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Remove some code duplication in TRANSLATE-PREDICATE
[maxima.git] / src / combin.lisp
blob4ef1a023455a04b0d33f1ce1792eda80b00f046f
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* *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 ($euler %k)))
372 (go a)))
373
374 (defun simpeuler (x vestigial z)
375 (declare (ignore vestigial))
376 (oneargcheck x)
377 (let ((u (simpcheck (cadr x) z)))
378 (if (and (fixnump u) (>= u 0))
379 ($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 ($bern %k))))
425 (* a (denom1 bb))))
426 (setq b (* b (denom1 bb)))
427 (setq a (*red a b) b (denom1 a) a (num1 a))
428 (go a)))
429
430 (defun simpbern (x vestigial z)
431 (declare (ignore vestigial))
432 (oneargcheck x)
433 (let ((u (simpcheck (cadr x) z)))
434 (if (and (fixnump u) (not (< u 0)))
435 ($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 (defmfun $bernpoly (x s)
457 (let ((%n 0) ($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 (defmfun $eulerpoly (x s)
491 (let ((n 0) ($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 (defmfun $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 (tau (bigfloat:imagpart s)))
647 (cond
648 ;; abs(s)^2 < epsilon, use the expansion zeta(s) = -1/2-1/2*log(2*%pi)*s
649 ((bigfloat:< (bigfloat:abs (bigfloat:* s s)) (bigfloat:epsilon s))
650 (bigfloat:+ -1/2
651 (bigfloat:* -1/2
652 (bigfloat:log (bigfloat:* 2 (bigfloat:%pi s)))
653 s)))
654
655 ;; Reflection formula:
656 ;; zeta(s) = 2^s*%pi^(s-1)*sin(%pi*s/2)*gamma(1-s)*zeta(1-s)
657 ((not (bigfloat:plusp sigma))
658 (let ((n (bigfloat:floor sigma)))
659 ;; If s is a negative even integer, zeta(s) is zero,
660 ;; from the reflection formula because sin(%pi*s/2) is 0.
661 (when (and (bigfloat:zerop tau) (bigfloat:= n sigma) (evenp n))
662 (return-from float-zeta (bigfloat:float 0.0 sigma))))
663 (bigfloat:* (bigfloat:expt 2 s)
664 (bigfloat:expt (bigfloat:%pi s)
665 (bigfloat:- s 1))
666 (bigfloat:sin (bigfloat:* (bigfloat:/ (bigfloat:%pi s)
667 2)
668 s))
669 (bigfloat:to ($gamma (to (bigfloat:- 1 s))))
670 (float-zeta (bigfloat:- 1 s))))
671
672 ;; The general formula from above. Call the imaginary part "tau" rather
673 ;; than the "t" above, because that isn't a CL keyword...
674 (t
675 (let* ((logh
676 (bigfloat:-
677 (if (bigfloat:zerop tau) 0
678 (bigfloat:+
679 (bigfloat:* 1.6 (bigfloat:abs tau))
680 (bigfloat:log (bigfloat:1+
681 (bigfloat:abs
682 (bigfloat:/ tau sigma))))))
683 (bigfloat:log
684 (bigfloat:abs
685 (bigfloat:- 1 (bigfloat:expt 2 (bigfloat:- 1 s)))))))
686
687 (logeps (bigfloat:log (bigfloat:epsilon s)))
688
689 (n (max (bigfloat:ceiling
690 (bigfloat:/ (bigfloat:- logh logeps) (bigfloat:log 8)))
691 1))
692
693 (sum1 0)
694 (sum2 0))
695 (flet ((binsum (k n)
696 ;; sum(binomial(n,j), j, k, n) = sum(binomial(n,j), j, n, k)
697 (let ((sum 0)
698 (term 1))
699 (loop for j from n downto k
700 do
701 (progn
702 (incf sum term)
703 (setf term (/ (* term j) (+ n 1 (- j))))))
704 sum)))
705 ;; (format t "n = ~D terms~%" n)
706 ;; sum1 = sum((-1)^(k-1)/k^s,k,1,n)
707 ;; sum2 = sum((-1)^(k-1)/e(n,k-n)/k^s, k, n+1, 2*n)
708 ;; = (-1)^n*sum((-1)^(m-1)*e(n,m)/(n+k)^s, k, 1, n)
709 (loop for k from 1 to n
710 for d = (bigfloat:expt k s)
711 do (progn
712 (bigfloat:incf sum1 (bigfloat:/ (cl:expt -1 (- k 1)) d))
713 (bigfloat:incf sum2 (bigfloat:/ (* (cl:expt -1 (- k 1))
714 (binsum k n))
715 (bigfloat:expt (+ k n) s))))
716 finally (return (values sum1 sum2)))
717 (when (oddp n)
718 (setq sum2 (bigfloat:- sum2)))
719 (bigfloat:/ (bigfloat:+ sum1
720 (bigfloat:/ sum2 (bigfloat:expt 2 n)))
721 (bigfloat:- 1 (bigfloat:expt 2 (bigfloat:- 1 s))))))))))
722
723 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
724
725 (defmfun $fib (n)
726 (cond ((fixnump n) (ffib n))
727 (t (setq $prevfib `(($fib) ,(add2* n -1)))
728 `(($fib) ,n))))
729
730 (defun ffib (%n)
731 (declare (fixnum %n))
732 (cond ((= %n -1)
733 (setq $prevfib -1)
734 1)
735 ((zerop %n)
736 (setq $prevfib 1)
737 0)
738 (t
739 (let* ((f2 (ffib (ash (logandc2 %n 1) -1))) ; f2 = fib(n/2) or fib((n-1)/2)
740 (x (+ f2 $prevfib))
741 (y (* $prevfib $prevfib))
742 (z (* f2 f2)))
743 (setq f2 (- (* x x) y)
744 $prevfib (+ y z))
745 (when (oddp %n)
746 (psetq $prevfib f2
747 f2 (+ f2 $prevfib)))
748 f2))))
749
750 (defmfun $lucas (n)
751 (cond
752 ((fixnump n) (lucas n))
753 (t (setq $next_lucas `(($lucas) ,(add2* n 1)))
754 `(($lucas) ,n) )))
755
756 (defun lucas (n)
757 (declare (fixnum n))
758 (let ((w 2) (x 2) (y 1) u v (sign (signum n))) (declare (fixnum sign))
759 (setq n (abs n))
760 (do ((i (1- (integer-length n)) (1- i)))
761 ((< i 0))
762 (declare (fixnum i))
763 (setq u (* x x) v (* y y))
764 (if (logbitp i n)
765 (setq y (+ v w) x (+ y (- u) w) w -2)
766 (setq x (- u w) y (+ v w (- x)) w 2) ))
767 (cond
768 ((or (= 1 sign) (not (logbitp 0 n)))
769 (setq $next_lucas y)
770 x )
771 (t
772 (setq $next_lucas (neg y))
773 (neg x) ))))
774
775 ;; continued fraction stuff
776
777 (defmfun $cfdisrep (a)
778 (cond ((not ($listp a))
779 (merror (intl:gettext "cfdisrep: argument must be a list; found ~M") a))
780 ((null (cddr a)) (cadr a))
781 ((equal (cadr a) 0)
782 (list '(mexpt) (cfdisrep1 (cddr a)) -1))
783 ((cfdisrep1 (cdr a)))))
784
785 (defun cfdisrep1 (a)
786 (cond ((cdr a)
787 (list '(mplus simp cf) (car a)
788 (prog2 (setq a (cfdisrep1 (cdr a)))
789 (cond ((integerp a) (list '(rat simp) 1 a))
790 (t (list '(mexpt simp) a -1))))))
791 ((car a))))
792
793 (defun cfmak (a)
794 (setq a (meval a))
795 (cond ((integerp a) (list a))
796 ((eq (caar a) 'mlist) (cdr a))
797 ((eq (caar a) 'rat) (ratcf (cadr a) (caddr a)))
798 ((merror (intl:gettext "cf: continued fractions must be lists or integers; found ~M") a))))
799
800 (defun makcf (a)
801 (cond ((null (cdr a)) (car a))
802 ((cons '(mlist simp cf) a))))
803
804 ;;; Translation properties for $CF defined in MAXSRC;TRANS5 >
805
806 (defmspec $cf (a)
807 (cfratsimp (let ($listarith)
808 (cfeval (meval (fexprcheck a))))))
809
810 ;; Definition of cfratsimp as given in SF bug report # 620928.
811 (defun cfratsimp (a)
812 (cond ((atom a) a)
813 ((member 'cf (car a) :test #'eq) a)
814 (t (cons '(mlist cf simp)
815 (apply 'find-cf (cf-back-recurrence (cdr a)))))))
816
817 ; Code to expand nth degree roots of integers into continued fraction
818 ; approximations. E.g. cf(2^(1/3))
819 ; Courtesy of Andrei Zorine (feniy@mail.nnov.ru) 2005/05/07
820
821 (defun cfnroot(b)
822 (let ((ans (list '(mlist xf))) ent ($algebraic $true))
823 (dotimes (i $cflength (nreverse ans))
824 (setq ent (meval `(($floor) ,b))
825 ans (cons ent ans)
826 b ($ratsimp (m// (m- b ent)))))))
827
828 (defun cfeval (a)
829 (let (temp $ratprint)
830 (cond ((integerp a) (list '(mlist cf) a))
831 ((floatp a)
832 (let ((a (maxima-rationalize a)))
833 (cons '(mlist cf) (ratcf (car a) (cdr a)))))
834 (($bfloatp a)
835 (let (($bftorat t))
836 (setq a (bigfloat2rat a))
837 (cons '(mlist cf) (ratcf (car a) (cdr a)))))
838 ((atom a)
839 (merror (intl:gettext "cf: ~:M is not a continued fraction.") a))
840 ((eq (caar a) 'rat)
841 (cons '(mlist cf) (ratcf (cadr a) (caddr a))))
842 ((eq (caar a) 'mlist)
843 (cfratsimp a))
844 ;;the following doesn't work for non standard form
845 ;; (cfplus a '((mlist) 0)))
846 ((and (mtimesp a) (cddr a) (null (cdddr a))
847 (fixnump (cadr a))
848 (mexptp (caddr a))
849 (fixnump (cadr (caddr a)))
850 (alike1 (caddr (caddr a)) '((rat) 1 2)))
851 (cfsqrt (cfeval (* (expt (cadr a) 2) (cadr (caddr a))))))
852 ((eq (caar a) 'mexpt)
853 (cond ((alike1 (caddr a) '((rat) 1 2))
854 (cfsqrt (cfeval (cadr a))))
855 ((integerp (m* 2 (caddr a))) ; a^(n/2) was sqrt(a^n)
856 (cfsqrt (cfeval (cfexpt (cadr a) (m* 2 (caddr a))))))
857 ((integerp (cadr a)) (cfnroot a)) ; <=== new case x
858 ((cfexpt (cfeval (cadr a)) (caddr a)))))
859 ((setq temp (assoc (caar a) '((mplus . cfplus) (mtimes . cftimes) (mquotient . cfquot)
860 (mdifference . cfdiff) (mminus . cfminus)) :test #'eq))
861 (cf (cfeval (cadr a)) (cddr a) (cdr temp)))
862 ((eq (caar a) 'mrat)
863 (cfeval ($ratdisrep a)))
864 (t (merror (intl:gettext "cf: ~:M is not a continued fraction.") a)))))
865
866 (defun cf (a l fun)
867 (cond ((null l) a)
868 ((cf (funcall fun a (meval (list '($cf) (car l)))) (cdr l) fun))))
869
870 (defun cfplus (a b)
871 (setq a (cfmak a) b (cfmak b))
872 (makcf (cffun '(0 1 1 0) '(0 0 0 1) a b)))
873
874 (defun cftimes (a b)
875 (setq a (cfmak a) b (cfmak b))
876 (makcf (cffun '(1 0 0 0) '(0 0 0 1) a b)))
877
878 (defun cfdiff (a b)
879 (setq a (cfmak a) b (cfmak b))
880 (makcf (cffun '(0 1 -1 0) '(0 0 0 1) a b)))
881
882 (defun cfmin (a)
883 (setq a (cfmak a))
884 (makcf (cffun '(0 0 -1 0) '(0 0 0 1) a '(0))))
885
886 (defun cfquot (a b)
887 (setq a (cfmak a) b (cfmak b))
888 (makcf (cffun '(0 1 0 0) '(0 0 1 0) a b)))
889
890 (defun cfexpt (b e)
891 (setq b (cfmak b))
892 (cond ((null (integerp e))
893 (merror (intl:gettext "cf: can't raise continued fraction to non-integral power ~M") e))
894 ((let ((n (abs e)))
895 (do ((n (ash n -1) (ash n -1))
896 (s (cond ((oddp n) b)
897 (t '(1)))))
898 ((zerop n)
899 (makcf
900 (cond ((signp g e)
901 s)
902 ((cffun '(0 0 0 1) '(0 1 0 0) b '(1))))))
903 (setq b (cffun '(1 0 0 0) '(0 0 0 1) b b))
904 (and (oddp n)
905 (setq s (cffun '(1 0 0 0) '(0 0 0 1) s b))))))))
906
907
908 (defun conf1 (f g a b &aux (den (conf2 g a b)))
909 (cond ((zerop den)
910 (* (signum (conf2 f a b )) ; (/ most-positive-fixnum (^ 2 4))
911 #.(expt 2 31)))
912 (t (truncate (conf2 f a b) den))))
913
914 (defun conf2 (n a b) ;2*(abn_0+an_1+bn_2+n_3)
915 (* 2 (+ (* (car n) a b)
916 (* (cadr n) a)
917 (* (caddr n) b)
918 (cadddr n))))
919
920 ;;(cffun '(0 1 1 0) '(0 0 0 1) '(1 2) '(1 1 1 2)) gets error
921 ;;should give (3 10)
922
923 (defun cf-convergents-p-q (cf &optional (n (length cf)) &aux pp qq)
924 "returns two lists such that pp_i/qq_i is the quotient of the first i terms
925 of cf"
926 (case (length cf)
927 (0 1)
928 (1 cf(list 1))
929 (t
930 (setq pp (list (1+ (* (first cf) (second cf))) (car cf)))
931 (setq qq (list (second cf) 1))
932 (show pp qq)
933 (setq cf (cddr cf))
934 (loop for i from 2 to n
935 while cf
936 do
937 (push (+ (* (car cf) (car pp))
938 (second pp)) pp)
939 (push (+ (* (car cf) (car qq))
940 (second qq)) qq)
941 (setq cf (cdr cf))
942 finally (return (list (reverse pp) (reverse qq)))))))
943
944
945 (defun find-cf1 (p q so-far)
946 (multiple-value-bind (quot rem) (truncate p q)
947 (cond ((< rem 0) (incf rem q) (incf quot -1))
948 ((zerop rem) (return-from find-cf1 (cons quot so-far))))
949 (setq so-far (cons quot so-far))
950 (find-cf1 q rem so-far)))
951
952 (defun find-cf (p q)
953 "returns the continued fraction for p and q integers, q not zero"
954 (cond ((zerop q) (maxima-error "find-cf: quotient by zero"))
955 ((< q 0) (setq p (- p)) (setq q (- q))))
956 (nreverse (find-cf1 p q ())))
957
958 (defun cf-back-recurrence (cf &aux tem (num-gg 0)(den-gg 1))
959 "converts CF (a continued fraction list) to a list of numerator
960 denominator using recurrence from end
961 and not calculating intermediate quotients.
962 The numerator and denom are relatively
963 prime"
964 (loop for v in (reverse cf)
965 do (setq tem (* den-gg v))
966 (setq tem (+ tem num-gg))
967 (setq num-gg den-gg)
968 (setq den-gg tem)
969 finally
970 (return
971 (cond ((and (<= den-gg 0) (< num-gg 0))
972 (list (- den-gg) (- num-gg)))
973 (t(list den-gg num-gg))))))
974
975 (declare-top (unspecial w))
976
977 ;;(cffun '(0 1 1 0) '(0 0 0 1) '(1 2) '(1 1 1 2)) gets error
978 ;;should give (3 10)
979
980 (defun cffun (f g a b)
981 (prog (c v w)
982 (declare (special v))
983 a (and (zerop (cadddr g))
984 (zerop (caddr g))
985 (zerop (cadr g))
986 (zerop (car g))
987 (return (reverse c)))
988 (and (equal (setq w (conf1 f g (car a) (1+ (car b))))
989 (setq v (conf1 f g (car a) (car b))))
990 (equal (conf1 f g (1+ (car a)) (car b)) v)
991 (equal (conf1 f g (1+ (car a)) (1+ (car b))) v)
992 (setq g (mapcar #'(lambda (a b)
993 (declare (special v))
994 (- a (* v b)))
995 f (setq f g)))
996 (setq c (cons v c))
997 (go a))
998 (cond ((< (abs (- (conf1 f g (1+ (car a)) (car b)) v))
999 (abs (- w v)))
1000 (cond ((setq v (cdr b))
1001 (setq f (conf6 f b))
1002 (setq g (conf6 g b))
1003 (setq b v))
1004 (t (setq f (conf7 f b)) (setq g (conf7 g b)))))
1005 (t
1006 (cond ((setq v (cdr a))
1007 (setq f (conf4 f a))
1008 (setq g (conf4 g a))
1009 (setq a v))
1010 (t (setq f (conf5 f a)) (setq g (conf5 g a))))))
1011 (go a)))
1012
1013 (defun conf4 (n a) ;n_0*a_0+n_2,n_1*a_0+n_3,n_0,n_1
1014 (list (+ (* (car n) (car a)) (caddr n))
1015 (+ (* (cadr n) (car a)) (cadddr n))
1016 (car n)
1017 (cadr n)))
1018
1019 (defun conf5 (n a) ;0,0, n_0*a_0,n_2
1020 (list 0 0
1021 (+ (* (car n) (car a)) (caddr n))
1022 (+ (* (cadr n) (car a)) (cadddr n))))
1023
1024 (defun conf6 (n b)
1025 (list (+ (* (car n) (car b)) (cadr n))
1026 (car n)
1027 (+ (* (caddr n) (car b)) (cadddr n))
1028 (caddr n)))
1029
1030 (defun conf7 (n b)
1031 (list 0 (+ (* (car n) (car b)) (cadr n))
1032 0 (+ (* (caddr n) (car b)) (cadddr n))))
1033
1034 (defun cfsqrt (n)
1035 (cond ((cddr n) ;A non integer
1036 (merror (intl:gettext "cf: argument of sqrt must be an integer; found ~M") n))
1037 ((setq n (cadr n))))
1038 (setq n (sqcont n))
1039 (cond ((= $cflength 1)
1040 (cons '(mlist simp) n))
1041 ((do ((i 2 (1+ i))
1042 (a (copy-tree (cdr n))))
1043 ((> i $cflength) (cons '(mlist simp) n))
1044 (setq n (nconc n (copy-tree a)))))))
1045
1046 (defmfun $qunit (n)
1047 (let ((isqrtn ($isqrt n)))
1048 (when (or (not (integerp n))
1049 (minusp n)
1050 (= (* isqrtn isqrtn) n))
1051 (merror
1052 (intl:gettext "qunit: Argument must be a positive non quadratic integer.")))
1053 (let ((l (sqcont n)))
1054 (list '(mplus) (pelso1 l 0 1)
1055 (list '(mtimes)
1056 (list '(mexpt) n '((rat) 1 2))
1057 (pelso1 l 1 0))))))
1058
1059 (defun pelso1 (l a b)
1060 (do ((i l (cdr i))) (nil)
1061 (and (null (cdr i)) (return b))
1062 (setq b (+ a (* (car i) (setq a b))))))
1063
1064 (defun sqcont (n)
1065 (prog (q q1 q2 m m1 a0 a l)
1066 (setq a0 ($isqrt n) a (list a0) q2 1 m1 a0
1067 q1 (- n (* m1 m1)) l (* 2 a0))
1068 a (setq a (cons (truncate (+ m1 a0) q1) a))
1069 (cond ((equal (car a) l)
1070 (return (nreverse a))))
1071 (setq m (- (* (car a) q1) m1)
1072 q (+ q2 (* (car a) (- m1 m)))
1073 q2 q1 q1 q m1 m)
1074 (go a)))
1075
1076 (defun ratcf (x y)
1077 (prog (a b)
1078 a (cond ((equal y 1) (return (nreverse (cons x a))))
1079 ((minusp x)
1080 (setq b (+ y (rem x y))
1081 a (cons (1- (truncate x y)) a)
1082 x y y b))
1083 ((> y x)
1084 (setq a (cons 0 a))
1085 (setq b x x y y b))
1086 ((equal x y) (return (nreverse (cons 1 a))))
1087 ((setq b (rem x y))
1088 (setq a (cons (truncate x y) a) x y y b)))
1089 (go a)))
1090
1091 (defmfun $cfexpand (x)
1092 (cond ((null ($listp x)) x)
1093 ((cons '($matrix) (cfexpand (cdr x))))))
1094
1095 (defun cfexpand (ll)
1096 (do ((p1 0 p2)
1097 (p2 1 (simplify (list '(mplus) (list '(mtimes) (car l) p2) p1)))
1098 (q1 1 q2)
1099 (q2 0 (simplify (list '(mplus) (list '(mtimes) (car l) q2) q1)))
1100 (l ll (cdr l)))
1101 ((null l) (list (list '(mlist) p2 p1) (list '(mlist) q2 q1)))))
1102
1103 ;; Summation stuff
1104
1105 (defun adsum (e)
1106 (push (simplify e) sum))
1107
1108 (defun adusum (e)
1109 (push (simplify e) usum))
1110
1111 (defun simpsum2 (exp i lo hi)
1112 (prog (*plus *times $simpsum u)
1113 (setq *plus (list 0) *times 1)
1114 (when (or (and (eq hi '$inf) (eq lo '$minf))
1115 (equal 0 (m+ hi lo)))
1116 (setq $simpsum t lo 0)
1117 (setq *plus (cons (m* -1 *times (maxima-substitute 0 i exp)) *plus))
1118 (setq exp (m+ exp (maxima-substitute (m- i) i exp))))
1119 (cond ((eq ($sign (setq u (m- hi lo))) '$neg)
1120 (if (equal u -1)
1121 (return 0)
1122 (merror (intl:gettext "sum: lower bound ~M greater than upper bound ~M") lo hi)))
1123 ((free exp i)
1124 (return (m+l (cons (freesum exp lo hi *times) *plus))))
1125
1126 ((setq exp (sumsum exp i lo hi))
1127 (setq exp (m* *times (dosum (cadr exp) (caddr exp)
1128 (cadddr exp) (cadr (cdddr exp)) t :evaluate-summand nil))))
1129 (t (return (m+l *plus))))
1130 (return (m+l (cons exp *plus)))))
1131
1132 (defun sumsum (e *var* lo hi)
1133 (let (sum usum)
1134 (cond ((eq hi '$inf)
1135 (cond (*infsumsimp (isum e lo))
1136 ((setq usum (list e)))))
1137 ((finite-sum e 1 lo hi)))
1138 (cond ((eq sum nil)
1139 (return-from sumsum (list '(%sum) e *var* lo hi))))
1140 (setq *plus
1141 (nconc (mapcar
1142 #'(lambda (q) (simptimes (list '(mtimes) *times q) 1 nil))
1143 sum)
1144 *plus))
1145 (and usum (setq usum (list '(%sum) (simplus (cons '(plus) usum) 1 t) *var* lo hi)))))
1146
1147 (defun finite-sum (e y lo hi)
1148 (cond ((null e))
1149 ((free e *var*)
1150 (adsum (m* y e (m+ hi 1 (m- lo)))))
1151 ((poly? e *var*)
1152 (adsum (m* y (fpolysum e lo hi))))
1153 ((eq (caar e) '%binomial) (fbino e y lo hi))
1154 ((eq (caar e) 'mplus)
1155 (mapc #'(lambda (q) (finite-sum q y lo hi)) (cdr e)))
1156 ((and (or (mtimesp e) (mexptp e) (mplusp e))
1157 (fsgeo e y lo hi)))
1158 (t
1159 (adusum e)
1160 nil)))
1161
1162 (defun isum-giveup (e)
1163 (cond ((atom e) nil)
1164 ((eq (caar e) 'mexpt)
1165 (not (or (free (cadr e) *var*)
1166 (ratp (caddr e) *var*))))
1167 ((member (caar e) '(mplus mtimes) :test #'eq)
1168 (some #'identity (mapcar #'isum-giveup (cdr e))))
1169 (t)))
1170
1171 (defun isum (e lo)
1172 (cond ((isum-giveup e)
1173 (setq sum nil usum (list e)))
1174 ((eq (catch 'isumout (isum1 e lo)) 'divergent)
1175 (merror (intl:gettext "sum: sum is divergent.")))))
1176
1177 (defun isum1 (e lo)
1178 (cond ((free e *var*)
1179 (unless (eq (asksign e) '$zero)
1180 (throw 'isumout 'divergent)))
1181 ((ratp e *var*)
1182 (adsum (ipolysum e lo)))
1183 ((eq (caar e) 'mplus)
1184 (mapc #'(lambda (x) (isum1 x lo)) (cdr e)))
1185 ( (isgeo e lo))
1186 ((adusum e))))
1187
1188 (defun ipolysum (e lo)
1189 (ipoly1 ($expand e) lo))
1190
1191 (defun ipoly1 (e lo)
1192 (cond ((smono e *var*)
1193 (ipoly2 *a *n lo (asksign (simplify (list '(mplus) *n 1)))))
1194 ((mplusp e)
1195 (cons '(mplus) (mapcar #'(lambda (x) (ipoly1 x lo)) (cdr e))))
1196 (t (adusum e)
1197 0)))
1198
1199 (defun ipoly2 (a n lo sign)
1200 (cond ((member (asksign lo) '($zero $negative) :test #'eq)
1201 (throw 'isumout 'divergent)))
1202 (unless (equal lo 1)
1203 (let (($simpsum t))
1204 (adsum `((%sum)
1205 ((mtimes) ,a -1 ((mexpt) ,*var* ,n))
1206 ,*var* 1 ((mplus) -1 ,lo)))))
1207 (cond ((eq sign '$negative)
1208 (list '(mtimes) a ($zeta (meval (list '(mtimes) -1 n)))))
1209 ((throw 'isumout 'divergent))))
1210
1211 (defun fsgeo (e y lo hi)
1212 (let ((r ($ratsimp (div* (maxima-substitute (list '(mplus) *var* 1) *var* e) e))))
1213 (cond ((equal r 1)
1214 (adsum
1215 (list '(mtimes)
1216 (list '(mplus) 1 hi (list '(mtimes) -1 lo))
1217 (maxima-substitute lo *var* e))))
1218 ((free r *var*)
1219 (adsum
1220 (list '(mtimes) y
1221 (maxima-substitute 0 *var* e)
1222 (list '(mplus)
1223 (list '(mexpt) r (list '(mplus) hi 1))
1224 (list '(mtimes) -1 (list '(mexpt) r lo)))
1225 (list '(mexpt) (list '(mplus) r -1) -1)))))))
1226
1227 (defun isgeo (e lo)
1228 (let ((r ($ratsimp (div* (maxima-substitute (list '(mplus) *var* 1) *var* e) e))))
1229 (and (free r *var*)
1230 (isgeo1 (maxima-substitute lo *var* e)
1231 r (asksign (simplify (list '(mplus) (list '(mabs) r) -1)))))))
1232
1233 (defun isgeo1 (a r sign)
1234 (cond ((eq sign '$positive)
1235 (throw 'isumout 'divergent))
1236 ((eq sign '$zero)
1237 (throw 'isumout 'divergent))
1238 ((eq sign '$negative)
1239 (adsum (list '(mtimes) a
1240 (list '(mexpt) (list '(mplus) 1 (list '(mtimes) -1 r)) -1))))))
1241
1242
1243 ;; Sums of polynomials using
1244 ;; bernpoly(x+1, n) - bernpoly(x, n) = n*x^(n-1)
1245 ;; which implies
1246 ;; sum(k^n, k, A, B) = 1/(n+1)*(bernpoly(B+1, n+1) - bernpoly(A, n+1))
1247 ;;
1248 ;; fpoly1 returns 1/(n+1)*(bernpoly(foo+1, n+1) - bernpoly(0, n+1)) for each power
1249 ;; in the polynomial e
1250
1251 (defun fpolysum (e lo hi) ;returns *ans*
1252 (let ((a (fpoly1 (setq e ($expand ($ratdisrep ($rat e *var*)))) lo))
1253 ($prederror))
1254 (cond ((null a) 0)
1255 ((member lo '(0 1))
1256 (maxima-substitute hi 'foo a))
1257 (t
1258 (list '(mplus) (maxima-substitute hi 'foo a)
1259 (list '(mtimes) -1 (maxima-substitute (list '(mplus) lo -1) 'foo a)))))))
1260
1261 (defun fpoly1 (e lo)
1262 (cond ((smono e *var*)
1263 (fpoly2 *a *n e lo))
1264 ((eq (caar e) 'mplus)
1265 (cons '(mplus) (mapcar #'(lambda (x) (fpoly1 x lo)) (cdr e))))
1266 (t (adusum e) 0)))
1267
1268 (defun fpoly2 (a n e lo)
1269 (cond ((null (and (integerp n) (> n -1))) (adusum e) 0)
1270 ((equal n 0)
1271 (m* (cond ((signp e lo)
1272 (m1+ 'foo))
1273 (t 'foo))
1274 a))
1275 (($ratsimp
1276 (m* a (list '(rat) 1 (1+ n))
1277 (m- ($bernpoly (m+ 'foo 1) (1+ n))
1278 ($bern (1+ n))))))))
1279
1280 ;; fbino can do these sums:
1281 ;; a) sum(binomial(n,k),k,0,n) -> 2^n
1282 ;; b) sum(binomial(n-k,k,k,0,n) -> fib(n+1)
1283 ;; c) sum(binomial(n,2k),k,0,n) -> 2^(n-1)
1284 ;; d) sum(binomial(a+k,b),k,l,h) -> binomial(h+a+1,b+1) - binomial(l+a,b+1)
1285 (defun fbino (e y lo hi)
1286 ;; e=binomial(n,d)
1287 (prog (n d l h)
1288 ;; check that n and d are linear in *var*
1289 (when (null (setq n (m2 (cadr e) (list 'n 'linear* *var*))))
1290 (return (adusum e)))
1291 (setq n (cdr (assoc 'n n :test #'eq)))
1292 (when (null (setq d (m2 (caddr e) (list 'd 'linear* *var*))))
1293 (return (adusum e)))
1294 (setq d (cdr (assoc 'd d :test #'eq)))
1295
1296 ;; binomial(a+b*k,c+b*k) -> binomial(a+b*k, a-c)
1297 (when (equal (cdr n) (cdr d))
1298 (setq d (cons (m- (car n) (car d)) 0)))
1299
1300 (cond
1301 ;; substitute k with -k in sum(binomial(a+b*k, c-d*k))
1302 ;; and sum(binomial(a-b*k,c))
1303 ((and (numberp (cdr d))
1304 (or (minusp (cdr d))
1305 (and (zerop (cdr d))
1306 (numberp (cdr n))
1307 (minusp (cdr n)))))
1308 (rplacd d (- (cdr d)))
1309 (rplacd n (- (cdr n)))
1310 (setq l (m- hi)
1311 h (m- lo)))
1312 (t (setq l lo h hi)))
1313
1314 (cond
1315
1316 ;; sum(binomial(a+k,c),k,l,h)
1317 ((and (equal 0 (cdr d)) (equal 1 (cdr n)))
1318 (adsum (m* y (m- (list '(%binomial) (m+ h (car n) 1) (m+ (car d) 1))
1319 (list '(%binomial) (m+ l (car n)) (m+ (car d) 1))))))
1320
1321 ;; sum(binomial(n,k),k,0,n)=2^n
1322 ((and (equal 1 (cdr d)) (equal 0 (cdr n)))
1323 ;; sum(binomial(n,k+c),k,l,h)=sum(binomial(n,k+c+l),k,0,h-l)
1324 (let ((h1 (m- h l))
1325 (c (m+ (car d) l)))
1326 (if (and (integerp (m- (car n) h1))
1327 (integerp c))
1328 (progn
1329 (adsum (m* y (m^ 2 (car n))))
1330 (when (member (asksign (m- (m+ h1 c) (car n))) '($zero $negative) :test #'eq)
1331 (adsum (m* -1 y (dosum (list '(%binomial) (car n) *var*)
1332 *var* (m+ h1 c 1) (car n) t :evaluate-summand nil))))
1333 (when (> c 0)
1334 (adsum (m* -1 y (dosum (list '(%binomial) (car n) *var*)
1335 *var* 0 (m- c 1) t :evaluate-summand nil)))))
1336 (adusum e))))
1337
1338 ;; sum(binomial(b-k,k),k,0,floor(b/2))=fib(b+1)
1339 ((and (equal -1 (cdr n)) (equal 1 (cdr d)))
1340 ;; sum(binomial(a-k,b+k),k,l,h)=sum(binomial(a+b-k,k),k,l+b,h+b)
1341 (let ((h1 (m+ h (car d)))
1342 (l1 (m+ l (car d)))
1343 (a1 (m+ (car n) (car d))))
1344 ;; sum(binomial(a1-k,k),k,0,floor(a1/2))=fib(a1+1)
1345 ;; we only do sums with h>floor(a1/2)
1346 (if (and (integerp l1)
1347 (member (asksign (m- h1 (m// a1 2))) '($zero $positive) :test #'eq))
1348 (progn
1349 (adsum (m* y ($fib (m+ a1 1))))
1350 (when (> l1 0)
1351 (adsum (m* -1 y (dosum (list '(%binomial) (m- a1 *var*) *var*)
1352 *var* 0 (m- l1 1) t :evaluate-summand nil)))))
1353 (adusum e))))
1354
1355 ;; sum(binomial(n,2*k),k,0,floor(n/2))=2^(n-1)
1356 ;; sum(binomial(n,2*k+1),k,0,floor((n-1)/2))=2^(n-1)
1357 ((and (equal 0 (cdr n)) (equal 2 (cdr d)))
1358 ;; sum(binomial(a,2*k+b),k,l,h)=sum(binomial(a,2*k),k,l+b/2,h+b/2), b even
1359 ;; 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
1360 (let ((a (car n))
1361 (r1 (if (oddp (car d)) 1 0))
1362 (l1 (if (oddp (car d))
1363 (m+ l (truncate (1- (car d)) 2))
1364 (m+ l (truncate (car d) 2)))))
1365 (when (and (integerp l1)
1366 (member (asksign (m- a hi)) '($zero $positive) :test #'eq))
1367 (adsum (m* y (m^ 2 (m- a 1))))
1368 (when (> l1 0)
1369 (adsum (m* -1 y (dosum (list '(%binomial) a (m+ *var* *var* r1))
1370 *var* 0 (m- l1 1) t :evaluate-summand nil)))))))
1371
1372 ;; other sums we can't do
1373 (t
1374 (adusum e)))))
1375
1376 ;; product routines
1377
1378 (defmspec $product (l)
1379 (setq l (cdr l))
1380 (cond ((not (= (length l) 4)) (merror (intl:gettext "product: expected exactly four arguments.")))
1381 ((dosum (car l) (cadr l) (meval (caddr l)) (meval (cadddr l)) nil :evaluate-summand t))))
1382
1383 (declare-top (special $ratsimpexpons))
1384
1385 ;; Is this guy actually looking at the value of its middle arg?
1386
1387 (defun simpprod (x y z)
1388 (let (($ratsimpexpons t))
1389 (cond ((equal y 1)
1390 (setq y (simplifya (cadr x) z)))
1391 ((setq y (simptimes (list '(mexpt) (cadr x) y) 1 z)))))
1392 (simpprod1 y (caddr x)
1393 (simplifya (cadddr x) z)
1394 (simplifya (cadr (cdddr x)) z)))
1395
1396 (defmfun $taytorat (e)
1397 (cond ((mbagp e) (cons (car e) (mapcar #'$taytorat (cdr e))))
1398 ((or (atom e) (not (member 'trunc (cdar e) :test #'eq))) (ratf e))
1399 ((catch 'srrat (srrat e)))
1400 (t (ratf ($ratdisrep e)))))
1401
1402 (defun srrat (e)
1403 (cons (list 'mrat 'simp (caddar e) (cadddr (car e)))
1404 (srrat2 (cdr e))))
1405
1406 (defun srrat2 (e)
1407 (if (pscoefp e) e (srrat3 (terms e) (gvar e))))
1408
1409 (defun srrat3 (l *var*)
1410 (cond ((null l) '(0 . 1))
1411 ((null (=1 (cdr (le l))))
1412 (throw 'srrat nil))
1413 ((null (n-term l))
1414 (rattimes (cons (list *var* (car (le l)) 1) 1)
1415 (srrat2 (lc l))
1416 t))
1417 ((ratplus
1418 (rattimes (cons (list *var* (car (le l)) 1) 1)
1419 (srrat2 (lc l))
1420 t)
1421 (srrat3 (n-term l) *var*)))))
1422
1423
1424 (declare-top (special $props *i))
1425
1426 (defmspec $deftaylor (l)
1427 (prog (fun series param op ops)
1428 a (when (null (setq l (cdr l))) (return (cons '(mlist) ops)))
1429 (setq fun (meval (car l)) series (meval (cadr l)) l (cdr l) param () )
1430 (when (or (atom fun)
1431 (if (eq (caar fun) 'mqapply)
1432 (or (cdddr fun) ; must be one parameter
1433 (null (cddr fun)) ; must have exactly one
1434 (do ((subs (cdadr fun) (cdr subs)))
1435 ((null subs)
1436 (setq op (caaadr fun))
1437 (when (cddr fun)
1438 (setq param (caddr fun)))
1439 '())
1440 (unless (atom (car subs)) (return 't))))
1441 (progn
1442 (setq op (caar fun))
1443 (when (cdr fun) (setq param (cadr fun)))
1444 (or (and (zl-get op 'op) (not (eq op 'mfactorial)))
1445 (not (atom (cadr fun)))
1446 (not (= (length fun) 2))))))
1447 (merror (intl:gettext "deftaylor: don't know how to handle this function: ~M") fun))
1448 (when (zl-get op 'sp2)
1449 (mtell (intl:gettext "deftaylor: redefining ~:M.~%") op))
1450 (when param (setq series (subst 'sp2var param series)))
1451 (setq series (subsum '*index series))
1452 (putprop op series 'sp2)
1453 (when (eq (caar fun) 'mqapply)
1454 (putprop op (cdadr fun) 'sp2subs))
1455 (add2lnc op $props)
1456 (push op ops)
1457 (go a)))
1458
1459 (defun subsum (*i e) (susum1 e))
1460
1461 (defun susum1 (e)
1462 (cond ((atom e) e)
1463 ((eq (caar e) '%sum)
1464 (if (null (smonop (cadr e) 'sp2var))
1465 (merror (intl:gettext "deftaylor: argument must be a power series at 0."))
1466 (subst *i (caddr e) e)))
1467 (t (recur-apply #'susum1 e))))
1468
1469 (declare-top (special varlist genvar $factorflag $ratfac *p* *var* *x*))
1470
1471 (defmfun $polydecomp (e v)
1472 (let ((varlist (list v))
1473 (genvar nil)
1474 *var* p den $factorflag $ratfac)
1475 (setq p (cdr (ratf (ratdisrep e)))
1476 *var* (cdr (ratf v)))
1477 (cond ((or (null (cdr *var*))
1478 (null (equal (cdar *var*) '(1 1))))
1479 (merror (intl:gettext "polydecomp: second argument must be an atom; found ~M") v))
1480 (t (setq *var* (caar *var*))))
1481 (cond ((or (pcoefp (cdr p))
1482 (null (eq (cadr p) *var*)))
1483 (setq den (cdr p)
1484 p (car p)))
1485 (t (merror (intl:gettext "polydecomp: cannot apply 'polydecomp' to a rational function."))))
1486 (cons '(mlist)
1487 (cond ((or (pcoefp p)
1488 (null (eq (car p) *var*)))
1489 (list (rdis (cons p den))))
1490 (t (setq p (pdecomp p *var*))
1491 (do ((l
1492 (setq p (mapcar #'(lambda (q) (cons q 1)) p))
1493 (cdr l))
1494 (a))
1495 ((null l)
1496 (cons (rdis (cons (caar p)
1497 (ptimes (cdar p) den)))
1498 (mapcar #'rdis (cdr p))))
1499 (cond ((setq a (pdecpow (car l) *var*))
1500 (rplaca l (car a))
1501 (cond ((cdr l)
1502 (rplacd l
1503 (cons (ratplus
1504 (rattimes
1505 (cadr l)
1506 (cons (ptterm (cdaadr a) 1)
1507 (cdadr a))
1508 t)
1509 (cons
1510 (ptterm (cdaadr a) 0)
1511 (cdadr a)))
1512 (cddr l))))
1513 ((equal (cadr a)
1514 (cons (list *var* 1 1) 1)))
1515 (t (rplacd l (list (cadr a)))))))))))))
1516
1517
1518 ;;; POLYDECOMP is like $POLYDECOMP except it takes a poly in *POLY* format (as
1519 ;;; defined in SOLVE) (numerator of a RAT form) and returns a list of
1520 ;;; headerless rat forms. In otherwords, it is $POLYDECOMP minus type checking
1521 ;;; and conversions to/from general representation which SOLVE doesn't
1522 ;;; want/need on a general basis.
1523 ;;; It is used in the SOLVE package and as such it should have an autoload
1524 ;;; property
1525
1526 (defun polydecomp (p *var*)
1527 (let ($factorflag $ratfac)
1528 (cond ((or (pcoefp p)
1529 (null (eq (car p) *var*)))
1530 (cons p nil))
1531 (t (setq p (pdecomp p *var*))
1532 (do ((l (setq p (mapcar #'(lambda (q) (cons q 1)) p))
1533 (cdr l))
1534 (a))
1535 ((null l)
1536 (cons (cons (caar p)
1537 (cdar p))
1538 (cdr p)))
1539 (cond ((setq a (pdecpow (car l) *var*))
1540 (rplaca l (car a))
1541 (cond ((cdr l)
1542 (rplacd l
1543 (cons (ratplus
1544 (rattimes
1545 (cadr l)
1546 (cons (ptterm (cdaadr a) 1)
1547 (cdadr a))
1548 t)
1549 (cons
1550 (ptterm (cdaadr a) 0)
1551 (cdadr a)))
1552 (cddr l))))
1553 ((equal (cadr a)
1554 (cons (list *var* 1 1) 1)))
1555 (t (rplacd l (list (cadr a))))))))))))
1556
1557
1558
1559 (defun pdecred (f h *var*) ;f = g(h(*var*))
1560 (cond ((or (pcoefp h) (null (eq (car h) *var*))
1561 (equal (cadr h) 1)
1562 (null (zerop (rem (cadr f) (cadr h))))
1563 (and (null (pzerop (caadr (setq f (pdivide f h)))))
1564 (equal (cdadr f) 1)))
1565 nil)
1566 (t (do ((q (pdivide (caar f) h) (pdivide (caar q) h))
1567 (i 1 (1+ i))
1568 (*ans*))
1569 ((pzerop (caar q))
1570 (cond ((and (equal (cdadr q) 1)
1571 (or (pcoefp (caadr q))
1572 (null (eq (caar (cadr q)) *var*))))
1573 (psimp *var* (cons i (cons (caadr q) *ans*))))))
1574 (cond ((and (equal (cdadr q) 1)
1575 (or (pcoefp (caadr q))
1576 (null (eq (caar (cadr q)) *var*))))
1577 (and (null (pzerop (caadr q)))
1578 (setq *ans* (cons i (cons (caadr q) *ans*)))))
1579 (t (return nil)))))))
1580
1581 (defun pdecomp (p *var*)
1582 (let ((c (ptterm (cdr p) 0))
1583 (a) (*x* (list *var* 1 1)))
1584 (cons (pcplus c (car (setq a (pdecomp* (pdifference p c)))))
1585 (cdr a))))
1586
1587 (defun pdecomp* (*p*)
1588 (let ((a)
1589 (l (pdecgdfrm (pfactor (pquotient *p* *x*)))))
1590 (cond ((or (pdecprimep (cadr *p*))
1591 (null (setq a (pdecomp1 *x* l))))
1592 (list *p*))
1593 (t (append (pdecomp* (car a)) (cdr a))))))
1594
1595 (defun pdecomp1 (prod l)
1596 (cond ((null l)
1597 (and (null (equal (cadr prod) (cadr *p*)))
1598 (setq l (pdecred *p* prod *var*))
1599 (list l prod)))
1600 ((pdecomp1 prod (cdr l)))
1601 (t (pdecomp1 (ptimes (car l) prod) (cdr l)))))
1602
1603 (defun pdecgdfrm (l) ;Get list of divisors
1604 (do ((l (copy-list l ))
1605 (ll (list (car l))
1606 (cons (car l) ll)))
1607 (nil)
1608 (rplaca (cdr l) (1- (cadr l)))
1609 (cond ((signp e (cadr l))
1610 (setq l (cddr l))))
1611 (cond ((null l) (return ll)))))
1612
1613 (defun pdecprimep (x)
1614 (setq x (cfactorw x))
1615 (and (null (cddr x)) (equal (cadr x) 1)))
1616
1617 (defun pdecpow (p *var*)
1618 (setq p (car p))
1619 (let ((p1 (pderivative p *var*))
1620 p2 p1p p1c a lin p2p)
1621 (setq p1p (oldcontent p1)
1622 p1c (car p1p) p1p (cadr p1p))
1623 (setq p2 (pderivative p1 *var*))
1624 (setq p2p (cadr (oldcontent p2)))
1625 (and (setq lin (testdivide p1p p2p))
1626 (null (pcoefp lin))
1627 (eq (car lin) *var*)
1628 (list (ratplus
1629 (rattimes (cons (list *var* (cadr p) 1) 1)
1630 (setq a (ratreduce p1c
1631 (ptimes (cadr p)
1632 (caddr lin))))
1633 t)
1634 (ratdif (cons p 1)
1635 (rattimes a (cons (pexpt lin (cadr p)) 1)
1636 t)))
1637 (cons lin 1)))))
1638
1639 (declare-top (unspecial *mfactl *factlist donel nn* dn*
1640 *var* *ans* *n *a*
1641 *infsumsimp *times *plus sum usum makef))
Maxima, a Computer Algebra System