+++ /dev/null
-;; Calculator for GNU Emacs, part II [calc-poly.el]
-;; Copyright (C) 1990, 1991, 1992, 1993 Free Software Foundation, Inc.
-;; Written by Dave Gillespie, daveg@synaptics.com.
-
-;; This file is part of GNU Emacs.
-
-;; GNU Emacs is distributed in the hope that it will be useful,
-;; but WITHOUT ANY WARRANTY. No author or distributor
-;; accepts responsibility to anyone for the consequences of using it
-;; or for whether it serves any particular purpose or works at all,
-;; unless he says so in writing. Refer to the GNU Emacs General Public
-;; License for full details.
-
-;; Everyone is granted permission to copy, modify and redistribute
-;; GNU Emacs, but only under the conditions described in the
-;; GNU Emacs General Public License. A copy of this license is
-;; supposed to have been given to you along with GNU Emacs so you
-;; can know your rights and responsibilities. It should be in a
-;; file named COPYING. Among other things, the copyright notice
-;; and this notice must be preserved on all copies.
-
-
-
-;; This file is autoloaded from calc-ext.el.
-(require 'calc-ext)
-
-(require 'calc-macs)
-
-(defun calc-Need-calc-poly () nil)
-
-
-(defun calcFunc-pcont (expr &optional var)
- (cond ((Math-primp expr)
- (cond ((Math-zerop expr) 1)
- ((Math-messy-integerp expr) (math-trunc expr))
- ((Math-objectp expr) expr)
- ((or (equal expr var) (not var)) 1)
- (t expr)))
- ((eq (car expr) '*)
- (math-mul (calcFunc-pcont (nth 1 expr) var)
- (calcFunc-pcont (nth 2 expr) var)))
- ((eq (car expr) '/)
- (math-div (calcFunc-pcont (nth 1 expr) var)
- (calcFunc-pcont (nth 2 expr) var)))
- ((and (eq (car expr) '^) (Math-natnump (nth 2 expr)))
- (math-pow (calcFunc-pcont (nth 1 expr) var) (nth 2 expr)))
- ((memq (car expr) '(neg polar))
- (calcFunc-pcont (nth 1 expr) var))
- ((consp var)
- (let ((p (math-is-polynomial expr var)))
- (if p
- (let ((lead (nth (1- (length p)) p))
- (cont (math-poly-gcd-list p)))
- (if (math-guess-if-neg lead)
- (math-neg cont)
- cont))
- 1)))
- ((memq (car expr) '(+ - cplx sdev))
- (let ((cont (calcFunc-pcont (nth 1 expr) var)))
- (if (eq cont 1)
- 1
- (let ((c2 (calcFunc-pcont (nth 2 expr) var)))
- (if (and (math-negp cont)
- (if (eq (car expr) '-) (math-posp c2) (math-negp c2)))
- (math-neg (math-poly-gcd cont c2))
- (math-poly-gcd cont c2))))))
- (var expr)
- (t 1))
-)
-
-(defun calcFunc-pprim (expr &optional var)
- (let ((cont (calcFunc-pcont expr var)))
- (if (math-equal-int cont 1)
- expr
- (math-poly-div-exact expr cont var)))
-)
-
-(defun math-div-poly-const (expr c)
- (cond ((memq (car-safe expr) '(+ -))
- (list (car expr)
- (math-div-poly-const (nth 1 expr) c)
- (math-div-poly-const (nth 2 expr) c)))
- (t (math-div expr c)))
-)
-
-(defun calcFunc-pdeg (expr &optional var)
- (if (Math-zerop expr)
- '(neg (var inf var-inf))
- (if var
- (or (math-polynomial-p expr var)
- (math-reject-arg expr "Expected a polynomial"))
- (math-poly-degree expr)))
-)
-
-(defun math-poly-degree (expr)
- (cond ((Math-primp expr)
- (if (eq (car-safe expr) 'var) 1 0))
- ((eq (car expr) 'neg)
- (math-poly-degree (nth 1 expr)))
- ((eq (car expr) '*)
- (+ (math-poly-degree (nth 1 expr))
- (math-poly-degree (nth 2 expr))))
- ((eq (car expr) '/)
- (- (math-poly-degree (nth 1 expr))
- (math-poly-degree (nth 2 expr))))
- ((and (eq (car expr) '^) (natnump (nth 2 expr)))
- (* (math-poly-degree (nth 1 expr)) (nth 2 expr)))
- ((memq (car expr) '(+ -))
- (max (math-poly-degree (nth 1 expr))
- (math-poly-degree (nth 2 expr))))
- (t 1))
-)
-
-(defun calcFunc-plead (expr var)
- (cond ((eq (car-safe expr) '*)
- (math-mul (calcFunc-plead (nth 1 expr) var)
- (calcFunc-plead (nth 2 expr) var)))
- ((eq (car-safe expr) '/)
- (math-div (calcFunc-plead (nth 1 expr) var)
- (calcFunc-plead (nth 2 expr) var)))
- ((and (eq (car-safe expr) '^) (math-natnump (nth 2 expr)))
- (math-pow (calcFunc-plead (nth 1 expr) var) (nth 2 expr)))
- ((Math-primp expr)
- (if (equal expr var)
- 1
- expr))
- (t
- (let ((p (math-is-polynomial expr var)))
- (if (cdr p)
- (nth (1- (length p)) p)
- 1))))
-)
-
-
-
-
-
-;;; Polynomial quotient, remainder, and GCD.
-;;; Originally by Ove Ewerlid (ewerlid@mizar.DoCS.UU.SE).
-;;; Modifications and simplifications by daveg.
-
-(setq math-poly-modulus 1)
-
-;;; Return gcd of two polynomials
-(defun calcFunc-pgcd (pn pd)
- (if (math-any-floats pn)
- (math-reject-arg pn "Coefficients must be rational"))
- (if (math-any-floats pd)
- (math-reject-arg pd "Coefficients must be rational"))
- (let ((calc-prefer-frac t)
- (math-poly-modulus (math-poly-modulus pn pd)))
- (math-poly-gcd pn pd))
-)
-
-;;; Return only quotient to top of stack (nil if zero)
-(defun calcFunc-pdiv (pn pd &optional base)
- (let* ((calc-prefer-frac t)
- (math-poly-modulus (math-poly-modulus pn pd))
- (res (math-poly-div pn pd base)))
- (setq calc-poly-div-remainder (cdr res))
- (car res))
-)
-
-;;; Return only remainder to top of stack
-(defun calcFunc-prem (pn pd &optional base)
- (let ((calc-prefer-frac t)
- (math-poly-modulus (math-poly-modulus pn pd)))
- (cdr (math-poly-div pn pd base)))
-)
-
-(defun calcFunc-pdivrem (pn pd &optional base)
- (let* ((calc-prefer-frac t)
- (math-poly-modulus (math-poly-modulus pn pd))
- (res (math-poly-div pn pd base)))
- (list 'vec (car res) (cdr res)))
-)
-
-(defun calcFunc-pdivide (pn pd &optional base)
- (let* ((calc-prefer-frac t)
- (math-poly-modulus (math-poly-modulus pn pd))
- (res (math-poly-div pn pd base)))
- (math-add (car res) (math-div (cdr res) pd)))
-)
-
-
-;;; Multiply two terms, expanding out products of sums.
-(defun math-mul-thru (lhs rhs)
- (if (memq (car-safe lhs) '(+ -))
- (list (car lhs)
- (math-mul-thru (nth 1 lhs) rhs)
- (math-mul-thru (nth 2 lhs) rhs))
- (if (memq (car-safe rhs) '(+ -))
- (list (car rhs)
- (math-mul-thru lhs (nth 1 rhs))
- (math-mul-thru lhs (nth 2 rhs)))
- (math-mul lhs rhs)))
-)
-
-(defun math-div-thru (num den)
- (if (memq (car-safe num) '(+ -))
- (list (car num)
- (math-div-thru (nth 1 num) den)
- (math-div-thru (nth 2 num) den))
- (math-div num den))
-)
-
-
-;;; Sort the terms of a sum into canonical order.
-(defun math-sort-terms (expr)
- (if (memq (car-safe expr) '(+ -))
- (math-list-to-sum
- (sort (math-sum-to-list expr)
- (function (lambda (a b) (math-beforep (car a) (car b))))))
- expr)
-)
-
-(defun math-list-to-sum (lst)
- (if (cdr lst)
- (list (if (cdr (car lst)) '- '+)
- (math-list-to-sum (cdr lst))
- (car (car lst)))
- (if (cdr (car lst))
- (math-neg (car (car lst)))
- (car (car lst))))
-)
-
-(defun math-sum-to-list (tree &optional neg)
- (cond ((eq (car-safe tree) '+)
- (nconc (math-sum-to-list (nth 1 tree) neg)
- (math-sum-to-list (nth 2 tree) neg)))
- ((eq (car-safe tree) '-)
- (nconc (math-sum-to-list (nth 1 tree) neg)
- (math-sum-to-list (nth 2 tree) (not neg))))
- (t (list (cons tree neg))))
-)
-
-;;; Check if the polynomial coefficients are modulo forms.
-(defun math-poly-modulus (expr &optional expr2)
- (or (math-poly-modulus-rec expr)
- (and expr2 (math-poly-modulus-rec expr2))
- 1)
-)
-
-(defun math-poly-modulus-rec (expr)
- (if (and (eq (car-safe expr) 'mod) (Math-natnump (nth 2 expr)))
- (list 'mod 1 (nth 2 expr))
- (and (memq (car-safe expr) '(+ - * /))
- (or (math-poly-modulus-rec (nth 1 expr))
- (math-poly-modulus-rec (nth 2 expr)))))
-)
-
-
-;;; Divide two polynomials. Return (quotient . remainder).
-(defun math-poly-div (u v &optional math-poly-div-base)
- (if math-poly-div-base
- (math-do-poly-div u v)
- (math-do-poly-div (calcFunc-expand u) (calcFunc-expand v)))
-)
-(setq math-poly-div-base nil)
-
-(defun math-poly-div-exact (u v &optional base)
- (let ((res (math-poly-div u v base)))
- (if (eq (cdr res) 0)
- (car res)
- (math-reject-arg (list 'vec u v) "Argument is not a polynomial")))
-)
-
-(defun math-do-poly-div (u v)
- (cond ((math-constp u)
- (if (math-constp v)
- (cons (math-div u v) 0)
- (cons 0 u)))
- ((math-constp v)
- (cons (if (eq v 1)
- u
- (if (memq (car-safe u) '(+ -))
- (math-add-or-sub (math-poly-div-exact (nth 1 u) v)
- (math-poly-div-exact (nth 2 u) v)
- nil (eq (car u) '-))
- (math-div u v)))
- 0))
- ((Math-equal u v)
- (cons math-poly-modulus 0))
- ((and (math-atomic-factorp u) (math-atomic-factorp v))
- (cons (math-simplify (math-div u v)) 0))
- (t
- (let ((base (or math-poly-div-base
- (math-poly-div-base u v)))
- vp up res)
- (if (or (null base)
- (null (setq vp (math-is-polynomial v base nil 'gen))))
- (cons 0 u)
- (setq up (math-is-polynomial u base nil 'gen)
- res (math-poly-div-coefs up vp))
- (cons (math-build-polynomial-expr (car res) base)
- (math-build-polynomial-expr (cdr res) base))))))
-)
-
-(defun math-poly-div-rec (u v)
- (cond ((math-constp u)
- (math-div u v))
- ((math-constp v)
- (if (eq v 1)
- u
- (if (memq (car-safe u) '(+ -))
- (math-add-or-sub (math-poly-div-rec (nth 1 u) v)
- (math-poly-div-rec (nth 2 u) v)
- nil (eq (car u) '-))
- (math-div u v))))
- ((Math-equal u v) math-poly-modulus)
- ((and (math-atomic-factorp u) (math-atomic-factorp v))
- (math-simplify (math-div u v)))
- (math-poly-div-base
- (math-div u v))
- (t
- (let ((base (math-poly-div-base u v))
- vp up res)
- (if (or (null base)
- (null (setq vp (math-is-polynomial v base nil 'gen))))
- (math-div u v)
- (setq up (math-is-polynomial u base nil 'gen)
- res (math-poly-div-coefs up vp))
- (math-add (math-build-polynomial-expr (car res) base)
- (math-div (math-build-polynomial-expr (cdr res) base)
- v))))))
-)
-
-;;; Divide two polynomials in coefficient-list form. Return (quot . rem).
-(defun math-poly-div-coefs (u v)
- (cond ((null v) (math-reject-arg nil "Division by zero"))
- ((< (length u) (length v)) (cons nil u))
- ((cdr u)
- (let ((q nil)
- (urev (reverse u))
- (vrev (reverse v)))
- (while
- (let ((qk (math-poly-div-rec (math-simplify (car urev))
- (car vrev)))
- (up urev)
- (vp vrev))
- (if (or q (not (math-zerop qk)))
- (setq q (cons qk q)))
- (while (setq up (cdr up) vp (cdr vp))
- (setcar up (math-sub (car up) (math-mul-thru qk (car vp)))))
- (setq urev (cdr urev))
- up))
- (while (and urev (Math-zerop (car urev)))
- (setq urev (cdr urev)))
- (cons q (nreverse (mapcar 'math-simplify urev)))))
- (t
- (cons (list (math-poly-div-rec (car u) (car v)))
- nil)))
-)
-
-;;; Perform a pseudo-division of polynomials. (See Knuth section 4.6.1.)
-;;; This returns only the remainder from the pseudo-division.
-(defun math-poly-pseudo-div (u v)
- (cond ((null v) nil)
- ((< (length u) (length v)) u)
- ((or (cdr u) (cdr v))
- (let ((urev (reverse u))
- (vrev (reverse v))
- up)
- (while
- (let ((vp vrev))
- (setq up urev)
- (while (setq up (cdr up) vp (cdr vp))
- (setcar up (math-sub (math-mul-thru (car vrev) (car up))
- (math-mul-thru (car urev) (car vp)))))
- (setq urev (cdr urev))
- up)
- (while up
- (setcar up (math-mul-thru (car vrev) (car up)))
- (setq up (cdr up))))
- (while (and urev (Math-zerop (car urev)))
- (setq urev (cdr urev)))
- (nreverse (mapcar 'math-simplify urev))))
- (t nil))
-)
-
-;;; Compute the GCD of two multivariate polynomials.
-(defun math-poly-gcd (u v)
- (cond ((Math-equal u v) u)
- ((math-constp u)
- (if (Math-zerop u)
- v
- (calcFunc-gcd u (calcFunc-pcont v))))
- ((math-constp v)
- (if (Math-zerop v)
- v
- (calcFunc-gcd v (calcFunc-pcont u))))
- (t
- (let ((base (math-poly-gcd-base u v)))
- (if base
- (math-simplify
- (calcFunc-expand
- (math-build-polynomial-expr
- (math-poly-gcd-coefs (math-is-polynomial u base nil 'gen)
- (math-is-polynomial v base nil 'gen))
- base)))
- (calcFunc-gcd (calcFunc-pcont u) (calcFunc-pcont u))))))
-)
-
-(defun math-poly-div-list (lst a)
- (if (eq a 1)
- lst
- (if (eq a -1)
- (math-mul-list lst a)
- (mapcar (function (lambda (x) (math-poly-div-exact x a))) lst)))
-)
-
-(defun math-mul-list (lst a)
- (if (eq a 1)
- lst
- (if (eq a -1)
- (mapcar 'math-neg lst)
- (and (not (eq a 0))
- (mapcar (function (lambda (x) (math-mul x a))) lst))))
-)
-
-;;; Run GCD on all elements in a list.
-(defun math-poly-gcd-list (lst)
- (if (or (memq 1 lst) (memq -1 lst))
- (math-poly-gcd-frac-list lst)
- (let ((gcd (car lst)))
- (while (and (setq lst (cdr lst)) (not (eq gcd 1)))
- (or (eq (car lst) 0)
- (setq gcd (math-poly-gcd gcd (car lst)))))
- (if lst (setq lst (math-poly-gcd-frac-list lst)))
- gcd))
-)
-
-(defun math-poly-gcd-frac-list (lst)
- (while (and lst (not (eq (car-safe (car lst)) 'frac)))
- (setq lst (cdr lst)))
- (if lst
- (let ((denom (nth 2 (car lst))))
- (while (setq lst (cdr lst))
- (if (eq (car-safe (car lst)) 'frac)
- (setq denom (calcFunc-lcm denom (nth 2 (car lst))))))
- (list 'frac 1 denom))
- 1)
-)
-
-;;; Compute the GCD of two monovariate polynomial lists.
-;;; Knuth section 4.6.1, algorithm C.
-(defun math-poly-gcd-coefs (u v)
- (let ((d (math-poly-gcd (math-poly-gcd-list u)
- (math-poly-gcd-list v)))
- (g 1) (h 1) (z 0) hh r delta ghd)
- (while (and u v (Math-zerop (car u)) (Math-zerop (car v)))
- (setq u (cdr u) v (cdr v) z (1+ z)))
- (or (eq d 1)
- (setq u (math-poly-div-list u d)
- v (math-poly-div-list v d)))
- (while (progn
- (setq delta (- (length u) (length v)))
- (if (< delta 0)
- (setq r u u v v r delta (- delta)))
- (setq r (math-poly-pseudo-div u v))
- (cdr r))
- (setq u v
- v (math-poly-div-list r (math-mul g (math-pow h delta)))
- 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)
-)
-
-
-;;; Return true if is a factor containing no sums or quotients.
-(defun math-atomic-factorp (expr)
- (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))
-)
-
-;;; 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.
-
-(defun math-poly-div-base (a b)
- (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))))))
-)
-
-;;; Same as above 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)
-
-(defun math-poly-gcd-base (a b)
- (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))))))))
-)
-
-;;; Sort a list of polynomial bases.
-(defun math-sort-poly-base-list (lst)
- (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) ... ).
-;;; Note dynamic scope of mpb-total-base.
-(defun math-total-polynomial-base (expr)
- (let ((mpb-total-base nil))
- (math-polynomial-base expr 'math-polynomial-p1)
- (math-sort-poly-base-list mpb-total-base))
-)
-
-(defun math-polynomial-p1 (subexpr)
- (or (assoc subexpr mpb-total-base)
- (memq (car subexpr) '(+ - * / neg))
- (and (eq (car subexpr) '^) (natnump (nth 2 subexpr)))
- (let* ((math-poly-base-variable subexpr)
- (exponent (math-polynomial-p mpb-top-expr subexpr)))
- (if exponent
- (setq mpb-total-base (cons (list subexpr exponent)
- mpb-total-base)))))
- nil
-)
-
-
-
-
-(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 (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))
- (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 (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 expr x) 1)
- (setq math-factored-vars (cons (list x) math-factored-vars))
- (math-factor-expr-try x)))
-)
-
-(defun math-factor-expr-try (x)
- (if (eq (car-safe expr) '*)
- (let ((res1 (catch 'factor (let ((expr (nth 1 expr)))
- (math-factor-expr-try x))))
- (res2 (catch 'factor (let ((expr (nth 2 expr)))
- (math-factor-expr-try x)))))
- (and (or res1 res2)
- (throw 'factor (math-accum-factors (or res1 (nth 1 expr)) 1
- (or res2 (nth 2 expr))))))
- (let* ((p (math-is-polynomial expr x 30 'gen))
- (math-poly-modulus (math-poly-modulus expr))
- res)
- (and (cdr p)
- (setq res (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) facs))
-)
-
-(defun math-factor-poly-coefs (p &optional square-free) ; uses "x"
- (let (t1 t2)
- (cond ((not (cdr p))
- (or (car p) 0))
-
- ;; Strip off multiples of 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 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 x.
- ((not (cdr (cdr p)))
- (math-add (math-factor-protect
- (math-sort-terms
- (math-factor-expr (car p))))
- (math-mul 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 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 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 kludgey 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)
- (unfac (nth 1 t1))
- (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 x 2))
- (math-mul (math-mul coef1 den) x))
- (math-mul coef0 den)))
- (let ((den (math-lcm-denoms coef0)))
- (setq scale (math-div scale den))
- (math-add (math-mul den 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)
- x)))))
- (math-build-polynomial-expr p 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)))
- ((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))
- (qr (math-poly-div num den))
- (q (car qr))
- (r (cdr qr)))
- (or var
- (setq var (math-polynomial-base den)))
- (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)
- (let ((p (math-is-polynomial expr var deg)))
- (append p (make-list (- deg (length p)) 0)))
-)
-
-(defun math-partial-fractions (r den var)
- (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 (> (nth 2 expr) 0)
- (or (and (or (> mmt-many 500000) (< mmt-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))
-)
-
-
-
projects / packages / blobdiff