src/polyrz.lisp

   1 ;;; -*-  Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10 -*- ;;;;
   2 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
   3 ;;;     The data in this file contains enhancements.                   ;;;;;
   4 ;;;                                                                    ;;;;;
   5 ;;;  Copyright (c) 1984,1987 by William Schelter,University of Texas   ;;;;;
   6 ;;;     All rights reserved                                            ;;;;;
   7 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
   8 ;;;     (c) Copyright 1980 Massachusetts Institute of Technology         ;;;
   9 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
  10
  11 (in-package :maxima)
  12
  13 (macsyma-module polyrz)
  14
  15 ;;(declare-top (special equations))
  16
  17 (load-macsyma-macros ratmac)
  18
  19 ;;      PACKAGE FOR FINDING REAL ZEROS OF UNIVARIATE POLYNOMIALS
  20 ;;      WITH INTEGER COEFFICIENTS USING STURM SEQUENCES.
  21
  22 (defmfun $realroots (exp &optional (eps $rootsepsilon) &aux exp1)
  23   (setq exp1 (meqhk exp))
  24   (when ($ratp exp1)
  25     (setq exp1 ($ratdisrep exp1)))
  26   (when (or (not (mnump eps)) (mnegp eps) (equal eps 0))
  27     (merror (intl:gettext "realroots: second argument must be a positive number; found: ~M") eps))
  28   (let (($keepfloat nil))
  29     (sturmseq exp exp1 eps)))
  30
  31 (defun unipoly (exp exp1)
  32   (setq exp1 (cadr (ratf exp1)))
  33   (cond ((and (not (atom exp1))
  34               (loop for v in (cdr exp1)
  35                      when (not (atom v))
  36                      do (return nil)
  37                      finally (return t)))
  38          ;;(EVERY #'ATOM (CDR EXP)))
  39          exp1)
  40         (t (merror (intl:gettext "UNIPOLY: argument must be a univariate polynomial with rational coefficients; found: ~M") exp))))
  41
  42 (defun makrat (pt)
  43   (cond ((floatp pt) (maxima-rationalize pt))
  44         ((numberp pt) (cons pt 1))
  45         (($bfloatp pt) (bigfloat2rat pt))
  46         ((atom pt) (merror (intl:gettext "MAKRAT: argument must be a number; found: ~M") pt))
  47         ((equal (caar pt) 'rat) (cons (cadr pt) (caddr pt)))
  48         (t (merror (intl:gettext "MAKRAT: argument must be a number; found: ~M") pt))))
  49
  50 ;;(declare-top (special equations))
  51
  52 (defun sturmseq (exp exp1 eps)
  53   (let (varlist $factorflag $ratprint $ratfac)
  54     (cond ($programmode
  55            (cons '(mlist)
  56                  (multout (findroots (psqfr (pabs (unipoly exp exp1)))
  57                                      (makrat eps)))))
  58           (t
  59            (multiple-value-bind (soln equations)
  60                (solve2 (findroots (psqfr (pabs (unipoly exp exp1)))
  61                                   (makrat eps))
  62                        nil)
  63              (cons '(mlist) equations))))))
  64
  65 ;;(declare-top (unspecial equations))
  66
  67 (defun sturm1 (poly eps &aux b llist)
  68   (setq b (cons (root-bound (cdr poly)) 1))
  69   (setq llist (isolat poly (cons (- (car b)) (cdr b)) b))
  70   (mapcar #'(lambda (int) (refine poly (car int) (cdr int) eps)) llist))
  71
  72 (defun root-bound (p)
  73   (prog (n lcf loglcf coef logb)
  74      (setq n (car p))
  75      (setq lcf (abs (cadr p)))
  76      (setq loglcf (- (integer-length lcf) 2))
  77      (setq logb 1)
  78      loop (cond ((null (setq p (cddr p))) (return (expt 2 logb)))
  79                 ((< (setq coef (abs (cadr p))) lcf) (go loop)))
  80      (setq logb (max logb (1+ (ceil (- (integer-length coef) loglcf 1) (- n (car p))))))
  81      (go loop)))
  82
  83 (defun ceil (a b)
  84   (+ (quotient a b)                     ;CEILING FOR POS A,B
  85      (signum (rem a b))))
  86
  87 (defun sturmapc (fn llist multiplicity)
  88   (cond ((null llist) nil)
  89         (t  (cons (funcall fn (car llist))
  90                   (cons  multiplicity
  91                          (sturmapc fn (cdr llist) multiplicity)))) ))
  92
  93 (defun findroots (l eps)
  94   (cond ((null l) nil)
  95         ((numberp (car l)) (findroots (cddr l) eps))
  96         (t (append (sturmapc 'sturmout (sturm1 (car l) eps)(cadr l))
  97                    (findroots (cddr l) eps) )) ))
  98
  99 (defun sturmout (int)
 100   (list '(mequal simp) (car varlist)
 101         (midout (rhalf (rplus* (car int) (cadr int)))) ))
 102
 103 (defun midout (pt)
 104   (cond ((equal (cdr pt) 1) (car pt))
 105         ($float (fpcofrat1 (car pt) (cdr pt)))
 106         (t (list '(rat simp) (car pt) (cdr pt))) ))
 107
 108 (defun uprimitive (p)
 109   (pquotient p (ucontent p)))           ;PRIMITIVE UNIVAR. POLY
 110
 111 (defun sturm (p)
 112   (prog (p1 p2 seq r)
 113      (setq p1 (uprimitive  p))
 114      (setq p2 (uprimitive (pderivative p1 (car p1))))
 115      (setq seq (list p2 p1))
 116      a    (setq r (prem p1 (pabs p2)))
 117      (cond ((pzerop r) (return (reverse seq))))
 118      (setq p1 p2)
 119      (setq p2 (pminus (uprimitive r)))
 120      (push p2 seq)
 121      (go a) ))
 122
 123 (defun signum(x)
 124   (cond ((zerop x) 0)
 125         ((minusp x) -1)
 126         (t 1)))
 127
 128 ;;      IVAR COUNTS SIGN CHANGES IN A STURM SEQUENCE
 129
 130 (defun ivar (seq pt)
 131   (prog (v s ls)
 132      (setq v 0)
 133      (setq ls 0)
 134      a    (cond ((null seq)(return v)))
 135      (setq s (reval (car seq) pt))
 136      (setq seq (cdr seq))
 137      (cond ((minusp (* s  ls))(setq v (1+ v)))
 138            ((not (zerop ls))(go a)))
 139      (setq ls s)
 140      (go a) ))
 141
 142 (defun ivar2 (seq pt)
 143   (cond ((not (atom pt)) (ivar seq pt))
 144         (t (setq seq (mapcar (function leadterm) seq))
 145            (ivar seq (cons pt 1)) )))
 146
 147 ;;      OUTPUT SIGN(P(R)) , R RATIONAL (A.B)
 148
 149 (defun reval (p r)
 150   (cond ((pcoefp p) (signum p))
 151         ((zerop (car r)) (signum (ptterm (cdr p) 0)))
 152         (t (prog (a b bi v m c)
 153               (setq bi 1)
 154               (setq v 0)
 155               (setq p (cdr p))
 156               (setq m (car p))
 157               (setq a (car r))
 158               (setq b (cdr r))
 159               a    (cond ((equal m (car p)) (setq c (cadr p))
 160                           (setq p (cddr p)))
 161                          (t (setq c 0)))
 162               (cond ((zerop m) (return (signum (+ v (* bi c))))))
 163               (setq v (* a (+ v (* bi c))))
 164               (setq bi (* bi b))
 165               (setq m (1- m))
 166               (go a) ))))
 167
 168 (defun makpoint (pt)
 169   (cond ((eq pt '$inf) 1)
 170         ((eq pt '$minf) -1)
 171         (t (makrat (let (($numer t))
 172                      (meval pt))))))
 173
 174 (defmfun $nroots (exp &optional (l '$minf) (r '$inf))
 175   (let (varlist $keepfloat $ratfac)
 176     (nroots (unipoly exp (meqhk exp)) (makpoint l) (makpoint r))))
 177
 178 (defun nroots (p l r)
 179   (rootaddup (psqfr p) l r))
 180
 181 (defun rootaddup (llist l r)
 182   (cond ((null llist) 0)
 183         ((numberp (car llist)) (rootaddup (cddr llist) l r))
 184         (t (+ (rootaddup (cddr llist) l r)
 185                  (* (cadr llist) (nroot1 (car llist) l r)))) ))
 186
 187 (defun nroot1 (p l r)
 188   (let ((seq (sturm p)))
 189     (- (ivar2 seq l) (ivar2 seq r))))
 190
 191 ;;      RETURNS ROOT IN INTERVAL OF FORM (A,B])
 192
 193 (defun isolat (p l r)
 194   (prog (seq lv rv mid midv tlist islist rts)
 195      (setq seq (sturm p))
 196      (setq lv (ivar seq l))
 197      (setq rv (ivar seq r))
 198      (setq tlist (setq islist nil))
 199      (cond ((equal lv rv) (return nil)))
 200      a  (cond ((> (setq rts (- lv rv)) 1)(go b))
 201               ((equal rts 1)(setq islist (cons (cons l r) islist))))
 202      (cond ((null tlist) (return islist)))
 203      (setq lv (car tlist))
 204      (setq rv (cadr tlist))
 205      (setq l (caddr tlist))
 206      (setq r (cadddr tlist))
 207      (setq tlist (cddddr tlist))
 208      (go a)
 209      b  (setq mid (rhalf (rplus* l r)))
 210      (setq midv (ivar seq mid))
 211      (cond ((not (equal lv midv))
 212             (setq tlist (append (list lv midv l mid) tlist))))
 213      (setq l mid)
 214      (setq lv midv)
 215      (go a)))
 216
 217 (defun refine (p l r eps)
 218   (prog (sr mid smid)
 219      (cond ((zerop (setq sr (reval p r)))
 220             (return (list r r))) )
 221      a  (cond ((rlessp (rdifference* r l) eps)
 222                (return (list l r))) )
 223      (setq mid (rhalf (rplus* l r)))
 224      (setq smid (reval p mid))
 225      (cond ((zerop smid)(return (list mid mid)))
 226            ((equal smid sr)(setq r mid))
 227            (t (setq l mid)) )
 228      (go a)))
 229
 230 (defun rhalf (r) (rreduce (car r) (* 2 (cdr r))))
 231
 232 (defun rreduce (a b)
 233   (let ((g (abs (gcd a b))))
 234     (cons (truncate a g) (truncate b g))) )
 235
 236 (defun rplus* (a b)
 237   (cons (+ (* (car a) (cdr b)) (* (car b) (cdr a)))
 238         (* (cdr a) (cdr b))))
 239
 240 (defun rdifference* (a b)
 241   (rplus* a (cons (- (car b)) (cdr b))) )
 242
 243 (defun rlessp (a b)
 244   (< (* (car a) (cdr b))
 245      (* (car b) (cdr a)) ))
 246
 247
 248 ;;; This next function is to do what SOLVE2 should do in programmode
 249 (defun multout (rootlist)
 250   (progn
 251     (setq rootlist (do ((rtlst)
 252                         (multlst)
 253                         (lunch rootlist))
 254                        ((null lunch) (cons (reverse rtlst)
 255                                            (reverse multlst)))
 256                      (setq rtlst (cons (car lunch) rtlst))
 257                      (setq multlst (cons (cadr lunch) multlst))
 258                      (setq lunch (cddr lunch))))
 259     (setq $multiplicities (cons '(mlist)  (cdr rootlist)))
 260     (car rootlist)))
 261
 262 ;;(declare-top (unspecial equations))