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7. Simplification

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7.1 Definitions for Simplification

System variable: askexp

When asksign is called, askexp is the expression asksign is testing.

At one time, it was possible for a user to inspect askexp by entering a Maxima break with control-A.

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.

Function: asksign (expr)

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.

Function: demoivre (expr)
Option variable: demoivre

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.

Option variable: domain

Default value: real

When domain is set to complex, sqrt (x^2) will remain sqrt (x^2) instead of returning abs(x).

Function: expand (expr)
Function: expand (expr, p, n)

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.

The expand flag used with ev causes expansion.

The file `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").

Function: expandwrt (expr, x_1, ..., x_n)

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'.

Option variable: expandwrt_denom

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.

Function: expandwrt_factored (expr, x_1, ..., x_n)

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'.

Option variable: expon

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.

Function: exponentialize (expr)
Option variable: exponentialize

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.

Option variable: expop

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.

Option variable: factlim

Default value: -1

factlim specifies the highest factorial which is automatically expanded. If it is -1 then all integers are expanded.

Function: intosum (expr)

Moves multiplicative factors outside a summation to inside. If the index is used in the outside expression, then the function tries to find a reasonable index, the same as it does for sumcontract. This is essentially the reverse idea of the outative property of summations, but note that it does not remove this property, it only bypasses it.

In some cases, a scanmap (multthru, expr) may be necessary before the intosum.

Declaration: lassociative

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).

Declaration: linear

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.

linear is equivalent to additive and outative. See also opproperties.

Declaration: mainvar

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.)

Option variable: maxapplydepth

Default value: 10000

maxapplydepth is the maximum depth to which apply1 and apply2 will delve.

Option variable: maxapplyheight

Default value: 10000

maxapplyheight is the maximum height to which applyb1 will reach before giving up.

Option variable: maxnegex

Default value: 1000

maxnegex is the largest negative exponent which will be expanded by the expand command (see also maxposex).

Option variable: maxposex

Default value: 1000

maxposex is the largest exponent which will be expanded with the expand command (see also maxnegex).

Declaration: multiplicative

declare (f, multiplicative) tells the Maxima simplifier that f is multiplicative.

  1. If 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).
  2. If 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).

This simplification does not occur when f is applied to expressions of the form product (x[i], i, m, n).

Option variable: negdistrib

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.

Option variable: negsumdispflag

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.

Special symbol: noeval

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.

Declaration: noun

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.

Option variable: noundisp

Default value: false

When noundisp is true, nouns display with a single quote. This switch is always true when displaying function definitions.

Special symbol: nouns

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.

Special symbol: numer

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.

Function: numerval (x_1, expr_1, ..., var_n, expr_n)

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.

System variable: opproperties

opproperties is the list of the special operator properties recognized by the Maxima simplifier: linear, additive, multiplicative, outative, evenfun, oddfun, commutative, symmetric, antisymmetric, nary, lassociative, rassociative.

Option variable: opsubst

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.

Declaration: outative

declare (f, outative) tells the Maxima simplifier that constant factors in the argument of f can be pulled out.

  1. If 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.
  2. If 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.

Declaration: posfun

declare (f, posfun) declares f to be a positive function. is (f(x) > 0) yields true.

Function: radcan (expr)

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.

When %e_to_numlog is true, %e^(r*log(expr)) simplifies to expr^r if r is a rational number.

When radexpand is false, certain transformations are inhibited. radcan (sqrt (1-x)) remains sqrt (1-x) and is not simplified to %i sqrt (x-1). radcan (sqrt (x^2 - 2*x + 11)) remains sqrt (x^2 - 2*x + 1) and is not simplified to x - 1.

example (radcan) displays some examples.

Option variable: radexpand

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).

Note that domain only matters when radexpand is true.

Option variable: radsubstflag

Default value: false

radsubstflag, if true, permits ratsubst to make substitutions such as u for sqrt (x) in x.

Declaration: rassociative

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))).

Function: scsimp (expr, rule_1, ..., rule_n)

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.

Option variable: simpsum

Default value: false

When simpsum is true, the result of a sum is simplified. This simplification may sometimes be able to produce a closed form. If simpsum is false or if the quoted form 'sum is used, the value is a sum noun form which is a representation of the sigma notation used in mathematics.

Function: sumcontract (expr)

Combines all sums of an addition that have upper and lower bounds that differ by constants. The result is an expression containing one summation for each set of such summations added to all appropriate extra terms that had to be extracted to form this sum. sumcontract combines all compatible sums and uses one of the indices from one of the sums if it can, and then try to form a reasonable index if it cannot use any supplied.

It may be necessary to do an intosum (expr) before the sumcontract.

Option variable: sumexpand

Default value: false

When sumexpand is true, products of sums and exponentiated sums simplify to nested sums.

See also cauchysum.


(%i1) sumexpand: true$
(%i2) sum (f (i), i, 0, m) * sum (g (j), j, 0, n);
                     m      n
                    ====   ====
                    \      \
(%o2)                >      >     f(i1) g(i2)
                    /      /
                    ====   ====
                    i1 = 0 i2 = 0
(%i3) sum (f (i), i, 0, m)^2;
                     m      m
                    ====   ====
                    \      \
(%o3)                >      >     f(i3) f(i4)
                    /      /
                    ====   ====
                    i3 = 0 i4 = 0
Option variable: sumsplitfact

Default value: true

When sumsplitfact is false, minfactorial is applied after a factcomb.

Declaration: symmetric

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.

Function: unknown (expr)

Returns true if and only if expr contains an operator or function not recognized by the Maxima simplifier.

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