present with
probability p and has a weight in the range [0,w]. The
function returns a list [network, source, sink].
Example:
(%i1) load (graphs)$ (%i2) [net, s, t] : random_network(50, 0.2, 10.0); (%o2) [DIGRAPH, 50, 51] (%i3) max_flow(net, s, t)$ (%i4) first(%); (%o4) 27.65981397932507
Categories: Package graphs · Package graphs - constructions
Returns a random tournament on n vertices.
Categories: Package graphs · Package graphs - constructions
Returns a random tree on n vertices.
Categories: Package graphs · Package graphs - constructions
Returns the small rhombicosidodecahedron graph.
Categories: Package graphs · Package graphs - constructions
Returns the small rhombicuboctahedron graph.
Categories: Package graphs · Package graphs - constructions
Returns the snub cube graph.
Categories: Package graphs · Package graphs - constructions
Returns the snub dodecahedron graph.
Categories: Package graphs · Package graphs - constructions
Returns the truncated cube graph.
Categories: Package graphs · Package graphs - constructions
Returns the truncated dodecahedron graph.
Categories: Package graphs · Package graphs - constructions
Returns the truncated icosahedron graph.
Categories: Package graphs · Package graphs - constructions
Returns the truncated tetrahedron graph.
Categories: Package graphs · Package graphs - constructions
Returns the Tutte graph.
Categories: Package graphs · Package graphs - constructions
Returns the underlying graph of the directed graph g.
Categories: Package graphs · Package graphs - constructions
Returns the wheel graph on n+1 vertices.
Categories: Package graphs · Package graphs - constructions
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Returns the adjacency matrix of the graph gr.
Example:
(%i1) load (graphs)$
(%i2) c5 : cycle_graph(4)$
(%i3) adjacency_matrix(c5);
[ 0 1 0 1 ]
[ ]
[ 1 0 1 0 ]
(%o3) [ ]
[ 0 1 0 1 ]
[ ]
[ 1 0 1 0 ]
Categories: Package graphs · Package graphs - properties
Returns the average degree of vertices in the graph gr.
Example:
(%i1) load (graphs)$
(%i2) average_degree(grotzch_graph());
40
(%o2) --
11
Categories: Package graphs · Package graphs - properties
Returns the (vertex sets of) 2-connected components of the graph gr.
Example:
(%i1) load (graphs)$
(%i2) g : create_graph(
[1,2,3,4,5,6,7],
[
[1,2],[2,3],[2,4],[3,4],
[4,5],[5,6],[4,6],[6,7]
])$
(%i3) biconnected_components(g);
(%o3) [[6, 7], [4, 5, 6], [1, 2], [2, 3, 4]]
Categories: Package graphs · Package graphs - properties
Returns a bipartition of the vertices of the graph gr or an empty list if gr is not bipartite.
Example:
(%i1) load (graphs)$ (%i2) h : heawood_graph()$ (%i3) [A,B]:bipartition(h); (%o3) [[8, 12, 6, 10, 0, 2, 4], [13, 5, 11, 7, 9, 1, 3]] (%i4) draw_graph(h, show_vertices=A, program=circular)$
Categories: Package graphs · Package graphs - properties
Returns the chromatic index of the graph gr.
Example:
(%i1) load (graphs)$ (%i2) p : petersen_graph()$ (%i3) chromatic_index(p); (%o3) 4
Categories: Package graphs · Package graphs - properties
Returns the chromatic number of the graph gr.
Example:
(%i1) load (graphs)$ (%i2) chromatic_number(cycle_graph(5)); (%o2) 3 (%i3) chromatic_number(cycle_graph(6)); (%o3) 2
Categories: Package graphs · Package graphs - properties
Removes the weight of the edge e in the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g : create_graph(3, [[[0,1], 1.5], [[1,2], 1.3]])$ (%i3) get_edge_weight([0,1], g); (%o3) 1.5 (%i4) clear_edge_weight([0,1], g)$ (%i5) get_edge_weight([0,1], g); (%o5) 1
Categories: Package graphs · Package graphs - properties
Removes the label of the vertex v in the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g : create_graph([[0,"Zero"], [1, "One"]], [[0,1]])$ (%i3) get_vertex_label(0, g); (%o3) Zero (%i4) clear_vertex_label(0, g); (%o4) done (%i5) get_vertex_label(0, g); (%o5) false
Categories: Package graphs · Package graphs - properties
Returns the (vertex sets of) connected components of the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g: graph_union(cycle_graph(5), path_graph(4))$ (%i3) connected_components(g); (%o3) [[1, 2, 3, 4, 0], [8, 7, 6, 5]]
Categories: Package graphs · Package graphs - properties
Returns the diameter of the graph gr.
Example:
(%i1) load (graphs)$ (%i2) diameter(dodecahedron_graph()); (%o2) 5
Categories: Package graphs · Package graphs - properties
Returns an optimal coloring of the edges of the graph gr.
The function returns the chromatic index and a list representing the coloring of the edges of gr.
Example:
(%i1) load (graphs)$ (%i2) p : petersen_graph()$ (%i3) [ch_index, col] : edge_coloring(p); (%o3) [4, [[[0, 5], 3], [[5, 7], 1], [[0, 1], 1], [[1, 6], 2], [[6, 8], 1], [[1, 2], 3], [[2, 7], 4], [[7, 9], 2], [[2, 3], 2], [[3, 8], 3], [[5, 8], 2], [[3, 4], 1], [[4, 9], 4], [[6, 9], 3], [[0, 4], 2]]] (%i4) assoc([0,1], col); (%o4) 1 (%i5) assoc([0,5], col); (%o5) 3
Categories: Package graphs · Package graphs - properties
Returns the list of vertex degrees of the graph gr.
Example:
(%i1) load (graphs)$ (%i2) degree_sequence(random_graph(10, 0.4)); (%o2) [2, 2, 2, 2, 2, 2, 3, 3, 3, 3]
Categories: Package graphs · Package graphs - properties
Returns the edge-connectivity of the graph gr.
See also min_edge_cut.
Categories: Package graphs · Package graphs - properties
Returns the list of edges (arcs) in a (directed) graph gr.
Example:
(%i1) load (graphs)$ (%i2) edges(complete_graph(4)); (%o2) [[2, 3], [1, 3], [1, 2], [0, 3], [0, 2], [0, 1]]
Categories: Package graphs · Package graphs - properties
Returns the weight of the edge e in the graph gr.
If there is no weight assigned to the edge, the function returns 1. If the edge is not present in the graph, the function signals an error or returns the optional argument ifnot.
Example:
(%i1) load (graphs)$ (%i2) c5 : cycle_graph(5)$ (%i3) get_edge_weight([1,2], c5); (%o3) 1 (%i4) set_edge_weight([1,2], 2.0, c5); (%o4) done (%i5) get_edge_weight([1,2], c5); (%o5) 2.0
Categories: Package graphs · Package graphs - properties
Returns the label of the vertex v in the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g : create_graph([[0,"Zero"], [1, "One"]], [[0,1]])$ (%i3) get_vertex_label(0, g); (%o3) Zero
Categories: Package graphs · Package graphs - properties
Returns the characteristic polynomial (in variable x) of the graph gr.
Example:
(%i1) load (graphs)$
(%i2) p : petersen_graph()$
(%i3) graph_charpoly(p, x), factor;
5 4
(%o3) (x - 3) (x - 1) (x + 2)
Categories: Package graphs · Package graphs - properties
Returns the center of the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g : grid_graph(5,5)$ (%i3) graph_center(g); (%o3) [12]
Categories: Package graphs · Package graphs - properties
Returns the eigenvalues of the graph gr. The function returns
eigenvalues in the same format as maxima eigenvalues function.
Example:
(%i1) load (graphs)$ (%i2) p : petersen_graph()$ (%i3) graph_eigenvalues(p); (%o3) [[3, - 2, 1], [1, 4, 5]]
Categories: Package graphs · Package graphs - properties
Returns the periphery of the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g : grid_graph(5,5)$ (%i3) graph_periphery(g); (%o3) [24, 20, 4, 0]
Categories: Package graphs · Package graphs - properties
Returns the number of edges in the graph gr.
Example:
(%i1) load (graphs)$ (%i2) p : petersen_graph()$ (%i3) graph_size(p); (%o3) 15
Categories: Package graphs · Package graphs - properties
Returns the number of vertices in the graph gr.
Example:
(%i1) load (graphs)$ (%i2) p : petersen_graph()$ (%i3) graph_order(p); (%o3) 10
Categories: Package graphs · Package graphs - properties
Returns the length of the shortest cycle in gr.
Example:
(%i1) load (graphs)$ (%i2) g : heawood_graph()$ (%i3) girth(g); (%o3) 6
Categories: Package graphs · Package graphs - properties
Returns the Hamilton cycle of the graph gr or an empty list if gr is not hamiltonian.
Example:
(%i1) load (graphs)$ (%i2) c : cube_graph(3)$ (%i3) hc : hamilton_cycle(c); (%o3) [7, 3, 2, 6, 4, 0, 1, 5, 7] (%i4) draw_graph(c, show_edges=vertices_to_cycle(hc))$
Categories: Package graphs · Package graphs - properties
Returns the Hamilton path of the graph gr or an empty list if gr does not have a Hamilton path.
Example:
(%i1) load (graphs)$ (%i2) p : petersen_graph()$ (%i3) hp : hamilton_path(p); (%o3) [0, 5, 7, 2, 1, 6, 8, 3, 4, 9] (%i4) draw_graph(p, show_edges=vertices_to_path(hp))$
Categories: Package graphs · Package graphs - properties
Returns a an isomorphism between graphs/digraphs gr1 and gr2. If gr1 and gr2 are not isomorphic, it returns an empty list.
Example:
(%i1) load (graphs)$
(%i2) clk5:complement_graph(line_graph(complete_graph(5)))$
(%i3) isomorphism(clk5, petersen_graph());
(%o3) [9 -> 0, 2 -> 1, 6 -> 2, 5 -> 3, 0 -> 4, 1 -> 5, 3 -> 6,
4 -> 7, 7 -> 8, 8 -> 9]
Categories: Package graphs · Package graphs - properties
Returns the list of in-neighbors of the vertex v in the directed graph gr.
Example:
(%i1) load (graphs)$ (%i2) p : path_digraph(3)$ (%i3) in_neighbors(2, p); (%o3) [1] (%i4) out_neighbors(2, p); (%o4) []
Categories: Package graphs · Package graphs - properties
Returns true if gr is 2-connected and false otherwise.
Example:
(%i1) load (graphs)$ (%i2) is_biconnected(cycle_graph(5)); (%o2) true (%i3) is_biconnected(path_graph(5)); (%o3) false
Categories: Package graphs · Package graphs - properties
Returns true if gr is bipartite (2-colorable) and false otherwise.
Example:
(%i1) load (graphs)$ (%i2) is_bipartite(petersen_graph()); (%o2) false (%i3) is_bipartite(heawood_graph()); (%o3) true
Categories: Package graphs · Package graphs - properties
Returns true if the graph gr is connected and false otherwise.
Example:
(%i1) load (graphs)$ (%i2) is_connected(graph_union(cycle_graph(4), path_graph(3))); (%o2) false
Categories: Package graphs · Package graphs - properties
Returns true if gr is a directed graph and false otherwise.
Example:
(%i1) load (graphs)$ (%i2) is_digraph(path_graph(5)); (%o2) false (%i3) is_digraph(path_digraph(5)); (%o3) true
Categories: Package graphs · Package graphs - properties
Returns true if e is an edge (arc) in the (directed) graph g
and false otherwise.
Example:
(%i1) load (graphs)$ (%i2) c4 : cycle_graph(4)$ (%i3) is_edge_in_graph([2,3], c4); (%o3) true (%i4) is_edge_in_graph([3,2], c4); (%o4) true (%i5) is_edge_in_graph([2,4], c4); (%o5) false (%i6) is_edge_in_graph([3,2], cycle_digraph(4)); (%o6) false
Categories: Package graphs · Package graphs - properties
Returns true if gr is a graph and false otherwise.
Example:
(%i1) load (graphs)$ (%i2) is_graph(path_graph(5)); (%o2) true (%i3) is_graph(path_digraph(5)); (%o3) false
Categories: Package graphs · Package graphs - properties
Returns true if gr is a graph or a directed graph and false otherwise.
Example:
(%i1) load (graphs)$ (%i2) is_graph_or_digraph(path_graph(5)); (%o2) true (%i3) is_graph_or_digraph(path_digraph(5)); (%o3) true
Categories: Package graphs · Package graphs - properties
Returns true if graphs/digraphs gr1 and gr2 are isomorphic
and false otherwise.
See also isomorphism.
Example:
(%i1) load (graphs)$ (%i2) clk5:complement_graph(line_graph(complete_graph(5)))$ (%i3) is_isomorphic(clk5, petersen_graph()); (%o3) true
Categories: Package graphs · Package graphs - properties
Returns true if gr is a planar graph and false otherwise.
The algorithm used is the Demoucron's algorithm, which is a quadratic time algorithm.
Example:
(%i1) load (graphs)$ (%i2) is_planar(dodecahedron_graph()); (%o2) true (%i3) is_planar(petersen_graph()); (%o3) false (%i4) is_planar(petersen_graph(10,2)); (%o4) true
Categories: Package graphs · Package graphs - properties
Returns true if the directed graph gr is strongly connected and
false otherwise.
Example:
(%i1) load (graphs)$ (%i2) is_sconnected(cycle_digraph(5)); (%o2) true (%i3) is_sconnected(path_digraph(5)); (%o3) false
Categories: Package graphs · Package graphs - properties
Returns true if v is a vertex in the graph g and false otherwise.
Example:
(%i1) load (graphs)$ (%i2) c4 : cycle_graph(4)$ (%i3) is_vertex_in_graph(0, c4); (%o3) true (%i4) is_vertex_in_graph(6, c4); (%o4) false
Categories: Package graphs · Package graphs - properties
Returns true if gr is a tree and false otherwise.
Example:
(%i1) load (graphs)$ (%i2) is_tree(random_tree(4)); (%o2) true (%i3) is_tree(graph_union(random_tree(4), random_tree(5))); (%o3) false
Categories: Package graphs · Package graphs - properties
Returns the laplacian matrix of the graph gr.
Example:
(%i1) load (graphs)$
(%i2) laplacian_matrix(cycle_graph(5));
[ 2 - 1 0 0 - 1 ]
[ ]
[ - 1 2 - 1 0 0 ]
[ ]
(%o2) [ 0 - 1 2 - 1 0 ]
[ ]
[ 0 0 - 1 2 - 1 ]
[ ]
[ - 1 0 0 - 1 2 ]
Categories: Package graphs · Package graphs - properties
Returns a maximum clique of the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g : random_graph(100, 0.5)$ (%i3) max_clique(g); (%o3) [6, 12, 31, 36, 52, 59, 62, 63, 80]
Categories: Package graphs · Package graphs - properties
Returns the maximal degree of vertices of the graph gr and a vertex of maximal degree.
Example:
(%i1) load (graphs)$ (%i2) g : random_graph(100, 0.02)$ (%i3) max_degree(g); (%o3) [6, 79] (%i4) vertex_degree(95, g); (%o4) 2
Categories: Package graphs · Package graphs - properties
Returns a maximum flow through the network net with the source s and the sink t.
The function returns the value of the maximal flow and a list representing the weights of the arcs in the optimal flow.
Example:
(%i1) load (graphs)$
(%i2) net : create_graph(
[1,2,3,4,5,6],
[[[1,2], 1.0],
[[1,3], 0.3],
[[2,4], 0.2],
[[2,5], 0.3],
[[3,4], 0.1],
[[3,5], 0.1],
[[4,6], 1.0],
[[5,6], 1.0]],
directed=true)$
(%i3) [flow_value, flow] : max_flow(net, 1, 6);
(%o3) [0.7, [[[1, 2], 0.5], [[1, 3], 0.2], [[2, 4], 0.2],
[[2, 5], 0.3], [[3, 4], 0.1], [[3, 5], 0.1], [[4, 6], 0.3],
[[5, 6], 0.4]]]
(%i4) fl : 0$
(%i5) for u in out_neighbors(1, net)
do fl : fl + assoc([1, u], flow)$
(%i6) fl;
(%o6) 0.7
Categories: Package graphs · Package graphs - properties
Returns a maximum independent set of the graph gr.
Example:
(%i1) load (graphs)$ (%i2) d : dodecahedron_graph()$ (%i3) mi : max_independent_set(d); (%o3) [0, 3, 5, 9, 10, 11, 18, 19] (%i4) draw_graph(d, show_vertices=mi)$
Categories: Package graphs · Package graphs - properties
Returns a maximum matching of the graph gr.
Example:
(%i1) load (graphs)$
(%i2) d : dodecahedron_graph()$
(%i3) m : max_matching(d);
(%o3) [[5, 7], [8, 9], [6, 10], [14, 19], [13, 18], [12, 17],
[11, 16], [0, 15], [3, 4], [1, 2]]
(%i4) draw_graph(d, show_edges=m)$
Categories: Package graphs · Package graphs - properties
Returns the minimum degree of vertices of the graph gr and a vertex of minimum degree.
Example:
(%i1) load (graphs)$ (%i2) g : random_graph(100, 0.1)$ (%i3) min_degree(g); (%o3) [3, 49] (%i4) vertex_degree(21, g); (%o4) 9
Categories: Package graphs · Package graphs - properties
Returns the minimum edge cut in the graph gr.
See also edge_connectivity.
Categories: Package graphs · Package graphs - properties
Returns the minimum vertex cover of the graph gr.
Categories: Package graphs · Package graphs - properties
Returns the minimum vertex cut in the graph gr.
See also vertex_connectivity.
Categories: Package graphs · Package graphs - properties
Returns the minimum spanning tree of the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g : graph_product(path_graph(10), path_graph(10))$ (%i3) t : minimum_spanning_tree(g)$ (%i4) draw_graph(g, show_edges=edges(t))$
Categories: Package graphs · Package graphs - properties
Returns the list of neighbors of the vertex v in the graph gr.
Example:
(%i1) load (graphs)$ (%i2) p : petersen_graph()$ (%i3) neighbors(3, p); (%o3) [4, 8, 2]
Categories: Package graphs · Package graphs - properties
Returns the length of the shortest odd cycle in the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g : graph_product(cycle_graph(4), cycle_graph(7))$ (%i3) girth(g); (%o3) 4 (%i4) odd_girth(g); (%o4) 7
Categories: Package graphs · Package graphs - properties
Returns the list of out-neighbors of the vertex v in the directed graph gr.
Example:
(%i1) load (graphs)$ (%i2) p : path_digraph(3)$ (%i3) in_neighbors(2, p); (%o3) [1] (%i4) out_neighbors(2, p); (%o4) []
Categories: Package graphs · Package graphs - properties
Returns the list of facial walks in a planar embedding of gr and
false if gr is not a planar graph.
The graph gr must be biconnected.
The algorithm used is the Demoucron's algorithm, which is a quadratic time algorithm.
Example:
(%i1) load (graphs)$
(%i2) planar_embedding(grid_graph(3,3));
(%o2) [[3, 6, 7, 8, 5, 2, 1, 0], [4, 3, 0, 1], [3, 4, 7, 6],
[8, 7, 4, 5], [1, 2, 5, 4]]
Categories: Package graphs · Package graphs - properties
Prints some information about the graph gr.
Example:
(%i1) load (graphs)$ (%i2) c5 : cycle_graph(5)$ (%i3) print_graph(c5)$ Graph on 5 vertices with 5 edges. Adjacencies: 4 : 0 3 3 : 4 2 2 : 3 1 1 : 2 0 0 : 4 1 (%i4) dc5 : cycle_digraph(5)$ (%i5) print_graph(dc5)$ Digraph on 5 vertices with 5 arcs. Adjacencies: 4 : 0 3 : 4 2 : 3 1 : 2 0 : 1 (%i6) out_neighbors(0, dc5); (%o6) [1]
Categories: Package graphs
Returns the radius of the graph gr.
Example:
(%i1) load (graphs)$ (%i2) radius(dodecahedron_graph()); (%o2) 5
Categories: Package graphs · Package graphs - properties
Assigns the weight w to the edge e in the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g : create_graph([1, 2], [[[1,2], 1.2]])$ (%i3) get_edge_weight([1,2], g); (%o3) 1.2 (%i4) set_edge_weight([1,2], 2.1, g); (%o4) done (%i5) get_edge_weight([1,2], g); (%o5) 2.1
Categories: Package graphs · Package graphs - properties
Assigns the label l to the vertex v in the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g : create_graph([[1, "One"], [2, "Two"]], [[1,2]])$ (%i3) get_vertex_label(1, g); (%o3) One (%i4) set_vertex_label(1, "oNE", g); (%o4) done (%i5) get_vertex_label(1, g); (%o5) oNE
Categories: Package graphs · Package graphs - properties
Returns the shortest path from u to v in the graph gr.
Example:
(%i1) load (graphs)$ (%i2) d : dodecahedron_graph()$ (%i3) path : shortest_path(0, 7, d); (%o3) [0, 1, 19, 13, 7] (%i4) draw_graph(d, show_edges=vertices_to_path(path))$
Categories: Package graphs · Package graphs - properties
Returns the length of the shortest weighted path and the shortest weighted path from u to v in the graph gr.
The length of a weighted path is the sum of edge weights of edges in the path. If an edge has no weight, then it has a default weight 1.
Example:
(%i1) load (graphs)$ (%i2) g: petersen_graph(20, 2)$ (%i3) for e in edges(g) do set_edge_weight(e, random(1.0), g)$ (%i4) shortest_weighted_path(0, 10, g); (%o4) [2.575143920268482, [0, 20, 38, 36, 34, 32, 30, 10]]
Categories: Package graphs · Package graphs - properties
Returns the strong components of a directed graph gr.
Example:
(%i1) load (graphs)$ (%i2) t : random_tournament(4)$ (%i3) strong_components(t); (%o3) [[1], [0], [2], [3]] (%i4) vertex_out_degree(3, t); (%o4) 3
Categories: Package graphs · Package graphs - properties
Returns a topological sorting of the vertices of a directed graph dag or an empty list if dag is not a directed acyclic graph.
Example:
(%i1) load (graphs)$
(%i2) g:create_graph(
[1,2,3,4,5],
[
[1,2], [2,5], [5,3],
[5,4], [3,4], [1,3]
],
directed=true)$
(%i3) topological_sort(g);
(%o3) [1, 2, 5, 3, 4]
Categories: Package graphs · Package graphs - properties
Returns the vertex connectivity of the graph g.
See also min_vertex_cut.
Categories: Package graphs · Package graphs - properties
Returns the degree of the vertex v in the graph gr.
Categories: Package graphs · Package graphs - properties
Returns the length of the shortest path between u and v in the (directed) graph gr.
Example:
(%i1) load (graphs)$ (%i2) d : dodecahedron_graph()$ (%i3) vertex_distance(0, 7, d); (%o3) 4 (%i4) shortest_path(0, 7, d); (%o4) [0, 1, 19, 13, 7]
Categories: Package graphs · Package graphs - properties
Returns the eccentricity of the vertex v in the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g:cycle_graph(7)$ (%i3) vertex_eccentricity(0, g); (%o3) 3
Categories: Package graphs · Package graphs - properties
Returns the in-degree of the vertex v in the directed graph gr.
Example:
(%i1) load (graphs)$ (%i2) p5 : path_digraph(5)$ (%i3) print_graph(p5)$ Digraph on 5 vertices with 4 arcs. Adjacencies: 4 : 3 : 4 2 : 3 1 : 2 0 : 1 (%i4) vertex_in_degree(4, p5); (%o4) 1 (%i5) in_neighbors(4, p5); (%o5) [3]
Categories: Package graphs · Package graphs - properties
Returns the out-degree of the vertex v in the directed graph gr.
Example:
(%i1) load (graphs)$ (%i2) t : random_tournament(10)$ (%i3) vertex_out_degree(0, t); (%o3) 2 (%i4) out_neighbors(0, t); (%o4) [7, 1]
Categories: Package graphs · Package graphs - properties
Returns the list of vertices in the graph gr.
Example:
(%i1) load (graphs)$ (%i2) vertices(complete_graph(4)); (%o2) [3, 2, 1, 0]
Categories: Package graphs · Package graphs - properties
Returns an optimal coloring of the vertices of the graph gr.
The function returns the chromatic number and a list representing the coloring of the vertices of gr.
Example:
(%i1) load (graphs)$
(%i2) p:petersen_graph()$
(%i3) vertex_coloring(p);
(%o3) [3, [[0, 2], [1, 3], [2, 2], [3, 3], [4, 1], [5, 3],
[6, 1], [7, 1], [8, 2], [9, 2]]]
Categories: Package graphs · Package graphs - properties
Returns the Wiener index of the graph gr.
Example:
(%i2) wiener_index(dodecahedron_graph()); (%o2) 500
Categories: Package graphs · Package graphs - properties
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Adds the edge e to the graph gr.
Example:
(%i1) load (graphs)$ (%i2) p : path_graph(4)$ (%i3) neighbors(0, p); (%o3) [1] (%i4) add_edge([0,3], p); (%o4) done (%i5) neighbors(0, p); (%o5) [3, 1]
Categories: Package graphs · Package graphs - modifications
Adds all edges in the list e_list to the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g : empty_graph(3)$ (%i3) add_edges([[0,1],[1,2]], g)$ (%i4) print_graph(g)$ Graph on 3 vertices with 2 edges. Adjacencies: 2 : 1 1 : 2 0 0 : 1
Categories: Package graphs · Package graphs - modifications
Adds the vertex v to the graph gr.
Example:
(%i1) load (graphs)$ (%i2) g : path_graph(2)$ (%i3) add_vertex(2, g)$ (%i4) print_graph(g)$ Graph on 3 vertices with 1 edges. Adjacencies: 2 : 1 : 0 0 : 1
Categories: Package graphs · Package graphs - modifications
Adds all vertices in the list v_list to the graph gr.
Categories: Package graphs · Package graphs - modifications
Connects all vertices from the list v_list with the vertices in the list u_list in the graph gr.
v_list and u_list can be single vertices or lists of vertices.
Example:
(%i1) load (graphs)$ (%i2) g : empty_graph(4)$ (%i3) connect_vertices(0, [1,2,3], g)$ (%i4) print_graph(g)$ Graph on 4 vertices with 3 edges. Adjacencies: 3 : 0 2 : 0 1 : 0 0 : 3 2 1
Categories: Package graphs · Package graphs - modifications
Contracts the edge e in the graph gr.
Example:
(%i1) load (graphs)$
(%i2) g: create_graph(
8, [[0,3],[1,3],[2,3],[3,4],[4,5],[4,6],[4,7]])$
(%i3) print_graph(g)$
Graph on 8 vertices with 7 edges.
Adjacencies:
7 : 4
6 : 4
5 : 4
4 : 7 6 5 3
3 : 4 2 1 0
2 : 3
1 : 3
0 : 3
(%i4) contract_edge([3,4], g)$
(%i5) print_graph(g)$
Graph on 7 vertices with 6 edges.
Adjacencies:
7 : 3
6 : 3
5 : 3
3 : 5 6 7 2 1 0
2 : 3
1 : 3
0 : 3
Categories: Package graphs · Package graphs - modifications
Removes the edge e from the graph gr.
Example:
(%i1) load (graphs)$ (%i2) c3 : cycle_graph(3)$ (%i3) remove_edge([0,1], c3)$ (%i4) print_graph(c3)$ Graph on 3 vertices with 2 edges. Adjacencies: 2 : 0 1 1 : 2 0 : 2
Categories: Package graphs · Package graphs - modifications
Removes the vertex v from the graph gr.
Categories: Package graphs
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Exports the graph into the file fl in the DIMACS format. Optional comments will be added to the top of the file.
Categories: Package graphs · Package graphs - io
Returns the graph from file fl in the DIMACS format.
Categories: Package graphs · Package graphs - io
Returns the graph encoded in the graph6 format in the string str.
Categories: Package graphs · Package graphs - io
Returns a string which encodes the graph gr in the graph6 format.
Categories: Package graphs · Package graphs - io
Exports graphs in the list gr_list to the file fl in the graph6 format.
Categories: Package graphs · Package graphs - io
Returns a list of graphs from the file fl in the graph6 format.
Categories: Package graphs · Package graphs - io
Returns the graph encoded in the sparse6 format in the string str.
Categories: Package graphs · Package graphs - io
Returns a string which encodes the graph gr in the sparse6 format.
Categories: Package graphs · Package graphs - io
Exports graphs in the list gr_list to the file fl in the sparse6 format.
Categories: Package graphs · Package graphs - io
Returns a list of graphs from the file fl in the sparse6 format.
Categories: Package graphs · Package graphs - io
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Draws the graph using the draw package.
The algorithm used to position vertices is specified by the optional
argument program. The default value is
program=spring_embedding. draw_graph can also use the
graphviz programs for positioning vertices, but graphviz must be
installed separately.
Example 1:
(%i1) load (graphs)$ (%i2) g:grid_graph(10,10)$ (%i3) m:max_matching(g)$ (%i4) draw_graph(g, spring_embedding_depth=100, show_edges=m, edge_type=dots, vertex_size=0)$
Example 2:
(%i1) load (graphs)$
(%i2) g:create_graph(16,
[
[0,1],[1,3],[2,3],[0,2],[3,4],[2,4],
[5,6],[6,4],[4,7],[6,7],[7,8],[7,10],[7,11],
[8,10],[11,10],[8,9],[11,12],[9,15],[12,13],
[10,14],[15,14],[13,14]
])$
(%i3) t:minimum_spanning_tree(g)$
(%i4) draw_graph(
g,
show_edges=edges(t),
show_edge_width=4,
show_edge_color=green,
vertex_type=filled_square,
vertex_size=2
)$
Example 3:
(%i1) load (graphs)$
(%i2) g:create_graph(16,
[
[0,1],[1,3],[2,3],[0,2],[3,4],[2,4],
[5,6],[6,4],[4,7],[6,7],[7,8],[7,10],[7,11],
[8,10],[11,10],[8,9],[11,12],[9,15],[12,13],
[10,14],[15,14],[13,14]
])$
(%i3) mi : max_independent_set(g)$
(%i4) draw_graph(
g,
show_vertices=mi,
show_vertex_type=filled_up_triangle,
show_vertex_size=2,
edge_color=cyan,
edge_width=3,
show_id=true,
text_color=brown
)$
Example 4:
(%i1) load (graphs)$
(%i2) net : create_graph(
[0,1,2,3,4,5],
[
[[0,1], 3], [[0,2], 2],
[[1,3], 1], [[1,4], 3],
[[2,3], 2], [[2,4], 2],
[[4,5], 2], [[3,5], 2]
],
directed=true
)$
(%i3) draw_graph(
net,
show_weight=true,
vertex_size=0,
show_vertices=[0,5],
show_vertex_type=filled_square,
head_length=0.2,
head_angle=10,
edge_color="dark-green",
text_color=blue
)$
Example 5:
(%i1) load(graphs)$ (%i2) g: petersen_graph(20, 2); (%o2) GRAPH (%i3) draw_graph(g, redraw=true, program=planar_embedding); (%o3) done
Example 6:
(%i1) load(graphs)$
(%i2) t: tutte_graph();
(%o2) GRAPH
(%i3) draw_graph(t, redraw=true,
fixed_vertices=[1,2,3,4,5,6,7,8,9]);
(%o3) done
Categories: Package graphs
Default value: spring_embedding
The default value for the program used to position vertices in
draw_graph program.
Categories: Package graphs · Package graphs - draw_graphs options
Default value: false
If true then ids of the vertices are displayed.
Categories: Package graphs · Package graphs - draw_graphs options
Default value: false
If true then labels of the vertices are displayed.
Categories: Package graphs · Package graphs - draw_graphs options
Default value: center
Determines how to align the labels/ids of the vertices. Can
be left, center or right.
Categories: Package graphs · Package graphs - draw_graphs options
Default value: false
If true then weights of the edges are displayed.
Categories: Package graphs · Package graphs - draw_graphs options
Default value: circle
Defines how vertices are displayed. See the point_type option for
the draw package for possible values.
Categories: Package graphs · Package graphs - draw_graphs options
The size of vertices.
Categories: Package graphs · Package graphs - draw_graphs options
The color used for displaying vertices.
Categories: Package graphs · Package graphs - draw_graphs options
Default value: []
Display selected vertices in the using a different color.
Categories: Package graphs · Package graphs - draw_graphs options
Defines how vertices specified in show_vertices are displayed.
See the point_type option for the draw package for possible
values.
Categories: Package graphs · Package graphs - draw_graphs options
The size of vertices in show_vertices.
Categories: Package graphs · Package graphs - draw_graphs options
The color used for displaying vertices in the show_vertices list.
Categories: Package graphs · Package graphs - draw_graphs options
Default value: []
A partition [[v1,v2,...],...,[vk,...,vn]] of the vertices of the
graph. The vertices of each list in the partition will be drawn in a
different color.
Categories: Package graphs · Package graphs - draw_graphs options
Specifies coloring of the vertices. The coloring col must be specified in the format as returned by vertex_coloring.
Categories: Package graphs · Package graphs - draw_graphs options
The color used for displaying edges.
Categories: Package graphs · Package graphs - draw_graphs options
The width of edges.
Categories: Package graphs · Package graphs - draw_graphs options
Defines how edges are displayed. See the line_type option for the
draw package.
Categories: Package graphs · Package graphs - draw_graphs options
Display edges specified in the list e_list using a different color.
Categories: Package graphs · Package graphs - draw_graphs options
The color used for displaying edges in the show_edges list.
Categories: Package graphs · Package graphs - draw_graphs options
The width of edges in show_edges.
Categories: Package graphs · Package graphs - draw_graphs options
Defines how edges in show_edges are displayed. See the
line_type option for the draw package.
Categories: Package graphs · Package graphs - draw_graphs options
A partition [[e1,e2,...],...,[ek,...,em]] of edges of the
graph. The edges of each list in the partition will be drawn using a
different color.
Categories: Package graphs · Package graphs - draw_graphs options
The coloring of edges. The coloring must be specified in the format as returned by the function edge_coloring.
Categories: Package graphs · Package graphs - draw_graphs options
Default value: false
If true, vertex positions are recomputed even if the positions
have been saved from a previous drawing of the graph.
Categories: Package graphs · Package graphs - draw_graphs options
Default value: 15
The angle for the arrows displayed on arcs (in directed graphs).
Categories: Package graphs · Package graphs - draw_graphs options
Default value: 0.1
The length for the arrows displayed on arcs (in directed graphs).
Categories: Package graphs · Package graphs - draw_graphs options
Default value: 50
The number of iterations in the spring embedding graph drawing algorithm.
Categories: Package graphs · Package graphs - draw_graphs options
The terminal used for drawing (see the terminal option in the
draw package).
Categories: Package graphs · Package graphs - draw_graphs options
The filename of the drawing if terminal is not screen.
Categories: Package graphs · Package graphs - draw_graphs options
Defines the program used for positioning vertices of the graph. Can be
one of the graphviz programs (dot, neato, twopi, circ, fdp),
circular, spring_embedding or
planar_embedding. planar_embedding is only available for
2-connected planar graphs. When program=spring_embedding, a set
of vertices with fixed position can be specified with the
fixed_vertices option.
Categories: Package graphs · Package graphs - draw_graphs options
Specifies a list of vertices which will have positions fixed along a regular polygon.
Can be used when program=spring_embedding.
Categories: Package graphs · Package graphs - draw_graphs options
Converts a list v_list of vertices to a list of edges of the path defined by v_list.
Categories: Package graphs
Converts a list v_list of vertices to a list of edges of the cycle defined by v_list.
Categories: Package graphs
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| 62.1 Introduction to grobner | ||
| 62.2 Functions and Variables for grobner |
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grobner is a package for working with Groebner bases in Maxima.
To use the following functions you must load the `grobner.lisp' package.
load(grobner);
A demo can be started by
demo("grobner.demo");
or
batch("grobner.demo")
Some of the calculation in the demo will take a lot of time therefore the output `grobner-demo.output' of the demo can be found in the same directory as the demo file.
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The package was written by
Marek Rychlik
http://alamos.math.arizona.edu
and is released 2002-05-24 under the terms of the General Public License(GPL) (see file `grobner.lisp'). This documentation was extracted from the files
`README', `grobner.lisp', `grobner.demo', `grobner-demo.output'
by Günter Nowak. Suggestions for improvement of the documentation can be discussed at the maxima-mailing-list maxima@math.utexas.edu. The code is a little bit out of date now. Modern implementation use the fast F4 algorithm described in
A new efficient algorithm for computing Gröbner bases (F4) Jean-Charles Faugère LIP6/CNRS Université Paris VI January 20, 1999 |
Categories: Groebner bases · Share packages · Package grobner
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lex
pure lexicographic, default order for monomial comparisons
grlex
total degree order, ties broken by lexicographic
grevlex
total degree, ties broken by reverse lexicographic
invlex
inverse lexicographic order
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Default value: lex
This global switch controls which monomial order is used in polynomial and Groebner Bases calculations. If not set, lex will be used.
Categories: Package grobner
Default value: expression_ring
This switch indicates the coefficient ring of the polynomials that
will be used in grobner calculations. If not set, maxima's general
expression ring will be used. This variable may be set to
ring_of_integers if desired.
Categories: Package grobner
Default value: false
Name of the default order for eliminated variables in
elimination-based functions. If not set, lex will be used.
Categories: Package grobner
Default value: false
Name of the default order for kept variables in elimination-based functions. If not set, lex will be used.
Categories: Package grobner
Default value: false
Name of the default elimination order used in elimination
calculations. If set, it overrides the settings in variables
poly_primary_elimination_order and poly_secondary_elimination_order.
The user must ensure that this is a true elimination order valid
for the number of eliminated variables.
Categories: Package grobner
Default value: false
If set to true, all functions in this package will return each
polynomial as a list of terms in the current monomial order rather
than a maxima general expression.
Categories: Package grobner
Default value: false
If set to true, produce debugging and tracing output.
Categories: Package grobner
Default value: buchberger
Possible values:
buchberger
parallel_buchberger
gebauer_moeller
The name of the algorithm used to find the Groebner Bases.
Categories: Package grobner
Default value: false
If not false, use top reduction only whenever possible. Top
reduction means that division algorithm stops after the first
reduction.
Categories: Package grobner
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poly_add, poly_subtract, poly_multiply and poly_expt
are the arithmetical operations on polynomials.
These are performed using the internal representation, but the results are converted back to the
maxima general form.
Adds two polynomials poly1 and poly2.
(%i1) poly_add(z+x^2*y,x-z,[x,y,z]);
2
(%o1) x y + x
Categories: Package grobner
Subtracts a polynomial poly2 from poly1.
(%i1) poly_subtract(z+x^2*y,x-z,[x,y,z]);
2
(%o1) 2 z + x y - x
Categories: Package grobner
Returns the product of polynomials poly1 and poly2.
(%i2) poly_multiply(z+x^2*y,x-z,[x,y,z])-(z+x^2*y)*(x-z),expand; (%o1) 0
Categories: Package grobner
Returns the syzygy polynomial (S-polynomial) of two polynomials poly1 and poly2.
Categories: Package grobner
Returns the polynomial poly divided by the GCD of its coefficients.
(%i1) poly_primitive_part(35*y+21*x,[x,y]); (%o1) 5 y + 3 x
Categories: Package grobner
Returns the polynomial poly divided by the leading coefficient. It assumes that the division is possible, which may not always be the case in rings which are not fields.
Categories: Package grobner
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This function parses polynomials to internal form and back. It
is equivalent to expand(poly) if poly parses correctly to
a polynomial. If the representation is not compatible with a
polynomial in variables varlist, the result is an error.
It can be used to test whether an expression correctly parses to the
internal representation. The following examples illustrate that
indexed and transcendental function variables are allowed.
(%i1) poly_expand((x-y)*(y+x),[x,y]);
2 2
(%o1) x - y
(%i2) poly_expand((y+x)^2,[x,y]);
2 2
(%o2) y + 2 x y + x
(%i3) poly_expand((y+x)^5,[x,y]);
5 4 2 3 3 2 4 5
(%o3) y + 5 x y + 10 x y + 10 x y + 5 x y + x
(%i4) poly_expand(-1-x*exp(y)+x^2/sqrt(y),[x]);
2
y x
(%o4) - x %e + ------- - 1
sqrt(y)
(%i5) poly_expand(-1-sin(x)^2+sin(x),[sin(x)]);
2
(%o5) - sin (x) + sin(x) - 1
Categories: Package grobner
exponentitates poly by a positive integer number. If number is not a positive integer number an error will be raised.
(%i1) poly_expt(x-y,3,[x,y])-(x-y)^3,expand; (%o1) 0
Categories: Package grobner
poly_content extracts the GCD of its coefficients
(%i1) poly_content(35*y+21*x,[x,y]); (%o1) 7
Categories: Package grobner
Pseudo-divide a polynomial poly by the list of n polynomials polylist. Return multiple values. The first value is a list of quotients a. The second value is the remainder r. The third argument is a scalar coefficient c, such that c*poly can be divided by polylist within the ring of coefficients, which is not necessarily a field. Finally, the fourth value is an integer count of the number of reductions performed. The resulting objects satisfy the equation:
c*poly=sum(a[i]*polylist[i],i=1...n)+r.
Categories: Package grobner
Divide a polynomial poly1 by another polynomial poly2. Assumes that exact division with no remainder is possible. Returns the quotient.
Categories: Package grobner
poly_normal_form finds the normal form of a polynomial poly with respect
to a set of polynomials polylist.
Categories: Package grobner
Returns true if polylist is a Groebner basis with respect to the current term
order, by using the Buchberger
criterion: for every two polynomials h1 and h2 in polylist the
S-polynomial S(h1,h2) reduces to 0 modulo polylist.
Categories: Package grobner
poly_buchberger performs the Buchberger algorithm on a list of
polynomials and returns the resulting Groebner basis.
Categories: Package grobner
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The k-th elimination Ideal I_k of an Ideal I over K[ x[1],...,x[n] ] is the ideal intersect(I, K[ x[k+1],...,x[n] ]).
The colon ideal I:J is the ideal {h|for all w in J: w*h in I}.
The ideal I:p^inf is the ideal {h| there is a n in N: p^n*h in I}.
The ideal I:J^inf is the ideal {h| there is a n in N and a p in J: p^n*h in I}.
The radical ideal sqrt(I) is the ideal
{h| there is a n in N : h^n in I }.
poly_reduction reduces a list of polynomials polylist, so that
each polynomial is fully reduced with respect to the other polynomials.
Categories: Package grobner
Returns a sublist of the polynomial list polylist spanning the same monomial ideal as polylist but minimal, i.e. no leading monomial of a polynomial in the sublist divides the leading monomial of another polynomial.
Categories: Package grobner
poly_normalize_list applies poly_normalize to each polynomial in the list.
That means it divides every polynomial in a list polylist by its leading coefficient.
Categories: Package grobner
Returns a Groebner basis of the ideal span by the polynomials polylist. Affected by the global flags.
Categories: Package grobner
Returns a reduced Groebner basis of the ideal span by the polynomials polylist. Affected by the global flags.
Categories: Package grobner
poly_depends tests whether a polynomial depends on a variable var.
Categories: Package grobner · Predicate functions
poly_elimination_ideal returns the grobner basis of the number-th elimination ideal of an
ideal specified as a list of generating polynomials (not necessarily Groebner basis).
Categories: Package grobner
Returns the reduced Groebner basis of the colon ideal
I(polylist1):I(polylist2)
where polylist1 and polylist2 are two lists of polynomials.
Categories: Package grobner
poly_ideal_intersection returns the intersection of two ideals.
Categories: Package grobner
Returns the lowest common multiple of poly1 and poly2.
Categories: Package grobner
Returns the greatest common divisor of poly1 and poly2.
See also ezgcd, gcd, gcdex, and
gcdivide.
Example:
(%i1) p1:6*x^3+19*x^2+19*x+6;
3 2
(%o1) 6 x + 19 x + 19 x + 6
(%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
5 4 3 2
(%o2) 6 x + 13 x + 12 x + 13 x + 6 x
(%i3) poly_gcd(p1, p2, [x]);
2
(%o3) 6 x + 13 x + 6
Categories: Package grobner
poly_grobner_equal tests whether two Groebner Bases generate the same ideal.
Returns true if two lists of polynomials polylist1 and polylist2, assumed to be Groebner Bases,
generate the same ideal, and false otherwise.
This is equivalent to checking that every polynomial of the first basis reduces to 0
modulo the second basis and vice versa. Note that in the example below the
first list is not a Groebner basis, and thus the result is false.
(%i1) poly_grobner_equal([y+x,x-y],[x,y],[x,y]); (%o1) false
Categories: Package grobner
poly_grobner_subsetp tests whether an ideal generated by polylist1
is contained in the ideal generated by polylist2. For this test to always succeed,
polylist2 must be a Groebner basis.
Categories: Package grobner · Predicate functions
Returns true if a polynomial poly belongs to the ideal generated by the
polynomial list polylist, which is assumed to be a Groebner basis. Returns false otherwise.
poly_grobner_member tests whether a polynomial belongs to an ideal generated by a list of polynomials,
which is assumed to be a Groebner basis. Equivalent to normal_form being 0.
Categories: Package grobner
Returns the reduced Groebner basis of the saturation of the ideal
I(polylist):poly^inf
Geometrically, over an algebraically closed field, this is the set of polynomials in the ideal generated by polylist which do not identically vanish on the variety of poly.
Categories: Package grobner
Returns the reduced Groebner basis of the saturation of the ideal
I(polylist1):I(polylist2)^inf
Geometrically, over an algebraically closed field, this is the set of polynomials in the ideal generated by polylist1 which do not identically vanish on the variety of polylist2.
Categories: Package grobner
polylist2 ist a list of n polynomials [poly1,...,polyn].
Returns the reduced Groebner basis of the ideal
I(polylist):poly1^inf:...:polyn^inf
obtained by a sequence of successive saturations in the polynomials of the polynomial list polylist2 of the ideal generated by the polynomial list polylist1.
Categories: Package grobner
polylistlist is a list of n list of polynomials [polylist1,...,polylistn].
Returns the reduced Groebner basis of the saturation of the ideal
I(polylist):I(polylist_1)^inf:...:I(polylist_n)^inf
Categories: Package grobner
poly_saturation_extension implements the famous Rabinowitz trick.
Categories: Package grobner
Categories: Package grobner
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| 63.1 Functions and Variables for impdiff |
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This subroutine computes implicit derivatives of multivariable functions. f is an array function, the indexes are the derivative degree in the indvarlist order; indvarlist is the independent variable list; orderlist is the order desired; and depvar is the dependent variable.
To use this function write first load("impdiff").
Categories: Differential calculus · Share packages · Package impdiff
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| 64.1 Introduction to interpol | ||
| 64.2 Functions and Variables for interpol |
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Package interpol defines the Lagrangian, the linear and the cubic
splines methods for polynomial interpolation.
For comments, bugs or suggestions, please contact me at 'mario AT edu DOT xunta DOT es'.
Categories: Numerical methods · Share packages · Package interpol
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Computes the polynomial interpolation by the Lagrangian method. Argument points must be either:
p:matrix([2,4],[5,6],[9,3]),
p: [[2,4],[5,6],[9,3]],
p: [4,6,3], in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
With the option argument it is possible to select the name for the independent variable, which is 'x by default; to define another one, write something like varname='z.
Note that when working with high degree polynomials, floating point evaluations are unstable.
See also linearinterpol, cspline, and ratinterpol.
Examples:
(%i1) load(interpol)$
(%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
(%i3) lagrange(p);
(x - 7) (x - 6) (x - 3) (x - 1)
(%o3) -------------------------------
35
(x - 8) (x - 6) (x - 3) (x - 1)
- -------------------------------
12
7 (x - 8) (x - 7) (x - 3) (x - 1)
+ ---------------------------------
30
(x - 8) (x - 7) (x - 6) (x - 1)
- -------------------------------
60
(x - 8) (x - 7) (x - 6) (x - 3)
+ -------------------------------
84
(%i4) f(x):=''%;
(x - 7) (x - 6) (x - 3) (x - 1)
(%o4) f(x) := -------------------------------
35
(x - 8) (x - 6) (x - 3) (x - 1)
- -------------------------------
12
7 (x - 8) (x - 7) (x - 3) (x - 1)
+ ---------------------------------
30
(x - 8) (x - 7) (x - 6) (x - 1)
- -------------------------------
60
(x - 8) (x - 7) (x - 6) (x - 3)
+ -------------------------------
84
(%i5) /* Evaluate the polynomial at some points */
expand(map(f,[2.3,5/7,%pi]));
4 3 2
919062 73 %pi 701 %pi 8957 %pi
(%o5) [- 1.567535, ------, ------- - -------- + ---------
84035 420 210 420
5288 %pi 186
- -------- + ---]
105 5
(%i6) %,numer;
(%o6) [- 1.567535, 10.9366573451538, 2.89319655125692]
(%i7) load(draw)$ /* load draw package */
(%i8) /* Plot the polynomial together with points */
draw2d(
color = red,
key = "Lagrange polynomial",
explicit(f(x),x,0,10),
point_size = 3,
color = blue,
key = "Sample points",
points(p))$
(%i9) /* Change variable name */
lagrange(p, varname=w);
(w - 7) (w - 6) (w - 3) (w - 1)
(%o9) -------------------------------
35
(w - 8) (w - 6) (w - 3) (w - 1)
- -------------------------------
12
7 (w - 8) (w - 7) (w - 3) (w - 1)
+ ---------------------------------
30
(w - 8) (w - 7) (w - 6) (w - 1)
- -------------------------------
60
(w - 8) (w - 7) (w - 6) (w - 3)
+ -------------------------------
84
Categories: Package interpol
Returns true if number x belongs to the interval [a, b), and false otherwise.
Categories: Package interpol
Computes the polynomial interpolation by the linear method. Argument points must be either:
p:matrix([2,4],[5,6],[9,3]),
p: [[2,4],[5,6],[9,3]],
p: [4,6,3], in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
With the option argument it is possible to select the name for the independent variable, which is 'x by default; to define another one, write something like varname='z.
See also lagrange, cspline, and ratinterpol.
Examples:
(%i1) load(interpol)$
(%i2) p: matrix([7,2],[8,3],[1,5],[3,2],[6,7])$
(%i3) linearinterpol(p);
13 3 x
(%o3) (-- - ---) charfun2(x, minf, 3)
2 2
+ (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
5 x
+ (--- - 3) charfun2(x, 3, 6)
3
(%i4) f(x):=''%;
13 3 x
(%o4) f(x) := (-- - ---) charfun2(x, minf, 3)
2 2
+ (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
5 x
+ (--- - 3) charfun2(x, 3, 6)
3
(%i5) /* Evaluate the polynomial at some points */
map(f,[7.3,25/7,%pi]);
62 5 %pi
(%o5) [2.3, --, ----- - 3]
21 3
(%i6) %,numer;
(%o6) [2.3, 2.952380952380953, 2.235987755982989]
(%i7) load(draw)$ /* load draw package */
(%i8) /* Plot the polynomial together with points */
draw2d(
color = red,
key = "Linear interpolator",
explicit(f(x),x,-5,20),
point_size = 3,
color = blue,
key = "Sample points",
points(args(p)))$
(%i9) /* Change variable name */
linearinterpol(p, varname='s);
13 3 s
(%o9) (-- - ---) charfun2(s, minf, 3)
2 2
+ (s - 5) charfun2(s, 7, inf) + (37 - 5 s) charfun2(s, 6, 7)
5 s
+ (--- - 3) charfun2(s, 3, 6)
3
Categories: Package interpol
Computes the polynomial interpolation by the cubic splines method. Argument points must be either:
p:matrix([2,4],[5,6],[9,3]),
p: [[2,4],[5,6],[9,3]],
p: [4,6,3], in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
There are three options to fit specific needs:
'd1, default 'unknown, is the first derivative at x_1; if it is 'unknown, the second derivative at x_1
is made equal to 0 (natural cubic spline); if it is equal to a number,
the second derivative is calculated based on this number.
'dn, default 'unknown, is the first derivative at x_n; if it is 'unknown, the second derivative at x_n
is made equal to 0 (natural cubic spline); if it is equal to a number,
the second derivative is calculated based on this number.
'varname, default 'x, is the name of the independent variable.
See also lagrange, linearinterpol, and ratinterpol.
Examples:
(%i1) load(interpol)$
(%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
(%i3) /* Unknown first derivatives at the extremes
is equivalent to natural cubic splines */
cspline(p);
3 2
1159 x 1159 x 6091 x 8283
(%o3) (------- - ------- - ------ + ----) charfun2(x, minf, 3)
3288 1096 3288 1096
3 2
2587 x 5174 x 494117 x 108928
+ (- ------- + ------- - -------- + ------) charfun2(x, 7, inf)
1644 137 1644 137
3 2
4715 x 15209 x 579277 x 199575
+ (------- - -------- + -------- - ------) charfun2(x, 6, 7)
1644 274 1644 274
3 2
3287 x 2223 x 48275 x 9609
+ (- ------- + ------- - ------- + ----) charfun2(x, 3, 6)
4932 274 1644 274
(%i4) f(x):=''%$
(%i5) /* Some evaluations */
map(f,[2.3,5/7,%pi]), numer;
(%o5) [1.991460766423356, 5.823200187269903, 2.227405312429507]
(%i6) load(draw)$ /* load draw package */
(%i7) /* Plotting interpolating function */
draw2d(
color = red,
key = "Cubic splines",
explicit(f(x),x,0,10),
point_size = 3,
color = blue,
key = "Sample points",
points(p))$
(%i8) /* New call, but giving values at the derivatives */
cspline(p,d1=0,dn=0);
3 2
1949 x 11437 x 17027 x 1247
(%o8) (------- - -------- + ------- + ----) charfun2(x, minf, 3)
2256 2256 2256 752
3 2
1547 x 35581 x 68068 x 173546
+ (- ------- + -------- - ------- + ------) charfun2(x, 7, inf)
564 564 141 141
3 2
607 x 35147 x 55706 x 38420
+ (------ - -------- + ------- - -----) charfun2(x, 6, 7)
188 564 141 47
3 2
3895 x 1807 x 5146 x 2148
+ (- ------- + ------- - ------ + ----) charfun2(x, 3, 6)
5076 188 141 47
(%i8) /* Defining new interpolating function */
g(x):=''%$
(%i9) /* Plotting both functions together */
draw2d(
color = black,
key = "Cubic splines (default)",
explicit(f(x),x,0,10),
color = red,
key = "Cubic splines (d1=0,dn=0)",
explicit(g(x),x,0,10),
point_size = 3,
color = blue,
key = "Sample points",
points(p))$
Categories: Package interpol
Generates a rational interpolator for data given by points and the degree of the numerator being equal to numdeg; the degree of the denominator is calculated automatically. Argument points must be either:
p:matrix([2,4],[5,6],[9,3]),
p: [[2,4],[5,6],[9,3]],
p: [4,6,3], in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
There is one option to fit specific needs:
'varname, default 'x, is the name of the independent variable.
See also lagrange, linearinterpol, cspline, minpack_lsquares, and lbfgs
Examples:
(%i1) load(interpol)$
(%i2) load(draw)$
(%i3) p:[[7.2,2.5],[8.5,2.1],[1.6,5.1],[3.4,2.4],[6.7,7.9]]$
(%i4) for k:0 thru length(p)-1 do
draw2d(
explicit(ratinterpol(p,k),x,0,9),
point_size = 3,
points(p),
title = concat("Degree of numerator = ",k),
yrange=[0,10])$
Categories: Package interpol
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| 65.1 Introduction to lapack | ||
| 65.2 Functions and Variables for lapack |
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lapack is a Common Lisp translation (via the program f2cl) of the Fortran library LAPACK,
as obtained from the SLATEC project.
Categories: Numerical methods · Share packages · Package lapack
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Computes the eigenvalues and, optionally, the eigenvectors of a matrix A. All elements of A must be integer or floating point numbers. A must be square (same number of rows and columns). A might or might not be symmetric.
dgeev(A) computes only the eigenvalues of A.
dgeev(A, right_p, left_p) computes the eigenvalues of A
and the right eigenvectors when right_p = true
and the left eigenvectors when left_p = true.
A list of three items is returned.
The first item is a list of the eigenvalues.
The second item is false or the matrix of right eigenvectors.
The third item is false or the matrix of left eigenvectors.
The right eigenvector v(j) (the j-th column of the right eigenvector matrix) satisfies
A . v(j) = lambda(j) . v(j)
where lambda(j) is the corresponding eigenvalue. The left eigenvector u(j) (the j-th column of the left eigenvector matrix) satisfies
u(j)**H . A = lambda(j) . u(j)**H
where u(j)**H denotes the conjugate transpose of u(j).
The Maxima function ctranspose computes the conjugate transpose.
The computed eigenvectors are normalized to have Euclidean norm equal to 1, and largest component has imaginary part equal to zero.
Example:
(%i1) load (lapack)$
(%i2) fpprintprec : 6;
(%o2) 6
(%i3) M : matrix ([9.5, 1.75], [3.25, 10.45]);
[ 9.5 1.75 ]
(%o3) [ ]
[ 3.25 10.45 ]
(%i4) dgeev (M);
(%o4) [[7.54331, 12.4067], false, false]
(%i5) [L, v, u] : dgeev (M, true, true);
[ - .666642 - .515792 ]
(%o5) [[7.54331, 12.4067], [ ],
[ .745378 - .856714 ]
[ - .856714 - .745378 ]
[ ]]
[ .515792 - .666642 ]
(%i6) D : apply (diag_matrix, L);
[ 7.54331 0 ]
(%o6) [ ]
[ 0 12.4067 ]
(%i7) M . v - v . D;
[ 0.0 - 8.88178E-16 ]
(%o7) [ ]
[ - 8.88178E-16 0.0 ]
(%i8) transpose (u) . M - D . transpose (u);
[ 0.0 - 4.44089E-16 ]
(%o8) [ ]
[ 0.0 0.0 ]
Categories: Package lapack
Computes the QR decomposition of the matrix A. All elements of A must be integer or floating point numbers. A may or may not have the same number of rows and columns.
A list of two items is returned.
The first item is the matrix Q, which is a square, orthonormal matrix
which has the same number of rows as A.
The second item is the matrix R, which is the same size as A,
and which has all elements equal to zero below the diagonal.
The product Q . R, where "." is the noncommutative multiplication operator,
is equal to A (ignoring floating point round-off errors).
(%i1) load (lapack) $
(%i2) fpprintprec : 6 $
(%i3) M : matrix ([1, -3.2, 8], [-11, 2.7, 5.9]) $
(%i4) [q, r] : dgeqrf (M);
[ - .0905357 .995893 ]
(%o4) [[ ],
[ .995893 .0905357 ]
[ - 11.0454 2.97863 5.15148 ]
[ ]]
[ 0 - 2.94241 8.50131 ]
(%i5) q . r - M;
[ - 7.77156E-16 1.77636E-15 - 8.88178E-16 ]
(%o5) [ ]
[ 0.0 - 1.33227E-15 8.88178E-16 ]
(%i6) mat_norm (%, 1);
(%o6) 3.10862E-15
Categories: Package lapack
Computes the solution x of the linear equation A x = b, where A is a square matrix, and b is a matrix of the same number of rows as A and any number of columns. The return value x is the same size as b.
The elements of A and b must evaluate to real floating point numbers via float;
thus elements may be any numeric type, symbolic numerical constants, or expressions which evaluate to floats.
The elements of x are always floating point numbers.
All arithmetic is carried out as floating point operations.
dgesv computes the solution via the LU decomposition of A.
Examples:
dgesv computes the solution of the linear equation A x = b.
(%i1) A : matrix ([1, -2.5], [0.375, 5]);
[ 1 - 2.5 ]
(%o1) [ ]
[ 0.375 5 ]
(%i2) b : matrix ([1.75], [-0.625]);
[ 1.75 ]
(%o2) [ ]
[ - 0.625 ]
(%i3) x : dgesv (A, b);
[ 1.210526315789474 ]
(%o3) [ ]
[ - 0.215789473684211 ]
(%i4) dlange (inf_norm, b - A.x);
(%o4) 0.0
b is a matrix with the same number of rows as A and any number of columns. x is the same size as b.
(%i1) A : matrix ([1, -0.15], [1.82, 2]);
[ 1 - 0.15 ]
(%o1) [ ]
[ 1.82 2 ]
(%i2) b : matrix ([3.7, 1, 8], [-2.3, 5, -3.9]);
[ 3.7 1 8 ]
(%o2) [ ]
[ - 2.3 5 - 3.9 ]
(%i3) x : dgesv (A, b);
[ 3.103827540695117 1.20985481742191 6.781786185657722 ]
(%o3) [ ]
[ -3.974483062032557 1.399032116146062 -8.121425428948527 ]
(%i4) dlange (inf_norm, b - A . x);
(%o4) 1.1102230246251565E-15
The elements of A and b must evaluate to real floating point numbers.
(%i1) A : matrix ([5, -%pi], [1b0, 11/17]);
[ 5 - %pi ]
[ ]
(%o1) [ 11 ]
[ 1.0b0 -- ]
[ 17 ]
(%i2) b : matrix ([%e], [sin(1)]);
[ %e ]
(%o2) [ ]
[ sin(1) ]
(%i3) x : dgesv (A, b);
[ 0.690375643155986 ]
(%o3) [ ]
[ 0.233510982552952 ]
(%i4) dlange (inf_norm, b - A . x);
(%o4) 2.220446049250313E-16
Categories: Package lapack · Linear equations
Computes the singular value decomposition (SVD) of a matrix A, comprising the singular values and, optionally, the left and right singular vectors. All elements of A must be integer or floating point numbers. A might or might not be square (same number of rows and columns).
Let m be the number of rows, and n the number of columns of A. The singular value decomposition of A comprises three matrices, U, Sigma, and V^T, such that
A = U . Sigma . V^T
where U is an m-by-m unitary matrix, Sigma is an m-by-n diagonal matrix, and V^T is an n-by-n unitary matrix.
Let sigma[i] be a diagonal element of Sigma,
that is, Sigma[i, i] = sigma[i].
The elements sigma[i] are the so-called singular values of A;
these are real and nonnegative, and returned in descending order.
The first min(m, n) columns of U and V are
the left and right singular vectors of A.
Note that dgesvd returns the transpose of V, not V itself.
dgesvd(A) computes only the singular values of A.
dgesvd(A, left_p, right_p) computes the singular values of A
and the left singular vectors when left_p = true
and the right singular vectors when right_p = true.
A list of three items is returned.
The first item is a list of the singular values.
The second item is false or the matrix of left singular vectors.
The third item is false or the matrix of right singular vectors.
Example:
(%i1) load (lapack)$
(%i2) fpprintprec : 6;
(%o2) 6
(%i3) M: matrix([1, 2, 3], [3.5, 0.5, 8], [-1, 2, -3], [4, 9, 7]);
[ 1 2 3 ]
[ ]
[ 3.5 0.5 8 ]
(%o3) [ ]
[ - 1 2 - 3 ]
[ ]
[ 4 9 7 ]
(%i4) dgesvd (M);
(%o4) [[14.4744, 6.38637, .452547], false, false]
(%i5) [sigma, U, VT] : dgesvd (M, true, true);
(%o5) [[14.4744, 6.38637, .452547],
[ - .256731 .00816168 .959029 - .119523 ]
[ ]
[ - .526456 .672116 - .206236 - .478091 ]
[ ],
[ .107997 - .532278 - .0708315 - 0.83666 ]
[ ]
[ - .803287 - .514659 - .180867 .239046 ]
[ - .374486 - .538209 - .755044 ]
[ ]
[ .130623 - .836799 0.5317 ]]
[ ]
[ - .917986 .100488 .383672 ]
(%i6) m : length (U);
(%o6) 4
(%i7) n : length (VT);
(%o7) 3
(%i8) Sigma:
genmatrix(lambda ([i, j], if i=j then sigma[i] else 0),
m, n);
[ 14.4744 0 0 ]
[ ]
[ 0 6.38637 0 ]
(%o8) [ ]
[ 0 0 .452547 ]
[ ]
[ 0 0 0 ]
(%i9) U . Sigma . VT - M;
[ 1.11022E-15 0.0 1.77636E-15 ]
[ ]
[ 1.33227E-15 1.66533E-15 0.0 ]
(%o9) [ ]
[ - 4.44089E-16 - 8.88178E-16 4.44089E-16 ]
[ ]
[ 8.88178E-16 1.77636E-15 8.88178E-16 ]
(%i10) transpose (U) . U;
[ 1.0 5.55112E-17 2.498E-16 2.77556E-17 ]
[ ]
[ 5.55112E-17 1.0 5.55112E-17 4.16334E-17 ]
(%o10) [ ]
[ 2.498E-16 5.55112E-17 1.0 - 2.08167E-16 ]
[ ]
[ 2.77556E-17 4.16334E-17 - 2.08167E-16 1.0 ]
(%i11) VT . transpose (VT);
[ 1.0 0.0 - 5.55112E-17 ]
[ ]
(%o11) [ 0.0 1.0 5.55112E-17 ]
[ ]
[ - 5.55112E-17 5.55112E-17 1.0 ]
Categories: Package lapack
Computes a norm or norm-like function of the matrix A.
maxCompute max(abs(A(i, j))) where i and j range over the rows and columns, respectively, of A. Note that this function is not a proper matrix norm.
one_normCompute the L[1] norm of A, that is, the maximum of the sum of the absolute value of elements in each column.
inf_normCompute the L[inf] norm of A, that is, the maximum of the sum of the absolute value of elements in each row.
frobeniusCompute the Frobenius norm of A, that is, the square root of the sum of squares of the matrix elements.
Categories: Package lapack
Compute the product of two matrices and optionally add the product to a third matrix.
In the simplest form, dgemm(A, B) computes the
product of the two real matrices, A and B.
In the second form, dgemm computes the alpha *
A * B + beta * C where A, B,
C are real matrices of the appropriate sizes and alpha and
beta are real numbers. Optionally, A and/or B can
be transposed before computing the product. The extra parameters are
specifed by optional keyword arguments: The keyword arguments are
optional and may be specified in any order. They all take the form
key=val. The keyword arguments are:
CThe matrix C that should be added. The default is false,
which means no matrix is added.
alphaThe product of A and B is multiplied by this value. The default is 1.
betaIf a matrix C is given, this value multiplies C before it is added. The default value is 0, which implies that C is not added, even if C is given. Hence, be sure to specify a non-zero value for beta.
transpose_aIf true, the transpose of A is used instead of A
for the product. The default is false.
transpose_bIf true, the transpose of B is used instead of B
for the product. The default is false.
(%i1) load (lapack)$
(%i2) A : matrix([1,2,3],[4,5,6],[7,8,9]);
[ 1 2 3 ]
[ ]
(%o2) [ 4 5 6 ]
[ ]
[ 7 8 9 ]
(%i3) B : matrix([-1,-2,-3],[-4,-5,-6],[-7,-8,-9]);
[ - 1 - 2 - 3 ]
[ ]
(%o3) [ - 4 - 5 - 6 ]
[ ]
[ - 7 - 8 - 9 ]
(%i4) C : matrix([3,2,1],[6,5,4],[9,8,7]);
[ 3 2 1 ]
[ ]
(%o4) [ 6 5 4 ]
[ ]
[ 9 8 7 ]
(%i5) dgemm(A,B);
[ - 30.0 - 36.0 - 42.0 ]
[ ]
(%o5) [ - 66.0 - 81.0 - 96.0 ]
[ ]
[ - 102.0 - 126.0 - 150.0 ]
(%i6) A . B;
[ - 30 - 36 - 42 ]
[ ]
(%o6) [ - 66 - 81 - 96 ]
[ ]
[ - 102 - 126 - 150 ]
(%i7) dgemm(A,B,transpose_a=true);
[ - 66.0 - 78.0 - 90.0 ]
[ ]
(%o7) [ - 78.0 - 93.0 - 108.0 ]
[ ]
[ - 90.0 - 108.0 - 126.0 ]
(%i8) transpose(A) . B;
[ - 66 - 78 - 90 ]
[ ]
(%o8) [ - 78 - 93 - 108 ]
[ ]
[ - 90 - 108 - 126 ]
(%i9) dgemm(A,B,c=C,beta=1);
[ - 27.0 - 34.0 - 41.0 ]
[ ]
(%o9) [ - 60.0 - 76.0 - 92.0 ]
[ ]
[ - 93.0 - 118.0 - 143.0 ]
(%i10) A . B + C;
[ - 27 - 34 - 41 ]
[ ]
(%o10) [ - 60 - 76 - 92 ]
[ ]
[ - 93 - 118 - 143 ]
(%i11) dgemm(A,B,c=C,beta=1, alpha=-1);
[ 33.0 38.0 43.0 ]
[ ]
(%o11) [ 72.0 86.0 100.0 ]
[ ]
[ 111.0 134.0 157.0 ]
(%i12) -A . B + C;
[ 33 38 43 ]
[ ]
(%o12) [ 72 86 100 ]
[ ]
[ 111 134 157 ]
Categories: Package lapack
Like dgeev, but the matrix A is complex.
Categories: Package lapack
Like zheev, but the matrix A is assumed to be a square
complex Hermitian matrix. If eigvec_p is true, then the
eigenvectors of the matrix are also computed.
No check is made that the matrix A is, in fact, Hermitian.
A list of two items is returned, as in dgeev: a list of
eigenvalues, and false or the matrix of the eigenvectors.
Categories: Package lapack
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 66.1 Introduction to lbfgs | ||
| 66.2 Functions and Variables for lbfgs |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
lbfgs is an implementation of the L-BFGS algorithm [1]
to solve unconstrained minimization problems via a limited-memory quasi-Newton (BFGS) algorithm.
It is called a limited-memory method because a low-rank approximation of the
Hessian matrix inverse is stored instead of the entire Hessian inverse.
The program was originally written in Fortran [2] by Jorge Nocedal,
incorporating some functions originally written by Jorge J. Moré and David J. Thuente,
and translated into Lisp automatically via the program f2cl.
The Maxima package lbfgs comprises the translated code plus
an interface function which manages some details.
References:
[1] D. Liu and J. Nocedal. "On the limited memory BFGS method for large scale optimization". Mathematical Programming B 45:503-528 (1989)
[2] http://netlib.org/opt/lbfgs_um.shar
Categories: Numerical methods · Optimization · Share packages · Package lbfgs
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Finds an approximate solution of the unconstrained minimization of the figure of merit FOM over the list of variables X, starting from initial estimates X0, such that norm(grad(FOM)) < epsilon*max(1, norm(X)).
grad, if present, is the gradient of FOM with respect to the variables X. grad may be a list or a function that returns a list, with one element for each element of X. If not present, the gradient is computed automatically by symbolic differentiation. If FOM is a function, the gradient grad must be supplied by the user.
The algorithm applied is a limited-memory quasi-Newton (BFGS) algorithm [1]. It is called a limited-memory method because a low-rank approximation of the Hessian matrix inverse is stored instead of the entire Hessian inverse. Each iteration of the algorithm is a line search, that is, a search along a ray in the variables X, with the search direction computed from the approximate Hessian inverse. The FOM is always decreased by a successful line search. Usually (but not always) the norm of the gradient of FOM also decreases.
iprint controls progress messages printed by lbfgs.
iprint[1]iprint[1] controls the frequency of progress messages.
iprint[1] < 0No progress messages.
iprint[1] = 0Messages at the first and last iterations.
iprint[1] > 0Print a message every iprint[1] iterations.
iprint[2]iprint[2] controls the verbosity of progress messages.
iprint[2] = 0Print out iteration count, number of evaluations of FOM, value of FOM, norm of the gradient of FOM, and step length.
iprint[2] = 1Same as iprint[2] = 0, plus X0 and the gradient of FOM evaluated at X0.
iprint[2] = 2Same as iprint[2] = 1, plus values of X at each iteration.
iprint[2] = 3Same as iprint[2] = 2, plus the gradient of FOM at each iteration.
The columns printed by lbfgs are the following.
INumber of iterations. It is incremented for each line search.
NFNNumber of evaluations of the figure of merit.
FUNCValue of the figure of merit at the end of the most recent line search.
GNORMNorm of the gradient of the figure of merit at the end of the most recent line search.
STEPLENGTHAn internal parameter of the search algorithm.
Additional information concerning details of the algorithm are found in the comments of the original Fortran code [2].
See also lbfgs_nfeval_max and lbfgs_ncorrections.
References:
[1] D. Liu and J. Nocedal. "On the limited memory BFGS method for large scale optimization". Mathematical Programming B 45:503-528 (1989)
[2] http://netlib.org/opt/lbfgs_um.shar
Examples:
The same FOM as computed by FGCOMPUTE in the program sdrive.f in the LBFGS package from Netlib. Note that the variables in question are subscripted variables. The FOM has an exact minimum equal to zero at u[k] = 1 for k = 1, ..., 8.
(%i1) load (lbfgs)$
(%i2) t1[j] := 1 - u[j];
(%o2) t1 := 1 - u
j j
(%i3) t2[j] := 10*(u[j + 1] - u[j]^2);
2
(%o3) t2 := 10 (u - u )
j j + 1 j
(%i4) n : 8;
(%o4) 8
(%i5) FOM : sum (t1[2*j - 1]^2 + t2[2*j - 1]^2, j, 1, n/2);
2 2 2 2 2 2
(%o5) 100 (u - u ) + (1 - u ) + 100 (u - u ) + (1 - u )
8 7 7 6 5 5
2 2 2 2 2 2
+ 100 (u - u ) + (1 - u ) + 100 (u - u ) + (1 - u )
4 3 3 2 1 1
(%i6) lbfgs (FOM, '[u[1],u[2],u[3],u[4],u[5],u[6],u[7],u[8]],
[-1.2, 1, -1.2, 1, -1.2, 1, -1.2, 1], 1e-3, [1, 0]);
*************************************************
N= 8 NUMBER OF CORRECTIONS=25
INITIAL VALUES
F= 9.680000000000000D+01 GNORM= 4.657353755084533D+02
*************************************************
I NFN FUNC GNORM STEPLENGTH 1 3 1.651479526340304D+01 4.324359291335977D+00 7.926153934390631D-04 2 4 1.650209316638371D+01 3.575788161060007D+00 1.000000000000000D+00 3 5 1.645461701312851D+01 6.230869903601577D+00 1.000000000000000D+00 4 6 1.636867301275588D+01 1.177589920974980D+01 1.000000000000000D+00 5 7 1.612153014409201D+01 2.292797147151288D+01 1.000000000000000D+00 6 8 1.569118407390628D+01 3.687447158775571D+01 1.000000000000000D+00 7 9 1.510361958398942D+01 4.501931728123679D+01 1.000000000000000D+00 8 10 1.391077875774293D+01 4.526061463810630D+01 1.000000000000000D+00 9 11 1.165625686278198D+01 2.748348965356907D+01 1.000000000000000D+00 10 12 9.859422687859144D+00 2.111494974231706D+01 1.000000000000000D+00 11 13 7.815442521732282D+00 6.110762325764183D+00 1.000000000000000D+00 12 15 7.346380905773044D+00 2.165281166715009D+01 1.285316401779678D-01 13 16 6.330460634066464D+00 1.401220851761508D+01 1.000000000000000D+00 14 17 5.238763939854303D+00 1.702473787619218D+01 1.000000000000000D+00 15 18 3.754016790406625D+00 7.981845727632704D+00 1.000000000000000D+00 16 20 3.001238402313225D+00 3.925482944745832D+00 2.333129631316462D-01 17 22 2.794390709722064D+00 8.243329982586480D+00 2.503577283802312D-01 18 23 2.563783562920545D+00 1.035413426522664D+01 1.000000000000000D+00 19 24 2.019429976373283D+00 1.065187312340952D+01 1.000000000000000D+00 20 25 1.428003167668592D+00 2.475962450735100D+00 1.000000000000000D+00 21 27 1.197874264859232D+00 8.441707983339661D+00 4.303451060697367D-01 22 28 9.023848942003913D-01 1.113189216665625D+01 1.000000000000000D+00 23 29 5.508226405855795D-01 2.380830599637816D+00 1.000000000000000D+00 24 31 3.902893258879521D-01 5.625595817143044D+00 4.834988416747262D-01 25 32 3.207542206881058D-01 1.149444645298493D+01 1.000000000000000D+00 26 33 1.874468266118200D-01 3.632482152347445D+00 1.000000000000000D+00 27 34 9.575763380282112D-02 4.816497449000391D+00 1.000000000000000D+00 28 35 4.085145106760390D-02 2.087009347116811D+00 1.000000000000000D+00 29 36 1.931106005512628D-02 3.886818624052740D+00 1.000000000000000D+00 30 37 6.894000636920714D-03 3.198505769992936D+00 1.000000000000000D+00 31 38 1.443296008850287D-03 1.590265460381961D+00 1.000000000000000D+00 32 39 1.571766574930155D-04 3.098257002223532D-01 1.000000000000000D+00 33 40 1.288011779655132D-05 1.207784334505595D-02 1.000000000000000D+00 34 41 1.806140190993455D-06 4.587890258846915D-02 1.000000000000000D+00 35 42 1.769004612050548D-07 1.790537363138099D-02 1.000000000000000D+00 36 43 3.312164244118216D-10 6.782068546986653D-04 1.000000000000000D+00
THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
IFLAG = 0
(%o6) [u = 1.000005339816132, u = 1.000009942840108,
1 2
u = 1.000005339816132, u = 1.000009942840108,
3 4
u = 1.000005339816132, u = 1.000009942840108,
5 6
u = 1.000005339816132, u = 1.000009942840108]
7 8
A regression problem.
The FOM is the mean square difference between the predicted value F(X[i])
and the observed value Y[i].
The function F is a bounded monotone function (a so-called "sigmoidal" function).
In this example, lbfgs computes approximate values for the parameters of F
and plot2d displays a comparison of F with the observed data.
(%i1) load (lbfgs)$
(%i2) FOM : '((1/length(X))*sum((F(X[i]) - Y[i])^2, i, 1,
length(X)));
2
sum((F(X ) - Y ) , i, 1, length(X))
i i
(%o2) -----------------------------------
length(X)
(%i3) X : [1, 2, 3, 4, 5];
(%o3) [1, 2, 3, 4, 5]
(%i4) Y : [0, 0.5, 1, 1.25, 1.5];
(%o4) [0, 0.5, 1, 1.25, 1.5]
(%i5) F(x) := A/(1 + exp(-B*(x - C)));
A
(%o5) F(x) := ----------------------
1 + exp((- B) (x - C))
(%i6) ''FOM;
A 2 A 2
(%o6) ((----------------- - 1.5) + (----------------- - 1.25)
- B (5 - C) - B (4 - C)
%e + 1 %e + 1
A 2 A 2
+ (----------------- - 1) + (----------------- - 0.5)
- B (3 - C) - B (2 - C)
%e + 1 %e + 1
2
A
+ --------------------)/5
- B (1 - C) 2
(%e + 1)
(%i7) estimates : lbfgs (FOM, '[A, B, C], [1, 1, 1], 1e-4, [1, 0]);
*************************************************
N= 3 NUMBER OF CORRECTIONS=25
INITIAL VALUES
F= 1.348738534246918D-01 GNORM= 2.000215531936760D-01
*************************************************
I NFN FUNC GNORM STEPLENGTH 1 3 1.177820636622582D-01 9.893138394953992D-02 8.554435968992371D-01 2 6 2.302653892214013D-02 1.180098521565904D-01 2.100000000000000D+01 3 8 1.496348495303004D-02 9.611201567691624D-02 5.257340567840710D-01 4 9 7.900460841091138D-03 1.325041647391314D-02 1.000000000000000D+00 5 10 7.314495451266914D-03 1.510670810312226D-02 1.000000000000000D+00 6 11 6.750147275936668D-03 1.914964958023037D-02 1.000000000000000D+00 7 12 5.850716021108202D-03 1.028089194579382D-02 1.000000000000000D+00 8 13 5.778664230657800D-03 3.676866074532179D-04 1.000000000000000D+00 9 14 5.777818823650780D-03 3.010740179797108D-04 1.000000000000000D+00
THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
IFLAG = 0
(%o7) [A = 1.461933911464101, B = 1.601593973254801,
C = 2.528933072164855]
(%i8) plot2d ([F(x), [discrete, X, Y]], [x, -1, 6]), ''estimates;
(%o8)
Gradient of FOM is specified (instead of computing it automatically).
Both the FOM and its gradient are passed as functions to lbfgs.
(%i1) load (lbfgs)$
(%i2) F(a, b, c) := (a - 5)^2 + (b - 3)^4 + (c - 2)^6$
(%i3) define(F_grad(a, b, c),
map (lambda ([x], diff (F(a, b, c), x)), [a, b, c]))$
(%i4) estimates : lbfgs ([F, F_grad],
[a, b, c], [0, 0, 0], 1e-4, [1, 0]);
*************************************************
N= 3 NUMBER OF CORRECTIONS=25
INITIAL VALUES
F= 1.700000000000000D+02 GNORM= 2.205175729958953D+02
*************************************************
I NFN FUNC GNORM STEPLENGTH 1 2 6.632967565917637D+01 6.498411132518770D+01 4.534785987412505D-03 2 3 4.368890936228036D+01 3.784147651974131D+01 1.000000000000000D+00 3 4 2.685298972775191D+01 1.640262125898520D+01 1.000000000000000D+00 4 5 1.909064767659852D+01 9.733664001790506D+00 1.000000000000000D+00 5 6 1.006493272061515D+01 6.344808151880209D+00 1.000000000000000D+00 6 7 1.215263596054292D+00 2.204727876126877D+00 1.000000000000000D+00 7 8 1.080252896385329D-02 1.431637116951845D-01 1.000000000000000D+00 8 9 8.407195124830860D-03 1.126344579730008D-01 1.000000000000000D+00 9 10 5.022091686198525D-03 7.750731829225275D-02 1.000000000000000D+00 10 11 2.277152808939775D-03 5.032810859286796D-02 1.000000000000000D+00 11 12 6.489384688303218D-04 1.932007150271009D-02 1.000000000000000D+00 12 13 2.075791943844547D-04 6.964319310814365D-03 1.000000000000000D+00 13 14 7.349472666162258D-05 4.017449067849554D-03 1.000000000000000D+00 14 15 2.293617477985238D-05 1.334590390856715D-03 1.000000000000000D+00 15 16 7.683645404048675D-06 6.011057038099202D-04 1.000000000000000D+00
THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
IFLAG = 0
(%o4) [a = 5.000086823042934, b = 3.052395429705181,
c = 1.927980629919583]
Categories: Package lbfgs
Default value: 100
lbfgs_nfeval_max is the maximum number of evaluations of the figure of merit (FOM) in lbfgs.
When lbfgs_nfeval_max is reached,
lbfgs returns the result of the last successful line search.
Categories: Package lbfgs
Default value: 25
lbfgs_ncorrections is the number of corrections applied
to the approximate inverse Hessian matrix which is maintained by lbfgs.
Categories: Package lbfgs
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| 67.1 Functions and Variables for lindstedt |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
This is a first pass at a Lindstedt code. It can solve problems with initial conditions entered, which can be arbitrary constants, (just not %k1 and %k2) where the initial conditions on the perturbation equations are z[i]=0, z'[i]=0 for i>0. ic is the list of initial conditions.
Problems occur when initial conditions are not given, as the constants in the perturbation equations are the same as the zero order equation solution. Also, problems occur when the initial conditions for the perturbation equations are not z[i]=0, z'[i]=0 for i>0, such as the Van der Pol equation.
Example:
(%i1) load("makeOrders")$
(%i2) load("lindstedt")$
(%i3) Lindstedt('diff(x,t,2)+x-(e*x^3)/6,e,2,[1,0]);
2
e (cos(5 T) - 24 cos(3 T) + 23 cos(T))
(%o3) [[[---------------------------------------
36864
e (cos(3 T) - cos(T))
- --------------------- + cos(T)],
192
2
7 e e
T = (- ---- - -- + 1) t]]
3072 16
To use this function write first load("makeOrders") and load("lindstedt").
Categories: Differential equations · Share packages · Package lindstedt
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| 68.1 Introduction to linearalgebra | ||
| 68.2 Functions and Variables for linearalgebra |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
linearalgebra is a collection of functions for linear algebra.
Example:
(%i1) M : matrix ([1, 2], [1, 2]);
[ 1 2 ]
(%o1) [ ]
[ 1 2 ]
(%i2) nullspace (M);
[ 1 ]
[ ]
(%o2) span([ 1 ])
[ - - ]
[ 2 ]
(%i3) columnspace (M);
[ 1 ]
(%o3) span([ ])
[ 1 ]
(%i4) ptriangularize (M - z*ident(2), z);
[ 1 2 - z ]
(%o4) [ ]
[ 2 ]
[ 0 3 z - z ]
(%i5) M : matrix ([1, 2, 3], [4, 5, 6], [7, 8, 9]) - z*ident(3);
[ 1 - z 2 3 ]
[ ]
(%o5) [ 4 5 - z 6 ]
[ ]
[ 7 8 9 - z ]
(%i6) MM : ptriangularize (M, z);
[ 4 5 - z 6 ]
[ ]
[ 2 ]
[ 66 z 102 z 132 ]
[ 0 -- - -- + ----- + --- ]
(%o6) [ 49 7 49 49 ]
[ ]
[ 3 2 ]
[ 49 z 245 z 147 z ]
[ 0 0 ----- - ------ - ----- ]
[ 264 88 44 ]
(%i7) algebraic : true;
(%o7) true
(%i8) tellrat (MM [3, 3]);
3 2
(%o8) [z - 15 z - 18 z]
(%i9) MM : ratsimp (MM);
[ 4 5 - z 6 ]
[ ]
[ 2 ]
(%o9) [ 66 7 z - 102 z - 132 ]
[ 0 -- - ------------------ ]
[ 49 49 ]
[ ]
[ 0 0 0 ]
(%i10) nullspace (MM);
[ 1 ]
[ ]
[ 2 ]
[ z - 14 z - 16 ]
[ -------------- ]
(%o10) span([ 8 ])
[ ]
[ 2 ]
[ z - 18 z - 12 ]
[ - -------------- ]
[ 12 ]
(%i11) M : matrix ([1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12],
[13, 14, 15, 16]);
[ 1 2 3 4 ]
[ ]
[ 5 6 7 8 ]
(%o11) [ ]
[ 9 10 11 12 ]
[ ]
[ 13 14 15 16 ]
(%i12) columnspace (M);
[ 1 ] [ 2 ]
[ ] [ ]
[ 5 ] [ 6 ]
(%o12) span([ ], [ ])
[ 9 ] [ 10 ]
[ ] [ ]
[ 13 ] [ 14 ]
(%i13) apply ('orthogonal_complement, args (nullspace (transpose (M))));
[ 0 ] [ 1 ]
[ ] [ ]
[ 1 ] [ 0 ]
(%o13) span([ ], [ ])
[ 2 ] [ - 1 ]
[ ] [ ]
[ 3 ] [ - 2 ]
Categories: Linear algebra · Share packages · Package linearalgebra
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Using the function f as the addition function, return the sum of the matrices M_1, …, M_n. The function f must accept any number of arguments (a Maxima nary function).
Examples:
(%i1) m1 : matrix([1,2],[3,4])$
(%i2) m2 : matrix([7,8],[9,10])$
(%i3) addmatrices('max,m1,m2);
(%o3) matrix([7,8],[9,10])
(%i4) addmatrices('max,m1,m2,5*m1);
(%o4) matrix([7,10],[15,20])
Categories: Package linearalgebra
Return true if and only if M is a matrix and every entry of M is a matrix.
Categories: Package linearalgebra · Predicate functions
If M is a matrix, return the matrix that results from doing the column
operation C_i <- C_i - theta * C_j. If M doesn't have a row
i or j, signal an error.
Categories: Package linearalgebra
If M is a matrix, swap columns i and j. If M doesn't have a column i or j, signal an error.
Categories: Package linearalgebra
If M is a matrix, return span (v_1, ..., v_n), where the set
{v_1, ..., v_n} is a basis for the column space of M. The span
of the empty set is {0}. Thus, when the column space has only
one member, return span ().
Categories: Package linearalgebra
Return a copy of the Maxima expression e. Although e can be any Maxima expression, the copy function is the most useful when e is either a list or a matrix; consider:
(%i1) m : [1,[2,3]]$ (%i2) mm : m$ (%i3) mm[2][1] : x$ (%i4) m; (%o4) [1,[x,3]] (%i5) mm; (%o5) [1,[x,3]]
Let's try the same experiment, but this time let mm be a copy of m
(%i6) m : [1,[2,3]]$ (%i7) mm : copy(m)$ (%i8) mm[2][1] : x$ (%i9) m; (%o9) [1,[2,3]] (%i10) mm; (%o10) [1,[x,3]]
This time, the assignment to mm does not change the value of m.
Categories: Package linearalgebra
Return the Cholesky factorization of the matrix selfadjoint (or hermitian)
matrix M. The second argument defaults to 'generalring.' For a
description of the possible values for field, see lu_factor.
Categories: Matrix decompositions · Package linearalgebra
Return the complex conjugate transpose of the matrix M. The function
ctranspose uses matrix_element_transpose to transpose each matrix
element.
Categories: Package linearalgebra
Return a diagonal matrix with diagonal entries d_1, d_2, …, d_n. When the diagonal entries are matrices, the zero entries of the returned matrix are zero matrices of the appropriate size; for example:
(%i1) diag_matrix(diag_matrix(1,2),diag_matrix(3,4));
[ [ 1 0 ] [ 0 0 ] ]
[ [ ] [ ] ]
[ [ 0 2 ] [ 0 0 ] ]
(%o1) [ ]
[ [ 0 0 ] [ 3 0 ] ]
[ [ ] [ ] ]
[ [ 0 0 ] [ 0 4 ] ]
(%i2) diag_matrix(p,q);
[ p 0 ]
(%o2) [ ]
[ 0 q ]
Categories: Package linearalgebra
Return the dotproduct of vectors u and v. This is the same as
conjugate (transpose (u)) . v. The arguments u and
v must be column vectors.
Categories: Package linearalgebra
Computes the eigenvalues and eigenvectors of A by the method of Jacobi
rotations. A must be a symmetric matrix (but it need not be positive
definite nor positive semidefinite). field_type indicates the
computational field, either floatfield or bigfloatfield.
If field_type is not specified, it defaults to floatfield.
The elements of A must be numbers or expressions which evaluate to numbers
via float or bfloat (depending on field_type).
Examples:
(%i1) S: matrix([1/sqrt(2), 1/sqrt(2)],[-1/sqrt(2), 1/sqrt(2)]);
[ 1 1 ]
[ ------- ------- ]
[ sqrt(2) sqrt(2) ]
(%o1) [ ]
[ 1 1 ]
[ - ------- ------- ]
[ sqrt(2) sqrt(2) ]
(%i2) L : matrix ([sqrt(3), 0], [0, sqrt(5)]);
[ sqrt(3) 0 ]
(%o2) [ ]
[ 0 sqrt(5) ]
(%i3) M : S . L . transpose (S);
[ sqrt(5) sqrt(3) sqrt(5) sqrt(3) ]
[ ------- + ------- ------- - ------- ]
[ 2 2 2 2 ]
(%o3) [ ]
[ sqrt(5) sqrt(3) sqrt(5) sqrt(3) ]
[ ------- - ------- ------- + ------- ]
[ 2 2 2 2 ]
(%i4) eigens_by_jacobi (M);
The largest percent change was 0.1454972243679
The largest percent change was 0.0
number of sweeps: 2
number of rotations: 1
(%o4) [[1.732050807568877, 2.23606797749979],
[ 0.70710678118655 0.70710678118655 ]
[ ]]
[ - 0.70710678118655 0.70710678118655 ]
(%i5) float ([[sqrt(3), sqrt(5)], S]);
(%o5) [[1.732050807568877, 2.23606797749979],
[ 0.70710678118655 0.70710678118655 ]
[ ]]
[ - 0.70710678118655 0.70710678118655 ]
(%i6) eigens_by_jacobi (M, bigfloatfield);
The largest percent change was 1.454972243679028b-1
The largest percent change was 0.0b0
number of sweeps: 2
number of rotations: 1
(%o6) [[1.732050807568877b0, 2.23606797749979b0],
[ 7.071067811865475b-1 7.071067811865475b-1 ]
[ ]]
[ - 7.071067811865475b-1 7.071067811865475b-1 ]
Categories: Matrix decompositions · Package linearalgebra
When x = lu_factor (A), then get_lu_factors returns a
list of the form [P, L, U], where P is a permutation matrix,
L is lower triangular with ones on the diagonal, and U is upper
triangular, and A = P L U.
Categories: Package linearalgebra
Return a Hankel matrix H. The first column of H is col; except for the first entry, the last row of H is row. The default for row is the zero vector with the same length as col.
Categories: Package linearalgebra
Returns the Hessian matrix of f with respect to the list of variables
x. The (i, j)-th element of the Hessian matrix is
diff(f, x[i], 1, x[j], 1).
Examples:
(%i1) hessian (x * sin (y), [x, y]);
[ 0 cos(y) ]
(%o1) [ ]
[ cos(y) - x sin(y) ]
(%i2) depends (F, [a, b]);
(%o2) [F(a, b)]
(%i3) hessian (F, [a, b]);
[ 2 2 ]
[ d F d F ]
[ --- ----- ]
[ 2 da db ]
[ da ]
(%o3) [ ]
[ 2 2 ]
[ d F d F ]
[ ----- --- ]
[ da db 2 ]
[ db ]
Categories: Differential calculus · Package linearalgebra
Return the n by n Hilbert matrix. When n isn't a positive integer, signal an error.
Categories: Package linearalgebra
Return an identity matrix that has the same shape as the matrix M. The diagonal entries of the identity matrix are the multiplicative identity of the field fld; the default for fld is generalring.
The first argument M should be a square matrix or a non-matrix. When M is a matrix, each entry of M can be a square matrix - thus M can be a blocked Maxima matrix. The matrix can be blocked to any (finite) depth.
See also zerofor
Categories: Package linearalgebra
Invert a matrix M by using the LU factorization. The LU factorization is done using the ring rng.
Categories: Package linearalgebra
Returns the Jacobian matrix of the list of functions f with respect to
the list of variables x. The (i, j)-th element of the Jacobian
matrix is diff(f[i], x[j]).
Examples:
(%i1) jacobian ([sin (u - v), sin (u * v)], [u, v]);
[ cos(v - u) - cos(v - u) ]
(%o1) [ ]
[ v cos(u v) u cos(u v) ]
(%i2) depends ([F, G], [y, z]);
(%o2) [F(y, z), G(y, z)]
(%i3) jacobian ([F, G], [y, z]);
[ dF dF ]
[ -- -- ]
[ dy dz ]
(%o3) [ ]
[ dG dG ]
[ -- -- ]
[ dy dz ]
Categories: Differential calculus · Package linearalgebra
Return the Kronecker product of the matrices A and B.
Categories: Package linearalgebra
Given an optional argument p, return true if e is a Maxima
list and p evaluates to true for every list element. When
listp is not given the optional argument, return true if e
is a Maxima list. In all other cases, return false.
Categories: Package linearalgebra · Predicate functions
The first argument must be a matrix; the arguments r_1 through c_2 determine a sub-matrix of M that consists of rows r_1 through r_2 and columns c_1 through c_2.
Find a entry in the sub-matrix M that satisfies some property. Three cases:
(1) rel = 'bool and f a predicate:
Scan the sub-matrix from left to right then top to bottom,
and return the index of the first entry that satisfies the
predicate f. If no matrix entry satisfies f, return false.
(2) rel = 'max and f real-valued:
Scan the sub-matrix looking for an entry that maximizes f. Return the index of a maximizing entry.
(3) rel = 'min and f real-valued:
Scan the sub-matrix looking for an entry that minimizes f. Return the index of a minimizing entry.
Categories: Package linearalgebra
When M = lu_factor (A, field),
then lu_backsub (M, b) solves the linear
system A x = b.
Categories: Package linearalgebra
Return a list of the form [LU, perm, fld], or
[LU, perm, fld, lower-cnd upper-cnd], where
(1) The matrix LU contains the factorization of M in a packed form.
Packed form means three things: First, the rows of LU are permuted
according to the list perm. If, for example, perm is the list
[3,2,1], the actual first row of the LU factorization is the
third row of the matrix LU. Second, the lower triangular factor of
m is the lower triangular part of LU with the diagonal entries
replaced by all ones. Third, the upper triangular factor of M is the
upper triangular part of LU.
(2) When the field is either floatfield or complexfield, the
numbers lower-cnd and upper-cnd are lower and upper bounds for
the infinity norm condition number of M. For all fields, the
condition number might not be estimated; for such fields, lu_factor
returns a two item list. Both the lower and upper bounds can differ from
their true values by arbitrarily large factors. (See also mat_cond.)
The argument M must be a square matrix.
The optional argument fld must be a symbol that determines a ring or field. The pre-defined fields and rings are:
(a) generalring - the ring of Maxima expressions,
(b) floatfield - the field of floating point numbers of the
type double,
(c) complexfield - the field of complex floating point numbers of
the type double,
(d) crering - the ring of Maxima CRE expressions,
(e) rationalfield - the field of rational numbers,
(f) runningerror - track the all floating point rounding errors,
(g) noncommutingring - the ring of Maxima expressions where
multiplication is the non-commutative dot
operator.
When the field is floatfield, complexfield, or
runningerror, the algorithm uses partial pivoting; for all
other fields, rows are switched only when needed to avoid a zero
pivot.
Floating point addition arithmetic isn't associative, so the meaning of 'field' differs from the mathematical definition.
A member of the field runningerror is a two member Maxima list
of the form [x,n],where x is a floating point number and
n is an integer. The relative difference between the 'true'
value of x and x is approximately bounded by the machine
epsilon times n. The running error bound drops some terms that
of the order the square of the machine epsilon.
There is no user-interface for defining a new field. A user that is
familiar with Common Lisp should be able to define a new field. To do
this, a user must define functions for the arithmetic operations and
functions for converting from the field representation to Maxima and
back. Additionally, for ordered fields (where partial pivoting will be
used), a user must define functions for the magnitude and for
comparing field members. After that all that remains is to define a
Common Lisp structure mring. The file mring has many
examples.
To compute the factorization, the first task is to convert each matrix
entry to a member of the indicated field. When conversion isn't
possible, the factorization halts with an error message. Members of
the field needn't be Maxima expressions. Members of the
complexfield, for example, are Common Lisp complex numbers. Thus
after computing the factorization, the matrix entries must be
converted to Maxima expressions.
See also get_lu_factors.
Examples:
(%i1) w[i,j] := random (1.0) + %i * random (1.0);
(%o1) w := random(1.) + %i random(1.)
i, j
(%i2) showtime : true$
Evaluation took 0.00 seconds (0.00 elapsed)
(%i3) M : genmatrix (w, 100, 100)$
Evaluation took 7.40 seconds (8.23 elapsed)
(%i4) lu_factor (M, complexfield)$
Evaluation took 28.71 seconds (35.00 elapsed)
(%i5) lu_factor (M, generalring)$
Evaluation took 109.24 seconds (152.10 elapsed)
(%i6) showtime : false$
(%i7) M : matrix ([1 - z, 3], [3, 8 - z]);
[ 1 - z 3 ]
(%o7) [ ]
[ 3 8 - z ]
(%i8) lu_factor (M, generalring);
[ 1 - z 3 ]
[ ]
(%o8) [[ 3 9 ], [1, 2], generalring]
[ ----- - z - ----- + 8 ]
[ 1 - z 1 - z ]
(%i9) get_lu_factors (%);
[ 1 0 ] [ 1 - z 3 ]
[ 1 0 ] [ ] [ ]
(%o9) [[ ], [ 3 ], [ 9 ]]
[ 0 1 ] [ ----- 1 ] [ 0 - z - ----- + 8 ]
[ 1 - z ] [ 1 - z ]
(%i10) %[1] . %[2] . %[3];
[ 1 - z 3 ]
(%o10) [ ]
[ 3 8 - z ]
Categories: Matrix decompositions · Package linearalgebra
Return the p-norm matrix condition number of the matrix
m. The allowed values for p are 1 and inf. This
function uses the LU factorization to invert the matrix m. Thus
the running time for mat_cond is proportional to the cube of
the matrix size; lu_factor determines lower and upper bounds
for the infinity norm condition number in time proportional to the
square of the matrix size.
Categories: Package linearalgebra
Return the matrix p-norm of the matrix M. The allowed values for
p are 1, inf, and frobenius (the Frobenius matrix norm).
The matrix M should be an unblocked matrix.
Categories: Package linearalgebra
Given an optional argument p, return true if e is
a matrix and p evaluates to true for every matrix element.
When matrixp is not given an optional argument, return true
if e is a matrix. In all other cases, return false.
See also blockmatrixp
Categories: Package linearalgebra · Predicate functions
Return a two member list that gives the number of rows and columns, respectively of the matrix M.
Categories: Package linearalgebra
If M is a block matrix, unblock the matrix to all levels. If M is a matrix, return M; otherwise, signal an error.
Categories: Package linearalgebra
Return the trace of the matrix M. If M isn't a matrix, return a
noun form. When M is a block matrix, mat_trace(M) returns
the same value as does mat_trace(mat_unblocker(m)).
Categories: Package linearalgebra
If M is a block matrix, unblock M one level. If M is a
matrix, mat_unblocker (M) returns M; otherwise, signal an error.
Thus if each entry of M is matrix, mat_unblocker (M) returns an
unblocked matrix, but if each entry of M is a block matrix,
mat_unblocker (M) returns a block matrix with one less level of blocking.
If you use block matrices, most likely you'll want to set
matrix_element_mult to "." and matrix_element_transpose to
'transpose. See also mat_fullunblocker.
Example:
(%i1) A : matrix ([1, 2], [3, 4]);
[ 1 2 ]
(%o1) [ ]
[ 3 4 ]
(%i2) B : matrix ([7, 8], [9, 10]);
[ 7 8 ]
(%o2) [ ]
[ 9 10 ]
(%i3) matrix ([A, B]);
[ [ 1 2 ] [ 7 8 ] ]
(%o3) [ [ ] [ ] ]
[ [ 3 4 ] [ 9 10 ] ]
(%i4) mat_unblocker (%);
[ 1 2 7 8 ]
(%o4) [ ]
[ 3 4 9 10 ]
Categories: Package linearalgebra
If M is a matrix, return span (v_1, ..., v_n), where the set
{v_1, ..., v_n} is a basis for the nullspace of M. The span of
the empty set is {0}. Thus, when the nullspace has only one member,
return span ().
Categories: Package linearalgebra
If M is a matrix, return the dimension of the nullspace of M.
Categories: Package linearalgebra
Return span (u_1, ..., u_m), where the set {u_1, ..., u_m} is a
basis for the orthogonal complement of the set (v_1, ..., v_n).
Each vector v_1 through v_n must be a column vector.
Categories: Package linearalgebra
Return true if p is a polynomial in the variables in the list
L. The predicate coeffp must evaluate to true for each
coefficient, and the predicate exponp must evaluate to true for all
exponents of the variables in L. If you want to use a non-default value
for exponp, you must supply coeffp with a value even if you want
to use the default for coeffp.
The command polynomialp (p, L, coeffp) is equivalent to
polynomialp (p, L, coeffp, 'nonnegintegerp) and the
command polynomialp (p, L) is equivalent to
polynomialp (p, L, 'constantp, 'nonnegintegerp).
The polynomial needn't be expanded:
(%i1) polynomialp ((x + 1)*(x + 2), [x]); (%o1) true (%i2) polynomialp ((x + 1)*(x + 2)^a, [x]); (%o2) false
An example using non-default values for coeffp and exponp:
(%i1) polynomialp ((x + 1)*(x + 2)^(3/2), [x], numberp, numberp);
(%o1) true
(%i2) polynomialp ((x^(1/2) + 1)*(x + 2)^(3/2), [x], numberp,
numberp);
(%o2) true
Polynomials with two variables:
(%i1) polynomialp (x^2 + 5*x*y + y^2, [x]); (%o1) false (%i2) polynomialp (x^2 + 5*x*y + y^2, [x, y]); (%o2) true
Categories: Package linearalgebra · Predicate functions
If p is a polynomial in x, return the companion matrix of p.
For a monic polynomial p of degree n, we have
p = (-1)^n charpoly (polytocompanion (p, x)).
When p isn't a polynomial in x, signal an error.
Categories: Package linearalgebra
If M is a matrix with each entry a polynomial in v, return a matrix M2 such that
(1) M2 is upper triangular,
(2) M2 = E_n ... E_1 M,
where E_1 through E_n are elementary matrices
whose entries are polynomials in v,
(3) |det (M)| = |det (M2)|,
Note: This function doesn't check that every entry is a polynomial in v.
Categories: Package linearalgebra
If M is a matrix, return the matrix that results from doing the
row operation R_i <- R_i - theta * R_j. If M doesn't have a row
i or j, signal an error.
Categories: Package linearalgebra
Return the rank of that matrix M. The rank is the dimension of the column space. Example:
(%i1) rank(matrix([1,2],[2,4]));
(%o1) 1
(%i2) rank(matrix([1,b],[c,d]));
Proviso: {d - b c # 0}
(%o2) 2
Categories: Package linearalgebra
If M is a matrix, swap rows i and j. If M doesn't have a row i or j, signal an error.
Categories: Package linearalgebra
Return a Toeplitz matrix T. The first first column of T is col; except for the first entry, the first row of T is row. The default for row is complex conjugate of col. Example:
(%i1) toeplitz([1,2,3],[x,y,z]);
[ 1 y z ]
[ ]
(%o1) [ 2 1 y ]
[ ]
[ 3 2 1 ]
(%i2) toeplitz([1,1+%i]);
[ 1 1 - %I ]
(%o2) [ ]
[ %I + 1 1 ]
Categories: Package linearalgebra
Return a n by n matrix whose i-th row is
[1, x_i, x_i^2, ... x_i^(n-1)].
Categories: Package linearalgebra
Return a zero matrix that has the same shape as the matrix M. Every entry of the zero matrix is the additive identity of the field fld; the default for fld is generalring.
The first argument M should be a square matrix or a non-matrix. When M is a matrix, each entry of M can be a square matrix - thus M can be a blocked Maxima matrix. The matrix can be blocked to any (finite) depth.
See also identfor
Categories: Package linearalgebra
If M is not a block matrix, return true if
is (equal (e, 0)) is true for each element e of the matrix
M. If M is a block matrix, return true if zeromatrixp
evaluates to true for each element of e.
Categories: Package linearalgebra · Predicate functions
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 69.1 Introduction to lsquares | ||
| 69.2 Functions and Variables for lsquares |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
lsquares is a collection of functions to implement the method of least squares
to estimate parameters for a model from numerical data.
Categories: Statistical estimation · Share packages · Package lsquares
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Estimate parameters a to best fit the equation e
in the variables x and a to the data D,
as determined by the method of least squares.
lsquares_estimates first seeks an exact solution,
and if that fails, then seeks an approximate solution.
The return value is a list of lists of equations of the form [a = ..., b = ..., c = ...].
Each element of the list is a distinct, equivalent minimum of the mean square error.
The data D must be a matrix.
Each row is one datum (which may be called a `record' or `case' in some contexts),
and each column contains the values of one variable across all data.
The list of variables x gives a name for each column of D,
even the columns which do not enter the analysis.
The list of parameters a gives the names of the parameters for which
estimates are sought.
The equation e is an expression or equation in the variables x and a;
if e is not an equation, it is treated the same as e = 0.
Additional arguments to lsquares_estimates
are specified as equations and passed on verbatim to the function lbfgs
which is called to find estimates by a numerical method
when an exact result is not found.
If some exact solution can be found (via solve),
the data D may contain non-numeric values.
However, if no exact solution is found,
each element of D must have a numeric value.
This includes numeric constants such as %pi and %e as well as literal numbers
(integers, rationals, ordinary floats, and bigfloats).
Numerical calculations are carried out with ordinary floating-point arithmetic,
so all other kinds of numbers are converted to ordinary floats for calculations.
load(lsquares) loads this function.
See also
lsquares_estimates_exact,
lsquares_estimates_approximate,
lsquares_mse,
lsquares_residuals,
and lsquares_residual_mse.
Examples:
A problem for which an exact solution is found.
(%i1) load (lsquares)$
(%i2) M : matrix (
[1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
[ 1 1 1 ]
[ ]
[ 3 ]
[ - 1 2 ]
[ 2 ]
[ ]
(%o2) [ 9 ]
[ - 2 1 ]
[ 4 ]
[ ]
[ 3 2 2 ]
[ ]
[ 2 2 1 ]
(%i3) lsquares_estimates (
M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
59 27 10921 107
(%o3) [[A = - --, B = - --, C = -----, D = - ---]]
16 16 1024 32
A problem for which no exact solution is found,
so lsquares_estimates resorts to numerical approximation.
(%i1) load (lsquares)$
(%i2) M : matrix ([1, 1], [2, 7/4], [3, 11/4], [4, 13/4]);
[ 1 1 ]
[ ]
[ 7 ]
[ 2 - ]
[ 4 ]
[ ]
(%o2) [ 11 ]
[ 3 -- ]
[ 4 ]
[ ]
[ 13 ]
[ 4 -- ]
[ 4 ]
(%i3) lsquares_estimates (
M, [x,y], y=a*x^b+c, [a,b,c], initial=[3,3,3], iprint=[-1,0]);
(%o3) [[a = 1.375751433061394, b = 0.7148891534417651,
c = - 0.4020908910062951]]
Exponential functions aren't well-conditioned for least min square fitting. In case that fitting to them fails it might be possible to get rid of the exponential function using an logarithm.
(%i1) load (lsquares)$
(%i2) yvalues:[1,3,5,60,200,203,80]$
(%i3) time:[1,2,4,5,6,8,10]$
(%i4) f:y=a*exp(b*t);
b t
(%o4) y = a %e
(%i5) yvalues_log:log(yvalues)$
(%i6) f_log:log(subst(y=exp(y),f));
b t
(%o6) y = log(a %e )
(%i7) lsquares_estimates(
transpose(matrix(yvalues_log,time)),
[y,t],
f_log,
[a,b]
);
*************************************************
N= 2 NUMBER OF CORRECTIONS=25
INITIAL VALUES
F= 6.802906290754687D+00 GNORM= 2.851243373781393D+01
*************************************************
I NFN FUNC GNORM STEPLENGTH
1 3 1.141838765593467D+00 1.067358003667488D-01 1.390943719972406D-02
2 5 1.141118195694385D+00 1.237977833033414D-01 5.000000000000000D+00
3 6 1.136945723147959D+00 3.806696991691383D-01 1.000000000000000D+00
4 7 1.133958243220262D+00 3.865103550379243D-01 1.000000000000000D+00
5 8 1.131725773805499D+00 2.292258231154026D-02 1.000000000000000D+00
6 9 1.131625585698168D+00 2.664440547017370D-03 1.000000000000000D+00
7 10 1.131620564856599D+00 2.519366958715444D-04 1.000000000000000D+00
THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
IFLAG = 0
(%o7) [[a = 1.155904145765554, b = 0.5772666876959847]]
Categories: Package lsquares · Numerical methods
Estimate parameters a to minimize the mean square error MSE,
by constructing a system of equations and attempting to solve them symbolically via solve.
The mean square error is an expression in the parameters a,
such as that returned by lsquares_mse.
The return value is a list of lists of equations of the form [a = ..., b = ..., c = ...].
The return value may contain zero, one, or two or more elements.
If two or more elements are returned,
each represents a distinct, equivalent minimum of the mean square error.
See also
lsquares_estimates,
lsquares_estimates_approximate,
lsquares_mse,
lsquares_residuals,
and lsquares_residual_mse.
Example:
(%i1) load (lsquares)$
(%i2) M : matrix (
[1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
[ 1 1 1 ]
[ ]
[ 3 ]
[ - 1 2 ]
[ 2 ]
[ ]
(%o2) [ 9 ]
[ - 2 1 ]
[ 4 ]
[ ]
[ 3 2 2 ]
[ ]
[ 2 2 1 ]
(%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
5
====
\ 2 2
> ((- B M ) - A M + (M + D) - C)
/ i, 3 i, 2 i, 1
====
i = 1
(%o3) -------------------------------------------------
5
(%i4) lsquares_estimates_exact (mse, [A, B, C, D]);
59 27 10921 107
(%o4) [[A = - --, B = - --, C = -----, D = - ---]]
16 16 1024 32
Categories: Package lsquares
Estimate parameters a to minimize the mean square error MSE,
via the numerical minimization function lbfgs.
The mean square error is an expression in the parameters a,
such as that returned by lsquares_mse.
The solution returned by lsquares_estimates_approximate is a local (perhaps global) minimum
of the mean square error.
For consistency with lsquares_estimates_exact,
the return value is a nested list which contains one element,
namely a list of equations of the form [a = ..., b = ..., c = ...].
Additional arguments to lsquares_estimates_approximate
are specified as equations and passed on verbatim to the function lbfgs.
MSE must evaluate to a number when the parameters are assigned numeric values.
This requires that the data from which MSE was constructed
comprise only numeric constants such as %pi and %e and literal numbers
(integers, rationals, ordinary floats, and bigfloats).
Numerical calculations are carried out with ordinary floating-point arithmetic,
so all other kinds of numbers are converted to ordinary floats for calculations.
load(lsquares) loads this function.
See also
lsquares_estimates,
lsquares_estimates_exact,
lsquares_mse,
lsquares_residuals,
and lsquares_residual_mse.
Example:
(%i1) load (lsquares)$
(%i2) M : matrix (
[1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
[ 1 1 1 ]
[ ]
[ 3 ]
[ - 1 2 ]
[ 2 ]
[ ]
(%o2) [ 9 ]
[ - 2 1 ]
[ 4 ]
[ ]
[ 3 2 2 ]
[ ]
[ 2 2 1 ]
(%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
5
====
\ 2 2
> ((- B M ) - A M + (M + D) - C)
/ i, 3 i, 2 i, 1
====
i = 1
(%o3) -------------------------------------------------
5
(%i4) lsquares_estimates_approximate (
mse, [A, B, C, D], iprint = [-1, 0]);
(%o4) [[A = - 3.678504947401971, B = - 1.683070351177937,
C = 10.63469950148714, D = - 3.340357993175297]]
Categories: Package lsquares · Numerical methods
Returns the mean square error (MSE), a summation expression, for the equation e in the variables x, with data D.
The MSE is defined as:
n
====
1 \ 2
- > (lhs(e ) - rhs(e ))
n / i i
====
i = 1
where n is the number of data and e[i] is the equation e
evaluated with the variables in x assigned values from the i-th datum, D[i].
load(lsquares) loads this function.
Example:
(%i1) load (lsquares)$
(%i2) M : matrix (
[1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
[ 1 1 1 ]
[ ]
[ 3 ]
[ - 1 2 ]
[ 2 ]
[ ]
(%o2) [ 9 ]
[ - 2 1 ]
[ 4 ]
[ ]
[ 3 2 2 ]
[ ]
[ 2 2 1 ]
(%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
5
====
\ 2 2
> ((- B M ) - A M + (M + D) - C)
/ i, 3 i, 2 i, 1
====
i = 1
(%o3) -------------------------------------------------
5
(%i4) diff (mse, D);
(%o4)
5
====
\ 2
4 > (M + D) ((- B M ) - A M + (M + D) - C)
/ i, 1 i, 3 i, 2 i, 1
====
i = 1
--------------------------------------------------------------
5
(%i5) ''mse, nouns;
2 2 9 2 2
(%o5) (((D + 3) - C - 2 B - 2 A) + ((D + -) - C - B - 2 A)
4
2 2 3 2 2
+ ((D + 2) - C - B - 2 A) + ((D + -) - C - 2 B - A)
2
2 2
+ ((D + 1) - C - B - A) )/5
(%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
5
====
\ 2 2
> ((D + M ) - C - M B - M A)
/ i, 1 i, 3 i, 2
====
i = 1
(%o3) ---------------------------------------------
5
(%i4) diff (mse, D);
5
====
\ 2
4 > (D + M ) ((D + M ) - C - M B - M A)
/ i, 1 i, 1 i, 3 i, 2
====
i = 1
(%o4) ----------------------------------------------------------
5
(%i5) ''mse, nouns;
2 2 9 2 2
(%o5) (((D + 3) - C - 2 B - 2 A) + ((D + -) - C - B - 2 A)
4
2 2 3 2 2
+ ((D + 2) - C - B - 2 A) + ((D + -) - C - 2 B - A)
2
2 2
+ ((D + 1) - C - B - A) )/5
Categories: Package lsquares
Returns the residuals for the equation e with specified parameters a and data D.
D is a matrix, x is a list of variables,
e is an equation or general expression;
if not an equation, e is treated as if it were e = 0.
a is a list of equations which specify values for any free parameters in e aside from x.
The residuals are defined as:
lhs(e ) - rhs(e )
i i
where e[i] is the equation e
evaluated with the variables in x assigned values from the i-th datum, D[i],
and assigning any remaining free variables from a.
load(lsquares) loads this function.
Example:
(%i1) load (lsquares)$
(%i2) M : matrix (
[1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
[ 1 1 1 ]
[ ]
[ 3 ]
[ - 1 2 ]
[ 2 ]
[ ]
(%o2) [ 9 ]
[ - 2 1 ]
[ 4 ]
[ ]
[ 3 2 2 ]
[ ]
[ 2 2 1 ]
(%i3) a : lsquares_estimates (
M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
59 27 10921 107
(%o3) [[A = - --, B = - --, C = -----, D = - ---]]
16 16 1024 32
(%i4) lsquares_residuals (
M, [z,x,y], (z+D)^2 = A*x+B*y+C, first(a));
13 13 13 13 13
(%o4) [--, - --, - --, --, --]
64 64 32 64 64
Categories: Package lsquares
Returns the residual mean square error (MSE) for the equation e with specified parameters a and data D.
The residual MSE is defined as:
n
====
1 \ 2
- > (lhs(e ) - rhs(e ))
n / i i
====
i = 1
where e[i] is the equation e
evaluated with the variables in x assigned values from the i-th datum, D[i],
and assigning any remaining free variables from a.
load(lsquares) loads this function.
Example:
(%i1) load (lsquares)$
(%i2) M : matrix (
[1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
[ 1 1 1 ]
[ ]
[ 3 ]
[ - 1 2 ]
[ 2 ]
[ ]
(%o2) [ 9 ]
[ - 2 1 ]
[ 4 ]
[ ]
[ 3 2 2 ]
[ ]
[ 2 2 1 ]
(%i3) a : lsquares_estimates (
M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
59 27 10921 107
(%o3) [[A = - --, B = - --, C = -----, D = - ---]]
16 16 1024 32
(%i4) lsquares_residual_mse (
M, [z,x,y], (z + D)^2 = A*x + B*y + C, first (a));
169
(%o4) ----
2560
Categories: Package lsquares
Multivariable polynomial adjustment of a data table by the "least squares"
method. Mat is a matrix containing the data, VarList is a list of variable names (one for each Mat column, but use "-" instead of varnames to ignore Mat columns), depvars is the name of a dependent variable or a list with one or more names of dependent variables (which names should be in VarList), maxexpon is the optional maximum exponent for each independent variable (1 by default), and maxdegree is the optional maximum polynomial degree (maxexpon by default); note that the sum of exponents of each term must be equal or smaller than maxdegree, and if maxdgree = 0 then no limit is applied.
If depvars is the name of a dependent variable (not in a list), plsquares returns the adjusted polynomial. If depvars is a list of one or more dependent variables, plsquares
returns a list with the adjusted polynomial(s). The Coefficients of
Determination are displayed in order to inform about the goodness of
fit, which ranges from 0 (no correlation) to 1 (exact correlation).
These values are also stored in the global variable DETCOEF (a list if depvars is a list).
A simple example of multivariable linear adjustment:
(%i1) load("plsquares")$
(%i2) plsquares(matrix([1,2,0],[3,5,4],[4,7,9],[5,8,10]),
[x,y,z],z);
Determination Coefficient for z = .9897039897039897
11 y - 9 x - 14
(%o2) z = ---------------
3
The same example without degree restrictions:
(%i3) plsquares(matrix([1,2,0],[3,5,4],[4,7,9],[5,8,10]),
[x,y,z],z,1,0);
Determination Coefficient for z = 1.0
x y + 23 y - 29 x - 19
(%o3) z = ----------------------
6
How many diagonals does a N-sides polygon have? What polynomial degree should be used?
(%i4) plsquares(matrix([3,0],[4,2],[5,5],[6,9],[7,14],[8,20]),
[N,diagonals],diagonals,5);
Determination Coefficient for diagonals = 1.0
2
N - 3 N
(%o4) diagonals = --------
2
(%i5) ev(%, N=9); /* Testing for a 9 sides polygon */
(%o5) diagonals = 27
How many ways do we have to put two queens without they are threatened into a n x n chessboard?
(%i6) plsquares(matrix([0,0],[1,0],[2,0],[3,8],[4,44]),
[n,positions],[positions],4);
Determination Coefficient for [positions] = [1.0]
4 3 2
3 n - 10 n + 9 n - 2 n
(%o6) [positions = -------------------------]
6
(%i7) ev(%[1], n=8); /* Testing for a (8 x 8) chessboard */
(%o7) positions = 1288
An example with six dependent variables:
(%i8) mtrx:matrix([0,0,0,0,0,1,1,1],[0,1,0,1,1,1,0,0],
[1,0,0,1,1,1,0,0],[1,1,1,1,0,0,0,1])$
(%i8) plsquares(mtrx,[a,b,_And,_Or,_Xor,_Nand,_Nor,_Nxor],
[_And,_Or,_Xor,_Nand,_Nor,_Nxor],1,0);
Determination Coefficient for
[_And, _Or, _Xor, _Nand, _Nor, _Nxor] =
[1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
(%o2) [_And = a b, _Or = - a b + b + a,
_Xor = - 2 a b + b + a, _Nand = 1 - a b,
_Nor = a b - b - a + 1, _Nxor = 2 a b - b - a + 1]
To use this function write first load("lsquares").
Categories: Package lsquares · Numerical methods
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| 70.1 Introduction to minpack | ||
| 70.2 Functions and Variables for minpack |
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Minpack is a Common Lisp translation (via f2cl) of the
Fortran library MINPACK, as obtained from Netlib.
Categories: Numerical methods · Optimization · Share packages · Package minpack
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Compute the point that minimizes the sum of the squares of the functions in the list flist. The variables are in the list varlist. An initial guess of the optimum point must be provided in guess.
The optional keyword arguments, tolerance and jacobian
provide some control over the algorithm. tolerance is the
estimated relative error desired in the sum of squares.
jacobian can be used to specify the Jacobian. If jacobian
is not given or is true (the default), the Jacobian is computed
from flist. If jacobian is false, a numerical
approximation is used.
minpack_lsquares returns a list. The first item is the
estimated solution; the second is the sum of squares, and the third
indicates the success of the algorithm. The possible values are
0improper input parameters.
1algorithm estimates that the relative error in the sum of squares is
at most tolerance.
2algorithm estimates that the relative error between x and the solution
is at most tolerance.
3conditions for info = 1 and info = 2 both hold.
4fvec is orthogonal to the columns of the jacobian to machine precision.
5number of calls to fcn with iflag = 1 has reached 100*(n+1).
6tol is too small. no further reduction in the sum of squares is possible.
7tol is too small. no further improvement in the approximate solution x is possible.
/* Problem 6: Powell singular function */
(%i1) powell(x1,x2,x3,x4) :=
[x1+10*x2, sqrt(5)*(x3-x4), (x2-2*x3)^2,
sqrt(10)*(x1-x4)^2]$
(%i2) minpack_lsquares(powell(x1,x2,x3,x4), [x1,x2,x3,x4],
[3,-1,0,1]);
(%o2) [[1.652117596168394e-17, - 1.652117596168393e-18,
2.643388153869468e-18, 2.643388153869468e-18],
6.109327859207777e-34, 4]
/* Same problem but use numerical approximation to Jacobian */
(%i3) minpack_lsquares(powell(x1,x2,x3,x4), [x1,x2,x3,x4],
[3,-1,0,1], jacobian = false);
(%o3) [[5.060282149485331e-11, - 5.060282149491206e-12,
2.179447843547218e-11, 2.179447843547218e-11],
3.534491794847031e-21, 5]
Solve a system of n equations in n unknowns.
The n equations are given in the list flist, and the
unknowns are in varlist. An initial guess of the solution must
be provided in guess.
The optional keyword arguments, tolerance and jacobian
provide some control over the algorithm. tolerance is the
estimated relative error desired in the sum of squares.
jacobian can be used to specify the Jacobian. If jacobian
is not given or is true (the default), the Jacobian is computed
from flist. If jacobian is false, a numerical
approximation is used.
minpack_solve returns a list. The first item is the
estimated solution; the second is the sum of squares, and the third
indicates the success of the algorithm. The possible values are
0improper input parameters.
1algorithm estimates that the relative error in the solution is
at most tolerance.
2number of calls to fcn with iflag = 1 has reached 100*(n+1).
3tol is too small. no further reduction in the sum of squares is possible.
4Iteration is not making good progress.
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| 71.1 Functions and Variables for makeOrders |
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Returns a list of all powers for a polynomial up to and including the arguments.
(%i1) load("makeOrders")$
(%i2) makeOrders([a,b],[2,3]);
(%o2) [[0, 0], [0, 1], [0, 2], [0, 3], [1, 0], [1, 1],
[1, 2], [1, 3], [2, 0], [2, 1], [2, 2], [2, 3]]
(%i3) expand((1+a+a^2)*(1+b+b^2+b^3));
2 3 3 3 2 2 2 2 2
(%o3) a b + a b + b + a b + a b + b + a b + a b
2
+ b + a + a + 1
where [0, 1] is associated with the term b and [2, 3] with a^2 b^3.
To use this function write first load("makeOrders").
Categories: Polynomials · Share packages · Package makeOrders
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| 72.1 Introduction to mnewton | ||
| 72.2 Functions and Variables for mnewton |
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mnewton is an implementation of Newton's method for solving nonlinear
equations in one or more variables.
Categories: Numerical methods · Share packages · Package mnewton
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Default value: 10.0^(-fpprec/2)
Precision to determine when the mnewton function has converged towards
the solution. If newtonepsilon is a bigfloat, then mnewton
computations are done with bigfloats. See also mnewton.
Categories: Package mnewton
Default value: 50
Maximum number of iterations to stop the mnewton function
if it does not converge or if it converges too slowly.
See also mnewton.
Categories: Package mnewton
Multiple nonlinear functions solution using the Newton method. FuncList is the list of functions to solve, VarList is the list of variable names, and GuessList is the list of initial approximations.
The solution is returned in the same format that solve() returns.
If the solution is not found, [] is returned.
This function is controlled by global variables newtonepsilon and
newtonmaxiter.
(%i1) load("mnewton")$
(%i2) mnewton([x1+3*log(x1)-x2^2, 2*x1^2-x1*x2-5*x1+1],
[x1, x2], [5, 5]);
(%o2) [[x1 = 3.756834008012769, x2 = 2.779849592817897]]
(%i3) mnewton([2*a^a-5],[a],[1]);
(%o3) [[a = 1.70927556786144]]
(%i4) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
(%o4) [[u = 1.066618389595407, v = 1.552564766841786]]
The variable newtonepsilon controls the precision of the
approximations. It also controls if computations are performed with
floats or bigfloats.
(%i1) load(mnewton)$
(%i2) (fpprec : 25, newtonepsilon : bfloat(10^(-fpprec+5)))$
(%i3) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
(%o3) [[u = 1.066618389595406772591173b0,
v = 1.552564766841786450100418b0]]
To use this function write first load("mnewton").
See also newtonepsilon and newtonmaxiter.
Categories: Package mnewton
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| 73.1 Introduction to numericalio | ||
| 73.2 Functions and Variables for plain-text input and output | ||
| 73.3 Functions and Variables for binary input and output |
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numericalio is a collection of functions to read and write files and streams.
Functions for plain-text input and output
can read and write numbers (integer, float, or bigfloat), symbols, and strings.
Functions for binary input and output
can read and write only floating-point numbers.
If there already exists a list, matrix, or array object to store input data,
numericalio input functions can write data into that object.
Otherwise, numericalio can guess, to some degree, the structure of an object
to store the data, and return that object.
Categories: File input · File output · Share packages · Package numericalio
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In plain-text input and output,
it is assumed that each item to read or write is an atom:
an integer, float, bigfloat, string, or symbol,
and not a rational or complex number or any other kind of nonatomic expression.
The numericalio functions may attempt to do something sensible faced with nonatomic expressions,
but the results are not specified here and subject to change.
Atoms in both input and output files have the same format as
in Maxima batch files or the interactive console.
In particular, strings are enclosed in double quotes,
backslash \ prevents any special interpretation of the next character,
and the question mark ? is recognized at the beginning of a symbol
to mean a Lisp symbol (as opposed to a Maxima symbol).
No continuation character (to join broken lines) is recognized.
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The functions for plain-text input and output take an optional argument, separator_flag, that tells what character separates data.
For plain-text input, these values of separator_flag are recognized:
comma for comma separated values,
pipe for values separated by the vertical bar character |,
semicolon for values separated by semicolon ;,
and space for values separated by space or tab characters.
If the file name ends in .csv and separator_flag is not specified,
comma is assumed.
If the file name ends in something other than .csv and separator_flag is not specified,
space is assumed.
In plain-text input, multiple successive space and tab characters count as a single separator.
However, multiple comma, pipe, or semicolon characters are significant.
Successive comma, pipe, or semicolon characters (with or without intervening spaces or tabs)
are considered to have false between the separators.
For example, 1234,,Foo is treated the same as 1234,false,Foo.
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For plain-text output, tab, for values separated by the tab character,
is recognized as a value of separator_flag,
as well as comma, pipe, semicolon, and space.
In plain-text output, false atoms are written as such;
a list [1234, false, Foo] is written 1234,false,Foo,
and there is no attempt to collapse the output to 1234,,Foo.
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numericalio functions can read and write 8-byte IEEE 754 floating-point numbers.
These numbers can be stored either least significant byte first or most significant byte first,
according to the global flag set by assume_external_byte_order.
If not specified, numericalio assumes the external byte order is most-significant byte first.
Other kinds of numbers are coerced to 8-byte floats;
numericalio cannot read or write binary non-numeric data.
Some Lisp implementations do not recognize IEEE 754 special values
(positive and negative infinity, not-a-number values, denormalized values).
The effect of reading such values with numericalio is undefined.
numericalio includes functions to open a stream for reading or writing a stream of bytes.
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read_matrix(S) reads the source S and returns its entire content as a matrix.
The size of the matrix is inferred from the input data;
each line of the file becomes one row of the matrix.
If some lines have different lengths, read_matrix complains.
read_matrix(S, M) read the source S into the matrix M,
until M is full or the source is exhausted.
Input data are read into the matrix in row-major order;
the input need not have the same number of rows and columns as M.
The source S may be a file name or a stream which for example allows skipping the very first line of a file (that may be useful, if you read CSV data, where the first line often contains the description of the columns):
s : openr("data.txt");
readline(s); /* skip the first line */
M : read_matrix(s, comma); /* read the following (comma-separated) lines into matrix M */
close(s);
The recognized values of separator_flag are
comma, pipe, semicolon, and space.
If separator_flag is not specified, the file is assumed space-delimited.
Categories: Package numericalio · File input
Reads the source S into the array A, until A is full or the source is exhausted. Input data are read into the array in row-major order; the input need not conform to the dimensions of A.
The source S may be a file name or a stream.
The recognized values of separator_flag are
comma, pipe, semicolon, and space.
If separator_flag is not specified, the file is assumed space-delimited.
Categories: Package numericalio · File input
Reads the source S and returns its entire content as a hashed array. The source S may be a file name or a stream.
read_hashed_array treats the first item on each line as a hash key,
and associates the remainder of the line (as a list) with the key.
For example,
the line 567 12 17 32 55 is equivalent to A[567]: [12, 17, 32, 55]$.
Lines need not have the same numbers of elements.
The recognized values of separator_flag are
comma, pipe, semicolon, and space.
If separator_flag is not specified, the file is assumed space-delimited.
Categories: Package numericalio · File input
Reads the source S and returns its entire content as a nested list. The source S may be a file name or a stream.
read_nested_list returns a list which has a sublist for each
line of input. Lines need not have the same numbers of elements.
Empty lines are not ignored: an empty line yields an empty sublist.
The recognized values of separator_flag are
comma, pipe, semicolon, and space.
If separator_flag is not specified, the file is assumed space-delimited.
Categories: Package numericalio · File input
read_list(S) reads the source S and returns its entire content as a flat list.
read_list(S, L) reads the source S into the list L,
until L is full or the source is exhausted.
The source S may be a file name or a stream.
The recognized values of separator_flag are
comma, pipe, semicolon, and space.
If separator_flag is not specified, the file is assumed space-delimited.
Categories: Package numericalio · File input
Writes the object X to the destination D.
write_data writes a matrix in row-major order,
with one line per row.
write_data writes an array created by array or make_array
in row-major order, with a new line at the end of every slab.
Higher-dimensional slabs are separated by additional new lines.
write_data writes a hashed array with each key followed by
its associated list on one line.
write_data writes a nested list with each sublist on one line.
write_data writes a flat list all on one line.
The destination D may be a file name or a stream.
When the destination is a file name,
the global variable file_output_append governs
whether the output file is appended or truncated.
When the destination is a stream,
no special action is taken by write_data after all the data are written;
in particular, the stream remains open.
The recognized values of separator_flag are
comma, pipe, semicolon, space, and tab.
If separator_flag is not specified, the file is assumed space-delimited.
Categories: Package numericalio · File output
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Tells numericalio the byte order for reading and writing binary data.
Two values of byte_order_flag are recognized:
lsb which indicates least-significant byte first, also called little-endian byte order;
and msb which indicates most-significant byte first, also called big-endian byte order.
If not specified, numericalio assumes the external byte order is most-significant byte first.
Categories: Package numericalio
Returns an input stream of 8-bit unsigned bytes to read the file named by file_name.
Categories: Package numericalio · File input
Returns an output stream of 8-bit unsigned bytes to write the file named by file_name.
Categories: Package numericalio · File output
Returns an output stream of 8-bit unsigned bytes to append the file named by file_name.
Categories: Package numericalio · File output
Reads binary 8-byte floating point numbers from the source S into the matrix M until M is full, or the source is exhausted. Elements of M are read in row-major order.
The source S may be a file name or a stream.
The byte order in elements of the source is specified by assume_external_byte_order.
Categories: Package numericalio · File input
Reads binary 8-byte floating point numbers from the source S into the array A
until A is full, or the source is exhausted.
A must be an array created by array or make_array.
Elements of A are read in row-major order.
The source S may be a file name or a stream.
The byte order in elements of the source is specified by assume_external_byte_order.
Categories: Package numericalio · File input
read_binary_list(S) reads the entire content of the source S
as a sequence of binary 8-byte floating point numbers, and returns it as a list.
The source S may be a file name or a stream.
read_binary_list(S, L) reads 8-byte binary floating point numbers
from the source S until the list L is full, or the source is exhausted.
The byte order in elements of the source is specified by assume_external_byte_order.
Categories: Package numericalio · File input
Writes the object X, comprising binary 8-byte IEEE 754 floating-point numbers,
to the destination D.
Other kinds of numbers are coerced to 8-byte floats.
write_binary_data cannot write non-numeric data.
The object X may be a list, a nested list, a matrix,
or an array created by array or make_array;
X cannot be an undeclared array or any other type of object.
write_binary_data writes nested lists, matrices, and arrays in row-major order.
The destination D may be a file name or a stream.
When the destination is a file name,
the global variable file_output_append governs
whether the output file is appended or truncated.
When the destination is a stream,
no special action is taken by write_binary_data after all the data are written;
in particular, the stream remains open.
The byte order in elements of the destination
is specified by assume_external_byte_order.
Categories: Package numericalio · File output
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| 74.1 Introduction to operatingsystem | ||
| 74.2 Directory operations | ||
| 74.3 File operations | ||
| 74.4 Environment operations |
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Package operatingsystem contains functions for operatingsystem-tasks, like file system operations.
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Change to directory dir
Create directory dir
remove directory dir
returns the current working directory.
See also directory.
Examples:
(%i1) load("operatingsystem")$
(%i2) mkdir("testdirectory")$
(%i3) chdir("testdirectory")$
(%i4) chdir("..")$
(%i5) rmdir("testdirectory")$
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copies file file1 to file2
renames file file1 to file2
deletes file file1
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Get the value of the environmentvariable env
Example:
(%i1) load("operatingsystem")$
(%i2) getenv("PATH");
(%o2) /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin
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| 75.1 Functions and Variables for opsubst |
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The function opsubst is similar to the function subst, except that
opsubst only makes substitutions for the operators in an expression. In general,
When f is an operator in the expression e, substitute g
for f in the expression e.
To determine the operator, opsubst sets inflag to true. This means
opsubst substitutes for the internal, not the displayed, operator
in the expression.
Examples:
(%i1) load ("opsubst")$
(%i2) opsubst(f,g,g(g(x)));
(%o2) f(f(x))
(%i3) opsubst(f,g,g(g));
(%o3) f(g)
(%i4) opsubst(f,g[x],g[x](z));
(%o4) f(z)
(%i5) opsubst(g[x],f, f(z));
(%o5) g (z)
x
(%i6) opsubst(tan, sin, sin(sin));
(%o6) tan(sin)
(%i7) opsubst([f=g,g=h],f(x));
(%o7) h(x)
Internally, Maxima does not use the unary negation, division, or the subtraction operators; thus:
(%i8) opsubst("+","-",a-b);
(%o8) a - b
(%i9) opsubst("f","-",-a);
(%o9) - a
(%i10) opsubst("^^","/",a/b);
a
(%o10) -
b
The internal representation of -a*b is *(-1,a,b); thus
(%i11) opsubst("[","*", -a*b);
(%o11) [- 1, a, b]
When either operator isn't a Maxima symbol, generally some other function will signal an error:
(%i12) opsubst(a+b,f, f(x)); Improper name or value in functional position: b + a -- an error. Quitting. To debug this try debugmode(true);
However, subscripted operators are allowed:
(%i13) opsubst(g[5],f, f(x));
(%o13) g (x)
5
To use this function write first load("opsubst").
Categories: Expressions · Share packages · Package opsubst
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| 76.1 Introduction to orthogonal polynomials | ||
| 76.2 Functions and Variables for orthogonal polynomials |
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orthopoly is a package for symbolic and numerical evaluation of
several kinds of orthogonal polynomials, including Chebyshev,
Laguerre, Hermite, Jacobi, Legendre, and ultraspherical (Gegenbauer)
polynomials. Additionally, orthopoly includes support for the spherical Bessel,
spherical Hankel, and spherical harmonic functions.
For the most part, orthopoly follows the conventions of Abramowitz and Stegun
Handbook of Mathematical Functions, Chapter 22 (10th printing, December 1972);
additionally, we use Gradshteyn and Ryzhik,
Table of Integrals, Series, and Products (1980 corrected and
enlarged edition), and Eugen Merzbacher Quantum Mechanics (2nd edition, 1970).
Barton Willis of the University of Nebraska at Kearney (UNK) wrote
the orthopoly package and its documentation. The package
is released under the GNU General Public License (GPL).
Categories: Orthogonal polynomials · Share packages · Package orthopoly
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load (orthopoly) loads the orthopoly package.
To find the third-order Legendre polynomial,
(%i1) legendre_p (3, x);
3 2
5 (1 - x) 15 (1 - x)
(%o1) - ---------- + ----------- - 6 (1 - x) + 1
2 2
To express this as a sum of powers of x, apply ratsimp or rat to the result.
(%i2) [ratsimp (%), rat (%)];
3 3
5 x - 3 x 5 x - 3 x
(%o2)/R/ [----------, ----------]
2 2
Alternatively, make the second argument to legendre_p (its "main" variable)
a canonical rational expression (CRE).
(%i1) legendre_p (3, rat (x));
3
5 x - 3 x
(%o1)/R/ ----------
2
For floating point evaluation, orthopoly uses a running error analysis
to estimate an upper bound for the error. For example,
(%i1) jacobi_p (150, 2, 3, 0.2); (%o1) interval(- 0.062017037936715, 1.533267919277521E-11)
Intervals have the form interval (c, r), where c is the
center and r is the radius of the interval. Since Maxima
does not support arithmetic on intervals, in some situations, such
as graphics, you want to suppress the error and output only the
center of the interval. To do this, set the option
variable orthopoly_returns_intervals to false.
(%i1) orthopoly_returns_intervals : false; (%o1) false (%i2) jacobi_p (150, 2, 3, 0.2); (%o2) - 0.062017037936715
Refer to the section see Floating point Evaluation for more information.
Most functions in orthopoly have a gradef property; thus
(%i1) diff (hermite (n, x), x);
(%o1) 2 n H (x)
n - 1
(%i2) diff (gen_laguerre (n, a, x), x);
(a) (a)
n L (x) - (n + a) L (x) unit_step(n)
n n - 1
(%o2) ------------------------------------------
x
The unit step function in the second example prevents an error that would otherwise arise by evaluating with n equal to 0.
(%i3) ev (%, n = 0); (%o3) 0
The gradef property only applies to the "main" variable; derivatives with
respect other arguments usually result in an error message; for example
(%i1) diff (hermite (n, x), x);
(%o1) 2 n H (x)
n - 1
(%i2) diff (hermite (n, x), n);
Maxima doesn't know the derivative of hermite with respect the first
argument
-- an error. Quitting. To debug this try debugmode(true);
Generally, functions in orthopoly map over lists and matrices. For
the mapping to fully evaluate, the option variables
doallmxops and listarith must both be true (the defaults).
To illustrate the mapping over matrices, consider
(%i1) hermite (2, x);
2
(%o1) - 2 (1 - 2 x )
(%i2) m : matrix ([0, x], [y, 0]);
[ 0 x ]
(%o2) [ ]
[ y 0 ]
(%i3) hermite (2, m);
[ 2 ]
[ - 2 - 2 (1 - 2 x ) ]
(%o3) [ ]
[ 2 ]
[ - 2 (1 - 2 y ) - 2 ]
In the second example, the i, j element of the value
is hermite (2, m[i,j]); this is not the same as computing
-2 + 4 m . m, as seen in the next example.
(%i4) -2 * matrix ([1, 0], [0, 1]) + 4 * m . m;
[ 4 x y - 2 0 ]
(%o4) [ ]
[ 0 4 x y - 2 ]
If you evaluate a function at a point outside its domain, generally
orthopoly returns the function unevaluated. For example,
(%i1) legendre_p (2/3, x);
(%o1) P (x)
2/3
orthopoly supports translation into TeX; it also does two-dimensional
output on a terminal.
(%i1) spherical_harmonic (l, m, theta, phi);
m
(%o1) Y (theta, phi)
l
(%i2) tex (%);
$$Y_{l}^{m}\left(\vartheta,\varphi\right)$$
(%o2) false
(%i3) jacobi_p (n, a, a - b, x/2);
(a, a - b) x
(%o3) P (-)
n 2
(%i4) tex (%);
$$P_{n}^{\left(a,a-b\right)}\left({{x}\over{2}}\right)$$
(%o4) false
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When an expression involves several orthogonal polynomials with symbolic orders, it's possible that the expression actually vanishes, yet Maxima is unable to simplify it to zero. If you divide by such a quantity, you'll be in trouble. For example, the following expression vanishes for integers n greater than 1, yet Maxima is unable to simplify it to zero.
(%i1) (2*n - 1) * legendre_p (n - 1, x) * x - n * legendre_p (n, x)
+ (1 - n) * legendre_p (n - 2, x);
(%o1) (2 n - 1) P (x) x - n P (x) + (1 - n) P (x)
n - 1 n n - 2
For a specific n, we can reduce the expression to zero.
(%i2) ev (% ,n = 10, ratsimp); (%o2) 0
Generally, the polynomial form of an orthogonal polynomial is ill-suited for floating point evaluation. Here's an example.
(%i1) p : jacobi_p (100, 2, 3, x)$ (%i2) subst (0.2, x, p); (%o2) 3.4442767023833592E+35 (%i3) jacobi_p (100, 2, 3, 0.2); (%o3) interval(0.18413609135169, 6.8990300925815987E-12) (%i4) float(jacobi_p (100, 2, 3, 2/10)); (%o4) 0.18413609135169
The true value is about 0.184; this calculation suffers from extreme subtractive cancellation error. Expanding the polynomial and then evaluating, gives a better result.
(%i5) p : expand(p)$ (%i6) subst (0.2, x, p); (%o6) 0.18413609766122982
This isn't a general rule; expanding the polynomial does not always result in an expression that is better suited for numerical evaluation. By far, the best way to do numerical evaluation is to make one or more of the function arguments floating point numbers. By doing that, specialized floating point algorithms are used for evaluation.
Maxima's float function is somewhat indiscriminate; if you apply
float to an expression involving an orthogonal polynomial with a
symbolic degree or order parameter, these parameters may be
converted into floats; after that, the expression will not evaluate
fully. Consider
(%i1) assoc_legendre_p (n, 1, x);
1
(%o1) P (x)
n
(%i2) float (%);
1.0
(%o2) P (x)
n
(%i3) ev (%, n=2, x=0.9);
1.0
(%o3) P (0.9)
2
The expression in (%o3) will not evaluate to a float; orthopoly doesn't
recognize floating point values where it requires an integer. Similarly,
numerical evaluation of the pochhammer function for orders that
exceed pochhammer_max_index can be troublesome; consider
(%i1) x : pochhammer (1, 10), pochhammer_max_index : 5;
(%o1) (1)
10
Applying float doesn't evaluate x to a float
(%i2) float (x);
(%o2) (1.0)
10.0
To evaluate x to a float, you'll need to bind
pochhammer_max_index to 11 or greater and apply float to x.
(%i3) float (x), pochhammer_max_index : 11; (%o3) 3628800.0
The default value of pochhammer_max_index is 100;
change its value after loading orthopoly.
Finally, be aware that reference books vary on the definitions of the orthogonal polynomials; we've generally used the conventions of Abramowitz and Stegun.
Before you suspect a bug in orthopoly, check some special cases
to determine if your definitions match those used by orthopoly.
Definitions often differ by a normalization; occasionally, authors
use "shifted" versions of the functions that makes the family
orthogonal on an interval other than (-1, 1). To define, for example,
a Legendre polynomial that is orthogonal on (0, 1), define
(%i1) shifted_legendre_p (n, x) := legendre_p (n, 2*x - 1)$
(%i2) shifted_legendre_p (2, rat (x));
2
(%o2)/R/ 6 x - 6 x + 1
(%i3) legendre_p (2, rat (x));
2
3 x - 1
(%o3)/R/ --------
2
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Most functions in orthopoly use a running error analysis to
estimate the error in floating point evaluation; the
exceptions are the spherical Bessel functions and the associated Legendre
polynomials of the second kind. For numerical evaluation, the spherical
Bessel functions call SLATEC functions. No specialized method is used
for numerical evaluation of the associated Legendre polynomials of the
second kind.
The running error analysis ignores errors that are second or higher order in the machine epsilon (also known as unit roundoff). It also ignores a few other errors. It's possible (although unlikely) that the actual error exceeds the estimate.
Intervals have the form interval (c, r), where c is the
center of the interval and r is its radius. The
center of an interval can be a complex number, and the radius is always a positive real number.
Here is an example.
(%i1) fpprec : 50$ (%i2) y0 : jacobi_p (100, 2, 3, 0.2); (%o2) interval(0.1841360913516871, 6.8990300925815987E-12) (%i3) y1 : bfloat (jacobi_p (100, 2, 3, 1/5)); (%o3) 1.8413609135168563091370224958913493690868904463668b-1
Let's test that the actual error is smaller than the error estimate
(%i4) is (abs (part (y0, 1) - y1) < part (y0, 2)); (%o4) true
Indeed, for this example the error estimate is an upper bound for the true error.
Maxima does not support arithmetic on intervals.
(%i1) legendre_p (7, 0.1) + legendre_p (8, 0.1);
(%o1) interval(0.18032072148437508, 3.1477135311021797E-15)
+ interval(- 0.19949294375000004, 3.3769353084291579E-15)
A user could define arithmetic operators that do interval math. To define interval addition, we can define
(%i1) infix ("@+")$
(%i2) "@+"(x,y) := interval (part (x, 1) + part (y, 1), part (x, 2)
+ part (y, 2))$
(%i3) legendre_p (7, 0.1) @+ legendre_p (8, 0.1);
(%o3) interval(- 0.019172222265624955, 6.5246488395313372E-15)
The special floating point routines get called when the arguments are complex. For example,
(%i1) legendre_p (10, 2 + 3.0*%i);
(%o1) interval(- 3.876378825E+7 %i - 6.0787748E+7,
1.2089173052721777E-6)
Let's compare this to the true value.
(%i1) float (expand (legendre_p (10, 2 + 3*%i))); (%o1) - 3.876378825E+7 %i - 6.0787748E+7
Additionally, when the arguments are big floats, the special floating point routines get called; however, the big floats are converted into double floats and the final result is a double.
(%i1) ultraspherical (150, 0.5b0, 0.9b0); (%o1) interval(- 0.043009481257265, 3.3750051301228864E-14)
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orthopoly To plot expressions that involve the orthogonal polynomials, you must do two things:
orthopoly_returns_intervals to false,
orthopoly functions.
If function calls aren't quoted, Maxima evaluates them to polynomials before plotting; consequently, the specialized floating point code doesn't get called. Here is an example of how to plot an expression that involves a Legendre polynomial.
(%i1) plot2d ('(legendre_p (5, x)), [x, 0, 1]),
orthopoly_returns_intervals : false;
(%o1)
The entire expression legendre_p (5, x) is quoted; this is
different than just quoting the function name using 'legendre_p (5, x).
Categories: Plotting
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The orthopoly package defines the
Pochhammer symbol and a unit step function. orthopoly uses
the Kronecker delta function and the unit step function in
gradef statements.
To convert Pochhammer symbols into quotients of gamma functions,
use makegamma.
(%i1) makegamma (pochhammer (x, n));
gamma(x + n)
(%o1) ------------
gamma(x)
(%i2) makegamma (pochhammer (1/2, 1/2));
1
(%o2) ---------
sqrt(%pi)
Derivatives of the Pochhammer symbol are given in terms of the psi
function.
(%i1) diff (pochhammer (x, n), x);
(%o1) (x) (psi (x + n) - psi (x))
n 0 0
(%i2) diff (pochhammer (x, n), n);
(%o2) (x) psi (x + n)
n 0
You need to be careful with the expression in (%o1); the difference of the
psi functions has polynomials when x = -1, -2, .., -n. These polynomials
cancel with factors in pochhammer (x, n) making the derivative a degree
n - 1 polynomial when n is a positive integer.
The Pochhammer symbol is defined for negative orders through its representation as a quotient of gamma functions. Consider
(%i1) q : makegamma (pochhammer (x, n));
gamma(x + n)
(%o1) ------------
gamma(x)
(%i2) sublis ([x=11/3, n= -6], q);
729
(%o2) - ----
2240
Alternatively, we can get this result directly.
(%i1) pochhammer (11/3, -6);
729
(%o1) - ----
2240
The unit step function is left-continuous; thus
(%i1) [unit_step (-1/10), unit_step (0), unit_step (1/10)]; (%o1) [0, 0, 1]
If you need a unit step function that is neither left or right continuous
at zero, define your own using signum; for example,
(%i1) xunit_step (x) := (1 + signum (x))/2$
(%i2) [xunit_step (-1/10), xunit_step (0), xunit_step (1/10)];
1
(%o2) [0, -, 1]
2
Do not redefine unit_step itself; some code in orthopoly
requires that the unit step function be left-continuous.
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Generally, orthopoly does symbolic evaluation by using a hypergeometic
representation of the orthogonal polynomials. The hypergeometic
functions are evaluated using the (undocumented) functions hypergeo11
and hypergeo21. The exceptions are the half-integer Bessel functions
and the associated Legendre function of the second kind. The half-integer Bessel functions are
evaluated using an explicit representation, and the associated Legendre
function of the second kind is evaluated using recursion.
For floating point evaluation, we again convert most functions into a hypergeometic form; we evaluate the hypergeometic functions using forward recursion. Again, the exceptions are the half-integer Bessel functions and the associated Legendre function of the second kind. Numerically, the half-integer Bessel functions are evaluated using the SLATEC code.
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The associated Legendre function of the first kind of degree n and order m.
Reference: Abramowitz and Stegun, equations 22.5.37, page 779, 8.6.6 (second equation), page 334, and 8.2.5, page 333.
Categories: Package orthopoly
The associated Legendre function of the second kind of degree n and order m.
Reference: Abramowitz and Stegun, equation 8.5.3 and 8.1.8.
Categories: Package orthopoly
The Chebyshev polynomial of the first kind of degree n.
Reference: Abramowitz and Stegun, equation 22.5.47, page 779.
Categories: Package orthopoly
The Chebyshev polynomial of the second kind of degree n.
Reference: Abramowitz and Stegun, equation 22.5.48, page 779.
Categories: Package orthopoly
The generalized Laguerre polynomial of degree n.
Reference: Abramowitz and Stegun, equation 22.5.54, page 780.
Categories: Package orthopoly
The Hermite polynomial of degree n.
Reference: Abramowitz and Stegun, equation 22.5.55, page 780.
Categories: Package orthopoly
Return true if the input is an interval and return false if it isn't.
Categories: Package orthopoly · Predicate functions
The Jacobi polynomial.
The Jacobi polynomials are actually defined for all
a and b; however, the Jacobi polynomial
weight (1 - x)^a (1 + x)^b isn't integrable for a <= -1 or
b <= -1.
Reference: Abramowitz and Stegun, equation 22.5.42, page 779.
Categories: Package orthopoly
The Laguerre polynomial of degree n.
Reference: Abramowitz and Stegun, equations 22.5.16 and 22.5.54, page 780.
Categories: Package orthopoly
The Legendre polynomial of the first kind of degree n.
Reference: Abramowitz and Stegun, equations 22.5.50 and 22.5.51, page 779.
Categories: Package orthopoly
The Legendre function of the second kind of degree n.
Reference: Abramowitz and Stegun, equations 8.5.3 and 8.1.8.
Categories: Package orthopoly
Returns a recursion relation for the orthogonal function family f with arguments args. The recursion is with respect to the polynomial degree.
(%i1) orthopoly_recur (legendre_p, [n, x]);
(2 n + 1) P (x) x - n P (x)
n n - 1
(%o1) P (x) = -------------------------------
n + 1 n + 1
The second argument to orthopoly_recur must be a list with the
correct number of arguments for the function f; if it isn't,
Maxima signals an error.
(%i1) orthopoly_recur (jacobi_p, [n, x]); Function jacobi_p needs 4 arguments, instead it received 2 -- an error. Quitting. To debug this try debugmode(true);
Additionally, when f isn't the name of one of the families of orthogonal polynomials, an error is signalled.
(%i1) orthopoly_recur (foo, [n, x]); A recursion relation for foo isn't known to Maxima -- an error. Quitting. To debug this try debugmode(true);
Categories: Package orthopoly
Default value: true
When orthopoly_returns_intervals is true, floating point results are returned in
the form interval (c, r), where c is the center of an interval
and r is its radius. The center can be a complex number; in that
case, the interval is a disk in the complex plane.
Categories: Package orthopoly
Returns a three element list; the first element is the formula of the weight for the orthogonal polynomial family f with arguments given by the list args; the second and third elements give the lower and upper endpoints of the interval of orthogonality. For example,
(%i1) w : orthopoly_weight (hermite, [n, x]);
2
- x
(%o1) [%e , - inf, inf]
(%i2) integrate(w[1]*hermite(3, x)*hermite(2, x), x, w[2], w[3]);
(%o2) 0
The main variable of f must be a symbol; if it isn't, Maxima signals an error.
Categories: Package orthopoly
The Pochhammer symbol. For nonnegative integers n with
n <= pochhammer_max_index, the expression pochhammer (x, n)
evaluates to the product x (x + 1) (x + 2) ... (x + n - 1)
when n > 0 and
to 1 when n = 0. For negative n,
pochhammer (x, n) is defined as (-1)^n / pochhammer (1 - x, -n).
Thus
(%i1) pochhammer (x, 3);
(%o1) x (x + 1) (x + 2)
(%i2) pochhammer (x, -3);
1
(%o2) - -----------------------
(1 - x) (2 - x) (3 - x)
To convert a Pochhammer symbol into a quotient of gamma functions,
(see Abramowitz and Stegun, equation 6.1.22) use makegamma; for example
(%i1) makegamma (pochhammer (x, n));
gamma(x + n)
(%o1) ------------
gamma(x)
When n exceeds pochhammer_max_index or when n
is symbolic, pochhammer returns a noun form.
(%i1) pochhammer (x, n);
(%o1) (x)
n
Categories: Package orthopoly · Gamma and factorial functions
Default value: 100
pochhammer (n, x) expands to a product if and only if
n <= pochhammer_max_index.
Examples:
(%i1) pochhammer (x, 3), pochhammer_max_index : 3;
(%o1) x (x + 1) (x + 2)
(%i2) pochhammer (x, 4), pochhammer_max_index : 3;
(%o2) (x)
4
Reference: Abramowitz and Stegun, equation 6.1.16, page 256.
Categories: Package orthopoly · Gamma and factorial functions
The spherical Bessel function of the first kind.
Reference: Abramowitz and Stegun, equations 10.1.8, page 437 and 10.1.15, page 439.
Categories: Package orthopoly · Bessel functions
The spherical Bessel function of the second kind.
Reference: Abramowitz and Stegun, equations 10.1.9, page 437 and 10.1.15, page 439.
Categories: Package orthopoly · Bessel functions
The spherical Hankel function of the first kind.
Reference: Abramowitz and Stegun, equation 10.1.36, page 439.
Categories: Package orthopoly · Bessel functions
The spherical Hankel function of the second kind.
Reference: Abramowitz and Stegun, equation 10.1.17, page 439.
Categories: Package orthopoly · Bessel functions
The spherical harmonic function.
Reference: Merzbacher 9.64.
Categories: Package orthopoly
The left-continuous unit step function; thus
unit_step (x) vanishes for x <= 0 and equals
1 for x > 0.
If you want a unit step function that
takes on the value 1/2 at zero, use (1 + signum (x))/2.
Categories: Package orthopoly · Mathematical functions
The ultraspherical polynomial (also known as the Gegenbauer polynomial).
Reference: Abramowitz and Stegun, equation 22.5.46, page 779.
Categories: Package orthopoly
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The package ratpow provides functions that find the exponents
of the denominator in a CRE polynomial. If the exponents in the denominator
are needed instead ratdenom can be used to extract this denominator
first. Returned coefficients are in CRE form except for numbers.
In order to get a list of vars in a CRE polynomial showratvars can be
used.
For information about CREs see also rat, ratdisrep and
showratvars.
| 77.1 Functions and Variables for ratpow |
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Finds the highest power of the main variable in ratnumer(expr)
(%i1) load("ratpow")$
(%i2) ratp_hipow( x^(5/2) + x^2 , x);
(%o2) 2
(%i3) ratp_hipow( x^(5/2) + x^2 , sqrt(x));
(%o3) 5
Categories: Rational expressions Package ratpow
Finds the lowest power of the main variable in ratnumer(expr)
(%i1) load("ratpow")$
(%i2) ratp_lopow( x^5 + x^2 , x);
(%o2) 2
The following example will return 0 since 1 equals x^0:
(%i1) load("ratpow")$
(%i2) ratp_lopow( x^5 + x^2 + 1, x);
(%o2) 0
The CRE form of the following equation contains sqrt(x) and
x. Since they are interpreted as independent variables
ratp_lopow returns 0 in this case:
(%i1) load("ratpow")$
(%i2) g:sqrt(x)^5 + sqrt(x)^2;
5/2
(%o2) x + x
(%i3) showratvars(g);
1/2
(%o3) [x , x]
(%i4) ratp_lopow( g, x);
(%o4) 0
(%i5) ratp_lopow( g, sqrt(x));
(%o5) 0
Categories: Rational expressions Package ratpow
Generates a list of powers and coefficients of the main variable
ratnumer(expr).
(%i1) load("ratpow")$
(%i2) ratp_coeffs( 4*x^3 + x + sqrt(x), x);
(%o2)/R/ [[3, 4], [1, 1], [0, sqrt(x)]]
Categories: Rational expressions Package ratpow
Generates a list of coefficients in ratnumer(expr); returned
coefficients are in CRE form except for numbers.
(%i1) load("ratpow")$
(%i2) ratp_dense_coeffs( 4*x^3 + x + sqrt(x), x);
(%o2)/R/ [4, 0, 1, sqrt(x)]
Categories: Rational expressions Package ratpow
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| 78.1 Functions and Variables for romberg |
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Computes a numerical integration by Romberg's method.
romberg(expr, x, a, b)
returns an estimate of the integral integrate(expr, x, a, b).
expr must be an expression which evaluates to a floating point value
when x is bound to a floating point value.
romberg(F, a, b)
returns an estimate of the integral integrate(F(x), x, a, b)
where x represents the unnamed, sole argument of F;
the actual argument is not named x.
F must be a Maxima or Lisp function which returns a floating point value
when the argument is a floating point value.
F may name a translated or compiled Maxima function.
The accuracy of romberg is governed by the global variables
rombergabs and rombergtol.
romberg terminates successfully when
the absolute difference between successive approximations is less than rombergabs,
or the relative difference in successive approximations is less than rombergtol.
Thus when rombergabs is 0.0 (the default)
only the relative error test has any effect on romberg.
romberg halves the stepsize at most rombergit times before it gives up;
the maximum number of function evaluations is therefore 2^rombergit.
If the error criterion established by rombergabs and rombergtol
is not satisfied, romberg prints an error message.
romberg always makes at least rombergmin iterations;
this is a heuristic intended to prevent spurious termination when the integrand is oscillatory.
romberg repeatedly evaluates the integrand after binding the variable
of integration to a specific value (and not before).
This evaluation policy makes it possible to nest calls to romberg,
to compute multidimensional integrals.
However, the error calculations do not take the errors of nested integrations
into account, so errors may be underestimated.
Also, methods devised especially for multidimensional problems may yield
the same accuracy with fewer function evaluations.
load(romberg) loads this function.
See also QUADPACK, a collection of numerical integration functions.
Examples:
A 1-dimensional integration.
(%i1) load (romberg);
(%o1) /usr/share/maxima/5.11.0/share/numeric/romberg.lisp
(%i2) f(x) := 1/((x - 1)^2 + 1/100) + 1/((x - 2)^2 + 1/1000)
+ 1/((x - 3)^2 + 1/200);
1 1 1
(%o2) f(x) := -------------- + --------------- + --------------
2 1 2 1 2 1
(x - 1) + --- (x - 2) + ---- (x - 3) + ---
100 1000 200
(%i3) rombergtol : 1e-6;
(%o3) 9.9999999999999995E-7
(%i4) rombergit : 15;
(%o4) 15
(%i5) estimate : romberg (f(x), x, -5, 5);
(%o5) 173.6730736617464
(%i6) exact : integrate (f(x), x, -5, 5);
(%o6) 10 sqrt(10) atan(70 sqrt(10))
+ 10 sqrt(10) atan(30 sqrt(10)) + 10 sqrt(2) atan(80 sqrt(2))
+ 10 sqrt(2) atan(20 sqrt(2)) + 10 atan(60) + 10 atan(40)
(%i7) abs (estimate - exact) / exact, numer;
(%o7) 7.5527060865060088E-11
A 2-dimensional integration, implemented by nested calls to romberg.
(%i1) load (romberg);
(%o1) /usr/share/maxima/5.11.0/share/numeric/romberg.lisp
(%i2) g(x, y) := x*y / (x + y);
x y
(%o2) g(x, y) := -----
x + y
(%i3) rombergtol : 1e-6;
(%o3) 9.9999999999999995E-7
(%i4) estimate : romberg (romberg (g(x, y), y, 0, x/2), x, 1, 3);
(%o4) 0.81930239628356
(%i5) assume (x > 0);
(%o5) [x > 0]
(%i6) integrate (integrate (g(x, y), y, 0, x/2), x, 1, 3);
3
2 log(-) - 1
9 2 9
(%o6) - 9 log(-) + 9 log(3) + ------------ + -
2 6 2
(%i7) exact : radcan (%);
26 log(3) - 26 log(2) - 13
(%o7) - --------------------------
3
(%i8) abs (estimate - exact) / exact, numer;
(%o8) 1.3711979871851024E-10
Categories: Package romberg · Numerical methods
Default value: 0.0
The accuracy of romberg is governed by the global variables
rombergabs and rombergtol.
romberg terminates successfully when
the absolute difference between successive approximations is less than rombergabs,
or the relative difference in successive approximations is less than rombergtol.
Thus when rombergabs is 0.0 (the default)
only the relative error test has any effect on romberg.
See also rombergit and rombergmin.
Categories: Package romberg
Default value: 11
romberg halves the stepsize at most rombergit times before it gives up;
the maximum number of function evaluations is therefore 2^rombergit.
romberg always makes at least rombergmin iterations;
this is a heuristic intended to prevent spurious termination when the integrand is oscillatory.
See also rombergabs and rombergtol.
Categories: Package romberg
Default value: 0
romberg always makes at least rombergmin iterations;
this is a heuristic intended to prevent spurious termination when the integrand is oscillatory.
See also rombergit, rombergabs, and rombergtol.
Categories: Package romberg
Default value: 1e-4
The accuracy of romberg is governed by the global variables
rombergabs and rombergtol.
romberg terminates successfully when
the absolute difference between successive approximations is less than rombergabs,
or the relative difference in successive approximations is less than rombergtol.
Thus when rombergabs is 0.0 (the default)
only the relative error test has any effect on romberg.
See also rombergit and rombergmin.
Categories: Package romberg
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| 79.1 Introduction to simplex | ||
| 79.2 Functions and Variables for simplex |
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simplex is a package for linear optimization using the simplex algorithm.
Example:
(%i1) load("simplex")$
(%i2) minimize_lp(x+y, [3*x+2*y>2, x+4*y>3]);
9 7 1
(%o2) [--, [y = --, x = -]]
10 10 5
Categories: Numerical methods · Optimization · Share packages · Package simplex
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There are some tests in the directory share/simplex/Tests.
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The function klee_minty produces input for linear_program, for which
exponential time for solving is required without scaling.
Example:
load(klee_minty)$ apply(linear_program, klee_minty(6));
A better approach:
epsilon_sx : 0$ scale_sx : true$ apply(linear_program, klee_minty(10));
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Some smaller problems from netlib (http://www.netlib.org/lp/data/) test
suite are converted to a format, readable by Maxima. Problems are adlittle,
afiro, kb2 and sc50a. Each problem has three input files in CSV format for
matrix A and vectors b and c.
Example:
A : read_matrix("adlittle_A.csv", 'csv)$
b : read_list("adlittle_b.csv", 'csv)$
c : read_list("adlittle_c.csv", 'csv)$
linear_program(A, b, c)$
%[2]
=> 225494.963126615
Results:
PROBLEM MINIMUM SCALING adlittle 225494.963126615 no afiro - 464.7531428571429 no kb2 - 1749.900129055996 yes sc50a - 64.5750770585645 no
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Default value: 10^-8
Epsilon used for numerical computations in linear_program.
See also: linear_program.
Categories: Package simplex
linear_program is an implementation of the simplex algorithm.
linear_program(A, b, c) computes a vector x for which
c.x is minimum possible among vectors for which A.x = b
and x >= 0. Argument A is a matrix and arguments b
and c are lists.
linear_program returns a list which contains the minimizing
vector x and the minimum value c.x. If the problem is not
bounded, it returns "Problem not bounded!" and if the problem is not
feasible, it returns "Problem not feasible!".
To use this function first load the simplex package with
load(simplex);.
Example:
(%i2) A: matrix([1,1,-1,0], [2,-3,0,-1], [4,-5,0,0])$
(%i3) b: [1,1,6]$
(%i4) c: [1,-2,0,0]$
(%i5) linear_program(A, b, c);
13 19 3
(%o5) [[--, 4, --, 0], - -]
2 2 2
See also: minimize_lp, scale_lp, and epsilon_lp.
Categories: Package simplex · Numerical methods
Maximizes linear objective function obj subject to some linear
constraints cond. See minimize_lp for detailed
description of arguments and return value.
See also: minimize_lp.
Categories: Package simplex · Numerical methods
Minimizes a linear objective function obj subject to some linear
constraints cond. cond a list of linear equations or
inequalities. In strict inequalities > is replaced by >=
and < by <=. The optional argument pos is a list
of decision variables which are assumed to be positive.
If the minimum exists, minimize_lp returns a list which
contains the minimum value of the objective function and a list of
decision variable values for which the minimum is attained. If the
problem is not bounded, minimize_lp returns "Problem not
bounded!" and if the problem is not feasible, it returns "Problem not
feasible!".
The decision variables are not assumed to be nonegative by default. If
all decision variables are nonegative, set nonegative_lp to
true. If only some of decision variables are positive, list
them in the optional argument pos (note that this is more
efficient than adding constraints).
minimize_lp uses the simplex algorithm which is implemented in
maxima linear_program function.
To use this function first load the simplex package with
load(simplex);.
Examples:
(%i1) minimize_lp(x+y, [3*x+y=0, x+2*y>2]);
4 6 2
(%o1) [-, [y = -, x = - -]]
5 5 5
(%i2) minimize_lp(x+y, [3*x+y>0, x+2*y>2]), nonegative_lp=true;
(%o2) [1, [y = 1, x = 0]]
(%i3) minimize_lp(x+y, [3*x+y=0, x+2*y>2]), nonegative_lp=true;
(%o3) Problem not feasible!
(%i4) minimize_lp(x+y, [3*x+y>0]);
(%o4) Problem not bounded!
See also: maximize_lp, nonegative_lp, epsilon_lp.
Categories: Package simplex · Numerical methods
Default value: false
If nonegative_lp is true all decision variables to
minimize_lp and maximize_lp are assumed to be positive.
See also: minimize_lp.
Categories: Package simplex
Default value: false
When scale_lp is true,
linear_program scales its input so that the maximum absolute value in each row or column is 1.
Categories: Package simplex
After linear_program returns,
pivot_count_sx is the number of pivots in last computation.
Categories: Package simplex
pivot_max_sx is the maximum number of pivots allowed by linear_program.
Categories: Package simplex
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| 80.1 Introduction to simplification | ||
| 80.2 Package absimp | ||
| 80.3 Package facexp | ||
| 80.4 Package functs | ||
| 80.5 Package ineq | ||
| 80.6 Package rducon | ||
| 80.7 Package scifac | ||
| 80.8 Package sqdnst |
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The directory maxima/share/simplification contains several scripts
which implement simplification rules and functions,
and also some functions not related to simplification.
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The absimp package contains pattern-matching rules that
extend the built-in simplification rules for the abs and signum
functions.
absimp respects relations
established with the built-in assume function and by declarations such
as modedeclare (m, even, n, odd) for even or odd integers.
absimp defines unitramp and unitstep functions
in terms of abs and signum.
load (absimp) loads this package.
demo (absimp) shows a demonstration of this package.
Examples:
(%i1) load (absimp)$
(%i2) (abs (x))^2;
2
(%o2) x
(%i3) diff (abs (x), x);
x
(%o3) ------
abs(x)
(%i4) cosh (abs (x));
(%o4) cosh(x)
Categories: Simplification functions · Rules and patterns · Share packages · Package absimp
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The facexp package contains several related functions that
provide the user with the ability to structure expressions by controlled
expansion. This capability is especially useful when the expression
contains variables that have physical meaning, because it is often true
that the most economical form of such an expression can be obtained by
fully expanding the expression with respect to those variables, and then
factoring their coefficients. While it is true that this procedure is
not difficult to carry out using standard Maxima functions, additional
fine-tuning may also be desirable, and these finishing touches can be
more difficult to apply.
The function facsum and its related forms
provide a convenient means for controlling the structure of expressions
in this way. Another function, collectterms, can be used to add two or
more expressions that have already been simplified to this form, without
resimplifying the whole expression again. This function may be
useful when the expressions are very large.
load (facexp) loads this package.
demo (facexp) shows a demonstration of this package.
Categories: Expressions · Share packages · Package facexp
Returns a form of expr which depends on the
arguments arg_1, ..., arg_n.
The arguments can be any form suitable for ratvars, or they can be
lists of such forms. If the arguments are not lists, then the form
returned is fully expanded with respect to the arguments, and the
coefficients of the arguments are factored. These coefficients are
free of the arguments, except perhaps in a non-rational sense.
If any of the arguments are lists, then all such lists are combined
into a single list, and instead of calling factor on the
coefficients of the arguments, facsum calls itself on these
coefficients, using this newly constructed single list as the new
argument list for this recursive call. This process can be repeated to
arbitrary depth by nesting the desired elements in lists.
It is possible that one may wish to facsum with respect to more
complicated subexpressions, such as log (x + y). Such arguments are
also permissible.
Occasionally the user may wish to obtain any of the above forms
for expressions which are specified only by their leading operators.
For example, one may wish to facsum with respect to all log's. In
this situation, one may include among the arguments either the specific
log's which are to be treated in this way, or alternatively, either
the expression operator (log) or 'operator (log). If one wished to
facsum the expression expr with respect to the operators op_1, ..., op_n,
one would evaluate facsum (expr, operator (op_1, ..., op_n)).
The operator form may also appear inside list arguments.
In addition, the setting of the switches facsum_combine and
nextlayerfactor may affect the result of facsum.
Categories: Package facexp · Expressions
Default value: false
When nextlayerfactor is true, recursive calls of facsum
are applied to the factors of the factored form of the
coefficients of the arguments.
When false, facsum is applied to
each coefficient as a whole whenever recusive calls to facsum occur.
Inclusion of the atom
nextlayerfactor in the argument list of facsum has the effect of
nextlayerfactor: true, but for the next level of the expression only.
Since nextlayerfactor is always bound to either true or false, it
must be presented single-quoted whenever it appears in the argument list of facsum.
Categories: Package facexp · Expressions
Default value: true
facsum_combine controls the form of the final result returned by
facsum when its argument is a quotient of polynomials. If
facsum_combine is false then the form will be returned as a fully
expanded sum as described above, but if true, then the expression
returned is a ratio of polynomials, with each polynomial in the form
described above.
The true setting of this switch is useful when one
wants to facsum both the numerator and denominator of a rational
expression, but does not want the denominator to be multiplied
through the terms of the numerator.
Categories: Package facexp · Expressions
Returns a form of expr which is
obtained by calling facsum on the factors of expr with arg_1, ... arg_n as
arguments. If any of the factors of expr is raised to a power, both
the factor and the exponent will be processed in this way.
Categories: Package facexp · Expressions
If several expressions have been simplified with the following functions
facsum, factorfacsum, factenexpand, facexpten or
factorfacexpten, and they are to be added together, it may be desirable
to combine them using the function collecterms. collecterms can
take as arguments all of the arguments that can be given to these other
associated functions with the exception of nextlayerfactor, which has no
effect on collectterms. The advantage of collectterms is that it
returns a form similar to facsum, but since it is adding forms that have
already been processed by facsum, it does not need to repeat that effort.
This capability is especially useful when the expressions to be summed are very
large.
Categories: Package facexp · Expressions
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Categories: Share packages · Package functs
Removes part n from the expression expr.
If n is a list of the form [l, m]
then parts l thru m are removed.
To use this function write first load(functs).
Categories: Package functs · Expressions
Returns the Wronskian matrix of the list of expressions [f_1, ..., f_n] in the variable x. The determinant of the Wronskian matrix is the Wronskian determinant of the list of expressions.
To use wronskian, first load(functs). Example:
(%i1) load(functs)$ (%i2) wronskian([f(x), g(x)],x); (%o2) matrix([f(x),g(x)],['diff(f(x),x,1),'diff(g(x),x,1)])
Categories: Package functs · Differential calculus
Returns the trace (sum of the diagonal elements) of matrix M.
To use this function write first load(functs).
Categories: Package functs · Matrices
Multiplies numerator and denominator of z by the complex conjugate of denominator, thus rationalizing the denominator. Returns canonical rational expression (CRE) form if given one, else returns general form.
To use this function write first load(functs).
Categories: Package functs · Expressions
Returns true if expr is nonzero and freeof (x, expr) returns true.
Returns false otherwise.
To use this function write first load(functs).
Categories: Package functs · Expressions
When expr is an expression of the form a*x + b
where a is nonzero, and a and b are free of x,
linear returns a list of three equations, one for each of the three formal
variables b, a, and x. Otherwise, linear returns false.
load(antid) loads this function.
Example:
(%i1) load (antid); (%o1) /usr/share/maxima/5.29.1/share/integration/antid.mac (%i2) linear ((1 - w)*(1 - x)*z, z); (%o2) [bargumentb = 0, aargumenta = (w - 1) x - w + 1, xargumentx = z] (%i3) linear (cos(u - v) + cos(u + v), u); (%o3) false
Categories: Package antid · Expressions
When the option variable takegcd is true which is the default,
gcdivide divides the polynomials p and q by their greatest
common divisor and returns the ratio of the results. gcdivde calls the
function ezgcd to divide the polynomials by the greatest common divisor.
When takegcd is false, gcdivide returns the ratio
p/q.
To use this function write first load(functs).
See also ezgcd, gcd, gcdex, and
poly_gcd.
Example:
(%i1) load(functs)$
(%i2) p1:6*x^3+19*x^2+19*x+6;
3 2
(%o2) 6 x + 19 x + 19 x + 6
(%i3) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
5 4 3 2
(%o3) 6 x + 13 x + 12 x + 13 x + 6 x
(%i4) gcdivide(p1, p2);
x + 1
(%o4) ------
3
x + x
(%i5) takegcd:false;
(%o5) false
(%i6) gcdivide(p1, p2);
3 2
6 x + 19 x + 19 x + 6
(%o6) ----------------------------------
5 4 3 2
6 x + 13 x + 12 x + 13 x + 6 x
(%i7) ratsimp(%);
x + 1
(%o7) ------
3
x + x
Categories: Package functs · Polynomials
Returns the n-th term of the arithmetic series
a, a + d, a + 2*d, ..., a + (n - 1)*d.
To use this function write first load(functs).
Categories: Package functs · Sums and products
Returns the n-th term of the geometric series
a, a*r, a*r^2, ..., a*r^(n - 1).
To use this function write first load(functs).
Categories: Package functs · Sums and products
Returns the n-th term of the harmonic series
a/b, a/(b + c), a/(b + 2*c), ..., a/(b + (n - 1)*c).
To use this function write first load(functs).
Categories: Package functs · Sums and products
Returns the sum of the arithmetic series from 1 to n.
To use this function write first load(functs).
Categories: Package functs · Sums and products
Returns the sum of the geometric series from 1 to n. If n is
infinity (inf) then a sum is finite only if the absolute value
of r is less than 1.
To use this function write first load(functs).
Categories: Package functs · Sums and products
Returns the Gaussian probability function
%e^(-x^2/2) / sqrt(2*%pi).
To use this function write first load(functs).
Categories: Package functs · Mathematical functions
Returns the Gudermannian function
2*atan(%e^x)-%pi/2.
To use this function write first load(functs).
Categories: Package functs · Mathematical functions
Returns the inverse Gudermannian function
log (tan (%pi/4 + x/2)).
To use this function write first load(functs).
Categories: Package functs · Mathematical functions
Returns the versed sine 1 - cos (x).
To use this function write first load(functs).
Categories: Package functs · Mathematical functions
Returns the coversed sine 1 - sin (x).
To use this function write first load(functs).
Categories: Package functs · Mathematical functions
Returns the exsecant sec (x) - 1.
To use this function write first load(functs).
Categories: Package functs · Mathematical functions
Returns the haversine (1 - cos(x))/2.
To use this function write first load(functs).
Categories: Package functs · Mathematical functions
Returns the number of combinations of n objects taken r at a time.
To use this function write first load(functs).
Categories: Package functs · Mathematical functions
Returns the number of permutations of r objects selected from a set of n objects.
To use this function write first load(functs).
Categories: Package functs · Mathematical functions
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The ineq package contains simplification rules
for inequalities.
Example session:
(%i1) load(ineq)$
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
Warning: Putting rules on '+' or '*' is inefficient, and may not work.
(%i2) a>=4; /* a sample inequality */
(%o2) a >= 4
(%i3) (b>c)+%; /* add a second, strict inequality */
(%o3) b + a > c + 4
(%i4) 7*(x<y); /* multiply by a positive number */
(%o4) 7 x < 7 y
(%i5) -2*(x>=3*z); /* multiply by a negative number */
(%o5) - 2 x <= - 6 z
(%i6) (1+a^2)*(1/(1+a^2)<=1); /* Maxima knows that 1+a^2 > 0 */
2
(%o6) 1 <= a + 1
(%i7) assume(x>0)$ x*(2<3); /* assuming x>0 */
(%o7) 2 x < 3 x
(%i8) a>=b; /* another inequality */
(%o8) a >= b
(%i9) 3+%; /* add something */
(%o9) a + 3 >= b + 3
(%i10) %-3; /* subtract it out */
(%o10) a >= b
(%i11) a>=c-b; /* yet another inequality */
(%o11) a >= c - b
(%i12) b+%; /* add b to both sides */
(%o12) b + a >= c
(%i13) %-c; /* subtract c from both sides */
(%o13) - c + b + a >= 0
(%i14) -%; /* multiply by -1 */
(%o14) c - b - a <= 0
(%i15) (z-1)^2>-2*z; /* determining truth of assertion */
2
(%o15) (z - 1) > - 2 z
(%i16) expand(%)+2*z; /* expand this and add 2*z to both sides */
2
(%o16) z + 1 > 0
(%i17) %,pred;
(%o17) true
Be careful about using parentheses
around the inequalities: when the user types in (A > B) + (C = 5) the
result is A + C > B + 5, but A > B + C = 5 is a syntax error,
and (A > B + C) = 5 is something else entirely.
Do disprule (all) to see a complete listing
of the rule definitions.
The user will be queried if Maxima is unable to decide the sign of a quantity multiplying an inequality.
The most common mis-feature is illustrated by:
(%i1) eq: a > b; (%o1) a > b (%i2) 2*eq; (%o2) 2 (a > b) (%i3) % - eq; (%o3) a > b
Another problem is 0 times an inequality; the default to have this
turn into 0 has been left alone. However, if you type
X*some_inequality and Maxima asks about the sign of X and you
respond zero (or z), the program returns X*some_inequality
and not use the information that X is 0. You should do ev (%, x: 0) in such
a case, as the database will only be used for comparison purposes
in decisions, and not for the purpose of evaluating X.
The user may note a slower response when this package is loaded, as
the simplifier is forced to examine more rules than without the
package, so you might wish to remove the rules after making use of
them. Do kill (rules) to eliminate all of the rules (including any
that you might have defined); or you may be more selective by
killing only some of them; or use remrule on a specific rule.
Note that if you load this package after defining your own
rules you will clobber your rules that have the same name. The
rules in this package are:
*rule1, ..., *rule8,
+rule1, ..., +rule18,
and you must enclose the rulename in quotes to refer to it, as
in remrule ("+", "+rule1") to specifically remove the first rule on "+"
or disprule ("*rule2") to display the definition of the second multiplicative rule.
Categories: Simplification functions · Rules and patterns · Share packages · Package ineq
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Categories: Expressions · Share packages · Package rducon
Replaces constant subexpressions of expr with
constructed constant atoms, saving the definition of all these
constructed constants in the list of equations const_eqns, and
returning the modified expr. Those parts of expr are constant which
return true when operated on by the function constantp. Hence,
before invoking reduce_consts, one should do
declare ([objects to be given the constant property], constant)$
to set up a database of the constant quantities occurring in your expressions.
If you are planning to generate Fortran output after these symbolic calculations, one of the first code sections should be the calculation of all constants. To generate this code segment, do
map ('fortran, const_eqns)$
Variables besides const_eqns which affect reduce_consts are:
const_prefix (default value: xx) is the string of characters used to prefix all
symbols generated by reduce_consts to represent constant subexpressions.
const_counter (default value: 1) is the integer index used to generate unique
symbols to represent each constant subexpression found by reduce_consts.
load (rducon) loads this function.
demo (rducon) shows a demonstration of this function.
Categories: Package rducon · Expressions
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Categories: Expressions · Share packages · Package scifac
gcfac is a factoring function that attempts to apply the same heuristics which
scientists apply in trying to make expressions simpler. gcfac is limited
to monomial-type factoring. For a sum, gcfac does the following:
Item (3) does not necessarily do an optimal job of pairwise factoring because of the combinatorially-difficult nature of finding which of all possible rearrangements of the pairs yields the most compact pair-factored result.
load (scifac) loads this function.
demo (scifac) shows a demonstration of this function.
Categories: Package scifac · Expressions
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Denests sqrt of simple, numerical, binomial surds, where possible. E.g.
(%i1) load (sqdnst)$
(%i2) sqrt(sqrt(3)/2+1)/sqrt(11*sqrt(2)-12);
sqrt(3)
sqrt(------- + 1)
2
(%o2) ---------------------
sqrt(11 sqrt(2) - 12)
(%i3) sqrtdenest(%);
sqrt(3) 1
------- + -
2 2
(%o3) -------------
1/4 3/4
3 2 - 2
Sometimes it helps to apply sqrtdenest more than once, on such as
(19601-13860 sqrt(2))^(7/4).
load (sqdnst) loads this function.
Categories: Expressions · Share packages · Package sqdnst
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| 81.1 Introduction to solve_rec | ||
| 81.2 Functions and Variables for solve_rec |
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solve_rec is a package for solving linear recurrences with polynomial
coefficients.
A demo is available with demo(solve_rec);.
Example:
(%i1) load("solve_rec")$
(%i2) solve_rec((n+4)*s[n+2] + s[n+1] - (n+1)*s[n], s[n]);
n
%k (2 n + 3) (- 1) %k
1 2
(%o2) s = -------------------- + ---------------
n (n + 1) (n + 2) (n + 1) (n + 2)
Categories: Linear recurrences · Share packages · Package solve_rec
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Reduces the order of linear recurrence rec when a particular solution sol is known. The reduced reccurence can be used to get other solutions.
Example:
(%i3) rec: x[n+2] = x[n+1] + x[n]/n;
x
n
(%o3) x = x + --
n + 2 n + 1 n
(%i4) solve_rec(rec, x[n]);
WARNING: found some hypergeometrical solutions!
(%o4) x = %k n
n 1
(%i5) reduce_order(rec, n, x[n]);
(%t5) x = n %z
n n
n - 1
====
\
(%t6) %z = > %u
n / %j
====
%j = 0
(%o6) (- n - 2) %u - %u
n + 1 n
(%i6) solve_rec((n+2)*%u[n+1] + %u[n], %u[n]);
n
%k (- 1)
1
(%o6) %u = ----------
n (n + 1)!
So the general solution is
n - 1
==== j
\ (- 1)
%k n > -------- + %k n
2 / (j + 1)! 1
====
j = 0
Categories: Package solve_rec
Default value: true
If simplify_products is true, solve_rec will try to
simplify products in result.
See also: solve_rec.
Categories: Package solve_rec
Tries to simplify all sums appearing in expr to a closed form.
To use this function first load the simplify_sum package with
load(simplify_sum).
Example:
(%i1) load("simplify_sum")$
(%i2) sum(binomial(n+k,k)/2^k,k,1,n)+sum(binomial(2*n,2*k),k,1,n);
n n
==== ====
\ binomial(n + k, k) \
(%o2) > ------------------ + > binomial(2 n, 2 k)
/ k /
==== 2 ====
k = 1 k = 1
(%i3) simplify_sum(%);
2 n - 1 n
(%o3) 2 + 2 - 2
Categories: Package solve_rec · Sums and products · Simplification functions
Solves for hypergeometrical solutions to linear recurrence eqn with polynomials coefficient in variable var. Optional arguments init are initial conditions.
solve_rec can solve linear recurrences with constant coefficients,
finds hypergeometrical solutions to homogeneous linear recurrences with
polynomial coefficients, rational solutions to linear recurrences with
polynomial coefficients and can solve Ricatti type recurrences.
Note that the running time of the algorithm used to find hypergeometrical solutions is exponential in the degree of the leading and trailing coefficient.
To use this function first load the solve_rec package with
load(solve_rec);.
Example of linear recurrence with constant coefficients:
(%i2) solve_rec(a[n]=a[n-1]+a[n-2]+n/2^n, a[n]);
n n
(sqrt(5) - 1) %k (- 1)
1 n
(%o2) a = ------------------------- - ----
n n n
2 5 2
n
(sqrt(5) + 1) %k
2 2
+ ------------------ - ----
n n
2 5 2
Example of linear recurrence with polynomial coefficients:
(%i7) 2*x*(x+1)*y[x] - (x^2+3*x-2)*y[x+1] + (x-1)*y[x+2];
2
(%o7) (x - 1) y - (x + 3 x - 2) y + 2 x (x + 1) y
x + 2 x + 1 x
(%i8) solve_rec(%, y[x], y[1]=1, y[3]=3);
x
3 2 x!
(%o9) y = ---- - --
x 4 2
Example of Ricatti type recurrence:
(%i2) x*y[x+1]*y[x] - y[x+1]/(x+2) + y[x]/(x-1) = 0;
y y
x + 1 x
(%o2) x y y - ------ + ----- = 0
x x + 1 x + 2 x - 1
(%i3) solve_rec(%, y[x], y[3]=5)$
(%i4) ratsimp(minfactorial(factcomb(%)));
3
30 x - 30 x
(%o4) y = - -------------------------------------------------
x 6 5 4 3 2
5 x - 3 x - 25 x + 15 x + 20 x - 12 x - 1584
See also: solve_rec_rat, simplify_products and product_use_gamma.
Categories: Package solve_rec
Solves for rational solutions to linear recurrences. See solve_rec for description of arguments.
To use this function first load the solve_rec package with
load(solve_rec);.
Example:
(%i1) (x+4)*a[x+3] + (x+3)*a[x+2] - x*a[x+1] + (x^2-1)*a[x];
(%o1) (x + 4) a + (x + 3) a - x a
x + 3 x + 2 x + 1
2
+ (x - 1) a
x
(%i2) solve_rec_rat(% = (x+2)/(x+1), a[x]);
1
(%o2) a = ---------------
x (x - 1) (x + 1)
See also: solve_rec.
Categories: Package solve_rec
Default value: true
When simplifying products, solve_rec introduces gamma function
into the expression if product_use_gamma is true.
See also: simplify_products, solve_rec.
Categories: Package solve_rec
Returns the recurrence sattisfied by the sum
hi
====
\
> summand
/
====
k = lo
where summand is hypergeometrical in k and n. If lo and hi
are omitted, they are assumed to be lo = -inf and hi = inf.
To use this function first load the simplify_sum package with
load(simplify_sum).
Example:
(%i1) load("simplify_sum")$
(%i2) summand: binom(n,k);
(%o2) binomial(n, k)
(%i3) summand_to_rec(summand,k,n);
(%o3) 2 sm - sm = 0
n n + 1
(%i7) summand: binom(n, k)/(k+1);
binomial(n, k)
(%o7) --------------
k + 1
(%i8) summand_to_rec(summand, [k, 0, n], n);
(%o8) 2 (n + 1) sm - (n + 2) sm = - 1
n n + 1
Categories: Package solve_rec
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| 82.1 Introduction to stats | ||
| 82.2 Functions and Variables for inference_result | ||
| 82.3 Functions and Variables for stats | ||
| 82.4 Functions and Variables for special distributions |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Package stats contains a set of classical statistical inference and
hypothesis testing procedures.
All these functions return an inference_result Maxima object which contains
the necessary results for population inferences and decision making.
Global variable stats_numer controls whether results are given in
floating point or symbolic and rational format; its default value is true
and results are returned in floating point format.
Package descriptive contains some utilities to manipulate data structures
(lists and matrices); for example, to extract subsamples. It also contains some
examples on how to use package numericalio to read data from plain text
files. See descriptive and numericalio for more details.
Package stats loads packages descriptive, distrib and
inference_result.
For comments, bugs or suggestions, please contact the author at
'mario AT edu DOT xunta DOT es'.
Categories: Statistical inference · Share packages · Package stats
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Constructs an inference_result object of the type returned by the
stats functions. Argument title is a
string with the name of the procedure; values is a list with
elements of the form symbol = value and numbers is a list
with positive integer numbers ranging from one to length(values),
indicating which values will be shown by default.
Example:
This is a simple example showing results concerning a rectangle. The title of
this object is the string "Rectangle", it stores five results, named
'base, 'height, 'diagonal, 'area,
and 'perimeter, but only the first, second, fifth, and fourth
will be displayed. The 'diagonal is stored in this object, but it is
not displayed; to access its value, make use of function take_inference.
(%i1) load(inference_result)$
(%i2) b: 3$ h: 2$
(%i3) inference_result("Rectangle",
['base=b,
'height=h,
'diagonal=sqrt(b^2+h^2),
'area=b*h,
'perimeter=2*(b+h)],
[1,2,5,4] );
| Rectangle
|
| base = 3
|
(%o3) | height = 2
|
| perimeter = 10
|
| area = 6
(%i4) take_inference('diagonal,%);
(%o4) sqrt(13)
See also take_inference.
Categories: Package stats
Returns true or false, depending on whether obj is an
inference_result object or not.
Categories: Package stats
Returns a list with the names of the items stored in obj, which must
be an inference_result object.
Example:
The inference_result object stores two values, named 'pi and 'e,
but only the second is displayed. The items_inference function returns the names
of all items, no matter they are displayed or not.
(%i1) load(inference_result)$
(%i2) inference_result("Hi", ['pi=%pi,'e=%e],[2]);
| Hi
(%o2) |
| e = %e
(%i3) items_inference(%);
(%o3) [pi, e]
Categories: Package stats
Returns the n-th value stored in obj if n is a positive integer,
or the item named name if this is the name of an item. If the first
argument is a list of numbers and/or symbols, function take_inference returns
a list with the corresponding results.
Example:
Given an inference_result object, function take_inference is
called in order to extract some information stored in it.
(%i1) load(inference_result)$
(%i2) b: 3$ h: 2$
(%i3) sol: inference_result("Rectangle",
['base=b,
'height=h,
'diagonal=sqrt(b^2+h^2),
'area=b*h,
'perimeter=2*(b+h)],
[1,2,5,4] );
| Rectangle
|
| base = 3
|
(%o3) | height = 2
|
| perimeter = 10
|
| area = 6
(%i4) take_inference('base,sol);
(%o4) 3
(%i5) take_inference(5,sol);
(%o5) 10
(%i6) take_inference([1,'diagonal],sol);
(%o6) [3, sqrt(13)]
(%i7) take_inference(items_inference(sol),sol);
(%o7) [3, 2, sqrt(13), 6, 10]
See also inference_result, and take_inference.
Categories: Package stats
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Default value: true
If stats_numer is true, inference statistical functions
return their results in floating point numbers. If it is false,
results are given in symbolic and rational format.
Categories: Package stats · Numerical evaluation
This is the mean t-test. Argument x is a list or a column matrix
containing a one dimensional sample. It also performs an asymptotic test
based on the Central Limit Theorem if option 'asymptotic is
true.
Options:
'mean, default 0, is the mean value to be checked.
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
'dev, default 'unknown, this is the value of the standard deviation when it is
known; valid values are: 'unknown or a positive expression.
'conflevel, default 95/100, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
'asymptotic, default false, indicates whether it performs an exact t-test or
an asymptotic one based on the Central Limit Theorem;
valid values are true and false.
The output of function test_mean is an inference_result Maxima object
showing the following results:
'mean_estimate: the sample mean.
'conf_level: confidence level selected by the user.
'conf_interval: confidence interval for the population mean.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameter(s).
'p_value: p-value of the test.
Examples:
Performs an exact t-test with unknown variance. The null hypothesis is H_0: mean=50 against the one sided alternative H_1: mean<50; according to the results, the p-value is too great, there are no evidence for rejecting H_0.
(%i1) load("stats")$
(%i2) data: [78,64,35,45,45,75,43,74,42,42]$
(%i3) test_mean(data,'conflevel=0.9,'alternative='less,'mean=50);
| MEAN TEST
|
| mean_estimate = 54.3
|
| conf_level = 0.9
|
| conf_interval = [minf, 61.51314273502712]
|
(%o3) | method = Exact t-test. Unknown variance.
|
| hypotheses = H0: mean = 50 , H1: mean < 50
|
| statistic = .8244705235071678
|
| distribution = [student_t, 9]
|
| p_value = .7845100411786889
This time Maxima performs an asymptotic test, based on the Central Limit Theorem.
The null hypothesis is H_0: equal(mean, 50) against the two sided alternative H_1: not equal(mean, 50);
according to the results, the p-value is very small, H_0 should be rejected in
favor of the alternative H_1. Note that, as indicated by the Method component,
this procedure should be applied to large samples.
(%i1) load("stats")$
(%i2) test_mean([36,118,52,87,35,256,56,178,57,57,89,34,25,98,35,
98,41,45,198,54,79,63,35,45,44,75,42,75,45,45,
45,51,123,54,151],
'asymptotic=true,'mean=50);
| MEAN TEST
|
| mean_estimate = 74.88571428571429
|
| conf_level = 0.95
|
| conf_interval = [57.72848600856194, 92.04294256286663]
|
(%o2) | method = Large sample z-test. Unknown variance.
|
| hypotheses = H0: mean = 50 , H1: mean # 50
|
| statistic = 2.842831192874313
|
| distribution = [normal, 0, 1]
|
| p_value = .004471474652002261
Categories: Package stats
This is the difference of means t-test for two samples.
Arguments x1 and x2 are lists or column matrices
containing two independent samples. In case of different unknown variances
(see options 'dev1, 'dev2 and 'varequal bellow),
the degrees of freedom are computed by means of the Welch approximation.
It also performs an asymptotic test
based on the Central Limit Theorem if option 'asymptotic is
set to true.
Options:
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
'dev1, default 'unknown, this is the value of the standard deviation
of the x1 sample when it is known; valid values are: 'unknown or a positive expression.
'dev2, default 'unknown, this is the value of the standard deviation
of the x2 sample when it is known; valid values are: 'unknown or a positive expression.
'varequal, default false, whether variances should be considered to be equal or not;
this option takes effect only when 'dev1 and/or 'dev2 are 'unknown.
'conflevel, default 95/100, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
'asymptotic, default false, indicates whether it performs an exact t-test or
an asymptotic one based on the Central Limit Theorem;
valid values are true and false.
The output of function test_means_difference is an inference_result Maxima object
showing the following results:
'diff_estimate: the difference of means estimate.
'conf_level: confidence level selected by the user.
'conf_interval: confidence interval for the difference of means.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameter(s).
'p_value: p-value of the test.
Examples:
The equality of means is tested with two small samples x and y, against the alternative H_1: m_1>m_2, being m_1 and m_2 the populations means; variances are unknown and supposed to be different.
(%i1) load("stats")$
(%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
(%i3) y: [1.2,6.9,38.7,20.4,17.2]$
(%i4) test_means_difference(x,y,'alternative='greater);
| DIFFERENCE OF MEANS TEST
|
| diff_estimate = 20.31999999999999
|
| conf_level = 0.95
|
| conf_interval = [- .04597417812882298, inf]
|
(%o4) | method = Exact t-test. Welch approx.
|
| hypotheses = H0: mean1 = mean2 , H1: mean1 > mean2
|
| statistic = 1.838004300728477
|
| distribution = [student_t, 8.62758740184604]
|
| p_value = .05032746527991905
The same test as before, but now variances are supposed to be equal.
(%i1) load("stats")$
(%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
(%i3) y: matrix([1.2],[6.9],[38.7],[20.4],[17.2])$
(%i4) test_means_difference(x,y,'alternative='greater,
'varequal=true);
| DIFFERENCE OF MEANS TEST
|
| diff_estimate = 20.31999999999999
|
| conf_level = 0.95
|
| conf_interval = [- .7722627696897568, inf]
|
(%o4) | method = Exact t-test. Unknown equal variances
|
| hypotheses = H0: mean1 = mean2 , H1: mean1 > mean2
|
| statistic = 1.765996124515009
|
| distribution = [student_t, 9]
|
| p_value = .05560320992529344
Categories: Package stats
This is the variance chi^2-test. Argument x is a list or a column matrix containing a one dimensional sample taken from a normal population.
Options:
'mean, default 'unknown, is the population's mean, when it is known.
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
'variance, default 1, this is the variance value (positive) to be checked.
'conflevel, default 95/100, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
The output of function test_variance is an inference_result Maxima object
showing the following results:
'var_estimate: the sample variance.
'conf_level: confidence level selected by the user.
'conf_interval: confidence interval for the population variance.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameter.
'p_value: p-value of the test.
Examples:
It is tested whether the variance of a population with unknown mean is equal to or greater than 200.
(%i1) load("stats")$
(%i2) x: [203,229,215,220,223,233,208,228,209]$
(%i3) test_variance(x,'alternative='greater,'variance=200);
| VARIANCE TEST
|
| var_estimate = 110.75
|
| conf_level = 0.95
|
| conf_interval = [57.13433376937479, inf]
|
(%o3) | method = Variance Chi-square test. Unknown mean.
|
| hypotheses = H0: var = 200 , H1: var > 200
|
| statistic = 4.43
|
| distribution = [chi2, 8]
|
| p_value = .8163948512777689
Categories: Package stats
This is the variance ratio F-test for two normal populations. Arguments x1 and x2 are lists or column matrices containing two independent samples.
Options:
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
'mean1, default 'unknown, when it is known, this is the mean of
the population from which x1 was taken.
'mean2, default 'unknown, when it is known, this is the mean of
the population from which x2 was taken.
'conflevel, default 95/100, confidence level for the confidence interval of the
ratio; it must be an expression which takes a value in (0,1).
The output of function test_variance_ratio is an inference_result Maxima object
showing the following results:
'ratio_estimate: the sample variance ratio.
'conf_level: confidence level selected by the user.
'conf_interval: confidence interval for the variance ratio.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameters.
'p_value: p-value of the test.
Examples:
The equality of the variances of two normal populations is checked against the alternative that the first is greater than the second.
(%i1) load("stats")$
(%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
(%i3) y: [1.2,6.9,38.7,20.4,17.2]$
(%i4) test_variance_ratio(x,y,'alternative='greater);
| VARIANCE RATIO TEST
|
| ratio_estimate = 2.316933391522034
|
| conf_level = 0.95
|
| conf_interval = [.3703504689507268, inf]
|
(%o4) | method = Variance ratio F-test. Unknown means.
|
| hypotheses = H0: var1 = var2 , H1: var1 > var2
|
| statistic = 2.316933391522034
|
| distribution = [f, 5, 4]
|
| p_value = .2179269692254457
Categories: Package stats
Inferences on a proportion. Argument x is the number of successes in n trials in a Bernoulli experiment with unknown probability.
Options:
'proportion, default 1/2, is the value of the proportion to be checked.
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
'conflevel, default 95/100, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
'asymptotic, default false, indicates whether it performs an exact test
based on the binomial distribution, or an asymptotic one based on the Central Limit Theorem;
valid values are true and false.
'correct, default true, indicates whether Yates correction is applied or not.
The output of function test_proportion is an inference_result Maxima object
showing the following results:
'sample_proportion: the sample proportion.
'conf_level: confidence level selected by the user.
'conf_interval: Wilson confidence interval for the proportion.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameters.
'p_value: p-value of the test.
Examples:
Performs an exact test. The null hypothesis is H_0: p=1/2 against the one sided alternative H_1: p<1/2.
(%i1) load("stats")$
(%i2) test_proportion(45, 103, alternative = less);
| PROPORTION TEST
|
| sample_proportion = .4368932038834951
|
| conf_level = 0.95
|
| conf_interval = [0, 0.522714149150231]
|
(%o2) | method = Exact binomial test.
|
| hypotheses = H0: p = 0.5 , H1: p < 0.5
|
| statistic = 45
|
| distribution = [binomial, 103, 0.5]
|
| p_value = .1184509388901454
A two sided asymptotic test. Confidence level is 99/100.
(%i1) load("stats")$
(%i2) fpprintprec:7$
(%i3) test_proportion(45, 103,
conflevel = 99/100, asymptotic=true);
| PROPORTION TEST
|
| sample_proportion = .43689
|
| conf_level = 0.99
|
| conf_interval = [.31422, .56749]
|
(%o3) | method = Asympthotic test with Yates correction.
|
| hypotheses = H0: p = 0.5 , H1: p # 0.5
|
| statistic = .43689
|
| distribution = [normal, 0.5, .048872]
|
| p_value = .19662
Categories: Package stats
Inferences on the difference of two proportions. Argument x1 is the number of successes in n1 trials in a Bernoulli experiment in the first population, and x2 and n2 are the corresponding values in the second population. Samples are independent and the test is asymptotic.
Options:
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided (p1 # p2), 'greater (p1 > p2)
and 'less (p1 < p2).
'conflevel, default 95/100, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
'correct, default true, indicates whether Yates correction is applied or not.
The output of function test_proportions_difference is an inference_result Maxima object
showing the following results:
'proportions: list with the two sample proportions.
'conf_level: confidence level selected by the user.
'conf_interval: Confidence interval for the difference of proportions p1 - p2.
'method: inference procedure and warning message in case of any of the samples sizes
is less than 10.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameters.
'p_value: p-value of the test.
Examples:
A machine produced 10 defective articles in a batch of 250.
After some maintenance work, it produces 4 defective in a batch of 150.
In order to know if the machine has improved, we test the null
hypothesis H0:p1=p2, against the alternative H0:p1>p2,
where p1 and p2 are the probabilities for one produced
article to be defective before and after maintenance. According to
the p value, there is not enough evidence to accept the alternative.
(%i1) load("stats")$
(%i2) fpprintprec:7$
(%i3) test_proportions_difference(10, 250, 4, 150,
alternative = greater);
| DIFFERENCE OF PROPORTIONS TEST
|
| proportions = [0.04, .02666667]
|
| conf_level = 0.95
|
| conf_interval = [- .02172761, 1]
|
(%o3) | method = Asymptotic test. Yates correction.
|
| hypotheses = H0: p1 = p2 , H1: p1 > p2
|
| statistic = .01333333
|
| distribution = [normal, 0, .01898069]
|
| p_value = .2411936
Exact standard deviation of the asymptotic normal distribution when the data are unknown.
(%i1) load("stats")$
(%i2) stats_numer: false$
(%i3) sol: test_proportions_difference(x1,n1,x2,n2)$
(%i4) last(take_inference('distribution,sol));
1 1 x2 + x1
(-- + --) (x2 + x1) (1 - -------)
n2 n1 n2 + n1
(%o4) sqrt(---------------------------------)
n2 + n1
Categories: Package stats
This is the non parametric sign test for the median of a continuous population. Argument x is a list or a column matrix containing a one dimensional sample.
Options:
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
'median, default 0, is the median value to be checked.
The output of function test_sign is an inference_result Maxima object
showing the following results:
'med_estimate: the sample median.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameter(s).
'p_value: p-value of the test.
Examples:
Checks whether the population from which the sample was taken has median 6, against the alternative H_1: median > 6.
(%i1) load("stats")$
(%i2) x: [2,0.1,7,1.8,4,2.3,5.6,7.4,5.1,6.1,6]$
(%i3) test_sign(x,'median=6,'alternative='greater);
| SIGN TEST
|
| med_estimate = 5.1
|
| method = Non parametric sign test.
|
(%o3) | hypotheses = H0: median = 6 , H1: median > 6
|
| statistic = 7
|
| distribution = [binomial, 10, 0.5]
|
| p_value = .05468749999999989
Categories: Package stats
This is the Wilcoxon signed rank test to make inferences about the median of a continuous population. Argument x is a list or a column matrix containing a one dimensional sample. Performs normal approximation if the sample size is greater than 20, or if there are zeroes or ties.
See also pdf_rank_test and cdf_rank_test
Options:
'median, default 0, is the median value to be checked.
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
The output of function test_signed_rank is an inference_result Maxima object
with the following results:
'med_estimate: the sample median.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameter(s).
'p_value: p-value of the test.
Examples:
Checks the null hypothesis H_0: median = 15 against the alternative H_1: median > 15. This is an exact test, since there are no ties.
(%i1) load("stats")$
(%i2) x: [17.1,15.9,13.7,13.4,15.5,17.6]$
(%i3) test_signed_rank(x,median=15,alternative=greater);
| SIGNED RANK TEST
|
| med_estimate = 15.7
|
| method = Exact test
|
(%o3) | hypotheses = H0: med = 15 , H1: med > 15
|
| statistic = 14
|
| distribution = [signed_rank, 6]
|
| p_value = 0.28125
Checks the null hypothesis H_0: equal(median, 2.5) against the alternative H_1: not equal(median, 2.5). This is an approximated test, since there are ties.
(%i1) load("stats")$
(%i2) y:[1.9,2.3,2.6,1.9,1.6,3.3,4.2,4,2.4,2.9,1.5,3,2.9,4.2,3.1]$
(%i3) test_signed_rank(y,median=2.5);
| SIGNED RANK TEST
|
| med_estimate = 2.9
|
| method = Asymptotic test. Ties
|
(%o3) | hypotheses = H0: med = 2.5 , H1: med # 2.5
|
| statistic = 76.5
|
| distribution = [normal, 60.5, 17.58195097251724]
|
| p_value = .3628097734643669
Categories: Package stats
This is the Wilcoxon-Mann-Whitney test for comparing the medians of two continuous populations. The first two arguments x1 and x2 are lists or column matrices with the data of two independent samples. Performs normal approximation if any of the sample sizes is greater than 10, or if there are ties.
Option:
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
The output of function test_rank_sum is an inference_result Maxima object
with the following results:
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameters.
'p_value: p-value of the test.
Examples:
Checks whether populations have similar medians. Samples sizes are small and an exact test is made.
(%i1) load("stats")$
(%i2) x:[12,15,17,38,42,10,23,35,28]$
(%i3) y:[21,18,25,14,52,65,40,43]$
(%i4) test_rank_sum(x,y);
| RANK SUM TEST
|
| method = Exact test
|
| hypotheses = H0: med1 = med2 , H1: med1 # med2
(%o4) |
| statistic = 22
|
| distribution = [rank_sum, 9, 8]
|
| p_value = .1995886466474702
Now, with greater samples and ties, the procedure makes normal approximation. The alternative hypothesis is H_1: median1 < median2.
(%i1) load("stats")$
(%i2) x: [39,42,35,13,10,23,15,20,17,27]$
(%i3) y: [20,52,66,19,41,32,44,25,14,39,43,35,19,56,27,15]$
(%i4) test_rank_sum(x,y,'alternative='less);
| RANK SUM TEST
|
| method = Asymptotic test. Ties
|
| hypotheses = H0: med1 = med2 , H1: med1 < med2
(%o4) |
| statistic = 48.5
|
| distribution = [normal, 79.5, 18.95419580097078]
|
| p_value = .05096985666598441
Categories: Package stats
Shapiro-Wilk test for normality. Argument x is a list of numbers, and sample
size must be greater than 2 and less or equal than 5000, otherwise, function
test_normality signals an error message.
Reference:
[1] Algorithm AS R94, Applied Statistics (1995), vol.44, no.4, 547-551
The output of function test_normality is an inference_result Maxima object
with the following results:
'statistic: value of the W statistic.
'p_value: p-value under normal assumption.
Examples:
Checks for the normality of a population, based on a sample of size 9.
(%i1) load("stats")$
(%i2) x:[12,15,17,38,42,10,23,35,28]$
(%i3) test_normality(x);
| SHAPIRO - WILK TEST
|
(%o3) | statistic = .9251055695162436
|
| p_value = .4361763918860381
Categories: Package stats
Multivariate linear regression, y_i = b0 + b1*x_1i + b2*x_2i + ... + bk*x_ki + u_i, where u_i are N(0,sigma) independent random variables. Argument x must be a matrix with more than one column. The last column is considered as the responses (y_i).
Option:
'conflevel, default 95/100, confidence level for the
confidence intervals; it must be an expression which takes a value
in (0,1).
The output of function linear_regression is an
inference_result Maxima object with the following results:
'b_estimation: regression coefficients estimates.
'b_covariances: covariance matrix of the regression
coefficients estimates.
b_conf_int: confidence intervals of the regression coefficients.
b_statistics: statistics for testing coefficient.
b_p_values: p-values for coefficient tests.
b_distribution: probability distribution for coefficient tests.
v_estimation: unbiased variance estimator.
v_conf_int: variance confidence interval.
v_distribution: probability distribution for variance test.
residuals: residuals.
adc: adjusted determination coefficient.
aic: Akaike's information criterion.
bic: Bayes's information criterion.
Only items 1, 4, 5, 6, 7, 8, 9 and 11 above, in this order,
are shown by default. The rest remain hidden until the user
makes use of functions items_inference and take_inference.
Example:
Fitting a linear model to a trivariate sample. The last column is considered as the responses (y_i).
(%i2) load("stats")$
(%i3) X:matrix(
[58,111,64],[84,131,78],[78,158,83],
[81,147,88],[82,121,89],[102,165,99],
[85,174,101],[102,169,102])$
(%i4) fpprintprec: 4$
(%i5) res: linear_regression(X);
| LINEAR REGRESSION MODEL
|
| b_estimation = [9.054, .5203, .2397]
|
| b_statistics = [.6051, 2.246, 1.74]
|
| b_p_values = [.5715, .07466, .1423]
|
(%o5) | b_distribution = [student_t, 5]
|
| v_estimation = 35.27
|
| v_conf_int = [13.74, 212.2]
|
| v_distribution = [chi2, 5]
|
| adc = .7922
(%i6) items_inference(res);
(%o6) [b_estimation, b_covariances, b_conf_int, b_statistics,
b_p_values, b_distribution, v_estimation, v_conf_int,
v_distribution, residuals, adc, aic, bic]
(%i7) take_inference('b_covariances, res);
[ 223.9 - 1.12 - .8532 ]
[ ]
(%o7) [ - 1.12 .05367 - .02305 ]
[ ]
[ - .8532 - .02305 .01898 ]
(%i8) take_inference('bic, res);
(%o8) 30.98
(%i9) load("draw")$
(%i10) draw2d(
points_joined = true,
grid = true,
points(take_inference('residuals, res)) )$
Categories: Package stats · Statistical estimation
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Probability density function of the exact distribution of the signed rank statistic. Argument x is a real number and n a positive integer.
See also test_signed_rank.
Categories: Package stats
Cumulative density function of the exact distribution of the signed rank statistic. Argument x is a real number and n a positive integer.
See also test_signed_rank.
Categories: Package stats
Probability density function of the exact distribution of the rank sum statistic. Argument x is a real number and n and m are both positive integers.
See also test_rank_sum.
Categories: Package stats
Cumulative density function of the exact distribution of the rank sum statistic. Argument x is a real number and n and m are both positive integers.
See also test_rank_sum.
Categories: Package stats
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| 83.1 Functions and Variables for stirling |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Replace gamma(x) with the O(1/x^(2n-1)) Stirling formula. when n isn't
a nonnegative integer, signal an error. With the optional third argument pred,
the Stirling formula is applied only when pred is true.
Reference: Abramowitz & Stegun, " Handbook of mathematical functions", 6.1.40.
Examples:
(%i1) load (stirling)$
(%i2) stirling(gamma(%alpha+x)/gamma(x),1);
1/2 - x x + %alpha - 1/2
(%o2) x (x + %alpha)
1 1
--------------- - ---- - %alpha
12 (x + %alpha) 12 x
%e
(%i3) taylor(%,x,inf,1);
%alpha 2 %alpha
%alpha x %alpha - x %alpha
(%o3)/T/ x + -------------------------------- + . . .
2 x
(%i4) map('factor,%);
%alpha - 1
%alpha (%alpha - 1) %alpha x
(%o4) x + -------------------------------
2
The function stirling knows the difference between the variable 'gamma' and
the function gamma:
(%i5) stirling(gamma + gamma(x),0);
x - 1/2 - x
(%o5) gamma + sqrt(2) sqrt(%pi) x %e
(%i6) stirling(gamma(y) + gamma(x),0);
y - 1/2 - y
(%o6) sqrt(2) sqrt(%pi) y %e
x - 1/2 - x
+ sqrt(2) sqrt(%pi) x %e
To apply the Stirling formula only to terms that involve the variable k,
use an optional third argument; for example
(%i7) makegamma(pochhammer(a,k)/pochhammer(b,k)); (%o7) (gamma(b)*gamma(k+a))/(gamma(a)*gamma(k+b)) (%i8) stirling(%,1, lambda([s], not(freeof(k,s)))); (%o8) (%e^(b-a)*gamma(b)*(k+a)^(k+a-1/2)*(k+b)^(-k-b+1/2))/gamma(a)
The terms gamma(a) and gamma(b) are free of k, so the Stirling formula
was not applied to these two terms.
To use this function write first load("stirling").
Categories: Gamma and factorial functions · Share packages · Package stirling
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 84.1 Introduction to String Processing | ||
| 84.2 Input and Output | ||
| 84.3 Characters | ||
| 84.4 String Processing | ||
| 84.5 Octets and Utilities for Cryptography |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
The package stringproc contains functions for processing strings
and characters including formatting, encoding and data streams.
This package is completed by some tools for cryptography, e.g. base64 and hash
functions.
It can be directly loaded via load(stringproc) or automatically by
using one of its functions.
For questions and bug reports please contact the author. The following command prints his e-mail-address.
printf(true, "~{~a~}@gmail.com", split(sdowncase("Volker van Nek")))$
A string is constructed by typing e.g. "Text".
When the option variable stringdisp is set to false, which is
the default, the double quotes won't be printed.
stringp is a test, if an object is a string.
(%i1) str: "Text"; (%o1) Text (%i2) stringp(str); (%o2) true
Characters are represented by a string of length 1. charp is the corresponding test.
(%i1) char: "e"; (%o1) e (%i2) charp(char); (%o2) true
In Maxima position indices in strings are like in list 1-indexed which results to the following consistency.
(%i1) is(charat("Lisp",1) = charlist("Lisp")[1]);
(%o1) true
A string may contain Maxima expressions. These can be parsed with parse_string.
(%i1) map(parse_string, ["42" ,"sqrt(2)", "%pi"]);
(%o1) [42, sqrt(2), %pi]
(%i2) map('float, %);
(%o2) [42.0, 1.414213562373095, 3.141592653589793]
Strings can be processed as characters or in binary form as octets. Functions for conversions are string_to_octets and octets_to_string. Usable encodings depend on the platform, the application and the underlying Lisp. (The following shows Maxima in GNU/Linux, compiled with SBCL.)
(%i1) obase: 16.$
(%i2) string_to_octets("$£€", "cp1252");
(%o2) [24, 0A3, 80]
(%i3) string_to_octets("$£€", "utf-8");
(%o3) [24, 0C2, 0A3, 0E2, 82, 0AC]
Strings may be written to character streams or as octets to binary streams. The following example demonstrates file in and output of characters.
openw returns an output stream to a file, printf writes formatted to that file and by e.g. close all characters contained in the stream are written to the file.
(%i1) s: openw("file.txt");
(%o1) #<output stream file.txt>
(%i2) printf(s, "~%~d ~f ~a ~a ~f ~e ~a~%",
42, 1.234, sqrt(2), %pi, 1.0e-2, 1.0e-2, 1.0b-2)$
(%i3) close(s)$
openr then returns an input stream from the previously used file and readline returns the line read as a string. The string may be tokenized by e.g. split or tokens and finally parsed by parse_string.
(%i4) s: openr("file.txt");
(%o4) #<input stream file.txt>
(%i5) readline(s);
(%o5) 42 1.234 sqrt(2) %pi 0.01 1.0E-2 1.0b-2
(%i6) map(parse_string, split(%));
(%o6) [42, 1.234, sqrt(2), %pi, 0.01, 0.01, 1.0b-2]
(%i7) close(s)$
Categories: Strings · Share packages · Package stringproc
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Example: Formatted printing to a file.
(%i1) s: openw("file.txt");
(%o1) #<output stream file.txt>
(%i2) control:
"~2tAn atom: ~20t~a~%~2tand a list: ~20t~{~r ~}~%~2t\
and an integer: ~20t~d~%"$
(%i3) printf( s,control, 'true,[1,2,3],42 )$
(%o3) false
(%i4) close(s);
(%o4) true
(%i5) s: openr("file.txt");
(%o5) #<input stream file.txt>
(%i6) while stringp( tmp:readline(s) ) do print(tmp)$
An atom: true
and a list: one two three
and an integer: 42
(%i7) close(s)$
Closes stream and returns true if stream had been open.
Categories: File input · File output · Package stringproc
stream has to be an open stream from or to a file.
flength then returns the number of bytes which are currently present in this file.
Example: See writebyte .
Categories: File input · File output · Package stringproc
Flushes stream where stream has to be an output stream to a file.
Example: See writebyte .
Categories: File output · Package stringproc
Returns the current position in stream, if pos is not used.
If pos is used, fposition sets the position in stream.
stream has to be a stream from or to a file and
pos has to be a positive number.
Positions in data streams are like in strings or lists 1-indexed, i.e. the first element in stream is in position 1.
Categories: File input · File output · Package stringproc
Writes a new line to the standard output stream
if the position is not at the beginning of a line und returns true.
Using the optional argument stream the new line is written to that stream.
There are some cases, where freshline() does not work as expected.
See also newline.
Categories: File output · Package stringproc
Returns a string containing all the characters currently present in stream which must be an open string-output stream. The returned characters are removed from stream.
Example: See make_string_output_stream .
Categories: Package stringproc
Returns an input stream which contains parts of string and an end of file. Without optional arguments the stream contains the entire string and is positioned in front of the first character. start and end define the substring contained in the stream. The first character is available at position 1.
(%i1) istream : make_string_input_stream("text", 1, 4);
(%o1) #<string-input stream from "text">
(%i2) (while (c : readchar(istream)) # false do sprint(c), newline())$
t e x
(%i3) close(istream)$
Categories: Package stringproc
Returns an output stream that accepts characters. Characters currently present in this stream can be retrieved by get_output_stream_string.
(%i1) ostream : make_string_output_stream(); (%o1) #<string-output stream 09622ea0> (%i2) printf(ostream, "foo")$ (%i3) printf(ostream, "bar")$ (%i4) string : get_output_stream_string(ostream); (%o4) foobar (%i5) printf(ostream, "baz")$ (%i6) string : get_output_stream_string(ostream); (%o6) baz (%i7) close(ostream)$
Categories: Package stringproc
Writes a new line to the standard output stream.
Using the optional argument stream the new line is written to that stream.
There are some cases, where newline() does not work as expected.
See sprint for an example of using newline().
Categories: File output · Package stringproc
Returns a character output stream to file.
If an existing file is opened, opena appends elements at the end of file.
For binary output see opena_binary .
Categories: File output · Package stringproc
Returns a character input stream to file.
openr assumes that file already exists.
For binary input see openr_binary .
Categories: File input · Package stringproc
Returns a character output stream to file.
If file does not exist, it will be created.
If an existing file is opened, openw destructively modifies file.
For binary output see openw_binary .
Categories: File output · Package stringproc
Produces formatted output by outputting the characters of control-string string and observing that a tilde introduces a directive. The character after the tilde, possibly preceded by prefix parameters and modifiers, specifies what kind of formatting is desired. Most directives use one or more elements of the arguments expr_1, ..., expr_n to create their output.
If dest is a stream or true, then printf returns false.
Otherwise, printf returns a string containing the output.
By default the streams stdin, stdout and stderr are defined.
If maxima is running as a server (which is the normal case if maxima communicating
with a graphical user interface) setup-client will define old_stdout and
old_stderr, too.
printf provides the Common Lisp function format in Maxima.
The following example illustrates the general relation between these two
functions.
(%i1) printf(true, "R~dD~d~%", 2, 2); R2D2 (%o1) false (%i2) :lisp (format t "R~dD~d~%" 2 2) R2D2 NIL
The following description is limited to a rough sketch of the possibilities of
printf.
The Lisp function format is described in detail in many reference books.
Of good help is e.g. the free available online-manual
"Common Lisp the Language" by Guy L. Steele. See chapter 22.3.3 there.
~% new line
~& fresh line
~t tab
~$ monetary
~d decimal integer
~b binary integer
~o octal integer
~x hexadecimal integer
~br base-b integer
~r spell an integer
~p plural
~f floating point
~e scientific notation
~g ~f or ~e, depending upon magnitude
~h bigfloat
~a uses Maxima function string
~s like ~a, but output enclosed in "double quotes"
~~ ~
~< justification, ~> terminates
~( case conversion, ~) terminates
~[ selection, ~] terminates
~{ iteration, ~} terminates
The directive ~h for bigfloat is no Lisp-standard and is therefore illustrated below.
Note that the directive ~* is not supported.
If dest is a stream or true, then printf returns false.
Otherwise, printf returns a string containing the output.
(%i1) printf( false, "~a ~a ~4f ~a ~@r",
"String",sym,bound,sqrt(12),144), bound = 1.234;
(%o1) String sym 1.23 2*sqrt(3) CXLIV
(%i2) printf( false,"~{~a ~}",["one",2,"THREE"] );
(%o2) one 2 THREE
(%i3) printf(true,"~{~{~9,1f ~}~%~}",mat ),
mat = args(matrix([1.1,2,3.33],[4,5,6],[7,8.88,9]))$
1.1 2.0 3.3
4.0 5.0 6.0
7.0 8.9 9.0
(%i4) control: "~:(~r~) bird~p ~[is~;are~] singing."$
(%i5) printf( false,control, n,n,if n=1 then 1 else 2 ), n=2;
(%o5) Two birds are singing.
The directive ~h has been introduced to handle bigfloats.
~w,d,e,x,o,p@H w : width d : decimal digits behind floating point e : minimal exponent digits x : preferred exponent o : overflow character p : padding character @ : display sign for positive numbers
(%i1) fpprec : 1000$ (%i2) printf(true, "|~h|~%", 2.b0^-64)$ |0.0000000000000000000542101086242752217003726400434970855712890625| (%i3) fpprec : 26$ (%i4) printf(true, "|~h|~%", sqrt(2))$ |1.4142135623730950488016887| (%i5) fpprec : 24$ (%i6) printf(true, "|~h|~%", sqrt(2))$ |1.41421356237309504880169| (%i7) printf(true, "|~28h|~%", sqrt(2))$ | 1.41421356237309504880169| (%i8) printf(true, "|~28,,,,,'*h|~%", sqrt(2))$ |***1.41421356237309504880169| (%i9) printf(true, "|~,18h|~%", sqrt(2))$ |1.414213562373095049| (%i10) printf(true, "|~,,,-3h|~%", sqrt(2))$ |1414.21356237309504880169b-3| (%i11) printf(true, "|~,,2,-3h|~%", sqrt(2))$ |1414.21356237309504880169b-03| (%i12) printf(true, "|~20h|~%", sqrt(2))$ |1.41421356237309504880169| (%i13) printf(true, "|~20,,,,'+h|~%", sqrt(2))$ |++++++++++++++++++++|
Categories: File output · Package stringproc
Removes and returns the first byte in stream which must be a binary input stream.
If the end of file is encountered readbyte returns false.
Example: Read the first 16 bytes from a file encrypted with AES in OpenSSL.
(%i1) ibase: obase: 16.$
(%i2) in: openr_binary("msg.bin");
(%o2) #<input stream msg.bin>
(%i3) (L:[], thru 16. do push(readbyte(in), L), L:reverse(L));
(%o3) [53, 61, 6C, 74, 65, 64, 5F, 5F, 88, 56, 0DE, 8A, 74, 0FD, 0AD, 0F0]
(%i4) close(in);
(%o4) true
(%i5) map(ascii, rest(L,-8));
(%o5) [S, a, l, t, e, d, _, _]
(%i6) salt: octets_to_number(rest(L,8));
(%o6) 8856de8a74fdadf0
Categories: File input · Package stringproc
Removes and returns the first character in stream.
If the end of file is encountered readchar returns false.
Example: See make_string_input_stream.
Categories: File input · Package stringproc
Returns a string containing all characters starting at the current position
in stream up to the end of the line or false
if the end of the file is encountered.
Categories: File input · Package stringproc
Evaluates and displays its arguments one after the other `on a line' starting at
the leftmost position. The expressions are printed with a space character right next
to the number, and it disregards line length.
newline() might be used for line breaking.
Example: Sequential printing with sprint.
Creating a new line with newline().
(%i1) for n:0 thru 19 do sprint(fib(n))$
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
(%i2) for n:0 thru 22 do (
sprint(fib(n)),
if mod(n,10) = 9 then newline() )$
0 1 1 2 3 5 8 13 21 34
55 89 144 233 377 610 987 1597 2584 4181
6765 10946 17711
Categories: Package stringproc
Writes byte to stream which must be a binary output stream.
writebyte returns byte.
Example: Write some bytes to a binary file output stream.
In this example all bytes correspond to printable characters and are printed
by printfile.
The bytes remain in the stream until flush_output or close have been called.
(%i1) ibase: obase: 16.$
(%i2) bytes: map(cint, charlist("GNU/Linux"));
(%o2) [47, 4E, 55, 2F, 4C, 69, 6E, 75, 78]
(%i3) out: openw_binary("test.bin");
(%o3) #<output stream test.bin>
(%i4) for i thru 3 do writebyte(bytes[i], out);
(%o4) done
(%i5) printfile("test.bin")$
(%i6) flength(out);
(%o6) 0
(%i7) flush_output(out);
(%o7) true
(%i8) flength(out);
(%o8) 3
(%i9) printfile("test.bin")$
GNU
(%i0A) for b in rest(bytes,3) do writebyte(b, out);
(%o0A) done
(%i0B) close(out);
(%o0B) true
(%i0C) printfile("test.bin")$
GNU/Linux
Categories: File output · Package stringproc
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Characters are strings of length 1.
Prints information about the current external format of the Lisp reader
and in case the external format encoding differs from the encoding of the
application which runs Maxima adjust_external_format tries to adjust
the encoding or prints some help or instruction.
adjust_external_format returns true when the external format has
been changed and false otherwise.
Functions like cint, unicode, octets_to_string and string_to_octets need UTF-8 as the external format of the Lisp reader to work properly over the full range of Unicode characters.
Examples (Maxima on Windows, March 2016):
Using adjust_external_format when the default external format
is not equal to the encoding provided by the application.
1. Command line Maxima
In case a terminal session is preferred it is recommended to use Maxima compiled
with SBCL. Here Unicode support is provided by default and calls to
adjust_external_format are unnecessary.
If Maxima is compiled with CLISP or GCL it is recommended to change
the terminal encoding from CP850 to CP1252.
adjust_external_format prints some help.
CCL reads UTF-8 while the terminal input is CP850 by default.
CP1252 is not supported by CCL. adjust_external_format
prints instructions for changing the terminal encoding and external format
both to iso-8859-1.
2. wxMaxima
In wxMaxima SBCL reads CP1252 by default but the input from the application is UTF-8 encoded. Adjustment is needed.
Calling adjust_external_format and restarting Maxima
permanently changes the default external format to UTF-8.
(%i1)adjust_external_format(); The line (setf sb-impl::*default-external-format* :utf-8) has been appended to the init file C:/Users/Username/.sbclrc Please restart Maxima to set the external format to UTF-8. (%i1) false
Restarting Maxima.
(%i1) adjust_external_format(); The external format is currently UTF-8 and has not been changed. (%i1) false
Categories: Package stringproc
Returns true if char is an alphabetic character.
To identify a non-US-ASCII character as an alphabetic character the underlying Lisp must provide full Unicode support. E.g. a German umlaut is detected as an alphabetic character with SBCL in GNU/Linux but not with GCL. (In Windows Maxima, when compiled with SBCL, must be set to UTF-8. See adjust_external_format for more.)
Example: Examination of non-US-ASCII characters.
The underlying Lisp (SBCL, GNU/Linux) is able to convert the typed character into a Lisp character and to examine.
(%i1) alphacharp("ü");
(%o1) true
In GCL this is not possible. An error break occurs.
(%i1) alphacharp("u");
(%o1) true
(%i2) alphacharp("ü");
package stringproc: ü cannot be converted into a Lisp character.
-- an error.
Categories: Predicate functions · Package stringproc
Returns true if char is an alphabetic character or a digit
(only corresponding US-ASCII characters are regarded as digits).
Note: See remarks on alphacharp.
Categories: Predicate functions · Package stringproc
Returns the US-ASCII character corresponding to the integer int
which has to be less than 128.
See unicode for converting code points larger than 127.
Examples:
(%i1) for n from 0 thru 127 do (
ch: ascii(n),
if alphacharp(ch) then sprint(ch),
if n = 96 then newline() )$
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
a b c d e f g h i j k l m n o p q r s t u v w x y z
Categories: Package stringproc
Returns true if char_1 and char_2 are the same character.
Categories: Predicate functions · Package stringproc
Like cequal but ignores case which is only possible for non-US-ASCII
characters when the underlying Lisp is able to recognize a character as an
alphabetic character. See remarks on alphacharp.
Categories: Predicate functions · Package stringproc
Returns true if the code point of char_1 is greater than the
code point of char_2.
Categories: Predicate functions · Package stringproc
Like cgreaterp but ignores case which is only possible for non-US-ASCII
characters when the underlying Lisp is able to recognize a character as an
alphabetic character. See remarks on alphacharp.
Categories: Predicate functions · Package stringproc
Returns true if obj is a Maxima-character.
See introduction for example.
Categories: Predicate functions · Package stringproc
Returns the Unicode code point of char which must be a
Maxima character, i.e. a string of length 1.
Examples: The hexadecimal code point of some characters (Maxima with SBCL on GNU/Linux).
(%i1) obase: 16.$ (%i2) map(cint, ["$","£","€"]); (%o2) [24, 0A3, 20AC]
Warning: It is not possible to enter characters corresponding to code points larger than 16 bit in wxMaxima with SBCL on Windows when the external format has not been set to UTF-8. See adjust_external_format.
CMUCL doesn't process these characters as one character.
cint then returns false.
Converting a character to a code point via UTF-8-octets may serve as a workaround:
utf8_to_unicode(string_to_octets(character));
See utf8_to_unicode, string_to_octets.
Categories: Package stringproc
Returns true if the code point of char_1 is less than the
code point of char_2.
Categories: Predicate functions · Package stringproc
Like clessp but ignores case which is only possible for non-US-ASCII
characters when the underlying Lisp is able to recognize a character as an
alphabetic character. See remarks on alphacharp.
Categories: Predicate functions · Package stringproc
Returns true if char is a graphic character but not a space character.
A graphic character is a character one can see, plus the space character.
(constituent is defined by Paul Graham.
See Paul Graham, ANSI Common Lisp, 1996, page 67.)
(%i1) for n from 0 thru 255 do (
tmp: ascii(n), if constituent(tmp) then sprint(tmp) )$
! " # % ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B
C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c
d e f g h i j k l m n o p q r s t u v w x y z { | } ~
Categories: Predicate functions · Package stringproc
Returns true if char is a digit where only the corresponding
US-ASCII-character is regarded as a digit.
Categories: Predicate functions · Package stringproc
Returns true if char is a lowercase character.
Note: See remarks on alphacharp.
Categories: Predicate functions · Package stringproc
The newline character (ASCII-character 10).
Categories: Global variables · Package stringproc
The space character.
Categories: Global variables · Package stringproc
The tab character.
Categories: Global variables · Package stringproc
Returns the character defined by arg which might be a Unicode code point or a name string if the underlying Lisp provides full Unicode support.
Example: Characters defined by hexadecimal code points (Maxima with SBCL on GNU/Linux).
(%i1) ibase: 16.$ (%i2) map(unicode, [24, 0A3, 20AC]); (%o2) [$, £, €]
Warning: In wxMaxima with SBCL on Windows it is not possible to convert code points larger than 16 bit to characters when the external format has not been set to UTF-8. See adjust_external_format for more information.
CMUCL doesn't process code points larger than 16 bit.
In these cases unicode returns false.
Converting a code point to a character via UTF-8 octets may serve as a workaround:
octets_to_string(unicode_to_utf8(code_point));
See octets_to_string, unicode_to_utf8.
In case the underlying Lisp provides full Unicode support the character might be
specified by its name. The following is possible in ECL, CLISP and SBCL,
where in SBCL on Windows the external format has to be set to UTF-8.
unicode(name) is supported by CMUCL too but again limited to 16 bit
characters.
The string argument to unicode is basically the same string returned by
printf using the "~@c" specifier.
But as shown below the prefix "#\" must be omitted.
Underlines might be replaced by spaces and uppercase letters by lowercase ones.
Example (continued): Characters defined by names (Maxima with SBCL on GNU/Linux).
(%i3) printf(false, "~@c", unicode(0DF));
(%o3) #\LATIN_SMALL_LETTER_SHARP_S
(%i4) unicode("LATIN_SMALL_LETTER_SHARP_S");
(%o4) ß
(%i5) unicode("Latin small letter sharp s");
(%o5) ß
Categories: Package stringproc
Returns a list containing the UTF-8 code corresponding to the Unicode code_point.
Examples: Converting Unicode code points to UTF-8 and vice versa.
(%i1) ibase: obase: 16.$ (%i2) map(cint, ["$","£","€"]); (%o2) [24, 0A3, 20AC] (%i3) map(unicode_to_utf8, %); (%o3) [[24], [0C2, 0A3], [0E2, 82, 0AC]] (%i4) map(utf8_to_unicode, %); (%o4) [24, 0A3, 20AC]
Categories: Package stringproc
Returns true if char is an uppercase character.
Note: See remarks on alphacharp.
Categories: Predicate functions · Package stringproc
This option variable affects Maxima when the character encoding provided by the application which runs Maxima is UTF-8 but the external format of the Lisp reader is not equal to UTF-8.
On GNU/Linux this is true when Maxima is built with GCL
and on Windows in wxMaxima with GCL- and SBCL-builds.
With SBCL it is recommended to change the external format to UTF-8.
Setting us_ascii_only is unnecessary then.
See adjust_external_format for details.
us_ascii_only is false by default.
Maxima itself then (i.e. in the above described situation) parses the UTF-8 encoding.
When us_ascii_only is set to true it is assumed that all strings
used as arguments to string processing functions do not contain Non-US-ASCII characters.
Given that promise, Maxima avoids parsing UTF-8 and strings can be processed more efficiently.
Categories: Global variables · Package stringproc
Returns a Unicode code point corresponding to the list which must contain the UTF-8 encoding of a single character.
Examples: See unicode_to_utf8.
Categories: Package stringproc
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Position indices in strings are 1-indexed like in Maxima lists. See example in charat.
Returns the n-th character of string. The first character in string is returned with n = 1.
(%i1) charat("Lisp",1);
(%o1) L
(%i2) charlist("Lisp")[1];
(%o2) L
Categories: Package stringproc
Returns the list of all characters in string.
(%i1) charlist("Lisp");
(%o1) [L, i, s, p]
Categories: Package stringproc
Parse the string str as a Maxima expression and evaluate it.
The string str may or may not have a terminator (dollar sign $ or semicolon ;).
Only the first expression is parsed and evaluated, if there is more than one.
Complain if str is not a string.
Examples:
(%i1) eval_string ("foo: 42; bar: foo^2 + baz");
(%o1) 42
(%i2) eval_string ("(foo: 42, bar: foo^2 + baz)");
(%o2) baz + 1764
See also parse_string.
Categories: Package stringproc
Parse the string str as a Maxima expression (do not evaluate it).
The string str may or may not have a terminator (dollar sign $ or semicolon ;).
Only the first expression is parsed, if there is more than one.
Complain if str is not a string.
Examples:
(%i1) parse_string ("foo: 42; bar: foo^2 + baz");
(%o1) foo : 42
(%i2) parse_string ("(foo: 42, bar: foo^2 + baz)");
2
(%o2) (foo : 42, bar : foo + baz)
See also eval_string.
Categories: Package stringproc
Returns a copy of string as a new string.
Categories: Package stringproc
Like supcase but uppercase characters are converted to lowercase.
Categories: Package stringproc
Returns true if string_1 and string_2 contain the same
sequence of characters.
Categories: Predicate functions · Package stringproc
Like sequal but ignores case which is only possible for non-US-ASCII
characters when the underlying Lisp is able to recognize a character as an
alphabetic character. See remarks on alphacharp.
Categories: Predicate functions · Package stringproc
sexplode is an alias for function charlist.
Categories: Package stringproc
simplode takes a list of expressions and concatenates them into a string.
If no delimiter delim is specified, simplode uses no delimiter.
delim can be any string.
Examples:
(%i1) simplode(["xx[",3,"]:",expand((x+y)^3)]);
(%o1) xx[3]:y^3+3*x*y^2+3*x^2*y+x^3
(%i2) simplode( sexplode("stars")," * " );
(%o2) s * t * a * r * s
(%i3) simplode( ["One","more","coffee."]," " );
(%o3) One more coffee.
Categories: Package stringproc
Returns a string that is a concatenation of substring(string, 1, pos-1),
the string seq and substring (string, pos).
Note that the first character in string is in position 1.
Examples:
(%i1) s: "A submarine."$
(%i2) concat( substring(s,1,3),"yellow ",substring(s,3) );
(%o2) A yellow submarine.
(%i3) sinsert("hollow ",s,3);
(%o3) A hollow submarine.
Categories: Package stringproc
Returns string except that each character from position start to end is inverted. If end is not given, all characters from start to the end of string are replaced.
Examples:
(%i1) sinvertcase("sInvertCase");
(%o1) SiNVERTcASE
Categories: Package stringproc
Returns the number of characters in string.
Categories: Package stringproc
Returns a new string with a number of num characters char.
Example:
(%i1) smake(3,"w"); (%o1) www
Categories: Package stringproc
Returns the position of the first character of string_1 at which string_1 and string_2 differ or false.
Default test function for matching is sequal.
If smismatch should ignore case, use sequalignore as test.
Example:
(%i1) smismatch("seven","seventh");
(%o1) 6
Categories: Package stringproc
Returns the list of all tokens in string.
Each token is an unparsed string.
split uses delim as delimiter.
If delim is not given, the space character is the default delimiter.
multiple is a boolean variable with true by default.
Multiple delimiters are read as one.
This is useful if tabs are saved as multiple space characters.
If multiple is set to false, each delimiter is noted.
Examples:
(%i1) split("1.2 2.3 3.4 4.5");
(%o1) [1.2, 2.3, 3.4, 4.5]
(%i2) split("first;;third;fourth",";",false);
(%o2) [first, , third, fourth]
Categories: Package stringproc
Returns the position of the first character in string which matches char. The first character in string is in position 1. For matching characters ignoring case see ssearch.
Categories: Package stringproc
Returns a string like string but without all substrings matching seq.
Default test function for matching is sequal.
If sremove should ignore case while searching for seq, use sequalignore as test.
Use start and end to limit searching.
Note that the first character in string is in position 1.
Examples:
(%i1) sremove("n't","I don't like coffee.");
(%o1) I do like coffee.
(%i2) sremove ("DO ",%,'sequalignore);
(%o2) I like coffee.
Categories: Package stringproc
Like sremove except that only the first substring that matches seq is removed.
Categories: Package stringproc
Returns a string with all the characters of string in reverse order.
Categories: Package stringproc
Returns the position of the first substring of string that matches the string seq.
Default test function for matching is sequal.
If ssearch should ignore case, use sequalignore as test.
Use start and end to limit searching.
Note that the first character in string is in position 1.
Example:
(%i1) ssearch("~s","~{~S ~}~%",'sequalignore);
(%o1) 4
Categories: Package stringproc
Returns a string that contains all characters from string in an order such there are no two successive characters c and d such that test (c, d) is false and test (d, c) is true.
Default test function for sorting is clessp.
The set of test functions is {clessp, clesspignore, cgreaterp, cgreaterpignore, cequal, cequalignore}.
Examples:
(%i1) ssort("I don't like Mondays.");
(%o1) '.IMaddeiklnnoosty
(%i2) ssort("I don't like Mondays.",'cgreaterpignore);
(%o2) ytsoonnMlkIiedda.'
Categories: Package stringproc
Returns a string like string except that all substrings matching old are replaced by new.
old and new need not to be of the same length.
Default test function for matching is sequal.
If ssubst should ignore case while searching for old, use sequalignore as test.
Use start and end to limit searching.
Note that the first character in string is in position 1.
Examples:
(%i1) ssubst("like","hate","I hate Thai food. I hate green tea.");
(%o1) I like Thai food. I like green tea.
(%i2) ssubst("Indian","thai",%,'sequalignore,8,12);
(%o2) I like Indian food. I like green tea.
Categories: Package stringproc
Like subst except that only the first substring that matches old is replaced.
Categories: Package stringproc
Returns a string like string, but with all characters that appear in seq removed from both ends.
Examples:
(%i1) "/* comment */"$
(%i2) strim(" /*",%);
(%o2) comment
(%i3) slength(%);
(%o3) 7
Categories: Package stringproc
Like strim except that only the left end of string is trimmed.
Categories: Package stringproc
Like strim except that only the right end of string is trimmed.
Categories: Package stringproc
Returns true if obj is a string.
See introduction for example.
Categories: Predicate functions · Package stringproc
Returns the substring of string beginning at position start and ending at position end. The character at position end is not included. If end is not given, the substring contains the rest of the string. Note that the first character in string is in position 1.
Examples:
(%i1) substring("substring",4);
(%o1) string
(%i2) substring(%,4,6);
(%o2) in
Categories: Package stringproc
Returns string except that lowercase characters from position start to end are replaced by the corresponding uppercase ones. If end is not given, all lowercase characters from start to the end of string are replaced.
Example:
(%i1) supcase("english",1,2);
(%o1) English
Categories: Package stringproc
Returns a list of tokens, which have been extracted from string.
The tokens are substrings whose characters satisfy a certain test function.
If test is not given, constituent is used as the default test.
{constituent, alphacharp, digitcharp, lowercasep, uppercasep, charp, characterp, alphanumericp} is the set of test functions.
(The Lisp-version of tokens is written by Paul Graham. ANSI Common Lisp, 1996, page 67.)
Examples:
(%i1) tokens("24 October 2005");
(%o1) [24, October, 2005]
(%i2) tokens("05-10-24",'digitcharp);
(%o2) [05, 10, 24]
(%i3) map(parse_string,%);
(%o3) [5, 10, 24]
Categories: Package stringproc
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Returns the base64-representation of arg as a string. The argument arg may be a string, a non-negative integer or a list of octets.
Examples:
(%i1) base64: base64("foo bar baz");
(%o1) Zm9vIGJhciBiYXo=
(%i2) string: base64_decode(base64);
(%o2) foo bar baz
(%i3) obase: 16.$
(%i4) integer: base64_decode(base64, 'number);
(%o4) 666f6f206261722062617a
(%i5) octets: base64_decode(base64, 'list);
(%o5) [66, 6F, 6F, 20, 62, 61, 72, 20, 62, 61, 7A]
(%i6) ibase: 16.$
(%i7) base64(octets);
(%o7) Zm9vIGJhciBiYXo=
Note that if arg contains umlauts (resp. octets larger than 127) the resulting base64-string is platform dependend. However the decoded string will be equal to the original.
Categories: Package stringproc
By default base64_decode decodes the base64-string back to the original string.
The optional argument return-type allows base64_decode to
alternatively return the corresponding number or list of octets.
return-type may be string, number or list.
Example: See base64.
Categories: Package stringproc
By default crc24sum returns the CRC24 checksum of an octet-list
as a string.
The optional argument return-type allows crc24sum to
alternatively return the corresponding number or list of octets.
return-type may be string, number or list.
Example:
-----BEGIN PGP SIGNATURE----- Version: GnuPG v2.0.22 (GNU/Linux) iQEcBAEBAgAGBQJVdCTzAAoJEG/1Mgf2DWAqCSYH/AhVFwhu1D89C3/QFcgVvZTM wnOYzBUURJAL/cT+IngkLEpp3hEbREcugWp+Tm6aw3R4CdJ7G3FLxExBH/5KnDHi rBQu+I7+3ySK2hpryQ6Wx5J9uZSa4YmfsNteR8up0zGkaulJeWkS4pjiRM+auWVe vajlKZCIK52P080DG7Q2dpshh4fgTeNwqCuCiBhQ73t8g1IaLdhDN6EzJVjGIzam /spqT/sTo6sw8yDOJjvU+Qvn6/mSMjC/YxjhRMaQt9EMrR1AZ4ukBF5uG1S7mXOH WdiwkSPZ3gnIBhM9SuC076gLWZUNs6NqTeE3UzMjDAFhH3jYk1T7mysCvdtIkms= =WmeC -----END PGP SIGNATURE-----
(%i1) ibase : obase : 16.$ (%i2) sig64 : sconcat( "iQEcBAEBAgAGBQJVdCTzAAoJEG/1Mgf2DWAqCSYH/AhVFwhu1D89C3/QFcgVvZTM", "wnOYzBUURJAL/cT+IngkLEpp3hEbREcugWp+Tm6aw3R4CdJ7G3FLxExBH/5KnDHi", "rBQu+I7+3ySK2hpryQ6Wx5J9uZSa4YmfsNteR8up0zGkaulJeWkS4pjiRM+auWVe", "vajlKZCIK52P080DG7Q2dpshh4fgTeNwqCuCiBhQ73t8g1IaLdhDN6EzJVjGIzam", "/spqT/sTo6sw8yDOJjvU+Qvn6/mSMjC/YxjhRMaQt9EMrR1AZ4ukBF5uG1S7mXOH", "WdiwkSPZ3gnIBhM9SuC076gLWZUNs6NqTeE3UzMjDAFhH3jYk1T7mysCvdtIkms=" )$ (%i3) octets: base64_decode(sig64, 'list)$ (%i4) crc24: crc24sum(octets, 'list); (%o4) [5A, 67, 82] (%i5) base64(crc24); (%o5) WmeC
Categories: Package stringproc
Returns the MD5 checksum of a string, a non-negative integer or
a list of octets. The default return value is a string containing 32 hex characters.
The optional argument return-type allows md5sum to alternatively
return the corresponding number or list of octets.
return-type may be string, number or list.
Examples:
(%i1) ibase: obase: 16.$
(%i2) msg: "foo bar baz"$
(%i3) string: md5sum(msg);
(%o3) ab07acbb1e496801937adfa772424bf7
(%i4) integer: md5sum(msg, 'number);
(%o4) 0ab07acbb1e496801937adfa772424bf7
(%i5) octets: md5sum(msg, 'list);
(%o5) [0AB,7,0AC,0BB,1E,49,68,1,93,7A,0DF,0A7,72,42,4B,0F7]
(%i6) sdowncase( printf(false, "~{~2,'0x~^:~}", octets) );
(%o6) ab:07:ac:bb:1e:49:68:01:93:7a:df:a7:72:42:4b:f7
Note that in case arg contains German umlauts or other non-ASCII
characters (resp. octets larger than 127) the MD5 checksum is platform dependend.
Categories: Package stringproc
Returns a pseudo random number of variable length. By default the returned value is a number with a length of len octets.
The optional argument return-type allows mgf1_sha1 to alternatively
return the corresponding list of len octets.
return-type may be number or list.
The computation of the returned value is described in RFC 3447,
appendix B.2.1 MGF1.
SHA1 ist used as hash function, i.e. the randomness of the computed number
relies on the randomness of SHA1 hashes.
Example:
(%i1) ibase: obase: 16.$ (%i2) number: mgf1_sha1(4711., 8); (%o2) 0e0252e5a2a42fea1 (%i3) octets: mgf1_sha1(4711., 8, 'list); (%o3) [0E0,25,2E,5A,2A,42,0FE,0A1]
Categories: Package stringproc
Returns an octet-representation of number as a list of octets. The number must be a non-negative integer.
Example:
(%i1) ibase : obase : 16.$ (%i2) octets: [0ca,0fe,0ba,0be]$ (%i3) number: octets_to_number(octets); (%o3) 0cafebabe (%i4) number_to_octets(number); (%o4) [0CA, 0FE, 0BA, 0BE]
Categories: Package stringproc
Returns a number by concatenating the octets in the list of octets.
Example: See number_to_octets.
Categories: Package stringproc
Computes an object identifier (OID) from the list of octets.
Example: RSA encryption OID
(%i1) ibase : obase : 16.$ (%i2) oid: octets_to_oid([2A,86,48,86,0F7,0D,1,1,1]); (%o2) 1.2.840.113549.1.1.1 (%i3) oid_to_octets(oid); (%o3) [2A, 86, 48, 86, 0F7, 0D, 1, 1, 1]
Categories: Package stringproc
Decodes the list of octets into a string according to current system defaults. When decoding octets corresponding to Non-US-ASCII characters the result depends on the platform, application and underlying Lisp.
Example: Using system defaults (Maxima compiled with GCL, which uses no format definition and simply passes through the UTF-8-octets encoded by the GNU/Linux terminal).
(%i1) octets: string_to_octets("abc");
(%o1) [61, 62, 63]
(%i2) octets_to_string(octets);
(%o2) abc
(%i3) ibase: obase: 16.$
(%i4) unicode(20AC);
(%o4) €
(%i5) octets: string_to_octets(%);
(%o5) [0E2, 82, 0AC]
(%i6) octets_to_string(octets);
(%o6) €
(%i7) utf8_to_unicode(octets);
(%o7) 20AC
In case the external format of the Lisp reader is equal to UTF-8 the optional argument encoding allows to set the encoding for the octet to string conversion. If necessary see adjust_external_format for changing the external format.
Some names of supported encodings (see corresponding Lisp manual for more):
CCL, CLISP, SBCL: utf-8, ucs-2be, ucs-4be, iso-8859-1, cp1252, cp850
CMUCL: utf-8, utf-16-be, utf-32-be, iso8859-1, cp1252
ECL: utf-8, ucs-2be, ucs-4be, iso-8859-1, windows-cp1252, dos-cp850
Example (continued): Using the optional encoding argument (Maxima compiled with SBCL, GNU/Linux terminal).
(%i8) string_to_octets("€", "ucs-2be");
(%o8) [20, 0AC]
Categories: Package stringproc
Convertes an object identifier (OID) to a list of octets.
Example: See octets_to_oid.
Categories: Package stringproc
Returns the SHA1 fingerprint of a string, a non-negative integer or
a list of octets. The default return value is a string containing 40 hex characters.
The optional argument return-type allows sha1sum to alternatively
return the corresponding number or list of octets.
return-type may be string, number or list.
Example:
(%i1) ibase: obase: 16.$
(%i2) msg: "foo bar baz"$
(%i3) string: sha1sum(msg);
(%o3) c7567e8b39e2428e38bf9c9226ac68de4c67dc39
(%i4) integer: sha1sum(msg, 'number);
(%o4) 0c7567e8b39e2428e38bf9c9226ac68de4c67dc39
(%i5) octets: sha1sum(msg, 'list);
(%o5) [0C7,56,7E,8B,39,0E2,42,8E,38,0BF,9C,92,26,0AC,68,0DE,4C,67,0DC,39]
(%i6) sdowncase( printf(false, "~{~2,'0x~^:~}", octets) );
(%o6) c7:56:7e:8b:39:e2:42:8e:38:bf:9c:92:26:ac:68:de:4c:67:dc:39
Note that in case arg contains German umlauts or other non-ASCII
characters (resp. octets larger than 127) the SHA1 fingerprint is platform dependend.
Categories: Package stringproc
Returns the SHA256 fingerprint of a string, a non-negative integer or
a list of octets. The default return value is a string containing 64 hex characters.
The optional argument return-type allows sha256sum to alternatively
return the corresponding number or list of octets (see sha1sum).
Example:
(%i1) string: sha256sum("foo bar baz");
(%o1) dbd318c1c462aee872f41109a4dfd3048871a03dedd0fe0e757ced57dad6f2d7
Note that in case arg contains German umlauts or other non-ASCII
characters (resp. octets larger than 127) the SHA256 fingerprint is platform dependend.
Categories: Package stringproc
Encodes a string into a list of octets according to current system defaults. When encoding strings containing Non-US-ASCII characters the result depends on the platform, application and underlying Lisp.
In case the external format of the Lisp reader is equal to UTF-8 the optional argument encoding allows to set the encoding for the string to octet conversion. If necessary see adjust_external_format for changing the external format.
See octets_to_string for examples and some more information.
Categories: Package stringproc
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| 85.1 Functions and Variables for to_poly_solve |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
The packages to_poly and to_poly_solve are experimental;
the specifications of the functions in these packages might change or
the some of the functions in these packages might be merged into other
Maxima functions.
Barton Willis (Professor of Mathematics, University of Nebraska at
Kearney) wrote the to_poly and to_poly_solve packages and the
English language user documentation for these packages.
The operator %and is a simplifying nonshort-circuited logical
conjunction. Maxima simplifies an %and expression to either true,
false, or a logically equivalent, but simplified, expression. The
operator %and is associative, commutative, and idempotent. Thus
when %and returns a noun form, the arguments of %and form
a non-redundant sorted list; for example
(%i1) a %and (a %and b); (%o1) a %and b
If one argument to a conjunction is the explicit the negation of another
argument, %and returns false:
(%i2) a %and (not a); (%o2) false
If any member of the conjunction is false, the conjunction simplifies to false even if other members are manifestly non-boolean; for example
(%i3) 42 %and false; (%o3) false
Any argument of an %and expression that is an inequation (that
is, an inequality or equation), is simplified using the Fourier
elimination package. The Fourier elimination simplifier has a
pre-processor that converts some, but not all, nonlinear inequations
into linear inequations; for example the Fourier elimination code
simplifies abs(x) + 1 > 0 to true, so
(%i4) (x < 1) %and (abs(x) + 1 > 0); (%o4) x < 1
Notes
prederror does not alter the
simplification %and expressions.
%and, %or, and not should be
fully parenthesized.
and and or are both
short-circuited. Thus and isn't associative or commutative.
Limitations The conjunction %and simplifies inequations
locally, not globally. This means that conjunctions such as
(%i5) (x < 1) %and (x > 1); (%o5) (x > 1) %and (x < 1)
do not simplify to false. Also, the Fourier elimination code ignores the fact database;
(%i6) assume(x > 5); (%o6) [x > 5] (%i7) (x > 1) %and (x > 2); (%o7) (x > 1) %and (x > 2)
Finally, nonlinear inequations that aren't easily converted into an equivalent linear inequation aren't simplified.
There is no support for distributing %and over %or;
neither is there support for distributing a logical negation over
%and.
To use `load(to_poly_solve)'
Related functions %or, %if, and, or, not
Status The operator %and is experimental; the
specifications of this function might change and its functionality
might be merged into other Maxima functions.
The operator %if is a simplifying conditional. The
conditional bool should be boolean-valued. When the
conditional is true, return the second argument; when the conditional is
false, return the third; in all other cases, return a noun form.
Maxima inequations (either an inequality or an equality) are not
boolean-valued; for example, Maxima does not simplify 5 < 6
to true, and it does not simplify 5 = 6 to false; however, in
the context of a conditional to an %if statement, Maxima
automatically attempts to determine the truth value of an
inequation. Examples:
(%i1) f : %if(x # 1, 2, 8); (%o1) %if(x - 1 # 0, 2, 8) (%i2) [subst(x = -1,f), subst(x=1,f)]; (%o2) [2, 8]
If the conditional involves an inequation, Maxima simplifies it using the Fourier elimination package.
Notes
(%i3) %if(42,1,2); (%o3) %if(42, 1, 2)
if is nary, the operator %if isn't
nary.
Limitations The Fourier elimination code only simplifies nonlinear inequations that are readily convertible to an equivalent linear inequation.
To use: `load(to_poly_solve)'
Status: The operator %if is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The operator %or is a simplifying nonshort-circuited logical
disjunction. Maxima simplifies an %or expression to either
true, false, or a logically equivalent, but simplified,
expression. The operator %or is associative, commutative, and
idempotent. Thus when %or returns a noun form, the arguments
of %or form a non-redundant sorted list; for example
(%i1) a %or (a %or b); (%o1) a %or b
If one member of the disjunction is the explicit the negation of another
member, %or returns true:
(%i2) a %or (not a); (%o2) true
If any member of the disjunction is true, the disjunction simplifies to true even if other members of the disjunction are manifestly non-boolean; for example
(%i3) 42 %or true; (%o3) true
Any argument of an %or expression that is an inequation (that
is, an inequality or equation), is simplified using the Fourier
elimination package. The Fourier elimination code simplifies
abs(x) + 1 > 0 to true, so we have
(%i4) (x < 1) %or (abs(x) + 1 > 0); (%o4) true
Notes
prederror does not alter the
simplification of %or expressions.
%and, %or, and not; the binding powers of these
operators might not match your expectations.
and and or are both short-circuited.
Thus or isn't associative or commutative.
Limitations The conjunction %or simplifies inequations
locally, not globally. This means that conjunctions such as
(%i1) (x < 1) %or (x >= 1); (%o1) (x > 1) %or (x >= 1)
do not simplify to true. Further, the Fourier elimination code ignores the fact database;
(%i2) assume(x > 5); (%o2) [x > 5] (%i3) (x > 1) %and (x > 2); (%o3) (x > 1) %and (x > 2)
Finally, nonlinear inequations that aren't easily converted into an equivalent linear inequation aren't simplified.
The algorithm that looks for terms that cannot both be false is weak;
also there is no support for distributing %or over %and;
neither is there support for distributing a logical negation over
%or.
To use `load(to_poly_solve)'
Related functions %or, %if, and, or, not
Status The operator %or is experimental; the
specifications of this function might change and its functionality
might be merged into other Maxima functions.
The predicate complex_number_p returns true if its argument is
either a + %i * b, a, %i b, or %i,
where a and b are either rational or floating point
numbers (including big floating point); for all other inputs,
complex_number_p returns false; for example
(%i1) map('complex_number_p,[2/3, 2 + 1.5 * %i, %i]);
(%o1) [true, true, true]
(%i2) complex_number_p((2+%i)/(5-%i));
(%o2) false
(%i3) complex_number_p(cos(5 - 2 * %i));
(%o3) false
Related functions isreal_p
To use `load(to_poly_solve)'
Status The operator complex_number_p is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function call compose_functions(l) returns a lambda form that is
the composition of the functions in the list l. The functions are
applied from right to left; for example
(%i1) compose_functions([cos, exp]);
%g151
(%o1) lambda([%g151], cos(%e ))
(%i2) %(x);
x
(%o2) cos(%e )
When the function list is empty, return the identity function:
(%i3) compose_functions([]); (%o3) lambda([%g152], %g152) (%i4) %(x); (%o4) x
Notes
funmake (not compose_functions)
signals an error:
(%i5) compose_functions([a < b]); funmake: first argument must be a symbol, subscripted symbol, string, or lambda expression; found: a < b #0: compose_functions(l=[a < b])(to_poly_solve.mac line 40) -- an error. To debug this try: debugmode(true);
new_variable.
(%i6) compose_functions([%g0]); (%o6) lambda([%g154], %g0(%g154)) (%i7) compose_functions([%g0]); (%o7) lambda([%g155], %g0(%g155))
Although the independent variables are different, Maxima is able to to deduce that these lambda forms are semantically equal:
(%i8) is(equal(%o6,%o7)); (%o8) true
To use `load(to_poly_solve)'
Status The function compose_functions is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function dfloat is a similar to float, but the function
dfloat applies rectform when float fails to evaluate
to an IEEE double floating point number; thus
(%i1) float(4.5^(1 + %i));
%i + 1
(%o1) 4.5
(%i2) dfloat(4.5^(1 + %i));
(%o2) 4.48998802962884 %i + .3000124893895671
Notes
float is both an option variable (default
value false) and a function name.
Related functions float, bfloat
To use `load(to_poly_solve)'
Status The function dfloat is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function elim eliminates the variables in the set or list
x from the equations in the set or list l. Each member
of x must be a symbol; the members of l can either be
equations, or expressions that are assumed to equal zero.
The function elim returns a list of two lists; the first is
the list of expressions with the variables eliminated; the second
is the list of pivots; thus, the second list is a list of
expressions that elim used to eliminate the variables.
Here is a example of eliminating between linear equations:
(%i1) elim(set(x + y + z = 1, x - y - z = 8, x - z = 1),
set(x,y));
(%o1) [[2 z - 7], [y + 7, z - x + 1]]
Eliminating x and y yields the single equation 2 z - 7 = 0;
the equations y + 7 = 0 and z - z + 1 = 1 were used as pivots.
Eliminating all three variables from these equations, triangularizes the linear
system:
(%i2) elim(set(x + y + z = 1, x - y - z = 8, x - z = 1),
set(x,y,z));
(%o2) [[], [2 z - 7, y + 7, z - x + 1]]
Of course, the equations needn't be linear:
(%i3) elim(set(x^2 - 2 * y^3 = 1, x - y = 5), [x,y]);
3 2
(%o3) [[], [2 y - y - 10 y - 24, y - x + 5]]
The user doesn't control the order the variables are eliminated. Instead, the algorithm uses a heuristic to attempt to choose the best pivot and the best elimination order.
Notes
eliminate, the function
elim does not invoke solve when the number of equations
equals the number of variables.
elim works by applying resultants; the option
variable resultant determines which algorithm Maxima
uses. Using sqfr, Maxima factors each resultant and suppresses
multiple zeros.
elim will triangularize a nonlinear set of polynomial
equations; the solution set of the triangularized set can be larger
than that solution set of the untriangularized set. Thus, the triangularized
equations can have spurious solutions.
Related functions elim_allbut, eliminate_using, eliminate
Option variables resultant
To use `load(to_poly)'
Status The function elim is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
This function is similar to elim, except that it eliminates all the
variables in the list of equations l except for those variables that
in in the list x
(%i1) elim_allbut([x+y = 1, x - 5*y = 1],[]); (%o1) [[], [y, y + x - 1]] (%i2) elim_allbut([x+y = 1, x - 5*y = 1],[x]); (%o2) [[x - 1], [y + x - 1]]
To use `load(to_poly)'
Option variables resultant
Related functions elim, eliminate_using, eliminate
Status The function elim_allbut is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
Using e as the pivot, eliminate the symbol x from the
list or set of equations in l. The function eliminate_using
returns a set.
(%i1) eq : [x^2 - y^2 - z^3 , x*y - z^2 - 5, x - y + z];
3 2 2 2
(%o1) [- z - y + x , - z + x y - 5, z - y + x]
(%i2) eliminate_using(eq,first(eq),z);
3 2 2 3 2
(%o2) {y + (1 - 3 x) y + 3 x y - x - x ,
4 3 3 2 2 4
y - x y + 13 x y - 75 x y + x + 125}
(%i3) eliminate_using(eq,second(eq),z);
2 2 4 3 3 2 2 4
(%o3) {y - 3 x y + x + 5, y - x y + 13 x y - 75 x y + x
+ 125}
(%i4) eliminate_using(eq, third(eq),z);
2 2 3 2 2 3 2
(%o4) {y - 3 x y + x + 5, y + (1 - 3 x) y + 3 x y - x - x }
Option variables resultant
Related functions elim, eliminate, elim_allbut
To use `load(to_poly)'
Status The function eliminate_using is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
Fourier elimination is the analog of Gauss elimination for linear inequations
(equations or inequalities). The function call fourier_elim([eq1, eq2,
...], [var1, var2, ...]) does Fourier elimination on a list of linear
inequations [eq1, eq2, ...] with respect to the variables
[var1, var2, ...]; for example
(%i1) fourier_elim([y-x < 5, x - y < 7, 10 < y],[x,y]); (%o1) [y - 5 < x, x < y + 7, 10 < y] (%i2) fourier_elim([y-x < 5, x - y < 7, 10 < y],[y,x]); (%o2) [max(10, x - 7) < y, y < x + 5, 5 < x]
Eliminating first with respect to x and second with respect to y yields lower and upper bounds for x that depend on y, and lower and upper bounds for y that are numbers. Eliminating in the other order gives x dependent lower and upper bounds for y, and numerical lower and upper bounds for x.
When necessary, fourier_elim returns a disjunction of lists of
inequations:
(%i3) fourier_elim([x # 6],[x]); (%o3) [x < 6] or [6 < x]
When the solution set is empty, fourier_elim returns emptyset,
and when the solution set is all reals, fourier_elim returns universalset;
for example
(%i4) fourier_elim([x < 1, x > 1],[x]); (%o4) emptyset (%i5) fourier_elim([minf < x, x < inf],[x]); (%o5) universalset
For nonlinear inequations, fourier_elim returns a (somewhat)
simplified list of inequations:
(%i6) fourier_elim([x^3 - 1 > 0],[x]);
2 2
(%o6) [1 < x, x + x + 1 > 0] or [x < 1, - (x + x + 1) > 0]
(%i7) fourier_elim([cos(x) < 1/2],[x]);
(%o7) [1 - 2 cos(x) > 0]
Instead of a list of inequations, the first argument to fourier_elim
may be a logical disjunction or conjunction:
(%i8) fourier_elim((x + y < 5) and (x - y >8),[x,y]);
3
(%o8) [y + 8 < x, x < 5 - y, y < - -]
2
(%i9) fourier_elim(((x + y < 5) and x < 1) or (x - y >8),[x,y]);
(%o9) [y + 8 < x] or [x < min(1, 5 - y)]
The function fourier_elim supports the inequation operators
<, <=, >, >=, #, and =.
The Fourier elimination code has a preprocessor that converts some nonlinear inequations that involve the absolute value, minimum, and maximum functions into linear in equations. Additionally, the preprocessor handles some expressions that are the product or quotient of linear terms:
(%i10) fourier_elim([max(x,y) > 6, x # 8, abs(y-1) > 12],[x,y]); (%o10) [6 < x, x < 8, y < - 11] or [8 < x, y < - 11] or [x < 8, 13 < y] or [x = y, 13 < y] or [8 < x, x < y, 13 < y] or [y < x, 13 < y] (%i11) fourier_elim([(x+6)/(x-9) <= 6],[x]); (%o11) [x = 12] or [12 < x] or [x < 9] (%i12) fourier_elim([x^2 - 1 # 0],[x]); (%o12) [- 1 < x, x < 1] or [1 < x] or [x < - 1]
To use `load(fourier_elim)'
The predicate isreal_p returns true when Maxima is able to
determine that e is real-valued on the entire real line; it
returns false when Maxima is able to determine that e isn't
real-valued on some nonempty subset of the real line; and it returns a
noun form for all other cases.
(%i1) map('isreal_p, [-1, 0, %i, %pi]);
(%o1) [true, true, false, true]
Maxima variables are assumed to be real; thus
(%i2) isreal_p(x); (%o2) true
The function isreal_p examines the fact database:
(%i3) declare(z,complex)$ (%i4) isreal_p(z); (%o4) isreal_p(z)
Limitations
Too often, isreal_p returns a noun form when it should be able
to return false; a simple example: the logarithm function isn't
real-valued on the entire real line, so isreal_p(log(x)) should
return false; however
(%i5) isreal_p(log(x)); (%o5) isreal_p(log(x))
To use `load(to_poly_solve)'
Related functions complex_number_p
Status The function isreal_p is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
Return a unique symbol of the form %[z,n,r,c,g]k, where
k is an integer. The allowed values for type are
integer, natural_number, real, natural_number, and general.
(By natural number, we mean the nonnegative integers; thus zero is
a natural number. Some, but not all,definitions of natural number
exclude zero.)
When type isn't one of the allowed values, type defaults to general. For integers, natural numbers, and complex numbers, Maxima automatically appends this information to the fact database.
(%i1) map('new_variable,
['integer, 'natural_number, 'real, 'complex, 'general]);
(%o1) [%z144, %n145, %r146, %c147, %g148]
(%i2) nicedummies(%);
(%o2) [%z0, %n0, %r0, %c0, %g0]
(%i3) featurep(%z0, 'integer);
(%o3) true
(%i4) featurep(%n0, 'integer);
(%o4) true
(%i5) is(%n0 >= 0);
(%o5) true
(%i6) featurep(%c0, 'complex);
(%o6) true
Note Generally, the argument to new_variable should be quoted. The quote
will protect against errors similar to
(%i7) integer : 12$
(%i8) new_variable(integer);
(%o8) %g149
(%i9) new_variable('integer);
(%o9) %z150
Related functions nicedummies
To use `load(to_poly_solve)'
Status The function new_variable is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
Starting with zero, the function nicedummies re-indexes the variables
in an expression that were introduced by new_variable;
(%i1) new_variable('integer) + 52 * new_variable('integer);
(%o1) 52 %z136 + %z135
(%i2) new_variable('integer) - new_variable('integer);
(%o2) %z137 - %z138
(%i3) nicedummies(%);
(%o3) %z0 - %z1
Related functions new_variable
To use `load(to_poly_solve)'
Status The function nicedummies is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function parg is a simplifying version of the complex argument function
carg; thus
(%i1) map('parg,[1,1+%i,%i, -1 + %i, -1]);
%pi %pi 3 %pi
(%o1) [0, ---, ---, -----, %pi]
4 2 4
Generally, for a non-constant input, parg returns a noun form; thus
(%i2) parg(x + %i * sqrt(x)); (%o2) parg(x + %i sqrt(x))
When sign can determine that the input is a positive or negative real
number, parg will return a non-noun form for a non-constant input.
Here are two examples:
(%i3) parg(abs(x)); (%o3) 0 (%i4) parg(-x^2-1); (%o4) %pi
Note The sign function mostly ignores the variables that are declared
to be complex (declare(x,complex)); for variables that are declared
to be complex, the parg can return incorrect values; for example
(%i1) declare(x,complex)$ (%i2) parg(x^2 + 1); (%o2) 0
Related function carg, isreal_p
To use `load(to_poly_solve)'
Status The function parg is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function real_imagpart_to_conjugate replaces all occurrences
of realpart and imagpart to algebraically equivalent expressions
involving the conjugate.
(%i1) declare(x, complex)$
(%i2) real_imagpart_to_conjugate(realpart(x) + imagpart(x) = 3);
conjugate(x) + x %i (x - conjugate(x))
(%o2) ---------------- - --------------------- = 3
2 2
To use `load(to_poly_solve)'
Status The function real_imagpart_to_conjugate is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function rectform_log_if_constant converts all terms of the form
log(c) to rectform(log(c)), where c is
either a declared constant expression or explicitly declared constant
(%i1) rectform_log_if_constant(log(1-%i) - log(x - %i));
log(2) %i %pi
(%o1) - log(x - %i) + ------ - ------
2 4
(%i2) declare(a,constant, b,constant)$
(%i3) rectform_log_if_constant(log(a + %i*b));
2 2
log(b + a )
(%o3) ------------ + %i atan2(b, a)
2
To use `load(to_poly_solve)'
Status The function rectform_log_if_constant is
experimental; the specifications of this function might change might change and its functionality
might be merged into other Maxima functions.
The function simp_inequality applies some simplifications to
conjunctions and disjunctions of inequations.
Limitations The function simp_inequality is limited in at least two ways;
first, the simplifications are local; thus
(%i1) simp_inequality((x > minf) %and (x < 0)); (%o1) (x>1) %and (x<1)
And second, simp_inequality doesn't consult the fact database:
(%i2) assume(x > 0)$ (%i3) simp_inequality(x > 0); (%o3) x > 0
To use `load(fourier_elim)'
Status The function simp_inequality is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
This function applies the identities cot(x) = atan(1/x),
acsc(x) = asin(1/x), and similarly for asec, acoth, acsch
and asech to an expression. See Abramowitz and Stegun,
Eqs. 4.4.6 through 4.4.8 and 4.6.4 through 4.6.6.
To use `load(to_poly_solve)'
Status The function standardize_inverse_trig is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
When l is a single equation or a list of equations, substitute
the right hand side of each equation for the left hand side. The
substitutions are made in parallel; for example
(%i1) load(to_poly_solve)$ (%i2) subst_parallel([x=y,y=x], [x,y]); (%o2) [y, x]
Compare this to substitutions made serially:
(%i3) subst([x=y,y=x],[x,y]); (%o3) [x, x]
The function subst_parallel is similar to sublis except that
subst_parallel allows for substitution of nonatoms; for example
(%i4) subst_parallel([x^2 = a, y = b], x^2 * y);
(%o4) a b
(%i5) sublis([x^2 = a, y = b], x^2 * y);
2
sublis: left-hand side of equation must be a symbol; found: x
-- an error. To debug this try: debugmode(true);
The substitutions made by subst_parallel are literal, not semantic; thus
subst_parallel does not recognize that x * y is a subexpression
of x^2 * y
(%i6) subst_parallel([x * y = a], x^2 * y);
2
(%o6) x y
The function subst_parallel completes all substitutions
before simplifications. This allows for substitutions into
conditional expressions where errors might occur if the
simplifications were made earlier:
(%i7) subst_parallel([x = 0], %if(x < 1, 5, log(x))); (%o7) 5 (%i8) subst([x = 0], %if(x < 1, 5, log(x))); log: encountered log(0). -- an error. To debug this try: debugmode(true);
Related functions subst, sublis, ratsubst
To use `load(to_poly_solve_extra.lisp)'
Status The function subst_parallel is experimental; the
specifications of this function might change might change and its
functionality might be merged into other Maxima functions.
The function to_poly attempts to convert the equation e
into a polynomial system along with inequality constraints; the
solutions to the polynomial system that satisfy the constraints are
solutions to the equation e. Informally, to_poly
attempts to polynomialize the equation e; an example might
clarify:
(%i1) load(to_poly_solve)$
(%i2) to_poly(sqrt(x) = 3, [x]);
2
(%o2) [[%g130 - 3, x = %g130 ],
%pi %pi
[- --- < parg(%g130), parg(%g130) <= ---], []]
2 2
The conditions -%pi/2<parg(%g130),parg(%g130)<=%pi/2 tell us that
%g130 is in the range of the square root function. When this is
true, the solution set to sqrt(x) = 3 is the same as the
solution set to %g130-3,x=%g130^2.
To polynomialize trigonometric expressions, it is necessary to
introduce a non algebraic substitution; these non algebraic substitutions
are returned in the third list returned by to_poly; for example
(%i3) to_poly(cos(x),[x]);
2 %i x
(%o3) [[%g131 + 1], [2 %g131 # 0], [%g131 = %e ]]
Constant terms aren't polynomializied unless the number one is a member of the variable list; for example
(%i4) to_poly(x = sqrt(5),[x]);
(%o4) [[x - sqrt(5)], [], []]
(%i5) to_poly(x = sqrt(5),[1,x]);
2
(%o5) [[x - %g132, 5 = %g132 ],
%pi %pi
[- --- < parg(%g132), parg(%g132) <= ---], []]
2 2
To generate a polynomial with sqrt(5) + sqrt(7) as one of its roots, use the commands
(%i6) first(elim_allbut(first(to_poly(x = sqrt(5) + sqrt(7),
[1,x])), [x]));
4 2
(%o6) [x - 24 x + 4]
Related functions to_poly_solve
To use `load(to_poly)'
Status: The function to_poly is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function to_poly_solve tries to solve the equations e
for the variables l. The equation(s) e can either be a
single expression or a set or list of expressions; similarly, l
can either be a single symbol or a list of set of symbols. When
a member of e isn't explicitly an equation, for example x^2 -1,
the solver asummes that the expression vanishes.
The basic strategy of to_poly_solve is to convert the input into a polynomial form and to
call algsys on the polynomial system. Internally to_poly_solve defaults algexact
to true. To change the default for algexact, append 'algexact=false to the to_poly_solve
argument list.
When to_poly_solve is able to determine the solution set, each
member of the solution set is a list in a %union object:
(%i1) load(to_poly_solve)$ (%i2) to_poly_solve(x*(x-1) = 0, x); (%o2) %union([x = 0], [x = 1])
When to_poly_solve is unable to determine the solution set, a
%solve nounform is returned (in this case, a warning is printed)
(%i3) to_poly_solve(x^k + 2* x + 1 = 0, x);
Nonalgebraic argument given to 'to_poly'
unable to solve
k
(%o3) %solve([x + 2 x + 1 = 0], [x])
Subsitution into a %solve nounform can sometimes result in the solution
(%i4) subst(k = 2, %); (%o4) %union([x = - 1])
Especially for trigonometric equations, the solver sometimes needs
to introduce an arbitrary integer. These arbitrary integers have the
form %zXXX, where XXX is an integer; for example
(%i5) to_poly_solve(sin(x) = 0, x); (%o5) %union([x = 2 %pi %z33 + %pi], [x = 2 %pi %z35])
To re-index these variables to zero, use nicedummies:
(%i6) nicedummies(%); (%o6) %union([x = 2 %pi %z0 + %pi], [x = 2 %pi %z1])
Occasionally, the solver introduces an arbitrary complex number of the
form %cXXX or an arbitrary real number of the form %rXXX.
The function nicedummies will re-index these identifiers to zero.
The solution set sometimes involves simplifing versions of various
of logical operators including %and, %or, or %if
for conjunction, disjuntion, and implication, respectively; for example
(%i7) sol : to_poly_solve(abs(x) = a, x);
(%o7) %union(%if(isnonnegative_p(a), [x = - a], %union()),
%if(isnonnegative_p(a), [x = a], %union()))
(%i8) subst(a = 42, sol);
(%o8) %union([x = - 42], [x = 42])
(%i9) subst(a = -42, sol);
(%o9) %union()
The empty set is represented by %union.
The function to_poly_solve is able to solve some, but not all,
equations involving rational powers, some nonrational powers, absolute
values, trigonometric functions, and minimum and maximum. Also, some it
can solve some equations that are solvable in in terms of the Lambert W
function; some examples:
(%i1) load(to_poly_solve)$
(%i2) to_poly_solve(set(max(x,y) = 5, x+y = 2), set(x,y));
(%o2) %union([x = - 3, y = 5], [x = 5, y = - 3])
(%i3) to_poly_solve(abs(1-abs(1-x)) = 10,x);
(%o3) %union([x = - 10], [x = 12])
(%i4) to_poly_solve(set(sqrt(x) + sqrt(y) = 5, x + y = 10),
set(x,y));
3/2 3/2
5 %i - 10 5 %i + 10
(%o4) %union([x = - ------------, y = ------------],
2 2
3/2 3/2
5 %i + 10 5 %i - 10
[x = ------------, y = - ------------])
2 2
(%i5) to_poly_solve(cos(x) * sin(x) = 1/2,x,
'simpfuncs = ['expand, 'nicedummies]);
%pi
(%o5) %union([x = %pi %z0 + ---])
4
(%i6) to_poly_solve(x^(2*a) + x^a + 1,x);
2 %i %pi %z81
-------------
1/a a
(sqrt(3) %i - 1) %e
(%o6) %union([x = -----------------------------------],
1/a
2
2 %i %pi %z83
-------------
1/a a
(- sqrt(3) %i - 1) %e
[x = -------------------------------------])
1/a
2
(%i7) to_poly_solve(x * exp(x) = a, x);
(%o7) %union([x = lambert_w(a)])
For linear inequalities, to_poly_solve automatically does Fourier
elimination:
(%i8) to_poly_solve([x + y < 1, x - y >= 8], [x,y]);
7
(%o8) %union([x = y + 8, y < - -],
2
7
[y + 8 < x, x < 1 - y, y < - -])
2
Each optional argument to to_poly_solve must be an equation;
generally, the order of these options does not matter.
simpfuncs = l, where l is a list of functions.
Apply the composition of the members of l to each solution.
(%i1) to_poly_solve(x^2=%i,x);
1/4 1/4
(%o1) %union([x = - (- 1) ], [x = (- 1) ])
(%i2) to_poly_solve(x^2= %i,x, 'simpfuncs = ['rectform]);
%i 1 %i 1
(%o2) %union([x = - ------- - -------], [x = ------- + -------])
sqrt(2) sqrt(2) sqrt(2) sqrt(2)
Sometimes additional simplification can revert a simplification; for example
(%i3) to_poly_solve(x^2=1,x);
(%o3) %union([x = - 1], [x = 1])
(%i4) to_poly_solve(x^2= 1,x, 'simpfuncs = [polarform]);
%i %pi
(%o4) %union([x = 1], [x = %e ]
Maxima doesn't try to check that each member of the function list l is
purely a simplification; thus
(%i5) to_poly_solve(x^2 = %i,x, 'simpfuncs = [lambda([s],s^2)]); (%o5) %union([x = %i])
To convert each solution to a double float, use simpfunc = ['dfloat]:
(%i6) to_poly_solve(x^3 +x + 1 = 0,x,
'simpfuncs = ['dfloat]), algexact : true;
(%o6) %union([x = - .6823278038280178],
[x = .3411639019140089 - 1.161541399997251 %i],
[x = 1.161541399997251 %i + .3411639019140089])
use_grobner = true With this option, the function
poly_reduced_grobner is applied to the equations before
attempting their solution. Primarily, this option provides a workaround
for weakness in the function algsys. Here is an example of
such a workaround:
(%i7) to_poly_solve([x^2+y^2=2^2,(x-1)^2+(y-1)^2=2^2],[x,y],
'use_grobner = true);
sqrt(7) - 1 sqrt(7) + 1
(%o7) %union([x = - -----------, y = -----------],
2 2
sqrt(7) + 1 sqrt(7) - 1
[x = -----------, y = - -----------])
2 2
(%i8) to_poly_solve([x^2+y^2=2^2,(x-1)^2+(y-1)^2=2^2],[x,y]);
(%o8) %union()
maxdepth = k, where k is a positive integer. This
function controls the maximum recursion depth for the solver. The
default value for maxdepth is five. When the recursions depth is
exceeded, the solver signals an error:
(%i9) to_poly_solve(cos(x) = x,x, 'maxdepth = 2); Unable to solve Unable to solve (%o9) %solve([cos(x) = x], [x], maxdepth = 2)
parameters = l, where l is a list of symbols. The solver
attempts to return a solution that is valid for all members of the list
l; for example:
(%i10) to_poly_solve(a * x = x, x);
(%o10) %union([x = 0])
(%i11) to_poly_solve(a * x = x, x, 'parameters = [a]);
(%o11) %union(%if(a - 1 = 0, [x = %c111], %union()),
%if(a - 1 # 0, [x = 0], %union()))
In (%o2), the solver introduced a dummy variable; to re-index the
these dummy variables, use the function nicedummies:
(%i12) nicedummies(%);
(%o12) %union(%if(a - 1 = 0, [x = %c0], %union()),
%if(a - 1 # 0, [x = 0], %union()))
The to_poly_solve uses data stored in the hashed array
one_to_one_reduce to solve equations of the form f(a) =
f(b). The assignment one_to_one_reduce['f,'f] : lambda([a,b],
a=b) tells to_poly_solve that the solution set of f(a)
= f(b) equals the solution set of a=b; for example
(%i13) one_to_one_reduce['f,'f] : lambda([a,b], a=b)$ (%i14) to_poly_solve(f(x^2-1) = f(0),x); (%o14) %union([x = - 1], [x = 1])
More generally, the assignment one_to_one_reduce['f,'g] : lambda([a,b],
w(a, b) = 0 tells to_poly_solve that the solution set of f(a)
= f(b) equals the solution set of w(a,b) = 0; for example
(%i15) one_to_one_reduce['f,'g] : lambda([a,b], a = 1 + b/2)$ (%i16) to_poly_solve(f(x) - g(x),x); (%o16) %union([x = 2])
Additionally, the function to_poly_solve uses data stored in the hashed array
function_inverse to solve equations of the form f(a) = b.
The assignment function_inverse['f] : lambda([s], g(s))
informs to_poly_solve that the solution set to f(x) = b equals
the solution set to x = g(b); two examples:
(%i17) function_inverse['Q] : lambda([s], P(s))$ (%i18) to_poly_solve(Q(x-1) = 2009,x); (%o18) %union([x = P(2009) + 1]) (%i19) function_inverse['G] : lambda([s], s+new_variable(integer)); (%o19) lambda([s], s + new_variable(integer)) (%i20) to_poly_solve(G(x - a) = b,x); (%o20) %union([x = b + a + %z125])
Notes
fullratsubst is
able to appropriately make substitutions, the solve variables can be nonsymbols:
(%i1) to_poly_solve([x^2 + y^2 + x * y = 5, x * y = 8],
[x^2 + y^2, x * y]);
2 2
(%o1) %union([x y = 8, y + x = - 3])
(%i1) declare(x,complex)$
(%i2) to_poly_solve(x + (5 + %i) * conjugate(x) = 1, x);
%i + 21
(%o2) %union([x = - -----------])
25 %i - 125
(%i3) declare(y,complex)$
(%i4) to_poly_solve(set(conjugate(x) - y = 42 + %i,
x + conjugate(y) = 0), set(x,y));
%i - 42 %i + 42
(%o4) %union([x = - -------, y = - -------])
2 2
to_poly_solve consults the fact database to decide if the
argument to the absolute value is complex valued. When
(%i1) to_poly_solve(abs(x) = 6, x);
(%o1) %union([x = - 6], [x = 6])
(%i2) declare(z,complex)$
(%i3) to_poly_solve(abs(z) = 6, z);
(%o3) %union(%if((%c11 # 0) %and (%c11 conjugate(%c11) - 36 =
0), [z = %c11], %union()))
This is the only situation that the solver consults the fact database. If
a solve variable is declared to be an integer, for example, to_poly_solve
ignores this declaration.
Relevant option variables algexact, resultant, algebraic
Related functions to_poly
To use `load(to_poly_solve)'
Status: The function to_poly_solve is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
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| 86.1 Introduction to Units | ||
| 86.2 Functions and Variables for Units |
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The unit package enables the user to convert between arbitrary units and work with dimensions in equations. The functioning of this package is radically different from the original Maxima units package - whereas the original was a basic list of definitions, this package uses rulesets to allow the user to chose, on a per dimension basis, what unit final answers should be rendered in. It will separate units instead of intermixing them in the display, allowing the user to readily identify the units associated with a particular answer. It will allow a user to simplify an expression to its fundamental Base Units, as well as providing fine control over simplifying to derived units. Dimensional analysis is possible, and a variety of tools are available to manage conversion and simplification options. In addition to customizable automatic conversion, units also provides a traditional manual conversion option.
Note - when unit conversions are inexact Maxima will make approximations resulting in fractions. This is a consequence of the techniques used to simplify units. The messages warning of this type of substitution are disabled by default in the case of units (normally they are on) since this situation occurs frequently and the warnings clutter the output. (The existing state of ratprint is restored after unit conversions, so user changes to that setting will be preserved otherwise.) If the user needs this information for units, they can set unitverbose:on to reactivate the printing of warnings from the unit conversion process.
unit is included in Maxima in the share/contrib/unit directory. It obeys normal Maxima package loading conventions:
(%i1) load("unit")$
*******************************************************************
* Units version 0.50 *
* Definitions based on the NIST Reference on *
* Constants, Units, and Uncertainty *
* Conversion factors from various sources including *
* NIST and the GNU units package *
*******************************************************************
Redefining necessary functions...
WARNING: DEFUN/DEFMACRO: redefining function TOPLEVEL-MACSYMA-EVAL ...
WARNING: DEFUN/DEFMACRO: redefining function MSETCHK ...
WARNING: DEFUN/DEFMACRO: redefining function KILL1 ...
WARNING: DEFUN/DEFMACRO: redefining function NFORMAT ...
Initializing unit arrays...
Done.
The WARNING messages are expected and not a cause for concern - they indicate the unit package is redefining functions already defined in Maxima proper. This is necessary in order to properly handle units. The user should be aware that if other changes have been made to these functions by other packages those changes will be overwritten by this loading process.
The unit.mac file also loads a lisp file unit-functions.lisp which contains the lisp functions needed for the package.
Clifford Yapp is the primary author. He has received valuable assistance from Barton Willis of the University of Nebraska at Kearney (UNK), Robert Dodier, and other intrepid folk of the Maxima mailing list.
There are probably lots of bugs. Let me know. float and numer
don't do what is expected.
TODO : dimension functionality, handling of temperature, showabbr and friends. Show examples with addition of quantities containing units.
Categories: Physical units · Share packages · Package unit
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By default, the unit package does not use any derived dimensions, but will convert all units to the seven fundamental dimensions using MKS units.
(%i2) N;
kg m
(%o2) ----
2
s
(%i3) dyn;
1 kg m
(%o3) (------) (----)
100000 2
s
(%i4) g;
1
(%o4) (----) (kg)
1000
(%i5) centigram*inch/minutes^2;
127 kg m
(%o5) (-------------) (----)
1800000000000 2
s
In some cases this is the desired behavior. If the user wishes to use other
units, this is achieved with the setunits command:
(%i6) setunits([centigram,inch,minute]);
(%o6) done
(%i7) N;
1800000000000 %in cg
(%o7) (-------------) (------)
127 2
%min
(%i8) dyn;
18000000 %in cg
(%o8) (--------) (------)
127 2
%min
(%i9) g;
(%o9) (100) (cg)
(%i10) centigram*inch/minutes^2;
%in cg
(%o10) ------
2
%min
The setting of units is quite flexible. For example, if we want to get back to kilograms, meters, and seconds as defaults for those dimensions we can do:
(%i11) setunits([kg,m,s]);
(%o11) done
(%i12) centigram*inch/minutes^2;
127 kg m
(%o12) (-------------) (----)
1800000000000 2
s
Derived units are also handled by this command:
(%i17) setunits(N);
(%o17) done
(%i18) N;
(%o18) N
(%i19) dyn;
1
(%o19) (------) (N)
100000
(%i20) kg*m/s^2;
(%o20) N
(%i21) centigram*inch/minutes^2;
127
(%o21) (-------------) (N)
1800000000000
Notice that the unit package recognized the non MKS combination of mass, length, and inverse time squared as a force, and converted it to Newtons. This is how Maxima works in general. If, for example, we prefer dyne to Newtons, we simply do the following:
(%i22) setunits(dyn);
(%o22) done
(%i23) kg*m/s^2;
(%o23) (100000) (dyn)
(%i24) centigram*inch/minutes^2;
127
(%o24) (--------) (dyn)
18000000
To discontinue simplifying to any force, we use the uforget command:
(%i26) uforget(dyn);
(%o26) false
(%i27) kg*m/s^2;
kg m
(%o27) ----
2
s
(%i28) centigram*inch/minutes^2;
127 kg m
(%o28) (-------------) (----)
1800000000000 2
s
This would have worked equally well with uforget(N) or
uforget(%force).
See also uforget. To use this function write first load("unit").
Categories: Package unit
By default, the unit package converts all units to the
seven fundamental dimensions using MKS units. This behavior can
be changed with the setunits command. After that, the
user can restore the default behavior for a particular dimension
by means of the uforget command:
(%i13) setunits([centigram,inch,minute]);
(%o13) done
(%i14) centigram*inch/minutes^2;
%in cg
(%o14) ------
2
%min
(%i15) uforget([cg,%in,%min]);
(%o15) [false, false, false]
(%i16) centigram*inch/minutes^2;
127 kg m
(%o16) (-------------) (----)
1800000000000 2
s
uforget operates on dimensions,
not units, so any unit of a particular dimension will work. The
dimension itself is also a legal argument.
See also setunits. To use this function write first load("unit").
Categories: Package unit
When resetting the global environment is overkill, there is the convert
command, which allows one time conversions. It can accept either a single
argument or a list of units to use in conversion. When a convert operation is
done, the normal global evaluation system is bypassed, in order to avoid the
desired result being converted again. As a consequence, for inexact calculations
"rat" warnings will be visible if the global environment controlling this behavior
(ratprint) is true. This is also useful for spot-checking the
accuracy of a global conversion. Another feature is convert will allow a
user to do Base Dimension conversions even if the global environment is set to
simplify to a Derived Dimension.
(%i2) kg*m/s^2;
kg m
(%o2) ----
2
s
(%i3) convert(kg*m/s^2,[g,km,s]);
g km
(%o3) ----
2
s
(%i4) convert(kg*m/s^2,[g,inch,minute]);
`rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
18000000000 %in g
(%o4) (-----------) (-----)
127 2
%min
(%i5) convert(kg*m/s^2,[N]);
(%o5) N
(%i6) convert(kg*m^2/s^2,[N]);
(%o6) m N
(%i7) setunits([N,J]);
(%o7) done
(%i8) convert(kg*m^2/s^2,[N]);
(%o8) m N
(%i9) convert(kg*m^2/s^2,[N,inch]);
`rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
5000
(%o9) (----) (%in N)
127
(%i10) convert(kg*m^2/s^2,[J]);
(%o10) J
(%i11) kg*m^2/s^2;
(%o11) J
(%i12) setunits([g,inch,s]);
(%o12) done
(%i13) kg*m/s^2;
(%o13) N
(%i14) uforget(N);
(%o14) false
(%i15) kg*m/s^2;
5000000 %in g
(%o15) (-------) (-----)
127 2
s
(%i16) convert(kg*m/s^2,[g,inch,s]);
`rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
5000000 %in g
(%o16) (-------) (-----)
127 2
s
See also setunits and uforget. To use this function write first load("unit").
Categories: Package unit
Default value: none
If a user wishes to have a default unit behavior other than that described,
they can make use of maxima-init.mac and the usersetunits
variable. The unit package will check on startup to see if this variable
has been assigned a list. If it has, it will use setunits on that list and take
the units from that list to be defaults. uforget will revert to the behavior
defined by usersetunits over its own defaults. For example, if we have a
maxima-init.mac file containing:
usersetunits : [N,J];
we would see the following behavior:
(%i1) load("unit")$
*******************************************************************
* Units version 0.50 *
* Definitions based on the NIST Reference on *
* Constants, Units, and Uncertainty *
* Conversion factors from various sources including *
* NIST and the GNU units package *
*******************************************************************
Redefining necessary functions...
WARNING: DEFUN/DEFMACRO: redefining function
TOPLEVEL-MACSYMA-EVAL ...
WARNING: DEFUN/DEFMACRO: redefining function MSETCHK ...
WARNING: DEFUN/DEFMACRO: redefining function KILL1 ...
WARNING: DEFUN/DEFMACRO: redefining function NFORMAT ...
Initializing unit arrays...
Done.
User defaults found...
User defaults initialized.
(%i2) kg*m/s^2;
(%o2) N
(%i3) kg*m^2/s^2;
(%o3) J
(%i4) kg*m^3/s^2;
(%o4) J m
(%i5) kg*m*km/s^2;
(%o5) (1000) (J)
(%i6) setunits([dyn,eV]);
(%o6) done
(%i7) kg*m/s^2;
(%o7) (100000) (dyn)
(%i8) kg*m^2/s^2;
(%o8) (6241509596477042688) (eV)
(%i9) kg*m^3/s^2;
(%o9) (6241509596477042688) (eV m)
(%i10) kg*m*km/s^2;
(%o10) (6241509596477042688000) (eV)
(%i11) uforget([dyn,eV]);
(%o11) [false, false]
(%i12) kg*m/s^2;
(%o12) N
(%i13) kg*m^2/s^2;
(%o13) J
(%i14) kg*m^3/s^2;
(%o14) J m
(%i15) kg*m*km/s^2;
(%o15) (1000) (J)
Without usersetunits, the initial inputs would have been converted
to MKS, and uforget would have resulted in a return to MKS rules. Instead,
the user preferences are respected in both cases. Notice these can still
be overridden if desired. To completely eliminate this simplification - i.e.
to have the user defaults reset to factory defaults - the dontusedimension
command can be used. uforget can restore user settings again, but
only if usedimension frees it for use. Alternately,
kill(usersetunits) will completely remove all knowledge of the user defaults
from the session. Here are some examples of how these various options work.
(%i2) kg*m/s^2;
(%o2) N
(%i3) kg*m^2/s^2;
(%o3) J
(%i4) setunits([dyn,eV]);
(%o4) done
(%i5) kg*m/s^2;
(%o5) (100000) (dyn)
(%i6) kg*m^2/s^2;
(%o6) (6241509596477042688) (eV)
(%i7) uforget([dyn,eV]);
(%o7) [false, false]
(%i8) kg*m/s^2;
(%o8) N
(%i9) kg*m^2/s^2;
(%o9) J
(%i10) dontusedimension(N);
(%o10) [%force]
(%i11) dontusedimension(J);
(%o11) [%energy, %force]
(%i12) kg*m/s^2;
kg m
(%o12) ----
2
s
(%i13) kg*m^2/s^2;
2
kg m
(%o13) -----
2
s
(%i14) setunits([dyn,eV]);
(%o14) done
(%i15) kg*m/s^2;
kg m
(%o15) ----
2
s
(%i16) kg*m^2/s^2;
2
kg m
(%o16) -----
2
s
(%i17) uforget([dyn,eV]);
(%o17) [false, false]
(%i18) kg*m/s^2;
kg m
(%o18) ----
2
s
(%i19) kg*m^2/s^2;
2
kg m
(%o19) -----
2
s
(%i20) usedimension(N);
Done. To have Maxima simplify to this dimension, use
setunits([unit]) to select a unit.
(%o20) true
(%i21) usedimension(J);
Done. To have Maxima simplify to this dimension, use
setunits([unit]) to select a unit.
(%o21) true
(%i22) kg*m/s^2;
kg m
(%o22) ----
2
s
(%i23) kg*m^2/s^2;
2
kg m
(%o23) -----
2
s
(%i24) setunits([dyn,eV]);
(%o24) done
(%i25) kg*m/s^2;
(%o25) (100000) (dyn)
(%i26) kg*m^2/s^2;
(%o26) (6241509596477042688) (eV)
(%i27) uforget([dyn,eV]);
(%o27) [false, false]
(%i28) kg*m/s^2;
(%o28) N
(%i29) kg*m^2/s^2;
(%o29) J
(%i30) kill(usersetunits);
(%o30) done
(%i31) uforget([dyn,eV]);
(%o31) [false, false]
(%i32) kg*m/s^2;
kg m
(%o32) ----
2
s
(%i33) kg*m^2/s^2;
2
kg m
(%o33) -----
2
s
Unfortunately this wide variety of options is a little confusing at first, but once the user grows used to them they should find they have very full control over their working environment.
Categories: Package unit
Rebuilds global unit lists automatically creating all desired metric units. x is a numerical argument which is used to specify how many metric prefixes the user wishes defined. The arguments are as follows, with each higher number defining all lower numbers' units:
0 - none. Only base units
1 - kilo, centi, milli
(default) 2 - giga, mega, kilo, hecto, deka, deci, centi, milli,
micro, nano
3 - peta, tera, giga, mega, kilo, hecto, deka, deci,
centi, milli, micro, nano, pico, femto
4 - all
Normally, Maxima will not define the full expansion since this results in a
very large number of units, but metricexpandall can be used to
rebuild the list in a more or less complete fashion. The relevant variable
in the unit.mac file is %unitexpand.
Categories: Package unit
Default value: 2
This is the value supplied to metricexpandall during the initial loading
of unit.
Categories: Package unit
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 87.1 Introduction to zeilberger | ||
| 87.2 Functions and Variables for zeilberger |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
zeilberger is a implementation of Zeilberger's algorithm
for definite hypergeometric summation, and also
Gosper's algorithm for indefinite hypergeometric
summation.
zeilberger makes use of the "filtering" optimization method developed by Axel Riese.
zeilberger was developed by Fabrizio Caruso.
load (zeilberger) loads this package.
Categories: Sums and products · Share packages · Package zeilberger
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
zeilberger implements Gosper's algorithm for indefinite hypergeometric summation.
Given a hypergeometric term F_k in k we want to find its hypergeometric
anti-difference, that is, a hypergeometric term f_k such that
F_k = f_(k+1) - f_k.
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
zeilberger implements Zeilberger's algorithm for definite hypergeometric summation.
Given a proper hypergeometric term (in n and k)
F_(n,k)
and a positive integer d we want to find a d-th order linear
recurrence with polynomial coefficients (in n) for
F_(n,k)
and a rational function R in n and k such that
a_0 F_(n,k) + ... + a_d F_(n+d),k = Delta_k(R(n,k) F_(n,k)),
where Delta_k is the k-forward difference operator, i.e., Delta_k(t_k) := t_(k+1) - t_k.
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
There are also verbose versions of the commands which are called by adding one of the following prefixes:
SummaryJust a summary at the end is shown
VerboseSome information in the intermidiate steps
VeryVerboseMore information
ExtraEven more information including information on the linear system in Zeilberger's algorithm
For example:
GosperVerbose, parGosperVeryVerbose,
ZeilbergerExtra, AntiDifferenceSummary.
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Returns the hypergeometric anti-difference of F_k, if it exists.
Otherwise AntiDifference returns no_hyp_antidifference.
Categories: Package zeilberger
Returns the rational certificate R(k) for F_k, that is,
a rational function such that
F_k = R(k+1) F_(k+1) - R(k) F_k,
if it exists.
Otherwise, Gosper returns no_hyp_sol.
Categories: Package zeilberger
Returns the summmation of F_k from k = a to k = b
if F_k has a hypergeometric anti-difference.
Otherwise, GosperSum returns nongosper_summable.
Examples:
(%i1) load (zeilberger)$
(%i2) GosperSum ((-1)^k*k / (4*k^2 - 1), k, 1, n);
Dependent equations eliminated: (1)
3 n + 1
(n + -) (- 1)
2 1
(%o2) - ------------------ - -
2 4
2 (4 (n + 1) - 1)
(%i3) GosperSum (1 / (4*k^2 - 1), k, 1, n);
3
- n - -
2 1
(%o3) -------------- + -
2 2
4 (n + 1) - 1
(%i4) GosperSum (x^k, k, 1, n);
n + 1
x x
(%o4) ------ - -----
x - 1 x - 1
(%i5) GosperSum ((-1)^k*a! / (k!*(a - k)!), k, 1, n);
n + 1
a! (n + 1) (- 1) a!
(%o5) - ------------------------- - ----------
a (- n + a - 1)! (n + 1)! a (a - 1)!
(%i6) GosperSum (k*k!, k, 1, n);
Dependent equations eliminated: (1)
(%o6) (n + 1)! - 1
(%i7) GosperSum ((k + 1)*k! / (k + 1)!, k, 1, n);
(n + 1) (n + 2) (n + 1)!
(%o7) ------------------------ - 1
(n + 2)!
(%i8) GosperSum (1 / ((a - k)!*k!), k, 1, n);
(%o8) NON_GOSPER_SUMMABLE
Categories: Package zeilberger
Attempts to find a d-th order recurrence for F_(n,k).
The algorithm yields a sequence [s_1, s_2, ..., s_m] of solutions. Each solution has the form
[R(n, k), [a_0, a_1, ..., a_d]].
parGosper returns [] if it fails to find a recurrence.
Categories: Package zeilberger
Attempts to compute the indefinite hypergeometric summation of F_(n,k).
Zeilberger first invokes Gosper, and if that fails to find a solution, then invokes
parGosper with order 1, 2, 3, ..., up to MAX_ORD.
If Zeilberger finds a solution before reaching MAX_ORD,
it stops and returns the solution.
The algorithms yields a sequence [s_1, s_2, ..., s_m] of solutions. Each solution has the form
[R(n,k), [a_0, a_1, ..., a_d]].
Zeilberger returns [] if it fails to find a solution.
Zeilberger invokes Gosper only if Gosper_in_Zeilberger is true.
Categories: Package zeilberger
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Default value: 5
MAX_ORD is the maximum recurrence order attempted by Zeilberger.
Categories: Package zeilberger
Default value: false
When simplified_output is true,
functions in the zeilberger package attempt
further simplification of the solution.
Categories: Package zeilberger
Default value: linsolve
linear_solver names the solver which is used to solve the system
of equations in Zeilberger's algorithm.
Categories: Package zeilberger
Default value: true
When warnings is true,
functions in the zeilberger package print
warning messages during execution.
Categories: Package zeilberger
Default value: true
When Gosper_in_Zeilberger is true,
the Zeilberger function calls Gosper before calling parGosper.
Otherwise, Zeilberger goes immediately to parGosper.
Categories: Package zeilberger
Default value: true
When trivial_solutions is true,
Zeilberger returns solutions
which have certificate equal to zero, or all coefficients equal to zero.
Categories: Package zeilberger
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Default value: false
When mod_test is true,
parGosper executes a
modular test for discarding systems with no solutions.
Categories: Package zeilberger
Default value: linsolve
modular_linear_solver names the linear solver used by the modular test in parGosper.
Categories: Package zeilberger
Default value: big_primes[10]
ev_point is the value at which the variable n is evaluated
when executing the modular test in parGosper.
Categories: Package zeilberger
Default value: big_primes[1]
mod_big_prime is the modulus used by the modular test in parGosper.
Categories: Package zeilberger
Default value: 4
mod_threshold is the
greatest order for which the modular test in parGosper is attempted.
Categories: Package zeilberger
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| 88.1 Error messages | ||
| 88.2 Warning messages |
This chapter provides detailed information about the meaning of some error messages or on how to recover from errors.
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| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
part() was used to access the nth item in something that has less than
n items.
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
A function could not be plotted since it still contained a variable maxima doesn't know the value of.
In order to find out which variable this could be it is sometimes helpful to
temporarily replace the name of the drawing command (draw2d, plot2d
or similar) by a random name (for example ddraw2d) that doesn't coincide
with the name of an existing function to make maxima print out what parameters
the drawing command sees.
(%i1) load("draw")$
(%i2) f(x):=sin(omega*t);
(%o2) f(x) := sin(omega t)
(%i3) draw2d(
explicit(
f(x),
x,1,10
)
);
draw2d (explicit): non defined variable
-- an error. To debug this try: debugmode(true);
(%i4) ddraw2d(
explicit(
f(x),
x,1,10
)
);
(%o4) ddraw2d(explicit(sin(omega t), x, 1, 10))
Categories: Error messages
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
This error message normally indicates that the file exists, but can not be read.
If the file is present and readable there is another possible for this error
message: Maxima can compile packages to native binary files in order to make them
run faster. If after compiling the file something in the system has changed in a
way that makes it incompatible with the binary the binary the file cannot be
loaded any more. Maxima normally puts binary files it creates from its own packages
in a folder named binary within the folder whose name it is printed after
typing:
(%i1) maxima_userdir; (%o1) /home/gunter/.maxima
If this directory is missing maxima will recreate it again as soon as it has to compile a package.
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The most probable cause for this error is that there was an attempt to either use a number or a variable whose numerical value is known as a loop counter.
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Lisp typically handles several types of memory containing at least one stack and a heap that contains user objects. To avoid running out of memory several approaches might be useful:
--dynamic-space-size <n> allows to tell
sbcl to reserve n megabytes for the heap. It is to note, though,
that sbcl has to handle several distinct types of memory and therefore
might be able to only reserve about half of the available physical
memory. Also note that 32-bit processes might only be able to access
2GB of physical memory.
Categories: Error messages
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Maxima supports several assignment operators. When trying to define a function
:= has to be used.
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
One common cause for this error message is that square brackets operator
([ ]) was used trying to access a list element that whose element
number was < 1 or > length(list).
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
This might be caused by a command starting with a comma (,) or by one comma
being directly followed by another one..
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
makelist expects the second argument to be the name of the variable whose value is to
be stepped. This time instead of the name of a still-undefined variable maxima has found
something else, possibly a list or the name of a list.
Categories: Error messages
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Common reasons for this error appearing are a closing parenthesis without an opening one or a closing parenthesis directly preceded by a comma.
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
This might either mean that VTK is actually not installed - or cannot be found by maxima - or that maxima has no write access to the directory whose name is output if the following maxima command is entered:
(%i1) maxima_tempdir; (%o1) /home/gunter
Categories: Error messages
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 88.2.1 Encountered undefined variable <x> in translation | ||
| 88.2.2 Rat: replaced <x> by <y> = <z> |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
A function was compiled but the type of the variable x was not known.
This means that the compiled command contains additional code that makes it
retain all the flexibility maxima provides in respect to this variable.
If x isn't meant as a variable name but just a named option to a
command prepending the named option by a single quote (') should
resolve this issue.
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Floating-point numbers provide a maximum number of digits that is typically high,
but still limited. Good examples that this limitation might be too low
even for harmless-looking examples include
Wilkinson's Polynomial,
The Rump polynomial and the fact that an exact 1/10 cannot be expressed as a binary
floating-point number.
In places where the floating-point error might add up or hinder terms from
cancelling each other out maxima therefore by default replaces them with exact
fractions.
See also ratprint, ratepsilon, bftorat, fpprintprec
and rationalize.
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| Jump to: | !
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A B C D E F G H I J K L M N O P Q R S T U V W X Y Z |
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Introduction to Special Functions airy_ai airy_bi airy_dai airy_dbi
%rnum_list algepsilon algexact algsys backsubst breakup dispflag eliminate bf_find_root find_root find_root_abs find_root_error find_root_rel funcsolve multiplicities newton programmode realonly solve solvedecomposes solveexplicit solvefactors solvenullwarn solveradcan solvetrigwarn
array arrayapply arrayinfo arraymake arrays arraysetapply fillarray listarray make_array rearray remarray subvar translate_fast_arrays use_fast_arrays
%f %s Introduction to Special Functions bessel_i bessel_j bessel_k bessel_simplify bessel_y besselexpand hankel_1 hankel_2 scaled_bessel_i scaled_bessel_i0 scaled_bessel_i1 spherical_bessel_j spherical_bessel_y spherical_hankel1 spherical_hankel2
Introduction to Numbers cabs carg cbffac conjugate demoivre exponentialize imagpart plog polarform polartorect realpart rectform recttopolar residue
% %th ? ?? Documentation Interrupts _ appendfile closefile demo describe entermatrix example ibase kill labels labels linenum myoptions obase optionset playback printfile prompt quit ratprint read readonly refcheck setcheck setcheckbreak setval to_lisp writefile
%e %gamma %i %phi %pi constant constantp false ind inf infinity minf true und zeroa zerob
Keyword Commands Source Level Debugging backtrace break bug_report build_info debugmode error_size error_syms room run_testsuite share_testsuite_files showtime testsuite_files time timer timer_devalue timer_info trace trace_options untimer untrace
activate activecontexts additive alias aliases alphabetic antisymmetric askexp askinteger asksign assume assume_pos assume_pos_pred assumescalar atvalue commutative compare constant context contexts deactivate declare decreasing increasing dependencies depends even odd facts featurep features forget get gradef gradefs infix infolists integer noninteger integervalued is killcontext lassociative linear mainvar matchdeclare maybe multiplicative newcontext nonscalar numerval outative posfun printprops properties props propvars put qput rassociative irrational rational complex imaginary real rem remove scalar sign supcontext symmetric
atomgrad del depends derivabbrev derivdegree derivlist derivsubst diff dscalar express gradef gradefs hessian implicit_derivative jacobian vect_cross wronskian
Introduction to contrib_ode Introduction to drawdf Introduction to numerical solution of differential equations Lindstedt at atvalue bc2 desolve ic1 ic2 laplace ode2 plotdf ploteq rk
%edispflag absboxchar dispflag display2d display_format_internal error_size error_syms exptdispflag fpprintprec grind inchar labels leftjust linechar linel linenum lispdisp lmxchar negsumdispflag noundisp obase outchar pfeformat powerdisp psexpand rmxchar showtime sqrtdispflag stardisp stringdisp ttyoff
disp dispcon dispfun display disprule dispterms engineering_format_floats grind labels ldisp ldisplay playback print printpois printprops reveal show showratvars
Introduction to Elliptic Functions and Integrals inverse_jacobi_cd inverse_jacobi_cn inverse_jacobi_cs inverse_jacobi_dc inverse_jacobi_dn inverse_jacobi_ds inverse_jacobi_nc inverse_jacobi_nd inverse_jacobi_ns inverse_jacobi_sc inverse_jacobi_sd inverse_jacobi_sn jacobi_cd jacobi_cn jacobi_cs jacobi_dc jacobi_dn jacobi_ds jacobi_nc jacobi_nd jacobi_ns jacobi_sc jacobi_sd jacobi_sn
elliptic_e elliptic_ec elliptic_eu elliptic_f elliptic_kc elliptic_pi
Comma is not a prefix operator Illegal use of delimiter No such list element VTK is not installed loadfile failed to load makelist second argument must evaluate to a number out of memory part fell off the end undefined variable during plotting
%enumer detout eval evflag evfun float infeval noeval nouns numer pred simp
expintegral_chi expintegral_ci expintegral_e expintegral_e1 expintegral_e_simplify expintegral_ei expintegral_li expintegral_shi expintegral_si expintexpand expintrep
%e_to_numlog %edispflag %emode %enumer exp exptsubst li log logabs logarc logconcoeffp logcontract logexpand lognegint logsimp plog polarform taylor_logexpand
Package facexp Package rducon Package scifac append args arrayapply arraymake arraysetapply assoc atom block box boxchar collapse collectterms combine concat cons delete denom derivdegree derivsubst disolate dispform distrib dontfactor dpart eighth endcons expand expandwrt expandwrt_denom expandwrt_factored expon expop exptdispflag exptisolate exptsubst facsum facsum_combine factorfacsum factorout factorsum fifth first fourth freeof fullmap fullmapl funmake gcfac gfactorsum hipow inflag inpart intosum isolate isolate_wrt_times last length lfreeof lhs linear listconstvars listdummyvars listofvars lopow lpart m1pbranch mainvar maxnegex maxposex member multiplicative multthru nextlayerfactor ninth nonarray nonzeroandfreeof nterms num numfactor op opsubst opsubst optimize optimprefix ordergreat orderless ordergreatp orderlessp part partition partswitch pickapart piece pop psubst push rational reduce_consts rembox rempart rest reveal reverse rhs rncombine rootsconmode sconcat second seventh sixth sqrtdenest sublis sublis_apply_lambda subnumsimp subst substinpart substpart subvar tenth third unorder xthru
Introduction to numericalio batch batchload close demo directory file_search file_search_demo file_search_lisp file_search_maxima file_search_tests file_search_usage file_type filename_merge flength fposition load load_pathname loadfile loadprint openr openr_binary pathname_directory pathname_name pathname_type printfile read_array read_binary_array read_binary_list read_binary_matrix read_hashed_array read_list read_matrix read_nested_list readbyte readchar readline setup_autoload
Introduction to numericalio appendfile close closefile draw draw2d draw3d draw_file file_output_append filename_merge flength flush_output fposition freshline multiplot_mode newline opena opena_binary openw openw_binary printf save stringout tex with_stdout write_binary_data write_data writebyte writefile
Introduction to Fourier series Introduction to fast Fourier transform
apply fullmap fullmapl funmake macroexpand macroexpand1 macroexpansion map maperror maplist mapprint maxapplydepth maxapplyheight outermap scanmap
::= := Function buildq define dispfun functions fundef lambda local macros remfunction setup_autoload splice
!! beta beta_args_sum_to_integer beta_expand beta_incomplete beta_incomplete_generalized beta_incomplete_regularized bffac bfpsi bfpsi0 cbffac factcomb factlim ! factorial factorial_expand gamma gamma_expand gamma_greek gamma_incomplete gamma_incomplete_generalized gamma_incomplete_regularized gammalim genfact log_gamma makefact makegamma maxpsifracdenom maxpsifracnum maxpsinegint maxpsiposint pochhammer pochhammer_max_index psi stirling sumsplitfact
debugmode derivabbrev engineering_format_floats file_output_append listarith loadprint logabs m1pbranch macroexpansion nolabels optionset ratvarswitch refcheck setcheck setcheckbreak timer_devalue use_fast_arrays
% %% _ __ adim aform aliases arrays asymbol dependencies file_search_demo file_search_lisp file_search_maxima file_search_tests file_search_usage functions infolists macros manual_demo maxima_tempdir maxima_userdir myoptions newline opproperties prompt props setval share_testsuite_files space structures tab testsuite_files us_ascii_only values
%iargs acosh acoth acsch asech asinh atanh cosh coth csch demoivre exponentialize sech sinh tanh
divisors gcfactor integer_partitions intfaclim modulus moebius multinomial_coeff num_distinct_partitions num_partitions stirling1 stirling2
Introduction to QUADPACK absint antid antidiff changevar dblint defint erfflag intanalysis integrate integrate_use_rootsof integration_constant integration_constant_counter ldefint logabs residue risch tldefint
augcoefmatrix coefmatrix dgesv echelon globalsolve linsolve linsolve_params linsolvewarn triangularize
[ ] append assoc cons copylist create_list delete eighth endcons fifth first flatten fourth fullsetify join last length listarith listp lmax lmin lreduce makelist member ninth permut permutations pop push random_permutation rest reverse rreduce second setify seventh sixth some sort sublist sublist_indices tenth third tree_reduce xreduce
abs agd ceiling charfun combination covers delta entier exsec fix floor gaussprob gd hav isqrt lmax lmin max min mod permutation round signum sqrt sqrtdispflag truncate unit_step vers
addcol addrow adjoint augcoefmatrix cauchy_matrix charpoly coefmatrix col columnvector covect copymatrix determinant detout diag diagmatrix doallmxops domxexpt domxmxops domxnctimes doscmxops doscmxplus echelon eigen ematrix entermatrix genmatrix ident invert list_matrix_entries lmxchar matrix matrix_element_add matrix_element_mult matrix_element_transpose matrixexp matrixmap matrixp mattrace minor ncharpoly newdet nonscalar nonscalarp permanent rank ratmx row scalarmatrixp scalarp setelmx sparse submatrix tracematrix transpose triangularize zeromatrix
bern bernpoly bfhzeta bfzeta binomial bit_and bit_length bit_lsh bit_not bit_onep bit_or bit_rsh bit_xor burn chinese divsum euler factors_only fib fibtophi ifactors igcdex inrt inv_mod jacobi lcm lucas minfactorial next_prime power_mod prev_prime primep primep_number_of_tests primes qunit totient zerobern zeta zeta%pi zn_add_table zn_carmichael_lambda zn_characteristic_factors zn_determinant zn_factor_generators zn_invert_by_lu zn_log zn_mult_table zn_nth_root zn_order zn_power_table zn_primroot zn_primroot_limit zn_primroot_p zn_primroot_pretest zn_primroot_verbose
bffac bfhzeta bfloat bfloatp bfpsi bfpsi0 bftorat bftrunc bfzeta cbffac float float2bf floatnump fpprec fpprintprec keepfloat numer numer_pbranch numerval ratepsilon rationalize ratprint stats_numer
Introduction to QUADPACK Introduction to cobyla Introduction to fast Fourier transform Introduction to interpol Introduction to lapack Introduction to lbfgs Introduction to minpack Introduction to mnewton Introduction to numerical solution of differential equations Introduction to simplex allroots augmented_lagrangian_method bfallroots bf_find_root find_root find_root_abs find_root_error find_root_rel horner linear_program lsquares_estimates lsquares_estimates_approximate maximize_lp minimize_lp newton nroots plotdf ploteq plsquares polyfactor quad_control quad_qag quad_qagi quad_qagp quad_qags quad_qawc quad_qawf quad_qawo quad_qaws random realroots rk romberg rootsepsilon
!! # ' " ** * + - / ^ . : :: ::= := < <= > >= = @ Introduction to operators [ ] ^^ additive and antisymmetric commutative equal express ! factorial infix lassociative linear matchfix nary nofix not notequal op operatorp opproperties or outative posfun postfix prefix rassociative symmetric | ~
Introduction to cobyla Introduction to lbfgs Introduction to minpack Introduction to simplex augmented_lagrangian_method
Introduction to Affine all_dotsimp_denoms check_overlaps declare_weights dotsimp extract_linear_equations fast_central_elements fast_linsolve grobner_basis list_nc_monomials mono monomial_dimensions nc_degree set_up_dot_simplifications
define_alt_display info_display mathml_display multi_display_for_texinfo reset_displays set_alt_display set_prompt tex_display
Introduction to atensor abasep adim af aform alg_type asymbol atensimp av init_atensor sf
%c %k1 %k2 Introduction to contrib_ode bessel_simplify contrib_ode dgauss_a dgauss_b dkummer_m dkummer_u expintegral_e_simplify gauss_a gauss_b kummer_m kummer_u method ode_check odelin
Introduction to ctensor bdvac bimetric cdisplay cframe_flag cgeodesic checkdiv christof cmetric cnonmet_flag cograd contortion contragrad csetup ct_coords ct_coordsys ctaylor ctaypov ctaypt ctayswitch ctayvar ctorsion_flag ctransform ctrgsimp deleten diagmatrixp diagmetric dim dscalar dscalar einstein fb findde frame_bracket gdet ic_convert init_ctensor invariant1 invariant2 kinvariant kt leinstein lfg lg lriem lriemann nm nmc nonmetricity np npi nptetrad ntermst petrov psi ratchristof rateinstein ratriemann ratweyl ric ricci riem riemann rinvariant scurvature symmetricp tensorkill tr ufg ug uric uricci uriem uriemann weyl weyl
Introduction to descriptive barsplot barsplot_description boxplot boxplot_description build_sample central_moment continuous_freq cor cov cov1 cv discrete_freq geometric_mean global_variances harmonic_mean histogram histogram_description kurtosis list_correlations mean mean_deviation median median_deviation noncentral_moment pearson_skewness piechart piechart_description principal_components qrange quantile quartile_skewness range scatterplot scatterplot_description skewness smax smin standardize starplot starplot_description std std1 stemplot subsample transform_sample var var1
JF ModeMatrix diag dispJordan jordan mat_function minimalPoly
Introduction to distrib cdf_bernoulli cdf_beta cdf_binomial cdf_cauchy cdf_chi2 cdf_continuous_uniform cdf_discrete_uniform cdf_exp cdf_f cdf_gamma cdf_general_finite_discrete cdf_geometric cdf_gumbel cdf_hypergeometric cdf_laplace cdf_logistic cdf_lognormal cdf_negative_binomial cdf_noncentral_chi2 cdf_noncentral_student_t cdf_normal cdf_pareto cdf_poisson cdf_rayleigh cdf_student_t cdf_weibull kurtosis_bernoulli kurtosis_beta kurtosis_binomial kurtosis_chi2 kurtosis_continuous_uniform kurtosis_discrete_uniform kurtosis_exp kurtosis_f kurtosis_gamma kurtosis_general_finite_discrete kurtosis_geometric kurtosis_gumbel kurtosis_gumbel kurtosis_hypergeometric kurtosis_laplace kurtosis_logistic kurtosis_lognormal kurtosis_negative_binomial kurtosis_noncentral_chi2 kurtosis_noncentral_student_t kurtosis_normal kurtosis_pareto kurtosis_poisson kurtosis_rayleigh kurtosis_student_t kurtosis_weibull mean_bernoulli mean_beta mean_binomial mean_chi2 mean_continuous_uniform mean_discrete_uniform mean_exp mean_f mean_gamma mean_general_finite_discrete mean_geometric mean_gumbel mean_hypergeometric mean_laplace mean_logistic mean_lognormal mean_negative_binomial mean_noncentral_chi2 mean_noncentral_student_t mean_normal mean_pareto mean_poisson mean_rayleigh mean_student_t mean_weibull pdf_bernoulli pdf_beta pdf_binomial pdf_cauchy pdf_chi2 pdf_continuous_uniform pdf_discrete_uniform pdf_exp pdf_f pdf_gamma pdf_general_finite_discrete pdf_geometric pdf_gumbel pdf_hypergeometric pdf_laplace pdf_logistic pdf_lognormal pdf_negative_binomial pdf_noncentral_chi2 pdf_noncentral_student_t pdf_normal pdf_pareto pdf_poisson pdf_rayleigh pdf_student_t pdf_weibull quantile_bernoulli quantile_beta quantile_binomial quantile_cauchy quantile_chi2 quantile_continuous_uniform quantile_discrete_uniform quantile_exp quantile_f quantile_gamma quantile_general_finite_discrete quantile_geometric quantile_gumbel quantile_hypergeometric quantile_laplace quantile_logistic quantile_lognormal quantile_negative_binomial quantile_noncentral_chi2 quantile_noncentral_student_t quantile_normal quantile_pareto quantile_poisson quantile_rayleigh quantile_student_t quantile_weibull random_bernoulli random_beta random_binomial random_cauchy random_chi2 random_continuous_uniform random_discrete_uniform random_exp random_f random_gamma random_general_finite_discrete random_geometric random_gumbel random_hypergeometric random_laplace random_logistic random_lognormal random_negative_binomial random_noncentral_chi2 random_noncentral_student_t random_normal random_pareto random_poisson random_rayleigh random_student_t random_weibull skewness_bernoulli skewness_beta skewness_binomial skewness_chi2 skewness_continuous_uniform skewness_discrete_uniform skewness_exp skewness_f skewness_gamma skewness_general_finite_discrete skewness_geometric skewness_gumbel skewness_hypergeometric skewness_laplace skewness_logistic skewness_lognormal skewness_negative_binomial skewness_noncentral_chi2 skewness_noncentral_student_t skewness_normal skewness_pareto skewness_poisson skewness_rayleigh skewness_student_t skewness_weibull std_bernoulli std_beta std_binomial std_chi2 std_continuous_uniform std_discrete_uniform std_exp std_f std_gamma std_general_finite_discrete std_geometric std_gumbel std_hypergeometric std_laplace std_logistic std_lognormal std_negative_binomial std_noncentral_chi2 std_noncentral_student_t std_normal std_pareto std_poisson std_rayleigh std_student_t std_weibull var_bernoulli var_beta var_binomial var_chi2 var_continuous_uniform var_discrete_uniform var_exp var_f var_gamma var_general_finite_discrete var_geometric var_gumbel var_hypergeometric var_laplace var_logistic var_lognormal var_negative_binomial var_noncentral_chi2 var_noncentral_student_t var_normal var_pareto var_poisson var_rayleigh var_student_t var_weibull
Introduction to draw Introduction to drawdf adapt_depth allocation axis_3d axis_bottom axis_left axis_right axis_top background_color bars border boundaries_array capping cbrange cbtics color colorbox columns contour contour_levels cylindrical data_file_name delay dimensions draw draw2d draw3d draw_file draw_realpart elevation_grid ellipse enhanced3d errors explicit file_name fill_color filled_func font font_size geomap get_pixel gnuplot_file_name gr2d gr3d grid head_angle head_both head_length head_type image implicit interpolate_color ip_grid ip_grid_in key key_pos label label_alignment label_orientation line_type line_width logcb logx logx_secondary logy logy_secondary logz make_level_picture make_poly_continent make_poly_country make_polygon make_rgb_picture mesh multiplot_mode negative_picture nticks numbered_boundaries palette parametric parametric_surface picture_equalp picturep point_size point_type points points_joined polar polygon proportional_axes quadrilateral read_xpm rectangle region_boundaries region_boundaries_plus rgb2level set_draw_defaults spherical surface_hide take_channel terminal title transform transparent triangle tube unit_vectors user_preamble vector view wired_surface x_voxel xaxis xaxis_color xaxis_secondary xaxis_type xaxis_width xlabel xlabel_secondary xrange xrange_secondary xtics xtics_axis xtics_rotate xtics_rotate_secondary xtics_secondary xtics_secondary_axis xu_grid xy_file xyplane y_voxel yaxis yaxis_color yaxis_secondary yaxis_type yaxis_width ylabel ylabel_secondary yrange yrange_secondary ytics ytics_axis ytics_rotate ytics_rotate_secondary ytics_secondary ytics_secondary_axis yv_grid z_voxel zaxis zaxis_color zaxis_type zaxis_width zlabel zlabel_rotate zrange ztics ztics_axis ztics_rotate
The dynamics package animation azimuth background capping center chaosgame color cone cube cylinder elevation endphi endtheta evolution evolution2d height height ifs julia linewidth mandelbrot opacity orbits orientation origin phiresolution points pointsize position radius resolution restart scale scene sphere staircase startphi starttheta surface thetaresolution track tstep width windowname windowtitle wireframe xlength ylength zlength
eigen eigenvalues eivals eigenvectors eivects gramschmidt innerproduct inprod similaritytransform simtran ueivects uniteigenvectors unitvector uvect
Introduction to ezunits ` `` constvalue declare_constvalue declare_dimensions declare_fundamental_dimensions fundamental_dimensions remove_fundamental_dimensions declare_fundamental_units remove_fundamental_units declare_qty declare_unit_conversion declare_units dimensionless dimensions dimensions_as_list fundamental_units natural_unit qty remove_constvalue remove_dimensions unitp units
Package facexp collectterms facsum facsum_combine factorfacsum nextlayerfactor
Introduction to fast Fourier transform bf_fft bf_inverse_fft bf_inverse_real_fft bf_real_fft fft inverse_fft inverse_real_fft polartorect real_fft recttopolar
amortization annuity_fv annuity_pv arit_amortization benefit_cost days360 fv geo_amortization geo_annuity_fv geo_annuity_pv graph_flow irr npv pv saving
Introduction to Fourier series absint cosnpiflag equalp fourcos fourexpand fourier fourint fourintcos fourintsin foursimp foursin funp remfun sinnpiflag totalfourier
fernfale hilbertmap julia_parameter julia_set julia_sin mandelbrot_set sierpinskiale sierpinskimap snowmap treefale
Package functs agd arithmetic arithsum combination covers exsec gaussprob gcdivide gd geometric geosum harmonic hav nonzeroandfreeof permutation rational rempart tracematrix vers wronskian
circulant_graph clebsch_graph complement_graph complete_bipartite_graph complete_graph copy_graph cube_graph cuboctahedron_graph cycle_digraph cycle_graph dodecahedron_graph empty_graph flower_snark from_adjacency_matrix frucht_graph graph_product graph_union great_rhombicosidodecahedron_graph great_rhombicuboctahedron_graph grid_graph grotzch_graph heawood_graph icosahedron_graph icosidodecahedron_graph induced_subgraph line_graph make_graph mycielski_graph new_graph path_digraph path_graph petersen_graph random_bipartite_graph random_digraph random_graph random_graph1 random_network random_regular_graph random_tournament random_tree small_rhombicosidodecahedron_graph small_rhombicuboctahedron_graph snub_cube_graph snub_dodecahedron_graph truncated_cube_graph truncated_dodecahedron_graph truncated_icosahedron_graph truncated_tetrahedron_graph tutte_graph underlying_graph wheel_graph
draw_graph_program edge_color edge_coloring edge_partition edge_type edge_width file_name fixed_vertices head_angle head_length label_alignment program redraw show_edge_color show_edge_type show_edge_width show_edges show_id show_label show_vertex_color show_vertex_size show_vertex_type show_vertices show_weight spring_embedding_depth terminal vertex_color vertex_coloring vertex_partition vertex_size vertex_type
dimacs_export dimacs_import graph6_decode graph6_encode graph6_export graph6_import sparse6_decode sparse6_encode sparse6_export sparse6_import
add_edge add_edges add_vertex add_vertices connect_vertices contract_edge remove_edge
adjacency_matrix average_degree biconnected_components bipartition chromatic_index chromatic_number clear_edge_weight clear_vertex_label connected_components degree_sequence diameter edge_coloring edge_connectivity edges get_edge_weight get_vertex_label girth graph_center graph_charpoly graph_eigenvalues graph_order graph_periphery graph_size hamilton_cycle hamilton_path in_neighbors is_biconnected is_bipartite is_connected is_digraph is_edge_in_graph is_graph is_graph_or_digraph is_isomorphic is_planar is_sconnected is_tree is_vertex_in_graph isomorphism laplacian_matrix max_clique max_degree max_flow max_independent_set max_matching min_degree min_edge_cut min_vertex_cover min_vertex_cut minimum_spanning_tree neighbors odd_girth out_neighbors planar_embedding radius set_edge_weight set_vertex_label shortest_path shortest_weighted_path strong_components topological_sort vertex_coloring vertex_connectivity vertex_degree vertex_distance vertex_eccentricity vertex_in_degree vertex_out_degree vertices wiener_index
Introduction to graphs add_edge add_edges add_vertex add_vertices adjacency_matrix average_degree biconnected_components bipartition chromatic_index chromatic_number circulant_graph clear_edge_weight clear_vertex_label clebsch_graph complement_graph complete_bipartite_graph complete_graph connect_vertices connected_components contract_edge copy_graph create_graph cube_graph cuboctahedron_graph cycle_digraph cycle_graph degree_sequence diameter dimacs_export dimacs_import dodecahedron_graph draw_graph draw_graph_program edge_color edge_coloring edge_coloring edge_connectivity edge_partition edge_type edge_width edges empty_graph file_name fixed_vertices flower_snark from_adjacency_matrix frucht_graph get_edge_weight get_vertex_label girth graph6_decode graph6_encode graph6_export graph6_import graph_center graph_charpoly graph_eigenvalues graph_order graph_periphery graph_product graph_size graph_union great_rhombicosidodecahedron_graph great_rhombicuboctahedron_graph grid_graph grotzch_graph hamilton_cycle hamilton_path head_angle head_length heawood_graph icosahedron_graph icosidodecahedron_graph in_neighbors induced_subgraph is_biconnected is_bipartite is_connected is_digraph is_edge_in_graph is_graph is_graph_or_digraph is_isomorphic is_planar is_sconnected is_tree is_vertex_in_graph isomorphism label_alignment laplacian_matrix line_graph make_graph max_clique max_degree max_flow max_independent_set max_matching min_degree min_edge_cut min_vertex_cover min_vertex_cut minimum_spanning_tree mycielski_graph neighbors new_graph odd_girth out_neighbors path_digraph path_graph petersen_graph planar_embedding print_graph program radius random_bipartite_graph random_digraph random_graph random_graph1 random_network random_regular_graph random_tournament random_tree redraw remove_edge remove_vertex set_edge_weight set_vertex_label shortest_path shortest_weighted_path show_edge_color show_edge_type show_edge_width show_edges show_id show_label show_vertex_color show_vertex_size show_vertex_type show_vertices show_weight small_rhombicosidodecahedron_graph small_rhombicuboctahedron_graph snub_cube_graph snub_dodecahedron_graph sparse6_decode sparse6_encode sparse6_export sparse6_import spring_embedding_depth strong_components terminal topological_sort truncated_cube_graph truncated_dodecahedron_graph truncated_icosahedron_graph truncated_tetrahedron_graph tutte_graph underlying_graph vertex_color vertex_coloring vertex_coloring vertex_connectivity vertex_degree vertex_distance vertex_eccentricity vertex_in_degree vertex_out_degree vertex_partition vertex_size vertex_type vertices vertices_to_cycle vertices_to_path wheel_graph wiener_index
Introduction to grobner poly_add poly_buchberger poly_buchberger_criterion poly_coefficient_ring poly_colon_ideal poly_content poly_depends_p poly_elimination_ideal poly_elimination_order poly_exact_divide poly_expand poly_expt poly_gcd poly_grobner poly_grobner_algorithm poly_grobner_debug poly_grobner_equal poly_grobner_member poly_grobner_subsetp poly_ideal_intersection poly_ideal_polysaturation poly_ideal_polysaturation1 poly_ideal_saturation poly_ideal_saturation1 poly_lcm poly_minimization poly_monomial_order poly_multiply poly_normal_form poly_normalize poly_normalize_list poly_polysaturation_extension poly_primary_elimination_order poly_primitive_part poly_pseudo_divide poly_reduced_grobner poly_reduction poly_return_term_list poly_s_polynomial poly_saturation_extension poly_secondary_elimination_order poly_subtract poly_top_reduction_only
Introduction to interpol charfun2 cspline lagrange linearinterpol ratinterpol
Introduction to itensor allsym canform canten changename components concan conmetderiv contract coord covdiff decsym defcon diff entertensor evundiff extdiff flipflag flush flush1deriv flushd flushnd hodge ic_convert icc1 icc2 ichr1 ichr2 icounter icurvature idiff idim idummy idummyx ifb ifc1 ifc2 ifg ifgi ifr iframe_bracket_form iframes ifri igeodesic_coords igeowedge_flag ikt1 ikt2 imetric indexed_tensor indices inm inmc1 inmc2 ishow itr kdels kdelta lc2kdt lc_l lc_u levi_civita liediff listoftens lorentz_gauge makebox rediff remcomps remcon remcoord remsym rename show showcomps simpmetderiv tentex undiff | ~
Introduction to lapack dgeev dgemm dgeqrf dgesv dgesvd dlange zlange zgeev zheev
Introduction to lbfgs lbfgs lbfgs_ncorrections lbfgs_nfeval_max
Introduction to linearalgebra addmatrices blockmatrixp cholesky columnop columnspace columnswap copy ctranspose diag_matrix dotproduct eigens_by_jacobi get_lu_factors hankel hessian hilbert_matrix identfor invert_by_lu jacobian kronecker_product listp locate_matrix_entry lu_backsub lu_factor mat_cond mat_fullunblocker mat_norm mat_trace mat_unblocker matrix_size matrixp nonnegintegerp nullity nullspace orthogonal_complement polynomialp polytocompanion ptriangularize rank rowop rowswap toeplitz vandermonde_matrix zerofor zeromatrixp
Introduction to lsquares lsquares_estimates lsquares_estimates_approximate lsquares_estimates_exact lsquares_mse lsquares_residual_mse lsquares_residuals plsquares
Introduction to numericalio assume_external_byte_order opena_binary openr_binary openw_binary read_array read_binary_array read_binary_list read_binary_matrix read_hashed_array read_list read_matrix read_nested_list write_binary_data write_data
Introduction to orthogonal polynomials assoc_legendre_p assoc_legendre_q chebyshev_t chebyshev_u gen_laguerre hermite intervalp jacobi_p laguerre legendre_p legendre_q orthopoly_recur orthopoly_returns_intervals orthopoly_weight pochhammer pochhammer_max_index spherical_bessel_j spherical_bessel_y spherical_hankel1 spherical_hankel2 spherical_harmonic ultraspherical unit_step
Introduction to QUADPACK quad_control quad_qag quad_qagi quad_qagp quad_qags quad_qawc quad_qawf quad_qawo quad_qaws
Introduction to simplex epsilon_lp linear_program maximize_lp minimize_lp nonegative_lp pivot_count_sx pivot_max_sx scale_lp
Introduction to solve_rec product_use_gamma reduce_order simplify_products simplify_sum solve_rec solve_rec_rat summand_to_rec
Introduction to stats cdf_rank_sum cdf_signed_rank inference_result inferencep items_inference linear_regression pdf_rank_sum pdf_signed_rank stats_numer take_inference test_mean test_means_difference test_normality test_proportion test_proportions_difference test_rank_sum test_sign test_signed_rank test_variance test_variance_ratio
Introduction to String Processing adjust_external_format alphacharp alphanumericp ascii base64 base64_decode cequal cequalignore cgreaterp cgreaterpignore charat charlist charp cint clessp clesspignore close constituent crc24sum digitcharp eval_string flength flush_output fposition freshline get_output_stream_string lowercasep make_string_input_stream make_string_output_stream md5sum mgf1_sha1 newline newline number_to_octets octets_to_number octets_to_oid octets_to_string oid_to_octets opena openr openw parse_string printf readbyte readchar readline scopy sdowncase sequal sequalignore sexplode sha1sum sha256sum simplode sinsert sinvertcase slength smake smismatch space split sposition sprint sremove sremovefirst sreverse ssearch ssort ssubst ssubstfirst strim striml strimr string_to_octets stringp substring supcase tab tokens unicode unicode_to_utf8 uppercasep us_ascii_only utf8_to_unicode writebyte
Introduction to Symmetries comp2pui cont2part contract direct ele2comp ele2polynome ele2pui elem explose kostka lgtreillis ltreillis mon2schur multi_elem multi_orbit multi_pui multinomial multsym orbit part2cont partpol permut polynome2ele prodrac pui pui2comp pui2ele pui2polynome pui_direct puireduc resolvante resolvante_alternee1 resolvante_bipartite resolvante_diedrale resolvante_klein resolvante_klein3 resolvante_produit_sym resolvante_unitaire resolvante_vierer schur2comp somrac tcontract tpartpol treillis treinat
%unitexpand Introduction to Units convert metricexpandall setunits uforget usersetunits
AntiDifference Gosper GosperSum Gosper_in_Zeilberger Introduction to zeilberger MAX_ORD Zeilberger Zeilberger ev_point linear_solver mod_big_prime mod_test mod_threshold modular_linear_solver parGosper parGosper simplified_output trivial_solutions warnings
Introduction to Units Introduction to ezunits Introduction to physical_constants
Introduction to Plotting Introduction to draw Introduction to drawdf Introduction to numerical solution of differential equations Introduction to orthogonal polynomials Plotting Formats adapt_depth animation axes azimuth azimuth background barsplot barsplot_description bode_gain bode_phase box boxplot boxplot_description capping center chaosgame color color color_bar color_bar_tics cone contour_plot cube cylinder elevation elevation endphi endtheta evolution evolution2d gnuplot_close gnuplot_command gnuplot_curve_styles gnuplot_curve_titles gnuplot_default_term_command gnuplot_dumb_term_command gnuplot_file_args gnuplot_out_file gnuplot_pdf_term_command gnuplot_pm3d gnuplot_png_term_command gnuplot_postamble gnuplot_preamble gnuplot_ps_term_command gnuplot_replot gnuplot_reset gnuplot_restart gnuplot_start gnuplot_svg_term_command gnuplot_term gnuplot_view_args grid grid2d height height histogram histogram_description ifs implicit_plot iterations julia label legend linewidth logx logy make_transform mandelbrot mesh_lines_color nticks opacity orbits orientation origin palette pdf_file phiresolution piechart piechart_description plot2d plot3d plot_format plot_options plot_realpart plotdf ploteq png_file point_type points pointsize polar_to_xy position ps_file radius remove_plot_option resolution restart run_viewer same_xy same_xyz scale scatterplot scatterplot_description scene set_plot_option sphere spherical_to_xyz staircase starplot starplot_description startphi starttheta stemplot style surface svg_file t thetaresolution title track transform_xy tstep width windowname windowtitle wireframe x xlabel xlength xtics xy_scale y ylabel ylength ytics yx_ratio z zlabel zlength zmin
intopois outofpois poisdiff poisexpt poisint poislim poismap poisplus poissimp poisson poissubst poistimes poistrim printpois
Introduction to Affine Introduction to Polynomials Introduction to Symmetries allroots berlefact bezout bfallroots bothcoef coeff content divide eliminate ezgcd facexpand factor factorflag fasttimes gcd gcdex gcdivide gfactor lratsubst makeOrders multiplicities nroots nthroot polydecomp polyfactor polymod programmode quotient ratcoef realroots remainder resultant resultant rootsepsilon savefactors sqfr tellrat untellrat
deftaylor maxtayorder pade powerseries revert revert2 taylor taylor_logexpand taylor_order_coefficients taylor_simplifier taylor_truncate_polynomials taylordepth taylorinfo taylorp taytorat trunc verbose
abasep alphacharp alphanumericp atom bfloatp blockmatrixp cequal cequalignore cgreaterp cgreaterpignore charp clessp clesspignore constantp constituent diagmatrixp digitcharp disjointp elementp emptyp evenp featurep floatnump if integerp intervalp is listp listp lowercasep mapatom matrixp matrixp maybe member nonnegintegerp nonscalarp numberp oddp operatorp ordergreatp orderlessp picture_equalp picturep poly_depends_p poly_grobner_subsetp polynomialp prederror primep ratnump ratp scalarp sequal sequalignore setequalp setp stringp subsetp subvarp symbolp symmetricp taylorp unknown uppercasep zeroequiv zeromatrixp zn_primroot_p
Function Lisp and Maxima block catch do in errcatch error errormsg errormsg for go if local prederror return sstatus status throw unless while
make_random_state random random_bernoulli random_beta random_binomial random_cauchy random_chi2 random_continuous_uniform random_discrete_uniform random_exp random_f random_gamma random_general_finite_discrete random_geometric random_gumbel random_hypergeometric random_laplace random_logistic random_lognormal random_negative_binomial random_noncentral_chi2 random_noncentral_student_t random_normal random_pareto random_poisson random_rayleigh random_student_t random_weibull set_random_state
Introduction to Polynomials fullratsimp fullratsubst gcd gcdex lratsubst rat ratcoef ratdenom ratdenomdivide ratdiff ratdisrep ratepsilon ratexpand ratfac ratmx ratnumer ratnump ratp ratp_coeffs ratp_dense_coeffs ratp_hipow ratp_lopow ratprint ratsimp ratsimpexpons ratsubst ratvars ratvarswitch ratweight ratweights ratwtlvl showratvars taytorat tellrat totaldisrep untellrat
Introduction to Rules and Patterns Package absimp Package ineq apply1 apply2 applyb1 clear_rules current_let_rule_package default_let_rule_package defmatch defrule disprule let let_rule_packages letrat letrules letsimp matchdeclare remlet remrule tellsimp tellsimpafter
Introduction for Runtime Environment batch batchload kill load loadfile myoptions nolabels optionset reset save stringout
Introduction to Sets adjoin belln cardinality cartesian_product disjoin disjointp elementp emptyp equiv_classes every extremal_subset flatten full_listify intersect intersection listify lmax lmin makeset partition_set permutations powerset random_permutation set_partitions setdifference setequalp setp some subset subsetp symmdifference tree_reduce union xreduce
Introduction to Affine Introduction to Fourier series Introduction to QUADPACK Introduction to String Processing Introduction to Symmetries Introduction to Units Introduction to atensor Introduction to cobyla Introduction to contrib_ode Introduction to ctensor Introduction to descriptive Introduction to distrib Introduction to draw Introduction to drawdf Introduction to ezunits Introduction to fast Fourier transform Introduction to graphs Introduction to grobner Introduction to interpol Introduction to itensor Introduction to lapack Introduction to lbfgs Introduction to linearalgebra Introduction to lsquares Introduction to minpack Introduction to mnewton Introduction to numericalio Introduction to orthogonal polynomials Introduction to physical_constants Introduction to simplex Introduction to solve_rec Introduction to stats Introduction to zeilberger Lindstedt Package absimp Package facexp Package functs Package ineq Package rducon Package scifac The dynamics package Vectors augmented_lagrangian_method diag dimension eigen engineering_format_floats f90 ggf implicit_derivative implicit_plot makeOrders opsubst sqrtdenest stirling
%e_to_numlog %emode %iargs %piargs algebraic besselexpand beta_args_sum_to_integer beta_expand ctrgsimp distribute_over domain dot0nscsimp dot0simp dot1simp dotassoc dotconstrules dotdistrib dotexptsimp dotident dotscrules evflag gamma_expand gammalim halfangles logarc logconcoeffp logexpand lognegint logsimp negdistrib radexpand radsubstflag ratalgdenom ratdenomdivide ratsimpexpons rootsconmode scalarmatrixp simpproduct simpsum sumexpand sumsplitfact trigexpandplus trigexpandtimes triginverses trigsign
Package absimp Package ineq atensimp foursimp fullratsimp hypergeometric_simp logarc radcan ratsimp rootscontract scsimp simplify_sum trigexpand trigrat trigreduce trigsimp unknown vectorsimp
Introduction to Rules and Patterns lassociative linear multiplicative
%f %m %s %w Introduction to Special Functions airy_ai airy_bi airy_dai airy_dbi bessel_i bessel_j bessel_k bessel_simplify bessel_y besselexpand erf erf_generalized erfc erfi expintegral_chi expintegral_ci expintegral_e expintegral_e1 expintegral_e_simplify expintegral_ei expintegral_li expintegral_shi expintegral_si fresnel_c fresnel_s gamma gamma_greek gamma_incomplete gamma_incomplete_generalized gamma_incomplete_regularized generalized_lambert_w hankel_1 hankel_2 hypergeometric_simp lambert_w log_gamma nzeta nzetai nzetar parabolic_cylinder_d scaled_bessel_i0 scaled_bessel_i1 struve_h struve_l
Introduction to zeilberger arithmetic arithsum bashindices cauchysum genindex gensumnum geometric geosum harmonic lsum niceindices niceindicespref nusum product simplify_sum simpproduct simpsum sum sumcontract sumexpand unsum
Comments Identifiers Introduction to Strings Introduction to operators infix matchfix nary nofix postfix prefix
get_tex_environment set_tex_environment get_tex_environment_default set_tex_environment_default tentex tex texput
Introduction to atensor Introduction to ctensor Introduction to itensor
absolute_real_time elapsed_real_time elapsed_run_time parse_timedate timedate
compfile compile compile_file declare_translated define_variable f90 fortindent fortran fortspaces mode_declare mode_identity tr_warnings_get translate translate_file
mode_check_errorp mode_check_warnp mode_checkp packagefile savedef tr_array_as_ref tr_bound_function_applyp tr_file_tty_messagesp tr_float_can_branch_complex tr_function_call_default tr_numer tr_optimize_max_loop tr_semicompile tr_state_vars tr_warn_bad_function_calls tr_warn_fexpr tr_warn_meval tr_warn_mode tr_warn_undeclared tr_warn_undefined_variable transcompile translate_fast_arrays transrun
%iargs %piargs Introduction to Trigonometric acos acot acsc asec asin atan atan2 atrig1 cos cot csc demoivre exponentialize foursimp halfangles ntrig sec sin tan trigexpand trigexpandplus trigexpandtimes triginverses trigrat trigreduce trigsign trigsimp
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Applied Mathematics and Programming Division, K.U. Leuven
Applied Mathematics and Programming Division, K.U. Leuven
Institut für Mathematik, T.U. Wien
National Bureau of Standards, Washington, D.C., U.S.A
http://www.netlib.org/quadpack
R. Piessens, E. de Doncker-Kapenga, C.W. Uberhuber, and D.K. Kahaner. QUADPACK: A Subroutine Package for Automatic Integration. Berlin: Springer-Verlag, 1983, ISBN 0387125531.
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