9a488fe9219b383ff2111a6bc0a808bf14d67a6e
[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 // Complex sign
128 //////////
129
130 static ex csgn_evalf(const ex & arg)
131 {
132         if (is_exactly_a<numeric>(arg))
133                 return csgn(ex_to<numeric>(arg));
134         
135         return csgn(arg).hold();
136 }
137
138 static ex csgn_eval(const ex & arg)
139 {
140         if (is_exactly_a<numeric>(arg))
141                 return csgn(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                                 // csgn(42*x) -> csgn(x)
149                                 return csgn(arg/oc).hold();
150                         else
151                                 // csgn(-42*x) -> -csgn(x)
152                                 return -csgn(arg/oc).hold();
153                 }
154                 if (oc.real().is_zero()) {
155                         if (oc.imag() > 0)
156                                 // csgn(42*I*x) -> csgn(I*x)
157                                 return csgn(I*arg/oc).hold();
158                         else
159                                 // csgn(-42*I*x) -> -csgn(I*x)
160                                 return -csgn(I*arg/oc).hold();
161                 }
162         }
163         
164         return csgn(arg).hold();
165 }
166
167 static ex csgn_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("csgn_series(): on imaginary axis"));
177         
178         epvector seq;
179         seq.push_back(expair(csgn(arg_pt), _ex0));
180         return pseries(rel,seq);
181 }
182
183 static ex csgn_conjugate(const ex& arg)
184 {
185         return csgn(arg);
186 }
187
188 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
189                         evalf_func(csgn_evalf).
190                         series_func(csgn_series).
191                         conjugate_func(csgn_conjugate));
192
193
194 //////////
195 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
196 // This function is closely related to the unwinding number K, sometimes found
197 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
198 //////////
199
200 static ex eta_evalf(const ex &x, const ex &y)
201 {
202         // It seems like we basically have to replicate the eval function here,
203         // since the expression might not be fully evaluated yet.
204         if (x.info(info_flags::positive) || y.info(info_flags::positive))
205                 return _ex0;
206
207         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
208                 const numeric nx = ex_to<numeric>(x);
209                 const numeric ny = ex_to<numeric>(y);
210                 const numeric nxy = ex_to<numeric>(x*y);
211                 int cut = 0;
212                 if (nx.is_real() && nx.is_negative())
213                         cut -= 4;
214                 if (ny.is_real() && ny.is_negative())
215                         cut -= 4;
216                 if (nxy.is_real() && nxy.is_negative())
217                         cut += 4;
218                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
219                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
220         }
221
222         return eta(x,y).hold();
223 }
224
225 static ex eta_eval(const ex &x, const ex &y)
226 {
227         // trivial:  eta(x,c) -> 0  if c is real and positive
228         if (x.info(info_flags::positive) || y.info(info_flags::positive))
229                 return _ex0;
230
231         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
232                 // don't call eta_evalf here because it would call Pi.evalf()!
233                 const numeric nx = ex_to<numeric>(x);
234                 const numeric ny = ex_to<numeric>(y);
235                 const numeric nxy = ex_to<numeric>(x*y);
236                 int cut = 0;
237                 if (nx.is_real() && nx.is_negative())
238                         cut -= 4;
239                 if (ny.is_real() && ny.is_negative())
240                         cut -= 4;
241                 if (nxy.is_real() && nxy.is_negative())
242                         cut += 4;
243                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
244                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
245         }
246         
247         return eta(x,y).hold();
248 }
249
250 static ex eta_series(const ex & x, const ex & y,
251                      const relational & rel,
252                      int order,
253                      unsigned options)
254 {
255         const ex x_pt = x.subs(rel, subs_options::no_pattern);
256         const ex y_pt = y.subs(rel, subs_options::no_pattern);
257         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
258             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
259             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
260                         throw (std::domain_error("eta_series(): on discontinuity"));
261         epvector seq;
262         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
263         return pseries(rel,seq);
264 }
265
266 static ex eta_conjugate(const ex & x, const ex & y)
267 {
268         return -eta(x,y);
269 }
270
271 REGISTER_FUNCTION(eta, eval_func(eta_eval).
272                        evalf_func(eta_evalf).
273                        series_func(eta_series).
274                        latex_name("\\eta").
275                        set_symmetry(sy_symm(0, 1)).
276                        conjugate_func(eta_conjugate));
277
278
279 //////////
280 // dilogarithm
281 //////////
282
283 static ex Li2_evalf(const ex & x)
284 {
285         if (is_exactly_a<numeric>(x))
286                 return Li2(ex_to<numeric>(x));
287         
288         return Li2(x).hold();
289 }
290
291 static ex Li2_eval(const ex & x)
292 {
293         if (x.info(info_flags::numeric)) {
294                 // Li2(0) -> 0
295                 if (x.is_zero())
296                         return _ex0;
297                 // Li2(1) -> Pi^2/6
298                 if (x.is_equal(_ex1))
299                         return power(Pi,_ex2)/_ex6;
300                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
301                 if (x.is_equal(_ex1_2))
302                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
303                 // Li2(-1) -> -Pi^2/12
304                 if (x.is_equal(_ex_1))
305                         return -power(Pi,_ex2)/_ex12;
306                 // Li2(I) -> -Pi^2/48+Catalan*I
307                 if (x.is_equal(I))
308                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
309                 // Li2(-I) -> -Pi^2/48-Catalan*I
310                 if (x.is_equal(-I))
311                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
312                 // Li2(float)
313                 if (!x.info(info_flags::crational))
314                         return Li2(ex_to<numeric>(x));
315         }
316         
317         return Li2(x).hold();
318 }
319
320 static ex Li2_deriv(const ex & x, unsigned deriv_param)
321 {
322         GINAC_ASSERT(deriv_param==0);
323         
324         // d/dx Li2(x) -> -log(1-x)/x
325         return -log(_ex1-x)/x;
326 }
327
328 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
329 {
330         const ex x_pt = x.subs(rel, subs_options::no_pattern);
331         if (x_pt.info(info_flags::numeric)) {
332                 // First special case: x==0 (derivatives have poles)
333                 if (x_pt.is_zero()) {
334                         // method:
335                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot 
336                         // simply substitute x==0.  The limit, however, exists: it is 1.
337                         // We also know all higher derivatives' limits:
338                         // (d/dx)^n Li2(x) == n!/n^2.
339                         // So the primitive series expansion is
340                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
341                         // and so on.
342                         // We first construct such a primitive series expansion manually in
343                         // a dummy symbol s and then insert the argument's series expansion
344                         // for s.  Reexpanding the resulting series returns the desired
345                         // result.
346                         const symbol s;
347                         ex ser;
348                         // manually construct the primitive expansion
349                         for (int i=1; i<order; ++i)
350                                 ser += pow(s,i) / pow(numeric(i), *_num2_p);
351                         // substitute the argument's series expansion
352                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
353                         // maybe that was terminating, so add a proper order term
354                         epvector nseq;
355                         nseq.push_back(expair(Order(_ex1), order));
356                         ser += pseries(rel, nseq);
357                         // reexpanding it will collapse the series again
358                         return ser.series(rel, order);
359                         // NB: Of course, this still does not allow us to compute anything
360                         // like sin(Li2(x)).series(x==0,2), since then this code here is
361                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
362                         // substitution x==0.  Probably limits *are* needed for the general
363                         // cases.  In case L'Hospital's rule is implemented for limits and
364                         // basic::series() takes care of this, this whole block is probably
365                         // obsolete!
366                 }
367                 // second special case: x==1 (branch point)
368                 if (x_pt.is_equal(_ex1)) {
369                         // method:
370                         // construct series manually in a dummy symbol s
371                         const symbol s;
372                         ex ser = zeta(_ex2);
373                         // manually construct the primitive expansion
374                         for (int i=1; i<order; ++i)
375                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
376                         // substitute the argument's series expansion
377                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
378                         // maybe that was terminating, so add a proper order term
379                         epvector nseq;
380                         nseq.push_back(expair(Order(_ex1), order));
381                         ser += pseries(rel, nseq);
382                         // reexpanding it will collapse the series again
383                         return ser.series(rel, order);
384                 }
385                 // third special case: x real, >=1 (branch cut)
386                 if (!(options & series_options::suppress_branchcut) &&
387                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
388                         // method:
389                         // This is the branch cut: assemble the primitive series manually
390                         // and then add the corresponding complex step function.
391                         const symbol &s = ex_to<symbol>(rel.lhs());
392                         const ex point = rel.rhs();
393                         const symbol foo;
394                         epvector seq;
395                         // zeroth order term:
396                         seq.push_back(expair(Li2(x_pt), _ex0));
397                         // compute the intermediate terms:
398                         ex replarg = series(Li2(x), s==foo, order);
399                         for (size_t i=1; i<replarg.nops()-1; ++i)
400                                 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));
401                         // append an order term:
402                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
403                         return pseries(rel, seq);
404                 }
405         }
406         // all other cases should be safe, by now:
407         throw do_taylor();  // caught by function::series()
408 }
409
410 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
411                        evalf_func(Li2_evalf).
412                        derivative_func(Li2_deriv).
413                        series_func(Li2_series).
414                        latex_name("\\mbox{Li}_2"));
415
416 //////////
417 // trilogarithm
418 //////////
419
420 static ex Li3_eval(const ex & x)
421 {
422         if (x.is_zero())
423                 return x;
424         return Li3(x).hold();
425 }
426
427 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
428                        latex_name("\\mbox{Li}_3"));
429
430 //////////
431 // Derivatives of Riemann's Zeta-function  zetaderiv(0,x)==zeta(x)
432 //////////
433
434 static ex zetaderiv_eval(const ex & n, const ex & x)
435 {
436         if (n.info(info_flags::numeric)) {
437                 // zetaderiv(0,x) -> zeta(x)
438                 if (n.is_zero())
439                         return zeta(x);
440         }
441         
442         return zetaderiv(n, x).hold();
443 }
444
445 static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
446 {
447         GINAC_ASSERT(deriv_param<2);
448         
449         if (deriv_param==0) {
450                 // d/dn zeta(n,x)
451                 throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
452         }
453         // d/dx psi(n,x)
454         return zetaderiv(n+1,x);
455 }
456
457 REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
458                                  derivative_func(zetaderiv_deriv).
459                                  latex_name("\\zeta^\\prime"));
460
461 //////////
462 // factorial
463 //////////
464
465 static ex factorial_evalf(const ex & x)
466 {
467         return factorial(x).hold();
468 }
469
470 static ex factorial_eval(const ex & x)
471 {
472         if (is_exactly_a<numeric>(x))
473                 return factorial(ex_to<numeric>(x));
474         else
475                 return factorial(x).hold();
476 }
477
478 static void factorial_print_dflt_latex(const ex & x, const print_context & c)
479 {
480         if (is_exactly_a<symbol>(x) ||
481             is_exactly_a<constant>(x) ||
482                 is_exactly_a<function>(x)) {
483                 x.print(c); c.s << "!";
484         } else {
485                 c.s << "("; x.print(c); c.s << ")!";
486         }
487 }
488
489 static ex factorial_conjugate(const ex & x)
490 {
491         return factorial(x);
492 }
493
494 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
495                              evalf_func(factorial_evalf).
496                              print_func<print_dflt>(factorial_print_dflt_latex).
497                              print_func<print_latex>(factorial_print_dflt_latex).
498                              conjugate_func(factorial_conjugate));
499
500 //////////
501 // binomial
502 //////////
503
504 static ex binomial_evalf(const ex & x, const ex & y)
505 {
506         return binomial(x, y).hold();
507 }
508
509 static ex binomial_sym(const ex & x, const numeric & y)
510 {
511         if (y.is_integer()) {
512                 if (y.is_nonneg_integer()) {
513                         const unsigned N = y.to_int();
514                         if (N == 0) return _ex0;
515                         if (N == 1) return x;
516                         ex t = x.expand();
517                         for (unsigned i = 2; i <= N; ++i)
518                                 t = (t * (x + i - y - 1)).expand() / i;
519                         return t;
520                 } else
521                         return _ex0;
522         }
523
524         return binomial(x, y).hold();
525 }
526
527 static ex binomial_eval(const ex & x, const ex &y)
528 {
529         if (is_exactly_a<numeric>(y)) {
530                 if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
531                         return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
532                 else
533                         return binomial_sym(x, ex_to<numeric>(y));
534         } else
535                 return binomial(x, y).hold();
536 }
537
538 // At the moment the numeric evaluation of a binomail function always
539 // gives a real number, but if this would be implemented using the gamma
540 // function, also complex conjugation should be changed (or rather, deleted).
541 static ex binomial_conjugate(const ex & x, const ex & y)
542 {
543         return binomial(x,y);
544 }
545
546 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
547                             evalf_func(binomial_evalf).
548                             conjugate_func(binomial_conjugate));
549
550 //////////
551 // Order term function (for truncated power series)
552 //////////
553
554 static ex Order_eval(const ex & x)
555 {
556         if (is_exactly_a<numeric>(x)) {
557                 // O(c) -> O(1) or 0
558                 if (!x.is_zero())
559                         return Order(_ex1).hold();
560                 else
561                         return _ex0;
562         } else if (is_exactly_a<mul>(x)) {
563                 const mul &m = ex_to<mul>(x);
564                 // O(c*expr) -> O(expr)
565                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
566                         return Order(x / m.op(m.nops() - 1)).hold();
567         }
568         return Order(x).hold();
569 }
570
571 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
572 {
573         // Just wrap the function into a pseries object
574         epvector new_seq;
575         GINAC_ASSERT(is_a<symbol>(r.lhs()));
576         const symbol &s = ex_to<symbol>(r.lhs());
577         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
578         return pseries(r, new_seq);
579 }
580
581 static ex Order_conjugate(const ex & x)
582 {
583         return Order(x);
584 }
585
586 // Differentiation is handled in function::derivative because of its special requirements
587
588 REGISTER_FUNCTION(Order, eval_func(Order_eval).
589                          series_func(Order_series).
590                          latex_name("\\mathcal{O}").
591                          conjugate_func(Order_conjugate));
592
593 //////////
594 // Solve linear system
595 //////////
596
597 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
598 {
599         // solve a system of linear equations
600         if (eqns.info(info_flags::relation_equal)) {
601                 if (!symbols.info(info_flags::symbol))
602                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
603                 const ex sol = lsolve(lst(eqns),lst(symbols));
604                 
605                 GINAC_ASSERT(sol.nops()==1);
606                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
607                 
608                 return sol.op(0).op(1); // return rhs of first solution
609         }
610         
611         // syntax checks
612         if (!eqns.info(info_flags::list)) {
613                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
614         }
615         for (size_t i=0; i<eqns.nops(); i++) {
616                 if (!eqns.op(i).info(info_flags::relation_equal)) {
617                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
618                 }
619         }
620         if (!symbols.info(info_flags::list)) {
621                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
622         }
623         for (size_t i=0; i<symbols.nops(); i++) {
624                 if (!symbols.op(i).info(info_flags::symbol)) {
625                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
626                 }
627         }
628         
629         // build matrix from equation system
630         matrix sys(eqns.nops(),symbols.nops());
631         matrix rhs(eqns.nops(),1);
632         matrix vars(symbols.nops(),1);
633         
634         for (size_t r=0; r<eqns.nops(); r++) {
635                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
636                 ex linpart = eq;
637                 for (size_t c=0; c<symbols.nops(); c++) {
638                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
639                         linpart -= co*symbols.op(c);
640                         sys(r,c) = co;
641                 }
642                 linpart = linpart.expand();
643                 rhs(r,0) = -linpart;
644         }
645         
646         // test if system is linear and fill vars matrix
647         for (size_t i=0; i<symbols.nops(); i++) {
648                 vars(i,0) = symbols.op(i);
649                 if (sys.has(symbols.op(i)))
650                         throw(std::logic_error("lsolve: system is not linear"));
651                 if (rhs.has(symbols.op(i)))
652                         throw(std::logic_error("lsolve: system is not linear"));
653         }
654         
655         matrix solution;
656         try {
657                 solution = sys.solve(vars,rhs,options);
658         } catch (const std::runtime_error & e) {
659                 // Probably singular matrix or otherwise overdetermined system:
660                 // It is consistent to return an empty list
661                 return lst();
662         }
663         GINAC_ASSERT(solution.cols()==1);
664         GINAC_ASSERT(solution.rows()==symbols.nops());
665         
666         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
667         lst sollist;
668         for (size_t i=0; i<symbols.nops(); i++)
669                 sollist.append(symbols.op(i)==solution(i,0));
670         
671         return sollist;
672 }
673
674 //////////
675 // Find real root of f(x) numerically
676 //////////
677
678 const numeric
679 fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
680 {
681         if (!x1.is_real() || !x2.is_real()) {
682                 throw std::runtime_error("fsolve(): interval not bounded by real numbers");
683         }
684         if (x1==x2) {
685                 throw std::runtime_error("fsolve(): vanishing interval");
686         }
687         // xx[0] == left interval limit, xx[1] == right interval limit.
688         // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
689         // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
690         numeric xx[2] = { x1<x2 ? x1 : x2,
691                           x1<x2 ? x2 : x1 };
692         ex f;
693         if (is_a<relational>(f_in)) {
694                 f = f_in.lhs()-f_in.rhs();
695         } else {
696                 f = f_in;
697         }
698         const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
699                             f.subs(x==xx[1]).evalf() };
700         if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
701                 throw std::runtime_error("fsolve(): function does not evaluate numerically");
702         }
703         numeric fx[2] = { ex_to<numeric>(fx_[0]),
704                           ex_to<numeric>(fx_[1]) };
705         if (!fx[0].is_real() || !fx[1].is_real()) {
706                 throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
707         }
708         if (fx[0]*fx[1]>=0) {
709                 throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
710         }
711
712         // The Newton-Raphson method has quadratic convergence!  Simply put, it
713         // replaces x with x-f(x)/f'(x) at each step.  -f/f' is the delta:
714         const ex ff = normal(-f/f.diff(x));
715         int side = 0;  // Start at left interval limit.
716         numeric xxprev;
717         numeric fxprev;
718         do {
719                 xxprev = xx[side];
720                 fxprev = fx[side];
721                 xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
722                 fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
723                 if ((side==0 && xx[0]<xxprev) || (side==1 && xx[1]>xxprev) || xx[0]>xx[1]) {
724                         // Oops, Newton-Raphson method shot out of the interval.
725                         // Restore, and try again with the other side instead!
726                         xx[side] = xxprev;
727                         fx[side] = fxprev;
728                         side = !side;
729                         xxprev = xx[side];
730                         fxprev = fx[side];
731                         xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
732                         fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
733                 }
734                 if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
735                         // Oops, the root isn't bracketed any more.
736                         // Restore, and perform a bisection!
737                         xx[side] = xxprev;
738                         fx[side] = fxprev;
739
740                         // Ah, the bisection! Bisections converge linearly. Unfortunately,
741                         // they occur pretty often when Newton-Raphson arrives at an x too
742                         // close to the result on one side of the interval and
743                         // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
744                         // precision errors! Recall that this function does not have a
745                         // precision goal as one of its arguments but instead relies on
746                         // x converging to a fixed point. We speed up the (safe but slow)
747                         // bisection method by mixing in a dash of the (unsafer but faster)
748                         // secant method: Instead of splitting the interval at the
749                         // arithmetic mean (bisection), we split it nearer to the root as
750                         // determined by the secant between the values xx[0] and xx[1].
751                         // Don't set the secant_weight to one because that could disturb
752                         // the convergence in some corner cases!
753                         static const double secant_weight = 0.984375;  // == 63/64 < 1
754                         numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
755                             + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
756                         numeric fxmid = ex_to<numeric>(f.subs(x==xxmid).evalf());
757                         if (fxmid.is_zero()) {
758                                 // Luck strikes...
759                                 return xxmid;
760                         }
761                         if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
762                                 side = !side;
763                         }
764                         xxprev = xx[side];
765                         fxprev = fx[side];
766                         xx[side] = xxmid;
767                         fx[side] = fxmid;
768                 }
769         } while (xxprev!=xx[side]);
770         return xxprev;
771 }
772
773
774 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
775  * for static lib (so ginsh will see them). */
776 unsigned force_include_tgamma = tgamma_SERIAL::serial;
777 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
778
779 } // namespace GiNaC