;;; -*- Mode: Lisp -*- ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;; ;;; Copyright (C) 1999, 2002, 2009, 2015 Marek Rychlik ;;; ;;; This program is free software; you can redistribute it and/or modify ;;; it under the terms of the GNU General Public License as published by ;;; the Free Software Foundation; either version 2 of the License, or ;;; (at your option) any later version. ;;; ;;; This program is distributed in the hope that it will be useful, ;;; but WITHOUT ANY WARRANTY; without even the implied warranty of ;;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ;;; GNU General Public License for more details. ;;; ;;; You should have received a copy of the GNU General Public License ;;; along with this program; if not, write to the Free Software ;;; Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. ;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; (defpackage "SYMBOLIC-POLYNOMIAL" (:use :cl :utils :ring :monom :order :term :polynomial :infix) (:export "SYMBOLIC-POLY" "READ-INFIX-FORM" "STRING->POLY" "+LIST-MARKER+") (:documentation "Implements symbolic polynomials. A symbolic polynomial is and object which uses symbolic variables for reading and printing in standard human-readable (infix) form.")) (in-package :symbolic-polynomial) (defparameter +list-marker+ :[ "A sexp with this head is considered a list of polynomials.") (defclass symbolic-poly (poly) ((vars :initform nil :initarg :vars :accessor symbolic-poly-vars) ) (:default-initargs :termlist nil :vars nil)) (defmethod print-object ((self symbolic-poly) stream) (print-unreadable-object (self stream :type t :identity t) (with-accessors ((dimension poly-dimension) (termlist poly-termlist) (order poly-term-order) (vars symbolic-poly-vars)) self (format stream "DIMENSION=~A TERMLIST=~A ORDER=~A VARS=~A" dimension termlist order vars)))) (defmethod r-equalp ((self symbolic-poly) (other symbolic-poly)) (when (r-equalp (symbolic-poly-vars self) (symbolic-poly-vars other)) (call-next-method))) (defmethod update-instance-for-different-class :after ((old poly) (new symbolic-poly) &key) "After adding variables to NEW, we need to make sure that the number of variables given by POLY-DIMENSION is consistent with VARS." (assert (= (length (symbolic-poly-vars new)) (poly-dimension new)))) (defgeneric poly-eval (expr vars order) (:documentation "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in variables VARS. Return the resulting polynomial or list of polynomials. Standard arithmetical operators in form EXPR are replaced with their analogues in the ring of polynomials, and the resulting expression is evaluated, resulting in a polynomial or a list of polynomials in internal form. A similar operation in another computer algebra system could be called 'expand' or so.") (:method ((expr symbolic-poly) vars order) expr) (:method (expr vars order) (labels ((p-eval (p) (poly-eval p vars order)) (p-eval-scalar (p) (poly-eval p '() order)) (p-eval-list (plist) (mapcar #'p-eval plist))) (cond ((eq expr 0) (make-instance 'symbolic-poly :dimension (length vars) :vars vars)) ((member expr vars :test #'equalp) (let ((pos (position expr vars :test #'equalp))) (make-monom-variable (length vars) pos))) ((atom expr) expr) ((eq (car expr) +list-marker+) (cons +list-marker+ (p-eval-list (cdr expr)))) (t (case (car expr) (+ (reduce #'r+ (p-eval-list (cdr expr)))) (- (case (length expr) (1 (make-poly-zero)) (2 (r- (p-eval (cadr expr)))) (3 (r- (p-eval (cadr expr)) (p-eval (caddr expr)))) (otherwise (r- (p-eval (cadr expr)) (reduce #'r+ (p-eval-list (cddr expr))))))) (* (if (endp (cddr expr)) ;unary (p-eval (cdr expr)) (reduce #'r* (p-eval-list (cdr expr))))) (/ ;; A polynomial can be divided by a scalar (cond ((endp (cddr expr)) ;; A special case (/ ?), the inverse (r/ (cdr expr))) (t (let ((num (p-eval (cadr expr))) (denom-inverse (r/ (mapcar #'p-eval-scalar (cddr expr))))) (r* denom-inverse num))))) (expt (cond ((member (cadr expr) vars :test #'equalp) ;;Special handling of (expt var pow) (let ((pos (position (cadr expr) vars :test #'equalp))) (make-poly-variable ring (length vars) pos (caddr expr)))) ((not (and (integerp (caddr expr)) (plusp (caddr expr)))) ;; Negative power means division in coefficient ring ;; Non-integer power means non-polynomial coefficient (coerce-coeff ring expr vars)) (t (r-expt (p-eval (cadr expr)) (caddr expr))))) (otherwise (coerce-coeff ring expr vars)))))))) #| (defun poly-eval-scalar (expr &optional (ring +ring-of-integers+) &aux (order #'lex>)) "Evaluate a scalar expression EXPR in ring RING." (declare (type ring ring)) (poly-lc (poly-eval expr nil ring order))) |# (defun read-infix-form (&key (stream t)) "Parser of infix expressions with integer/rational coefficients The parser will recognize two kinds of polynomial expressions: - polynomials in fully expanded forms with coefficients written in front of symbolic expressions; constants can be optionally enclosed in (); for example, the infix form X^2-Y^2+(-4/3)*U^2*W^3-5 parses to (+ (- (EXPT X 2) (EXPT Y 2)) (* (- (/ 4 3)) (EXPT U 2) (EXPT W 3)) (- 5)) - lists of polynomials; for example [X-Y, X^2+3*Z] parses to (:[ (- X Y) (+ (EXPT X 2) (* 3 Z))) where the first symbol [ marks a list of polynomials. -other infix expressions, for example [(X-Y)*(X+Y)/Z,(X+1)^2] parses to: (:[ (/ (* (- X Y) (+ X Y)) Z) (EXPT (+ X 1) 2)) Currently this function is implemented using M. Kantrowitz's INFIX package." (read-from-string (concatenate 'string "#I(" (with-output-to-string (s) (loop (multiple-value-bind (line eof) (read-line stream t) (format s "~A" line) (when eof (return))))) ")"))) (defun read-poly (vars &key (stream t) (order #'lex>)) "Reads an expression in prefix form from a stream STREAM. The expression read from the strem should represent a polynomial or a list of polynomials in variables VARS, over the ring RING. The polynomial or list of polynomials is returned, with terms in each polynomial ordered according to monomial order ORDER." (poly-eval (read-infix-form :stream stream) vars order)) (defun string->poly (str vars &optional (order #'lex>)) "Converts a string STR to a polynomial in variables VARS." (with-input-from-string (s str) (read-poly vars :stream s :order order))) (defun poly->alist (p) "Convert a polynomial P to an association list. Thus, the format of the returned value is ((MONOM[0] . COEFF[0]) (MONOM[1] . COEFF[1]) ...), where MONOM[I] is a list of exponents in the monomial and COEFF[I] is the corresponding coefficient in the ring." (cond ((poly-p p) (mapcar #'term->cons (poly-termlist p))) ((and (consp p) (eq (car p) :[)) (cons :[ (mapcar #'poly->alist (cdr p)))))) (defun string->alist (str vars &optional (ring +ring-of-integers+) (order #'lex>)) "Convert a string STR representing a polynomial or polynomial list to an association list (... (MONOM . COEFF) ...)." (poly->alist (string->poly str vars ring order))) (defun poly-equal-no-sugar-p (p q) "Compare polynomials for equality, ignoring sugar." (declare (type poly p q)) (equalp (poly-termlist p) (poly-termlist q))) (defun poly-set-equal-no-sugar-p (p q) "Compare polynomial sets P and Q for equality, ignoring sugar." (null (set-exclusive-or p q :test #'poly-equal-no-sugar-p ))) (defun poly-list-equal-no-sugar-p (p q) "Compare polynomial lists P and Q for equality, ignoring sugar." (every #'poly-equal-no-sugar-p p q))