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From: Raymond Toy <rtoy@us...>  20050221 17:47:21

Update of /cvsroot/maxima/maxima/src In directory sc8prcvs1.sourceforge.net:/tmp/cvsserv12483/src Modified Files: hyp.lisp hypgeo.lisp Log Message: o Rename PAR to *PAR* to make it clear that it's a special variable. o Fix typo in hypgeo.lisp. Index: hyp.lisp =================================================================== RCS file: /cvsroot/maxima/maxima/src/hyp.lisp,v retrieving revision 1.59 retrieving revision 1.60 diff u d r1.59 r1.60  hyp.lisp 8 Feb 2005 15:55:05 0000 1.59 +++ hyp.lisp 21 Feb 2005 17:46:31 0000 1.60 @@ 12,7 +12,7 @@ (declaretop (special fun #w b l# $true $false # n c # #l1 l2#)) (declaretop (special var par zerosigntest productcase +(declaretop (special var *par* zerosigntest productcase fldeg flgkum checkcoefsignlist serieslist $exponentialize $bestriglim $radexpand)) @@ 100,7 +100,7 @@ (defun $hgfred (argl1 argl2 arg) (let (($radexpand '$all) (var arg)  (par arg)) + (*par* arg)) (hgfsimpexec (cdr argl1) (cdr argl2) arg))) @@ 427,7 +427,7 @@ (format t "l1 = ~A~%" argl1) (format t "vfvp arg = ~A~%" (div (add (cadr argl1) n) 2)) (format t "var = ~A~%" var)  (format t "par = ~A~%" par) + (format t "*par* = ~A~%" *par*) # (setq l (vfvp (div (add (cadr argl1) n) 2))) @@ 453,7 +453,7 @@ (inv (factf (mul 2 v) (* 1 n)))))) (gegenpol (mul 1 n) v  (sub 1 (mul 2 par))))))) + (sub 1 (mul 2 *par*))))))) ;; A&S 15.4.6 says ;; F(n, n + a + 1 + b; a + 1; x) = n!*jacobi_p(n,a,b,12*x)/pochhammer(a+1,n); ;; @@ 466,7 +466,7 @@ (jacobpol (mul 1 n) (add (car argl2) 1) (sub (mul 2 v) (car argl2))  (sub 1 (mul 2 par))))))) + (sub 1 (mul 2 *par*))))))) #+nil @@ 500,7 +500,7 @@ ;; polynomial. (defun $hgfpoly (argl1 argl2 arg) (let ((var arg)  (par arg) + (*par* arg) (n (hypnegpinl (cdr argl1)))) (createanypoly (cdr argl1) (cdr argl2) ( n)))) @@ 2098,9 +2098,9 @@ (cond ((freevar exp) (freepar exp)) (t nil))) (declaretop (special serieslist var par zerosigntest productcase)) +(declaretop (special serieslist var *par* zerosigntest productcase)) (setq par '$p) +(setq *par* '$p) ;;(DEFUN FREEVAR (A) ;; (COND ((ATOM A) (NOT (EQ A VAR))) @@ 2114,7 +2114,7 @@ (defun freepar (exp) (cond ((atom exp)  (not (eq exp par))) + (not (eq exp *par*))) (t (and (freepar (car exp)) (freepar (cdr exp)))))) @@ 2609,7 +2609,7 @@ ;; (COND ((EQ SIGN '$POSITIVE) '$NEGATIVE) (T '$POSITIVE))) (setq par '$p) +(setq *par* '$p) (defun vfvp (exp) (m2 exp '(v freevarpar) nil)) @@ 3222,7 +3222,7 @@ (evalwhen #+gcl (compile) #gcl (:compiletoplevel)  (declaretop (unspecial serieslist var par zerosigntest productcase + (declaretop (unspecial serieslist var *par* zerosigntest productcase fldeg flgkum listcmdiff checkcoefsignlist )) (declaretop (unspecial fun #w b l n c#))) Index: hypgeo.lisp =================================================================== RCS file: /cvsroot/maxima/maxima/src/hypgeo.lisp,v retrieving revision 1.11 retrieving revision 1.12 diff u d r1.11 r1.12  hypgeo.lisp 19 Feb 2005 21:17:00 0000 1.11 +++ hypgeo.lisp 21 Feb 2005 17:46:44 0000 1.12 @@ 13,7 +13,7 @@ (macsymamodule hypgeo) (declaretop (special var par zerosigntest productcase checkcoefsignlist +(declaretop (special var *par* zerosigntest productcase checkcoefsignlist $exponentialize $radexpand)) (loadmacsymamacros rzmac) @@ 60,7 +60,7 @@ (defun expor1p(exp)(or (equal exp 1)(eq exp '$%e))) (defun parp(a)(eq a par)) +(defun parp(a)(eq a *par*)) @@ 770,9 +770,9 @@ (return 'otherdefinttofollownegtest))) (defun ltscale  (exp var par c par0 e f) + (exp var *par* c par0 e f) (mul* (power '$%e c)  (substl (sub par par0) par (ltexec exp e f)))) + (substl (sub *par* par0) *par* (ltexec exp e f)))) (defun defltep (exp) @@ 793,7 +793,7 @@ ;;it dispatches according to the kind of transform it matches (defun hypgeoexec (exp var par) +(defun hypgeoexec (exp var *par*) (prog (l u a c e f) (setq exp (simplifya exp nil)) (cond ((setq l (ltep exp)) @@ 802,7 +802,7 @@ c (cdras 'c l) e (cdras 'e l) f (cdras 'f l))  (return (ltscale u var par c a e f)))) + (return (ltscale u var *par* c a e f)))) (return 'othertranstofollow))) (defun substl @@ 878,7 +878,7 @@ (defun f1p137 (pow)  (mul* (gm (add pow 1))(power par (sub (mul 1 pow) 1)))) + (mul* (gm (add pow 1))(power *par* (sub (mul 1 pow) 1)))) ;; Table of Integral Transforms ;; @@ 893,8 +893,8 @@ (gm v) (power 2 v) (power a (div v 2))  (power '$%e (mul* a par par))  (dtford (mul* 2 par (power a (1//2)))(mul 1 v)))) + (power '$%e (mul* a *par* *par*)) + (dtford (mul* 2 *par* (power a (1//2)))(mul 1 v)))) ;; Table of Integral Transforms ;; @@ 908,9 +908,9 @@ (mul* c (gm (add v v)) (power 2 (sub 1 v)) ; Is this supposed to be here?  (power par (mul 1 v))  (power '$%e (mul* a (1//2) (inv par)))  (dtford (power (mul* 2 a (inv par)) + (power *par* (mul 1 v)) + (power '$%e (mul* a (1//2) (inv *par*))) + (dtford (power (mul* 2 a (inv *par*)) (1//2)) (mul 2 v)))) @@ 924,7 +924,7 @@ ;; Re(a) > 0 (defun f29p146 (v a) (mul* 2  (power (mul* a (inv 4) (inv par)) + (power (mul* a (inv 4) (inv *par*)) (div v 2)) (ktfork a v))) @@ 937,7 +937,7 @@ (cond ((maximaintegerp v) (kmodbes z v)) (t (simpktf z v))))  (power (mul* a par)(1//2)))) + (power (mul* a *par*)(1//2)))) (defun dtford (z v) @@ 1738,12 +1738,12 @@ (neginp (sub (sub (1//2) i2) i1)) (neginp (sub (add (1//2) i2) i1)))) ;; Compute r*exp(var*par). +;; Compute r*exp(var**par*). ;; ;; (Probably r*exp(p*t), where t is the variable of integration and p ;; is the parameter of the Laplace transform.) (defun init (r)  (mul* r (power '$%e (mul* 1 var par)))) + (mul* r (power '$%e (mul* 1 var *par*)))) ;; (1)^n*n!*laguerre(n,a,x) = U(n,a+1,x) ;; @@ 1873,7 +1873,7 @@ (defun distrexecinit (fun) (cond ((equal (caar fun) 'mplus) (distrexec (cdr fun)))  (t (hypgeoexec fun var par)))) + (t (hypgeoexec fun var *par*)))) (defun distrdefexecinit (fun) (cond ((equal (caar fun) 'mplus) (distrdefexec (cdr fun))) @@ 1881,7 +1881,7 @@ (defun distrexec (fun) (cond ((null fun) 0)  (t (add (hypgeoexec (car fun) var par) + (t (add (hypgeoexec (car fun) var *par*) (distrexec (cdr fun)))))) (defun distrdefexec (fun) @@ 2211,8 +2211,8 @@ (cond ((equal q 1)(setq a 0)(go loop))) (setq a (cdras 'a l)) loop  (return (substl (sub par a)  par + (return (substl (sub *par* a) + *par* (execf19 l (cadr l1)))))) ;; Dispatch function to convert the given function to a hypergeometric @@ 2295,7 +2295,7 @@ (list (mul* (gm (add n a 1)) (inv (gm (add a 1))) (inv (factorial n)))  (reffp1 (list (mul 1 n) (add* n a b 1)) + (reffpq (list (mul 1 n) (add* n a b 1)) (list (add a 1)) (sub (1//2) (div x 2))))) @@ 2576,12 +2576,12 @@ ;; The args below are s, [a's], [p's], c^k, k. (defun f19p220simp (s l1 l2 cf k) (mul* (gm s)  (inv (power par s)) + (inv (power *par* s)) (hgfsimpexec (append l1 (addarglist s k)) l2 (mul* cf (power k k)  (power (inv par) k))))) + (power (inv *par*) k))))) ;; Bessel function expressed as a hypergeometric function. ;; @@ 2650,24 +2650,24 @@ (defun f50p188simp (d u v a) (mul d (power a (inv 2))  (power par (mul 1 u))  (power '$%e (div a (mul 2 par))) + (power *par* (mul 1 u)) + (power '$%e (div a (mul 2 *par*))) (sub (mul (tan% (mul '$%pi (sub u v))) (gm (add u v (inv 2))) (inv (gm (add v v 1)))  (mwhit (div a par) u v)) + (mwhit (div a *par*) u v)) (mul `((%sec) ,(mul '$%pi (sub u v)))  (wwhit (div a par) u v))))) + (wwhit (div a *par*) u v))))) (defun f2p105v2condsimp (m v a) (mul 2. (power '$%pi 1.) (gm (add m v))  (power (add (mul a a) (mul par par)) (mul 1. (inv 2.) m)) + (power (add (mul a a) (mul *par* *par*)) (mul 1. (inv 2.) m)) (leg2fsimp (sub m 1.) (mul 1. v)  (mul par  (power (add (mul a a) (mul par par)) + (mul *par* + (power (add (mul a a) (mul *par* *par*)) (inv 2.)))))) (defun leg1fsimp (m v z) 
From: Robert Dodier <robert_dodier@us...>  20050221 06:06:46

Update of /cvsroot/maxima/maxima/doc/info In directory sc8prcvs1.sourceforge.net:/tmp/cvsserv374 Modified Files: Polynomials.texi Log Message: Reworked a number of items: rewording, enclosing names in @code or @var as appropriate, additional examples in some cases. This file still needs work. Index: Polynomials.texi =================================================================== RCS file: /cvsroot/maxima/maxima/doc/info/Polynomials.texi,v retrieving revision 1.13 retrieving revision 1.14 diff u d r1.13 r1.14  Polynomials.texi 20 Feb 2005 00:19:46 0000 1.13 +++ Polynomials.texi 21 Feb 2005 06:06:37 0000 1.14 @@ 1,3 +1,4 @@ +@c FOR THE FUNCTIONS WHICH RETURN A CRE, BE SURE TO MENTION THAT @menu * Introduction to Polynomials:: * Definitions for Polynomials:: @@ 281,7 +282,6 @@ be attempted. The user's setting of @code{intfaclim} is used for internal calls to @code{factor}. Thus, @code{intfaclim} may be reset to prevent Maxima from taking an inordinately long time factoring large integers. @...{newfac} may be set to true to use the new factoring routines. Examples: @c EXAMPLES BELOW ADAPTED FROM examples (factor) @@ 983,40 +983,103 @@ @end defun @defun ratdenom (@var{expr}) Returns the denominator of the rational expression @var{expr}. If @var{expr} is in general form then the @code{denom} function should be used instead, unless one wishes to get a CRE result. +Returns the denominator of @var{expr}, +after coercing @var{expr} to a canonical rational expression (CRE). +The return value is a CRE. + +@c ACTUALLY THE CONVERSION IS CARRIED OUT BY ratf BUT THAT'S WHAT $rat CALLS +@var{expr} is coerced to a CRE by @code{rat} +if it is not already a CRE. +This conversion may change the form of @var{expr} by putting all terms +over a common denominator. + +@code{denom} is similar, but returns an ordinary expression instead of a CRE. +Also, @code{denom} does not attempt to place all terms over a common denominator, +and thus some expressions which are considered ratios by @code{ratdenom} +are not considered ratios by @code{denom}. +@c NEEDS AN EXAMPLE HERE @end defun @defvar ratdenomdivide Default value: @code{true} When @code{ratdenomdivide} is @code{false}, stops the splitting up of the terms of the numerator of @code{ratexpand}'ed expressions from occurring. +When @code{ratdenomdivide} is @code{true}, +@code{ratexpand} expands a ratio in which the numerator is a sum +into a sum of ratios, +all having a common denominator. +Otherwise, @code{ratexpand} collapses a sum of ratios into a single ratio, +the numerator of which is the sum of the numerators of each ratio. + +Examples: + +@example +(%i1) expr: (x^2 + x + 1)/(y^2 + 7); + 2 + x + x + 1 +(%o1)  + 2 + y + 7 +(%i2) ratdenomdivide: true$ +(%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 +@end example @end defvar @defun ratdiff (@var{expr}, @var{x}) Differentiates the rational expression @var{expr} (which must be a ratio of polynomials or a polynomial in the variable @var{x}) with respect to @var{x}. For rational expressions this is much faster than @code{diff}. @... IS THE RESULT EQUIVALENT TO diff ?? +Differentiates the rational expression @var{expr} with respect to @var{x}. +@var{expr} must be a ratio of polynomials or a polynomial in @var{x}. +The argument @var{x} may be a variable or a subexpression of @var{expr}. +@c NOT CLEAR (FROM READING CODE) HOW x OTHER THAN A VARIABLE IS HANDLED  +@c LOOKS LIKE (a+b), 10*(a+b), (a+b)^2 ARE ALL TREATED LIKE (a+b); +@c HOW TO DESCRIBE THAT ?? The result is returned in canonical rational expression (CRE) form. @... "SHOULD NOT"  WHAT IS THIS ABOUT ?? @... (GREP ALL TEXI FILES FOR "SHOULD" AT SOME POINT) However, @code{ratdiff} should not be used on factored CRE forms; use @code{diff} instead for such expressions. +The result is equivalent to @code{diff}, although perhaps in a different form. +@code{ratdiff} may be faster than @code{diff}, for rational expressions. + +@code{ratdiff} returns a canonical rational expression (CRE) if @code{expr} is a CRE. +Otherwise, @code{ratdiff} returns a general expression. + +@code{ratdiff} considers only the dependence of @var{expr} on @var{x}, +and ignores any dependencies established by @code{depends}. + +@c WHAT THIS IS ABOUT  ratdiff (rat (factor (expr)), x) AND ratdiff (factor (rat (expr)), x) BOTH SUCCEED +@c COMMENTING THIS OUT UNTIL SOMEONE CAN ESTABLISH SOME CRE'S FOR WHICH ratdiff FAILS +@c However, @code{ratdiff} should not be used on factored CRE forms; +@c use @code{diff} instead for such expressions. Example: @c FOLLOWING GENERATED FROM THESE EXPRESSIONS @c expr: (4*x^3 + 10*x  11)/(x^5 + 5); @... modulus: 3$ @... mod (expr); @c ratdiff (expr, x); +@c expr: f(x)^3  f(x)^2 + 7; +@c ratdiff (expr, f(x)); +@c expr: (a + b)^3 + (a + b)^2; +@c ratdiff (expr, a + b); @example (%i1) expr: (4*x^3 + 10*x  11)/(x^5 + 5); @@ 1025,30 +1088,39 @@ (%o1)  5 x + 5 (%i2) modulus: 3$ (%i3) mod (expr);  2  x + x  1 (%o3)   4 3 2  x + x + x + x + 1 (%i4) ratdiff (expr, x);  5 4 3  x  x  x + x  1 (%o4)   8 7 5 4 3  x  x + x  x + x  x + 1 +(%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 @end example @end defun @defun ratdisrep (@var{expr}) changes its argument from CRE form to general form. +Returns its argument as a general expression. +If @var{expr} is a general expression, it is returned unchanged. + +Typically @code{ratdisrep} is called to convert a canonical rational expression (CRE) +into a general expression. +@c NOT REALLY FOND OF YOUCANDOTHISYOUCANDOTHAT STATEMENTS This is sometimes convenient if one wishes to stop the "contagion", or use rational functions in nonrational contexts. Most CRE functions will work on either CRE or nonCRE expressions, but the answers may take different forms. If @code{ratdisrep} is given a nonCRE for an argument, it returns its argument unchanged. See also @code{totaldisrep}. +use rational functions in nonrational contexts. + +See also @code{totaldisrep}. @end defun @@ 1058,28 +1130,37 @@ @code{ratepsilon} is the tolerance used in the conversion of floating point numbers to rational numbers. +@c NEED EXAMPLES HERE @end defvar @... ratexpand IS ALSO A VARIABLE  MAKE SURE THAT IS CLEAR @... ratexpand (@var{expr}) expands @var{expr} by multiplying out products of sums and +@defvr {Function} ratexpand (@var{expr}) +@defvrx {Variable} ratexpand +Expands @var{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. This is accomplished by converting @var{expr} to CRE form and then back to general form. +respective terms divided by the denominator. + +The return value of @code{ratexpand} is a general expression, +even if @var{expr} is a canonical rational expression (CRE). +@c WHAT DOES THE FOLLOWING MEAN EXACTLY ?? The switch @code{ratexpand} if @code{true} will cause CRE expressions to be fully expanded when they are converted back to general form or displayed, while if it is @code{false} then they will be put into a recursive form. (see @code{ratsimp}) +into a recursive form. +See also @code{ratsimp}. @...{ratdenomdivide}  if @code{false} will stop the splitting up of the terms of the numerator of @code{ratexpand}'ed expressions from occurring. +When @code{ratdenomdivide} is @code{true}, +@code{ratexpand} expands a ratio in which the numerator is a sum +into a sum of ratios, +all having a common denominator. +Otherwise, @code{ratexpand} collapses a sum of ratios into a single ratio, +the numerator of which is the sum of the numerators of each ratio. @...{keepfloat} if set to @code{true} will prevent floating point numbers from being rationalized when expressions which contain them are converted to CRE form. +When @code{keepfloat} is @code{true}, prevents floating +point numbers from being rationalized when expressions which contain +them are converted to canonical rational expression (CRE) form. Examples: @c FOLLOWING GENERATED FROM THESE EXPRESSIONS @@ 1110,73 +1191,96 @@ x + x  x  1 x + x  x  1 @end example @... defun +@end defvr @defvar ratfac Default value: @code{false} When @code{ratfac} is @code{true} invokes a partially factored form for CRE rational expressions. 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, for example RAT((X^2 1)^4/(X+1)^2); yields (X1)^4*(X+1)^2), +When @code{ratfac} is @code{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 @code{factor}. +This should always save space and may save time in some computations. +The numerator and denominator are made relatively prime, for example +@code{rat ((x^2  1)^4/(x + 1)^2)} yields @code{(x  1)^4 (x + 1)^2)}, but the factors within each part may not be relatively prime. In the @code{ctensr} (Component Tensor Manipulation) Package, if @code{ratfac} is @...{true}, it causes the Ricci, Einstein, Riemann, and Weyl tensors and the Scalar Curvature to be factored automatically. @i{This should only be + +In the @code{ctensr} (Component Tensor Manipulation) package, +Ricci, Einstein, Riemann, and Weyl tensors and the scalar curvature +are factored automatically when @code{ratfac} is @code{true}. +@i{@code{ratfac} should only be set for cases where the tensorial components are known to consist of few terms.} Note: The @code{ratfac} and @code{ratweight} schemes are incompatible and may not +The @code{ratfac} and @code{ratweight} schemes are incompatible and may not both be used at the same time. +@c NEED EXAMPLES HERE @end defvar @defun ratnumer (@var{expr}) obtains the numerator of the rational expression @var{expr}. If @var{expr} is in general form then the @code{num} function should be used instead, unless one wishes to get a CRE result. +Returns the numerator of @var{expr}, +after coercing @var{expr} to a canonical rational expression (CRE). +The return value is a CRE. + +@c ACTUALLY THE CONVERSION IS CARRIED OUT BY ratf BUT THAT'S WHAT $rat CALLS +@var{expr} is coerced to a CRE by @code{rat} +if it is not already a CRE. +This conversion may change the form of @var{expr} by putting all terms +over a common denominator. + +@code{num} is similar, but returns an ordinary expression instead of a CRE. +Also, @code{num} does not attempt to place all terms over a common denominator, +and thus some expressions which are considered ratios by @code{ratnumer} +are not considered ratios by @code{num}. +@c NEEDS AN EXAMPLE HERE @end defun @defun ratnump (@var{expr}) is @code{true} if @var{expr} is a rational number (includes integers) else @code{false}. +Returns @code{true} if @var{expr} is a literal integer or ratio of literal integers, +otherwise @code{false}. @end defun @defun ratp (@var{expr}) is @code{true} if @var{expr} is in CRE or extended CRE form else @code{false}. +Returns @code{true} if @var{expr} is a canonical rational expression (CRE) or extended CRE, +otherwise @code{false}. + +CRE are created by @code{rat} and related functions. +Extended CRE are created by @code{taylor} and related functions. @end defun @defvar ratprint Default value: @code{true} When @code{ratprint} is @code{false}, suppresses the printout of the message informing the user of the conversion of floating point numbers to rational numbers. +When @code{ratprint} is @code{true}, +a message informing the user of the conversion of floating point numbers +to rational numbers is displayed. @end defvar @defun ratsimp (@var{expr}) rationally simplifies (similar to @code{ratexpand}) the expression @var{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, i.e. the coefficients of the main variable are polynomials in the other variables. Variables may, as in @code{ratexpand}, include nonrational functions (e.g. SIN(X**2+1) ) but with @code{ratsimp}, the arguments to nonrational functions are rationally simplified. Note that @code{ratsimp} is affected by some of the variables which affect @code{ratexpand}. +@defunx ratsimp (@var{expr}, @var{x_1}, ..., @var{x_n}) +Simplifies the expression @var{expr} and all of its subexpressions, +including the arguments to nonrational 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 nonrational functions (e.g., @code{sin (x^2 + 1)}) +and the arguments to any such functions are also rationally simplified. RATSIMP(@var{expr},v1,v2,...,vn)  enables rational simplification with the +@code{ratsimp (@var{expr}, @var{x_1}, ..., @var{x_n})} +enables rational simplification with the specification of variable ordering as in @code{ratvars}. @...{ratsimpexpons}  if @code{true} will cause exponents of expressions to be @code{ratsimp}'ed automatically during simplification. +When @code{ratsimpexpons} is @code{true}, +@code{ratsimp} is applied to the exponents of expressions during simplification. + +See also @code{ratexpand}. +Note that @code{ratsimp} is affected by some of the +flags which affect @code{ratexpand}. Examples: @c FOLLOWING GENERATED FROM THESE EXPRESSIONS @@ 1220,25 +1324,38 @@ @defvar ratsimpexpons Default value: @code{false} When @code{ratsimpexpons} is @code{true}, causes exponents of expressions to be @code{ratsimp}'ed automatically during simplification. +When @code{ratsimpexpons} is @code{true}, +@code{ratsimp} is applied to the exponents of expressions during simplification. +@c NEED AN EXAMPLE HERE  RECYCLE THE ratsimpexpons EXAMPLE FROM ratsimp ABOVE @end defvar @... ratsubst (a, b, c) substitutes a for b in c. b may be a sum, product, power, etc. @code{ratsubst} knows something of the meaning of expressions whereas @code{subst} does a purely syntactic substitution. Thus SUBST(A,X+Y,X+Y+Z) returns X+Y+Z whereas @code{ratsubst} would return Z+A. @...{radsubstflag} if @code{true} permits @code{ratsubst} to make substitutions like U for SQRT(X) in X. +@defun ratsubst (@var{a}, @var{b}, @var{c}) +Substitutes @var{a} for @var{b} in @var{c} and returns the resulting expression. +@c "ETC" SUGGESTS THE READER KNOWS WHAT ELSE GOES THERE  NOT LIKELY THE CASE +@var{b} may be a sum, product, power, etc. + +@c WHAT, EXACTLY, DOES ratsubst KNOW ?? +@code{ratsubst} knows something of the meaning of expressions +whereas @code{subst} does a purely syntactic substitution. +Thus @code{subst (a, x + y, x + y + z)} returns @code{x + y + z} +whereas @code{ratsubst} returns @code{z + a}. + +When @code{radsubstflag} is @code{true}, +@code{ratsubst} makes substitutions for radicals in expressions +which don't explicitly contain them. Examples: @c EXAMPLES BELOW ADAPTED FROM examples (ratsubst) +@c WITH SOME ADDITIONAL STUFF @c ratsubst (a, x*y^2, x^4*y^3 + x^4*y^8); @c cos(x)^4 + cos(x)^3 + cos(x)^2 + cos(x) + 1; @c ratsubst (1  sin(x)^2, cos(x)^2, %); @c ratsubst (1  cos(x)^2, sin(x)^2, sin(x)^4); +@c radsubstflag: false$ +@c ratsubst (u, sqrt(x), x); +@c radsubstflag: true$ +@c ratsubst (u, sqrt(x), x); @example (%i1) ratsubst (a, x*y^2, x^4*y^3 + x^4*y^8); @@ 1253,30 +1370,55 @@ (%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 @end example @end defun @... ratvars (var1, var2, ..., varn) forms its n arguments into a list in which the rightmost variable varn will be the main variable of future rational expressions in which it occurs, and the other variables will follow in sequence. If a variable is missing from the @code{ratvars} list, it will be given lower priority than the leftmost variable var1. The arguments to @code{ratvars} can be either variables or nonrational functions (e.g. SIN(X)). The variable @code{ratvars} is a list of the arguments which have been given to this function. +@defvr {Function} ratvars (@var{x_1}, ..., @var{x_n}) +@defvrx {Function} ratvars () +@defvrx {Variable} ratvars +Declares main variables @var{x_1}, ..., @var{x_n} for rational expressions. +@var{x_n}, if present in a rational expression, is considered the main variable. +Otherwise, @var{x_[n1]} is considered the main variable if present, +and so on through the preceding variables to @var{x_1}, +which is considered the main variable only if none of the succeeding variables are present. @... defun +If a variable in a rational expression is not present in the @code{ratvars} list, +it is given a lower priority than @var{x_1}. @... ratweight (v1, w1, ..., vn, wn) assigns a weight of wi to the variable vi. This causes a term to be replaced by 0 if its weight exceeds the value of the variable @code{ratwtlvl} (default is no truncation). +The arguments to @code{ratvars} can be either variables or nonrational functions +such as @code{sin(x)}. + +The variable @code{ratvars} is a list of the arguments of +the function @code{ratvars} when it was called most recently. +Each call to the function @code{ratvars} resets the list. +@code{ratvars ()} clears the list. + +@c NEED EXAMPLES HERE +@end defvr + +@defun ratweight (@var{x_1}, @var{w_1}, ..., @var{x_n}, @var{w_n}) +@defunx ratweight () +Assigns a weight @var{w_i} to the variable @var{x_i}. +This causes a term to be replaced by 0 if its weight exceeds the +value of the variable @code{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. Thus the weight of 3*v1**2*v2 is 2*w1+w2. This truncation occurs only when multiplying or exponentiating CRE forms of expressions. +weight of a variable in the term times its power. +For example, the weight of @code{3 x_1^2 x_2} is @code{2 w_1 + w_2}. +Truncation according to @code{ratwtlvl} is carried out only when multiplying +or exponentiating canonical rational expressions (CRE). + +@code{ratweight ()} returns the cumulative list of weight assignments. + +Note: The @code{ratfac} and @code{ratweight} schemes are incompatible and may not +both be used at the same time. Examples: @c FOLLOWING GENERATED FROM THESE EXPRESSIONS @@ 1298,16 +1440,14 @@ (%o5)/R/ 2 b + 2 a + 1 @end example Note: The @code{ratfac} and @code{ratweight} schemes are incompatible and may not both be used at the same time.  @end defun @defvar ratweights @... DEFAULT VALUE ?? +Default value: @code{[]} @...{ratweights} is a list of weight assignments (set up by the @code{ratweight} function), @code{ratweights;} or @code{ratweight();} will show you the list. +@code{ratweights} is the list of weights assigned by @code{ratweight}. +The list is cumulative: +each call to @code{ratweight} places additional items in the list. @c DO WE REALLY NEED TO MENTION THIS ?? @code{kill (ratweights)} and @code{save (ratweights)} both work as expected. @@ 1318,39 +1458,54 @@ Default value: @code{false} @code{ratwtlvl} is used in combination with the @code{ratweight} function to control the truncation of rational (CRE form) expressions (for the default value of @code{false}, no truncation occurs). +function to control the truncation of canonical rational expressions (CRE). +For the default value of @code{false}, no truncation occurs. @end defvar @... remainder (p1, p2, var1, ...) computes the remainder of the polynomial p1 divided by the polynomial p2. +@defun remainder (@var{p_1}, @var{p_2}) +@defunx remainder (@var{p_1}, @var{p_2}, @var{x_1}, ..., @var{x_n}) +Returns the remainder of the polynomial @var{p_1} divided by the polynomial @var{p_2}. +The arguments @var{x_1}, ..., @var{x_n} are interpreted as in @code{ratvars}. + +@code{remainder} returns the second element +of the twoelement list returned by @code{divide}. +@c NEED SOME EXAMPLES HERE @end defun @... BOTH A VARIABLE AND A FUNCTION @... resultant (p1, p2, var) computes the resultant of the two polynomials p1 and p2, eliminating the variable var. The resultant is a determinant of the coefficients of var in p1 and p2 which equals zero if and only if p1 and p2 have a nonconstant factor in common. If p1 or p2 can be factored, it may be desirable to call @code{factor} before calling @code{resultant}. +@c CAN PROBABLY BE CLARIFIED +@defvr {Function} resultant (@var{p_1}, @var{p_2}, @var{x}) +@defvrx {Variable} resultant +Computes the resultant of the two polynomials @var{p_1} and @var{p_2}, +eliminating the variable @var{x}. +The resultant is a determinant of the coefficients of @var{x} +in @var{p_1} and @var{p_2}, +which equals zero +if and only if @var{p_1} and @var{p_2} have a nonconstant factor in common. @...{resultant}  controls which algorithm will be used to compute the resultant. @code{subres} for subresultant prs [the default], @code{mod} for modular resultant algorithm, and @code{red} for reduced prs. On most problems @code{subres} should be best. On some large degree univariate or bivariate problems @code{mod} may be better. Another alternative is the @...{bezout} command which takes the same arguments as @code{resultant} and returns a matrix. @code{determinant} of this matrix is the desired resultant. +If @var{p_1} or @var{p_2} can be factored, +it may be desirable to call @code{factor} before calling @code{resultant}. @... defun +The variable @code{resultant} controls which algorithm will be used to compute +the resultant. +@c WHAT DOES THE FOLLOWING MEAN EXACTLY ?? +@code{subres} for subresultant prs, +@code{mod} for modular resultant algorithm, +and @code{red} for reduced prs. +On most problems @code{subres} should be best. +On some large degree univariate or bivariate problems @code{mod} may be better. + +The function @code{bezout} takes the same arguments as @code{resultant} and returns +a matrix. The determinant of the return value is the desired resultant. + +@c NEED AN EXAMPLE HERE +@end defvr @defvar savefactors Default value: @code{false} +@c "certain functions"  WHICH ONES ?? When @code{savefactors} is @code{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 @@ 1358,14 +1513,16 @@ @end defvar +@c I CAN'T TELL WHAT THIS IS SUPPOSED TO BE ABOUT @defun sqfr (@var{expr}) is similar to @code{factor} except that the polynomial factors are "squarefree." That is, they have factors only of degree one. This algorithm, which is also used by the first stage of @code{factor}, utilizes the fact that a polynomial has in common with its nth derivative all its factors of degree > n. Thus by taking gcds with the polynomial of +is similar to @code{factor} except that the polynomial factors are "squarefree." +That is, they have factors only of degree one. +This algorithm, which is also used by the first stage of @code{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 > 1 can be found. +factors of degree greater than 1 can be found. Example: @c FOLLOWING GENERATED FROM THIS EXPRESSION @@ 1379,26 +1536,42 @@ @end defun @... tellrat (poly) adds to the ring of algebraic integers known to Maxima, the element which is the solution of the polynomial with integer coefficients. Maxima initially knows about %I and all roots of integers. TELLRAT(X); means substitute 0 for X in rational functions. There is a command @code{untellrat} which takes kernels and removes @code{tellrat} properties. When @code{tellrat}'ing a multivariate polynomial, e.g. TELLRAT(X^2Y^2);, there would be an ambiguity as to whether to substitute Y^2 for X^2 or vice versa. The system will pick 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. @...{tellrat} and @code{untellrat} both can take any number of arguments, and TELLRAT(); returns a list of the current substitutions. Note: When you @code{tellrat} reducible polynomials, you want to be careful not to attempt to rationalize a denominator with a zero divisor. E.g. TELLRAT(W^31)$ ALGEBRAIC:TRUE$ RAT(1/(W^2W)); will give "quotient by zero". This error can be avoided by setting @code{ratalgdenom} to @code{false}. +@c THIS ITEM STILL NEEDS WORK +@defun tellrat (@var{p_1}, ..., @var{p_n}) +@defunx tellrat () +Adds to the ring of algebraic integers known to Maxima +the elements which are the solutions of the polynomials @var{p_1}, ..., @var{p_n}. +Each argument @var{p_i} is a polynomial with integer coefficients. + +@code{tellrat (@var{x})} effectively means substitute 0 for @var{x} in rational +functions. + +@code{tellrat ()} returns a list of the current substitutions. + @code{algebraic} must be set to @code{true} in order for the simplification of algebraic integers to take effect. +Maxima initially knows about the imaginary unit @code{%i} +and all roots of integers. + +There is a command @code{untellrat} which takes kernels and +removes @code{tellrat} properties. + +When @code{tellrat}'ing a multivariate +polynomial, e.g., @code{tellrat (x^2  y^2)}, there would be an ambiguity as to +whether to substitute @code{@var{y}^2} for @code{@var{x}^2} +or vice versa. +Maxima picks a particular ordering, but if the user wants to specify which, e.g. +@code{tellrat (y^2 = x^2)} provides a syntax which says replace +@code{@var{y}^2} by @code{@var{x}^2}. + +@c CAN'T TELL WHAT THIS IS ABOUT  tellrat(w^31)$ algebraic:true$ rat(1/(w^2w)); +@c DOES NOT YIELD AN ERROR, SO WHAT IS THE POINT ABOUT ratalgdenom ?? +@c When you @code{tellrat} reducible polynomials, you want to be careful not to +@c attempt to rationalize a denominator with a zero divisor. E.g. +@c tellrat(w^31)$ algebraic:true$ rat(1/(w^2w)); will give "quotient by +@c zero". This error can be avoided by setting @code{ratalgdenom} to @code{false}. + Examples: @c EXAMPLE ADAPTED FROM example (tellrat) @c 10*(%i + 1)/(%i + 3^(1/3)); @@ 1436,16 +1609,20 @@ @end defun @defun totaldisrep (@var{expr}) converts every subexpression of @var{expr} from CRE to general form. If @var{expr} is itself in CRE form then this is identical to @...{ratdisrep} but if not then @code{ratdisrep} would return @var{expr} unchanged while @...{totaldisrep} would "totally disrep" it. This is useful for ratdisrepping expressions e.g., equations, lists, matrices, etc. which +Converts every subexpression of @var{expr} from canonical rational expressions (CRE) to +general form and returns the result. +If @var{expr} is itself in CRE form then @code{totaldisrep} is identical to +@code{ratdisrep}. + +@code{totaldisrep} may be useful for +ratdisrepping expressions such as equations, lists, matrices, etc., which have some subexpressions in CRE form. +@c NEED EXAMPLES HERE @end defun @... untellrat (x) takes kernels and removes @code{tellrat} properties. +@defun untellrat (@var{x_1}, ..., @var{x_n}) +Removes @code{tellrat} properties from @var{x_1}, ..., @var{x_n}. +@c NEED EXAMPLES HERE @end defun 