More evaluation rules: abs(x^n) => abs(x)^n (x > 0, n is real).
[ginac.git] / ginac / inifcns.cpp
1 /** @file inifcns.cpp
2  *
3  *  Implementation of GiNaC's initially known functions. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2011 Johannes Gutenberg University Mainz, Germany
7  *
8  *  This program is free software; you can redistribute it and/or modify
9  *  it under the terms of the GNU General Public License as published by
10  *  the Free Software Foundation; either version 2 of the License, or
11  *  (at your option) any later version.
12  *
13  *  This program is distributed in the hope that it will be useful,
14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
16  *  GNU General Public License for more details.
17  *
18  *  You should have received a copy of the GNU General Public License
19  *  along with this program; if not, write to the Free Software
20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
21  */
22
23 #include "inifcns.h"
24 #include "ex.h"
25 #include "constant.h"
26 #include "lst.h"
27 #include "matrix.h"
28 #include "mul.h"
29 #include "power.h"
30 #include "operators.h"
31 #include "relational.h"
32 #include "pseries.h"
33 #include "symbol.h"
34 #include "symmetry.h"
35 #include "utils.h"
36
37 #include <stdexcept>
38 #include <vector>
39
40 namespace GiNaC {
41
42 //////////
43 // complex conjugate
44 //////////
45
46 static ex conjugate_evalf(const ex & arg)
47 {
48         if (is_exactly_a<numeric>(arg)) {
49                 return ex_to<numeric>(arg).conjugate();
50         }
51         return conjugate_function(arg).hold();
52 }
53
54 static ex conjugate_eval(const ex & arg)
55 {
56         return arg.conjugate();
57 }
58
59 static void conjugate_print_latex(const ex & arg, const print_context & c)
60 {
61         c.s << "\\bar{"; arg.print(c); c.s << "}";
62 }
63
64 static ex conjugate_conjugate(const ex & arg)
65 {
66         return arg;
67 }
68
69 static ex conjugate_real_part(const ex & arg)
70 {
71         return arg.real_part();
72 }
73
74 static ex conjugate_imag_part(const ex & arg)
75 {
76         return -arg.imag_part();
77 }
78
79 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
80                                       evalf_func(conjugate_evalf).
81                                       print_func<print_latex>(conjugate_print_latex).
82                                       conjugate_func(conjugate_conjugate).
83                                       real_part_func(conjugate_real_part).
84                                       imag_part_func(conjugate_imag_part).
85                                       set_name("conjugate","conjugate"));
86
87 //////////
88 // real part
89 //////////
90
91 static ex real_part_evalf(const ex & arg)
92 {
93         if (is_exactly_a<numeric>(arg)) {
94                 return ex_to<numeric>(arg).real();
95         }
96         return real_part_function(arg).hold();
97 }
98
99 static ex real_part_eval(const ex & arg)
100 {
101         return arg.real_part();
102 }
103
104 static void real_part_print_latex(const ex & arg, const print_context & c)
105 {
106         c.s << "\\Re"; arg.print(c); c.s << "";
107 }
108
109 static ex real_part_conjugate(const ex & arg)
110 {
111         return real_part_function(arg).hold();
112 }
113
114 static ex real_part_real_part(const ex & arg)
115 {
116         return real_part_function(arg).hold();
117 }
118
119 static ex real_part_imag_part(const ex & arg)
120 {
121         return 0;
122 }
123
124 REGISTER_FUNCTION(real_part_function, eval_func(real_part_eval).
125                                       evalf_func(real_part_evalf).
126                                       print_func<print_latex>(real_part_print_latex).
127                                       conjugate_func(real_part_conjugate).
128                                       real_part_func(real_part_real_part).
129                                       imag_part_func(real_part_imag_part).
130                                       set_name("real_part","real_part"));
131
132 //////////
133 // imag part
134 //////////
135
136 static ex imag_part_evalf(const ex & arg)
137 {
138         if (is_exactly_a<numeric>(arg)) {
139                 return ex_to<numeric>(arg).imag();
140         }
141         return imag_part_function(arg).hold();
142 }
143
144 static ex imag_part_eval(const ex & arg)
145 {
146         return arg.imag_part();
147 }
148
149 static void imag_part_print_latex(const ex & arg, const print_context & c)
150 {
151         c.s << "\\Im"; arg.print(c); c.s << "";
152 }
153
154 static ex imag_part_conjugate(const ex & arg)
155 {
156         return imag_part_function(arg).hold();
157 }
158
159 static ex imag_part_real_part(const ex & arg)
160 {
161         return imag_part_function(arg).hold();
162 }
163
164 static ex imag_part_imag_part(const ex & arg)
165 {
166         return 0;
167 }
168
169 REGISTER_FUNCTION(imag_part_function, eval_func(imag_part_eval).
170                                       evalf_func(imag_part_evalf).
171                                       print_func<print_latex>(imag_part_print_latex).
172                                       conjugate_func(imag_part_conjugate).
173                                       real_part_func(imag_part_real_part).
174                                       imag_part_func(imag_part_imag_part).
175                                       set_name("imag_part","imag_part"));
176
177 //////////
178 // absolute value
179 //////////
180
181 static ex abs_evalf(const ex & arg)
182 {
183         if (is_exactly_a<numeric>(arg))
184                 return abs(ex_to<numeric>(arg));
185         
186         return abs(arg).hold();
187 }
188
189 static ex abs_eval(const ex & arg)
190 {
191         if (is_exactly_a<numeric>(arg))
192                 return abs(ex_to<numeric>(arg));
193
194         if (arg.info(info_flags::nonnegative))
195                 return arg;
196
197         if (is_ex_the_function(arg, abs))
198                 return arg;
199
200         if (is_ex_the_function(arg, exp))
201                 return exp(arg.op(0).real_part());
202
203         if (is_exactly_a<power>(arg)) {
204                 const ex& base = arg.op(0);
205                 const ex& exponent = arg.op(1);
206                 if (base.info(info_flags::positive) || exponent.info(info_flags::real))
207                         return pow(abs(base), exponent.real_part());
208         }
209
210         return abs(arg).hold();
211 }
212
213 static void abs_print_latex(const ex & arg, const print_context & c)
214 {
215         c.s << "{|"; arg.print(c); c.s << "|}";
216 }
217
218 static void abs_print_csrc_float(const ex & arg, const print_context & c)
219 {
220         c.s << "fabs("; arg.print(c); c.s << ")";
221 }
222
223 static ex abs_conjugate(const ex & arg)
224 {
225         return abs(arg).hold();
226 }
227
228 static ex abs_real_part(const ex & arg)
229 {
230         return abs(arg).hold();
231 }
232
233 static ex abs_imag_part(const ex& arg)
234 {
235         return 0;
236 }
237
238 static ex abs_power(const ex & arg, const ex & exp)
239 {
240         if (arg.is_equal(arg.conjugate()) && ((is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
241                                                 || exp.info(info_flags::even)))
242                 return power(arg, exp);
243         else
244                 return power(abs(arg), exp).hold();
245 }
246
247 REGISTER_FUNCTION(abs, eval_func(abs_eval).
248                        evalf_func(abs_evalf).
249                        print_func<print_latex>(abs_print_latex).
250                        print_func<print_csrc_float>(abs_print_csrc_float).
251                        print_func<print_csrc_double>(abs_print_csrc_float).
252                        conjugate_func(abs_conjugate).
253                        real_part_func(abs_real_part).
254                        imag_part_func(abs_imag_part).
255                        power_func(abs_power));
256
257 //////////
258 // Step function
259 //////////
260
261 static ex step_evalf(const ex & arg)
262 {
263         if (is_exactly_a<numeric>(arg))
264                 return step(ex_to<numeric>(arg));
265         
266         return step(arg).hold();
267 }
268
269 static ex step_eval(const ex & arg)
270 {
271         if (is_exactly_a<numeric>(arg))
272                 return step(ex_to<numeric>(arg));
273         
274         else if (is_exactly_a<mul>(arg) &&
275                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
276                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
277                 if (oc.is_real()) {
278                         if (oc > 0)
279                                 // step(42*x) -> step(x)
280                                 return step(arg/oc).hold();
281                         else
282                                 // step(-42*x) -> step(-x)
283                                 return step(-arg/oc).hold();
284                 }
285                 if (oc.real().is_zero()) {
286                         if (oc.imag() > 0)
287                                 // step(42*I*x) -> step(I*x)
288                                 return step(I*arg/oc).hold();
289                         else
290                                 // step(-42*I*x) -> step(-I*x)
291                                 return step(-I*arg/oc).hold();
292                 }
293         }
294         
295         return step(arg).hold();
296 }
297
298 static ex step_series(const ex & arg,
299                       const relational & rel,
300                       int order,
301                       unsigned options)
302 {
303         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
304         if (arg_pt.info(info_flags::numeric)
305             && ex_to<numeric>(arg_pt).real().is_zero()
306             && !(options & series_options::suppress_branchcut))
307                 throw (std::domain_error("step_series(): on imaginary axis"));
308         
309         epvector seq;
310         seq.push_back(expair(step(arg_pt), _ex0));
311         return pseries(rel,seq);
312 }
313
314 static ex step_conjugate(const ex& arg)
315 {
316         return step(arg).hold();
317 }
318
319 static ex step_real_part(const ex& arg)
320 {
321         return step(arg).hold();
322 }
323
324 static ex step_imag_part(const ex& arg)
325 {
326         return 0;
327 }
328
329 REGISTER_FUNCTION(step, eval_func(step_eval).
330                         evalf_func(step_evalf).
331                         series_func(step_series).
332                         conjugate_func(step_conjugate).
333                         real_part_func(step_real_part).
334                         imag_part_func(step_imag_part));
335
336 //////////
337 // Complex sign
338 //////////
339
340 static ex csgn_evalf(const ex & arg)
341 {
342         if (is_exactly_a<numeric>(arg))
343                 return csgn(ex_to<numeric>(arg));
344         
345         return csgn(arg).hold();
346 }
347
348 static ex csgn_eval(const ex & arg)
349 {
350         if (is_exactly_a<numeric>(arg))
351                 return csgn(ex_to<numeric>(arg));
352         
353         else if (is_exactly_a<mul>(arg) &&
354                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
355                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
356                 if (oc.is_real()) {
357                         if (oc > 0)
358                                 // csgn(42*x) -> csgn(x)
359                                 return csgn(arg/oc).hold();
360                         else
361                                 // csgn(-42*x) -> -csgn(x)
362                                 return -csgn(arg/oc).hold();
363                 }
364                 if (oc.real().is_zero()) {
365                         if (oc.imag() > 0)
366                                 // csgn(42*I*x) -> csgn(I*x)
367                                 return csgn(I*arg/oc).hold();
368                         else
369                                 // csgn(-42*I*x) -> -csgn(I*x)
370                                 return -csgn(I*arg/oc).hold();
371                 }
372         }
373         
374         return csgn(arg).hold();
375 }
376
377 static ex csgn_series(const ex & arg,
378                       const relational & rel,
379                       int order,
380                       unsigned options)
381 {
382         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
383         if (arg_pt.info(info_flags::numeric)
384             && ex_to<numeric>(arg_pt).real().is_zero()
385             && !(options & series_options::suppress_branchcut))
386                 throw (std::domain_error("csgn_series(): on imaginary axis"));
387         
388         epvector seq;
389         seq.push_back(expair(csgn(arg_pt), _ex0));
390         return pseries(rel,seq);
391 }
392
393 static ex csgn_conjugate(const ex& arg)
394 {
395         return csgn(arg).hold();
396 }
397
398 static ex csgn_real_part(const ex& arg)
399 {
400         return csgn(arg).hold();
401 }
402
403 static ex csgn_imag_part(const ex& arg)
404 {
405         return 0;
406 }
407
408 static ex csgn_power(const ex & arg, const ex & exp)
409 {
410         if (is_a<numeric>(exp) && exp.info(info_flags::positive) && ex_to<numeric>(exp).is_integer()) {
411                 if (ex_to<numeric>(exp).is_odd())
412                         return csgn(arg).hold();
413                 else
414                         return power(csgn(arg), _ex2).hold();
415         } else
416                 return power(csgn(arg), exp).hold();
417 }
418
419
420 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
421                         evalf_func(csgn_evalf).
422                         series_func(csgn_series).
423                         conjugate_func(csgn_conjugate).
424                         real_part_func(csgn_real_part).
425                         imag_part_func(csgn_imag_part).
426                         power_func(csgn_power));
427
428
429 //////////
430 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
431 // This function is closely related to the unwinding number K, sometimes found
432 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
433 //////////
434
435 static ex eta_evalf(const ex &x, const ex &y)
436 {
437         // It seems like we basically have to replicate the eval function here,
438         // since the expression might not be fully evaluated yet.
439         if (x.info(info_flags::positive) || y.info(info_flags::positive))
440                 return _ex0;
441
442         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
443                 const numeric nx = ex_to<numeric>(x);
444                 const numeric ny = ex_to<numeric>(y);
445                 const numeric nxy = ex_to<numeric>(x*y);
446                 int cut = 0;
447                 if (nx.is_real() && nx.is_negative())
448                         cut -= 4;
449                 if (ny.is_real() && ny.is_negative())
450                         cut -= 4;
451                 if (nxy.is_real() && nxy.is_negative())
452                         cut += 4;
453                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
454                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
455         }
456
457         return eta(x,y).hold();
458 }
459
460 static ex eta_eval(const ex &x, const ex &y)
461 {
462         // trivial:  eta(x,c) -> 0  if c is real and positive
463         if (x.info(info_flags::positive) || y.info(info_flags::positive))
464                 return _ex0;
465
466         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
467                 // don't call eta_evalf here because it would call Pi.evalf()!
468                 const numeric nx = ex_to<numeric>(x);
469                 const numeric ny = ex_to<numeric>(y);
470                 const numeric nxy = ex_to<numeric>(x*y);
471                 int cut = 0;
472                 if (nx.is_real() && nx.is_negative())
473                         cut -= 4;
474                 if (ny.is_real() && ny.is_negative())
475                         cut -= 4;
476                 if (nxy.is_real() && nxy.is_negative())
477                         cut += 4;
478                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
479                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
480         }
481         
482         return eta(x,y).hold();
483 }
484
485 static ex eta_series(const ex & x, const ex & y,
486                      const relational & rel,
487                      int order,
488                      unsigned options)
489 {
490         const ex x_pt = x.subs(rel, subs_options::no_pattern);
491         const ex y_pt = y.subs(rel, subs_options::no_pattern);
492         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
493             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
494             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
495                         throw (std::domain_error("eta_series(): on discontinuity"));
496         epvector seq;
497         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
498         return pseries(rel,seq);
499 }
500
501 static ex eta_conjugate(const ex & x, const ex & y)
502 {
503         return -eta(x, y).hold();
504 }
505
506 static ex eta_real_part(const ex & x, const ex & y)
507 {
508         return 0;
509 }
510
511 static ex eta_imag_part(const ex & x, const ex & y)
512 {
513         return -I*eta(x, y).hold();
514 }
515
516 REGISTER_FUNCTION(eta, eval_func(eta_eval).
517                        evalf_func(eta_evalf).
518                        series_func(eta_series).
519                        latex_name("\\eta").
520                        set_symmetry(sy_symm(0, 1)).
521                        conjugate_func(eta_conjugate).
522                        real_part_func(eta_real_part).
523                        imag_part_func(eta_imag_part));
524
525
526 //////////
527 // dilogarithm
528 //////////
529
530 static ex Li2_evalf(const ex & x)
531 {
532         if (is_exactly_a<numeric>(x))
533                 return Li2(ex_to<numeric>(x));
534         
535         return Li2(x).hold();
536 }
537
538 static ex Li2_eval(const ex & x)
539 {
540         if (x.info(info_flags::numeric)) {
541                 // Li2(0) -> 0
542                 if (x.is_zero())
543                         return _ex0;
544                 // Li2(1) -> Pi^2/6
545                 if (x.is_equal(_ex1))
546                         return power(Pi,_ex2)/_ex6;
547                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
548                 if (x.is_equal(_ex1_2))
549                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
550                 // Li2(-1) -> -Pi^2/12
551                 if (x.is_equal(_ex_1))
552                         return -power(Pi,_ex2)/_ex12;
553                 // Li2(I) -> -Pi^2/48+Catalan*I
554                 if (x.is_equal(I))
555                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
556                 // Li2(-I) -> -Pi^2/48-Catalan*I
557                 if (x.is_equal(-I))
558                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
559                 // Li2(float)
560                 if (!x.info(info_flags::crational))
561                         return Li2(ex_to<numeric>(x));
562         }
563         
564         return Li2(x).hold();
565 }
566
567 static ex Li2_deriv(const ex & x, unsigned deriv_param)
568 {
569         GINAC_ASSERT(deriv_param==0);
570         
571         // d/dx Li2(x) -> -log(1-x)/x
572         return -log(_ex1-x)/x;
573 }
574
575 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
576 {
577         const ex x_pt = x.subs(rel, subs_options::no_pattern);
578         if (x_pt.info(info_flags::numeric)) {
579                 // First special case: x==0 (derivatives have poles)
580                 if (x_pt.is_zero()) {
581                         // method:
582                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot 
583                         // simply substitute x==0.  The limit, however, exists: it is 1.
584                         // We also know all higher derivatives' limits:
585                         // (d/dx)^n Li2(x) == n!/n^2.
586                         // So the primitive series expansion is
587                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
588                         // and so on.
589                         // We first construct such a primitive series expansion manually in
590                         // a dummy symbol s and then insert the argument's series expansion
591                         // for s.  Reexpanding the resulting series returns the desired
592                         // result.
593                         const symbol s;
594                         ex ser;
595                         // manually construct the primitive expansion
596                         for (int i=1; i<order; ++i)
597                                 ser += pow(s,i) / pow(numeric(i), *_num2_p);
598                         // substitute the argument's series expansion
599                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
600                         // maybe that was terminating, so add a proper order term
601                         epvector nseq;
602                         nseq.push_back(expair(Order(_ex1), order));
603                         ser += pseries(rel, nseq);
604                         // reexpanding it will collapse the series again
605                         return ser.series(rel, order);
606                         // NB: Of course, this still does not allow us to compute anything
607                         // like sin(Li2(x)).series(x==0,2), since then this code here is
608                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
609                         // substitution x==0.  Probably limits *are* needed for the general
610                         // cases.  In case L'Hospital's rule is implemented for limits and
611                         // basic::series() takes care of this, this whole block is probably
612                         // obsolete!
613                 }
614                 // second special case: x==1 (branch point)
615                 if (x_pt.is_equal(_ex1)) {
616                         // method:
617                         // construct series manually in a dummy symbol s
618                         const symbol s;
619                         ex ser = zeta(_ex2);
620                         // manually construct the primitive expansion
621                         for (int i=1; i<order; ++i)
622                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
623                         // substitute the argument's series expansion
624                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
625                         // maybe that was terminating, so add a proper order term
626                         epvector nseq;
627                         nseq.push_back(expair(Order(_ex1), order));
628                         ser += pseries(rel, nseq);
629                         // reexpanding it will collapse the series again
630                         return ser.series(rel, order);
631                 }
632                 // third special case: x real, >=1 (branch cut)
633                 if (!(options & series_options::suppress_branchcut) &&
634                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
635                         // method:
636                         // This is the branch cut: assemble the primitive series manually
637                         // and then add the corresponding complex step function.
638                         const symbol &s = ex_to<symbol>(rel.lhs());
639                         const ex point = rel.rhs();
640                         const symbol foo;
641                         epvector seq;
642                         // zeroth order term:
643                         seq.push_back(expair(Li2(x_pt), _ex0));
644                         // compute the intermediate terms:
645                         ex replarg = series(Li2(x), s==foo, order);
646                         for (size_t i=1; i<replarg.nops()-1; ++i)
647                                 seq.push_back(expair((replarg.op(i)/power(s-foo,i)).series(foo==point,1,options).op(0).subs(foo==s, subs_options::no_pattern),i));
648                         // append an order term:
649                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
650                         return pseries(rel, seq);
651                 }
652         }
653         // all other cases should be safe, by now:
654         throw do_taylor();  // caught by function::series()
655 }
656
657 static ex Li2_conjugate(const ex & x)
658 {
659         // conjugate(Li2(x))==Li2(conjugate(x)) unless on the branch cuts which
660         // run along the positive real axis beginning at 1.
661         if (x.info(info_flags::negative)) {
662                 return Li2(x).hold();
663         }
664         if (is_exactly_a<numeric>(x) &&
665             (!x.imag_part().is_zero() || x < *_num1_p)) {
666                 return Li2(x.conjugate());
667         }
668         return conjugate_function(Li2(x)).hold();
669 }
670
671 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
672                        evalf_func(Li2_evalf).
673                        derivative_func(Li2_deriv).
674                        series_func(Li2_series).
675                        conjugate_func(Li2_conjugate).
676                        latex_name("\\mathrm{Li}_2"));
677
678 //////////
679 // trilogarithm
680 //////////
681
682 static ex Li3_eval(const ex & x)
683 {
684         if (x.is_zero())
685                 return x;
686         return Li3(x).hold();
687 }
688
689 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
690                        latex_name("\\mathrm{Li}_3"));
691
692 //////////
693 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
694 //////////
695
696 static ex zetaderiv_eval(const ex & n, const ex & x)
697 {
698         if (n.info(info_flags::numeric)) {
699                 // zetaderiv(0,x) -> zeta(x)
700                 if (n.is_zero())
701                         return zeta(x).hold();
702         }
703         
704         return zetaderiv(n, x).hold();
705 }
706
707 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
708 {
709         GINAC_ASSERT(deriv_param<2);
710         
711         if (deriv_param==0) {
712                 // d/dn zeta(n,x)
713                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
714         }
715         // d/dx psi(n,x)
716         return zetaderiv(n+1,x);
717 }
718
719 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
720                                  derivative_func(zetaderiv_deriv).
721                                  latex_name("\\zeta^\\prime"));
722
723 //////////
724 // factorial
725 //////////
726
727 static ex factorial_evalf(const ex & x)
728 {
729         return factorial(x).hold();
730 }
731
732 static ex factorial_eval(const ex & x)
733 {
734         if (is_exactly_a<numeric>(x))
735                 return factorial(ex_to<numeric>(x));
736         else
737                 return factorial(x).hold();
738 }
739
740 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
741 {
742         if (is_exactly_a<symbol>(x) ||
743             is_exactly_a<constant>(x) ||
744                 is_exactly_a<function>(x)) {
745                 x.print(c); c.s << "!";
746         } else {
747                 c.s << "("; x.print(c); c.s << ")!";
748         }
749 }
750
751 static ex factorial_conjugate(const ex & x)
752 {
753         return factorial(x).hold();
754 }
755
756 static ex factorial_real_part(const ex & x)
757 {
758         return factorial(x).hold();
759 }
760
761 static ex factorial_imag_part(const ex & x)
762 {
763         return 0;
764 }
765
766 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
767                              evalf_func(factorial_evalf).
768                              print_func<print_dflt>(factorial_print_dflt_latex).
769                              print_func<print_latex>(factorial_print_dflt_latex).
770                              conjugate_func(factorial_conjugate).
771                              real_part_func(factorial_real_part).
772                              imag_part_func(factorial_imag_part));
773
774 //////////
775 // binomial
776 //////////
777
778 static ex binomial_evalf(const ex & x, const ex & y)
779 {
780         return binomial(x, y).hold();
781 }
782
783 static ex binomial_sym(const ex & x, const numeric & y)
784 {
785         if (y.is_integer()) {
786                 if (y.is_nonneg_integer()) {
787                         const unsigned N = y.to_int();
788                         if (N == 0) return _ex1;
789                         if (N == 1) return x;
790                         ex t = x.expand();
791                         for (unsigned i = 2; i <= N; ++i)
792                                 t = (t * (x + i - y - 1)).expand() / i;
793                         return t;
794                 } else
795                         return _ex0;
796         }
797
798         return binomial(x, y).hold();
799 }
800
801 static ex binomial_eval(const ex & x, const ex &y)
802 {
803         if (is_exactly_a<numeric>(y)) {
804                 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
805                         return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
806                 else
807                         return binomial_sym(x, ex_to<numeric>(y));
808         } else
809                 return binomial(x, y).hold();
810 }
811
812 // At the moment the numeric evaluation of a binomail function always
813 // gives a real number, but if this would be implemented using the gamma
814 // function, also complex conjugation should be changed (or rather, deleted).
815 static ex binomial_conjugate(const ex & x, const ex & y)
816 {
817         return binomial(x,y).hold();
818 }
819
820 static ex binomial_real_part(const ex & x, const ex & y)
821 {
822         return binomial(x,y).hold();
823 }
824
825 static ex binomial_imag_part(const ex & x, const ex & y)
826 {
827         return 0;
828 }
829
830 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
831                             evalf_func(binomial_evalf).
832                             conjugate_func(binomial_conjugate).
833                             real_part_func(binomial_real_part).
834                             imag_part_func(binomial_imag_part));
835
836 //////////
837 // Order term function (for truncated power series)
838 //////////
839
840 static ex Order_eval(const ex & x)
841 {
842         if (is_exactly_a<numeric>(x)) {
843                 // O(c) -> O(1) or 0
844                 if (!x.is_zero())
845                         return Order(_ex1).hold();
846                 else
847                         return _ex0;
848         } else if (is_exactly_a<mul>(x)) {
849                 const mul &m = ex_to<mul>(x);
850                 // O(c*expr) -> O(expr)
851                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
852                         return Order(x / m.op(m.nops() - 1)).hold();
853         }
854         return Order(x).hold();
855 }
856
857 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
858 {
859         // Just wrap the function into a pseries object
860         epvector new_seq;
861         GINAC_ASSERT(is_a<symbol>(r.lhs()));
862         const symbol &s = ex_to<symbol>(r.lhs());
863         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
864         return pseries(r, new_seq);
865 }
866
867 static ex Order_conjugate(const ex & x)
868 {
869         return Order(x).hold();
870 }
871
872 static ex Order_real_part(const ex & x)
873 {
874         return Order(x).hold();
875 }
876
877 static ex Order_imag_part(const ex & x)
878 {
879         if(x.info(info_flags::real))
880                 return 0;
881         return Order(x).hold();
882 }
883
884 // Differentiation is handled in function::derivative because of its special requirements
885
886 REGISTER_FUNCTION(Order, eval_func(Order_eval).
887                          series_func(Order_series).
888                          latex_name("\\mathcal{O}").
889                          conjugate_func(Order_conjugate).
890                          real_part_func(Order_real_part).
891                          imag_part_func(Order_imag_part));
892
893 //////////
894 // Solve linear system
895 //////////
896
897 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
898 {
899         // solve a system of linear equations
900         if (eqns.info(info_flags::relation_equal)) {
901                 if (!symbols.info(info_flags::symbol))
902                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
903                 const ex sol = lsolve(lst(eqns),lst(symbols));
904                 
905                 GINAC_ASSERT(sol.nops()==1);
906                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
907                 
908                 return sol.op(0).op(1); // return rhs of first solution
909         }
910         
911         // syntax checks
912         if (!eqns.info(info_flags::list)) {
913                 throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
914         }
915         for (size_t i=0; i<eqns.nops(); i++) {
916                 if (!eqns.op(i).info(info_flags::relation_equal)) {
917                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
918                 }
919         }
920         if (!symbols.info(info_flags::list)) {
921                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
922         }
923         for (size_t i=0; i<symbols.nops(); i++) {
924                 if (!symbols.op(i).info(info_flags::symbol)) {
925                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
926                 }
927         }
928         
929         // build matrix from equation system
930         matrix sys(eqns.nops(),symbols.nops());
931         matrix rhs(eqns.nops(),1);
932         matrix vars(symbols.nops(),1);
933         
934         for (size_t r=0; r<eqns.nops(); r++) {
935                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
936                 ex linpart = eq;
937                 for (size_t c=0; c<symbols.nops(); c++) {
938                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
939                         linpart -= co*symbols.op(c);
940                         sys(r,c) = co;
941                 }
942                 linpart = linpart.expand();
943                 rhs(r,0) = -linpart;
944         }
945         
946         // test if system is linear and fill vars matrix
947         for (size_t i=0; i<symbols.nops(); i++) {
948                 vars(i,0) = symbols.op(i);
949                 if (sys.has(symbols.op(i)))
950                         throw(std::logic_error("lsolve: system is not linear"));
951                 if (rhs.has(symbols.op(i)))
952                         throw(std::logic_error("lsolve: system is not linear"));
953         }
954         
955         matrix solution;
956         try {
957                 solution = sys.solve(vars,rhs,options);
958         } catch (const std::runtime_error & e) {
959                 // Probably singular matrix or otherwise overdetermined system:
960                 // It is consistent to return an empty list
961                 return lst();
962         }
963         GINAC_ASSERT(solution.cols()==1);
964         GINAC_ASSERT(solution.rows()==symbols.nops());
965         
966         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
967         lst sollist;
968         for (size_t i=0; i<symbols.nops(); i++)
969                 sollist.append(symbols.op(i)==solution(i,0));
970         
971         return sollist;
972 }
973
974 //////////
975 // Find real root of f(x) numerically
976 //////////
977
978 const numeric
979 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
980 {
981         if (!x1.is_real() || !x2.is_real()) {
982                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
983         }
984         if (x1==x2) {
985                 throw std::runtime_error("fsolve(): vanishing interval");
986         }
987         // xx[0] == left interval limit, xx[1] == right interval limit.
988         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
989         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
990         numeric xx[2] = { x1<x2 ? x1 : x2,
991                           x1<x2 ? x2 : x1 };
992         ex f;
993         if (is_a<relational>(f_in)) {
994                 f = f_in.lhs()-f_in.rhs();
995         } else {
996                 f = f_in;
997         }
998         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
999                             f.subs(x==xx[1]).evalf() };
1000         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
1001                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1002         }
1003         numeric fx[2] = { ex_to<numeric>(fx_[0]),
1004                           ex_to<numeric>(fx_[1]) };
1005         if (!fx[0].is_real() || !fx[1].is_real()) {
1006                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
1007         }
1008         if (fx[0]*fx[1]>=0) {
1009                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
1010         }
1011
1012         // The Newton-Raphson method has quadratic convergence!  Simply put, it
1013         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
1014         const ex ff = normal(-f/f.diff(x));
1015         int side = 0;  // Start at left interval limit.
1016         numeric xxprev;
1017         numeric fxprev;
1018         do {
1019                 xxprev = xx[side];
1020                 fxprev = fx[side];
1021                 ex dx_ = ff.subs(x == xx[side]).evalf();
1022                 if (!is_a<numeric>(dx_))
1023                         throw std::runtime_error("fsolve(): function derivative does not evaluate numerically");
1024                 xx[side] += ex_to<numeric>(dx_);
1025                 // Now check if Newton-Raphson method shot out of the interval 
1026                 bool bad_shot = (side == 0 && xx[0] < xxprev) || 
1027                                 (side == 1 && xx[1] > xxprev) || xx[0] > xx[1];
1028                 if (!bad_shot) {
1029                         // Compute f(x) only if new x is inside the interval.
1030                         // The function might be difficult to compute numerically
1031                         // or even ill defined outside the interval. Also it's
1032                         // a small optimization. 
1033                         ex f_x = f.subs(x == xx[side]).evalf();
1034                         if (!is_a<numeric>(f_x))
1035                                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
1036                         fx[side] = ex_to<numeric>(f_x);
1037                 }
1038                 if (bad_shot) {
1039                         // Oops, Newton-Raphson method shot out of the interval.
1040                         // Restore, and try again with the other side instead!
1041                         xx[side] = xxprev;
1042                         fx[side] = fxprev;
1043                         side = !side;
1044                         xxprev = xx[side];
1045                         fxprev = fx[side];
1046
1047                         ex dx_ = ff.subs(x == xx[side]).evalf();
1048                         if (!is_a<numeric>(dx_))
1049                                 throw std::runtime_error("fsolve(): function derivative does not evaluate numerically [2]");
1050                         xx[side] += ex_to<numeric>(dx_);
1051
1052                         ex f_x = f.subs(x==xx[side]).evalf();
1053                         if (!is_a<numeric>(f_x))
1054                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [2]");
1055                         fx[side] = ex_to<numeric>(f_x);
1056                 }
1057                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
1058                         // Oops, the root isn't bracketed any more.
1059                         // Restore, and perform a bisection!
1060                         xx[side] = xxprev;
1061                         fx[side] = fxprev;
1062
1063                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
1064                         // they occur pretty often when Newton-Raphson arrives at an x too
1065                         // close to the result on one side of the interval and
1066                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
1067                         // precision errors! Recall that this function does not have a
1068                         // precision goal as one of its arguments but instead relies on
1069                         // x converging to a fixed point. We speed up the (safe but slow)
1070                         // bisection method by mixing in a dash of the (unsafer but faster)
1071                         // secant method: Instead of splitting the interval at the
1072                         // arithmetic mean (bisection), we split it nearer to the root as
1073                         // determined by the secant between the values xx[0] and xx[1].
1074                         // Don't set the secant_weight to one because that could disturb
1075                         // the convergence in some corner cases!
1076                         static const double secant_weight = 0.984375;  // == 63/64 < 1
1077                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
1078                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
1079                         ex fxmid_ = f.subs(x == xxmid).evalf();
1080                         if (!is_a<numeric>(fxmid_))
1081                                 throw std::runtime_error("fsolve(): function does not evaluate numerically [3]");
1082                         numeric fxmid = ex_to<numeric>(fxmid_);
1083                         if (fxmid.is_zero()) {
1084                                 // Luck strikes...
1085                                 return xxmid;
1086                         }
1087                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
1088                                 side = !side;
1089                         }
1090                         xxprev = xx[side];
1091                         fxprev = fx[side];
1092                         xx[side] = xxmid;
1093                         fx[side] = fxmid;
1094                 }
1095         } while (xxprev!=xx[side]);
1096         return xxprev;
1097 }
1098
1099
1100 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
1101  * for static lib (so ginsh will see them). */
1102 unsigned force_include_tgamma = tgamma_SERIAL::serial;
1103 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
1104
1105 } // namespace GiNaC