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# Simplification

## Definitions for Simplification

Function: APPLY_NOUNS (exp)
  causes the application of noun forms in an expression. E.g. EXP:'DIFF(X^2/2,X); APPLY_NOUNS(EXP); gives X. This gives the same result as EV(EXP,NOUNS); except that it can be faster and use less storage. It also can be used in translated code, where EV may cause problems. Note that it is called APPLY_NOUNS, not EV_NOUNS, because what it does is to APPLY the rules corresponding to the noun-form operators, which is not evaluation.

Variable: ASKEXP
  default: [] contains the expression upon which ASKSIGN is called. A user may enter a MACSYMA break with ^A and inspect this expression in order to answer questions asked by ASKSIGN.

Function: ASKINTEGER (exp,<optional-arg>)
  exp is any valid macsyma expression and optional-arg is EVEN,ODD,INTEGER and defaults to INTEGER if not given. This function attempts to determine from the data-base whether exp is EVEN, ODD or just an INTEGER. It will ask the user if it cannot tell otherwise and attempt to install the information in the data-base if possible.

Function: ASKSIGN (exp)
  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 (one "C-line"). The value of ASKSIGN is one of POS, NEG, or ZERO.

Variable: DEMOIVRE
  default: [FALSE] if TRUE will cause

      %E^(A+B*%I) ==> %E^A*(COS(B)+%I*SIN(B))

  if B is free of %I. A and B are not expanded. DEMOIVRE:TRUE; is the way to reverse the effect of EXPONENTIALIZE:TRUE;

  DEMOIVRE(exp) will cause the conversion without setting the switch or having to re-evaluate the expression with EV.

Variable: DOMAIN
  default: [REAL] - if set to COMPLEX, SQRT(X^2) will remain SQRT(X^2) instead of returning ABS(X). The notion of a "domain" of simplification is still in its infancy, and controls little more than this at the moment.

Function: EXPAND (exp)
  will cause products of sums and exponentiated sums to be multiplied out, numerators of rational expressions which are sums to be split into their respective terms, and multiplication (commutative and non-commutative) to be distributed over addition at all levels of exp. For polynomials one should usually use RATEXPAND which uses a more efficient algorithm (see DESCRIBE(RATEXPAND);). MAXNEGEX[1000] and MAXPOSEX[1000] control the maximum negative and positive exponents, respectively, which will expand. EXPAND(exp,p,n) expands exp, using p for MAXPOSEX and n for MAXNEGEX. This is useful in order to expand part but not all of an expression. EXPON[0] - 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[0] - 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 (see 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 (exp,var1,var2,...)
  expands exp with respect to the vari. All products involving the vari appear explicitly. The form returned will be free of products of sums of expressions that are not free of the vari. The vari 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'`.

Variable: EXPANDWRT_DENOM
  default:[FALSE] 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 (exp, var1, var2, ..., varN)
  is similar to EXPANDWRT, but treats expressions that are products somewhat differently. EXPANDWRT_FACTORED will perform the required expansion only on those factors of exp that contain the variables in its argument list argument list. This function is autoloaded from ``simplification/stopex.mac'`.

Variable: EXPON
  default: [0] - 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.

Variable: EXPONENTIALIZE
  default: [FALSE] if TRUE will cause all circular and hyperbolic functions to be converted to exponential form. (Setting DEMOIVRE:TRUE; will reverse the effect.) EXPONENTIALIZE(exp) will cause the conversion to exponential form of an expression without setting the switch or having to re-evaluate the expression with EV.

Variable: EXPOP
  default: [0] - 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.

Variable: FACTLIM
  default: [-1] gives the highest factorial which is automatically expanded. If it is -1 then all integers are expanded.

Function: INTOSUM (expr)
  will take all things that a summation is multiplied by, and put them inside the summation. 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
  - If DECLARE(G,LASSOCIATIVE); is done, this tells the 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 MACSYMA's OPPROPERTIES. For univariate f so declared, "expansion" F(X+Y) -> F(X)+F(Y), F(A*X) -> A*F(X) takes place where A is a "constant". For functions F of >=2 args, "linearity" is defined to be as in the case of 'SUM or 'INTEGRATE, i.e. F(A*X+B,X) -> A*F(X,X)+B*F(1,X) for A,B FREEOF X. (LINEAR is just ADDITIVE + OUTATIVE.)

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(expression,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.)

Variable: MAXAPPLYDEPTH
  default: [10000] - the maximum depth to which APPLY1 and APPLY2 will delve.

Variable: MAXAPPLYHEIGHT
  default: [10000] - the maximum height to which APPLYB1 will reach before giving up.

Variable: MAXNEGEX
  default: [1000] - the largest negative exponent which will be expanded by the EXPAND command (see also MAXPOSEX).

Variable: MAXPOSEX
  default: [1000] - the largest exponent which will be expanded with the EXPAND command (see also MAXNEGEX).

declaration: MULTIPLICATIVE
  - If DECLARE(F,MULTIPLICATIVE) has been executed, then: (1) If F is univariate, whenever the simplifier encounters F applied to a product, F will be distributed over that product. I.e. F(X*Y); will simplify 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, i.e. F(G(X)*H(X),X); will simplify 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,lower-limit,upper-limit).

Variable: NEGDISTRIB
  default: [TRUE] - when TRUE allows -1 to be distributed 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 MACSYMA.

Variable: NEGSUMDISPFLAG
  default: [TRUE] - when 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
  - 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
  - One of the options of the DECLARE command. It makes a function so DECLAREd a "noun", meaning that it won't be evaluated automatically.

Variable: NOUNDISP
  default: [FALSE] - if TRUE will cause NOUNs to display with a single quote. This switch is always TRUE when displaying function definitions.

special symbol: NOUNS
  (EVFLAG) when used as an option to the EV command, 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
  causes some mathematical functions (including exponentiation) with numerical arguments to be evaluated in floating point. It causes variables in exp which have been given numervals to be replaced by their values. It also sets the FLOAT switch on.

Function: NUMERVAL (var1, exp1, var2, exp2, ...)
  declares vari to have a numerval of expi which is evaluated and substituted for the variable in any expressions in which the variable occurs if the NUMER flag is TRUE. (see the EV function).

Variable: OPPROPERTIES
  - the list of the special operator-properties handled by the MACSYMA simplifier: LINEAR, ADDITIVE, MULTIPLICATIVE, OUTATIVE, EVENFUN, ODDFUN, COMMUTATIVE, SYMMETRIC, ANTISYMMETRIC, NARY, LASSOCIATIVE, and RASSOCIATIVE.

Variable: OPSUBST
  default:[TRUE] - if 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.

declaration: OUTATIVE
  - If DECLARE(F,OUTATIVE) has been executed, then: (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. I.e. 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. Initially, 'SUM, 'INTEGRATE, and 'LIMIT are declared to be OUTATIVE.

declaration: POSFUN
  - POSitive FUNction, e.g. DECLARE(F,POSFUN); IS(F(X)>0); -> TRUE.

Variable: PRODHACK
  default: [FALSE] - if set to TRUE then PRODUCT(F(I),I,3,1); will yield 1/F(2), by the identity PRODUCT(F(I),I,A,B) = 1/PRODUCT(F(I),I,B+1,A-1) when A>B.

Function: RADCAN (exp)
  simplifies exp, 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 will not necessarily have the same appearance, but their difference will be simplified by RADCAN to zero. For some expressions RADCAN can be 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. %E_TO_NUMLOG (default: [FALSE]) - when set to TRUE, for "r" some rational number, and "x" some expression, %E^(r*LOG(x)) will be simplified into x^r . RADEXPAND[TRUE] when set to FALSE will inhibit certain transformations: RADCAN(SQRT(1-X)) will remain SQRT(1-X) and will not become %I SQRT(X-1). RADCAN(SQRT(X^2-2*X+1)) will remain SQRT(X^2-2*X + 1) and will not be transformed to X- 1. Do EXAMPLE(RADCAN); for examples.

Variable: RADEXPAND
  default: [TRUE] - if set to ALL will cause 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) will become 4*X . More particularly, consider SQRT(X^2). (a) If RADEXPAND is ALL or ASSUME(X>0) has been done, SQRT(X^2) will become X. (b) If RADEXPAND is TRUE and DOMAIN is REAL (its default), SQRT(X^2) will become ABS(X). (c) If RADEXPAND is FALSE, or RADEXPAND is TRUE and DOMAIN is COMPLEX, SQRT(X^2) will be returned. (The notion of DOMAIN with settings of REAL or COMPLEX is still in its infancy. Note that its setting here only matters when RADEXPAND is TRUE.)

Variable: RADPRODEXPAND
  - this switch has been renamed RADEXPAND.

Variable: RADSUBSTFLAG
  default: [FALSE] - if TRUE permits RATSUBST to make substitutions such as U for SQRT(X) in X.

declaration: RASSOCIATIVE
  - If DECLARE(G,RASSOCIATIVE); is done, this tells the simplifier that G is right-associative. E.g. G(G(A,B),G(C,D)) will simplify to G(A,G(B,G(C,D))).

Function: SCSIMP (exp,rule1, rule2,...,rulen)
  Sequential Comparative Simplification [Stoute]) takes an expression (its first argument) and a set of identities, or rules (its other arguments) and tries simplifying. If a smaller expression is obtained, the process repeats. Otherwise after all simplifications are tried, it returns the original answer. For examples, try EXAMPLE(SCSIMP); .

Function: SIMP
  causes exp to be simplified regardless of the setting of the switch SIMP which inhibits simplification if FALSE.

Variable: SIMPSUM
  default: [FALSE] - if 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 'SUM is used, the value is a sum noun form which is a representation of the sigma notation used in mathematics.

Function: SUMCONTRACT (expr)
  will combine all sums of an addition that have upper and lower bounds that differ by constants. The result will be 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 will combine all compatible sums and use 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.

Variable: SUMEXPAND
  default: [FALSE] if TRUE, products of sums and exponentiated sums are converted into nested sums. For example:

              SUMEXPAND:TRUE$
              SUM(F(I),I,0,M)*SUM(G(J),J,0,N); ->
                              'SUM('SUM(F(I1)*G(I2),I2,0,N),I1,0,M)
              SUM(F(I),I,0,M)^2; -> 'SUM('SUM(F(I3)*F(I4),I4,0,M),I3,0,M)

  If FALSE, they are left alone. See also CAUCHYSUM.

Variable: SUMHACK
  default: [FALSE] - if set to TRUE then SUM(F(I),I,3,1); will yield -F(2), by the identity SUM(F(I),I,A,B) = - SUM(F(I),I,B+1,A-1) when A>B.

Variable: SUMSPLITFACT
  default: [TRUE] - if set to FALSE will cause MINFACTORIAL to be applied after a FACTCOMB.

declaration: SYMMETRIC
  - If DECLARE(H,SYMMETRIC); is done, this tells the simplifier that H is a symmetric function. E.g. H(X,Z,Y) will simplify to H(X, Y, Z). This is the same as COMMUTATIVE.

Function: UNKNOWN (exp)
  returns TRUE iff exp contains an operator or function not known to the built-in simplifier.

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