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| 2 | $Id: svpoly.lisp,v 1.4 2009/01/23 10:37:28 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 "SVPOLY"
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| 12 | (:export))
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| 13 |
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| 14 | (in-package "SVPOLY")
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| 15 |
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| 16 | #+debug(proclaim '(optimize (speed 0) (debug 3)))
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| 17 | #-debug(proclaim '(optimize (speed 3) (debug 0)))
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| 18 |
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| 19 | (defstruct (svpoly (:constructor make-svpoly-raw))
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| 20 | (nvars "Number of variables." (:type fixnum))
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| 21 | (coefficient-type "Type of the coefficient.")
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| 22 | (order "Monomial order.")
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| 23 | (terms "The array of terms."))
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| 24 |
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| 25 | (defun make-svpoly (alist order
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| 26 | &aux (nterms (length alist))
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| 27 | (nvars (length (caar alist)))
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| 28 | (coefficient-type (type-of (cdar alist))))
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| 29 | "Construct svpolynomial from ALIST."
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| 30 | (let ((svp (make-svpoly-raw
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| 31 | :nvars nvars
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| 32 | :coefficient-type (type-of (cdar alist))
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| 33 | :order order
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| 34 | :terms (make-array nterms :element-type 'cons))))
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| 35 | (dotimes (i nterms)
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| 36 | (let ((term (nth i alist)))
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| 37 | (setf (svref (svpoly-terms svp) i)
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| 38 | (cons (make-array nvars :initial-contents (car term))
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| 39 | (cdr term)))))
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| 40 | (svpoly-sort svp)))
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| 41 |
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| 42 | (defun svpoly-sort (svp)
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| 43 | "Destructively sorts an sv-polynomial SVP."
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| 44 | (setf (svpoly-terms svp) (sort (svpoly-terms svp) (svpoly-order svp) :key #'car)) svp)
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| 45 |
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| 46 | (defun make-order (nvars order)
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| 47 | "Returns an order function with two parameters P and Q such that if called
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| 48 | on a pair of monomials with exactly NVARS variables, this function
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| 49 | will return T if P is greater than Q and NIL otherwise. The keyword
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| 50 | ORDER indicates one of several standard orders (:LEX, etc)."
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| 51 | (declare (fixnum nvars))
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| 52 | (ecase order
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| 53 | (:lex #'(lambda (p q)
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| 54 | (dotimes (i nvars (values NIL T))
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| 55 | (declare (fixnum i))
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| 56 | (cond
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| 57 | ((> (svref p i) (svref q i))
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| 58 | (return (values t nil)))
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| 59 | ((< (svref p i) (svref q i))
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| 60 | (return (values nil nil)))))))))
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| 61 |
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| 62 | (defun svpoly-add (p q)
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| 63 | "Adds polynomials P and Q, where P and Q are assumed to be ordered
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| 64 | by the same monomial order. Destructive to P and Q."
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| 65 | (setf (svpoly-terms p) (add-terms (svpoly-terms p) (svpoly-terms q)
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| 66 | (svpoly-order p)))
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| 67 | p)
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| 68 |
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| 69 | (defun add-terms (p q pred &aux (lp (length p)) (lq (length q)))
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| 70 | (do ((r (make-array (+ lp lq) :element-type 'cons))
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| 71 | (i 0) (j 0) (k 0) done)
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| 72 | (done (coerce (adjust-array r k) 'simple-vector))
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| 73 | (cond
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| 74 | ((= i lp)
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| 75 | (do nil
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| 76 | ((>= j lq) (setf done t))
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| 77 | (setf (aref r k) (svref q j))
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| 78 | (incf j) (incf k)))
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| 79 | ((= j lq)
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| 80 | (do nil
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| 81 | ((>= i lp) (setf done t))
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| 82 | (setf (aref r k) (svref p i))
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| 83 | (incf i) (incf k)))
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| 84 | (t
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| 85 | (multiple-value-bind
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| 86 | (mgreater mequal)
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| 87 | (funcall pred (car (svref p i)) (car (svref q j)))
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| 88 | (cond
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| 89 | (mequal
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| 90 | (let ((s (+ (cdr (svref p i)) (cdr (svref q j)))))
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| 91 | (unless (zerop s) ;check for cancellation
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| 92 | (setf (aref r k) (cons (car (svref p i)) s))
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| 93 | (incf k))
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| 94 | (incf i)
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| 95 | (incf j)))
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| 96 | (mgreater
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| 97 | (setf (aref r k) (svref p i))
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| 98 | (incf i)
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| 99 | (incf k))
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| 100 | (t
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| 101 | (setf (aref r k) (svref q j))
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| 102 | (incf j)
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| 103 | (incf k))))))))
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| 104 |
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| 105 | #|
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| 106 |
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| 107 | (defun scalar-times-poly (c p)
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| 108 | "Return product of a scalar C by a polynomial P with coefficient ring RING."
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| 109 | (unless (funcall (ring-zerop ring) c)
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| 110 | (mapcar
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| 111 | #'(lambda (term) (cons (car term) (funcall (ring-* ring) c (cdr term))))
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| 112 | p)))
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| 113 |
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| 114 | (defun term-times-poly (term f &optional (ring *coefficient-ring*))
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| 115 | "Return product of a term TERM by a polynomial F with coefficient ring RING."
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| 116 | (mapcar #'(lambda (x) (term* term x ring)) f))
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| 117 |
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| 118 | (defun monom-times-poly (m f)
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| 119 | "Return product of a monomial M by a polynomial F with coefficient ring RING."
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| 120 | (mapcar #'(lambda (x) (monom-times-term m x)) f))
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| 121 |
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| 122 | (defun minus-poly (f &optional (ring *coefficient-ring*))
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| 123 | "Changes the sign of a polynomial F with coefficients in coefficient ring
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| 124 | RING, and returns the result."
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| 125 | (mapcar #'(lambda (x) (cons (car x) (funcall (ring-- ring) (cdr x)))) f))
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| 126 |
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| 127 |
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| 128 | (defun poly+ (p q &optional (pred #'lex>) (ring *coefficient-ring*))
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| 129 | "Returns the sum of two polynomials P and Q with coefficients in
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| 130 | ring RING, with terms ordered according to monomial order PRED."
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| 131 | (do (r done)
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| 132 | (done r)
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| 133 | (cond
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| 134 | ((endp p) (setf r (append (nreverse r) q) done t))
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| 135 | ((endp q) (setf r (append (nreverse r) p) done t))
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| 136 | (t
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| 137 | (multiple-value-bind
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| 138 | (mgreater mequal)
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| 139 | (funcall pred (caar p) (caar q))
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| 140 | (cond
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| 141 | (mequal
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| 142 | (let ((s (funcall (ring-+ ring) (cdar p) (cdar q))))
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| 143 | (unless (funcall (ring-zerop ring) s) ;check for cancellation
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| 144 | (setf r (cons (cons (caar p) s) r)))
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| 145 | (setf p (cdr p) q (cdr q))))
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| 146 | (mgreater
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| 147 | (setf r (cons (car p) r)
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| 148 | p (cdr p)))
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| 149 | (t (setf r (cons (car q) r)
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| 150 | q (cdr q)))))))))
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| 151 |
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| 152 | (defun poly- (p q &optional (pred #'lex>) (ring *coefficient-ring*))
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| 153 | "Returns the difference of two polynomials P and Q with coefficients
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| 154 | in ring RING, with terms ordered according to monomial order PRED."
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| 155 | (do (r done)
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| 156 | (done r)
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| 157 | (cond
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| 158 | ((endp p) (setf r (append (nreverse r) (minus-poly q ring)) done t))
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| 159 | ((endp q) (setf r (append (nreverse r) p) done t))
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| 160 | (t
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| 161 | (multiple-value-bind
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| 162 | (mgreater mequal)
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| 163 | (funcall pred (caar p) (caar q))
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| 164 | (cond
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| 165 | (mequal
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| 166 | (let ((s (funcall (ring-- ring) (cdar p) (cdar q))))
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| 167 | (unless (zerop s) ;check for cancellation
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| 168 | (setf r (cons (cons (caar p) s) r)))
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| 169 | (setf p (cdr p) q (cdr q))))
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| 170 | (mgreater
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| 171 | (setf r (cons (car p) r)
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| 172 | p (cdr p)))
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| 173 | (t (setf r (cons (cons (caar q) (funcall (ring-- ring) (cdar q))) r)
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| 174 | q (cdr q)))))))))
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| 175 |
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| 176 | ;; Multiplication of polynomials
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| 177 | ;; Non-destructive version
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| 178 | (defun poly* (p q &optional (pred #'lex>) (ring *coefficient-ring*))
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| 179 | "Returns the product of two polynomials P and Q with coefficients in
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| 180 | ring RING, with terms ordered according to monomial order PRED."
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| 181 | (cond ((or (endp p) (endp q)) nil) ;p or q is 0 (represented by NIL)
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| 182 | ;; If p=p0+p1 and q=q0+q1 then pq=p0q0+p0q1+p1q
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| 183 | (t (cons (cons (monom* (caar p) (caar q))
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| 184 | (funcall (ring-* ring) (cdar p) (cdar q)))
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| 185 | (poly+ (term-times-poly (car p) (cdr q) ring)
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| 186 | (poly* (cdr p) q pred ring)
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| 187 | pred ring)))))
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| 188 |
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| 189 | (defun poly-op (f m g pred ring)
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| 190 | "Returns F-M*G, where F and G are polynomials with coefficients in
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| 191 | ring RING, ordered according to monomial order PRED and M is a
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| 192 | monomial."
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| 193 | (poly- f (term-times-poly m g ring) pred ring))
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| 194 |
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| 195 | (defun poly-expt (poly n &optional (pred #'lex>) (ring *coefficient-ring*))
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| 196 | "Exponentiate a polynomial POLY to power N. The terms of the
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| 197 | polynomial are assumed to be ordered by monomial order PRED and with
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| 198 | coefficients in ring RING. Use the Chinese algorithm; assume N>=0 and
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| 199 | POLY is non-zero (not NIL)."
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| 200 | (labels ((poly-one ()
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| 201 | (list
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| 202 | (cons (make-list (length (caar poly)) :initial-element 0) (ring-unit ring)))))
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| 203 | (cond
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| 204 | ((minusp n) (error "Negative exponent."))
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| 205 | ((endp poly) (if (zerop n) (poly-one) nil))
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| 206 | (t
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| 207 | (do ((k 1 (ash k 1))
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| 208 | (q poly (poly* q q pred ring)) ;keep squaring
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| 209 | (p (poly-one) (if (not (zerop (logand k n))) (poly* p q pred ring) p)))
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| 210 | ((> k n) p))))))
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| 211 |
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| 212 | (defun poly-mexpt (plist monom &optional (pred #'lex>) (ring *coefficient-ring*))
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| 213 | "Raise a polynomial vector represented ad a list of polynomials
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| 214 | PLIST to power MULTIINDEX. Every polynomial has its terms ordered by
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| 215 | predicate PRED and coefficients in the ring RING."
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| 216 | (reduce
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| 217 | #'(lambda (u v) (poly* u v pred ring))
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| 218 | (mapcan #'(lambda (y i)
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| 219 | (cond ((endp y) (if (zerop i) nil (list nil)))
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| 220 | (t (list (poly-expt y i pred ring)))))
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| 221 | plist monom)))
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| 222 |
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| 223 | (defun poly-constant-p (p)
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| 224 | "Returns T if P is a constant polynomial."
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| 225 | (and (= (length p) 1)
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| 226 | (every #'zerop (caar p))))
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| 227 |
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| 228 |
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| 229 | (defun poly-extend (p &optional (m (list 0)))
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| 230 | "Given a polynomial P in k[x[r+1],...,xn], it returns the same
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| 231 | polynomial as an element of k[x1,...,xn], optionally multiplying it by
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| 232 | a monomial x1^m1*x2^m2*...*xr^mr, where m=(m1,m2,...,mr) is a
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| 233 | multiindex."
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| 234 | (mapcar #'(lambda (term) (cons (append m (car term)) (cdr term))) p))
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| 235 |
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| 236 | (defun poly-extend-end (p &optional (m (list 0)))
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| 237 | "Similar to POLY-EXTEND, but it adds new variables at the end."
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| 238 | (mapcar #'(lambda (term) (cons (append (car term) m) (cdr term))) p))
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| 239 |
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| 240 | (defun poly-zerop (p)
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| 241 | "Returns T if P is a zero polynomial."
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| 242 | (null p))
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| 243 |
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| 244 | (defun lt (p)
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| 245 | "Returns the leading term of a polynomial P."
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| 246 | #+debugging(assert (consp p))
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| 247 | (first p))
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| 248 |
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| 249 | (defun lm (p)
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| 250 | "Returns the leading monomial of a polynomial P."
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| 251 | #+debugging(assert (consp (lt p)))
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| 252 | (car (lt p)))
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| 253 |
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| 254 | (defun lc (p)
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| 255 | "Returns the leading coefficient of a polynomial P."
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| 256 | #+debugging(assert (consp (lt p)))
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| 257 | (cdr (lt p)))
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| 258 |
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| 259 | |#
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