src/sinint.lisp

   1 ;;; -*-  Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10 -*- ;;;;
   2 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
   3 ;;;     The data in this file contains enhancments.                    ;;;;;
   4 ;;;                                                                    ;;;;;
   5 ;;;  Copyright (c) 1984,1987 by William Schelter,University of Texas   ;;;;;
   6 ;;;     All rights reserved                                            ;;;;;
   7 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
   8 ;;;     (c) Copyright 1982 Massachusetts Institute of Technology         ;;;
   9 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
  10
  11 (in-package :maxima)
  12
  13 (macsyma-module sinint)
  14
  15 (load-macsyma-macros ratmac)
  16
  17 (declare-top (special genvar checkfactors
  18                       exp var $factorflag $logabs $expop $expon
  19                       $keepfloat ratform rootfactor pardenom $algebraic
  20                       wholepart parnumer varlist logptdx switch1))
  21
  22 (defun rootfac (q)
  23   (prog (nthdq nthdq1 simproots ans n-loops)
  24      (setq nthdq (pgcd q (pderivative q var)))
  25      (setq simproots (pquotient q nthdq))
  26      (setq ans (list (pquotient simproots (pgcd nthdq simproots))))
  27      (setq n-loops 0)
  28      amen (cond
  29             ((= n-loops $factor_max_degree)
  30              (return (list q)))
  31             ((or (pcoefp nthdq) (pointergp var (car nthdq)))
  32              (return (reverse ans))))
  33      (setq nthdq1 (pgcd (pderivative nthdq var) nthdq))
  34      (push (pquotient (pgcd nthdq simproots) (pgcd nthdq1 simproots)) ans)
  35      (setq nthdq nthdq1)
  36      (incf n-loops)
  37      (go amen)))
  38
  39 (defun aprog (q)
  40   (setq q (oldcontent q))
  41   (setq rootfactor (rootfac (cadr q)))
  42   (setq rootfactor
  43         (cons (ptimes (car q) (car rootfactor)) (cdr rootfactor)))
  44   (do ((pd (list (car rootfactor)))
  45        (rf (cdr rootfactor) (cdr rf))
  46        (n 2 (1+ n)))
  47       ((null rf) (setq pardenom (reverse pd)))
  48     (push (pexpt (car rf) n) pd))
  49   rootfactor)
  50
  51 (defun cprog (top bottom)
  52   (prog (frpart pardenomc ppdenom thebpg)
  53      (setq frpart (pdivide top bottom))
  54      (setq wholepart (car frpart))
  55      (setq frpart (cadr frpart))
  56      (if (= (length pardenom) 1)
  57          (return (setq parnumer (list frpart))))
  58      (setq pardenomc (cdr pardenom))
  59      (setq ppdenom (list (car pardenom)))
  60      dseq (if (= (length pardenomc) 1) (go ok))
  61      (setq ppdenom (cons (ptimes (car ppdenom) (car pardenomc)) ppdenom))
  62      (setq pardenomc (cdr pardenomc))
  63      (go dseq)
  64      ok   (setq pardenomc (reverse pardenom))
  65      numc (setq thebpg (bprog (car pardenomc) (car ppdenom)))
  66      (setq parnumer
  67            (cons (cdr (ratdivide (ratti frpart (cdr thebpg) t)
  68                                  (car pardenomc)))
  69                  parnumer))
  70      (setq frpart
  71            (cdr (ratdivide (ratti frpart (car thebpg) t)
  72                            (car ppdenom))))
  73      (setq pardenomc (cdr pardenomc))
  74      (setq ppdenom (cdr ppdenom))
  75      (if (null ppdenom)
  76          (return (setq parnumer (cons frpart parnumer))))
  77      (go numc)))
  78
  79 (defun polyint (p) (ratqu (polyint1 (ratnumerator p)) (ratdenominator p)))
  80
  81 (defun polyint1 (p)
  82   (cond ((or (null p) (equal p 0)) (cons 0 1))
  83         ((atom p) (list var 1 p))
  84         ((not (numberp (car p)))
  85          (if (pointergp var (car p)) (list var 1 p) (polyint1 (cdr p))))
  86         (t (ratplus (polyint2 p) (polyint1 (cddr p))))))
  87
  88 (defun polyint2 (p) (cons (list var (1+ (car p)) (cadr p)) (1+ (car p))))
  89
  90 (defun dprog (ratarg)
  91   (prog (klth kx arootf deriv thebpg thetop thebot prod1 prod2 ans)
  92      (setq ans (cons 0 1))
  93      (if (or (pcoefp (cdr ratarg)) (pointergp var (cadr ratarg)))
  94          (return (disrep (polyint ratarg))))
  95      (aprog (ratdenominator ratarg))
  96      (cprog (ratnumerator ratarg) (ratdenominator ratarg))
  97      (setq rootfactor (reverse rootfactor))
  98      (setq parnumer (reverse parnumer))
  99      (setq klth (length rootfactor))
 100      intg (if (= klth 1) (go simp))
 101      (setq arootf (car rootfactor))
 102      (if (zerop (pdegree arootf var)) (go reset))
 103      (setq deriv (pderivative arootf var))
 104      (setq thebpg (bprog arootf deriv))
 105      (setq kx (1- klth))
 106      (setq thetop (car parnumer))
 107      iter (setq prod1 (ratti thetop (car thebpg) t))
 108      (setq prod2 (ratti thetop (cdr thebpg) t))
 109      (setq thebot (pexpt arootf kx))
 110      (setq ans
 111            (ratplus ans (ratqu (ratminus prod2) (ratti kx thebot t))))
 112      (setq thetop
 113            (ratplus prod1
 114                     (ratqu (ratreduce (pderivative (car prod2) var)
 115                                       (cdr prod2))
 116                            kx)))
 117      (setq thetop (cdr (ratdivide thetop thebot)))
 118      (cond ((= kx 1) (setq logptdx (cons (ratqu thetop arootf) logptdx))
 119             (go reset)))
 120      (setq kx (1- kx))
 121      (go iter)
 122      reset(setq rootfactor (cdr rootfactor))
 123      (setq parnumer (cdr parnumer))
 124      (decf klth)
 125      (go intg)
 126      simp (push (ratqu (car parnumer) (car rootfactor)) logptdx)
 127      (if (equal ans 0) (return (disrep (polyint wholepart))))
 128      (setq thetop
 129            (cadr (pdivide (ratnumerator ans) (ratdenominator ans))))
 130      (return (list '(mplus)
 131                    (disrep (polyint wholepart))
 132                    (disrep (ratqu thetop (ratdenominator ans)))))))
 133
 134 (defun logmabs (x)
 135   (list '(%log) (if $logabs (simplify (list '(mabs) x)) x)))
 136
 137 (defun npask (exp)
 138   (cond ((freeof '$%i exp)
 139          (learn `((mnotequal) ,exp 0) t) (asksign exp))
 140         (t '$positive)))
 141
 142 (defvar $integrate_use_rootsof nil "Use the rootsof form for integrals when denominator does not factor")
 143
 144 (defun integrate-use-rootsof (f q variable)
 145   (let ((dummy (make-param))
 146         (qprime (disrep (pderivative q (p-var q))))
 147         (ff (disrep f))
 148         (qq (disrep q)))
 149     ;; This basically does a partial fraction expansion and integrates
 150     ;; the result.  Let r be one (simple) root of the denominator
 151     ;; polynomial q.  Then the partial fraction expansion is
 152     ;;
 153     ;;   f(x)/q(x) = A/(x-r) + similar terms.
 154     ;;
 155     ;; Then
 156     ;;
 157     ;;   f(x) = A*q(x)/(x-r) + others
 158     ;;
 159     ;; Take the limit as x -> r.
 160     ;;
 161     ;;   f(r) = A*limit(q(x)/(x-r),x,r) + others
 162     ;;        = A*at(diff(q(x),r), [x=r])
 163     ;;
 164     ;; Hence, A = f(r)/at(diff(q(x),x),[x=r])
 165     ;;
 166     ;; Then it follows that the integral is
 167     ;;
 168     ;;    A*log(x-r)
 169         ;;
 170         ;; Note that we don't express the polynomial in terms of the
 171         ;; variable of integration, but in our dummy variable instead.
 172         ;; Using the variable of integration results in a wrong answer
 173         ;; when a substitution was done previously, since when the
 174         ;; substitution is finally undone, that modifies the polynomial.
 175     `((%lsum) ((mtimes)
 176                ,(div* (subst dummy variable ff)
 177                       (subst dummy variable qprime))
 178                ((%log) ,(sub* variable dummy)))
 179       ,dummy
 180       (($rootsof) ,(subst dummy variable qq) ,dummy))))
 181
 182 (defun eprog (p)
 183   (prog (p1e p2e a1e a2e a3e discrim repart sign ncc dcc allcc xx deg)
 184      (if (or (equal p 0) (equal (car p) 0)) (return 0))
 185      (setq p1e (ratnumerator p) p2e (ratdenominator p))
 186      (cond ((or switch1
 187                 (and (not (atom p2e))
 188                      (eq (car (setq xx (cadr (oldcontent p2e)))) var)
 189                      (member (setq deg (pdegree xx var)) '(5 6) :test #'equal)
 190                      (zerocoefl xx deg)
 191                      (or (equal deg 5) (not (pminusp (car (last xx)))))))
 192             (go efac)))
 193      (setq a1e (intfactor p2e))
 194      (if (> (length a1e) 1) (go e40))
 195      efac       (setq ncc (oldcontent p1e))
 196      (setq p1e (cadr ncc))
 197      (setq dcc (oldcontent p2e))
 198      (setq p2e (cadr dcc))
 199      (setq allcc (ratqu (car ncc) (car dcc)))
 200      (setq deg (pdegree p2e var))
 201      (setq a1e (pderivative p2e var))
 202      (setq a2e (ratqu (polcoef p1e (pdegree p1e var))
 203                       (polcoef a1e (pdegree a1e var))))
 204      (cond ((equal (ratti a2e a1e t) (cons p1e 1))
 205             (return (list '(mtimes)
 206                           (disrep (ratti allcc a2e t))
 207                           (logmabs (disrep p2e))))))
 208      (cond ((equal deg 1) (go e10))
 209            ((equal deg 2) (go e20))
 210            ((and (equal deg 3) (equal (polcoef p2e 2) 0)
 211                  (equal (polcoef p2e 1) 0))
 212             (return (e3prog p1e p2e allcc)))
 213            ((and (member deg '(4 5 6) :test #'equal) (zerocoefl p2e deg))
 214             (return (enprog p1e p2e allcc deg))))
 215      (cond ((and $integrate_use_rootsof (equal (car (psqfr p2e)) p2e))
 216             (return (list '(mtimes) (disrep allcc)
 217                           (integrate-use-rootsof p1e p2e
 218                                                  (car (last varlist)))))))
 219      (return (list '(mtimes)
 220                    (disrep allcc)
 221                    (list '(%integrate)
 222                          (list '(mquotient) (disrep p1e) (disrep p2e))
 223                          (car (last varlist)))))
 224      e10        (return (list '(mtimes)
 225                               (disrep (ratti allcc
 226                                              (ratqu (polcoef p1e (pdegree p1e var))
 227                                                     (polcoef p2e 1))
 228                                              t))
 229                               (logmabs (disrep p2e))))
 230      e20        (setq discrim
 231                       (ratdifference (cons (pexpt (polcoef p2e 1) 2) 1)
 232                                      (ratti 4 (ratti (polcoef p2e 2) (polcoef p2e 0) t) t)))
 233      (setq a2e (ratti (polcoef p2e (pdegree p2e var)) 2 t))
 234      (setq xx (simplify (disrep discrim)))
 235      (when (equal ($imagpart xx) 0)
 236        (setq sign (npask xx))
 237        (cond ((eq sign '$negative) (go e30))
 238              ((eq sign '$zero) (go zip))))
 239      (setq a1e (ratsqrt discrim))
 240      (setq a3e (logmabs
 241                 (list '(mquotient)
 242                       (list '(mplus)
 243                             (list '(mtimes)
 244                                   (disrep a2e) (disrep (list var 1 1)))
 245                             (disrep (polcoef p2e 1))
 246                             (list '(mminus) a1e))
 247                       (list '(mplus)
 248                             (list '(mtimes)
 249                                   (disrep a2e) (disrep (list var 1 1)))
 250                             (disrep (polcoef p2e 1))
 251                             a1e))))
 252      (cond ((zerop (pdegree p1e var))
 253             (return (list '(mtimes)
 254                           (disrep allcc)
 255                           (list '(mquotient) (disrep (polcoef p1e 0)) a1e)
 256                           a3e))))
 257      (return
 258        (list
 259         '(mplus)
 260         (list '(mtimes)
 261               (disrep (ratti allcc (ratqu (polcoef p1e (pdegree p1e var)) a2e) t))
 262               (logmabs (disrep p2e)))
 263         (list
 264          '(mtimes)
 265          (list
 266           '(mquotient)
 267           (disrep (ratti allcc (ratqu (eprogratd a2e p1e p2e) a2e) t))
 268           a1e)
 269          a3e)))
 270      e30        (setq a1e (ratsqrt (ratminus discrim)))
 271      (setq
 272       repart
 273       (ratqu (cond ((zerop (pdegree p1e var)) (ratti a2e (polcoef p1e 0) t))
 274                    (t (eprogratd a2e p1e p2e)))
 275              (polcoef p2e (pdegree p2e var))))
 276      (setq a3e (cond ((equal 0 (car repart)) 0)
 277                      (t `((mtimes) ((mquotient)
 278                                     ,(disrep (ratti allcc repart t))
 279                                     ,a1e)
 280                           ((%atan)
 281                            ((mquotient)
 282                             ,(disrep (pderivative p2e var))
 283                             ,a1e))))))
 284      (if (zerop (pdegree p1e var)) (return a3e))
 285      (return (list '(mplus)
 286                    (list '(mtimes)
 287                          (disrep (ratti allcc
 288                                         (ratqu (polcoef p1e (pdegree p1e var)) a2e)
 289                                         t))
 290                          (logmabs (disrep p2e)))
 291                    a3e))
 292      zip        (setq
 293                  p2e
 294                  (ratqu
 295                   (psimp
 296                    (p-var p2e)
 297                    (pcoefadd 2
 298                              (pexpt (ptimes 2 (polcoef p2e 2)) 2)
 299                              (pcoefadd 1 (ptimes 4 (ptimes (polcoef p2e 2)
 300                                                            (polcoef p2e 1)))
 301                                        (pcoefadd 0 (pexpt (polcoef p2e 1) 2) ()))))
 302                   (ptimes 4 (polcoef p2e 2))))
 303      (return (fprog (ratti allcc (ratqu p1e p2e) t)))
 304      e40        (setq parnumer nil pardenom a1e switch1 t)
 305      (cprog p1e p2e)
 306      (setq a2e
 307            (mapcar #'(lambda (j k) (eprog (ratqu j k))) parnumer pardenom))
 308      (setq switch1 nil)
 309      (return (cons '(mplus) a2e))))
 310
 311 (defun e3prog (num denom cont)
 312   (prog (a b c d e r ratr var* x)
 313      (setq a (polcoef num 2) b (polcoef num 1) c (polcoef num 0)
 314            d (polcoef denom 3) e (polcoef denom 0))
 315      (setq r (cond ((eq (npask (simplify (disrep (ratqu e d)))) '$negative)
 316                     (simpnrt (disrep (ratqu (ratti -1 e t) d)) 3))
 317                    (t (neg (simpnrt (disrep (ratqu e d)) 3)))))
 318      (setq var* (list var 1 1))
 319      (newvar r)
 320      (orderpointer varlist)
 321      (setq x (ratf r))
 322      (setq ratform (car x) ratr (cdr x))
 323      (return
 324        (simplify
 325         (list '(mplus)
 326               (list '(mtimes)
 327                     (disrep (ratqu (r* cont (r+ (r* a ratr ratr) (r* b ratr) c))
 328                                    (r* ratr ratr 3 d)))
 329                     (logmabs (disrep (ratpl (ratti -1 ratr t) var*))))
 330               (eprog (r* cont (ratqu (r+ (r* (r+ (r* 2 a ratr ratr)
 331                                                  (r* -1 b ratr)
 332                                                  (r* -1 c))
 333                                              var*)
 334                                          (r+ (ratqu (r* -1 a e) d)
 335                                              (r* b ratr ratr)
 336                                              (r* -1 2 c ratr)))
 337                                      (r* 3 d ratr ratr
 338                                          (r+ (ratti var* var* t)
 339                                              (ratti ratr var* t)
 340                                              (ratti ratr ratr t))))))
 341               )))))
 342
 343 (defun eprogratd (a2e p1e p2e)
 344   (ratdifference (ratti a2e (polcoef p1e (1- (pdegree p1e var))) t)
 345                  (ratti (polcoef p2e (1- (pdegree p2e var)))
 346                         (polcoef p1e (pdegree p1e var))
 347                         t)))
 348
 349 (defun enprog (num denom cont deg)
 350   ;; Denominator is (A*VAR^4+B) =
 351   ;;   if B<0 then (SQRT(A)*VAR^2 - SQRT(-B)) (SQRT(A)*VAR^2 + SQRT(-B))
 352   ;;         else
 353   ;;    (SQRT(A)*VAR^2 - SQRT(2)*A^(1/4)*B^(1/4)*VAR + SQRT(B)) *
 354   ;;    (SQRT(A)*VAR^2 + SQRT(2)*A^(1/4)*B^(1/4)*VAR + SQRT(B))
 355   ;; or (A*VAR^5+B) =
 356   ;;    (1/4) * (A^(1/5)*VAR + B^(1/5)) *
 357   ;;    (2*A^(2/5)*VAR^2 + (-SQRT(5)-1)*A^(1/5)*B^(1/5)*VAR + 2*B^(2/5)) *
 358   ;;    (2*A^(2/5)*VAR^2 + (+SQRT(5)-1)*A^(1/5)*B^(1/5)*VAR + 2*B^(2/5))
 359   ;; or (A*VAR^6+B) =
 360   ;;   if B<0 then (SQRT(A)*VAR^3 - SQRT(-B)) (SQRT(A)*VAR^3 + SQRT(-B))
 361   ;;         else
 362   ;;    (A^(1/3)*VAR^2 + B^(1/3)) *
 363   ;;    (A^(1/3)*VAR^2 - SQRT(3)*A^(1/6)*B^(1/6)*VAR + B^(1/3)) *
 364   ;;    (A^(1/3)*VAR^2 + SQRT(3)*A^(1/6)*B^(1/6)*VAR + B^(1/3))
 365   (prog ($expop $expon a b term disvar $algebraic)
 366      (setq $expop 0 $expon 0)
 367      (setq a (simplify (disrep (polcoef denom deg)))
 368            b (simplify (disrep (polcoef denom 0)))
 369            disvar (simplify (get var 'disrep))
 370            num (simplify (disrep num))
 371            cont (simplify (disrep cont)))
 372      (cond ((= deg 4)
 373             (if (eq '$neg ($asksign b))
 374                 (setq denom
 375                       (mul2 (add2 (mul2 (power a '((rat simp) 1 2)) (power disvar 2))
 376                                   (power (mul -1 b) '((rat simp) 1 2)))
 377                             (add2 (mul2 (power a '((rat simp) 1 2)) (power disvar 2))
 378                                   (mul -1 (power (mul -1 b) '((rat simp) 1 2))))))
 379                 (progn
 380                   (setq denom (add2 (mul2 (power a '((rat simp) 1 2)) (power disvar 2))
 381                                     (power b '((rat simp) 1 2)))
 382                         term (muln (list (power 2 '((rat simp) 1 2))
 383                                          (power a '((rat simp) 1 4))
 384                                          (power b '((rat simp) 1 4))
 385                                          disvar)
 386                                    t))
 387                   (setq denom (mul2 (add2 denom term) (sub denom term))))))
 388            ((= deg 5)
 389             (setq term (mul3 (power a '((rat simp) 1 5))
 390                              (power b '((rat simp) 1 5))
 391                              disvar))
 392             (setq denom (add2 (mul3 2 (power a '((rat simp) 2 5))
 393                                     (power disvar 2))
 394                               (sub (mul2 2 (power b '((rat simp) 2 5))) term)))
 395             (setq term (mul2 (power 5 '((rat simp) 1 2)) term))
 396             (setq denom (muln (list '((rat simp) 1 4)
 397                                     (add2 (mul2 (power a '((rat simp) 1 5)) disvar)
 398                                           (power b '((rat simp) 1 5)))
 399                                     (add2 denom term) (sub denom term))
 400                               t)))
 401            (t
 402             (if (eq '$neg ($asksign b))
 403                 (setq denom
 404                       (mul2 (add2 (mul2 (power a '((rat simp) 1 2)) (power disvar 3))
 405                                   (power (mul -1 b) '((rat simp) 1 2)))
 406                             (add2 (mul2 (power a '((rat simp) 1 2)) (power disvar 3))
 407                                   (mul -1 (power (mul -1 b) '((rat simp) 1 2))))))
 408                 (progn
 409                   (setq denom (add2 (mul2 (power a '((rat simp) 1 3)) (power disvar 2))
 410                                     (power b '((rat simp) 1 3)))
 411                         term (muln (list (power 3 '((rat simp) 1 2))
 412                                          (power a '((rat simp) 1 6))
 413                                          (power b '((rat simp) 1 6))
 414                                          disvar)
 415                                    t))
 416                   (setq denom (mul3 denom (add2 denom term) (sub denom term))))
 417                 )))
 418      ;;Needs $ALGEBRAIC NIL so next call to RATF will preserve factorization.
 419      (return (mul2 cont (ratint (div num denom) disvar)))))
 420
 421 (defun zerocoefl (e n)
 422   (do ((i 1 (1+ i))) ((= i n) t)
 423     (if (not (equal (polcoef e i) 0)) (return nil))))
 424
 425 (defun ratsqrt (a) (let (varlist) (simpnrt (disrep a) 2)))
 426
 427 (defun fprog (rat*)
 428   (prog (rootfactor pardenom parnumer logptdx wholepart switch1)
 429      (return (addn (cons (dprog rat*) (mapcar #'eprog logptdx)) nil))))
 430
 431 (defun ratint (exp var)
 432   (prog (genvar checkfactors varlist ratarg ratform $keepfloat)
 433      (setq varlist (list var))
 434      (setq ratarg (ratf exp))
 435      (setq ratform (car ratarg))
 436      (setq var (caadr (ratf var)))
 437      (return (fprog (cdr ratarg)))))
 438
 439 (defun intfactor (l)
 440   (prog ($factorflag a b)
 441      (setq a (oldcontent l) b (everysecond (pfactor (cadr a))))
 442      (return (if (equal (car a) 1) b (cons (car a) b)))))
 443
 444 (defun everysecond (a)
 445   (if a (cons (if (numberp (car a))
 446                   (pexpt (car a) (cadr a))
 447                   (car a))
 448               (everysecond (cddr a)))))