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1992-12-23
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;;; JACAL: Symbolic Mathematics System. -*-scheme-*-
;;; Copyright 1989, 1990, 1991, 1992 Aubrey Jaffer.
;;; See the file "COPYING" for terms applying to this program.
;;;Functions which operate on polynomials as polynomials
;;;have prefix POLY_
;;;Functions which operate on polynomials with the same major variable
;;;have prefix UNIV_
;;;Functions which operate on coefficients (without major varible)
;;;have prefix COES_
;(proclaim '(optimize (speed 3) (compilation-speed 0)))
(define (one? n) (eqv? 1 n))
(define (poly_0? p) (eqv? 0 p))
(define (divides? a b) (zero? (remainder b a)))
;;; poly is the internal workhorse data type in the form
;;; of numeric or list (var coeff0 coeff1 ...)
;;; where var is a variable and coeffn is the coefficient of var^n.
;;; coeffn is poly. The variables currently are arranged
;;; with the reverse alphabetically with z higher order than A.
(define (poly_find-var? poly var)
(poly_find-var-if? poly (lambda (x) (eqv? var x))))
(define (poly_find-var-if? poly proc)
(cond ((number? poly) #f)
((proc (car poly)))
(else (some (lambda (x) (poly_find-var-if? x proc)) (cdr poly)))))
;;;POLY_VARS returns a list of all vars used in POLY
(define (poly_vars poly)
(let ((elts '()))
(poly_for-each-var (lambda (v) (set! elts (adjoin v elts))) poly)
elts))
;;;; the following functions are for internal use on the poly data type
;;; this normalizes short polys.
(define (univ_norm0 var col)
(cond ((null? col) 0)
((null? (cdr col)) (car col))
(else (cons var col))))
(define (map-no-end-0s proc l)
(if (null? l)
l
(let ((res (proc (car l)))
(rest (map-no-end-0s proc (cdr l))))
(if (and (null? rest) (eqv? 0 res))
rest
(cons res rest)))))
(define (map2c-no-end-0s proc l1 l2)
(cond ((null? l1) l2)
((null? l2) l1)
(else
(let ((res (proc (car l1) (car l2)))
(rest (map2c-no-end-0s proc (cdr l1) (cdr l2))))
(if (and (null? rest) (eqv? 0 res))
rest
(cons res rest))))))
(define (univ-w/o-lt p)
(univ_norm0 (car p) (map-no-end-0s identity (butlast (cdr p) 1))))
(define (make-list-length lst len fill)
(let ((ld (- len (length lst))))
(cond ((negative? ld) (butlast lst (- ld)))
(else (append lst (make-list ld fill))))))
;;; replaces each var in poly with (proc var)
(define (poly_do-vars proc poly)
(if (number? poly)
poly
(cons (proc (car poly))
(map (lambda (b) (poly_do-vars proc b))
(cdr poly)))))
(define (polys_do-vars proc polys)
(bunch_map (lambda (poly) (poly_do-vars proc poly)) polys))
(define (poleqns_do-vars proc licits)
(bunch_map (lambda (poly) (poly_do-vars proc (licit->poleqn poly))) licits))
;;; This can call proc more than once per var
(define (poly_for-each-var proc poly)
(cond ((number? poly))
(else
(proc (car poly))
(for-each (lambda (b) (poly_for-each-var proc b))
(cdr poly)))))
(define (polys_for-each-var proc polys)
(bunch_for-each (lambda (poly) (poly_for-each-var proc poly)) polys))
(define (ipow-by-squaring x n acc proc)
(cond ((zero? n) acc)
((one? n) (proc acc x))
(else (ipow-by-squaring (proc x x)
(quotient n 2)
(if (even? n) acc (proc acc x))
proc))))
(define (poly_add-const term p2)
(cons (car p2) (cons (poly_+ term (cadr p2)) (cddr p2))))
(define (poly_+ p1 p2)
(cond ((and (number? p1) (number? p2)) (+ p1 p2))
((number? p1) (if (zero? p1) p2 (poly_add-const p1 p2)))
((number? p2) (if (zero? p2) p1 (poly_add-const p2 p1)))
((eq? (car p1) (car p2))
(univ_norm0 (car p1) (map2c-no-end-0s poly_+ (cdr p1) (cdr p2))))
((var_> (car p2) (car p1)) (poly_add-const p1 p2))
(else (poly_add-const p2 p1))))
(define (univ_* p1 p2)
(let ((res (make-list (+ (length (cdr p1)) (length (cdr p2)) -1) 0)))
(do ((rpl res (cdr rpl))
(a (cdr p1) (cdr a)))
((null? a) (cons (car p1) res))
(do ((b (cdr p2) (cdr b))
(rp rpl (cdr rp)))
((null? b))
(set-car! rp (poly_+ (poly_* (car a) (car b)) (car rp)))))))
(define (poly_times-const term p2)
(cons (car p2) (map (lambda (x) (poly_* term x)) (cdr p2))))
(define (poly_* p1 p2)
(cond ((and (number? p1) (number? p2)) (* p1 p2))
((number? p1) (cond ((zero? p1) 0)
((one? p1) p2)
(else (poly_times-const p1 p2))))
((number? p2) (cond ((zero? p2) 0)
((one? p2) p1)
(else (poly_times-const p2 p1))))
((eq? (car p1) (car p2)) (univ_* p1 p2))
((var_> (car p2) (car p1)) (poly_times-const p1 p2))
(else (poly_times-const p2 p1))))
(define (poly_negate p) (poly_* -1 p))
(define (poly_- p1 p2) (poly_+ p1 (poly_negate p2)))
(define (univ_/? u v)
(let ((r (list->vector (cdr u)))
(m (length (cddr u)))
(n (length (cddr v)))
(vn (car (last-pair v)))
(q '()))
(do ((k (- m n) (- k 1))
(qk (poly_/? (vector-ref r m) vn)
(and (> k 0) (poly_/? (vector-ref r (+ n k -1)) vn))))
((not qk)
(and (< k 0)
(do ((k (- n 2) (- k 1)))
((or (< k 0) (not (poly_0? (vector-ref r k))))
(< k 0)))
(univ_norm0 (car u) q)))
(set! q (cons qk q))
(let ((qk- (poly_negate qk)))
(do ((j (+ n k -1) (- j 1)))
((< j k))
(vector-set! r j (poly_+ (vector-ref r j)
(poly_* (list-ref v (+ (- j k) 1)) qk-))))))))
;;; POLY_/? returns U / V if V divides U, otherwise returns #f
(define (poly_/? u v)
(cond ((equal? u v) 1)
((eqv? 0 u) 0)
((eqv? 0 v) #f)
((unit? v) (poly_* v u))
((and (number? u) (number? v)) (and (divides? v u) (quotient u v)))
((number? v) (univ_/? u (const_promote (car u) v)))
((number? u) #f)
((eq? (car u) (car v)) (univ_/? u v))
((var_> (car u) (car v))
(univ_/? u (const_promote (car u) v)))
(else #f)))
(define (univ_/ dividend divisor)
(or (univ_/? dividend divisor)
(math-error divisor " does not udivide " dividend)))
(define (poly_/ dividend divisor)
(or (poly_/? dividend divisor)
(math-error divisor " does not divide " dividend)))
(define (univ_monomial coeff n var)
(cond ((eq? 0 coeff) 0)
((>= 0 n) coeff)
(else
(cons var
((lambda (x) (set-car! (last-pair x) coeff) x)
(make-list (+ 1 n) 0))))))
(define (poly_degree p var)
(cond ((number? p) 0)
((eq? var (car p)) (length (cddr p)))
((var_> var (car p)) 0)
(else (reduce-init (lambda (m c) (max m (poly_degree c var)))
0
(cdr p)))))
;(define (leading-coeff p var) (poly_coeff p var (poly_degree p var)))
;(define (monic? u var) (one? (leading-coeff u var)))
(define (poly_^ x n)
(if (number? x)
; (expt x n)
(ipow-by-squaring x n 1 *)
(ipow-by-squaring x n 1 poly_*)))
;;;; Routines used in normalizing IMPL polynomials
(define (leading-number p)
(if (number? p) p (leading-number (car (last-pair p)))))
;;; This canonicalizes polys with respect to sign by forcing the
;;; numerical coefficient of the a certain term to always be positive.
(define (signcan p)
(if (> (leading-number p) 0) (poly_negate p) p))
(define (shorter? x y) (< (length x) (length y)))
(define (univ_degree p var)
(if (or (number? p) (not (eq? (car p) var))) 0 (length (cddr p))))
;;; THE NEXT 3 ROUTINES FOR SUBRESULTANT GCD ASSUME THAT THE ARGUMENTS
;;; ARE POLYNOMIALS WITH THE SAME MAJOR VARIABLE.
;;; THESE TWO ROUTINES ASSUME THAT THE FIRST ARGUMENT IS OF GREATER
;;; OR EQUAL ORDER THAN THE SECOND.
;;; These algorithms taken from:
;;; Knuth, D. E.,
;;; The Art Of Computer Programming, Vol. 2: Seminumerical Algorithms,
;;; Addison Wesley, Reading, MA 1969.
;;; Pseudo Remainder
;;; This returns a list of the pseudo quotient and pseudo remainder.
(define (univ_pdiv u v)
(let* ((r (list->vector (cdr u)))
(m (length (cddr u)))
(n (length (cddr v)))
(vn (car (last-pair v)))
(q (make-vector (+ (- m n) 1) 1)))
(do ((tt (- (- m n) 1) (- tt 1))
(k 1 (+ 1 k))
(vnp 1))
((< tt 0))
(set! vnp (poly_* vnp vn))
(vector-set! q k vnp)
(vector-set! r tt (poly_* (vector-ref r tt) vnp)))
(do ((k (- m n) (- k 1))
(rnk 0))
((< k 0))
(set! rnk (poly_negate (vector-ref r (+ n k))))
(do ((j (+ n k -1) (- j 1)))
((< j k))
(vector-set! r j (poly_+ (poly_* (vector-ref r j) vn)
(poly_* (list-ref v (+ (- j k) 1)) rnk)))))
(list
(do ((k (- m n) (+ -1 k))
(end '() (cons (poly_* (vector-ref r (+ n k))
(vector-ref q k)) end)))
((zero? k) (univ_norm0 (car u) (cons (vector-ref r n) end))))
(do ((j (- n 1) (- j 1))
(end '()))
((< j 0) (univ_norm0 (car u) end))
(if (not (and (null? end) (eqv? 0 (vector-ref r j))))
(set! end (cons (vector-ref r j) end)))))))
(define (poly_pdiv dividend divisor var)
(let ((pd1 (poly_degree dividend var))
(pd2 (poly_degree divisor var)))
(cond ((< pd1 pd2) (list 0 dividend))
((zero? (+ pd1 pd2))
(list (quotient dividend divisor) (remainder dividend divisor)))
((zero? pd1) (list 0 dividend))
((zero? pd2)
;;; This should work but doesn't.
;;; (map univ_demote (univ_pdiv (promote var dividend)
;;; (const_promote var divisor)))
(list 0 dividend))
(else
(map univ_demote (univ_pdiv (promote var dividend)
(promote var divisor)))))))
(define (univ_prem u v)
(let* ((r (list->vector (cdr u)))
(m (length (cddr u)))
(n (length (cddr v)))
(vn (car (last-pair v))))
(do ((k (- (- m n) 1) (- k 1))
(vnp 1))
((< k 0))
(set! vnp (poly_* vnp vn))
(vector-set! r k (poly_* (vector-ref r k) vnp)))
(do ((k (- m n) (- k 1))
(rnk 0))
((< k 0))
(set! rnk (poly_negate (vector-ref r (+ n k))))
(do ((j (+ n k -1) (- j 1)))
((< j k))
(vector-set! r j (poly_+ (poly_* (vector-ref r j) vn)
(poly_* (list-ref v (+ (- j k) 1)) rnk)))))
(do ((j (- n 1) (- j 1))
(end '()))
((< j 0) (univ_norm0 (car u) end))
(if (and (null? end) (eqv? 0 (vector-ref r j)))
#f
(set! end (cons (vector-ref r j) end))))))
;;; Pseudo Remainder Sequence
(define (univ_prs u v)
(let ((var (car u))
(g 1)
(h 1)
(delta 0))
(do ((r (univ_prem u v) (univ_prem u v)))
((eqv? 0 (univ_degree r var))
(if (eqv? 0 r) v r))
(set! delta (- (univ_degree u var) (univ_degree v var)))
(set! u v)
(set! v (univ_/ r (const_promote (car r) (poly_* g (poly_^ h delta)))))
(set! g (car (last-pair u)))
(set! h (cond ((one? delta) g)
((zero? delta) h)
(else (poly_/ (poly_^ g delta)
(poly_^ h (+ -1 delta)))))))))
(define (univ_gcd u v)
(let* ((cu (univ_cont u))
(cv (univ_cont v))
(c (poly_gcd cu cv))
(ppu (poly_/ u cu))
(ppv (poly_/ v cv))
(ans (if (shorter? ppv ppu)
(univ_prs ppu ppv)
(univ_prs ppv ppu))))
(if (zero? (univ_degree ans (car u)))
c
(poly_* c (univ_primpart ans)))))
(define (poly_gcd p1 p2)
(cond ((equal? p1 p2) p1)
((and (number? p1) (number? p2)) (gcd p1 p2))
((number? p1) (if (zero? p1)
p2
(apply poly_gcd* p1 (cdr p2))))
((number? p2) (if (zero? p2)
p1
(apply poly_gcd* p2 (cdr p1))))
((eq? (car p1) (car p2)) (univ_gcd p1 p2))
((var_> (car p2) (car p1)) (apply poly_gcd* p1 (cdr p2)))
(else (apply poly_gcd* p2 (cdr p1)))))
(define (poly_gcd* . li)
(let ((nums (remove-if-not number? li)))
(if (null? nums)
(reduce poly_gcd li)
(let ((gnum (reduce gcd nums)))
(if (= 1 gnum) 1
(reduce-init poly_gcd gnum (remove-if number? li)))))))
(define (univ_cont p) (apply poly_gcd* (cdr p)))
(define (univ_primpart p) (poly_/ p (univ_cont p)))
(define (poly_num-cont p)
(if (number? p) p
(do ((l (cdr p) (cdr l))
(n (poly_num-cont (cadr p))
(gcd n (poly_num-cont (cadr l)))))
((or (= 1 n) (null? (cdr l))) n))))
(define (list-ref? l n)
(cond ((null? l) #f)
((zero? n) (car l))
(else (list-ref? (cdr l) (- n 1)))))
(define (univ_coeff p ord) (or (list-ref? (cdr p) ord) 0))
(define (poly_coeff p var ord)
(cond ((or (number? p) (var_> var (car p)))
(if (zero? ord) p 0))
((eq? var (car p)) (univ_coeff p ord))
(else
(univ_norm0 (car p)
(map-no-end-0s (lambda (c) (poly_coeff c var ord))
(cdr p))))))
(define (poly_subst0 old e) (poly_coeff e old 0))
(define const_promote list)
(define (promote var p)
(if (eq? var (car p))
p
(let ((dgr (poly_degree p var)))
(do ((i dgr (+ -1 i))
(ol (list (poly_coeff p var dgr))
(cons (poly_coeff p var (+ -1 i)) ol)))
((zero? i) (cons var ol))))))
;;;this is bummed if v has higher priority than any variable in (cdr p)
(define (univ_demote p)
(if (number? p)
p
(let ((v (car p)))
(if (every (lambda (cof) (or (number? cof) (var_> v (car cof))))
(cdr p))
p
(poly_+ (cadr p)
(do ((trms (cddr p) (cdr trms))
(sum 0)
(mon (list v 0 1) (cons v (cons 0 (cdr mon)))))
((null? trms) sum)
(set! sum (poly_+ sum (poly_* mon (car trms))))))))))
(define (sylvester p1 p2 var)
(set! p1 (promote var p1))
(set! p2 (promote var p2))
(let ((d1 (univ_degree p1 var))
(d2 (univ_degree p2 var))
(m (list)))
(do ((i d1 (+ -1 i))
(row (nconc (make-list (+ -1 d1) 0) (reverse (cdr p2)))
(append (cdr row) (list 0))))
((<= i 1) (set! m (cons row m)))
(set! m (cons row m)))
(do ((i d2 (+ -1 i))
(row (nconc (make-list (+ -1 d2) 0) (reverse (cdr p1)))
(append (cdr row) (list 0))))
((<= i 1) (set! m (cons row m)))
(set! m (cons row m)))
m))
;;; Bareiss's integer preserving gaussian elimination.
;;; Bareiss, E.H.: Sylvester's identity and multistep
;;; integer-preserving Gaussian elimination. Mathematics of
;;; Computation 22, 565-578, 1968.
;;; as related by:
;;; Akritas, A.G.: Exact Algorithms for the Matrix-Triangulation
;;; Subresultant PRS Method. Computers and Mathematics, 145-155.
;;; Springer Verlag, 1989.
(define (bareiss m)
4)
(define (poly_resultant p1 p2 var)
(let ((u1 (promote var p1))
(u2 (promote var p2)))
(or (not (zero? (univ_degree u1 var)))
(not (zero? (univ_degree u2 var)))
(math-error var " does not appear in " p1 " or " p2))
(let ((res (cond ((zero? (univ_degree u1 var)) p1)
((zero? (univ_degree u2 var)) p2)
((shorter? u1 u2) (univ_prs u2 u1))
(else (univ_prs u1 u2)))))
(if (zero? (univ_degree res var)) res
0))))
(define (poly_mod_number poly modulus)
(if (number? poly)
(modulo poly modulus)
(cons (car poly)
(map-no-end-0s (lambda (x) (poly_mod_number x modulus))
(cdr poly)))))
(define (poly_prem dividend divisor)
(if (number? divisor)
(poly_mod_number dividend divisor)
(let ((var (car divisor)))
(if (> (poly_degree divisor var) (poly_degree dividend var))
dividend
(univ_demote (univ_prem (promote var dividend)
(promote var divisor)))))))
;;;; VERIFICATION TESTS
(define (poly_test)
(define a (sexp->var 'A))
(define b (sexp->var 'B))
(define c (sexp->var 'C))
(define x (sexp->var 'X))
(define y (sexp->var 'Y))
(test (list a 0 -2)
poly_gcd
(list a 0 -2)
(list a 0 0 -2))
(test (list x (list a 0 1) 1)
poly_gcd
(list x (list a 0 0 -1) 0 1)
(list x (list a 0 0 1) (list a 0 2) 1))
(test (list x 0 (list a 0 1))
poly_gcd
(list x 0 (list a 0 0 1))
(list x 0 0 (list a 0 1)))
(test (list x (list b 0 0 1) 0 (list b 1 2) (list a 0 1) 1)
poly_resultant
(list y (list x (list b 0 1) 0 1) (list x 0 1))
(list y (list x 1 (list a 0 1)) 0 1)
y)
(test (list y (list b 0 0 1) 0 (list b 1 2) (list a 0 1) 1)
poly_resultant
(list y (list b 0 1) (list x 0 1) 1)
(list y (list x 1 0 1) (list a 0 1))
x)
'(test 1
poly_gcd
(list x -5 2 8 -3 -3 1 1)
(list x 21 -9 -4 5 3))
'(test 1
poly_gcd
(list x -5 2 8 -3 -3 0 1 0 1)
(list x 21 -9 -4 0 5 0 3))
'done)