47252a8f0d85724a4ccccd02d11382ccbadfd265
[ginac.git] / ginac / inifcns.cpp
1 /** @file inifcns.cpp
2  *
3  *  Implementation of GiNaC's initially known functions. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2005 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 <vector>
24 #include <stdexcept>
25
26 #include "inifcns.h"
27 #include "ex.h"
28 #include "constant.h"
29 #include "lst.h"
30 #include "matrix.h"
31 #include "mul.h"
32 #include "power.h"
33 #include "operators.h"
34 #include "relational.h"
35 #include "pseries.h"
36 #include "symbol.h"
37 #include "symmetry.h"
38 #include "utils.h"
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 REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
70                                       evalf_func(conjugate_evalf).
71                                       print_func<print_latex>(conjugate_print_latex).
72                                       conjugate_func(conjugate_conjugate).
73                                       set_name("conjugate","conjugate"));
74
75 //////////
76 // absolute value
77 //////////
78
79 static ex abs_evalf(const ex & arg)
80 {
81         if (is_exactly_a<numeric>(arg))
82                 return abs(ex_to<numeric>(arg));
83         
84         return abs(arg).hold();
85 }
86
87 static ex abs_eval(const ex & arg)
88 {
89         if (is_exactly_a<numeric>(arg))
90                 return abs(ex_to<numeric>(arg));
91         else
92                 return abs(arg).hold();
93 }
94
95 static void abs_print_latex(const ex & arg, const print_context & c)
96 {
97         c.s << "{|"; arg.print(c); c.s << "|}";
98 }
99
100 static void abs_print_csrc_float(const ex & arg, const print_context & c)
101 {
102         c.s << "fabs("; arg.print(c); c.s << ")";
103 }
104
105 static ex abs_conjugate(const ex & arg)
106 {
107         return abs(arg);
108 }
109
110 static ex abs_power(const ex & arg, const ex & exp)
111 {
112         if (arg.is_equal(arg.conjugate()) && is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
113                 return power(arg, exp);
114         else
115                 return power(abs(arg), exp).hold();
116 }
117
118 REGISTER_FUNCTION(abs, eval_func(abs_eval).
119                        evalf_func(abs_evalf).
120                        print_func<print_latex>(abs_print_latex).
121                        print_func<print_csrc_float>(abs_print_csrc_float).
122                        print_func<print_csrc_double>(abs_print_csrc_float).
123                        conjugate_func(abs_conjugate).
124                        power_func(abs_power));
125
126 //////////
127 // Step function
128 //////////
129
130 static ex step_evalf(const ex & arg)
131 {
132         if (is_exactly_a<numeric>(arg))
133                 return step(ex_to<numeric>(arg));
134         
135         return step(arg).hold();
136 }
137
138 static ex step_eval(const ex & arg)
139 {
140         if (is_exactly_a<numeric>(arg))
141                 return step(ex_to<numeric>(arg));
142         
143         else if (is_exactly_a<mul>(arg) &&
144                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
145                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
146                 if (oc.is_real()) {
147                         if (oc > 0)
148                                 // step(42*x) -> step(x)
149                                 return step(arg/oc).hold();
150                         else
151                                 // step(-42*x) -> step(-x)
152                                 return step(-arg/oc).hold();
153                 }
154                 if (oc.real().is_zero()) {
155                         if (oc.imag() > 0)
156                                 // step(42*I*x) -> step(I*x)
157                                 return step(I*arg/oc).hold();
158                         else
159                                 // step(-42*I*x) -> step(-I*x)
160                                 return step(-I*arg/oc).hold();
161                 }
162         }
163         
164         return step(arg).hold();
165 }
166
167 static ex step_series(const ex & arg,
168                       const relational & rel,
169                       int order,
170                       unsigned options)
171 {
172         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
173         if (arg_pt.info(info_flags::numeric)
174             && ex_to<numeric>(arg_pt).real().is_zero()
175             && !(options & series_options::suppress_branchcut))
176                 throw (std::domain_error("step_series(): on imaginary axis"));
177         
178         epvector seq;
179         seq.push_back(expair(step(arg_pt), _ex0));
180         return pseries(rel,seq);
181 }
182
183 static ex step_conjugate(const ex& arg)
184 {
185         return step(arg);
186 }
187
188 REGISTER_FUNCTION(step, eval_func(step_eval).
189                         evalf_func(step_evalf).
190                         series_func(step_series).
191                         conjugate_func(step_conjugate));
192
193 //////////
194 // Complex sign
195 //////////
196
197 static ex csgn_evalf(const ex & arg)
198 {
199         if (is_exactly_a<numeric>(arg))
200                 return csgn(ex_to<numeric>(arg));
201         
202         return csgn(arg).hold();
203 }
204
205 static ex csgn_eval(const ex & arg)
206 {
207         if (is_exactly_a<numeric>(arg))
208                 return csgn(ex_to<numeric>(arg));
209         
210         else if (is_exactly_a<mul>(arg) &&
211                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
212                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
213                 if (oc.is_real()) {
214                         if (oc > 0)
215                                 // csgn(42*x) -> csgn(x)
216                                 return csgn(arg/oc).hold();
217                         else
218                                 // csgn(-42*x) -> -csgn(x)
219                                 return -csgn(arg/oc).hold();
220                 }
221                 if (oc.real().is_zero()) {
222                         if (oc.imag() > 0)
223                                 // csgn(42*I*x) -> csgn(I*x)
224                                 return csgn(I*arg/oc).hold();
225                         else
226                                 // csgn(-42*I*x) -> -csgn(I*x)
227                                 return -csgn(I*arg/oc).hold();
228                 }
229         }
230         
231         return csgn(arg).hold();
232 }
233
234 static ex csgn_series(const ex & arg,
235                       const relational & rel,
236                       int order,
237                       unsigned options)
238 {
239         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
240         if (arg_pt.info(info_flags::numeric)
241             && ex_to<numeric>(arg_pt).real().is_zero()
242             && !(options & series_options::suppress_branchcut))
243                 throw (std::domain_error("csgn_series(): on imaginary axis"));
244         
245         epvector seq;
246         seq.push_back(expair(csgn(arg_pt), _ex0));
247         return pseries(rel,seq);
248 }
249
250 static ex csgn_conjugate(const ex& arg)
251 {
252         return csgn(arg);
253 }
254
255 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
256                         evalf_func(csgn_evalf).
257                         series_func(csgn_series).
258                         conjugate_func(csgn_conjugate));
259
260
261 //////////
262 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
263 // This function is closely related to the unwinding number K, sometimes found
264 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
265 //////////
266
267 static ex eta_evalf(const ex &x, const ex &y)
268 {
269         // It seems like we basically have to replicate the eval function here,
270         // since the expression might not be fully evaluated yet.
271         if (x.info(info_flags::positive) || y.info(info_flags::positive))
272                 return _ex0;
273
274         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
275                 const numeric nx = ex_to<numeric>(x);
276                 const numeric ny = ex_to<numeric>(y);
277                 const numeric nxy = ex_to<numeric>(x*y);
278                 int cut = 0;
279                 if (nx.is_real() && nx.is_negative())
280                         cut -= 4;
281                 if (ny.is_real() && ny.is_negative())
282                         cut -= 4;
283                 if (nxy.is_real() && nxy.is_negative())
284                         cut += 4;
285                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
286                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
287         }
288
289         return eta(x,y).hold();
290 }
291
292 static ex eta_eval(const ex &x, const ex &y)
293 {
294         // trivial:  eta(x,c) -> 0  if c is real and positive
295         if (x.info(info_flags::positive) || y.info(info_flags::positive))
296                 return _ex0;
297
298         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
299                 // don't call eta_evalf here because it would call Pi.evalf()!
300                 const numeric nx = ex_to<numeric>(x);
301                 const numeric ny = ex_to<numeric>(y);
302                 const numeric nxy = ex_to<numeric>(x*y);
303                 int cut = 0;
304                 if (nx.is_real() && nx.is_negative())
305                         cut -= 4;
306                 if (ny.is_real() && ny.is_negative())
307                         cut -= 4;
308                 if (nxy.is_real() && nxy.is_negative())
309                         cut += 4;
310                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
311                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
312         }
313         
314         return eta(x,y).hold();
315 }
316
317 static ex eta_series(const ex & x, const ex & y,
318                      const relational & rel,
319                      int order,
320                      unsigned options)
321 {
322         const ex x_pt = x.subs(rel, subs_options::no_pattern);
323         const ex y_pt = y.subs(rel, subs_options::no_pattern);
324         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
325             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
326             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
327                         throw (std::domain_error("eta_series(): on discontinuity"));
328         epvector seq;
329         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
330         return pseries(rel,seq);
331 }
332
333 static ex eta_conjugate(const ex & x, const ex & y)
334 {
335         return -eta(x,y);
336 }
337
338 REGISTER_FUNCTION(eta, eval_func(eta_eval).
339                        evalf_func(eta_evalf).
340                        series_func(eta_series).
341                        latex_name("\\eta").
342                        set_symmetry(sy_symm(0, 1)).
343                        conjugate_func(eta_conjugate));
344
345
346 //////////
347 // dilogarithm
348 //////////
349
350 static ex Li2_evalf(const ex & x)
351 {
352         if (is_exactly_a<numeric>(x))
353                 return Li2(ex_to<numeric>(x));
354         
355         return Li2(x).hold();
356 }
357
358 static ex Li2_eval(const ex & x)
359 {
360         if (x.info(info_flags::numeric)) {
361                 // Li2(0) -> 0
362                 if (x.is_zero())
363                         return _ex0;
364                 // Li2(1) -> Pi^2/6
365                 if (x.is_equal(_ex1))
366                         return power(Pi,_ex2)/_ex6;
367                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
368                 if (x.is_equal(_ex1_2))
369                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
370                 // Li2(-1) -> -Pi^2/12
371                 if (x.is_equal(_ex_1))
372                         return -power(Pi,_ex2)/_ex12;
373                 // Li2(I) -> -Pi^2/48+Catalan*I
374                 if (x.is_equal(I))
375                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
376                 // Li2(-I) -> -Pi^2/48-Catalan*I
377                 if (x.is_equal(-I))
378                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
379                 // Li2(float)
380                 if (!x.info(info_flags::crational))
381                         return Li2(ex_to<numeric>(x));
382         }
383         
384         return Li2(x).hold();
385 }
386
387 static ex Li2_deriv(const ex & x, unsigned deriv_param)
388 {
389         GINAC_ASSERT(deriv_param==0);
390         
391         // d/dx Li2(x) -> -log(1-x)/x
392         return -log(_ex1-x)/x;
393 }
394
395 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
396 {
397         const ex x_pt = x.subs(rel, subs_options::no_pattern);
398         if (x_pt.info(info_flags::numeric)) {
399                 // First special case: x==0 (derivatives have poles)
400                 if (x_pt.is_zero()) {
401                         // method:
402                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot 
403                         // simply substitute x==0.  The limit, however, exists: it is 1.
404                         // We also know all higher derivatives' limits:
405                         // (d/dx)^n Li2(x) == n!/n^2.
406                         // So the primitive series expansion is
407                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
408                         // and so on.
409                         // We first construct such a primitive series expansion manually in
410                         // a dummy symbol s and then insert the argument's series expansion
411                         // for s.  Reexpanding the resulting series returns the desired
412                         // result.
413                         const symbol s;
414                         ex ser;
415                         // manually construct the primitive expansion
416                         for (int i=1; i<order; ++i)
417                                 ser += pow(s,i) / pow(numeric(i), *_num2_p);
418                         // substitute the argument's series expansion
419                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
420                         // maybe that was terminating, so add a proper order term
421                         epvector nseq;
422                         nseq.push_back(expair(Order(_ex1), order));
423                         ser += pseries(rel, nseq);
424                         // reexpanding it will collapse the series again
425                         return ser.series(rel, order);
426                         // NB: Of course, this still does not allow us to compute anything
427                         // like sin(Li2(x)).series(x==0,2), since then this code here is
428                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
429                         // substitution x==0.  Probably limits *are* needed for the general
430                         // cases.  In case L'Hospital's rule is implemented for limits and
431                         // basic::series() takes care of this, this whole block is probably
432                         // obsolete!
433                 }
434                 // second special case: x==1 (branch point)
435                 if (x_pt.is_equal(_ex1)) {
436                         // method:
437                         // construct series manually in a dummy symbol s
438                         const symbol s;
439                         ex ser = zeta(_ex2);
440                         // manually construct the primitive expansion
441                         for (int i=1; i<order; ++i)
442                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
443                         // substitute the argument's series expansion
444                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
445                         // maybe that was terminating, so add a proper order term
446                         epvector nseq;
447                         nseq.push_back(expair(Order(_ex1), order));
448                         ser += pseries(rel, nseq);
449                         // reexpanding it will collapse the series again
450                         return ser.series(rel, order);
451                 }
452                 // third special case: x real, >=1 (branch cut)
453                 if (!(options & series_options::suppress_branchcut) &&
454                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
455                         // method:
456                         // This is the branch cut: assemble the primitive series manually
457                         // and then add the corresponding complex step function.
458                         const symbol &s = ex_to<symbol>(rel.lhs());
459                         const ex point = rel.rhs();
460                         const symbol foo;
461                         epvector seq;
462                         // zeroth order term:
463                         seq.push_back(expair(Li2(x_pt), _ex0));
464                         // compute the intermediate terms:
465                         ex replarg = series(Li2(x), s==foo, order);
466                         for (size_t i=1; i<replarg.nops()-1; ++i)
467                                 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));
468                         // append an order term:
469                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
470                         return pseries(rel, seq);
471                 }
472         }
473         // all other cases should be safe, by now:
474         throw do_taylor();  // caught by function::series()
475 }
476
477 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
478                        evalf_func(Li2_evalf).
479                        derivative_func(Li2_deriv).
480                        series_func(Li2_series).
481                        latex_name("\\mbox{Li}_2"));
482
483 //////////
484 // trilogarithm
485 //////////
486
487 static ex Li3_eval(const ex & x)
488 {
489         if (x.is_zero())
490                 return x;
491         return Li3(x).hold();
492 }
493
494 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
495                        latex_name("\\mbox{Li}_3"));
496
497 //////////
498 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
499 //////////
500
501 static ex zetaderiv_eval(const ex & n, const ex & x)
502 {
503         if (n.info(info_flags::numeric)) {
504                 // zetaderiv(0,x) -> zeta(x)
505                 if (n.is_zero())
506                         return zeta(x);
507         }
508         
509         return zetaderiv(n, x).hold();
510 }
511
512 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
513 {
514         GINAC_ASSERT(deriv_param<2);
515         
516         if (deriv_param==0) {
517                 // d/dn zeta(n,x)
518                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
519         }
520         // d/dx psi(n,x)
521         return zetaderiv(n+1,x);
522 }
523
524 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
525                                  derivative_func(zetaderiv_deriv).
526                                  latex_name("\\zeta^\\prime"));
527
528 //////////
529 // factorial
530 //////////
531
532 static ex factorial_evalf(const ex & x)
533 {
534         return factorial(x).hold();
535 }
536
537 static ex factorial_eval(const ex & x)
538 {
539         if (is_exactly_a<numeric>(x))
540                 return factorial(ex_to<numeric>(x));
541         else
542                 return factorial(x).hold();
543 }
544
545 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
546 {
547         if (is_exactly_a<symbol>(x) ||
548             is_exactly_a<constant>(x) ||
549                 is_exactly_a<function>(x)) {
550                 x.print(c); c.s << "!";
551         } else {
552                 c.s << "("; x.print(c); c.s << ")!";
553         }
554 }
555
556 static ex factorial_conjugate(const ex & x)
557 {
558         return factorial(x);
559 }
560
561 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
562                              evalf_func(factorial_evalf).
563                              print_func<print_dflt>(factorial_print_dflt_latex).
564                              print_func<print_latex>(factorial_print_dflt_latex).
565                              conjugate_func(factorial_conjugate));
566
567 //////////
568 // binomial
569 //////////
570
571 static ex binomial_evalf(const ex & x, const ex & y)
572 {
573         return binomial(x, y).hold();
574 }
575
576 static ex binomial_sym(const ex & x, const numeric & y)
577 {
578         if (y.is_integer()) {
579                 if (y.is_nonneg_integer()) {
580                         const unsigned N = y.to_int();
581                         if (N == 0) return _ex0;
582                         if (N == 1) return x;
583                         ex t = x.expand();
584                         for (unsigned i = 2; i <= N; ++i)
585                                 t = (t * (x + i - y - 1)).expand() / i;
586                         return t;
587                 } else
588                         return _ex0;
589         }
590
591         return binomial(x, y).hold();
592 }
593
594 static ex binomial_eval(const ex & x, const ex &y)
595 {
596         if (is_exactly_a<numeric>(y)) {
597                 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
598                         return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
599                 else
600                         return binomial_sym(x, ex_to<numeric>(y));
601         } else
602                 return binomial(x, y).hold();
603 }
604
605 // At the moment the numeric evaluation of a binomail function always
606 // gives a real number, but if this would be implemented using the gamma
607 // function, also complex conjugation should be changed (or rather, deleted).
608 static ex binomial_conjugate(const ex & x, const ex & y)
609 {
610         return binomial(x,y);
611 }
612
613 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
614                             evalf_func(binomial_evalf).
615                             conjugate_func(binomial_conjugate));
616
617 //////////
618 // Order term function (for truncated power series)
619 //////////
620
621 static ex Order_eval(const ex & x)
622 {
623         if (is_exactly_a<numeric>(x)) {
624                 // O(c) -> O(1) or 0
625                 if (!x.is_zero())
626                         return Order(_ex1).hold();
627                 else
628                         return _ex0;
629         } else if (is_exactly_a<mul>(x)) {
630                 const mul &m = ex_to<mul>(x);
631                 // O(c*expr) -> O(expr)
632                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
633                         return Order(x / m.op(m.nops() - 1)).hold();
634         }
635         return Order(x).hold();
636 }
637
638 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
639 {
640         // Just wrap the function into a pseries object
641         epvector new_seq;
642         GINAC_ASSERT(is_a<symbol>(r.lhs()));
643         const symbol &s = ex_to<symbol>(r.lhs());
644         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
645         return pseries(r, new_seq);
646 }
647
648 static ex Order_conjugate(const ex & x)
649 {
650         return Order(x);
651 }
652
653 // Differentiation is handled in function::derivative because of its special requirements
654
655 REGISTER_FUNCTION(Order, eval_func(Order_eval).
656                          series_func(Order_series).
657                          latex_name("\\mathcal{O}").
658                          conjugate_func(Order_conjugate));
659
660 //////////
661 // Solve linear system
662 //////////
663
664 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
665 {
666         // solve a system of linear equations
667         if (eqns.info(info_flags::relation_equal)) {
668                 if (!symbols.info(info_flags::symbol))
669                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
670                 const ex sol = lsolve(lst(eqns),lst(symbols));
671                 
672                 GINAC_ASSERT(sol.nops()==1);
673                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
674                 
675                 return sol.op(0).op(1); // return rhs of first solution
676         }
677         
678         // syntax checks
679         if (!eqns.info(info_flags::list)) {
680                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
681         }
682         for (size_t i=0; i<eqns.nops(); i++) {
683                 if (!eqns.op(i).info(info_flags::relation_equal)) {
684                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
685                 }
686         }
687         if (!symbols.info(info_flags::list)) {
688                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
689         }
690         for (size_t i=0; i<symbols.nops(); i++) {
691                 if (!symbols.op(i).info(info_flags::symbol)) {
692                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
693                 }
694         }
695         
696         // build matrix from equation system
697         matrix sys(eqns.nops(),symbols.nops());
698         matrix rhs(eqns.nops(),1);
699         matrix vars(symbols.nops(),1);
700         
701         for (size_t r=0; r<eqns.nops(); r++) {
702                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
703                 ex linpart = eq;
704                 for (size_t c=0; c<symbols.nops(); c++) {
705                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
706                         linpart -= co*symbols.op(c);
707                         sys(r,c) = co;
708                 }
709                 linpart = linpart.expand();
710                 rhs(r,0) = -linpart;
711         }
712         
713         // test if system is linear and fill vars matrix
714         for (size_t i=0; i<symbols.nops(); i++) {
715                 vars(i,0) = symbols.op(i);
716                 if (sys.has(symbols.op(i)))
717                         throw(std::logic_error("lsolve: system is not linear"));
718                 if (rhs.has(symbols.op(i)))
719                         throw(std::logic_error("lsolve: system is not linear"));
720         }
721         
722         matrix solution;
723         try {
724                 solution = sys.solve(vars,rhs,options);
725         } catch (const std::runtime_error & e) {
726                 // Probably singular matrix or otherwise overdetermined system:
727                 // It is consistent to return an empty list
728                 return lst();
729         }
730         GINAC_ASSERT(solution.cols()==1);
731         GINAC_ASSERT(solution.rows()==symbols.nops());
732         
733         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
734         lst sollist;
735         for (size_t i=0; i<symbols.nops(); i++)
736                 sollist.append(symbols.op(i)==solution(i,0));
737         
738         return sollist;
739 }
740
741 //////////
742 // Find real root of f(x) numerically
743 //////////
744
745 const numeric
746 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
747 {
748         if (!x1.is_real() || !x2.is_real()) {
749                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
750         }
751         if (x1==x2) {
752                 throw std::runtime_error("fsolve(): vanishing interval");
753         }
754         // xx[0] == left interval limit, xx[1] == right interval limit.
755         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
756         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
757         numeric xx[2] = { x1<x2 ? x1 : x2,
758                           x1<x2 ? x2 : x1 };
759         ex f;
760         if (is_a<relational>(f_in)) {
761                 f = f_in.lhs()-f_in.rhs();
762         } else {
763                 f = f_in;
764         }
765         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
766                             f.subs(x==xx[1]).evalf() };
767         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
768                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
769         }
770         numeric fx[2] = { ex_to<numeric>(fx_[0]),
771                           ex_to<numeric>(fx_[1]) };
772         if (!fx[0].is_real() || !fx[1].is_real()) {
773                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
774         }
775         if (fx[0]*fx[1]>=0) {
776                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
777         }
778
779         // The Newton-Raphson method has quadratic convergence!  Simply put, it
780         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
781         const ex ff = normal(-f/f.diff(x));
782         int side = 0;  // Start at left interval limit.
783         numeric xxprev;
784         numeric fxprev;
785         do {
786                 xxprev = xx[side];
787                 fxprev = fx[side];
788                 xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
789                 fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
790                 if ((side==0 && xx[0]<xxprev) || (side==1 && xx[1]>xxprev) || xx[0]>xx[1]) {
791                         // Oops, Newton-Raphson method shot out of the interval.
792                         // Restore, and try again with the other side instead!
793                         xx[side] = xxprev;
794                         fx[side] = fxprev;
795                         side = !side;
796                         xxprev = xx[side];
797                         fxprev = fx[side];
798                         xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
799                         fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
800                 }
801                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
802                         // Oops, the root isn't bracketed any more.
803                         // Restore, and perform a bisection!
804                         xx[side] = xxprev;
805                         fx[side] = fxprev;
806
807                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
808                         // they occur pretty often when Newton-Raphson arrives at an x too
809                         // close to the result on one side of the interval and
810                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
811                         // precision errors! Recall that this function does not have a
812                         // precision goal as one of its arguments but instead relies on
813                         // x converging to a fixed point. We speed up the (safe but slow)
814                         // bisection method by mixing in a dash of the (unsafer but faster)
815                         // secant method: Instead of splitting the interval at the
816                         // arithmetic mean (bisection), we split it nearer to the root as
817                         // determined by the secant between the values xx[0] and xx[1].
818                         // Don't set the secant_weight to one because that could disturb
819                         // the convergence in some corner cases!
820                         static const double secant_weight = 0.984375;  // == 63/64 < 1
821                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
822                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
823                         numeric fxmid = ex_to<numeric>(f.subs(x==xxmid).evalf());
824                         if (fxmid.is_zero()) {
825                                 // Luck strikes...
826                                 return xxmid;
827                         }
828                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
829                                 side = !side;
830                         }
831                         xxprev = xx[side];
832                         fxprev = fx[side];
833                         xx[side] = xxmid;
834                         fx[side] = fxmid;
835                 }
836         } while (xxprev!=xx[side]);
837         return xxprev;
838 }
839
840
841 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
842  * for static lib (so ginsh will see them). */
843 unsigned force_include_tgamma = tgamma_SERIAL::serial;
844 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
845
846 } // namespace GiNaC