[1969] | 1 | ;;; -*- Mode: Lisp -*-
|
---|
| 2 | ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
---|
| 3 | ;;;
|
---|
| 4 | ;;; Copyright (C) 1999, 2002, 2009, 2015 Marek Rychlik <rychlik@u.arizona.edu>
|
---|
| 5 | ;;;
|
---|
| 6 | ;;; This program is free software; you can redistribute it and/or modify
|
---|
| 7 | ;;; it under the terms of the GNU General Public License as published by
|
---|
| 8 | ;;; the Free Software Foundation; either version 2 of the License, or
|
---|
| 9 | ;;; (at your option) any later version.
|
---|
| 10 | ;;;
|
---|
| 11 | ;;; This program is distributed in the hope that it will be useful,
|
---|
| 12 | ;;; but WITHOUT ANY WARRANTY; without even the implied warranty of
|
---|
| 13 | ;;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
|
---|
| 14 | ;;; GNU General Public License for more details.
|
---|
| 15 | ;;;
|
---|
| 16 | ;;; You should have received a copy of the GNU General Public License
|
---|
| 17 | ;;; along with this program; if not, write to the Free Software
|
---|
| 18 | ;;; Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
|
---|
| 19 | ;;;
|
---|
| 20 | ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
---|
| 21 |
|
---|
| 22 | ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
---|
| 23 | ;;
|
---|
[1997] | 24 | ;; Polynomials implemented in CLOS
|
---|
[1998] | 25 | ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
---|
[1969] | 26 | ;;
|
---|
[1998] | 27 | ;; A polynomial is an collection of terms. A
|
---|
| 28 | ;; term has a monomial and a coefficient.
|
---|
| 29 | ;;
|
---|
| 30 | ;; A polynomial can be represented by an s-expp
|
---|
| 31 | ;; (EXPR . VARS) where EXPR is an arithmetical formula
|
---|
| 32 | ;; recursively built of the arithmetical operations,
|
---|
| 33 | ;; and VARS are the variables of the polynomial.
|
---|
| 34 | ;; If a subtree of this s-exp is not an arithmetical
|
---|
| 35 | ;; operator +, -, *, expt, and is not a member
|
---|
| 36 | ;; of VARS then it represents a scalar expression
|
---|
| 37 | ;; which the Lisp reader must know how to convert
|
---|
| 38 | ;; into an object for which can be multiplied by a variable,
|
---|
| 39 | ;; subject to commutativity and associativity rules.
|
---|
| 40 | ;;
|
---|
[1969] | 41 | ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
---|
| 42 |
|
---|
[1970] | 43 | (defpackage "POL"
|
---|
[1969] | 44 | (:use :cl :ring :ring-and-order :monom :order :term :termlist :infix)
|
---|
| 45 | (:export "POLY"
|
---|
| 46 | "POLY-TERMLIST"
|
---|
| 47 | "POLY-SUGAR"
|
---|
| 48 | "POLY-RESET-SUGAR"
|
---|
| 49 | "POLY-LT"
|
---|
| 50 | "MAKE-POLY-FROM-TERMLIST"
|
---|
| 51 | "MAKE-POLY-ZERO"
|
---|
| 52 | "MAKE-POLY-VARIABLE"
|
---|
| 53 | "POLY-UNIT"
|
---|
| 54 | "POLY-LM"
|
---|
| 55 | "POLY-SECOND-LM"
|
---|
| 56 | "POLY-SECOND-LT"
|
---|
| 57 | "POLY-LC"
|
---|
| 58 | "POLY-SECOND-LC"
|
---|
| 59 | "POLY-ZEROP"
|
---|
| 60 | "POLY-LENGTH"
|
---|
| 61 | "SCALAR-TIMES-POLY"
|
---|
| 62 | "SCALAR-TIMES-POLY-1"
|
---|
| 63 | "MONOM-TIMES-POLY"
|
---|
| 64 | "TERM-TIMES-POLY"
|
---|
| 65 | "POLY-ADD"
|
---|
| 66 | "POLY-SUB"
|
---|
| 67 | "POLY-UMINUS"
|
---|
| 68 | "POLY-MUL"
|
---|
| 69 | "POLY-EXPT"
|
---|
| 70 | "POLY-APPEND"
|
---|
| 71 | "POLY-NREVERSE"
|
---|
| 72 | "POLY-REVERSE"
|
---|
| 73 | "POLY-CONTRACT"
|
---|
| 74 | "POLY-EXTEND"
|
---|
| 75 | "POLY-ADD-VARIABLES"
|
---|
| 76 | "POLY-LIST-ADD-VARIABLES"
|
---|
| 77 | "POLY-STANDARD-EXTENSION"
|
---|
| 78 | "SATURATION-EXTENSION"
|
---|
| 79 | "POLYSATURATION-EXTENSION"
|
---|
| 80 | "SATURATION-EXTENSION-1"
|
---|
| 81 | "COERCE-COEFF"
|
---|
| 82 | "POLY-EVAL"
|
---|
| 83 | "POLY-EVAL-SCALAR"
|
---|
| 84 | "SPOLY"
|
---|
| 85 | "POLY-PRIMITIVE-PART"
|
---|
| 86 | "POLY-CONTENT"
|
---|
| 87 | "READ-INFIX-FORM"
|
---|
| 88 | "READ-POLY"
|
---|
| 89 | "STRING->POLY"
|
---|
| 90 | "POLY->ALIST"
|
---|
| 91 | "STRING->ALIST"
|
---|
| 92 | "POLY-EQUAL-NO-SUGAR-P"
|
---|
| 93 | "POLY-SET-EQUAL-NO-SUGAR-P"
|
---|
| 94 | "POLY-LIST-EQUAL-NO-SUGAR-P"
|
---|
| 95 | ))
|
---|
| 96 |
|
---|
[1972] | 97 | (in-package :pol)
|
---|
[1969] | 98 |
|
---|
[1973] | 99 | (proclaim '(optimize (speed 0) (space 0) (safety 3) (debug 3)))
|
---|
[1969] | 100 |
|
---|
[1979] | 101 | (defclass poly ()
|
---|
[2007] | 102 | ((expr :initarg :expr))
|
---|
| 103 | ((vars :initarg :vars))
|
---|
[2004] | 104 | (:default-initargs :expr 0 :vars nil))
|
---|
[1969] | 105 |
|
---|
[2000] | 106 | (defmethod print-object ((self poly) stream)
|
---|
[2004] | 107 | (princ (slot-value self 'expr)))
|
---|
[1986] | 108 |
|
---|
[2000] | 109 | (defmethod poly-add ((p poly) (q poly)))
|
---|
[1969] | 110 |
|
---|
[2000] | 111 | (defmethod poly-sub ((p poly) (q poly)))
|
---|
[1969] | 112 |
|
---|
[2000] | 113 | (defmethod poly-uminus ((self poly)))
|
---|
[1969] | 114 |
|
---|
[2002] | 115 | (defmethod poly-mul ((p poly) (q poly)))
|
---|
[1969] | 116 |
|
---|
[2000] | 117 | (defmethod poly-expt ((self poly) n))
|
---|
[1969] | 118 |
|
---|
[2005] | 119 | (defun poly-eval (expr vars))
|
---|
[1969] | 120 | "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
|
---|
| 121 | variables VARS. Return the resulting polynomial or list of
|
---|
| 122 | polynomials. Standard arithmetical operators in form EXPR are
|
---|
| 123 | replaced with their analogues in the ring of polynomials, and the
|
---|
| 124 | resulting expression is evaluated, resulting in a polynomial or a list
|
---|
| 125 | of polynomials in internal form. A similar operation in another computer
|
---|
| 126 | algebra system could be called 'expand' or so."
|
---|
[2002] | 127 | (cond
|
---|
[2003] | 128 | ((null expr)
|
---|
[2006] | 129 | ;; Do nothing, nil is a representation of 0
|
---|
| 130 | ;; in all polynomial rings
|
---|
| 131 | )
|
---|
[2002] | 132 | ((member expr vars :test #'equalp)
|
---|
[2006] | 133 | (let* ((pos (position expr vars :test #'equalp))
|
---|
| 134 | (monom (let ((m (make-list (length vars) :initial-element 0)))
|
---|
| 135 | (setf (nth m pos) 1) m)))
|
---|
| 136 | (make-instance 'poly :expr (list (cons monom 1)))))
|
---|
[2002] | 137 | ((atom expr)
|
---|
| 138 | (scalar->poly ring expr vars))
|
---|
| 139 | ((eq (car expr) list-marker)
|
---|
| 140 | (cons list-marker (p-eval-list (cdr expr))))
|
---|
| 141 | (t
|
---|
| 142 | (case (car expr)
|
---|
| 143 | (+ (reduce #'p-add (p-eval-list (cdr expr))))
|
---|
| 144 | (- (case (length expr)
|
---|
| 145 | (1 (make-poly-zero))
|
---|
| 146 | (2 (poly-uminus ring (p-eval (cadr expr))))
|
---|
| 147 | (3 (poly-sub ring-and-order (p-eval (cadr expr)) (p-eval (caddr expr))))
|
---|
| 148 | (otherwise (poly-sub ring-and-order (p-eval (cadr expr))
|
---|
| 149 | (reduce #'p-add (p-eval-list (cddr expr)))))))
|
---|
| 150 | (*
|
---|
| 151 | (if (endp (cddr expr)) ;unary
|
---|
| 152 | (p-eval (cdr expr))
|
---|
| 153 | (reduce #'(lambda (p q) (poly-mul ring-and-order p q)) (p-eval-list (cdr expr)))))
|
---|
| 154 | (/
|
---|
| 155 | ;; A polynomial can be divided by a scalar
|
---|
| 156 | (cond
|
---|
| 157 | ((endp (cddr expr))
|
---|
| 158 | ;; A special case (/ ?), the inverse
|
---|
| 159 | (scalar->poly ring (apply (ring-div ring) (cdr expr)) vars))
|
---|
| 160 | (t
|
---|
| 161 | (let ((num (p-eval (cadr expr)))
|
---|
| 162 | (denom-inverse (apply (ring-div ring)
|
---|
| 163 | (cons (funcall (ring-unit ring))
|
---|
| 164 | (mapcar #'p-eval-scalar (cddr expr))))))
|
---|
| 165 | (scalar-times-poly ring denom-inverse num)))))
|
---|
| 166 | (expt
|
---|
| 167 | (cond
|
---|
| 168 | ((member (cadr expr) vars :test #'equalp)
|
---|
| 169 | ;;Special handling of (expt var pow)
|
---|
| 170 | (let ((pos (position (cadr expr) vars :test #'equalp)))
|
---|
| 171 | (make-poly-variable ring (length vars) pos (caddr expr))))
|
---|
| 172 | ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
|
---|
| 173 | ;; Negative power means division in coefficient ring
|
---|
| 174 | ;; Non-integer power means non-polynomial coefficient
|
---|
| 175 | (scalar->poly ring expr vars))
|
---|
| 176 | (t (poly-expt ring-and-order (p-eval (cadr expr)) (caddr expr)))))
|
---|
| 177 | (otherwise
|
---|
| 178 | (scalar->poly ring expr vars))))))
|
---|
[1969] | 179 |
|
---|
| 180 | (defun poly-eval-scalar (expr
|
---|
| 181 | &optional
|
---|
| 182 | (ring +ring-of-integers+)
|
---|
| 183 | &aux
|
---|
| 184 | (order #'lex>))
|
---|
| 185 | "Evaluate a scalar expression EXPR in ring RING."
|
---|
| 186 | (declare (type ring ring))
|
---|
| 187 | (poly-lc (poly-eval expr nil ring order)))
|
---|
| 188 |
|
---|
| 189 |
|
---|
| 190 |
|
---|
| 191 |
|
---|