[1] | 1 | #|
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| 2 | $Id: poly-gcd.lisp,v 1.4 2009/01/22 04:05:32 marek Exp $
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| 3 | *--------------------------------------------------------------------------*
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| 4 | | Copyright (C) 1994, Marek Rychlik (e-mail: rychlik@math.arizona.edu) |
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| 5 | | Department of Mathematics, University of Arizona, Tucson, AZ 85721 |
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| 6 | | |
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| 7 | | Everyone is permitted to copy, distribute and modify the code in this |
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| 8 | | directory, as long as this copyright note is preserved verbatim. |
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| 9 | *--------------------------------------------------------------------------*
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| 10 | |#
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| 11 | (defpackage "POLY-GCD"
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| 12 | (:export poly-gcd poly-content poly-pseudo-divide poly-primitive-part
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| 13 | poly-pseudo-remainder)
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| 14 | (:use "ORDER" "POLY" "DIVISION" "COMMON-LISP"))
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| 15 |
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| 16 | (in-package "POLY-GCD")
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| 17 |
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[55] | 18 | (proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
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[1] | 19 |
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| 20 | ;; This package calculates GCD of polynomials over integers.
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| 21 | ;; They are assumed to be ordered lexicographically.
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| 22 | ;;
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| 23 | ;; The algorithm is that on p. 57 of Geddes, Czapor, Labahn
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| 24 | ;; Given polynomials a(x),b(x) in D[x] where D is a UFD, we
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| 25 | ;; compute g(x)=GCD(a(x),b(x))
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| 26 | ;; Assume that the poly's are sorted lexicographically
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| 27 |
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| 28 | (defun poly-gcd (a b)
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| 29 | (cond
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| 30 | ((endp a) b)
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| 31 | ((endp b) a)
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| 32 | ((endp (caar a)) ;scalar
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| 33 | (list (cons nil (gcd (cdar a) (cdar b)))))
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| 34 | (t
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| 35 | (do* ((r nil (poly-pseudo-remainder c d))
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| 36 | (c (poly-primitive-part a) d)
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| 37 | (d (poly-primitive-part b) (poly-primitive-part r)))
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| 38 | ((endp d)
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| 39 | (poly* (poly-extend
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| 40 | (poly-gcd (poly-content a) (poly-content b)))
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| 41 | c))))))
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| 42 |
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| 43 | ;; Perform a pseudo-division in k[x2,....,xn][x1]
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| 44 | ;; Assume that the terms are sorted according to decreasing powers of x1
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| 45 | ;; p.297 of Cox
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| 46 | (defun poly-pseudo-divide (f g)
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| 47 | (multiple-value-bind (lg grest)
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| 48 | (lpart g)
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| 49 | (do* ((m (mdeg g))
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| 50 | (dm (poly-extend lg))
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| 51 | (lpart nil (multiple-value-list (lpart r)))
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| 52 | (lr nil (car lpart))
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| 53 | (lrest nil (cadr lpart))
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| 54 | (term nil (poly-extend lr (list (- (mdeg r) m))))
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| 55 | (r f (poly- (poly* dm lrest)
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| 56 | (poly* term grest)))
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| 57 | (q nil (append (poly* dm q) term)))
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| 58 | ((or (endp r) (< (mdeg r) m))
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| 59 | (values q r)))))
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| 60 |
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| 61 |
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| 62 | (defun poly-pseudo-remainder (f g)
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| 63 | (second (multiple-value-list (poly-pseudo-divide f g))))
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| 64 |
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| 65 | ;; Degree in main variable
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| 66 | (defun mdeg (b) (caaar b))
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| 67 |
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| 68 | ;; Leading coefficient in the first variable; a poly in k[x2,...,xn]
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| 69 | (defun lcoeff (b)
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| 70 | (first (multiple-value-list (lpart b))))
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| 71 |
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| 72 | (defun lrest (b)
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| 73 | (second (multiple-value-list (lpart b))))
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| 74 |
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| 75 | (defun lpart (b)
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| 76 | (do ((mdeg (mdeg b))
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| 77 | (b b (rest b))
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| 78 | (b1 nil (cons (cons (cdaar b) (cdar b)) b1)))
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| 79 | ((or (endp b) (/= (caaar b) mdeg))
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| 80 | (values (reverse b1) b))))
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| 81 |
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| 82 | ;; Divide f by its content
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| 83 | (defun poly-primitive-part (f)
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| 84 | (cond
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| 85 | ((endp (caar f)) ;scalar
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| 86 | f)
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| 87 | (t
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| 88 | (poly-exact-divide f (poly-extend (poly-content f))))))
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| 89 |
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| 90 |
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| 91 | (defun poly-content (f)
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| 92 | (cond
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| 93 | ((endp f) (error "Zero argument"))
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| 94 | (t (multiple-value-bind (lc r)
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| 95 | (lpart f)
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| 96 | (cond
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| 97 | ((endp r) lc)
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| 98 | ((and (= (length lc) 1)
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| 99 | (every #'zerop (caar lc))
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| 100 | (or (= (cdar lc) 1)
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| 101 | (= (cdar lc) -1))) ;lc is 1 or -1
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| 102 | lc)
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| 103 | (t (poly-gcd lc (poly-content r))))))))
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