src/combin.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 combin)
  14
  15 (declare-top (special *mfactl *factlist donel nn* dn* *ans* *var*
  16                       $zerobern *n $cflength *a* $prevfib $next_lucas
  17                       *infsumsimp *times *plus sum usum makef
  18                       varlist genvar $sumsplitfact $ratfac $simpsum
  19                       $prederror $listarith
  20                       $ratprint $zeta%pi $bftorat))
  21
  22 (load-macsyma-macros mhayat rzmac ratmac)
  23
  24 ;; minfactorial and factcomb stuff
  25
  26 (defmfun $makefact (e)
  27   (let ((makef t)) (if (atom e) e (simplify (makefact1 e)))))
  28
  29 (defun makefact1 (e)
  30   (cond ((atom e) e)
  31         ((eq (caar e) '%binomial)
  32          (subst (makefact1 (cadr e)) 'x
  33                 (subst (makefact1 (caddr e)) 'y
  34                        '((mtimes) ((mfactorial) x)
  35                          ((mexpt) ((mfactorial) y) -1)
  36                          ((mexpt) ((mfactorial) ((mplus) x ((mtimes) -1 y)))
  37                           -1)))))
  38         ((eq (caar e) '%gamma)
  39          (list '(mfactorial) (list '(mplus) -1 (makefact1 (cadr e)))))
  40         ((eq (caar e) '$beta)
  41          (makefact1 (subst (cadr e) 'x
  42                            (subst (caddr e) 'y
  43                                   '((mtimes) ((%gamma) x)
  44                                     ((%gamma) y)
  45                                     ((mexpt) ((%gamma) ((mplus) x y)) -1))))))
  46         (t (recur-apply #'makefact1 e))))
  47
  48 (defmfun $makegamma (e)
  49   (if (atom e) e (simplify (makegamma1 ($makefact e)))))
  50
  51 (defmfun $minfactorial (e)
  52   (let (*mfactl *factlist)
  53     (if (specrepp e) (setq e (specdisrep e)))
  54     (getfact e)
  55     (mapl #'evfac1 *factlist)
  56     (setq e (evfact e))))
  57
  58 (defun evfact (e)
  59   (cond ((atom e) e)
  60         ((eq (caar e) 'mfactorial)
  61          ;; Replace factorial with simplified expression from *factlist.
  62          (simplifya (cdr (assoc (cadr e) *factlist :test #'equal)) nil))
  63         ((member  (caar e) '(%sum %derivative %integrate %product) :test #'eq)
  64          (cons (list (caar e)) (cons (evfact (cadr e)) (cddr e))))
  65         (t (recur-apply #'evfact e))))
  66
  67 (defun adfactl (e l)
  68   (let (n)
  69     (cond ((null l) (push (list e) *mfactl))
  70           ((numberp (setq n ($ratsimp `((mplus) ,e ((mtimes) -1 ,(caar l))))))
  71            (cond ((plusp n)
  72                   (rplacd (car l) (cons e (cdar l))))
  73                  ((rplaca l (cons e (car l))))))
  74           ((adfactl e (cdr l))))))
  75
  76 (defun getfact (e)
  77   (cond ((atom e) nil)
  78         ((eq (caar e) 'mfactorial)
  79          (and (null (member (cadr e) *factlist :test #'equal))
  80               (prog2
  81                   (push (cadr e) *factlist)
  82                   (adfactl (cadr e) *mfactl))))
  83         ((member (caar e) '(%sum %derivative %integrate %product) :test #'eq)
  84          (getfact (cadr e)))
  85         ((mapc #'getfact (cdr e)))))
  86
  87 (defun evfac1 (e)
  88   (do ((al *mfactl (cdr al)))
  89       ((member (car e) (car al) :test #'equal)
  90        (rplaca e
  91                (cons (car e)
  92                      (list '(mtimes)
  93                            (gfact (car e)
  94                                   ($ratsimp (list '(mplus) (car e)
  95                                                   (list '(mtimes) -1 (caar al)))) 1)
  96                            (list '(mfactorial) (caar al))))))))
  97
  98 (defmfun $factcomb (e)
  99   (let ((varlist varlist ) genvar $ratfac (ratrep (and (not (atom e)) (eq (caar e) 'mrat))))
 100     (and ratrep (setq e (ratdisrep e)))
 101     (setq e (factcomb e)
 102           e (cond ((atom e) e)
 103                   (t (simplify (cons (list (caar e))
 104                                      (mapcar #'factcomb1 (cdr e)))))))
 105     (or $sumsplitfact (setq e ($minfactorial e)))
 106     (if ratrep (ratf e) e)))
 107
 108 (defun factcomb1 (e)
 109   (cond ((free e 'mfactorial) e)
 110         ((member (caar e) '(mplus mtimes mexpt) :test #'eq)
 111          (cons (list (caar e)) (mapcar #'factcomb1 (cdr e))))
 112         (t (setq e (factcomb e))
 113            (if (atom e)
 114                e
 115                (cons (list (caar e)) (mapcar #'factcomb1 (cdr e)))))))
 116
 117 (defun factcomb (e)
 118   (cond ((atom e) e)
 119         ((free e 'mfactorial) e)
 120         ((member (caar e) '(mplus mtimes) :test #'eq)
 121          (factpluscomb (factcombplus e)))
 122         ((eq (caar e) 'mexpt)
 123          (simpexpt (list '(mexpt) (factcomb (cadr e))
 124                          (factcomb (caddr e)))
 125                    1 nil))
 126         ((eq (caar e) 'mrat)
 127          (factrat e))
 128         (t (cons (car e) (mapcar #'factcomb (cdr e))))))
 129
 130 (defun factrat (e)
 131   (let (nn* dn*)
 132     (setq e (factqsnt ($ratdisrep (cons (car e) (cons (cadr e) 1)))
 133                       ($ratdisrep (cons (car e) (cons (cddr e) 1)))))
 134     (numden e)
 135     (div* (factpluscomb nn*) (factpluscomb dn*))))
 136
 137 (defun factqsnt (num den)
 138   (if (equal num 0) 0
 139       (let (nn* dn* (e (factpluscomb (div* den num))))
 140         (numden e)
 141         (factpluscomb (div* dn* nn*)))))
 142
 143 (defun factcombplus (e)
 144   (let (nn* dn*)
 145     (do ((l1 (nplus e) (cdr l1))
 146          (l2))
 147         ((null l1)
 148          (simplus (cons '(mplus)
 149                         (mapcar #'(lambda (q) (factqsnt (car q) (cdr q))) l2))
 150                   1 nil))
 151       (numden (car l1))
 152       (do ((l3 l2 (cdr l3))
 153            (l4))
 154           ((null l3) (setq l2 (nconc l2 (list (cons nn* dn*)))))
 155         (setq l4 (car l3))
 156         (cond ((not (free ($gcd dn* (cdr l4)) 'mfactorial))
 157                (numden (list '(mplus) (div* nn* dn*)
 158                              (div* (car l4) (cdr l4))))
 159                (setq l2 (delete l4 l2 :count 1 :test #'eq))))))))
 160
 161 (defun factpluscomb (e)
 162   (prog (donel fact indl tt)
 163    tag (setq e (factexpand e)
 164              fact (getfactorial e))
 165    (or fact (return e))
 166    (setq indl (mapcar #'(lambda (q) (factplusdep q fact))
 167                       (nplus e))
 168          tt (factpowerselect indl (nplus e) fact)
 169          e (cond ((cdr tt)
 170                   (cons '(mplus) (mapcar #'(lambda (q) (factplus2 q fact))
 171                                          tt)))
 172                  (t (factplus2 (car tt) fact))))
 173    (go tag)))
 174
 175 (defun nplus (e)
 176   (if (eq (caar e) 'mplus)
 177       (cdr e)
 178       (list e)))
 179
 180 (defun factexpand (e)
 181   (cond ((atom e) e)
 182         ((eq (caar e) 'mplus)
 183          (simplus (cons '(mplus) (mapcar #'factexpand (cdr e)))
 184                   1 nil))
 185         ((free e 'mfactorial) e)
 186         (t ($expand e))))
 187
 188 (defun getfactorial (e)
 189   (cond ((atom e) nil)
 190         ((member (caar e) '(mplus mtimes) :test #'eq)
 191          (do ((e (cdr e) (cdr e))
 192               (a))
 193              ((null e) nil)
 194            (setq a (getfactorial (car e)))
 195            (and a (return a))))
 196         ((eq (caar e) 'mexpt)
 197          (getfactorial (cadr e)))
 198         ((eq (caar e) 'mfactorial)
 199          (and (null (memalike (cadr e) donel))
 200               (list '(mfactorial)
 201                     (car (setq donel (cons (cadr e) donel))))))))
 202
 203 (defun factplusdep (e fact)
 204   (cond ((alike1 e fact) 1)
 205         ((atom e) nil)
 206         ((eq (caar e) 'mtimes)
 207          (do ((l (cdr e) (cdr l))
 208               (e) (out))
 209              ((null l) nil)
 210            (setq e (car l))
 211            (and (setq out (factplusdep e fact))
 212                 (return out))))
 213         ((eq (caar e) 'mexpt)
 214          (let ((fto (factplusdep (cadr e) fact)))
 215            (and fto (simptimes (list '(mtimes) fto
 216                                      (caddr e)) 1 t))))
 217         ((eq (caar e) 'mplus)
 218          (same (mapcar #'(lambda (q) (factplusdep q fact))
 219                        (cdr e))))))
 220
 221 (defun same (l)
 222   (do ((ca (car l))
 223        (cd (cdr l) (cdr cd))
 224        (cad))
 225       ((null cd) ca)
 226     (setq cad (car cd))
 227     (or (alike1 ca cad)
 228         (return nil))))
 229
 230 (defun factpowerselect (indl e fact)
 231   (let (l fl)
 232     (do ((i indl (cdr i))
 233          (j e (cdr j))
 234          (expt) (exp))
 235         ((null i) l)
 236       (setq expt (car i)
 237             exp (cond (expt
 238                        (setq exp ($divide (car j) `((mexpt) ,fact ,expt)))
 239                        ;; (car j) need not involve fact^expt since
 240                        ;; fact^expt may be the gcd of the num and denom
 241                        ;; of (car j) and $divide will cancel this out.
 242                        (if (not (equal (cadr exp) 0))
 243                            (cadr exp)
 244                            (progn
 245                              (setq expt '())
 246                              (caddr exp))))
 247                       (t (car j))))
 248       (cond ((null l) (setq l (list (list expt exp))))
 249             ((setq fl (assolike expt l))
 250              (nconc fl (list exp)))
 251             (t (nconc l (list (list expt exp))))))))
 252
 253 (defun factplus2 (l fact)
 254   (let ((expt (car l)))
 255     (cond (expt (factplus0 (cond ((cddr l) (rplaca l '(mplus)))
 256                                  (t (cadr l)))
 257                            expt (cadr fact)))
 258           (t (rplaca l '(mplus))))))
 259
 260 (defun factplus0 (r e fact)
 261   (do ((i -1 (1- i))
 262        (fpn fact (list '(mplus) fact i))
 263        (j -1) (exp) (rfpn) (div))
 264       (nil)
 265     (setq rfpn (simpexpt (list '(mexpt) fpn -1) 1 nil))
 266     (setq div (dypheyed r (simpexpt (list '(mexpt) rfpn e) 1 nil)))
 267     (cond ((or (null (or $sumsplitfact (equal (cadr div) 0)))
 268                (equal (car div) 0))
 269            (return (simplus (cons '(mplus) (mapcar
 270                                             #'(lambda (q)
 271                                                 (incf j)
 272                                                 (list '(mtimes) q (list '(mexpt)
 273                                                                         (list '(mfactorial) (list '(mplus) fpn j)) e)))
 274                                             (factplus1 (cons r exp) e fpn)))
 275                             1 nil)))
 276           (t (setq r (car div))
 277              (setq exp (cons (cadr div) exp))))))
 278
 279 (defun factplus1 (exp e fact)
 280   (do ((l exp (cdr l))
 281        (i 2 (1+ i))
 282        (fpn (list '(mplus) fact 1) (list '(mplus) fact i))
 283        (div))
 284       ((null l) exp)
 285     (setq div (dypheyed (car l) (list '(mexpt) fpn e)))
 286     (and (or $sumsplitfact (equal (cadr div) 0))
 287          (null (equal (car div) 0))
 288          (rplaca l (cadr div))
 289          (rplacd l (cons (cond ((cadr l)
 290                                 (simplus (list '(mplus) (car div) (cadr l))
 291                                          1 nil))
 292                                (t
 293                                 (setq donel
 294                                       (cons (simplus fpn 1 nil) donel))
 295                                 (car div)))
 296                          (cddr l))))))
 297
 298 (defun dypheyed (r f)
 299   (let (r1 p1 p2)
 300     (newvar r)
 301     (setq r1 (ratf f)
 302           p1 (pdegreevector (cadr r1))
 303           p2 (pdegreevector (cddr r1)))
 304     (do ((i p1 (cdr i))
 305          (j p2 (cdr j))
 306          (k (caddar r1) (cdr k)))
 307         ((null k) (kansel r (cadr r1) (cddr r1)))
 308       (cond ((> (car i) (car j))
 309              (return (cdr ($divide r f (car k)))))))))
 310
 311 (defun kansel (r n d)
 312   (let (r1 p1 p2)
 313     (setq r1 (ratf r)
 314           p1 (testdivide (cadr r1) n)
 315           p2 (testdivide (cddr r1) d))
 316     (if (and p1 p2)
 317         (cons (rdis (cons p1 p2)) '(0))
 318         (cons '0 (list r)))))
 319
 320 ;; euler and bernoulli stuff
 321
 322 (defvar *bn* (make-array 17 :adjustable t :element-type 'integer
 323                          :initial-contents '(0 -1 1 -1 5. -691. 7. -3617. 43867. -174611. 854513.
 324                                              -236364091. 8553103. -23749461029. 8615841276005.
 325                                              -7709321041217. 2577687858367.)))
 326
 327 (defvar *bd* (make-array 17 :adjustable t :element-type 'integer
 328                          :initial-contents '(0 30. 42. 30. 66. 2730. 6. 510. 798. 330. 138. 2730.
 329                                              6. 870. 14322.  510. 6.)))
 330
 331 (defvar *eu* (make-array 11 :adjustable t :element-type 'integer
 332                          :initial-contents '(-1 5. -61. 1385. -50521. 2702765. -199360981. 19391512145.
 333                                              -2404879675441. 370371188237525. -69348874393137901.)))
 334
 335 (putprop '*eu* 11 'lim)
 336 (putprop 'bern 16 'lim)
 337
 338 (defmfun $euler (s)
 339   (setq s
 340         (let ((%n 0) $float)
 341           (cond ((or (not (fixnump s)) (< s 0)) (list '($euler) s))
 342                 ((zerop (setq %n s)) 1)
 343                 ($zerobern
 344                  (cond ((oddp %n) 0)
 345                        ((null (> (ash %n -1) (get '*eu* 'lim)))
 346                         (aref *eu* (1- (ash %n -1))))
 347                        ((eq $zerobern '%$/#&)
 348                         (euler %n))
 349                        ((setq *eu* (adjust-array *eu* (1+ (ash %n -1))))
 350                         (euler %n))))
 351                 ((<= %n (get '*eu* 'lim))
 352                  (aref *eu* (1- %n)))
 353                 ((setq *eu* (adjust-array *eu* (1+ %n)))
 354                  (euler (* 2 %n))))))
 355   (simplify s))
 356
 357 (defun nxtbincoef (m nom)
 358   (truncate (* nom (- *a* m)) m))
 359
 360 (defun euler (%a*)
 361   (prog (nom %k e fl $zerobern *a*)
 362      (setq nom 1 %k %a* fl nil e 0 $zerobern '%$/#& *a* (1+ %a*))
 363      a  (cond ((zerop %k)
 364                (setq e (- e))
 365                (setf (aref *eu* (1- (ash %a* -1))) e)
 366                (putprop '*eu* (ash %a* -1) 'lim)
 367                (return e)))
 368      (setq nom (nxtbincoef (1+ (- %a* %k)) nom) %k (1- %k))
 369      (cond ((setq fl (null fl))
 370             (go a)))
 371      (incf e (* nom ($euler %k)))
 372      (go a)))
 373
 374 (defmfun simpeuler (x vestigial z)
 375   (declare (ignore vestigial))
 376   (oneargcheck x)
 377   (let ((u (simpcheck (cadr x) z)))
 378     (if (and (fixnump u) (>= u 0))
 379         ($euler u)
 380         (eqtest (list '($euler) u) x))))
 381
 382 (defmfun $bern (s)
 383   (setq s
 384         (let ((%n 0) $float)
 385           (cond ((or (not (fixnump s)) (< s 0)) (list '($bern) s))
 386                 ((= (setq %n s) 0) 1)
 387                 ((= %n 1) '((rat) -1 2))
 388                 ((= %n 2) '((rat) 1 6))
 389                 ($zerobern
 390                  (cond ((oddp %n) 0)
 391                        ((null (> (setq %n (1- (ash %n -1))) (get 'bern 'lim)))
 392                         (list '(rat) (aref *bn* %n) (aref *bd* %n)))
 393                        ((eq $zerobern '$/#&) (bern  (* 2 (1+ %n))))
 394                        (t
 395                         (setq *bn* (adjust-array *bn* (setq %n (1+ %n))))
 396                         (setq *bd* (adjust-array *bd* %n))
 397                         (bern  (* 2 %n)))))
 398                 ((null (> %n (get 'bern 'lim)))
 399                  (list '(rat) (aref *bn* (- %n 2)) (aref *bd* (- %n 2))))
 400                 (t
 401                  (setq *bn* (adjust-array *bn* (1+ %n)))
 402                  (setq *bd* (adjust-array *bd* (1+ %n)))
 403                  (bern (* 2 (1- %n)))))))
 404   (simplify s))
 405
 406 (defun bern (%a*)
 407   (prog (nom %k bb a b $zerobern l *a*)
 408      (setq %k 0
 409            l (1- %a*)
 410            %a* (1+ %a*)
 411            nom 1
 412            $zerobern '$/#&
 413            a 1
 414            b 1
 415            *a* (1+ %a*))
 416      a  (cond ((= %k l)
 417                (setq bb (*red a (* -1 b %a*)))
 418                (putprop 'bern (setq %a* (1- (ash %a* -1))) 'lim)
 419                (setf (aref *bn* %a*) (cadr bb))
 420                (setf (aref *bd* %a*) (caddr bb))
 421                (return bb)))
 422      (incf %k)
 423      (setq a (+ (* b (setq nom (nxtbincoef %k nom))
 424                           (num1 (setq bb ($bern %k))))
 425                    (* a (denom1 bb))))
 426      (setq b (* b (denom1 bb)))
 427      (setq a (*red a b) b (denom1 a) a (num1 a))
 428      (go a)))
 429
 430 (defmfun simpbern (x vestigial z)
 431   (declare (ignore vestigial))
 432   (oneargcheck x)
 433   (let ((u (simpcheck (cadr x) z)))
 434     (if (and (fixnump u) (not (< u 0)))
 435         ($bern u)
 436         (eqtest (list '($bern) u) x))))
 437
 438 ;;; ----------------------------------------------------------------------------
 439 ;;; Bernoulli polynomials
 440 ;;;
 441 ;;; The following explicit formula is directly implemented:
 442 ;;;
 443 ;;;              n
 444 ;;;             ====
 445 ;;;             \                        n - k
 446 ;;;     B (x) =  >    b  binomial(n, k) x
 447 ;;;      n      /      k
 448 ;;;             ====
 449 ;;;             k = 0
 450 ;;;
 451 ;;; The coeffizients b[k] are the Bernoulli numbers. The algorithm does not
 452 ;;; skip over Beroulli numbers, which are zero. We have to ensure that
 453 ;;; $zerobern is bound to true.
 454 ;;; ----------------------------------------------------------------------------
 455
 456 (defun $bernpoly (x s)
 457   (let ((%n 0) ($zerobern t))
 458     (cond ((not (fixnump s)) (list '($bernpoly) x s))
 459           ((> (setq %n s) -1)
 460            (do ((sum (cons (if (and (= %n 0) (zerop1 x))
 461                                (add 1 x)
 462                                (power x %n))
 463                            nil)
 464                      (cons (mul (binocomp %n %k)
 465                                 ($bern %k)
 466                                 (if (and (= %n %k) (zerop1 x))
 467                                     (add 1 x)
 468                                     (power x (- %n %k))))
 469                            sum))
 470                 (%k 1 (1+ %k)))
 471                ((> %k %n) (addn sum t))))
 472           (t (list '($bernpoly) x %n)))))
 473
 474 ;;; ----------------------------------------------------------------------------
 475 ;;; Euler polynomials
 476 ;;;
 477 ;;; The following explicit formula is directly implemented:
 478 ;;;
 479 ;;;              n                           1 n - k
 480 ;;;             ====  E  binomial(n, k) (x - -)
 481 ;;;             \      k                     2
 482 ;;;     E (x) =  >    ------------------------------
 483 ;;;      n      /                    k
 484 ;;;             ====                2
 485 ;;;             k = 0
 486 ;;;
 487 ;;; The coeffizients E[k] are the Euler numbers.
 488 ;;; ----------------------------------------------------------------------------
 489
 490 (defun $eulerpoly (x s)
 491   (let ((n 0) ($zerobern t) (y 0))
 492     (cond ((not (fixnump s)) (list '($eulerpoly) x s))
 493           ((> (setq n s) -1)
 494            (do ((sum (cons (if (and (zerop1 (setq y (sub x (div 1 2))))
 495                                     (= n 0))
 496                                (add 1 y)
 497                                (power y n))
 498                            nil)
 499                      (cons (mul (binocomp n k)
 500                                 ($euler k)
 501                                 (power 2 (mul -1 k))
 502                                 (if (and (zerop1 (setq y (sub x (div 1 2))))
 503                                          (= n k))
 504                                     (add 1 y)
 505                                     (power y (- n k))))
 506                            sum))
 507                 (k 1 (1+ k)))
 508                ((> k n) ($expand (addn sum t)))))
 509           (t (list '($eulerpoly) x n)))))
 510
 511 ;; zeta and fibonacci stuff
 512
 513 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
 514 ;;;
 515 ;;; Implementation of the Riemann Zeta function as a simplifying function
 516 ;;;
 517 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
 518
 519 (defun $zeta (z)
 520   (simplify (list '(%zeta) z)))
 521
 522 ;;; Set properties to give full support to the parser and display
 523
 524 (defprop $zeta %zeta alias)
 525 (defprop $zeta %zeta verb)
 526
 527 (defprop %zeta $zeta reversealias)
 528 (defprop %zeta $zeta noun)
 529
 530 ;;; The Riemann Zeta function is a simplifying function
 531
 532 (defprop %zeta simp-zeta operators)
 533
 534 ;;; The Riemann Zeta function has mirror symmetry
 535
 536 (defprop %zeta t commutes-with-conjugate)
 537
 538 ;;; The Riemann Zeta function distributes over lists, matrices, and equations
 539
 540 (defprop %zeta (mlist $matrix mequal) distribute_over)
 541
 542 ;;; We support a simplim%function. The function is looked up in simplimit and
 543 ;;; handles specific values of the function.
 544
 545 (defprop %zeta simplim%zeta simplim%function)
 546
 547 (defun simplim%zeta (expr var val)
 548   ;; Look for the limit of the argument
 549   (let* ((arg (limit (cadr expr) var val 'think))
 550          (dir (limit (add (cadr expr) (neg arg)) var val 'think)))
 551   (cond
 552     ;; Handle an argument 1 at this place
 553     ((onep1 arg)
 554      (cond ((eq dir '$zeroa)
 555             '$inf)
 556            ((eq dir '$zerob)
 557             '$minf)
 558            (t '$infinity)))
 559     (t
 560      ;; All other cases are handled by the simplifier of the function.
 561      (simplify (list '(%zeta) arg))))))
 562
 563 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
 564
 565 (defun simp-zeta (expr z simpflag)
 566   (oneargcheck expr)
 567   (setq z (simpcheck (cadr expr) simpflag))
 568   (cond
 569
 570     ;; Check for special values
 571     ((eq z '$inf) 1)
 572     ((alike1 z '((mtimes) -1 $minf)) 1)
 573     ((zerop1 z)
 574      (cond (($bfloatp z) ($bfloat '((rat) -1 2)))
 575            ((floatp z) -0.5)
 576            (t '((rat simp) -1 2))))
 577     ((onep1 z)
 578      (simp-domain-error (intl:gettext "zeta: zeta(~:M) is undefined.") z))
 579
 580     ;; Check for numerical evaluation
 581     ((or (bigfloat-numerical-eval-p z)
 582          (complex-bigfloat-numerical-eval-p z)
 583          (float-numerical-eval-p z)
 584          (complex-float-numerical-eval-p z))
 585      (to (float-zeta z)))
 586     ;; Check for transformations and argument simplifications
 587     ((integerp z)
 588      (cond
 589        ((oddp z)
 590         (cond ((> z 1)
 591                (eqtest (list '(%zeta) z) expr))
 592               ((setq z (sub 1 z))
 593                (mul -1 (div ($bern z) z)))))
 594        ((minusp z) 0)
 595        ((not $zeta%pi) (eqtest (list '(%zeta) z) expr))
 596        (t (let ($numer $float)
 597             (mul (power '$%pi z)
 598                  (mul (div (power 2 (1- z))
 599                            (take '(mfactorial) z))
 600                       (take '(mabs) ($bern z))))))))
 601     (t
 602      (eqtest (list '(%zeta) z) expr))))
 603
 604 ;; See http://numbers.computation.free.fr/Constants/constants.html
 605 ;; and, in particular,
 606 ;; http://numbers.computation.free.fr/Constants/Miscellaneous/zetaevaluations.pdf.
 607 ;; We use the algorithm from Proposition 2:
 608 ;;
 609 ;; zeta(s) = 1/(1-2^(1-s)) *
 610 ;;            (sum((-1)^(k-1)/k^s,k,1,n) +
 611 ;;             1/2^n*sum((-1)^(k-1)*e(k-n)/k^s,k,n+1,2*n))
 612 ;;           + g(n,s)
 613 ;;
 614 ;; where e(k) = sum(binomial(n,j), j, k, n). Writing s = sigma + %i*t, when
 615 ;; sigma is positive you get an error estimate of
 616 ;;
 617 ;;   |g(n,s)| <= 1/8^n * h(s)
 618 ;;
 619 ;; where
 620 ;;
 621 ;;   h(s) = ((1 + abs (t / sigma)) exp (abs (t) * %pi / 2)) / abs (1 - 2^(1 - s))
 622 ;;
 623 ;; We need to figure out how many terms are required to make |g(n,s)|
 624 ;; sufficiently small. The answer is
 625 ;;
 626 ;;      n = (log h(s) - log (eps)) / log (8)
 627 ;;
 628 ;; and
 629 ;;
 630 ;;   log (h (s)) = (%pi/2 * abs (t)) + log (1 + t/sigma) - log (abs (1 - 2^(1 - s)))
 631 ;;
 632 ;; Notice that this bound is a bit rubbish when sigma is near zero. In that
 633 ;; case, use the expansion zeta(s) = -1/2-1/2*log(2*pi)*s.
 634 (defun float-zeta (s)
 635   ;; If s is a rational (real or complex), convert to a float.  This
 636   ;; is needed so we can compute a sensible epsilon value.  (What is
 637   ;; the epsilon value for an exact rational?)
 638   (setf s (bigfloat:to s))
 639   (typecase s
 640     (rational
 641      (setf s (float s)))
 642     ((complex rational)
 643      (setf s (coerce s '(complex flonum)))))
 644
 645   (let ((sigma (bigfloat:realpart s)))
 646     (cond
 647       ;; abs(s)^2 < epsilon, use the expansion zeta(s) = -1/2-1/2*log(2*%pi)*s
 648       ((bigfloat:< (bigfloat:abs (bigfloat:* s s)) (bigfloat:epsilon s))
 649        (bigfloat:+ -1/2
 650                    (bigfloat:* -1/2
 651                                (bigfloat:log (bigfloat:* 2 (bigfloat:%pi s)))
 652                                s)))
 653
 654       ;; Reflection formula
 655       ((bigfloat:minusp sigma)
 656        (bigfloat:* (bigfloat:expt 2 s)
 657                    (bigfloat:expt (bigfloat:%pi s)
 658                                   (bigfloat:- s 1))
 659                    (bigfloat:sin (bigfloat:* (bigfloat:/ (bigfloat:%pi s)
 660                                                          2)
 661                                              s))
 662                    (bigfloat:to ($gamma (to (bigfloat:- 1 s))))
 663                    (float-zeta (bigfloat:- 1 s))))
 664
 665       ;; The general formula from above. Call the imaginary part "tau" rather
 666       ;; than the "t" above, because that isn't a CL keyword...
 667       (t
 668        (let* ((tau (bigfloat:imagpart s))
 669               (logh
 670                (bigfloat:-
 671                 (if (bigfloat:zerop tau) 0
 672                     (bigfloat:+
 673                      (bigfloat:* 1.6 (bigfloat:abs tau))
 674                      (bigfloat:log (bigfloat:1+
 675                                     (bigfloat:abs
 676                                      (bigfloat:/ tau sigma))))))
 677                 (bigfloat:log
 678                  (bigfloat:abs
 679                   (bigfloat:- 1 (bigfloat:expt 2 (bigfloat:- 1 s)))))))
 680
 681               (logeps (bigfloat:log (bigfloat:epsilon s)))
 682
 683               (n (max (bigfloat:ceiling
 684                        (bigfloat:/ (bigfloat:- logh logeps) (bigfloat:log 8)))
 685                       1))
 686
 687               (sum1 0)
 688               (sum2 0))
 689          (flet ((binsum (k n)
 690                   ;; sum(binomial(n,j), j, k, n) = sum(binomial(n,j), j, n, k)
 691                   (let ((sum 0)
 692                         (term 1))
 693                     (loop for j from n downto k
 694                        do
 695                          (progn
 696                            (incf sum term)
 697                            (setf term (/ (* term j) (+ n 1 (- j))))))
 698                     sum)))
 699            ;; (format t "n = ~D terms~%" n)
 700            ;; sum1 = sum((-1)^(k-1)/k^s,k,1,n)
 701            ;; sum2 = sum((-1)^(k-1)/e(n,k-n)/k^s, k, n+1, 2*n)
 702            ;;      = (-1)^n*sum((-1)^(m-1)*e(n,m)/(n+k)^s, k, 1, n)
 703            (loop for k from 1 to n
 704               for d = (bigfloat:expt k s)
 705               do (progn
 706                    (bigfloat:incf sum1 (bigfloat:/ (cl:expt -1 (- k 1)) d))
 707                    (bigfloat:incf sum2 (bigfloat:/ (* (cl:expt -1 (- k 1))
 708                                                       (binsum k n))
 709                                                    (bigfloat:expt (+ k n) s))))
 710               finally (return (values sum1 sum2)))
 711            (when (oddp n)
 712              (setq sum2 (bigfloat:- sum2)))
 713            (bigfloat:/ (bigfloat:+ sum1
 714                                    (bigfloat:/ sum2 (bigfloat:expt 2 n)))
 715                        (bigfloat:- 1 (bigfloat:expt 2 (bigfloat:- 1 s))))))))))
 716
 717 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
 718
 719 (defmfun $fib (n)
 720   (cond ((fixnump n) (ffib n))
 721         (t (setq $prevfib `(($fib) ,(add2* n -1)))
 722            `(($fib) ,n))))
 723
 724 (defun ffib (%n)
 725   (declare (fixnum %n))
 726   (cond ((= %n -1)
 727          (setq $prevfib -1)
 728          1)
 729         ((zerop %n)
 730          (setq $prevfib 1)
 731          0)
 732         (t
 733          (let* ((f2 (ffib (ash (logandc2 %n 1) -1))) ; f2 = fib(n/2) or fib((n-1)/2)
 734                 (x (+ f2 $prevfib))
 735                 (y (* $prevfib $prevfib))
 736                 (z (* f2 f2)))
 737            (setq f2 (- (* x x) y)
 738                  $prevfib (+ y z))
 739            (when (oddp %n)
 740              (psetq $prevfib f2
 741                     f2 (+ f2 $prevfib)))
 742            f2))))
 743
 744 (defmfun $lucas (n)
 745   (cond
 746     ((fixnump n) (lucas n))
 747          (t (setq $next_lucas `(($lucas) ,(add2* n 1)))
 748            `(($lucas) ,n) )))
 749
 750 (defun lucas (n)
 751   (declare (fixnum n))
 752   (let ((w 2) (x 2) (y 1) u v (sign (signum n))) (declare (fixnum sign))
 753     (setq n (abs n))
 754     (do ((i (1- (integer-length n)) (1- i)))
 755         ((< i 0))
 756         (declare (fixnum i))
 757       (setq u (* x x) v (* y y))
 758       (if (logbitp i n)
 759         (setq y (+ v w) x (+ y (- u) w) w -2)
 760         (setq x (- u w) y (+ v w (- x)) w 2) ))
 761     (cond
 762       ((or (= 1 sign) (not (logbitp 0 n)))
 763         (setq $next_lucas y)
 764         x )
 765       (t
 766         (setq $next_lucas (neg y))
 767         (neg x) ))))
 768
 769 ;; continued fraction stuff
 770
 771 (defmfun $cfdisrep (a)
 772   (cond ((not ($listp a))
 773          (merror (intl:gettext "cfdisrep: argument must be a list; found ~M") a))
 774         ((null (cddr a)) (cadr a))
 775         ((equal (cadr a) 0)
 776          (list '(mexpt) (cfdisrep1 (cddr a)) -1))
 777         ((cfdisrep1 (cdr a)))))
 778
 779 (defun cfdisrep1 (a)
 780   (cond ((cdr a)
 781          (list '(mplus simp cf) (car a)
 782                (prog2 (setq a (cfdisrep1 (cdr a)))
 783                    (cond ((integerp a) (list '(rat simp) 1 a))
 784                          (t (list '(mexpt simp) a -1))))))
 785         ((car a))))
 786
 787 (defun cfmak (a)
 788   (setq a (meval a))
 789   (cond ((integerp a) (list a))
 790         ((eq (caar a) 'mlist) (cdr a))
 791         ((eq (caar a) 'rat) (ratcf (cadr a) (caddr a)))
 792         ((merror (intl:gettext "cf: continued fractions must be lists or integers; found ~M") a))))
 793
 794 (defun makcf (a)
 795   (cond ((null (cdr a)) (car a))
 796         ((cons '(mlist simp cf) a))))
 797
 798 ;;; Translation properties for $CF defined in MAXSRC;TRANS5 >
 799
 800 (defmspec $cf (a)
 801   (cfratsimp  (let ($listarith)
 802                 (cfeval (meval (fexprcheck a))))))
 803
 804 ;; Definition of cfratsimp as given in SF bug report # 620928.
 805 (defun cfratsimp (a)
 806   (cond ((atom a) a)
 807         ((member 'cf (car a) :test #'eq) a)
 808         (t (cons '(mlist cf simp)
 809                  (apply 'find-cf (cf-back-recurrence (cdr a)))))))
 810
 811 ; Code to expand nth degree roots of integers into continued fraction
 812 ; approximations. E.g. cf(2^(1/3))
 813 ; Courtesy of Andrei Zorine (feniy@mail.nnov.ru) 2005/05/07
 814
 815 (defun cfnroot(b)
 816   (let ((ans (list '(mlist xf))) ent ($algebraic $true))
 817     (dotimes (i $cflength (nreverse ans))
 818       (setq ent (meval `(($floor) ,b))
 819             ans (cons ent ans)
 820             b ($ratsimp (m// (m- b ent)))))))
 821
 822 (defun cfeval (a)
 823   (let (temp $ratprint)
 824     (cond ((integerp a) (list '(mlist cf) a))
 825           ((floatp a)
 826            (let ((a (maxima-rationalize a)))
 827              (cons '(mlist cf) (ratcf (car a) (cdr a)))))
 828           (($bfloatp a)
 829            (let (($bftorat t))
 830              (setq a (bigfloat2rat a))
 831              (cons '(mlist cf) (ratcf (car a) (cdr a)))))
 832           ((atom a)
 833            (merror (intl:gettext "cf: ~:M is not a continued fraction.") a))
 834           ((eq (caar a) 'rat)
 835            (cons '(mlist cf) (ratcf (cadr a) (caddr a))))
 836           ((eq (caar a) 'mlist)
 837            (cfratsimp a))
 838           ;;the following doesn't work for non standard form
 839           ;;            (cfplus a '((mlist) 0)))
 840           ((and (mtimesp a) (cddr a) (null (cdddr a))
 841                 (fixnump (cadr a))
 842                 (mexptp (caddr a))
 843                 (fixnump (cadr (caddr a)))
 844                 (alike1 (caddr (caddr a)) '((rat) 1 2)))
 845            (cfsqrt (cfeval (* (expt (cadr a) 2) (cadr (caddr a))))))
 846           ((eq (caar a) 'mexpt)
 847            (cond ((alike1 (caddr a) '((rat) 1 2))
 848                   (cfsqrt (cfeval (cadr a))))
 849                  ((integerp (m* 2 (caddr a))) ; a^(n/2) was sqrt(a^n)
 850                   (cfsqrt (cfeval (cfexpt (cadr a) (m* 2 (caddr a))))))
 851                  ((integerp (cadr a)) (cfnroot a)) ; <=== new case x
 852                  ((cfexpt (cfeval (cadr a)) (caddr a)))))
 853           ((setq temp (assoc (caar a) '((mplus . cfplus) (mtimes . cftimes) (mquotient . cfquot)
 854                                        (mdifference . cfdiff) (mminus . cfminus)) :test #'eq))
 855            (cf (cfeval (cadr a)) (cddr a) (cdr temp)))
 856           ((eq (caar a) 'mrat)
 857            (cfeval ($ratdisrep a)))
 858           (t (merror (intl:gettext "cf: ~:M is not a continued fraction.") a)))))
 859
 860 (defun cf (a l fun)
 861   (cond ((null l) a)
 862         ((cf (funcall fun a (meval (list '($cf) (car l)))) (cdr l) fun))))
 863
 864 (defun cfplus (a b)
 865   (setq a (cfmak a) b (cfmak b))
 866   (makcf (cffun '(0 1 1 0) '(0 0 0 1) a b)))
 867
 868 (defun cftimes (a b)
 869   (setq a (cfmak a) b (cfmak b))
 870   (makcf (cffun '(1 0 0 0) '(0 0 0 1) a b)))
 871
 872 (defun cfdiff (a b)
 873   (setq a (cfmak a) b (cfmak b))
 874   (makcf (cffun '(0 1 -1 0) '(0 0 0 1) a b)))
 875
 876 (defun cfmin (a)
 877   (setq a (cfmak a))
 878   (makcf (cffun '(0 0 -1 0) '(0 0 0 1) a '(0))))
 879
 880 (defun cfquot (a b)
 881   (setq a (cfmak a) b (cfmak b))
 882   (makcf (cffun '(0 1 0 0) '(0 0 1 0) a b)))
 883
 884 (defun cfexpt (b e)
 885   (setq b (cfmak b))
 886   (cond ((null (integerp e))
 887          (merror (intl:gettext "cf: can't raise continued fraction to non-integral power ~M") e))
 888         ((let ((n (abs e)))
 889            (do ((n (ash n -1) (ash n -1))
 890                 (s (cond ((oddp n) b)
 891                          (t '(1)))))
 892                ((zerop n)
 893                 (makcf
 894                  (cond ((signp g e)
 895                         s)
 896                        ((cffun '(0 0 0 1) '(0 1 0 0) b '(1))))))
 897              (setq b (cffun '(1 0 0 0) '(0 0 0 1) b b))
 898              (and (oddp n)
 899                   (setq s (cffun '(1 0 0 0) '(0 0 0 1) s b))))))))
 900
 901
 902 (defun conf1 (f g a b &aux (den (conf2 g a b)))
 903   (cond ((zerop den)
 904          (* (signum (conf2 f a b )) ; (/ most-positive-fixnum (^ 2 4))
 905             #.(expt 2 31)))
 906         (t (truncate (conf2 f a b) den))))
 907
 908 (defun conf2 (n a b)                    ;2*(abn_0+an_1+bn_2+n_3)
 909   (* 2 (+ (* (car n) a b)
 910                  (* (cadr n) a)
 911                  (* (caddr n) b)
 912                  (cadddr n))))
 913
 914 ;;(cffun '(0 1 1 0) '(0 0 0 1) '(1 2) '(1 1 1 2)) gets error
 915 ;;should give (3 10)
 916
 917 (defun cf-convergents-p-q (cf &optional (n (length cf)) &aux pp qq)
 918   "returns two lists such that pp_i/qq_i is the quotient of the first i terms
 919    of cf"
 920   (case (length cf)
 921     (0 1)
 922     (1  cf(list 1))
 923     (t
 924      (setq pp (list (1+ (* (first cf) (second cf))) (car cf)))
 925      (setq qq (list (second cf) 1))
 926      (show pp qq)
 927      (setq cf (cddr cf))
 928      (loop for i from 2 to n
 929             while cf
 930             do
 931             (push (+  (* (car cf) (car pp))
 932                       (second pp)) pp)
 933             (push (+  (* (car cf) (car qq))
 934                       (second qq)) qq)
 935             (setq cf (cdr cf))
 936             finally (return (list (reverse pp) (reverse qq)))))))
 937
 938
 939 (defun find-cf1 (p q so-far)
 940   (multiple-value-bind (quot rem) (truncate p q)
 941     (cond ((< rem 0) (incf rem q) (incf quot -1))
 942           ((zerop rem) (return-from find-cf1 (cons quot so-far))))
 943     (setq so-far (cons quot so-far))
 944     (find-cf1 q rem so-far)))
 945
 946 (defun find-cf (p q)
 947   "returns the continued fraction for p and q integers, q not zero"
 948   (cond  ((zerop q) (maxima-error "find-cf: quotient by zero"))
 949          ((< q 0) (setq p (- p)) (setq q (- q))))
 950   (nreverse (find-cf1 p q ())))
 951
 952 (defun cf-back-recurrence (cf &aux tem (num-gg 0)(den-gg 1))
 953   "converts CF (a continued fraction list) to a list of numerator
 954   denominator using  recurrence from end
 955   and not calculating intermediate quotients.
 956   The numerator and denom are relatively
 957    prime"
 958   (loop for v in (reverse cf)
 959          do (setq tem (* den-gg v))
 960          (setq tem (+ tem num-gg))
 961          (setq num-gg den-gg)
 962          (setq den-gg tem)
 963          finally
 964          (return
 965            (cond ((and (<= den-gg 0) (< num-gg 0))
 966                   (list  (- den-gg) (- num-gg)))
 967                  (t(list den-gg num-gg))))))
 968
 969 (declare-top (unspecial w))
 970
 971 ;;(cffun '(0 1 1 0) '(0 0 0 1) '(1 2) '(1 1 1 2)) gets error
 972 ;;should give (3 10)
 973
 974 (defun cffun (f g a b)
 975   (prog (c v w)
 976      (declare (special v))
 977      a   (and (zerop (cadddr g))
 978               (zerop (caddr g))
 979               (zerop (cadr g))
 980               (zerop (car g))
 981               (return (reverse c)))
 982      (and (equal (setq w (conf1 f g (car a) (1+ (car b))))
 983                  (setq v (conf1 f g (car a) (car b))))
 984           (equal (conf1 f g (1+ (car a)) (car b)) v)
 985           (equal (conf1 f g (1+ (car a)) (1+ (car b))) v)
 986           (setq g (mapcar #'(lambda (a b)
 987                               (declare (special v))
 988                               (- a (* v b)))
 989                           f (setq f g)))
 990           (setq c (cons v c))
 991           (go a))
 992      (cond ((< (abs (- (conf1 f g (1+ (car a)) (car b)) v))
 993                    (abs (- w v)))
 994             (cond ((setq v (cdr b))
 995                    (setq f (conf6 f b))
 996                    (setq g (conf6 g b))
 997                    (setq b v))
 998                   (t (setq f (conf7 f b)) (setq g  (conf7 g b)))))
 999            (t
1000             (cond ((setq v (cdr a))
1001                    (setq f (conf4 f a))
1002                    (setq g (conf4 g a))
1003                    (setq a v))
1004                   (t (setq f (conf5 f a)) (setq g  (conf5 g a))))))
1005      (go a)))
1006
1007 (defun conf4 (n a)                    ;n_0*a_0+n_2,n_1*a_0+n_3,n_0,n_1
1008   (list (+ (* (car n) (car a)) (caddr n))
1009         (+ (* (cadr n) (car a)) (cadddr n))
1010         (car n)
1011         (cadr n)))
1012
1013 (defun conf5 (n a)                      ;0,0, n_0*a_0,n_2
1014   (list 0 0
1015         (+ (* (car n) (car a)) (caddr n))
1016         (+ (* (cadr n) (car a)) (cadddr n))))
1017
1018 (defun conf6 (n b)
1019   (list (+ (* (car n) (car b)) (cadr n))
1020         (car n)
1021         (+ (* (caddr n) (car b)) (cadddr n))
1022         (caddr n)))
1023
1024 (defun conf7 (n b)
1025   (list 0 (+ (* (car n) (car b)) (cadr n))
1026         0 (+ (* (caddr n) (car b)) (cadddr n))))
1027
1028 (defun cfsqrt (n)
1029   (cond ((cddr n)                       ;A non integer
1030          (merror (intl:gettext "cf: argument of sqrt must be an integer; found ~M") n))
1031         ((setq n (cadr n))))
1032   (setq n (sqcont n))
1033   (cond ((= $cflength 1)
1034          (cons '(mlist simp) n))
1035         ((do ((i 2 (1+ i))
1036               (a (copy-tree (cdr n))))
1037              ((> i $cflength) (cons '(mlist simp) n))
1038            (setq n (nconc n (copy-tree a)))))))
1039
1040 (defmfun $qunit (n)
1041   (let ((isqrtn ($isqrt n)))
1042     (when (or (not (integerp n))
1043               (minusp n)
1044               (= (* isqrtn isqrtn) n))
1045       (merror
1046         (intl:gettext "qunit: Argument must be a positive non quadratic integer.")))
1047     (let ((l (sqcont n)))
1048       (list '(mplus) (pelso1 l 0 1)
1049             (list '(mtimes)
1050                   (list '(mexpt) n '((rat) 1 2))
1051                   (pelso1 l 1 0))))))
1052
1053 (defun pelso1 (l a b)
1054   (do ((i l (cdr i))) (nil)
1055     (and (null (cdr i)) (return b))
1056     (setq b (+ a (* (car i) (setq a b))))))
1057
1058 (defun sqcont (n)
1059   (prog (q q1 q2 m m1 a0 a l)
1060      (setq a0 ($isqrt n) a (list a0) q2 1 m1 a0
1061            q1 (- n (* m1 m1)) l (* 2 a0))
1062      a  (setq a (cons (truncate (+ m1 a0) q1) a))
1063      (cond ((equal (car a) l)
1064             (return (nreverse a))))
1065      (setq m (- (* (car a) q1) m1)
1066            q (+ q2 (* (car a) (- m1 m)))
1067            q2 q1 q1 q m1 m)
1068      (go a)))
1069
1070 (defun ratcf (x y)
1071   (prog (a b)
1072    a    (cond ((equal  y 1) (return (nreverse (cons x a))))
1073               ((minusp x)
1074                (setq b (+ y (rem x y))
1075                      a (cons (1- (truncate x y)) a)
1076                      x y y b))
1077               ((> y x)
1078                (setq a (cons 0 a))
1079                (setq b x x y y b))
1080               ((equal x y) (return (nreverse (cons 1 a))))
1081               ((setq b (rem x y))
1082                (setq a (cons (truncate x y) a) x y y b)))
1083    (go a)))
1084
1085 (defmfun $cfexpand (x)
1086   (cond ((null ($listp x)) x)
1087         ((cons '($matrix) (cfexpand (cdr x))))))
1088
1089 (defun cfexpand (ll)
1090   (do ((p1 0 p2)
1091        (p2 1 (simplify (list '(mplus) (list '(mtimes) (car l) p2) p1)))
1092        (q1 1 q2)
1093        (q2 0 (simplify (list '(mplus) (list '(mtimes) (car l) q2) q1)))
1094        (l ll (cdr l)))
1095       ((null l) (list (list '(mlist) p2 p1) (list '(mlist) q2 q1)))))
1096
1097 ;; Summation stuff
1098
1099 (defun adsum (e)
1100   (push (simplify e) sum))
1101
1102 (defun adusum (e)
1103   (push (simplify e) usum))
1104
1105 (defmfun simpsum2 (exp i lo hi)
1106   (prog (*plus *times $simpsum u)
1107      (setq *plus (list 0) *times 1)
1108      (when (or (and (eq hi '$inf) (eq lo '$minf))
1109                (equal 0 (m+ hi lo)))
1110        (setq $simpsum t lo 0)
1111        (setq *plus (cons (m* -1 *times (maxima-substitute 0 i exp)) *plus))
1112        (setq exp (m+ exp (maxima-substitute (m- i) i exp))))
1113      (cond ((eq ($sign (setq u (m- hi lo))) '$neg)
1114             (if (equal u -1)
1115                 (return 0)
1116                 (merror (intl:gettext "sum: lower bound ~M greater than upper bound ~M") lo hi)))
1117            ((free exp i)
1118             (return (m+l (cons (freesum exp lo hi *times) *plus))))
1119
1120            ((setq exp (sumsum exp i lo hi))
1121             (setq exp (m* *times (dosum (cadr exp) (caddr exp)
1122                                         (cadddr exp) (cadr (cdddr exp)) t :evaluate-summand nil))))
1123            (t (return (m+l *plus))))
1124      (return (m+l (cons exp *plus)))))
1125
1126 (defun sumsum (e *var* lo hi)
1127   (let (sum usum)
1128     (cond ((eq hi '$inf)
1129            (cond (*infsumsimp (isum e lo))
1130                  ((setq usum (list e)))))
1131           ((finite-sum e 1 lo hi)))
1132     (cond ((eq sum nil)
1133            (return-from sumsum (list '(%sum) e *var* lo hi))))
1134     (setq *plus
1135           (nconc (mapcar
1136                   #'(lambda (q) (simptimes (list '(mtimes) *times q) 1 nil))
1137                   sum)
1138                  *plus))
1139     (and usum (setq usum (list '(%sum) (simplus (cons '(plus) usum) 1 t) *var* lo hi)))))
1140
1141 (defun finite-sum (e y lo hi)
1142   (cond ((null e))
1143         ((free e *var*)
1144          (adsum (m* y e (m+ hi 1 (m- lo)))))
1145         ((poly? e *var*)
1146          (adsum (m* y (fpolysum e lo hi))))
1147         ((eq (caar e) '%binomial) (fbino e y lo hi))
1148         ((eq (caar e) 'mplus)
1149          (mapc #'(lambda (q) (finite-sum q y lo hi)) (cdr e)))
1150         ((and (or (mtimesp e) (mexptp e) (mplusp e))
1151               (fsgeo e y lo hi)))
1152         (t
1153          (adusum e)
1154          nil)))
1155
1156 (defun isum-giveup (e)
1157   (cond ((atom e) nil)
1158         ((eq (caar e) 'mexpt)
1159          (not (or (free (cadr e) *var*)
1160                   (ratp (caddr e) *var*))))
1161         ((member (caar e) '(mplus mtimes) :test #'eq)
1162          (some #'identity (mapcar #'isum-giveup (cdr e))))
1163         (t)))
1164
1165 (defun isum (e lo)
1166   (cond ((isum-giveup e)
1167          (setq sum nil usum (list e)))
1168         ((eq (catch 'isumout (isum1 e lo)) 'divergent)
1169          (merror (intl:gettext "sum: sum is divergent.")))))
1170
1171 (defun isum1 (e lo)
1172   (cond ((free e *var*)
1173          (unless (eq (asksign e) '$zero)
1174            (throw 'isumout 'divergent)))
1175         ((ratp e *var*)
1176          (adsum (ipolysum e lo)))
1177         ((eq (caar e) 'mplus)
1178          (mapc #'(lambda (x) (isum1 x lo)) (cdr e)))
1179         ( (isgeo e lo))
1180         ((adusum e))))
1181
1182 (defun ipolysum (e lo)
1183   (ipoly1 ($expand e) lo))
1184
1185 (defun ipoly1 (e lo)
1186   (cond ((smono e *var*)
1187          (ipoly2 *a *n lo (asksign (simplify (list '(mplus) *n 1)))))
1188         ((mplusp e)
1189          (cons '(mplus) (mapcar #'(lambda (x) (ipoly1 x lo)) (cdr e))))
1190         (t (adusum e)
1191            0)))
1192
1193 (defun ipoly2 (a n lo sign)
1194   (cond ((member (asksign lo) '($zero $negative) :test #'eq)
1195          (throw 'isumout 'divergent)))
1196   (and (null (equal lo 1))
1197        (let ((sign sign) ($simpsum t))
1198          (adsum `((%sum)
1199                   ((mtimes) ,a -1 ((mexpt) ,*var* ,n))
1200                   ,*var* 1 ((mplus) -1 ,lo)))))
1201   (cond ((eq sign '$negative)
1202          (list '(mtimes) a ($zeta (meval (list '(mtimes) -1 n)))))
1203         ((throw 'isumout 'divergent))))
1204
1205 (defun fsgeo (e y lo hi)
1206   (let ((r ($ratsimp (div* (maxima-substitute (list '(mplus) *var* 1) *var* e) e))))
1207     (cond ((equal r 1)
1208            (adsum
1209             (list '(mtimes)
1210                   (list '(mplus) 1 hi (list '(mtimes) -1 lo))
1211                   (maxima-substitute lo *var* e))))
1212           ((free r *var*)
1213            (adsum
1214             (list '(mtimes) y
1215                   (maxima-substitute 0 *var* e)
1216                   (list '(mplus)
1217                         (list '(mexpt) r (list '(mplus) hi 1))
1218                         (list '(mtimes) -1 (list '(mexpt) r lo)))
1219                   (list '(mexpt) (list '(mplus) r -1) -1)))))))
1220
1221 (defun isgeo (e lo)
1222   (let ((r ($ratsimp (div* (maxima-substitute (list '(mplus) *var* 1) *var* e) e))))
1223     (and (free r *var*)
1224          (isgeo1 (maxima-substitute lo *var* e)
1225                  r (asksign (simplify (list '(mplus) (list '(mabs) r) -1)))))))
1226
1227 (defun isgeo1 (a r sign)
1228   (cond ((eq sign '$positive)
1229          (throw 'isumout 'divergent))
1230         ((eq sign '$zero)
1231          (throw 'isumout 'divergent))
1232         ((eq sign '$negative)
1233          (adsum (list '(mtimes) a
1234                       (list '(mexpt) (list '(mplus) 1 (list '(mtimes) -1 r)) -1))))))
1235
1236
1237 ;; Sums of polynomials using
1238 ;;   bernpoly(x+1, n) - bernpoly(x, n) = n*x^(n-1)
1239 ;; which implies
1240 ;;   sum(k^n, k, A, B) = 1/(n+1)*(bernpoly(B+1, n+1) - bernpoly(A, n+1))
1241 ;;
1242 ;; fpoly1 returns 1/(n+1)*(bernpoly(foo+1, n+1) - bernpoly(0, n+1)) for each power
1243 ;; in the polynomial e
1244
1245 (defun fpolysum (e lo hi)                       ;returns *ans*
1246   (let ((a (fpoly1 (setq e ($expand ($ratdisrep ($rat e *var*)))) lo))
1247         ($prederror))
1248     (cond ((null a) 0)
1249           ((member lo '(0 1))
1250            (maxima-substitute hi 'foo a))
1251           (t
1252            (list '(mplus) (maxima-substitute hi 'foo a)
1253                  (list '(mtimes) -1 (maxima-substitute (list '(mplus) lo -1) 'foo a)))))))
1254
1255 (defun fpoly1 (e lo)
1256   (cond ((smono e *var*)
1257          (fpoly2 *a *n e lo))
1258         ((eq (caar e) 'mplus)
1259          (cons '(mplus) (mapcar #'(lambda (x) (fpoly1 x lo)) (cdr e))))
1260         (t (adusum e) 0)))
1261
1262 (defun fpoly2 (a n e lo)
1263   (cond ((null (and (integerp n) (> n -1))) (adusum e) 0)
1264         ((equal n 0)
1265          (m* (cond ((signp e lo)
1266                     (m1+ 'foo))
1267                    (t 'foo))
1268              a))
1269         (($ratsimp
1270           (m* a (list '(rat) 1 (1+ n))
1271               (m- ($bernpoly (m+ 'foo 1) (1+ n))
1272                   ($bern (1+ n))))))))
1273
1274 ;; fbino can do these sums:
1275 ;;  a) sum(binomial(n,k),k,0,n) -> 2^n
1276 ;;  b) sum(binomial(n-k,k,k,0,n) -> fib(n+1)
1277 ;;  c) sum(binomial(n,2k),k,0,n) -> 2^(n-1)
1278 ;;  d) sum(binomial(a+k,b),k,l,h) -> binomial(h+a+1,b+1) - binomial(l+a,b+1)
1279 (defun fbino (e y lo hi)
1280   ;; e=binomial(n,d)
1281   (prog (n d l h)
1282      ;; check that n and d are linear in *var*
1283      (when (null (setq n (m2 (cadr e) (list 'n 'linear* *var*))))
1284        (return (adusum e)))
1285      (setq n (cdr (assoc 'n n :test #'eq)))
1286      (when (null (setq d (m2 (caddr e) (list 'd 'linear* *var*))))
1287        (return (adusum e)))
1288      (setq d (cdr (assoc 'd d :test #'eq)))
1289
1290      ;; binomial(a+b*k,c+b*k) -> binomial(a+b*k, a-c)
1291      (when (equal (cdr n) (cdr d))
1292        (setq d (cons (m- (car n) (car d)) 0)))
1293
1294      (cond
1295        ;; substitute k with -k in sum(binomial(a+b*k, c-d*k))
1296        ;; and sum(binomial(a-b*k,c))
1297        ((and (numberp (cdr d))
1298              (or (minusp (cdr d))
1299                  (and (zerop (cdr d))
1300                       (numberp (cdr n))
1301                       (minusp (cdr n)))))
1302         (rplacd d (- (cdr d)))
1303         (rplacd n (- (cdr n)))
1304         (setq l (m- hi)
1305               h (m- lo)))
1306        (t (setq l lo  h hi)))
1307
1308      (cond
1309
1310        ;; sum(binomial(a+k,c),k,l,h)
1311        ((and (equal 0 (cdr d)) (equal 1 (cdr n)))
1312         (adsum (m* y (m- (list '(%binomial) (m+ h (car n) 1) (m+ (car d) 1))
1313                          (list '(%binomial) (m+ l (car n)) (m+ (car d) 1))))))
1314
1315        ;; sum(binomial(n,k),k,0,n)=2^n
1316        ((and (equal 1 (cdr d)) (equal 0 (cdr n)))
1317         ;; sum(binomial(n,k+c),k,l,h)=sum(binomial(n,k+c+l),k,0,h-l)
1318         (let ((h1 (m- h l))
1319               (c (m+ (car d) l)))
1320           (if (and (integerp (m- (car n) h1))
1321                    (integerp c))
1322               (progn
1323                 (adsum (m* y (m^ 2 (car n))))
1324                 (when (member (asksign (m- (m+ h1 c) (car n))) '($zero $negative) :test #'eq)
1325                   (adsum (m* -1 y (dosum (list '(%binomial) (car n) *var*)
1326                                          *var* (m+ h1 c 1) (car n) t :evaluate-summand nil))))
1327                 (when (> c 0)
1328                   (adsum (m* -1 y (dosum (list '(%binomial) (car n) *var*)
1329                                          *var* 0 (m- c 1) t :evaluate-summand nil)))))
1330               (adusum e))))
1331
1332        ;; sum(binomial(b-k,k),k,0,floor(b/2))=fib(b+1)
1333        ((and (equal -1 (cdr n)) (equal 1 (cdr d)))
1334         ;; sum(binomial(a-k,b+k),k,l,h)=sum(binomial(a+b-k,k),k,l+b,h+b)
1335         (let ((h1 (m+ h (car d)))
1336               (l1 (m+ l (car d)))
1337               (a1 (m+ (car n) (car d))))
1338           ;; sum(binomial(a1-k,k),k,0,floor(a1/2))=fib(a1+1)
1339           ;; we only do sums with h>floor(a1/2)
1340           (if (and (integerp l1)
1341                    (member (asksign (m- h1 (m// a1 2))) '($zero $positive) :test #'eq))
1342               (progn
1343                 (adsum (m* y ($fib (m+ a1 1))))
1344                 (when (> l1 0)
1345                   (adsum (m* -1 y (dosum (list '(%binomial) (m- a1 *var*) *var*)
1346                                          *var* 0 (m- l1 1) t :evaluate-summand nil)))))
1347               (adusum e))))
1348
1349        ;; sum(binomial(n,2*k),k,0,floor(n/2))=2^(n-1)
1350        ;; sum(binomial(n,2*k+1),k,0,floor((n-1)/2))=2^(n-1)
1351        ((and (equal 0 (cdr n)) (equal 2 (cdr d)))
1352         ;; sum(binomial(a,2*k+b),k,l,h)=sum(binomial(a,2*k),k,l+b/2,h+b/2), b even
1353         ;; sum(binomial(a,2*k+b),k,l,h)=sum(binomial(a,2*k+1),k,l+(b-1)/2,h+(b-1)/2), b odd
1354         (let ((a (car n))
1355               (r1 (if (oddp (car d)) 1 0))
1356               (l1 (if (oddp (car d))
1357                       (m+ l (truncate (1- (car d)) 2))
1358                       (m+ l (truncate (car d) 2)))))
1359           (when (and (integerp l1)
1360                      (member (asksign (m- a hi)) '($zero $positive) :test #'eq))
1361             (adsum (m* y (m^ 2 (m- a 1))))
1362             (when (> l1 0)
1363               (adsum (m* -1 y (dosum (list '(%binomial) a (m+ *var* *var* r1))
1364                                      *var* 0 (m- l1 1) t :evaluate-summand nil)))))))
1365
1366        ;; other sums we can't do
1367        (t
1368         (adusum e)))))
1369
1370 ;; product routines
1371
1372 (defmspec $product (l)
1373   (setq l (cdr l))
1374   (cond ((not (= (length l) 4)) (merror (intl:gettext "product: expected exactly four arguments.")))
1375         ((dosum (car l) (cadr l) (meval (caddr l)) (meval (cadddr l)) nil :evaluate-summand t))))
1376
1377 (declare-top (special $ratsimpexpons))
1378
1379 ;; Is this guy actually looking at the value of its middle arg?
1380
1381 (defun simpprod (x y z)
1382   (let (($ratsimpexpons t))
1383     (cond ((equal y 1)
1384            (setq y (simplifya (cadr x) z)))
1385           ((setq y (simptimes (list '(mexpt) (cadr x) y) 1 z)))))
1386   (simpprod1 y (caddr x)
1387              (simplifya (cadddr x) z)
1388              (simplifya (cadr (cdddr x)) z)))
1389
1390 (defmfun $taytorat (e)
1391   (cond ((mbagp e) (cons (car e) (mapcar #'$taytorat (cdr e))))
1392         ((or (atom e) (not (member 'trunc (cdar e) :test #'eq))) (ratf e))
1393         ((catch 'srrat (srrat e)))
1394         (t (ratf ($ratdisrep e)))))
1395
1396 (defun srrat (e)
1397   (cons (list 'mrat 'simp (caddar e) (cadddr (car e)))
1398         (srrat2 (cdr e))))
1399
1400 (defun srrat2 (e)
1401   (if (pscoefp e) e (srrat3 (terms e) (gvar e))))
1402
1403 (defun srrat3 (l *var*)
1404   (cond ((null l) '(0 . 1))
1405         ((null (=1 (cdr (le l))))
1406          (throw 'srrat nil))
1407         ((null (n-term l))
1408          (rattimes (cons (list *var* (car (le l)) 1) 1)
1409                    (srrat2 (lc l))
1410                    t))
1411         ((ratplus
1412           (rattimes (cons (list *var* (car (le l)) 1) 1)
1413                     (srrat2 (lc l))
1414                     t)
1415           (srrat3 (n-term l) *var*)))))
1416
1417
1418 (declare-top (special $props *i))
1419
1420 (defmspec $deftaylor (l)
1421   (prog (fun series param op ops)
1422    a    (when (null (setq l (cdr l))) (return (cons '(mlist) ops)))
1423    (setq fun (meval (car l)) series (meval (cadr l)) l (cdr l) param () )
1424    (when (or (atom fun)
1425              (if (eq (caar fun) 'mqapply)
1426                  (or (cdddr fun)        ; must be one parameter
1427                      (null (cddr fun))  ; must have exactly one
1428                      (do ((subs (cdadr fun) (cdr subs)))
1429                          ((null subs)
1430                           (setq op (caaadr fun))
1431                           (when (cddr fun)
1432                             (setq param (caddr fun)))
1433                           '())
1434                        (unless (atom (car subs)) (return 't))))
1435                  (progn
1436                    (setq op (caar fun))
1437                    (when (cdr fun) (setq param (cadr fun)))
1438                    (or (and (oldget op 'op) (not (eq op 'mfactorial)))
1439                        (not (atom (cadr fun)))
1440                        (not (= (length fun) 2))))))
1441      (merror (intl:gettext "deftaylor: don't know how to handle this function: ~M") fun))
1442    (when (oldget op 'sp2)
1443      (mtell (intl:gettext "deftaylor: redefining ~:M.~%") op))
1444    (when param (setq series (subst 'sp2var param series)))
1445    (setq series (subsum '*index series))
1446    (putprop op series 'sp2)
1447    (when (eq (caar fun) 'mqapply)
1448      (putprop op (cdadr fun) 'sp2subs))
1449    (add2lnc op $props)
1450    (push op ops)
1451    (go a)))
1452
1453 (defun subsum (*i e) (susum1 e))
1454
1455 (defun susum1 (e)
1456   (cond ((atom e) e)
1457         ((eq (caar e) '%sum)
1458          (if (null (smonop (cadr e) 'sp2var))
1459              (merror (intl:gettext "deftaylor: argument must be a power series at 0."))
1460              (subst *i (caddr e) e)))
1461         (t (recur-apply #'susum1 e))))
1462
1463 (declare-top (special varlist genvar $factorflag $ratfac *p* *var* *l* *x*))
1464
1465 (defmfun $polydecomp (e v)
1466   (let ((varlist (list v))
1467         (genvar nil)
1468         *var* p den $factorflag $ratfac)
1469     (setq p (cdr (ratf (ratdisrep e)))
1470           *var* (cdr (ratf v)))
1471     (cond ((or (null (cdr *var*))
1472                (null (equal (cdar *var*) '(1 1))))
1473            (merror (intl:gettext "polydecomp: second argument must be an atom; found ~M") v))
1474           (t (setq *var* (caar *var*))))
1475     (cond ((or (pcoefp (cdr p))
1476                (null (eq (cadr p) *var*)))
1477            (setq den (cdr p)
1478                  p (car p)))
1479           (t (merror (intl:gettext "polydecomp: cannot apply 'polydecomp' to a rational function."))))
1480     (cons '(mlist)
1481           (cond ((or (pcoefp p)
1482                      (null (eq (car p) *var*)))
1483                  (list (rdis (cons p den))))
1484                 (t (setq p (pdecomp p *var*))
1485                    (do ((l
1486                          (setq p (mapcar #'(lambda (q) (cons q 1)) p))
1487                          (cdr l))
1488                         (a))
1489                        ((null l)
1490                         (cons (rdis (cons (caar p)
1491                                           (ptimes (cdar p) den)))
1492                               (mapcar #'rdis (cdr p))))
1493                      (cond ((setq a (pdecpow (car l) *var*))
1494                             (rplaca l (car a))
1495                             (cond ((cdr l)
1496                                    (rplacd l
1497                                            (cons (ratplus
1498                                                   (rattimes
1499                                                    (cadr l)
1500                                                    (cons (ptterm (cdaadr a) 1)
1501                                                          (cdadr a))
1502                                                    t)
1503                                                   (cons
1504                                                    (ptterm (cdaadr a) 0)
1505                                                    (cdadr a)))
1506                                                  (cddr l))))
1507                                   ((equal (cadr a)
1508                                           (cons (list *var* 1 1) 1)))
1509                                   (t (rplacd l (list (cadr a)))))))))))))
1510
1511
1512 ;;; POLYDECOMP is like $POLYDECOMP except it takes a poly in *POLY* format (as
1513 ;;; defined in SOLVE) (numerator of a RAT form) and returns a list of
1514 ;;; headerless rat forms.  In otherwords, it is $POLYDECOMP minus type checking
1515 ;;; and conversions to/from general representation which SOLVE doesn't
1516 ;;; want/need on a general basis.
1517 ;;; It is used in the SOLVE package and as such it should have an autoload
1518 ;;; property
1519
1520 (defun polydecomp (p *var*)
1521   (let ($factorflag $ratfac)
1522     (cond ((or (pcoefp p)
1523                (null (eq (car p) *var*)))
1524            (cons p nil))
1525           (t (setq p (pdecomp p *var*))
1526              (do ((l (setq p (mapcar #'(lambda (q) (cons q 1)) p))
1527                      (cdr l))
1528                   (a))
1529                  ((null l)
1530                   (cons (cons (caar p)
1531                               (cdar p))
1532                         (cdr p)))
1533                (cond ((setq a (pdecpow (car l) *var*))
1534                       (rplaca l (car a))
1535                       (cond ((cdr l)
1536                              (rplacd l
1537                                      (cons (ratplus
1538                                             (rattimes
1539                                              (cadr l)
1540                                              (cons (ptterm (cdaadr a) 1)
1541                                                    (cdadr a))
1542                                              t)
1543                                             (cons
1544                                              (ptterm (cdaadr a) 0)
1545                                              (cdadr a)))
1546                                            (cddr l))))
1547                             ((equal (cadr a)
1548                                     (cons (list *var* 1 1) 1)))
1549                             (t (rplacd l (list (cadr a))))))))))))
1550
1551
1552
1553 (defun pdecred (f h *var*)              ;f = g(h(*var*))
1554   (cond ((or (pcoefp h) (null (eq (car h) *var*))
1555              (equal (cadr h) 1)
1556              (null (zerop (rem (cadr f) (cadr h))))
1557              (and (null (pzerop (caadr (setq f (pdivide f h)))))
1558                   (equal (cdadr f) 1)))
1559          nil)
1560         (t (do ((q (pdivide (caar f) h) (pdivide (caar q) h))
1561                 (i 1 (1+ i))
1562                 (*ans*))
1563                ((pzerop (caar q))
1564                 (cond ((and (equal (cdadr q) 1)
1565                             (or (pcoefp (caadr q))
1566                                 (null (eq (caar (cadr q)) *var*))))
1567                        (psimp *var* (cons i (cons (caadr q) *ans*))))))
1568              (cond ((and (equal (cdadr q) 1)
1569                          (or (pcoefp (caadr q))
1570                              (null (eq (caar (cadr q)) *var*))))
1571                     (and (null (pzerop (caadr q)))
1572                          (setq *ans* (cons i (cons (caadr q) *ans*)))))
1573                    (t (return nil)))))))
1574
1575 (defun pdecomp (p *var*)
1576   (let ((c (ptterm (cdr p) 0))
1577         (a) (*x* (list *var* 1 1)))
1578     (cons (pcplus c (car (setq a (pdecomp* (pdifference p c)))))
1579           (cdr a))))
1580
1581 (defun pdecomp* (*p*)
1582   (let ((a)
1583         (l (pdecgdfrm (pfactor (pquotient *p* *x*)))))
1584     (cond ((or (pdecprimep (cadr *p*))
1585                (null (setq a (pdecomp1 *x* l))))
1586            (list *p*))
1587           (t (append (pdecomp* (car a)) (cdr a))))))
1588
1589 (defun pdecomp1 (prod l)
1590   (cond ((null l)
1591          (and (null (equal (cadr prod) (cadr *p*)))
1592               (setq l (pdecred *p* prod *var*))
1593               (list l prod)))
1594         ((pdecomp1 prod (cdr l)))
1595         (t (pdecomp1 (ptimes (car l) prod) (cdr l)))))
1596
1597 (defun pdecgdfrm (l)                    ;Get list of divisors
1598   (do ((l (copy-list l ))
1599        (ll (list (car l))
1600            (cons (car l) ll)))
1601       (nil)
1602     (rplaca (cdr l) (1- (cadr l)))
1603     (cond ((signp e (cadr l))
1604            (setq l (cddr l))))
1605     (cond ((null l) (return ll)))))
1606
1607 (defun pdecprimep (x)
1608   (setq x (cfactorw x))
1609   (and (null (cddr x)) (equal (cadr x) 1)))
1610
1611 (defun pdecpow (p *var*)
1612   (setq p (car p))
1613   (let ((p1 (pderivative p *var*))
1614         p2 p1p p1c a lin p2p)
1615     (setq p1p (oldcontent p1)
1616           p1c (car p1p) p1p (cadr p1p))
1617     (setq p2 (pderivative p1 *var*))
1618     (setq p2p (cadr (oldcontent p2)))
1619     (and (setq lin (testdivide p1p p2p))
1620          (null (pcoefp lin))
1621          (eq (car lin) *var*)
1622          (list (ratplus
1623                 (rattimes (cons (list *var* (cadr p) 1) 1)
1624                           (setq a (ratreduce p1c
1625                                              (ptimes (cadr p)
1626                                                      (caddr lin))))
1627                           t)
1628                 (ratdif (cons p 1)
1629                         (rattimes a (cons (pexpt lin (cadr p)) 1)
1630                                   t)))
1631                (cons lin 1)))))
1632
1633 (declare-top (unspecial *mfactl *factlist donel nn* dn*
1634                         *var* *ans* *n *a*
1635                         *infsumsimp *times *plus sum usum makef))