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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18 | (proclaim '(optimize (speed 0) (debug 3)))
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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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