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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_all
Display all tests. Normally, the tests are not displayed, unless the test
fails. (Defaults to false
).
display_known_bugs
Displays tests that are marked as known bugs. (Default is false
).
tests
This 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
.
time
Display 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_tests
Load 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:
labels
All bound %i
, %o
, and %t
labels.
values
All bound atoms which are user variables, not Maxima options or switches,
created by :
or ::
or functional binding.
functions
arrays
All declared and undeclared arrays, created by :
, ::
, or
:=
.
macros
All user-defined macro functions, created by ::=
.
myoptions
All options ever reset by the user (whether or not they are later reset to their default values).
rules
All user-defined pattern matching and simplification rules, created
by tellsimp
, tellsimpafter
, defmatch
, or
defrule
.
aliases
All atoms which have a user-defined alias, created by the alias
,
ordergreat
, orderless
functions or by declaring the atom as a
noun
with declare
.
dependencies
All atoms which have functional dependencies, created by the
depends
, dependencies
, or gradef
functions.
gradefs
All functions which have user-defined derivatives, created by the
gradef
function.
props
All 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_packages
All 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 |
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5.1.1 Introduction to Numbers | ||
5.1.2 Functions and Variables for Numbers |
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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 |
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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 n
th 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
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
5.5.1 Functions and Variables for Arrays |
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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.
prefix
rbp=180
postfix
lbp=180
infix
lbp=180, rbp=180
nary
lbp=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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9.1 Functions and Variables for Simplification |
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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 ratsimp
ing 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
ratsimp
ed.
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:
logexpand
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 b
integer, always simplifies.) If it is set to
false
, all of these simplifications will be turned off.
logsimp
if false
then no simplification of %e
to a power containing
log
's is done.
lognegint
if true
implements the rule log(-n)
-> log(n)+%i*%pi
for
n
a positive integer.
%e_to_numlog
when 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 |
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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:
trigexpand
If true
causes expansion of all
expressions containing sin's and cos's occurring subsequently.
halfangles
If true
causes half-angles to be simplified
away.
trigexpandplus
Controls the "sum" rule for trigexpand
,
expansion of sums (e.g. sin(x + y)
) will take place only if
trigexpandplus
is true
.
trigexpandtimes
Controls 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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[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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:
additive
Tells 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.
alphabetic
Tells Maxima to recognize all characters in a_i (which must be a string) as alphabetic characters.
antisymmetric, commutative, symmetric
Tells Maxima to recognize a_i as a symmetric or antisymmetric
function. commutative
is the same as symmetric
.
bindtest
Tells Maxima to trigger an error when a_i is evaluated unbound.
constant
Tells Maxima to consider a_i a symbolic constant.
even, odd
Tells Maxima to recognize a_i as an even or odd integer variable.
evenfun, oddfun
Tells Maxima to recognize a_i as an odd or even function.
evflag
Makes 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
.
evfun
Makes 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
.
feature
Tells Maxima to recognize a_i as the name of a feature. Other atoms may then be declared to have the a_i property.
increasing, decreasing
Tells Maxima to recognize a_i as an increasing or decreasing function.
integer, noninteger
Tells Maxima to recognize a_i as an integer or noninteger variable.
integervalued
Tells Maxima to recognize a_i as an integer-valued function.
lassociative, rassociative
Tells Maxima to recognize a_i as a right-associative or left-associative function.
linear
Equivalent to declaring a_i both outative
and
additive
.
mainvar
Tells 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
.
multiplicative
Tells 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.
nary
Tells 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)
.
nonarray
Tells Maxima to consider a_i not an array. This declaration prevents multiple evaluation of a subscripted variable name.
nonscalar
Tells Maxima to consider a_i a nonscalar variable. The usual application is to declare a variable as a symbolic vector or matrix.
noun
Tells 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.
outative
Tells 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.
posfun
Tells Maxima to recognize a_i as a positive function.
rational, irrational
Tells Maxima to recognize a_i as a rational or irrational real variable.
real, imaginary, complex
Tells Maxima to recognize a_i as a real, pure imaginary, or complex variable.
scalar
Tells 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
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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:
subres
for the subresultant polynomial remainder sequence (PRS) algorithm,
mod
for the modular resultant algorithm, and
red
for 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