;;; -*- Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10 -*- ;;;;
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;;; The data in this file contains enhancments. ;;;;;
;;; ;;;;;
;;; Copyright (c) 1984,1987 by William Schelter,University of Texas ;;;;;
;;; All rights reserved ;;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; (c) Copyright 1981 Massachusetts Institute of Technology ;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(in-package :maxima)
(macsyma-module rat3c)
;; THIS IS THE NEW RATIONAL FUNCTION PACKAGE PART 3.
;; IT INCLUDES THE GCD ROUTINES AND THEIR SUPPORTING FUNCTIONS
(load-macsyma-macros ratmac)
(declare-top (special $float $keepfloat $algebraic $ratfac *alpha genvar))
;; List of GCD algorithms. Default one is first.
(defmvar *gcdl* '($spmod $subres $ez $red $mod $algebraic))
(defmvar $gcd (car *gcdl*)) ;Sparse Modular
(defun cgcd (a b)
(cond (modulus 1)
((and $keepfloat (or (floatp a) (floatp b))) 1)
(t (gcd a b))))
(defmfun pquotientchk (a b)
(if (eqn b 1) a (pquotient a b)))
(defun ptimeschk (a b)
(cond ((eqn a 1) b)
((eqn b 1) a)
(t (ptimes a b))))
(defun pfloatp (x)
(catch 'float (if (pcoefp x) (floatp x) (pfloatp1 x))))
(defun pfloatp1 (x)
(mapc #'(lambda (q) (cond ((pcoefp q) (when (floatp q) (throw 'float t)))
((pfloatp1 q))))
(cdr x))
nil)
(defmfun pgcd (x y)
(setq x (car (pgcda x y nil)))
(cond ((pminusp x) (pminus x))
(modulus (monize x))
(t x)))
(defmfun plcm (x y)
(setq x (pgcdcofacts x y))
(ptimes (car x) (ptimes (cadr x) (caddr x))))
(defun plcmcofacts (x y)
(setq x (pgcdcofacts x y))
(list (ptimes (car x) (ptimes (cadr x) (caddr x)))
(caddr x) (cadr x)))
; returns list (gcd xx yy alg)
; where x * y = gcd^2 * xx * yy / alg^2
; and alg is non-nil only when $algebraic is true
(defun pgcdcofacts (x y)
(let ((a (pgcda x y t)))
(cond ((cdr a) a)
((equal (setq a (car a)) 1) (list 1 x y))
((and $algebraic (not (pcoefp a)))
(cons a (prog2 (setq x (rquotient x a)
y (rquotient y a)
a (pgcdcofacts (cdr x) (cdr y)))
(list (ptimes (car x) (caddr a))
(ptimes (car y) (cadr a))
(ptimes (cadr a) (cdr y))))))
((eq a x) (list x 1 (pquotient y x)))
((eq a y) (list a (pquotient x y) 1))
(t (list a (pquotient x a) (pquotient y a))))))
(defun pgcda (x y cofac? &aux a c)
(cond ((not $gcd) (list 1 x y))
((and $keepfloat (or (pfloatp x) (pfloatp y)))
(cond ((or (pcoefp x) (pcoefp y)
(pcoefp (setq a (car (ptermcont x))))
(pcoefp (setq a (pgcd a (car (ptermcont y))))))
(list 1 x y))
(t (list a))))
((pcoefp x)
(cond ((pcoefp y)
(cons (setq a (cgcd x y))
(and cofac?
(list (cquotient x a) ;(CQUOTIENT 0 0) = 0
(cquotient y a)))))
((zerop x) (list y x 1))
(t (list (pcontent1 (cdr y) x)))))
((pcoefp y) (cond ((zerop y) (list x 1 y))
(t (list (pcontent1 (cdr x) y)))))
((equal x y) (list x 1 1))
($ratfac (fpgcdco x y))
((not (eq (p-var x) (p-var y)))
(list (if (pointergp (p-var x) (p-var y))
(oldcontent1 (p-terms x) y)
(oldcontent1 (p-terms y) x))))
((progn (desetq (a x) (ptermcont x))
(desetq (c y) (ptermcont y))
(not (and (equal a 1) (equal c 1))))
(mapcar #'ptimes (monomgcdco a c cofac?) (pgcda x y cofac?)))
((and (not $algebraic) (not modulus)
(desetq (a . c) (lin-var-find (nreverse (pdegreevector x))
(nreverse (pdegreevector y))
(reverse genvar))))
(cond ((= a 1) (linhack x y (car c) (cadr c) cofac?))
(t (setq a (linhack y x a (cadr c) cofac?))
(if (cdr a) (rplacd a (nreverse (cdr a))))
a)))
((eq $gcd '$spmod) (list (zgcd x y)))
((eq $gcd '$subres) (list (oldgcd x y)))
((eq $gcd '$algebraic)
(if (or (palgp x) (palgp y))
(let (($gcd '$subres)) (list (oldgcd x y)))
(let (($gcd '$spmod)) (list (zgcd x y)))))
((eq $gcd '$ez) (ezgcd2 x y))
((eq $gcd '$red) (list (oldgcd x y)))
((eq $gcd '$mod) (newgcd x y modulus))
((not (member $gcd *gcdl* :test #'eq))
(merror "`gcd' set incorrectly:~%~M" $gcd))
(t (list 1 x y))))
(defun monomgcdco (p q cofac?)
(let ((gcd (monomgcd p q)))
(cons gcd (if cofac? (list (pquotient p gcd) (pquotient q gcd)) ()))))
(defun monomgcd (p q)
(cond ((or (pcoefp p) (pcoefp q)) 1)
((eq (p-var p) (p-var q))
(make-poly (p-var p) (min (p-le p) (p-le q))
(monomgcd (p-lc p) (p-lc q))))
((pointergp (car p) (car q)) (monomgcd (p-lc p) q))
(t (monomgcd p (p-lc q)))))
(defun linhack (pol1 pol2 nonlindeg var cofac?)
(prog (coeff11 coeff12 gcdab rpol1 rpol2 gcdcd gcdcoef)
(desetq (coeff11 . coeff12) (bothprodcoef (make-poly var) pol1))
(setq gcdab (if (pzerop coeff12) coeff11
(pgcd coeff11 coeff12)))
(cond ((equal gcdab 1)
(cond ((setq coeff11 (testdivide pol2 pol1))
(return (list pol1 1 coeff11)))
(t (return (list 1 pol1 pol2))))))
(setq rpol1 (pquotient pol1 gcdab))
(desetq (gcdcd rpol2) (linhackcontent var pol2 nonlindeg))
(cond ((equal gcdcd 1)
(cond ((setq coeff12 (testdivide rpol2 rpol1))
(return (list rpol1 gcdab coeff12)))
(t (return (list 1 pol1 pol2))))))
(cond (cofac? (desetq (gcdcoef coeff11 coeff12)
(pgcdcofacts gcdab gcdcd))
(cond ((setq gcdcd (testdivide rpol2 rpol1))
(return (list (ptimes gcdcoef rpol1)
coeff11
(ptimes coeff12 gcdcd))))
(t (return (list gcdcoef
(ptimes coeff11 rpol1)
(ptimes coeff12 rpol2))))))
(t (setq gcdcoef (pgcd gcdcd gcdab))
(cond ((testdivide rpol2 rpol1)
(return (list (ptimes gcdcoef rpol1))))
(t (return (list gcdcoef))))))))
(defun lin-var-find (a b c)
(do ((varl c (cdr varl))
(degl1 a (cdr degl1))
(degl2 b (cdr degl2)))
((or (null degl1) (null degl2)) nil)
(if (equal (min (car degl1) (car degl2)) 1)
(return (list (car degl1) (car degl2) (car varl))))))
(defun linhackcontent (var pol nonlindeg &aux (npol pol) coef gcd)
(do ((i nonlindeg (1- i)))
((= i 0) (list (setq gcd (pgcd gcd npol)) (pquotient pol gcd)))
(desetq (coef . npol) (bothprodcoef (make-poly var i 1) npol))
(unless (pzerop coef)
(setq gcd (if (null gcd) coef (pgcd coef gcd)))
(if (equal gcd 1) (return (list 1 pol))))))
;;*** THIS IS THE REDUCED POLYNOMIAL REMAINDER SEQUENCE GCD (COLLINS')
(defun oldgcd (x y &aux u v s egcd) ;only called from pgcda
(desetq (x u) (oldcontent x))
(desetq (y v) (oldcontent y))
(setq egcd (gcd (pgcdexpon u) (pgcdexpon v)))
(if (> egcd 1)
(setq u (pexpon*// u egcd nil)
v (pexpon*// v egcd nil)))
(if (> (p-le v) (p-le u)) (exch u v))
(setq s (case $gcd
($red (redgcd u v))
($subres (subresgcd u v))
(t (merror "Illegal `gcd' algorithm"))))
(unless (equal s 1)
(setq s (pexpon*// (primpart
(if $algebraic s
(pquotient s (pquotient (p-lc s)
(pgcd (p-lc u) (p-lc v))))))
egcd t)))
(setq s (ptimeschk s (pgcd x y)))
(and $algebraic (not (pcoefp (setq u (leadalgcoef s))))
(not (equal u s)) (setq s (algnormal s)))
(cond (modulus (monize s))
((pminusp s) (pminus s))
(t s)))
(defun pgcdexpon (p)
(if (pcoefp p) 0
(do ((d (cadr p) (gcd d (car l)))
(l (cdddr p) (cddr l)))
((or (null l) (= d 1)) d))))
(defun pexpon*// (p n *?)
(if (or (pcoefp p) (= n 1)) p
(do ((ans (list (car p))
(cons (cadr l)
(cons (if *? (* (car l) n)
(truncate (car l) n))
ans)))
(l (cdr p) (cddr l)))
((null l) (nreverse ans)))))
;;polynomial gcd using reduced prs
(defun redgcd (p q &aux (d 0))
(loop until (zerop (pdegree q (p-var p)))
do (psetq p q
q (pquotientchk (prem p q) (pexpt (p-lc p) d))
d (+ (p-le p) 1 (- (p-le q))))
finally (return (if (pzerop q) p 1))))
;;computes gcd's using subresultant prs TOMS Sept. 1978
(defun subresgcd (p q)
(loop for g = 1 then (p-lc p)
for h = 1 then (pquotient (pexpt g d) h^1-d)
for d = (- (p-le p) (p-le q))
for h^1-d = (if (equal h 1) 1 (pexpt h (1- d)))
do (psetq p q
q (pquotientchk (prem p q) (ptimes g (ptimes h h^1-d))))
if (zerop (pdegree q (p-var p))) return (if (pzerop q) p 1)))
;;*** THIS COMPUTES PSEUDO REMAINDERS
(defun psquorem1 (u v quop)
(prog (k (m 0) lcu lcv quo lc)
(declare (special k lcu lcv))
(setq lcv (pt-lc v))
(setq k (- (pt-le u) (pt-le v)))
(cond ((minusp k) (return (list 1 '(0 0) u))))
(if quop (setq lc (pexpt (pt-lc v) (1+ k))))
a (setq lcu (pminus (pt-lc u)))
(if quop (setq quo (cons (ptimes (pt-lc u) (pexpt (pt-lc v) k))
(cons k quo))))
(cond ((null (setq u (pgcd2 (pt-red u) (pt-red v))))
(return (list lc (nreverse quo) '(0 0))))
((minusp (setq m (- (pt-le u) (pt-le v))))
(setq u (cond ((zerop k) u)
(t (pctimes1 (pexpt lcv k) u))))
(return (list lc (nreverse quo) u)))
((> (1- k) m)
(setq u (pctimes1 (pexpt lcv (- (1- k) m)) u))))
(setq k m)
(go a)))
(defun prem (p q)
(cond ((pcoefp p) (if (pcoefp q) (cremainder p q) p))
((pcoefp q) (pzero))
(t (psimp (p-var p) (pgcd1 (p-terms p) (p-terms q))))))
(defmfun pgcd1 (u v) (caddr (psquorem1 u v nil)))
(defun pgcd2 (u v &aux (i 0))
(declare (special k lcu lcv) (fixnum k i))
(cond ((null u) (pcetimes1 v k lcu))
((null v) (pctimes1 lcv u))
((zerop (setq i (+ (pt-le u) (- k) (- (car v)))))
(pcoefadd (pt-le u) (pplus (ptimes lcv (pt-lc u))
(ptimes lcu (pt-lc v)))
(pgcd2 (pt-red u) (pt-red v))))
((minusp i)
(list* (+ (pt-le v) k) (ptimes lcu (pt-lc v)) (pgcd2 u (pt-red v))))
(t (list* (pt-le u) (ptimes lcv (pt-lc u)) (pgcd2 (pt-red u) v)))))
;;;*** OLDCONTENT REMOVES ALL BUT MAIN VARIABLE AND PUTS THAT IN CONTENT
;;;*** OLDCONTENT OF 3*A*X IS 3*A (WITH MAINVAR=X)
(defun rcontent (p) ;RETURNS RAT-FORMS
(let ((q (oldcontenta p)))
(list (cons q 1) (cond ($algebraic (rquotient p q))
(t (cons (pquotient p q) 1))))))
(defun oldcontenta (x)
(cond ((pcoefp x) x)
(t (setq x (contsort (cdr x)))
(oldcontent2 (cdr x) (car x)))))
(defmfun oldcontent (x)
(cond ((pcoefp x) (list x 1))
((null (p-red x))
(list (p-lc x) (make-poly (p-var x) (p-le x) 1)))
(t (let ((u (contsort (cdr x))) v)
(setq u (oldcontent2 (cdr u) (car u))
v (cond ($algebraic (car (rquotient x u)))
(t (pcquotient x u))))
(cond ((pminusp v) (list (pminus u) (pminus v)))
(t (list u v)))))))
(defun oldcontent1 (x gcd)
(cond ((equal gcd 1) 1)
((null x) gcd)
(t (oldcontent2 (contsort x) gcd))))
(defun oldcontent2 (x gcd)
(do ((x x (cdr x))
(gcd gcd (pgcd (car x) gcd)))
((or (null x) (equal gcd 1)) gcd)))
(defun contsort (x)
(setq x (coefl x))
(cond ((member 1 x) '(1))
((null (cdr x)) x)
(t (sort x #'contodr))))
(defun coefl (x)
(do ((x x (cddr x))
(ans nil (cons (cadr x) ans)))
((null x) ans)))
(defun contodr (a b)
(cond ((pcoefp a) t)
((pcoefp b) nil)
((eq (car a) (car b)) (not (> (cadr a) (cadr b))))
(t (pointergp (car b)(car a)))))
;;;*** PCONTENT COMPUTES INTEGER CONTENT
;;;*** PCONTENT OF 3*A*X IS 3 IF MODULUS = NIL 1 OTHERWISE
(defun pcontent (x)
(cond ((pcoefp x) (list x 1))
(t (let ((u (pcontentz x)))
(if (eqn u 1) (list 1 x)
(list u (pcquotient x u)))))))
(defun pcontent1 (x gcd)
(do ((x x (cddr x))
(gcd gcd (cgcd gcd (pcontentz (cadr x)))))
((or (null x) (equal gcd 1)) gcd)))
(defun pcontentz (p)
(cond ((pcoefp p) p)
(t (pcontent1 (p-red p) (pcontentz (p-lc p))))))
(defun ucontent (p) ;CONTENT OF UNIV. POLY
(cond ((pcoefp p) (abs p))
(t (setq p (mapcar #'abs (coefl (cdr p))))
(let ((m (apply #'min p)))
(oldcontent2 (delete m p :test #'equal) m)))))
;;*** PGCDU CORRESPONDS TO BROWN'S ALGORITHM U
;;;PGCDU IS NOT NOW IN RAT;UFACT >
(defmfun pgcdu (p q)
(do () ((pzerop q) (monize p))
(psetq p q q (pmodrem p q))))
(defun pmodrem (x y)
(cond ((null modulus)
(merror "Illegal use of `pmodrem'"))
((pacoefp y) (if (pzerop y) x 0))
((pacoefp x) x)
((eq (p-var x) (p-var y))
(psimp (car x) (pgcdu1 (p-terms x) (p-terms y) nil)))
(t (merror "Illegal use of `pmodrem'"))))
(defun pmodquo (u v &aux quo)
(declare (special quo))
(cond ((null modulus)
(merror "Illegal use of `pmodquo'"))
((pcoefp v) (cons (ptimes (crecip v) u) 0))
((alg v) (cons (ptimes (painvmod v) u) 0))
((pacoefp u) (cons 0 u))
((not (eq (p-var u) (p-var v)))
(merror "Illegal use of `pmodquo'"))
(t (xcons (psimp (car u) (pgcdu1 (cdr u) (cdr v) t))
(psimp (car u) quo)))))
(defun pgcdu1 (u v pquo*)
(let ((invv (painvmod (pt-lc v))) (k 0) q*)
(declare (special k quo q*) (fixnum k))
(loop until (minusp (setq k (- (pt-le u) (pt-le v))))
do (setq q* (ptimes invv (pt-lc u)))
if pquo* do (setq quo (nconc quo (list k q*)))
when (ptzerop (setq u (pquotient2 (pt-red u) (pt-red v))))
return (ptzero)
finally (return u))))
(defun newprime (p)
(declare (special bigprimes)) ;defined later on
(cond ((null p) (car bigprimes))
(t (do ((pl bigprimes (cdr pl)))
((null pl) (setq p (fnewprime p))
(setq bigprimes (nconc bigprimes (list p)))
p)
(if (< (car pl) p) (return (car pl)))))))
(defun fnewprime (p) ; Finds biggest prime less than fixnum P.
(do ((pp (if (oddp p) (- p 2) (- p 1)) (- pp 2))) ((< pp 0))
(if (primep pp) (return pp))))
;; #O