| 1 | ;;----------------------------------------------------------------
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| 2 | ;; File: polynomial.lisp
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| 3 | ;;----------------------------------------------------------------
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| 4 | ;;
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| 5 | ;; Author: Marek Rychlik (rychlik@u.arizona.edu)
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| 6 | ;; Date: Thu Aug 27 09:41:24 2015
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| 7 | ;; Copying: (C) Marek Rychlik, 2010. All rights reserved.
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| 8 | ;;
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| 9 | ;;----------------------------------------------------------------
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| 10 | ;;; -*- Mode: Lisp -*-
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| 11 | ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
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| 12 | ;;;
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| 13 | ;;; Copyright (C) 1999, 2002, 2009, 2015 Marek Rychlik <rychlik@u.arizona.edu>
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| 14 | ;;;
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| 15 | ;;; This program is free software; you can redistribute it and/or modify
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| 16 | ;;; it under the terms of the GNU General Public License as published by
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| 17 | ;;; the Free Software Foundation; either version 2 of the License, or
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| 18 | ;;; (at your option) any later version.
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| 19 | ;;;
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| 20 | ;;; This program is distributed in the hope that it will be useful,
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| 21 | ;;; but WITHOUT ANY WARRANTY; without even the implied warranty of
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| 22 | ;;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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| 23 | ;;; GNU General Public License for more details.
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| 24 | ;;;
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| 25 | ;;; You should have received a copy of the GNU General Public License
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| 26 | ;;; along with this program; if not, write to the Free Software
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| 27 | ;;; Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
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| 28 | ;;;
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| 29 | ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
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| 30 |
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| 31 | (defpackage "POLYNOMIAL"
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| 32 | (:use :cl :utils :monom)
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| 33 | (:export "POLY"
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| 34 | "POLY-DIMENSION"
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| 35 | "POLY-TERMLIST"
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| 36 | "POLY-TERM-ORDER"
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| 37 | "CHANGE-TERM-ORDER"
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| 38 | "STANDARD-EXTENSION"
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| 39 | "STANDARD-EXTENSION-1"
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| 40 | "STANDARD-SUM"
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| 41 | "SATURATION-EXTENSION"
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| 42 | "ALIST->POLY")
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| 43 | (:documentation "Implements polynomials. A polynomial is essentially
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| 44 | a mapping of monomials of the same degree to coefficients. The
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| 45 | momomials are ordered according to a monomial order."))
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| 46 |
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| 47 | (in-package :polynomial)
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| 48 |
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| 49 | (proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
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| 50 |
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| 51 | (defclass poly ()
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| 52 | ((dimension :initform nil
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| 53 | :initarg :dimension
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| 54 | :accessor poly-dimension
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| 55 | :documentation "Shared dimension of all terms, the number of variables")
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| 56 | (termlist :initform nil :initarg :termlist :accessor poly-termlist
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| 57 | :documentation "List of terms.")
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| 58 | (order :initform #'lex> :initarg :order :accessor poly-term-order
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| 59 | :documentation "Monomial/term order."))
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| 60 | (:default-initargs :dimension nil :termlist nil :order #'lex>)
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| 61 | (:documentation "A polynomial with a list of terms TERMLIST, ordered
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| 62 | according to term order ORDER, which defaults to LEX>."))
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| 63 |
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| 64 | (defmethod print-object ((self poly) stream)
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| 65 | (print-unreadable-object (self stream :type t :identity t)
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| 66 | (with-accessors ((dimension poly-dimension)
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| 67 | (termlist poly-termlist)
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| 68 | (order poly-term-order))
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| 69 | self
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| 70 | (format stream "DIMENSION=~A TERMLIST=~A ORDER=~A"
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| 71 | dimension termlist order))))
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| 72 |
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| 73 | (defgeneric change-term-order (self other)
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| 74 | (:documentation "Change term order of SELF to the term order of OTHER.")
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| 75 | (:method ((self poly) (other poly))
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| 76 | (unless (eq (poly-term-order self) (poly-term-order other))
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| 77 | (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other))
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| 78 | (poly-term-order self) (poly-term-order other)))
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| 79 | self))
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| 80 |
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| 81 | (defun alist->poly (alist &aux (poly (make-instance 'poly)))
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| 82 | "It reads polynomial from an alist formatted as ( ... (exponents . coeff) ...).
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| 83 | It can be used to enter simple polynomials by hand, e.g the polynomial
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| 84 | in two variables, X and Y, given in standard notation as:
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| 85 |
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| 86 | 3*X^2*Y^3+2*Y+7
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| 87 |
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| 88 | can be entered as
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| 89 | (ALIST->POLY '(((2 3) . 3) ((0 1) . 2) ((0 0) . 7))).
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| 90 |
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| 91 | NOTE: The primary use is for low-level debugging of the package."
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| 92 | (dolist (x alist poly)
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| 93 | (poly-insert-term poly (make-instance 'monom :exponents (car x)) (cdr x))))
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| 94 |
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| 95 |
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| 96 | (defmethod update-instance-for-different-class :after ((old term) (new poly) &key)
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| 97 | "Converts OLD of class TERM to a NEW of class POLY, by making it into a 1-element TERMLIST."
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| 98 | (reinitialize-instance new
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| 99 | :dimension (monom-dimension old)
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| 100 | :termlist (list old)))
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| 101 |
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| 102 | (defmethod update-instance-for-different-class :after ((old monom) (new poly) &key)
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| 103 | "Converts OLD of class MONOM to a NEW of class POLY, by making it into a 1-element TERMLIST."
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| 104 | (reinitialize-instance new
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| 105 | :dimension (monom-dimension old)
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| 106 | :termlist (list (cons monom 1))))
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| 107 |
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| 108 | (defmethod r-equalp ((self poly) (other poly))
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| 109 | "POLY instances are R-EQUALP if they have the same
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| 110 | order and if all terms are R-EQUALP."
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| 111 | (and (every #'r-equalp (poly-termlist self) (poly-termlist other))
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| 112 | (eq (poly-term-order self) (poly-term-order other))))
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| 113 |
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| 114 | (defgeneric poly-insert-term (self monom coeff)
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| 115 | (:method ((self poly) (monom monom) coeff)
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| 116 | (cond ((null (poly-dimension self))
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| 117 | (setf (poly-dimension self) (monom-dimension monom)))
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| 118 | (t (assert (= (poly-dimension self) (monom-dimension monom)))))
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| 119 | (push (cons monom coeff) (poly-termlist self))
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| 120 | self))
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| 121 |
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| 122 | (defmethod append-item ((self poly) (item term))
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| 123 | (cond ((null (poly-dimension self))
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| 124 | (setf (poly-dimension self) (monom-dimension item)))
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| 125 | (t (assert (= (poly-dimension self) (monom-dimension item)))))
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| 126 | (setf (cdr (last (poly-termlist self))) (list item))
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| 127 | self)
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| 128 |
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| 129 | ;; Leading term
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| 130 | (defgeneric leading-term (object)
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| 131 | (:method ((self poly))
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| 132 | (car (poly-termlist self)))
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| 133 | (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
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| 134 |
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| 135 | ;; Second term
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| 136 | (defgeneric second-leading-term (object)
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| 137 | (:method ((self poly))
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| 138 | (cadar (poly-termlist self)))
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| 139 | (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
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| 140 |
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| 141 | ;; Leading coefficient
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| 142 | (defgeneric leading-coefficient (object)
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| 143 | (:method ((self poly))
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| 144 | (scalar-coeff (leading-term self)))
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| 145 | (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
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| 146 |
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| 147 | ;; Second coefficient
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| 148 | (defgeneric second-leading-coefficient (object)
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| 149 | (:method ((self poly))
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| 150 | (scalar-coeff (second-leading-term self)))
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| 151 | (:documentation "The second leading coefficient of a polynomial. It
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| 152 | signals error for a polynomial with at most one term."))
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| 153 |
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| 154 | ;; Testing for a zero polynomial
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| 155 | (defmethod r-zerop ((self poly))
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| 156 | (null (poly-termlist self)))
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| 157 |
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| 158 | ;; The number of terms
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| 159 | (defmethod r-length ((self poly))
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| 160 | (length (poly-termlist self)))
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| 161 |
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| 162 | (defmethod multiply-by ((self poly) (other monom))
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| 163 | (mapc #'(lambda (term) (multiply-by term other))
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| 164 | (poly-termlist self))
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| 165 | self)
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| 166 |
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| 167 | (defmethod multiply-by ((self poly) (other term))
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| 168 | (mapc #'(lambda (term) (multiply-by term other))
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| 169 | (poly-termlist self))
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| 170 | self)
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| 171 |
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| 172 | (defmethod multiply-by ((self poly) (other scalar))
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| 173 | (mapc #'(lambda (term) (multiply-by term other))
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| 174 | (poly-termlist self))
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| 175 | self)
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| 176 |
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| 177 |
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| 178 | (defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
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| 179 | "Return an expression which will efficiently adds/subtracts two
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| 180 | polynomials, P and Q. The addition/subtraction of coefficients is
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| 181 | performed by calling ADD/SUBTRACT-METHOD-NAME. If UMINUS-METHOD-NAME
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| 182 | is supplied, it is used to negate the coefficients of Q which do not
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| 183 | have a corresponding coefficient in P. The code implements an
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| 184 | efficient algorithm to add two polynomials represented as sorted lists
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| 185 | of terms. The code destroys both arguments, reusing the terms to build
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| 186 | the result."
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| 187 | `(macrolet ((lc (x) `(scalar-coeff (car ,x))))
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| 188 | (do ((p ,p)
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| 189 | (q ,q)
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| 190 | r)
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| 191 | ((or (endp p) (endp q))
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| 192 | ;; NOTE: R contains the result in reverse order. Can it
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| 193 | ;; be more efficient to produce the terms in correct order?
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| 194 | (unless (endp q)
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| 195 | ;; Upon subtraction, we must change the sign of
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| 196 | ;; all coefficients in q
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| 197 | ,@(when uminus-fn
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| 198 | `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
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| 199 | (setf r (nreconc r q)))
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| 200 | r)
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| 201 | (multiple-value-bind
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| 202 | (greater-p equal-p)
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| 203 | (funcall ,order-fn (car p) (car q))
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| 204 | (cond
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| 205 | (greater-p
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| 206 | (rotatef (cdr p) r p)
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| 207 | )
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| 208 | (equal-p
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| 209 | (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
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| 210 | (cond
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| 211 | ((r-zerop s)
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| 212 | (setf p (cdr p))
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| 213 | )
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| 214 | (t
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| 215 | (setf (lc p) s)
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| 216 | (rotatef (cdr p) r p))))
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| 217 | (setf q (cdr q))
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| 218 | )
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| 219 | (t
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| 220 | ;;Negate the term of Q if UMINUS provided, signallig
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| 221 | ;;that we are doing subtraction
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| 222 | ,(when uminus-fn
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| 223 | `(setf (lc q) (funcall ,uminus-fn (lc q))))
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| 224 | (rotatef (cdr q) r q)))))))
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| 225 |
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| 226 |
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| 227 | (defmacro def-add/subtract-method (add/subtract-method-name
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| 228 | uminus-method-name
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| 229 | &optional
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| 230 | (doc-string nil doc-string-supplied-p))
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| 231 | "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
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| 232 | `(defmethod ,add/subtract-method-name ((self poly) (other poly))
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| 233 | ,@(when doc-string-supplied-p `(,doc-string))
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| 234 | ;; Ensure orders are compatible
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| 235 | (change-term-order other self)
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| 236 | (setf (poly-termlist self) (fast-add/subtract
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| 237 | (poly-termlist self) (poly-termlist other)
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| 238 | (poly-term-order self)
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| 239 | #',add/subtract-method-name
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| 240 | ,(when uminus-method-name `(function ,uminus-method-name))))
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| 241 | self))
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| 242 |
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| 243 | (eval-when (:compile-toplevel :load-toplevel :execute)
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| 244 |
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| 245 | (def-add/subtract-method add-to nil
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| 246 | "Adds to polynomial SELF another polynomial OTHER.
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| 247 | This operation destructively modifies both polynomials.
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| 248 | The result is stored in SELF. This implementation does
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| 249 | no consing, entirely reusing the sells of SELF and OTHER.")
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| 250 |
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| 251 | (def-add/subtract-method subtract-from unary-minus
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| 252 | "Subtracts from polynomial SELF another polynomial OTHER.
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| 253 | This operation destructively modifies both polynomials.
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| 254 | The result is stored in SELF. This implementation does
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| 255 | no consing, entirely reusing the sells of SELF and OTHER.")
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| 256 | )
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| 257 |
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| 258 | (defmethod unary-minus ((self poly))
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| 259 | "Destructively modifies the coefficients of the polynomial SELF,
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| 260 | by changing their sign."
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| 261 | (mapc #'unary-minus (poly-termlist self))
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| 262 | self)
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| 263 |
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| 264 | (defun add-termlists (p q order-fn)
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| 265 | "Destructively adds two termlists P and Q ordered according to ORDER-FN."
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| 266 | (fast-add/subtract p q order-fn #'add-to nil))
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| 267 |
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| 268 | (defmacro multiply-term-by-termlist-dropping-zeros (term termlist
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| 269 | &optional (reverse-arg-order-P nil))
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| 270 | "Multiplies term TERM by a list of term, TERMLIST.
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| 271 | Takes into accound divisors of zero in the ring, by
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| 272 | deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
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| 273 | is T, change the order of arguments; this may be important
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| 274 | if we extend the package to non-commutative rings."
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| 275 | `(mapcan #'(lambda (other-term)
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| 276 | (let ((prod (r*
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| 277 | ,@(cond
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| 278 | (reverse-arg-order-p
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| 279 | `(other-term ,term))
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| 280 | (t
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| 281 | `(,term other-term))))))
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| 282 | (cond
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| 283 | ((r-zerop prod) nil)
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| 284 | (t (list prod)))))
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| 285 | ,termlist))
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| 286 |
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| 287 | (defun multiply-termlists (p q order-fn)
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| 288 | "A version of polynomial multiplication, operating
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| 289 | directly on termlists."
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| 290 | (cond
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| 291 | ((or (endp p) (endp q))
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| 292 | ;;p or q is 0 (represented by NIL)
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| 293 | nil)
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| 294 | ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
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| 295 | ((endp (cdr p))
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| 296 | (multiply-term-by-termlist-dropping-zeros (car p) q))
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| 297 | ((endp (cdr q))
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| 298 | (multiply-term-by-termlist-dropping-zeros (car q) p t))
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| 299 | (t
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| 300 | (cons (r* (car p) (car q))
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| 301 | (add-termlists
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| 302 | (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
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| 303 | (multiply-termlists (cdr p) q order-fn)
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| 304 | order-fn)))))
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| 305 |
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| 306 | (defmethod multiply-by ((self poly) (other poly))
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| 307 | (change-term-order other self)
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| 308 | (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
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| 309 | (poly-termlist other)
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| 310 | (poly-term-order self)))
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| 311 | self)
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| 312 |
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| 313 | (defmethod r+ ((poly1 poly) poly2)
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| 314 | "Non-destructively add POLY1 by POLY2."
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| 315 | (add-to (copy-instance POLY1) (change-class (copy-instance POLY2) 'poly)))
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| 316 |
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| 317 | (defmethod r- ((minuend poly) &rest subtrahends)
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| 318 | "Non-destructively subtract MINUEND and SUBTRAHENDS."
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| 319 | (subtract-from (copy-instance minuend)
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| 320 | (change-class (reduce #'r+ subtrahends) 'poly)))
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| 321 |
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| 322 | (defmethod r+ ((poly1 term) poly2)
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| 323 | "Non-destructively add POLY1 by POLY2."
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| 324 | (add-to (change-class (copy-instance poly1) 'poly)
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| 325 | (change-class (copy-instance poly2) 'poly)))
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| 326 |
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| 327 | (defmethod r- ((minuend term) &rest subtrahends)
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| 328 | "Non-destructively subtract MINUEND and SUBTRAHENDS."
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| 329 | (subtract-from (change-class (copy-instance minuend) 'poly)
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| 330 | (change-class (reduce #'r+ subtrahends) 'poly)))
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| 331 |
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| 332 | (defmethod r+ ((poly1 monom) poly2)
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| 333 | "Non-destructively add POLY1 by POLY2."
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| 334 | (add-to (change-class (copy-instance poly1) 'poly)
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| 335 | (change-class (copy-instance poly2) 'poly)))
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| 336 |
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| 337 | (defmethod r- ((minuend monom) &rest subtrahends)
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| 338 | "Non-destructively subtract MINUEND and SUBTRAHENDS."
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| 339 | (subtract-from (change-class (copy-instance minuend) 'poly)
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| 340 | (change-class (reduce #'r+ subtrahends) 'poly)))
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| 341 |
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| 342 | (defmethod r* ((poly1 poly) (poly2 poly))
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| 343 | "Non-destructively multiply POLY1 by POLY2."
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| 344 | (multiply-by (copy-instance poly1) (copy-instance poly2)))
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| 345 |
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| 346 | (defmethod left-tensor-product-by ((self poly) (other term))
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| 347 | (setf (poly-termlist self)
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| 348 | (mapcan #'(lambda (term)
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| 349 | (let ((prod (left-tensor-product-by term other)))
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| 350 | (cond
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| 351 | ((r-zerop prod) nil)
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| 352 | (t (list prod)))))
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| 353 | (poly-termlist self)))
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| 354 | self)
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| 355 |
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| 356 | (defmethod right-tensor-product-by ((self poly) (other term))
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| 357 | (setf (poly-termlist self)
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| 358 | (mapcan #'(lambda (term)
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| 359 | (let ((prod (right-tensor-product-by term other)))
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| 360 | (cond
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| 361 | ((r-zerop prod) nil)
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| 362 | (t (list prod)))))
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| 363 | (poly-termlist self)))
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| 364 | self)
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| 365 |
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| 366 | (defmethod left-tensor-product-by ((self poly) (other monom))
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| 367 | (setf (poly-termlist self)
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| 368 | (mapcan #'(lambda (term)
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| 369 | (let ((prod (left-tensor-product-by term other)))
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| 370 | (cond
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| 371 | ((r-zerop prod) nil)
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| 372 | (t (list prod)))))
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| 373 | (poly-termlist self)))
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| 374 | (incf (poly-dimension self) (monom-dimension other))
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| 375 | self)
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| 376 |
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| 377 | (defmethod right-tensor-product-by ((self poly) (other monom))
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| 378 | (setf (poly-termlist self)
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| 379 | (mapcan #'(lambda (term)
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| 380 | (let ((prod (right-tensor-product-by term other)))
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| 381 | (cond
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| 382 | ((r-zerop prod) nil)
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| 383 | (t (list prod)))))
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| 384 | (poly-termlist self)))
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| 385 | (incf (poly-dimension self) (monom-dimension other))
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| 386 | self)
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| 387 |
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| 388 |
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| 389 | (defun standard-extension (plist &aux (k (length plist)) (i 0))
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| 390 | "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
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| 391 | is a list of polynomials. Destructively modifies PLIST elements."
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| 392 | (mapc #'(lambda (poly)
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| 393 | (left-tensor-product-by
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| 394 | poly
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| 395 | (prog1
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| 396 | (make-monom-variable k i)
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| 397 | (incf i))))
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| 398 | plist))
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| 399 |
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| 400 | (defun standard-extension-1 (plist
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| 401 | &aux
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| 402 | (plist (standard-extension plist))
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| 403 | (nvars (poly-dimension (car plist))))
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| 404 | "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].
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| 405 | Firstly, new K variables U1, U2, ..., UK, are inserted into each
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| 406 | polynomial. Subsequently, P1, P2, ..., PK are destructively modified
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| 407 | tantamount to replacing PI with UI*PI-1. It assumes that all
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| 408 | polynomials have the same dimension, and only the first polynomial
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| 409 | is examined to determine this dimension."
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| 410 | ;; Implementation note: we use STANDARD-EXTENSION and then subtract
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| 411 | ;; 1 from each polynomial; since UI*PI has no constant term,
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| 412 | ;; we just need to append the constant term at the end
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| 413 | ;; of each termlist.
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| 414 | (flet ((subtract-1 (p)
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| 415 | (append-item p (make-instance 'term :coeff -1 :dimension nvars))))
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| 416 | (setf plist (mapc #'subtract-1 plist)))
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| 417 | plist)
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| 418 |
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| 419 |
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| 420 | (defun standard-sum (plist
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| 421 | &aux
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| 422 | (plist (standard-extension plist))
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| 423 | (nvars (poly-dimension (car plist))))
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| 424 | "Calculate the polynomial U1*P1+U2*P2+...+UK*PK-1, where PLIST=[P1,P2,...,PK].
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| 425 | Firstly, new K variables, U1, U2, ..., UK, are inserted into each
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| 426 | polynomial. Subsequently, P1, P2, ..., PK are destructively modified
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| 427 | tantamount to replacing PI with UI*PI, and the resulting polynomials
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| 428 | are added. Finally, 1 is subtracted. It should be noted that the term
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| 429 | order is not modified, which is equivalent to using a lexicographic
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| 430 | order on the first K variables."
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| 431 | (flet ((subtract-1 (p)
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| 432 | (append-item p (make-instance 'term :coeff -1 :dimension nvars))))
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| 433 | (subtract-1
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| 434 | (make-instance
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| 435 | 'poly
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| 436 | :termlist (apply #'nconc (mapcar #'poly-termlist plist))))))
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| 437 |
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| 438 | #|
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| 439 |
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| 440 | (defun saturation-extension-1 (ring f p)
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| 441 | "Calculate [F, U*P-1]. It destructively modifies F."
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| 442 | (declare (type ring ring))
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| 443 | (polysaturation-extension ring f (list p)))
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| 444 |
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| 445 |
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| 446 |
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| 447 |
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| 448 | (defun spoly (ring-and-order f g
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| 449 | &aux
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| 450 | (ring (ro-ring ring-and-order)))
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| 451 | "It yields the S-polynomial of polynomials F and G."
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| 452 | (declare (type ring-and-order ring-and-order) (type poly f g))
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| 453 | (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))
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| 454 | (mf (monom-div lcm (poly-lm f)))
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| 455 | (mg (monom-div lcm (poly-lm g))))
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| 456 | (declare (type monom mf mg))
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| 457 | (multiple-value-bind (c cf cg)
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| 458 | (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))
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| 459 | (declare (ignore c))
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| 460 | (poly-sub
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| 461 | ring-and-order
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| 462 | (scalar-times-poly ring cg (monom-times-poly mf f))
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| 463 | (scalar-times-poly ring cf (monom-times-poly mg g))))))
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| 464 |
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| 465 |
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| 466 | (defun poly-primitive-part (ring p)
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| 467 | "Divide polynomial P with integer coefficients by gcd of its
|
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| 468 | coefficients and return the result."
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| 469 | (declare (type ring ring) (type poly p))
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| 470 | (if (poly-zerop p)
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| 471 | (values p 1)
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| 472 | (let ((c (poly-content ring p)))
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| 473 | (values (make-poly-from-termlist
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| 474 | (mapcar
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| 475 | #'(lambda (x)
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| 476 | (make-term :monom (term-monom x)
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| 477 | :coeff (funcall (ring-div ring) (term-coeff x) c)))
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| 478 | (poly-termlist p))
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| 479 | (poly-sugar p))
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| 480 | c))))
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| 481 |
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| 482 | (defun poly-content (ring p)
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| 483 | "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure
|
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| 484 | to compute the greatest common divisor."
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| 485 | (declare (type ring ring) (type poly p))
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| 486 | (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))
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| 487 |
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| 488 | |#
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