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 risch) 14 15(load-macsyma-macros rzmac ratmac) 16 17(declare-top (special parnumer pardenom logptdx wholepart 18 $ratalgdenom expexpflag $logsimp switch1 degree cary 19 $ratfac $logexpand ratform genvar *var var rootfactor 20 expint $keepfloat trigint operator $exponentialize $gcd 21 $logarc changevp klth r s beta gamma b mainvar expflag 22 expstuff liflag intvar switch varlist nogood genvar 23 $erfflag $liflag rischp $factorflag alphar m 24 genpairs hypertrigint *mosesflag *exp y $algebraic 25 implicit-real $%e_to_numlog generate-atan2 26 context rp-polylogp *in-risch-p*)) 27 28(defmvar $liflag t "Controls whether `risch' generates polylogs") 29 30(defmvar $erfflag t "Controls whether `risch' generates `erfs'") 31 32(defvar changevp t "When nil prevents changevar hack") 33 34(defmacro pair (al bl) `(mapcar #'cons ,al ,bl)) 35 36;; internal representation of risch expressions: list with canonical rational 37;; expression (CRE) as first element, standard maxima expressions as remaining 38;; elements. risch expression is sum of CRE and remaining elements. 39(defmacro rischzero () ''((0 . 1) 0)) 40 41(defun rischnoun (exp1 &optional (exp2 exp1 exp2p)) 42 (unless exp2p (setq exp1 (rzero))) 43 `(,exp1 ((%integrate) ,(disrep exp2) ,intvar))) 44 45(defun getrischvar () 46 (do ((vl varlist (cdr vl)) 47 (gl genvar (cdr gl))) 48 ((null (cdr vl)) (car gl)))) 49 50;; test whether CRE p is constant with respect to variable of integration. 51;; requires variables in varlist and genvar 52;; to be ordered as by intsetup, with var of integration ordered before 53;; any other expressions that contain it. 54(defun risch-pconstp (p) 55 (or (pcoefp p) (pointergp mainvar (car p)))) 56 57(defun risch-constp (r) 58 (setq r (ratfix r)) 59 (and (risch-pconstp (car r)) (risch-pconstp (cdr r)))) 60 61;; adds two risch expressions (defined above). 62(defun rischadd (x y) 63 (destructuring-let (((a . b) x) ((c . d) y)) 64 (cons (r+ a c) (append b d)))) 65 66(defmfun $risch (exp var) 67 (let ((*integrator-level* 0)) 68 (declare (special *integrator-level*)) 69 (with-new-context (context) 70 (rischint exp var)))) 71 72(defun spderivative (p var) 73 (cond ((pcoefp p) '(0 . 1)) 74 ((null (cdr p)) '(0 . 1)) 75 ((or (not (atom (car p))) (numberp (car p))) ;P IS A RATFORM 76 (let ((denprime (spderivative (cdr p) var))) 77 (cond ((rzerop denprime) 78 (ratqu (spderivative (car p) var) (cdr p))) 79 (t (ratqu (r- (r* (spderivative (car p) var) 80 (cdr p)) 81 (r* (car p) denprime)) 82 (r* (cdr p) (cdr p))))))) 83 (t (r+ (spderivative1 (car p) 84 (cadr p) 85 (caddr p) 86 var) 87 (spderivative (cons (car p) (cdddr p)) 88 var))))) 89 90(defun spderivative1 (var1 deg coeff var) 91 (cond ((eq var1 var) 92 (r* (ratexpt (cons (list var 1 1) 1) (1- deg)) 93 (pctimes deg coeff))) 94 ((pointergp var var1) '(0 . 1)) 95 ((equal deg 0) (spderivative coeff var)) 96 (t (r+ (r* (ratexpt (cons (list var1 1 1) 1) deg) 97 (spderivative coeff var)) 98 (r* (cond ((equal deg 1) coeff) 99 (t (r* deg 100 coeff 101 (ratexpt (cons (list var1 1 1) 1) 102 (1- deg))))) 103 (get var1 'rischdiff) ))))) 104 105(defun polylogp (exp &optional sub) 106 (and (mqapplyp exp) (eq (subfunname exp) '$li) 107 (or (null sub) (equal sub (car (subfunsubs exp)))))) 108 109(defun rischint (exp intvar &aux ($logarc nil) ($exponentialize nil) 110 ($gcd '$algebraic) ($algebraic t) (implicit-real t) 111 ($float nil) ($numer nil) 112 ;; The risch integrator expects $logexpand T. Otherwise, 113 ;; the integrator hangs for special types of integrals 114 ;; (See bug report ID:3039452) 115 ($logexpand t)) 116 (prog ($%e_to_numlog $logsimp trigint operator y z var ratform liflag 117 mainvar varlist genvar hypertrigint $ratfac $ratalgdenom ) 118 (if (specrepp exp) (setq exp (specdisrep exp))) 119 (if (specrepp intvar) (setq intvar (specdisrep intvar))) 120 (if (mnump intvar) 121 (merror (intl:gettext "risch: attempt to integrate wrt a number: ~:M") intvar)) 122 (if (and (atom intvar) (isinop exp intvar)) (go noun)) 123 (rischform exp) 124 (cond (trigint (return (trigin1 exp intvar))) 125 (hypertrigint (return (hypertrigint1 exp intvar t))) 126 (operator (go noun))) 127 (setq y (intsetup exp intvar)) 128 (if operator (go noun)) 129 (setq ratform (car y)) 130 (setq varlist (caddr ratform)) 131 (setq mainvar (caadr (ratf intvar))) 132 (setq genvar (cadddr ratform)) 133 (unless (some #'algpget varlist) 134 (setq $algebraic nil) 135 (setq $gcd (car *gcdl*))) 136 (setq var (getrischvar)) 137 (setq z (tryrisch (cdr y) mainvar)) 138 (setf (caddr ratform) varlist) 139 (setf (cadddr ratform) genvar) 140 (return (cond ((atom (cdr z)) (disrep (car z))) 141 (t (let (($logsimp t) ($%e_to_numlog t)) 142 (simplify (list* '(mplus) 143 (disrep (car z)) 144 (cdr z))))))) 145 noun (return (list '(%integrate) exp intvar)))) 146 147(defun rischform (l) 148 (cond ((or (atom l) (alike1 intvar l) (freeof intvar l)) nil) 149 ((polylogp l) 150 (if (and (integerp (car (subfunsubs l))) 151 (signp g (car (subfunsubs l)))) 152 (rischform (car (subfunargs l))) 153 (setq operator t))) 154 ((atom (caar l)) 155 (case (caar l) 156 ((%sin %cos %tan %cot %sec %csc) 157 (setq trigint t $exponentialize t) 158 (rischform (cadr l))) 159 ((%asin %acos %atan %acot %asec %acsc) 160 (setq trigint t $logarc t) 161 (rischform (cadr l))) 162 ((%sinh %cosh %tanh %coth %sech %csch) 163 (setq hypertrigint t $exponentialize t) 164 (rischform (cadr l))) 165 ((%asinh %acosh %atanh %acoth %asech %acsch) 166 (setq hypertrigint t $logarc t) 167 (rischform (cadr l))) 168 ((mtimes mplus mexpt rat %erf %log) 169 (mapc #'rischform (cdr l))) 170 (t (setq operator (caar l))))) 171 (t (setq operator (caar l))))) 172 173(defun hypertrigint1 (exp var hyperfunc) 174 (let ((result (if hyperfunc 175 (sinint (resimplify exp) var) 176 (rischint (resimplify exp) var)))) 177 ;; The result can contain solveable integrals. Look for this case. 178 (if (isinop result '%integrate) 179 ;; Found an integral. Evaluate the result again. 180 ;; Set the flag *in-risch-p* to make sure that we do not call 181 ;; rischint again from the integrator. This avoids endless loops. 182 (let ((*in-risch-p* t)) 183 (meval (list '($ev) result '$nouns))) 184 result))) 185 186(defun trigin1 (*exp var) 187 (let ((yyy (hypertrigint1 *exp var nil))) 188 (setq yyy (div ($expand ($num yyy)) 189 ($expand ($denom yyy)))) 190 (let ((rischp var) (rp-polylogp t) $logarc $exponentialize result) 191 (setq result (sratsimp (if (and (freeof '$%i *exp) (freeof '$li yyy)) 192 ($realpart yyy) 193 ($rectform yyy)))) 194 ;; The result can contain solveable integrals. Look for this case. 195 (if (isinop result '%integrate) 196 ;; Found an integral. Evaluate the result again. 197 ;; Set the flag *in-risch-p* to make sure that we do not call 198 ;; rischint again from the integrator. This avoids endless loops. 199 (let ((*in-risch-p* t)) 200 (meval (list '($ev) result '$nouns))) 201 result)))) 202 203(defun tryrisch (exp mainvar) 204 (prog (wholepart rootfactor parnumer pardenom 205 switch1 logptdx expflag expstuff expint y) 206 (setq expstuff '(0 . 1)) 207 (cond ((eq mainvar var) 208 (return (rischfprog exp))) 209 ((eq (get var 'leadop) 210 'mexpt) 211 (setq expflag t))) 212 (setq y (rischlogdprog exp)) 213 (dolist (rat logptdx) 214 (setq y (rischadd (rischlogeprog rat) y))) 215 (if varlist (setq y (rischadd (tryrisch1 expstuff mainvar) y))) 216 (return (if expint (rischadd (rischexppoly expint var) y) 217 y)))) 218 219(defun tryrisch1 (exp mainvar) 220 (let* ((varlist (reverse (cdr (reverse varlist)))) 221 (var (getrischvar))) 222 (tryrisch exp mainvar))) 223 224(defun rischfprog (rat) 225 (let (rootfactor pardenom parnumer logptdx wholepart switch1) 226 (cons (cdr (ratrep* (dprog rat))) 227 (let ((varlist varlist) 228 (genvar (subseq genvar 0 (length varlist)))) 229 (mapcar #'eprog logptdx))))) 230 231(defun rischlogdprog (ratarg) 232 (prog (klth arootf deriv thebpg thetop thebot prod1 prod2 ans) 233 (setq ans '(0 . 1)) 234 (cond ((or (pcoefp (cdr ratarg)) 235 (pointergp var (cadr ratarg))) 236 (return (rischlogpoly ratarg)))) 237 (aprog (ratdenominator ratarg)) 238 (cprog (ratnumerator ratarg) (ratdenominator ratarg)) 239 (do ((rootfactor (reverse rootfactor) (cdr rootfactor)) 240 (parnumer (reverse parnumer) (cdr parnumer)) 241 (klth (length rootfactor) (1- klth))) 242 ((= klth 1)) 243 (setq arootf (car rootfactor)) 244 (cond 245 ((pcoefp arootf)) 246 ((and (eq (get (car arootf) 'leadop) 'mexpt) 247 (null (cdddr arootf))) 248 (setq 249 expint 250 (append 251 (cond ((and (not (atom (car parnumer))) 252 (not (atom (caar parnumer))) 253 (eq (caaar parnumer) (car arootf))) 254 (gennegs arootf (cdaar parnumer) (cdar parnumer))) 255 (t (list 256 (list 'neg (car parnumer) 257 (car arootf) klth (cadr arootf))))) 258 expint))) 259 ((not (zerop (pdegree arootf var))) 260 (setq deriv (spderivative arootf mainvar)) 261 (setq thebpg (bprog arootf (ratnumerator deriv))) 262 (setq thetop (car parnumer)) 263 (do ((kx (1- klth) (1- kx))) ((= kx 0)) 264 (setq prod1 (r* thetop (car thebpg))) 265 (setq prod2 (r* thetop (cdr thebpg) (ratdenominator deriv))) 266 (setq thebot (pexpt arootf kx)) 267 (setq ans (r+ ans (ratqu (r- prod2) (r* kx thebot)))) 268 (setq thetop 269 (r+ prod1 (ratqu (spderivative prod2 mainvar) kx))) 270 (setq thetop (cdr (ratdivide thetop thebot)))) 271 (push (ratqu thetop arootf) logptdx)))) 272 (push (ratqu (car parnumer) (car rootfactor)) logptdx) 273 (cond ((or (pzerop ans) (pzerop (car ans))) 274 (return (rischlogpoly wholepart)))) 275 (setq thetop (cadr (pdivide (ratnumerator ans) 276 (ratdenominator ans)))) 277 (return (rischadd (ncons (ratqu thetop (ratdenominator ans))) 278 (rischlogpoly wholepart))))) 279 280(defun gennegs (denom num numdenom) 281 (cond ((null num) nil) 282 (t (cons (list 'neg (cadr num) 283 (car denom) 284 (- klth (car num)) 285 (r* numdenom (caddr denom) )) 286 (gennegs denom (cddr num) numdenom))))) 287 288(defun rischlogeprog (p) 289 (prog (p1e p2e p2deriv logcoef ncc dcc allcc expcoef my-divisor) 290 (if (or (pzerop p) (pzerop (car p))) (return (rischzero))) 291 (setq p1e (ratnumerator p)) 292 (desetq (dcc p2e) (oldcontent (ratdenominator p))) 293 (cond ((and (not switch1) 294 (cdr (setq pardenom (intfactor p2e)))) 295 (setq parnumer nil) 296 (setq switch1 t) 297 (desetq (ncc p1e) (oldcontent p1e)) 298 (cprog p1e p2e) 299 (setq allcc (ratqu ncc dcc)) 300 (return (do ((pnum parnumer (cdr pnum)) 301 (pden pardenom (cdr pden)) 302 (ans (rischzero))) 303 ((or (null pnum) (null pden)) 304 (setq switch1 nil) ans) 305 (setq ans (rischadd 306 (rischlogeprog 307 (r* allcc (ratqu (car pnum) (car pden)))) 308 ans)))))) 309 (when (and expflag (null (p-red p2e))) 310 (push (cons 'neg p) expint) 311 (return (rischzero))) 312 (if expflag (setq expcoef (r* (p-le p2e) (ratqu (get var 'rischdiff) 313 (make-poly var))))) 314 (setq p1e (ratqu p1e (ptimes dcc (p-lc p2e))) 315 p2e (ratqu p2e (p-lc p2e))) ;MAKE DENOM MONIC 316 (setq p2deriv (spderivative p2e mainvar)) 317 (setq my-divisor (if expflag (r- p2deriv (r* p2e expcoef)) p2deriv)) 318 (when (equal my-divisor '(0 . 1)) 319 ;; (format t "HEY RISCHLOGEPROG, FOUND ZERO DIVISOR; GIVE UP.~%") 320 (return (rischnoun p))) 321 (setq logcoef (ratqu p1e my-divisor)) 322 (when (risch-constp logcoef) 323 (if expflag 324 (setq expstuff (r- expstuff (r* expcoef logcoef)))) 325 (return 326 (list 327 '(0 . 1) 328 (list '(mtimes) 329 (disrep logcoef) 330 (logmabs (disrep p2e)))))) 331 (if (and expflag $liflag changevp) 332 (let* ((newvar (gensym)) 333 (new-int ($changevar 334 `((%integrate) ,(simplify (disrep p)) ,intvar) 335 (sub newvar (get var 'rischexpr)) 336 newvar intvar)) 337 (changevp nil)) ;prevents recursive changevar 338 (if (and (freeof intvar new-int) 339 (freeof '%integrate 340 (setq new-int (rischint (sdiff new-int newvar) 341 newvar)))) 342 (return 343 (list (rzero) 344 (maxima-substitute (get var 'rischexpr) newvar new-int)))))) 345 (return (rischnoun p)))) 346 347 348(defun findint (exp) 349 (cond ((atom exp) nil) 350 ((atom (car exp)) (findint (cdr exp))) 351 ((eq (caaar exp) '%integrate) t) 352 (t (findint (cdr exp))))) 353 354(defun logequiv (fn1 fn2) 355 (freeof intvar ($ratsimp (div* (remabs (leadarg fn1)) 356 (remabs (leadarg fn2)))))) 357 358(defun remabs (exp) 359 (cond ((atom exp) exp) 360 ((eq (caar exp) 'mabs) (cadr exp)) 361 (t exp))) 362 363(declare-top (special vlist lians degree)) 364 365(defun getfnsplit (l) 366 (let (coef fn) 367 (dolist (x l (values (muln coef nil) (muln fn nil))) 368 (if (free x intvar) 369 (push x coef) 370 (push x fn))))) 371 372(defun getfncoeff (a form) 373 (cond ((null a) 0) 374 ((equal (car a) 0) (getfncoeff (cdr a) form)) 375 ((and (listp (car a)) 376 (eq (caaar a) 'mplus) (ratpl (getfncoeff (cdar a) form) 377 (getfncoeff (cdr a) form)))) 378 ((and (listp (car a)) 379 (eq (caaar a) 'mtimes)) 380 (multiple-value-bind (coef newfn) 381 (getfnsplit (cdar a)) 382 ;; (car a) is a mtimes expression. We insert coef and newfn as the 383 ;; new arguments to the mtimes expression. This causes problems if 384 ;; (1) coef is a mtimes expression too and 385 ;; (2) (car a) has already a simp flag 386 ;; We get a nested mtimes expression, which does not simplify. 387 ;; We comment out the following code (DK 09/2009): 388 ;; (setf (cdar a) (list coef newfn)) 389 390 ;; Insert a complete mtimes expression without simpflag. 391 ;; Nested mtimes expressions simplify further. 392 (setf (car a) (list '(mtimes) coef newfn)) 393 394 (setf (cdar a) (list coef newfn)) 395 (cond ((zerop1 coef) (getfncoeff (cdr a) form)) 396 ((and (matanp newfn) (member '$%i varlist :test #'eq)) 397 (let (($logarc t) ($logexpand '$all)) 398 (rplaca a ($expand (resimplify (car a))))) 399 (getfncoeff a form)) 400 ((and (alike1 (leadop newfn) (leadop form)) 401 (or (alike1 (leadarg newfn) (leadarg form)) 402 (and (mlogp newfn) 403 (logequiv form newfn)))) 404 (ratpl (rform coef) 405 (prog2 (rplaca a 0) 406 (getfncoeff (cdr a) form)))) 407 ((do ((vl varlist (cdr vl))) ((null vl)) 408 (and (not (atom (car vl))) 409 (alike1 (leadop (car vl)) (leadop newfn)) 410 (if (mlogp newfn) 411 (logequiv (car vl) newfn) 412 (alike1 (car vl) newfn)) 413 (rplaca (cddar a) (car vl)) 414 (return nil)))) 415 ((let (vlist) (newvar1 (car a)) (null vlist)) 416 (setq cary 417 (ratpl (cdr (ratrep* (car a))) 418 cary)) 419 (rplaca a 0) 420 (getfncoeff (cdr a) form)) 421 ((and liflag 422 (mlogp form) 423 (mlogp newfn)) 424 (push (dilog (cons (car a) form)) lians) 425 (rplaca a 0) 426 (getfncoeff (cdr a) form)) 427 ((and liflag 428 (polylogp form) 429 (mlogp newfn) 430 (logequiv form newfn)) 431 (push (mul* (cadar a) (make-li (1+ (car (subfunsubs form))) 432 (leadarg form))) 433 lians) 434 (rplaca a 0) 435 (getfncoeff (cdr a) form)) 436 (t (setq nogood t) 0)))) 437 (t (rplaca a (list '(mtimes) 1 (car a))) 438 (getfncoeff a form)))) 439 440 441(defun rischlogpoly (exp) 442 (cond ((equal exp '(0 . 1)) (rischzero)) 443 (expflag (push (cons 'poly exp) expint) 444 (rischzero)) 445 ((not (among var exp)) (tryrisch1 exp mainvar)) 446 (t (do ((degree (pdegree (car exp) var) (1- degree)) 447 (p (car exp)) 448 (den (cdr exp)) 449 (lians ()) 450 (sum (rzero)) 451 (cary (rzero)) 452 (y) (z) (ak) (nogood) (lbkpl1)) 453 ((minusp degree) (cons sum (append lians (cdr y)))) 454 (setq ak (r- (ratqu (polcoef p degree) den) 455 (r* (cons (1+ degree) 1) 456 cary 457 (get var 'rischdiff)))) 458 (if (not (pzerop (polcoef p degree))) 459 (setq p (if (pcoefp p) (pzero) (psimp var (p-red p))))) 460 (setq y (tryrisch1 ak mainvar)) 461 (setq cary (car y)) 462 (and (> degree 0) (setq liflag $liflag)) 463 (setq z (getfncoeff (cdr y) (get var 'rischexpr))) 464 (setq liflag nil) 465 (cond ((and (> degree 0) 466 (or nogood (findint (cdr y)))) 467 (return (rischnoun sum (r+ (r* ak 468 (make-poly var degree 1)) 469 (ratqu p den)))))) 470 (setq lbkpl1 (ratqu z (cons (1+ degree) 1))) 471 (setq sum (r+ (r* lbkpl1 (make-poly var (1+ degree) 1)) 472 (r* cary (if (zerop degree) 1 473 (make-poly var degree 1))) 474 sum)))))) 475 476(defun make-li (sub arg) 477 (subfunmake '$li (ncons sub) (ncons arg))) 478 479;;integrates log(ro)^degree*log(rn)' in terms of polylogs 480;;finds constants c,d and integers j,k such that 481;;c*ro^j+d=rn^k If ro and rn are poly's then can assume either j=1 or k=1 482(defun dilog (l) 483 (destructuring-let* ((((nil coef nlog) . olog) l) 484 (narg (remabs (cadr nlog))) 485 (varlist varlist) 486 (genvar genvar) 487 (rn (rform narg)) ;; can add new vars to varlist 488 (ro (rform (cadr olog))) 489 (var (caar ro)) 490 ((j . k) (ratreduce (pdegree (car rn) var) (pdegree (car ro) var))) 491 (idx (gensym)) 492 (rc) (rd)) 493 (cond ((and (= j 1) (> k 1)) 494 (setq rn (ratexpt rn k) 495 coef (div coef k) 496 narg (rdis rn))) 497 ((and (= k 1) (> j 1)) 498 (setq ro (ratexpt ro j) 499 coef (div coef (f* j degree)) 500 olog (mul j olog)))) 501 (desetq (rc . rd) (ratdivide rn ro)) 502 (cond ((and (freeof intvar (rdis rc)) ;; can't use risch-constp because varlist 503 (freeof intvar (rdis rd))) ;; is not set up with vars in correct order. 504 (setq narg ($ratsimp (sub 1 (div narg (rdis rd))))) 505 (mul* coef (power -1 (1+ degree)) 506 `((mfactorial) ,degree) 507 (dosum (mul* (power -1 idx) 508 (div* (power olog idx) 509 `((mfactorial) ,idx)) 510 (make-li (add degree (neg idx) 1) narg)) 511 idx 0 degree t))) 512 (t (setq nogood t) 0)))) 513 514(defun exppolycontrol (flag f a expg n) 515 (let (y l var (varlist varlist) (genvar genvar)) 516 (setq varlist (reverse (cdr (reverse varlist)))) 517 (setq var (getrischvar)) 518 (setq y (get var 'leadop)) 519 (cond ((and (not (pzerop (ratnumerator f))) 520 (risch-constp (setq l (ratqu a f)))) 521 (cond (flag ;; multiply in expg^n - n may be negative 522 (list (r* l (ratexpt (cons (list expg 1 1) 1) n)) 523 0)) 524 (t l))) 525 ((eq y intvar) 526 (rischexpvar nil flag (list f a expg n))) 527 (t (rischexplog (eq y 'mexpt) flag f a 528 (list expg n (get var 'rischarg) 529 var (get var 'rischdiff))))))) 530 531(defun rischexppoly (expint var) 532 (let (y w num denom type (ans (rischzero)) 533 (expdiff (ratqu (get var 'rischdiff) (list var 1 1)))) 534 (do ((expint expint (cdr expint))) 535 ((null expint) ans) 536 (desetq (type . y) (car expint)) 537 (desetq (num . denom) (ratfix y)) 538 (cond ((eq type 'neg) 539 (setq w (exppolycontrol t 540 (r* (- (cadr denom)) 541 expdiff) 542 (ratqu num (caddr denom)) 543 var 544 (- (cadr denom))))) 545 ((or (numberp num) (not (eq (car num) var))) 546 (setq w (tryrisch1 y mainvar))) 547 (t (setq w (rischzero)) 548 (do ((num (cdr num) (cddr num))) ((null num)) 549 (cond ((equal (car num) 0) 550 (setq w (rischadd 551 (tryrisch1 (ratqu (cadr num) denom) mainvar) 552 w))) 553 (t (setq w (rischadd (exppolycontrol 554 t 555 (r* (car num) expdiff) 556 (ratqu (cadr num) denom) 557 var 558 (car num)) 559 w))))))) 560 (setq ans (rischadd w ans))))) 561 562(defun rischexpvar (expexpflag flag l) 563 (prog (lcm y m p alphar beta gamma delta r s 564 tt denom k wl wv i ytemp ttemp yalpha f a expg n yn yd) 565 (desetq (f a expg n) l) 566 (cond ((or (pzerop a) (pzerop (car a))) 567 (return (cond ((null flag) (rzero)) 568 (t (rischzero)))))) 569 (setq denom (ratdenominator f)) 570 (setq p (findpr (cdr (partfrac a mainvar)) 571 (cdr (partfrac f mainvar)))) 572 (setq lcm (plcm (ratdenominator a) p)) 573 (setq y (ratpl (spderivative (cons 1 p) mainvar) 574 (ratqu f p))) 575 (setq lcm (plcm lcm (ratdenominator y))) 576 (setq r (car (ratqu lcm p))) 577 (setq s (car (r* lcm y))) 578 (setq tt (car (r* a lcm))) 579 (setq beta (pdegree r mainvar)) 580 (setq gamma (pdegree s mainvar)) 581 (setq delta (pdegree tt mainvar)) 582 (setq alphar (max (- (1+ delta) beta) 583 (- delta gamma))) 584 (setq m 0) 585 (cond ((equal (1- beta) gamma) 586 (setq y (r* -1 587 (ratqu (polcoef s gamma) 588 (polcoef r beta)))) 589 (and (equal (cdr y) 1) 590 (numberp (car y)) 591 (setq m (car y))))) 592 (setq alphar (max alphar m)) 593 (if (minusp alphar) 594 (return (if flag (cxerfarg (rzero) expg n a) nil))) 595 (cond ((not (and (equal alphar m) (not (zerop m)))) 596 (go down2))) 597 (setq k (+ alphar beta -2)) 598 (setq wl nil) 599 l2 (setq wv (list (cons (polcoef tt k) 1))) 600 (setq i alphar) 601 l1 (setq wv 602 (cons (r+ (r* (cons i 1) 603 (polcoef r (+ k 1 (- i)))) 604 (cons (polcoef s (+ k (- i))) 1)) 605 wv)) 606 (decf i) 607 (cond ((> i -1) (go l1))) 608 (setq wl (cons wv wl)) 609 (decf k) 610 (cond ((> k -1) (go l2))) 611 (setq y (lsa wl)) 612 (if (or (eq y 'singular) (eq y 'inconsistent)) 613 (cond ((null flag) (return nil)) 614 (t (return (cxerfarg (rzero) expg n a))))) 615 (setq k 0) 616 (setq lcm 0) 617 (setq y (cdr y)) 618 l3 (setq lcm 619 (r+ (r* (car y) (pexpt (list mainvar 1 1) k)) 620 lcm)) 621 (incf k) 622 (setq y (cdr y)) 623 (cond ((null y) 624 (return (cond ((null flag) (ratqu lcm p)) 625 (t (list (r* (ratqu lcm p) 626 (cons (list expg n 1) 1)) 627 0)))))) 628 (go l3) 629 down2 (cond ((> (1- beta) gamma) 630 (setq k (+ alphar (1- beta))) 631 (setq denom '(ratti alphar (polcoef r beta) t))) 632 ((< (1- beta) gamma) 633 (setq k (+ alphar gamma)) 634 (setq denom '(polcoef s gamma))) 635 (t (setq k (+ alphar gamma)) 636 (setq denom 637 '(ratpl (ratti alphar (polcoef r beta) t) 638 (polcoef s gamma))))) 639 (setq y 0) 640 loop (setq yn (polcoef (ratnumerator tt) k) 641 yd (r* (ratdenominator tt) ;DENOM MAY BE 0 642 (cond ((zerop alphar) (polcoef s gamma)) 643 (t (eval denom))) )) 644 (cond ((rzerop yd) 645 (cond ((pzerop yn) (setq k (1- k) alphar (1- alphar)) 646 (go loop)) ;need more constraints? 647 (t (cond 648 ((null flag) (return nil)) 649 (t (return (cxerfarg (rzero) expg n a))))))) 650 (t (setq yalpha (ratqu yn yd)))) 651 (setq ytemp (r+ y (r* yalpha 652 (cons (list mainvar alphar 1) 1) ))) 653 (setq ttemp (r- tt (r* yalpha 654 (r+ (r* s (cons (list mainvar alphar 1) 1)) 655 (r* r alphar 656 (list mainvar (1- alphar) 1)))))) 657 (decf k) 658 (decf alphar) 659 (cond ((< alphar 0) 660 (cond 661 ((rzerop ttemp) 662 (cond 663 ((null flag) (return (ratqu ytemp p))) 664 (t (return (list (ratqu (r* ytemp (cons (list expg n 1) 1)) 665 p) 666 0))))) 667 ((null flag) (return nil)) 668 ((and (risch-constp (setq ttemp (ratqu ttemp lcm))) 669 $erfflag 670 (equal (pdegree (car (get expg 'rischarg)) mainvar) 2) 671 (equal (pdegree (cdr (get expg 'rischarg)) mainvar) 0)) 672 (return (list (ratqu (r* ytemp (cons (list expg n 1) 1)) p) 673 (erfarg2 (r* n (get expg 'rischarg)) ttemp)))) 674 (t (return 675 (cxerfarg 676 (ratqu (r* y (cons (list expg n 1) 1)) p) 677 expg 678 n 679 (ratqu tt lcm))))))) 680 (setq y ytemp) 681 (setq tt ttemp) 682 (go loop))) 683 684 685;; *JM should be declared as an array, although it is not created 686;; by this file. -- cwh 687 688(defun lsa (mm) 689 (prog (d *mosesflag m m2) 690 (setq d (length (car mm))) 691 ;; MTOA stands for MATRIX-TO-ARRAY. An array is created and 692 ;; associated functionally with the symbol *JM. The elements 693 ;; of the array are initialized from the matrix MM. 694 (mtoa '*jm* (length mm) d mm) 695 (setq m (tfgeli '*jm* (length mm) d)) 696 (cond ((or (and (null (car m)) (null (cadr m))) 697 (and (car m) 698 (> (length (car m)) (- (length mm) (1- d))))) 699 (return 'singular)) 700 ((cadr m) (return 'inconsistent))) 701 (setq *mosesflag t) 702 (ptorat '*jm* (1- d) d) 703 (setq m2 (xrutout '*jm* (1- d) d nil nil)) 704 (setq m2 (lsafix (cdr m2) (caddr m))) 705 (return m2))) 706 707(defun lsafix (l n) 708 (declare (special *jm*)) 709 (do ((n n (cdr n)) 710 (l l (cdr l))) 711 ((null l)) 712 (setf (aref *jm* 1 (car n)) (car l))) 713 (do ((s (length l) (1- s)) 714 (ans)) 715 ((= s 0) (cons '(list) ans)) 716 (setq ans (cons (aref *jm* 1 s) ans)))) 717 718 719(defun findpr (alist flist &aux (p 1) alphar fterm) 720 (do ((alist alist (cdr alist))) ((null alist)) 721 (setq fterm (findflist (cadar alist) flist)) 722 (if fterm (setq flist (remove y flist :count 1 :test #'eq))) 723 (setq alphar 724 (cond ((null fterm) (caddar alist)) 725 ((equal (caddr fterm) 1) 726 (fpr-dif (car flist) (caddar alist))) 727 (t (max (- (caddar alist) (caddr fterm)) 0)))) 728 (if (not (zerop alphar)) 729 (setq p (ptimes p (pexpt (cadar alist) alphar))))) 730 (do ((flist flist (cdr flist))) 731 ((null flist)) 732 (when (equal (caddar flist) 1) 733 (setq alphar (fpr-dif (car flist) 0)) 734 (setq p (ptimes p (pexpt (cadar flist) alphar))))) 735 p) 736 737(defun fpr-dif (fterm alpha) 738 (destructuring-let* (((num den mult) fterm) 739 (m (spderivative den mainvar)) 740 (n)) 741 (cond ((rzerop m) alpha) 742 (t (setq n (ratqu (cdr (ratdivide num den)) 743 m)) 744 (if (and (equal (cdr n) 1) (numberp (car n))) 745 (max (car n) alpha) 746 alpha))))) 747 748(defun findflist (a llist) 749 (cond ((null llist) nil) 750 ((equal (cadar llist) a) (car llist)) 751 (t (findflist a (cdr llist))))) 752 753 754(defun rischexplog (expexpflag flag f a l) 755 (declare (special var)) 756 (prog (lcm y yy m p alphar beta gamma delta 757 mu r s tt denom ymu rbeta expg n eta logeta logdiff 758 temp cary nogood vector aarray rmu rrmu rarray) 759 (desetq (expg n eta logeta logdiff) l) 760 (cond ((or (pzerop a) (pzerop (car a))) 761 (return (cond ((null flag) (rzero)) 762 (t (rischzero)))))) 763 (setq p (findpr (cdr (partfrac a var)) (cdr (partfrac f var)))) 764 (setq lcm (plcm (ratdenominator a) p)) 765 (setq y (ratpl (spderivative (cons 1 p) mainvar) 766 (ratqu f p))) 767 (setq lcm (plcm lcm (ratdenominator y))) 768 (setq r (car (ratqu lcm p))) 769 (setq s (car (r* lcm y))) 770 (setq tt (car (r* a lcm))) 771 (setq beta (pdegree r var)) 772 (setq gamma (pdegree s var)) 773 (setq delta (pdegree tt var)) 774 (cond (expexpflag (setq mu (max (- delta beta) 775 (- delta gamma))) 776 (go expcase))) 777 (setq mu (max (- (1+ delta) beta) 778 (- (1+ delta) gamma))) 779 (cond ((< beta gamma) (go back)) 780 ((= (1- beta) gamma) (go down1))) 781 (setq y (tryrisch1 (ratqu (r- (r* (polcoef r (1- beta)) 782 (polcoef s gamma)) 783 (r* (polcoef r beta) 784 (polcoef s (1- gamma)))) 785 (r* (polcoef r beta) 786 (polcoef r beta) )) 787 mainvar)) 788 (setq cary (car y)) 789 (setq yy (getfncoeff (cdr y) (get var 'rischexpr))) 790 (cond ((and (not (findint (cdr y))) 791 (not nogood) 792 (not (atom yy)) 793 (equal (cdr yy) 1) 794 (numberp (car yy)) 795 (> (car yy) mu)) 796 (setq mu (car yy)))) 797 (go back) 798 expcase 799 (cond ((not (equal beta gamma)) (go back))) 800 (setq y (tryrisch1 (ratqu (polcoef s gamma) (polcoef r beta)) 801 mainvar)) 802 (cond ((findint (cdr y)) (go back))) 803 (setq yy (ratqu (r* -1 (car y)) eta)) 804 (cond ((and (equal (cdr yy) 1) 805 (numberp (car yy)) 806 (> (car yy) mu)) 807 (setq mu (car yy)))) 808 (go back) 809 down1(setq y (tryrisch1 (ratqu (polcoef s gamma) (polcoef r beta)) 810 mainvar)) 811 (setq cary (car y)) 812 (setq yy (getfncoeff (cdr y) (get var 'rischexpr))) 813 (cond ((and (not (findint (cdr y))) 814 (not nogood) 815 (equal (cdr yy) 1) 816 (numberp (car yy)) 817 (> (- (car yy)) mu)) 818 (setq mu (- (car yy))))) 819 back (if (minusp mu) 820 (return (if flag (cxerfarg (rzero) expg n a) nil))) 821 (cond ((> beta gamma)(go lsacall)) 822 ((= beta gamma) 823 (go recurse))) 824 (setq denom (polcoef s gamma)) 825 (setq y '(0 . 1)) 826 linearloop 827 (setq ymu (ratqu (polcoef (ratnumerator tt) (+ mu gamma)) 828 (r* (ratdenominator tt) denom))) 829 (setq y (r+ y (setq ymu (r* ymu (pexpt (list logeta 1 1) mu) )))) 830 (setq tt (r- tt 831 (r* s ymu) 832 (r* r (spderivative ymu mainvar)))) 833 (decf mu) 834 (cond ((not (< mu 0)) (go linearloop)) 835 ((not flag) (return (if (rzerop tt) (ratqu y p) nil))) 836 ((rzerop tt) 837 (return (cons (ratqu (r* y (cons (list expg n 1) 1)) p) '(0)))) 838 (t (return (cxerfarg (ratqu (r* y (cons (list expg n 1) 1)) p) 839 expg 840 n 841 (ratqu tt lcm))))) 842 recurse 843 (setq rbeta (polcoef r beta)) 844 (setq y '(0 . 1)) 845 recurseloop 846 (setq f (r+ (ratqu (polcoef s gamma) rbeta) 847 (if expexpflag 848 (r* mu (spderivative eta mainvar)) 849 0))) 850 (setq ymu (exppolycontrol nil 851 f 852 (ratqu (polcoef (ratnumerator tt) 853 (+ beta mu)) 854 (r* (ratdenominator tt) rbeta)) 855 expg n)) 856 (when (null ymu) 857 (return (cond ((null flag) nil) 858 (t (return (cxerfarg (ratqu (r* y (cons (list expg n 1) 1)) p) 859 expg n (ratqu tt lcm))))))) 860 (setq y (r+ y (setq ymu (r* ymu (pexpt (list logeta 1 1) mu))))) 861 (setq tt (r- tt 862 (r* s ymu) 863 (r* r (spderivative ymu mainvar)))) 864 (decf mu) 865 (cond 866 ((not (< mu 0)) (go recurseloop)) 867 ((not flag) 868 (return (cond ((rzerop tt) (ratqu y p)) (t nil)))) 869 ((rzerop tt) 870 (return (cons (ratqu (r* y (cons (list expg n 1) 1)) p) '(0)))) 871 (t (return (cxerfarg (ratqu (r* y (cons (list expg n 1) 1)) p) 872 expg 873 n 874 (ratqu tt lcm))))) 875 lsacall 876 (setq rrmu mu) 877 muloop 878 (setq temp (r* (ratexpt (cons (list logeta 1 1) 1) (1- mu)) 879 (r+ (r* s (cons (list logeta 1 1) 1)) 880 (r* mu r logdiff )))) 881 mu1 (setq vector nil) 882 (setq rmu (+ rrmu beta)) 883 rmuloop 884 (setq vector (cons (ratqu (polcoef (ratnumerator temp) rmu) 885 (ratdenominator temp)) vector)) 886 (decf rmu) 887 (unless (< rmu 0) (go rmuloop)) 888 (decf mu) 889 (setq aarray (append aarray (list (reverse vector)))) 890 (cond ((not (< mu 0)) (go muloop)) 891 ((equal mu -2) (go skipmu))) 892 (setq temp tt) 893 (go mu1) 894 skipmu 895 (setq rarray nil) 896 arrayloop 897 (setq vector nil) 898 (setq vector (mapcar 'car aarray)) 899 (setq aarray (mapcar 'cdr aarray)) 900 (setq rarray (append rarray (list vector))) 901 (unless (null (car aarray)) (go arrayloop)) 902 (setq rmu (1+ rrmu)) 903 (setq vector nil) 904 array1loop 905 (setq vector (cons '(0 . 1) vector)) 906 (decf rmu) 907 (unless (< rmu 0) (go array1loop)) 908 (setq aarray nil) 909 array2loop 910 (cond ((equal (car rarray) vector) nil) 911 (t (setq aarray (cons (car rarray) aarray)))) 912 (setq rarray (cdr rarray)) 913 (when rarray (go array2loop)) 914 (setq rarray (reverse aarray)) 915 (setq temp (lsa rarray)) 916 (when (or (eq temp 'singular) (eq temp 'inconsistent)) 917 (return (if (null flag) nil (cxerfarg (rzero) expg n a)))) 918 (setq temp (reverse (cdr temp))) 919 (setq rmu 0) 920 (setq y 0) 921 l3 (setq y (r+ y (r* (car temp) (pexpt (list logeta 1 1) rmu)))) 922 (setq temp (cdr temp)) 923 (incf rmu) 924 (unless (> rmu rrmu) (go l3)) 925 (return (if (null flag) 926 (ratqu y p) 927 (cons (r* (list expg n 1) (ratqu y p)) '(0)))))) 928 929 930(defun erfarg (exparg coef) 931 (prog (num denom erfarg) 932 (setq exparg (r- exparg)) 933 (unless (and (setq num (pnthrootp (ratnumerator exparg) 2)) 934 (setq denom (pnthrootp (ratdenominator exparg) 2))) 935 (return nil)) 936 (setq erfarg (cons num denom)) 937 (if (risch-constp 938 (setq coef (ratqu coef (spderivative erfarg mainvar)))) 939 (return (simplify `((mtimes) ((rat) 1 2) 940 ((mexpt) $%pi ((rat) 1 2)) 941 ,(disrep coef) 942 ((%erf) ,(disrep erfarg)))))))) 943 944(defun erfarg2 (exparg coeff &aux (var mainvar) a b c d) 945 (when (and (= (pdegree (car exparg) var) 2) 946 (eq (caar exparg) var) 947 (risch-pconstp (cdr exparg)) 948 (risch-constp coeff)) 949 (setq a (ratqu (r* -1 (caddar exparg)) 950 (cdr exparg))) 951 (setq b (disrep (ratqu (r* -1 (polcoef (car exparg) 1)) 952 (cdr exparg)))) 953 (setq c (disrep (ratqu (r* (polcoef (car exparg) 0)) 954 (cdr exparg)))) 955 (setq d (ratsqrt a)) 956 (setq a (disrep a)) 957 (simplify `((mtimes) 958 ((mtimes) 959 ((mexpt) $%e ((mplus) ,c 960 ((mquotient) ((mexpt) ,b 2) 961 ((mtimes) 4 ,a)))) 962 ((rat) 1 2) 963 ,(disrep coeff) 964 ((mexpt) ,d -1) 965 ((mexpt) $%pi ((rat) 1 2))) 966 ((%erf) ((mplus) 967 ((mtimes) ,d ,intvar) 968 ((mtimes) ,b ((rat) 1 2) ((mexpt) ,d -1)))))))) 969 970 971(defun cxerfarg (ans expg n numdenom &aux (arg (r* n (get expg 'rischarg))) 972 (fails 0)) 973 (prog (denom erfans num nerf) 974 (desetq (num . denom) numdenom) 975 (unless $erfflag (setq fails num) (go lose)) 976 (if (setq erfans (erfarg arg numdenom)) 977 (return (list ans erfans))) 978 again (when (and (not (pcoefp denom)) 979 (null (p-red denom)) 980 (eq (get (car denom) 'leadop) 'mexpt)) 981 (setq arg (r+ arg (r* (- (p-le denom)) 982 (get (p-var denom) 'rischarg))) 983 denom (p-lc denom)) 984 (go again)) 985 (loop for (coef exparg exppoly) in (explist num arg 1) 986 do (setq coef (ratqu coef denom) 987 nerf (or (erfarg2 exparg coef) (erfarg exparg coef))) 988 (if nerf (push nerf erfans) (setq fails 989 (pplus fails exppoly)))) 990 lose (return 991 (if (pzerop fails) (cons ans erfans) 992 (rischadd (cons ans erfans) 993 (rischnoun (r* (ratexpt (cons (make-poly expg) 1) n) 994 (ratqu fails (cdr numdenom))))))))) 995 996(defun explist (p oarg exps) 997 (cond ((or (pcoefp p) (not (eq 'mexpt (get (p-var p) 'leadop)))) 998 (list (list p oarg (ptimes p exps)))) 999 (t (loop with narg = (get (p-var p) 'rischarg) 1000 for (exp coef) on (p-terms p) by #'cddr 1001 nconc (explist coef 1002 (r+ oarg (r* exp narg)) 1003 (ptimes exps 1004 (make-poly (p-var p) exp 1))))))) 1005 1006 1007(declare-top (special *fnewvarsw)) 1008 1009(defun intsetup (exp *var) 1010 (prog (varlist clist $factorflag dlist genpairs old y z $ratfac $keepfloat 1011 *fnewvarsw) 1012 y (setq exp (radcan1 exp)) 1013 (fnewvar exp) 1014 (setq *fnewvarsw t) 1015 a (setq clist nil) 1016 (setq dlist nil) 1017 (setq z varlist) 1018 up (setq y (pop z)) 1019 (cond ((freeof *var y) (push y clist)) 1020 ((eq y *var) nil) 1021 ((and (mexptp y) 1022 (not (eq (cadr y) '$%e))) 1023 (cond ((not (freeof *var (caddr y))) 1024 (setq dlist `((mexpt simp) 1025 $%e 1026 ,(mul2* (caddr y) 1027 `((%log) ,(cadr y))))) 1028 (setq exp (maxima-substitute dlist y exp)) 1029 (setq varlist nil) (go y)) 1030 ((atom (caddr y)) 1031 (cond ((numberp (caddr y)) (push y dlist)) 1032 (t (setq operator t)(return nil)))) 1033 (t (push y dlist)))) 1034 (t (push y dlist))) 1035 (if z (go up)) 1036 (if (member '$%i clist :test #'eq) (setq clist (cons '$%i (delete '$%i clist :test #'equal)))) 1037 (setq varlist (append clist 1038 (cons *var 1039 (nreverse (sort (append dlist nil) #'intgreat))))) 1040 (orderpointer varlist) 1041 (setq old varlist) 1042 (mapc #'intset1 (cons *var dlist)) 1043 (cond ((alike old varlist) (return (ratrep* exp))) 1044 (t (go a))))) 1045 1046(defun leadop (exp) 1047 (cond ((atom exp) exp) 1048 ((mqapplyp exp) (cadr exp)) 1049 (t (caar exp)))) 1050 1051(defun leadarg (exp) 1052 (cond ((atom exp) 0) 1053 ((and (mexptp exp) (eq (cadr exp) '$%e)) (caddr exp)) 1054 ((mqapplyp exp) (car (subfunargs exp))) 1055 (t (cadr exp)))) 1056 1057(defun intset1 (b) 1058 (let (e c d) 1059 (fnewvar 1060 (setq d (if (mexptp b) ;needed for radicals 1061 `((mtimes simp) 1062 ,b 1063 ,(radcan1 (sdiff (simplify (caddr b)) *var))) 1064 (radcan1 (sdiff (simplify b) *var))))) 1065 (setq d (ratrep* d)) 1066 (setq c (ratrep* (leadarg b))) 1067 (setq e (cdr (assoc b (pair varlist genvar) :test #'equal))) 1068 (putprop e (leadop b) 'leadop) 1069 (putprop e b 'rischexpr) 1070 (putprop e (cdr d) 'rischdiff) 1071 (putprop e (cdr c) 'rischarg))) 1072 1073;; order of expressions for risch. 1074;; expressions containing erf and li last. 1075;; then order by size of expression to guarantee that 1076;; any subexpressions are considered smaller. 1077;; this relation should be transitive, since it is called by sort. 1078(defun intgreat (a b) 1079 (cond ((and (not (atom a)) (not (atom b))) 1080 (cond ((and (not (freeof '%erf a)) (freeof '%erf b)) t) 1081 ((and (not (freeof '$li a)) (freeof '$li b)) t) 1082 ((and (freeof '$li a) (not (freeof '$li b))) nil) 1083 ((and (freeof '%erf a) (not (freeof '%erf b))) nil) 1084 ((> (conssize a) (conssize b)) t) 1085 ((< (conssize a) (conssize b)) nil) 1086 (t (great (resimplify (fixintgreat a)) 1087 (resimplify (fixintgreat b)))))) 1088 (t (great (resimplify (fixintgreat a)) 1089 (resimplify (fixintgreat b)))))) 1090 1091(defun fixintgreat (a) 1092 (subst '/_101x *var a)) 1093 1094(declare-top (unspecial b beta cary context *exp degree gamma 1095 klth liflag m nogood operator 1096 r s switch switch1 *var var y)) 1097