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1200 lines
39 KiB
EmacsLisp
1200 lines
39 KiB
EmacsLisp
;;; calc-poly.el --- polynomial functions for Calc -*- lexical-binding:t -*-
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;; Copyright (C) 1990-1993, 2001-2021 Free Software Foundation, Inc.
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;; Author: David Gillespie <daveg@synaptics.com>
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;; This file is part of GNU Emacs.
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;; GNU Emacs is free software: you can redistribute it and/or modify
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;; it under the terms of the GNU General Public License as published by
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;; the Free Software Foundation, either version 3 of the License, or
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;; (at your option) any later version.
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;; GNU Emacs is distributed in the hope that it will be useful,
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;; but WITHOUT ANY WARRANTY; without even the implied warranty of
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;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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;; GNU General Public License for more details.
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;; You should have received a copy of the GNU General Public License
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;; along with GNU Emacs. If not, see <https://www.gnu.org/licenses/>.
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;;; Commentary:
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;;; Code:
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;; This file is autoloaded from calc-ext.el.
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(require 'calc-ext)
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(require 'calc-macs)
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(defun calcFunc-pcont (expr &optional var)
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(cond ((Math-primp expr)
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(cond ((Math-zerop expr) 1)
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((Math-messy-integerp expr) (math-trunc expr))
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((Math-objectp expr) expr)
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((or (equal expr var) (not var)) 1)
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(t expr)))
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((eq (car expr) '*)
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(math-mul (calcFunc-pcont (nth 1 expr) var)
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(calcFunc-pcont (nth 2 expr) var)))
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((eq (car expr) '/)
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(math-div (calcFunc-pcont (nth 1 expr) var)
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(calcFunc-pcont (nth 2 expr) var)))
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((and (eq (car expr) '^) (Math-natnump (nth 2 expr)))
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(math-pow (calcFunc-pcont (nth 1 expr) var) (nth 2 expr)))
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((memq (car expr) '(neg polar))
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(calcFunc-pcont (nth 1 expr) var))
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((consp var)
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(let ((p (math-is-polynomial expr var)))
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(if p
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(let ((lead (nth (1- (length p)) p))
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(cont (math-poly-gcd-list p)))
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(if (math-guess-if-neg lead)
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(math-neg cont)
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cont))
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1)))
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((memq (car expr) '(+ - cplx sdev))
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(let ((cont (calcFunc-pcont (nth 1 expr) var)))
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(if (eq cont 1)
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1
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(let ((c2 (calcFunc-pcont (nth 2 expr) var)))
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(if (and (math-negp cont)
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(if (eq (car expr) '-) (math-posp c2) (math-negp c2)))
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(math-neg (math-poly-gcd cont c2))
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(math-poly-gcd cont c2))))))
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(var expr)
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(t 1)))
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(defun calcFunc-pprim (expr &optional var)
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(let ((cont (calcFunc-pcont expr var)))
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(if (math-equal-int cont 1)
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expr
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(math-poly-div-exact expr cont var))))
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(defun math-div-poly-const (expr c)
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(cond ((memq (car-safe expr) '(+ -))
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(list (car expr)
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(math-div-poly-const (nth 1 expr) c)
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(math-div-poly-const (nth 2 expr) c)))
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(t (math-div expr c))))
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(defun calcFunc-pdeg (expr &optional var)
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(if (Math-zerop expr)
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'(neg (var inf var-inf))
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(if var
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(or (math-polynomial-p expr var)
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(math-reject-arg expr "Expected a polynomial"))
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(math-poly-degree expr))))
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(defun math-poly-degree (expr)
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(cond ((Math-primp expr)
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(if (eq (car-safe expr) 'var) 1 0))
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((eq (car expr) 'neg)
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(math-poly-degree (nth 1 expr)))
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((eq (car expr) '*)
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(+ (math-poly-degree (nth 1 expr))
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(math-poly-degree (nth 2 expr))))
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((eq (car expr) '/)
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(- (math-poly-degree (nth 1 expr))
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(math-poly-degree (nth 2 expr))))
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((and (eq (car expr) '^) (natnump (nth 2 expr)))
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(* (math-poly-degree (nth 1 expr)) (nth 2 expr)))
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((memq (car expr) '(+ -))
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(max (math-poly-degree (nth 1 expr))
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(math-poly-degree (nth 2 expr))))
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(t 1)))
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(defun calcFunc-plead (expr var)
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(cond ((eq (car-safe expr) '*)
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(math-mul (calcFunc-plead (nth 1 expr) var)
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(calcFunc-plead (nth 2 expr) var)))
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((eq (car-safe expr) '/)
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(math-div (calcFunc-plead (nth 1 expr) var)
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(calcFunc-plead (nth 2 expr) var)))
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((and (eq (car-safe expr) '^) (math-natnump (nth 2 expr)))
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(math-pow (calcFunc-plead (nth 1 expr) var) (nth 2 expr)))
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((Math-primp expr)
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(if (equal expr var)
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1
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expr))
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(t
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(let ((p (math-is-polynomial expr var)))
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(if (cdr p)
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(nth (1- (length p)) p)
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1)))))
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;;; Polynomial quotient, remainder, and GCD.
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;;; Originally by Ove Ewerlid (ewerlid@mizar.DoCS.UU.SE).
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;;; Modifications and simplifications by daveg.
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(defvar math-poly-modulus 1)
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;;; Return gcd of two polynomials
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(defun calcFunc-pgcd (pn pd)
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(if (math-any-floats pn)
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(math-reject-arg pn "Coefficients must be rational"))
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(if (math-any-floats pd)
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(math-reject-arg pd "Coefficients must be rational"))
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(let ((calc-prefer-frac t)
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(math-poly-modulus (math-poly-modulus pn pd)))
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(math-poly-gcd pn pd)))
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;;; Return only quotient to top of stack (nil if zero)
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;; calc-poly-div-remainder is a local variable for
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;; calc-poly-div (in calc-alg.el), but is used by
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;; calcFunc-pdiv, which is called by calc-poly-div.
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(defvar calc-poly-div-remainder)
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(defun calcFunc-pdiv (pn pd &optional base)
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(let* ((calc-prefer-frac t)
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(math-poly-modulus (math-poly-modulus pn pd))
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(res (math-poly-div pn pd base)))
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(setq calc-poly-div-remainder (cdr res))
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(car res)))
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;;; Return only remainder to top of stack
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(defun calcFunc-prem (pn pd &optional base)
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(let ((calc-prefer-frac t)
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(math-poly-modulus (math-poly-modulus pn pd)))
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(cdr (math-poly-div pn pd base))))
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(defun calcFunc-pdivrem (pn pd &optional base)
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(let* ((calc-prefer-frac t)
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(math-poly-modulus (math-poly-modulus pn pd))
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(res (math-poly-div pn pd base)))
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(list 'vec (car res) (cdr res))))
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(defun calcFunc-pdivide (pn pd &optional base)
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(let* ((calc-prefer-frac t)
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(math-poly-modulus (math-poly-modulus pn pd))
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(res (math-poly-div pn pd base)))
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(math-add (car res) (math-div (cdr res) pd))))
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(defun math-mul-thru (lhs rhs)
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"Multiply two terms, expanding out products of sums."
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(if (memq (car-safe lhs) '(+ -))
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(list (car lhs)
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(math-mul-thru (nth 1 lhs) rhs)
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(math-mul-thru (nth 2 lhs) rhs))
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(if (memq (car-safe rhs) '(+ -))
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(list (car rhs)
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(math-mul-thru lhs (nth 1 rhs))
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(math-mul-thru lhs (nth 2 rhs)))
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(math-mul lhs rhs))))
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(defun math-div-thru (num den)
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(if (memq (car-safe num) '(+ -))
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(list (car num)
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(math-div-thru (nth 1 num) den)
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(math-div-thru (nth 2 num) den))
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(math-div num den)))
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(defun math-sort-terms (expr)
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"Sort the terms of a sum into canonical order."
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(if (memq (car-safe expr) '(+ -))
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(math-list-to-sum
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(sort (math-sum-to-list expr)
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(function (lambda (a b) (math-beforep (car a) (car b))))))
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expr))
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(defun math-list-to-sum (lst)
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(if (cdr lst)
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(list (if (cdr (car lst)) '- '+)
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(math-list-to-sum (cdr lst))
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(car (car lst)))
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(if (cdr (car lst))
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(math-neg (car (car lst)))
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(car (car lst)))))
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(defun math-sum-to-list (tree &optional neg)
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(cond ((eq (car-safe tree) '+)
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(nconc (math-sum-to-list (nth 1 tree) neg)
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(math-sum-to-list (nth 2 tree) neg)))
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((eq (car-safe tree) '-)
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(nconc (math-sum-to-list (nth 1 tree) neg)
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(math-sum-to-list (nth 2 tree) (not neg))))
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(t (list (cons tree neg)))))
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(defun math-poly-modulus (expr &optional expr2)
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"Check if the polynomial coefficients are modulo forms."
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(or (math-poly-modulus-rec expr)
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(and expr2 (math-poly-modulus-rec expr2))
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1))
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(defun math-poly-modulus-rec (expr)
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(if (and (eq (car-safe expr) 'mod) (Math-natnump (nth 2 expr)))
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(list 'mod 1 (nth 2 expr))
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(and (memq (car-safe expr) '(+ - * /))
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(or (math-poly-modulus-rec (nth 1 expr))
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(math-poly-modulus-rec (nth 2 expr))))))
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(defvar math-poly-div-base nil)
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(defun math-poly-div (u v &optional div-base)
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"Divide two polynomials. Return (quotient . remainder)."
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(let ((math-poly-div-base div-base))
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(if div-base
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(math-do-poly-div u v)
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(math-do-poly-div (calcFunc-expand u) (calcFunc-expand v)))))
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(defun math-poly-div-exact (u v &optional base)
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(let ((res (math-poly-div u v base)))
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(if (eq (cdr res) 0)
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(car res)
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(math-reject-arg (list 'vec u v) "Argument is not a polynomial"))))
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(defun math-do-poly-div (u v)
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(cond ((math-constp u)
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(if (math-constp v)
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(cons (math-div u v) 0)
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(cons 0 u)))
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((math-constp v)
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(cons (if (eq v 1)
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u
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(if (memq (car-safe u) '(+ -))
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(math-add-or-sub (math-poly-div-exact (nth 1 u) v)
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(math-poly-div-exact (nth 2 u) v)
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nil (eq (car u) '-))
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(math-div u v)))
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0))
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((Math-equal u v)
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(cons math-poly-modulus 0))
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((and (math-atomic-factorp u) (math-atomic-factorp v))
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(cons (math-simplify (math-div u v)) 0))
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(t
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(let ((base (or math-poly-div-base
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(math-poly-div-base u v)))
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vp up res)
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(if (or (null base)
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(null (setq vp (math-is-polynomial v base nil 'gen))))
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(cons 0 u)
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(setq up (math-is-polynomial u base nil 'gen)
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res (math-poly-div-coefs up vp))
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(cons (math-build-polynomial-expr (car res) base)
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(math-build-polynomial-expr (cdr res) base)))))))
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(defun math-poly-div-rec (u v)
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(cond ((math-constp u)
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(math-div u v))
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((math-constp v)
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(if (eq v 1)
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u
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(if (memq (car-safe u) '(+ -))
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(math-add-or-sub (math-poly-div-rec (nth 1 u) v)
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(math-poly-div-rec (nth 2 u) v)
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nil (eq (car u) '-))
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(math-div u v))))
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((Math-equal u v) math-poly-modulus)
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((and (math-atomic-factorp u) (math-atomic-factorp v))
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(math-simplify (math-div u v)))
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(math-poly-div-base
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(math-div u v))
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(t
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(let ((base (math-poly-div-base u v))
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vp up res)
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(if (or (null base)
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(null (setq vp (math-is-polynomial v base nil 'gen))))
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(math-div u v)
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(setq up (math-is-polynomial u base nil 'gen)
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res (math-poly-div-coefs up vp))
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(math-add (math-build-polynomial-expr (car res) base)
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(math-div (math-build-polynomial-expr (cdr res) base)
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v)))))))
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(defun math-poly-div-coefs (u v)
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"Divide two polynomials in coefficient-list form. Return (quot . rem)."
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(cond ((null v) (math-reject-arg nil "Division by zero"))
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((< (length u) (length v)) (cons nil u))
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((cdr u)
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(let ((q nil)
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(urev (reverse u))
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(vrev (reverse v)))
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(while
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(let ((qk (math-poly-div-rec (math-simplify (car urev))
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(car vrev)))
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(up urev)
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(vp vrev))
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(if (or q (not (math-zerop qk)))
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(setq q (cons qk q)))
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(while (setq up (cdr up) vp (cdr vp))
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(setcar up (math-sub (car up) (math-mul-thru qk (car vp)))))
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(setq urev (cdr urev))
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up))
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(while (and urev (Math-zerop (car urev)))
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(setq urev (cdr urev)))
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(cons q (nreverse (mapcar 'math-simplify urev)))))
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(t
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(cons (list (math-poly-div-rec (car u) (car v)))
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nil))))
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(defun math-poly-pseudo-div (u v)
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"Perform a pseudo-division of polynomials. (See Knuth section 4.6.1.)
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This returns only the remainder from the pseudo-division."
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(cond ((null v) nil)
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((< (length u) (length v)) u)
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((or (cdr u) (cdr v))
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(let ((urev (reverse u))
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(vrev (reverse v))
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up)
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(while
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(let ((vp vrev))
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(setq up urev)
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(while (setq up (cdr up) vp (cdr vp))
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(setcar up (math-sub (math-mul-thru (car vrev) (car up))
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(math-mul-thru (car urev) (car vp)))))
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(setq urev (cdr urev))
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up)
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(while up
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(setcar up (math-mul-thru (car vrev) (car up)))
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(setq up (cdr up))))
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(while (and urev (Math-zerop (car urev)))
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(setq urev (cdr urev)))
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(nreverse (mapcar 'math-simplify urev))))
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(t nil)))
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(defun math-poly-gcd (u v)
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"Compute the GCD of two multivariate polynomials."
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(cond ((Math-equal u v) u)
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((math-constp u)
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(if (Math-zerop u)
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v
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(calcFunc-gcd u (calcFunc-pcont v))))
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((math-constp v)
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(if (Math-zerop v)
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v
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(calcFunc-gcd v (calcFunc-pcont u))))
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(t
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(let ((base (math-poly-gcd-base u v)))
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(if base
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(math-simplify
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(calcFunc-expand
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(math-build-polynomial-expr
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(math-poly-gcd-coefs (math-is-polynomial u base nil 'gen)
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(math-is-polynomial v base nil 'gen))
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base)))
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(calcFunc-gcd (calcFunc-pcont u) (calcFunc-pcont u)))))))
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(defun math-poly-div-list (lst a)
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(if (eq a 1)
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lst
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(if (eq a -1)
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(math-mul-list lst a)
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(mapcar (function (lambda (x) (math-poly-div-exact x a))) lst))))
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(defun math-mul-list (lst a)
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(if (eq a 1)
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lst
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(if (eq a -1)
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(mapcar 'math-neg lst)
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(and (not (eq a 0))
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(mapcar (function (lambda (x) (math-mul x a))) lst)))))
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;;; Run GCD on all elements in a list.
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(defun math-poly-gcd-list (lst)
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(if (or (memq 1 lst) (memq -1 lst))
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(math-poly-gcd-frac-list lst)
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(let ((gcd (car lst)))
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(while (and (setq lst (cdr lst)) (not (eq gcd 1)))
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(or (eq (car lst) 0)
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(setq gcd (math-poly-gcd gcd (car lst)))))
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(if lst (setq lst (math-poly-gcd-frac-list lst)))
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gcd)))
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(defun math-poly-gcd-frac-list (lst)
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(while (and lst (not (eq (car-safe (car lst)) 'frac)))
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(setq lst (cdr lst)))
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(if lst
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(let ((denom (nth 2 (car lst))))
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(while (setq lst (cdr lst))
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(if (eq (car-safe (car lst)) 'frac)
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(setq denom (calcFunc-lcm denom (nth 2 (car lst))))))
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(list 'frac 1 denom))
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1))
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;;; Compute the GCD of two univariate polynomial lists.
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;;; Knuth section 4.6.1, algorithm C.
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(defun math-poly-gcd-coefs (u v)
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(let ((d (math-poly-gcd (math-poly-gcd-list u)
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(math-poly-gcd-list v)))
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(g 1) (h 1) (z 0) r delta)
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(while (and u v (Math-zerop (car u)) (Math-zerop (car v)))
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(setq u (cdr u) v (cdr v) z (1+ z)))
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(or (eq d 1)
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(setq u (math-poly-div-list u d)
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v (math-poly-div-list v d)))
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(while (progn
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(setq delta (- (length u) (length v)))
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(if (< delta 0)
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(setq r u u v v r delta (- delta)))
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(setq r (math-poly-pseudo-div u v))
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(cdr r))
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(setq u v
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v (math-poly-div-list r (math-mul g (math-pow h delta)))
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g (nth (1- (length u)) u)
|
|
h (if (<= delta 1)
|
|
(math-mul (math-pow g delta) (math-pow h (- 1 delta)))
|
|
(math-poly-div-exact (math-pow g delta)
|
|
(math-pow h (1- delta))))))
|
|
(setq v (if r
|
|
(list d)
|
|
(math-mul-list (math-poly-div-list v (math-poly-gcd-list v)) d)))
|
|
(if (math-guess-if-neg (nth (1- (length v)) v))
|
|
(setq v (math-mul-list v -1)))
|
|
(while (>= (setq z (1- z)) 0)
|
|
(setq v (cons 0 v)))
|
|
v))
|
|
|
|
|
|
(defun math-atomic-factorp (expr)
|
|
"Return true if is a factor containing no sums or quotients."
|
|
(cond ((eq (car-safe expr) '*)
|
|
(and (math-atomic-factorp (nth 1 expr))
|
|
(math-atomic-factorp (nth 2 expr))))
|
|
((memq (car-safe expr) '(+ - /))
|
|
nil)
|
|
((memq (car-safe expr) '(^ neg))
|
|
(math-atomic-factorp (nth 1 expr)))
|
|
(t t)))
|
|
|
|
(defun math-poly-div-base (a b)
|
|
"Find a suitable base for dividing a by b.
|
|
The base must exist in both expressions.
|
|
The degree in the numerator must be higher or equal than the
|
|
degree in the denominator.
|
|
If the above conditions are not met the quotient is just a remainder.
|
|
Return nil if this is the case."
|
|
(let (a-base b-base)
|
|
(and (setq a-base (math-total-polynomial-base a))
|
|
(setq b-base (math-total-polynomial-base b))
|
|
(catch 'return
|
|
(while a-base
|
|
(let ((maybe (assoc (car (car a-base)) b-base)))
|
|
(if maybe
|
|
(if (>= (nth 1 (car a-base)) (nth 1 maybe))
|
|
(throw 'return (car (car a-base))))))
|
|
(setq a-base (cdr a-base)))))))
|
|
|
|
(defun math-poly-gcd-base (a b)
|
|
"Same as `math-poly-div-base' but for gcd algorithm.
|
|
Here there is no requirement that degree(a) > degree(b).
|
|
Take the base that has the highest degree considering both a and b.
|
|
(\"a^20+b^21+x^3+a+b\", \"a+b^2+x^5+a^22+b^10\") --> (a 22)"
|
|
(let (a-base b-base)
|
|
(and (setq a-base (math-total-polynomial-base a))
|
|
(setq b-base (math-total-polynomial-base b))
|
|
(catch 'return
|
|
(while (and a-base b-base)
|
|
(if (> (nth 1 (car a-base)) (nth 1 (car b-base)))
|
|
(if (assoc (car (car a-base)) b-base)
|
|
(throw 'return (car (car a-base)))
|
|
(setq a-base (cdr a-base)))
|
|
(if (assoc (car (car b-base)) a-base)
|
|
(throw 'return (car (car b-base)))
|
|
(setq b-base (cdr b-base)))))))))
|
|
|
|
(defun math-sort-poly-base-list (lst)
|
|
"Sort a list of polynomial bases."
|
|
(sort lst (function (lambda (a b)
|
|
(or (> (nth 1 a) (nth 1 b))
|
|
(and (= (nth 1 a) (nth 1 b))
|
|
(math-beforep (car a) (car b))))))))
|
|
|
|
;;; Given an expression find all variables that are polynomial bases.
|
|
;;; Return list in the form '( (var1 degree1) (var2 degree2) ... ).
|
|
|
|
;; The variable math-poly-base-total-base and math-poly-base-top-expr are local
|
|
;; to math-total-polynomial-base, but used by math-polynomial-p1, which is
|
|
;; called by math-total-polynomial-base.
|
|
(defvar math-poly-base-total-base)
|
|
(defvar math-poly-base-top-expr)
|
|
|
|
(defun math-total-polynomial-base (expr)
|
|
(let ((math-poly-base-total-base nil)
|
|
(math-poly-base-top-expr expr))
|
|
(math-polynomial-base expr #'math-polynomial-p1)
|
|
(math-sort-poly-base-list math-poly-base-total-base)))
|
|
|
|
(defun math-polynomial-p1 (subexpr)
|
|
(or (assoc subexpr math-poly-base-total-base)
|
|
(memq (car subexpr) '(+ - * / neg))
|
|
(and (eq (car subexpr) '^) (natnump (nth 2 subexpr)))
|
|
(let* ((math-poly-base-variable subexpr)
|
|
(exponent (math-polynomial-p math-poly-base-top-expr subexpr)))
|
|
(if exponent
|
|
(setq math-poly-base-total-base (cons (list subexpr exponent)
|
|
math-poly-base-total-base)))))
|
|
nil)
|
|
|
|
;; The variable math-factored-vars is local to calcFunc-factors and
|
|
;; calcFunc-factor, but is used by math-factor-expr and
|
|
;; math-factor-expr-part, which are called (directly and indirectly) by
|
|
;; calcFunc-factor and calcFunc-factors.
|
|
(defvar math-factored-vars)
|
|
|
|
;; The variable math-fact-expr is local to calcFunc-factors,
|
|
;; calcFunc-factor and math-factor-expr, but is used by math-factor-expr-try
|
|
;; and math-factor-expr-part, which are called (directly and indirectly) by
|
|
;; calcFunc-factor, calcFunc-factors and math-factor-expr.
|
|
(defvar math-fact-expr)
|
|
|
|
;; The variable math-to-list is local to calcFunc-factors and
|
|
;; calcFunc-factor, but is used by math-accum-factors, which is
|
|
;; called (indirectly) by calcFunc-factors and calcFunc-factor.
|
|
(defvar math-to-list)
|
|
|
|
(defun calcFunc-factors (expr &optional var)
|
|
(let ((math-factored-vars (if var t nil))
|
|
(math-to-list t)
|
|
(calc-prefer-frac t))
|
|
(or var
|
|
(setq var (math-polynomial-base expr)))
|
|
(let ((res (math-factor-finish
|
|
(or (catch 'factor
|
|
(let ((math-fact-expr expr)) (math-factor-expr-try var)))
|
|
expr))))
|
|
(math-simplify (if (math-vectorp res)
|
|
res
|
|
(list 'vec (list 'vec res 1)))))))
|
|
|
|
(defun calcFunc-factor (expr &optional var)
|
|
(let ((math-factored-vars nil)
|
|
(math-to-list nil)
|
|
(calc-prefer-frac t))
|
|
(math-simplify (math-factor-finish
|
|
(if var
|
|
(let ((math-factored-vars t)
|
|
(math-fact-expr expr))
|
|
(or (catch 'factor (math-factor-expr-try var)) expr))
|
|
(math-factor-expr expr))))))
|
|
|
|
(defun math-factor-finish (x)
|
|
(if (Math-primp x)
|
|
x
|
|
(if (eq (car x) 'calcFunc-Fac-Prot)
|
|
(math-factor-finish (nth 1 x))
|
|
(cons (car x) (mapcar 'math-factor-finish (cdr x))))))
|
|
|
|
(defun math-factor-protect (x)
|
|
(if (memq (car-safe x) '(+ -))
|
|
(list 'calcFunc-Fac-Prot x)
|
|
x))
|
|
|
|
(defun math-factor-expr (expr)
|
|
(cond ((eq math-factored-vars t) expr)
|
|
((or (memq (car-safe expr) '(* / ^ neg))
|
|
(assq (car-safe expr) calc-tweak-eqn-table))
|
|
(cons (car expr) (mapcar 'math-factor-expr (cdr expr))))
|
|
((memq (car-safe expr) '(+ -))
|
|
(let* ((math-factored-vars math-factored-vars)
|
|
(y (catch 'factor (let ((math-fact-expr expr))
|
|
(math-factor-expr-part expr)))))
|
|
(if y
|
|
(math-factor-expr y)
|
|
expr)))
|
|
(t expr)))
|
|
|
|
(defun math-factor-expr-part (x) ; uses "expr"
|
|
(if (memq (car-safe x) '(+ - * / ^ neg))
|
|
(while (setq x (cdr x))
|
|
(math-factor-expr-part (car x)))
|
|
(and (not (Math-objvecp x))
|
|
(not (assoc x math-factored-vars))
|
|
(> (math-factor-contains math-fact-expr x) 1)
|
|
(setq math-factored-vars (cons (list x) math-factored-vars))
|
|
(math-factor-expr-try x))))
|
|
|
|
;; The variable math-fet-x is local to math-factor-expr-try, but is
|
|
;; used by math-factor-poly-coefs, which is called by math-factor-expr-try.
|
|
(defvar math-fet-x)
|
|
|
|
(defun math-factor-expr-try (x)
|
|
(if (eq (car-safe math-fact-expr) '*)
|
|
(let ((res1 (catch 'factor (let ((math-fact-expr (nth 1 math-fact-expr)))
|
|
(math-factor-expr-try x))))
|
|
(res2 (catch 'factor (let ((math-fact-expr (nth 2 math-fact-expr)))
|
|
(math-factor-expr-try x)))))
|
|
(and (or res1 res2)
|
|
(throw 'factor (math-accum-factors (or res1 (nth 1 math-fact-expr)) 1
|
|
(or res2 (nth 2 math-fact-expr))))))
|
|
(let* ((p (math-is-polynomial math-fact-expr x 30 'gen))
|
|
(math-poly-modulus (math-poly-modulus math-fact-expr))
|
|
res)
|
|
(and (cdr p)
|
|
(setq res (let ((math-fet-x x)) (math-factor-poly-coefs p)))
|
|
(throw 'factor res)))))
|
|
|
|
(defun math-accum-factors (fac pow facs)
|
|
(if math-to-list
|
|
(if (math-vectorp fac)
|
|
(progn
|
|
(while (setq fac (cdr fac))
|
|
(setq facs (math-accum-factors (nth 1 (car fac))
|
|
(* pow (nth 2 (car fac)))
|
|
facs)))
|
|
facs)
|
|
(if (and (eq (car-safe fac) '^) (natnump (nth 2 fac)))
|
|
(setq pow (* pow (nth 2 fac))
|
|
fac (nth 1 fac)))
|
|
(if (eq fac 1)
|
|
facs
|
|
(or (math-vectorp facs)
|
|
(setq facs (if (eq facs 1) '(vec)
|
|
(list 'vec (list 'vec facs 1)))))
|
|
(let ((found facs))
|
|
(while (and (setq found (cdr found))
|
|
(not (equal fac (nth 1 (car found))))))
|
|
(if found
|
|
(progn
|
|
(setcar (cdr (cdr (car found))) (+ pow (nth 2 (car found))))
|
|
facs)
|
|
;; Put constant term first.
|
|
(if (and (cdr facs) (Math-ratp (nth 1 (nth 1 facs))))
|
|
(cons 'vec (cons (nth 1 facs) (cons (list 'vec fac pow)
|
|
(cdr (cdr facs)))))
|
|
(cons 'vec (cons (list 'vec fac pow) (cdr facs))))))))
|
|
(math-mul (math-pow fac pow) (math-factor-protect facs))))
|
|
|
|
(defun math-factor-poly-coefs (p &optional square-free) ; uses "x"
|
|
(let (t1 t2 temp)
|
|
(cond ((not (cdr p))
|
|
(or (car p) 0))
|
|
|
|
;; Strip off multiples of math-fet-x.
|
|
((Math-zerop (car p))
|
|
(let ((z 0))
|
|
(while (and p (Math-zerop (car p)))
|
|
(setq z (1+ z) p (cdr p)))
|
|
(if (cdr p)
|
|
(setq p (math-factor-poly-coefs p square-free))
|
|
(setq p (math-sort-terms (math-factor-expr (car p)))))
|
|
(math-accum-factors math-fet-x z (math-factor-protect p))))
|
|
|
|
;; Factor out content.
|
|
((and (not square-free)
|
|
(not (eq 1 (setq t1 (math-mul (math-poly-gcd-list p)
|
|
(if (math-guess-if-neg
|
|
(nth (1- (length p)) p))
|
|
-1 1))))))
|
|
(math-accum-factors t1 1 (math-factor-poly-coefs
|
|
(math-poly-div-list p t1) 'cont)))
|
|
|
|
;; Check if linear in math-fet-x.
|
|
((not (cdr (cdr p)))
|
|
(math-sort-terms
|
|
(math-add (math-factor-protect
|
|
(math-sort-terms
|
|
(math-factor-expr (car p))))
|
|
(math-mul math-fet-x (math-factor-protect
|
|
(math-sort-terms
|
|
(math-factor-expr (nth 1 p))))))))
|
|
|
|
;; If symbolic coefficients, use FactorRules.
|
|
((let ((pp p))
|
|
(while (and pp (or (Math-ratp (car pp))
|
|
(and (eq (car (car pp)) 'mod)
|
|
(Math-integerp (nth 1 (car pp)))
|
|
(Math-integerp (nth 2 (car pp))))))
|
|
(setq pp (cdr pp)))
|
|
pp)
|
|
(let ((res (math-rewrite
|
|
(list 'calcFunc-thecoefs math-fet-x (cons 'vec p))
|
|
'(var FactorRules var-FactorRules))))
|
|
(or (and (eq (car-safe res) 'calcFunc-thefactors)
|
|
(= (length res) 3)
|
|
(math-vectorp (nth 2 res))
|
|
(let ((facs 1)
|
|
(vec (nth 2 res)))
|
|
(while (setq vec (cdr vec))
|
|
(setq facs (math-accum-factors (car vec) 1 facs)))
|
|
facs))
|
|
(math-build-polynomial-expr p math-fet-x))))
|
|
|
|
;; Check if rational coefficients (i.e., not modulo a prime).
|
|
((eq math-poly-modulus 1)
|
|
|
|
;; Check if there are any squared terms, or a content not = 1.
|
|
(if (or (eq square-free t)
|
|
(equal (setq t1 (math-poly-gcd-coefs
|
|
p (setq t2 (math-poly-deriv-coefs p))))
|
|
'(1)))
|
|
|
|
;; We now have a square-free polynomial with integer coefs.
|
|
;; For now, we use a kludgy method that finds linear and
|
|
;; quadratic terms using floating-point root-finding.
|
|
(if (setq t1 (let ((calc-symbolic-mode nil))
|
|
(math-poly-all-roots nil p t)))
|
|
(let ((roots (car t1))
|
|
(csign (if (math-negp (nth (1- (length p)) p)) -1 1))
|
|
(expr 1)
|
|
(scale (nth 2 t1)))
|
|
(while roots
|
|
(let ((coef0 (car (car roots)))
|
|
(coef1 (cdr (car roots))))
|
|
(setq expr (math-accum-factors
|
|
(if coef1
|
|
(let ((den (math-lcm-denoms
|
|
coef0 coef1)))
|
|
(setq scale (math-div scale den))
|
|
(math-add
|
|
(math-add
|
|
(math-mul den (math-pow math-fet-x 2))
|
|
(math-mul (math-mul coef1 den)
|
|
math-fet-x))
|
|
(math-mul coef0 den)))
|
|
(let ((den (math-lcm-denoms coef0)))
|
|
(setq scale (math-div scale den))
|
|
(math-add (math-mul den math-fet-x)
|
|
(math-mul coef0 den))))
|
|
1 expr)
|
|
roots (cdr roots))))
|
|
(setq expr (math-accum-factors
|
|
expr 1
|
|
(math-mul csign
|
|
(math-build-polynomial-expr
|
|
(math-mul-list (nth 1 t1) scale)
|
|
math-fet-x)))))
|
|
(math-build-polynomial-expr p math-fet-x)) ; can't factor it.
|
|
|
|
;; Separate out the squared terms (Knuth exercise 4.6.2-34).
|
|
;; This step also divides out the content of the polynomial.
|
|
(let* ((cabs (math-poly-gcd-list p))
|
|
(csign (if (math-negp (nth (1- (length p)) p)) -1 1))
|
|
(t1s (math-mul-list t1 csign))
|
|
(uu nil)
|
|
(v (car (math-poly-div-coefs p t1s)))
|
|
(w (car (math-poly-div-coefs t2 t1s))))
|
|
(while
|
|
(not (math-poly-zerop
|
|
(setq t2 (math-poly-simplify
|
|
(math-poly-mix
|
|
w 1 (math-poly-deriv-coefs v) -1)))))
|
|
(setq t1 (math-poly-gcd-coefs v t2)
|
|
uu (cons t1 uu)
|
|
v (car (math-poly-div-coefs v t1))
|
|
w (car (math-poly-div-coefs t2 t1))))
|
|
(setq t1 (length uu)
|
|
t2 (math-accum-factors (math-factor-poly-coefs v t)
|
|
(1+ t1) 1))
|
|
(while uu
|
|
(setq t2 (math-accum-factors (math-factor-poly-coefs
|
|
(car uu) t)
|
|
t1 t2)
|
|
t1 (1- t1)
|
|
uu (cdr uu)))
|
|
(math-accum-factors (math-mul cabs csign) 1 t2))))
|
|
|
|
;; Factoring modulo a prime.
|
|
((and (= (length (setq temp (math-poly-gcd-coefs
|
|
p (math-poly-deriv-coefs p))))
|
|
(length p)))
|
|
(setq p (car temp))
|
|
(while (cdr temp)
|
|
(setq temp (nthcdr (nth 2 math-poly-modulus) temp)
|
|
p (cons (car temp) p)))
|
|
(and (setq temp (math-factor-poly-coefs p))
|
|
(math-pow temp (nth 2 math-poly-modulus))))
|
|
(t
|
|
(math-reject-arg nil "*Modulo factorization not yet implemented")))))
|
|
|
|
(defun math-poly-deriv-coefs (p)
|
|
(let ((n 1)
|
|
(dp nil))
|
|
(while (setq p (cdr p))
|
|
(setq dp (cons (math-mul (car p) n) dp)
|
|
n (1+ n)))
|
|
(nreverse dp)))
|
|
|
|
(defun math-factor-contains (x a)
|
|
(if (equal x a)
|
|
1
|
|
(if (memq (car-safe x) '(+ - * / neg))
|
|
(let ((sum 0))
|
|
(while (setq x (cdr x))
|
|
(setq sum (+ sum (math-factor-contains (car x) a))))
|
|
sum)
|
|
(if (and (eq (car-safe x) '^)
|
|
(natnump (nth 2 x)))
|
|
(* (math-factor-contains (nth 1 x) a) (nth 2 x))
|
|
0))))
|
|
|
|
|
|
|
|
|
|
|
|
;;; Merge all quotients and expand/simplify the numerator
|
|
(defun calcFunc-nrat (expr)
|
|
(if (math-any-floats expr)
|
|
(setq expr (calcFunc-pfrac expr)))
|
|
(if (or (math-vectorp expr)
|
|
(assq (car-safe expr) calc-tweak-eqn-table))
|
|
(cons (car expr) (mapcar 'calcFunc-nrat (cdr expr)))
|
|
(let* ((calc-prefer-frac t)
|
|
(res (math-to-ratpoly expr))
|
|
(num (math-simplify (math-sort-terms (calcFunc-expand (car res)))))
|
|
(den (math-simplify (math-sort-terms (calcFunc-expand (cdr res)))))
|
|
(g (math-poly-gcd num den)))
|
|
(or (eq g 1)
|
|
(let ((num2 (math-poly-div num g))
|
|
(den2 (math-poly-div den g)))
|
|
(and (eq (cdr num2) 0) (eq (cdr den2) 0)
|
|
(setq num (car num2) den (car den2)))))
|
|
(math-simplify (math-div num den)))))
|
|
|
|
;;; Returns expressions (num . denom).
|
|
(defun math-to-ratpoly (expr)
|
|
(let ((res (math-to-ratpoly-rec expr)))
|
|
(cons (math-simplify (car res)) (math-simplify (cdr res)))))
|
|
|
|
(defun math-to-ratpoly-rec (expr)
|
|
(cond ((Math-primp expr)
|
|
(cons expr 1))
|
|
((memq (car expr) '(+ -))
|
|
(let ((r1 (math-to-ratpoly-rec (nth 1 expr)))
|
|
(r2 (math-to-ratpoly-rec (nth 2 expr))))
|
|
(if (equal (cdr r1) (cdr r2))
|
|
(cons (list (car expr) (car r1) (car r2)) (cdr r1))
|
|
(if (eq (cdr r1) 1)
|
|
(cons (list (car expr)
|
|
(math-mul (car r1) (cdr r2))
|
|
(car r2))
|
|
(cdr r2))
|
|
(if (eq (cdr r2) 1)
|
|
(cons (list (car expr)
|
|
(car r1)
|
|
(math-mul (car r2) (cdr r1)))
|
|
(cdr r1))
|
|
(let ((g (math-poly-gcd (cdr r1) (cdr r2))))
|
|
(let ((d1 (and (not (eq g 1)) (math-poly-div (cdr r1) g)))
|
|
(d2 (and (not (eq g 1)) (math-poly-div
|
|
(math-mul (car r1) (cdr r2))
|
|
g))))
|
|
(if (and (eq (cdr d1) 0) (eq (cdr d2) 0))
|
|
(cons (list (car expr) (car d2)
|
|
(math-mul (car r2) (car d1)))
|
|
(math-mul (car d1) (cdr r2)))
|
|
(cons (list (car expr)
|
|
(math-mul (car r1) (cdr r2))
|
|
(math-mul (car r2) (cdr r1)))
|
|
(math-mul (cdr r1) (cdr r2)))))))))))
|
|
((eq (car expr) '*)
|
|
(let* ((r1 (math-to-ratpoly-rec (nth 1 expr)))
|
|
(r2 (math-to-ratpoly-rec (nth 2 expr)))
|
|
(g (math-mul (math-poly-gcd (car r1) (cdr r2))
|
|
(math-poly-gcd (cdr r1) (car r2)))))
|
|
(if (eq g 1)
|
|
(cons (math-mul (car r1) (car r2))
|
|
(math-mul (cdr r1) (cdr r2)))
|
|
(cons (math-poly-div-exact (math-mul (car r1) (car r2)) g)
|
|
(math-poly-div-exact (math-mul (cdr r1) (cdr r2)) g)))))
|
|
((eq (car expr) '/)
|
|
(let* ((r1 (math-to-ratpoly-rec (nth 1 expr)))
|
|
(r2 (math-to-ratpoly-rec (nth 2 expr))))
|
|
(if (and (eq (cdr r1) 1) (eq (cdr r2) 1))
|
|
(cons (car r1) (car r2))
|
|
(let ((g (math-mul (math-poly-gcd (car r1) (car r2))
|
|
(math-poly-gcd (cdr r1) (cdr r2)))))
|
|
(if (eq g 1)
|
|
(cons (math-mul (car r1) (cdr r2))
|
|
(math-mul (cdr r1) (car r2)))
|
|
(cons (math-poly-div-exact (math-mul (car r1) (cdr r2)) g)
|
|
(math-poly-div-exact (math-mul (cdr r1) (car r2))
|
|
g)))))))
|
|
((and (eq (car expr) '^) (integerp (nth 2 expr)))
|
|
(let ((r1 (math-to-ratpoly-rec (nth 1 expr))))
|
|
(if (> (nth 2 expr) 0)
|
|
(cons (math-pow (car r1) (nth 2 expr))
|
|
(math-pow (cdr r1) (nth 2 expr)))
|
|
(cons (math-pow (cdr r1) (- (nth 2 expr)))
|
|
(math-pow (car r1) (- (nth 2 expr)))))))
|
|
((eq (car expr) 'neg)
|
|
(let ((r1 (math-to-ratpoly-rec (nth 1 expr))))
|
|
(cons (math-neg (car r1)) (cdr r1))))
|
|
(t (cons expr 1))))
|
|
|
|
|
|
(defun math-ratpoly-p (expr &optional var)
|
|
(cond ((equal expr var) 1)
|
|
((Math-primp expr) 0)
|
|
((memq (car expr) '(+ -))
|
|
(let ((p1 (math-ratpoly-p (nth 1 expr) var))
|
|
p2)
|
|
(and p1 (setq p2 (math-ratpoly-p (nth 2 expr) var))
|
|
(max p1 p2))))
|
|
((eq (car expr) '*)
|
|
(let ((p1 (math-ratpoly-p (nth 1 expr) var))
|
|
p2)
|
|
(and p1 (setq p2 (math-ratpoly-p (nth 2 expr) var))
|
|
(+ p1 p2))))
|
|
((eq (car expr) 'neg)
|
|
(math-ratpoly-p (nth 1 expr) var))
|
|
((eq (car expr) '/)
|
|
(let ((p1 (math-ratpoly-p (nth 1 expr) var))
|
|
p2)
|
|
(and p1 (setq p2 (math-ratpoly-p (nth 2 expr) var))
|
|
(- p1 p2))))
|
|
((and (eq (car expr) '^)
|
|
(integerp (nth 2 expr)))
|
|
(let ((p1 (math-ratpoly-p (nth 1 expr) var)))
|
|
(and p1 (* p1 (nth 2 expr)))))
|
|
((not var) 1)
|
|
((math-poly-depends expr var) nil)
|
|
(t 0)))
|
|
|
|
|
|
(defun calcFunc-apart (expr &optional var)
|
|
(cond ((Math-primp expr) expr)
|
|
((eq (car expr) '+)
|
|
(math-add (calcFunc-apart (nth 1 expr) var)
|
|
(calcFunc-apart (nth 2 expr) var)))
|
|
((eq (car expr) '-)
|
|
(math-sub (calcFunc-apart (nth 1 expr) var)
|
|
(calcFunc-apart (nth 2 expr) var)))
|
|
((and var (not (math-ratpoly-p expr var)))
|
|
(math-reject-arg expr "Expected a rational function"))
|
|
(t
|
|
(let* ((calc-prefer-frac t)
|
|
(rat (math-to-ratpoly expr))
|
|
(num (car rat))
|
|
(den (cdr rat)))
|
|
(or var
|
|
(setq var (math-polynomial-base den)))
|
|
(if (not (math-ratpoly-p expr var))
|
|
(math-reject-arg expr "Expected a rational function")
|
|
(let* ((qr (math-poly-div num den))
|
|
(q (car qr))
|
|
(r (cdr qr)))
|
|
(math-add q (or (and var
|
|
(math-expr-contains den var)
|
|
(math-partial-fractions r den var))
|
|
(math-div r den)))))))))
|
|
|
|
|
|
(defun math-padded-polynomial (expr var deg)
|
|
"Return a polynomial as list of coefficients.
|
|
If EXPR is of the form \"a + bx + cx^2 + ...\" in the variable VAR, return
|
|
the list (a b c ...) with at least DEG elements, else return NIL."
|
|
(let ((p (math-is-polynomial expr var deg)))
|
|
(append p (make-list (- deg (length p)) 0))))
|
|
|
|
(defun math-partial-fractions (r den var)
|
|
"Return R divided by DEN expressed in partial fractions of VAR.
|
|
All whole factors of DEN have already been split off from R.
|
|
If no partial fraction representation can be found, return nil."
|
|
(let* ((fden (calcFunc-factors den var))
|
|
(tdeg (math-polynomial-p den var))
|
|
(fp fden)
|
|
(dlist nil)
|
|
(eqns 0)
|
|
(lz nil)
|
|
(tz (make-list (1- tdeg) 0))
|
|
(calc-matrix-mode 'scalar))
|
|
(and (not (and (= (length fden) 2) (eq (nth 2 (nth 1 fden)) 1)))
|
|
(progn
|
|
(while (setq fp (cdr fp))
|
|
(let ((rpt (nth 2 (car fp)))
|
|
(deg (math-polynomial-p (nth 1 (car fp)) var))
|
|
dnum dvar deg2)
|
|
(while (> rpt 0)
|
|
(setq deg2 deg
|
|
dnum 0)
|
|
(while (> deg2 0)
|
|
(setq dvar (append '(vec) lz '(1) tz)
|
|
lz (cons 0 lz)
|
|
tz (cdr tz)
|
|
deg2 (1- deg2)
|
|
dnum (math-add dnum (math-mul dvar
|
|
(math-pow var deg2)))
|
|
dlist (cons (and (= deg2 (1- deg))
|
|
(math-pow (nth 1 (car fp)) rpt))
|
|
dlist)))
|
|
(let ((fpp fden)
|
|
(mult 1))
|
|
(while (setq fpp (cdr fpp))
|
|
(or (eq fpp fp)
|
|
(setq mult (math-mul mult
|
|
(math-pow (nth 1 (car fpp))
|
|
(nth 2 (car fpp)))))))
|
|
(setq dnum (math-mul dnum mult)))
|
|
(setq eqns (math-add eqns (math-mul dnum
|
|
(math-pow
|
|
(nth 1 (car fp))
|
|
(- (nth 2 (car fp))
|
|
rpt))))
|
|
rpt (1- rpt)))))
|
|
(setq eqns (math-div (cons 'vec (math-padded-polynomial r var tdeg))
|
|
(math-transpose
|
|
(cons 'vec
|
|
(mapcar
|
|
(function
|
|
(lambda (x)
|
|
(cons 'vec (math-padded-polynomial
|
|
x var tdeg))))
|
|
(cdr eqns))))))
|
|
(and (math-vectorp eqns)
|
|
(let ((res 0)
|
|
(num nil))
|
|
(setq eqns (nreverse eqns))
|
|
(while eqns
|
|
(setq num (cons (car eqns) num)
|
|
eqns (cdr eqns))
|
|
(if (car dlist)
|
|
(setq num (math-build-polynomial-expr
|
|
(nreverse num) var)
|
|
res (math-add res (math-div num (car dlist)))
|
|
num nil))
|
|
(setq dlist (cdr dlist)))
|
|
(math-normalize res)))))))
|
|
|
|
|
|
|
|
(defun math-expand-term (expr)
|
|
(cond ((and (eq (car-safe expr) '*)
|
|
(memq (car-safe (nth 1 expr)) '(+ -)))
|
|
(math-add-or-sub (list '* (nth 1 (nth 1 expr)) (nth 2 expr))
|
|
(list '* (nth 2 (nth 1 expr)) (nth 2 expr))
|
|
nil (eq (car (nth 1 expr)) '-)))
|
|
((and (eq (car-safe expr) '*)
|
|
(memq (car-safe (nth 2 expr)) '(+ -)))
|
|
(math-add-or-sub (list '* (nth 1 expr) (nth 1 (nth 2 expr)))
|
|
(list '* (nth 1 expr) (nth 2 (nth 2 expr)))
|
|
nil (eq (car (nth 2 expr)) '-)))
|
|
((and (eq (car-safe expr) '/)
|
|
(memq (car-safe (nth 1 expr)) '(+ -)))
|
|
(math-add-or-sub (list '/ (nth 1 (nth 1 expr)) (nth 2 expr))
|
|
(list '/ (nth 2 (nth 1 expr)) (nth 2 expr))
|
|
nil (eq (car (nth 1 expr)) '-)))
|
|
((and (eq (car-safe expr) '^)
|
|
(memq (car-safe (nth 1 expr)) '(+ -))
|
|
(integerp (nth 2 expr))
|
|
(if (and
|
|
(or (math-known-matrixp (nth 1 (nth 1 expr)))
|
|
(math-known-matrixp (nth 2 (nth 1 expr)))
|
|
(and
|
|
calc-matrix-mode
|
|
(not (eq calc-matrix-mode 'scalar))
|
|
(not (and (math-known-scalarp (nth 1 (nth 1 expr)))
|
|
(math-known-scalarp (nth 2 (nth 1 expr)))))))
|
|
(> (nth 2 expr) 1))
|
|
(if (= (nth 2 expr) 2)
|
|
(math-add-or-sub (list '* (nth 1 (nth 1 expr)) (nth 1 expr))
|
|
(list '* (nth 2 (nth 1 expr)) (nth 1 expr))
|
|
nil (eq (car (nth 1 expr)) '-))
|
|
(math-add-or-sub (list '* (nth 1 (nth 1 expr))
|
|
(list '^ (nth 1 expr)
|
|
(1- (nth 2 expr))))
|
|
(list '* (nth 2 (nth 1 expr))
|
|
(list '^ (nth 1 expr)
|
|
(1- (nth 2 expr))))
|
|
nil (eq (car (nth 1 expr)) '-)))
|
|
(if (> (nth 2 expr) 0)
|
|
(or (and (or (> math-mt-many 500000) (< math-mt-many -500000))
|
|
(math-expand-power (nth 1 expr) (nth 2 expr)
|
|
nil t))
|
|
(list '*
|
|
(nth 1 expr)
|
|
(list '^ (nth 1 expr) (1- (nth 2 expr)))))
|
|
(if (< (nth 2 expr) 0)
|
|
(list '/ 1 (list '^ (nth 1 expr) (- (nth 2 expr)))))))))
|
|
(t expr)))
|
|
|
|
(defun calcFunc-expand (expr &optional many)
|
|
(math-normalize (math-map-tree #'math-expand-term expr many)))
|
|
|
|
(defun math-expand-power (x n &optional var else-nil)
|
|
(or (and (natnump n)
|
|
(memq (car-safe x) '(+ -))
|
|
(let ((terms nil)
|
|
(cterms nil))
|
|
(while (memq (car-safe x) '(+ -))
|
|
(setq terms (cons (if (eq (car x) '-)
|
|
(math-neg (nth 2 x))
|
|
(nth 2 x))
|
|
terms)
|
|
x (nth 1 x)))
|
|
(setq terms (cons x terms))
|
|
(if var
|
|
(let ((p terms))
|
|
(while p
|
|
(or (math-expr-contains (car p) var)
|
|
(setq terms (delq (car p) terms)
|
|
cterms (cons (car p) cterms)))
|
|
(setq p (cdr p)))
|
|
(if cterms
|
|
(setq terms (cons (apply 'calcFunc-add cterms)
|
|
terms)))))
|
|
(if (= (length terms) 2)
|
|
(let ((i 0)
|
|
(accum 0))
|
|
(while (<= i n)
|
|
(setq accum (list '+ accum
|
|
(list '* (calcFunc-choose n i)
|
|
(list '*
|
|
(list '^ (nth 1 terms) i)
|
|
(list '^ (car terms)
|
|
(- n i)))))
|
|
i (1+ i)))
|
|
accum)
|
|
(if (= n 2)
|
|
(let ((accum 0)
|
|
(p1 terms)
|
|
p2)
|
|
(while p1
|
|
(setq accum (list '+ accum
|
|
(list '^ (car p1) 2))
|
|
p2 p1)
|
|
(while (setq p2 (cdr p2))
|
|
(setq accum (list '+ accum
|
|
(list '* 2 (list '*
|
|
(car p1)
|
|
(car p2))))))
|
|
(setq p1 (cdr p1)))
|
|
accum)
|
|
(if (= n 3)
|
|
(let ((accum 0)
|
|
(p1 terms)
|
|
p2 p3)
|
|
(while p1
|
|
(setq accum (list '+ accum (list '^ (car p1) 3))
|
|
p2 p1)
|
|
(while (setq p2 (cdr p2))
|
|
(setq accum (list '+
|
|
(list '+
|
|
accum
|
|
(list '* 3
|
|
(list
|
|
'*
|
|
(list '^ (car p1) 2)
|
|
(car p2))))
|
|
(list '* 3
|
|
(list
|
|
'* (car p1)
|
|
(list '^ (car p2) 2))))
|
|
p3 p2)
|
|
(while (setq p3 (cdr p3))
|
|
(setq accum (list '+ accum
|
|
(list '* 6
|
|
(list '*
|
|
(car p1)
|
|
(list
|
|
'* (car p2)
|
|
(car p3))))))))
|
|
(setq p1 (cdr p1)))
|
|
accum))))))
|
|
(and (not else-nil)
|
|
(list '^ x n))))
|
|
|
|
(defun calcFunc-expandpow (x n)
|
|
(math-normalize (math-expand-power x n)))
|
|
|
|
(provide 'calc-poly)
|
|
|
|
;;; calc-poly.el ends here
|