cleaned up some is_a<> vs. is_exactly_a<> stuff
[ginac.git] / ginac / inifcns.cpp
1 /** @file inifcns.cpp
2  *
3  *  Implementation of GiNaC's initially known functions. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2002 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., 59 Temple Place, Suite 330, Boston, MA  02111-1307  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 "relational.h"
34 #include "pseries.h"
35 #include "symbol.h"
36 #include "symmetry.h"
37 #include "utils.h"
38
39 namespace GiNaC {
40
41 //////////
42 // absolute value
43 //////////
44
45 static ex abs_evalf(const ex & arg)
46 {
47         if (is_exactly_a<numeric>(arg))
48                 return abs(ex_to<numeric>(arg));
49         
50         return abs(arg).hold();
51 }
52
53 static ex abs_eval(const ex & arg)
54 {
55         if (is_ex_exactly_of_type(arg, numeric))
56                 return abs(ex_to<numeric>(arg));
57         else
58                 return abs(arg).hold();
59 }
60
61 REGISTER_FUNCTION(abs, eval_func(abs_eval).
62                        evalf_func(abs_evalf));
63
64
65 //////////
66 // Complex sign
67 //////////
68
69 static ex csgn_evalf(const ex & arg)
70 {
71         if (is_exactly_a<numeric>(arg))
72                 return csgn(ex_to<numeric>(arg));
73         
74         return csgn(arg).hold();
75 }
76
77 static ex csgn_eval(const ex & arg)
78 {
79         if (is_ex_exactly_of_type(arg, numeric))
80                 return csgn(ex_to<numeric>(arg));
81         
82         else if (is_ex_exactly_of_type(arg, mul) &&
83                  is_ex_of_type(arg.op(arg.nops()-1),numeric)) {
84                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
85                 if (oc.is_real()) {
86                         if (oc > 0)
87                                 // csgn(42*x) -> csgn(x)
88                                 return csgn(arg/oc).hold();
89                         else
90                                 // csgn(-42*x) -> -csgn(x)
91                                 return -csgn(arg/oc).hold();
92                 }
93                 if (oc.real().is_zero()) {
94                         if (oc.imag() > 0)
95                                 // csgn(42*I*x) -> csgn(I*x)
96                                 return csgn(I*arg/oc).hold();
97                         else
98                                 // csgn(-42*I*x) -> -csgn(I*x)
99                                 return -csgn(I*arg/oc).hold();
100                 }
101         }
102         
103         return csgn(arg).hold();
104 }
105
106 static ex csgn_series(const ex & arg,
107                       const relational & rel,
108                       int order,
109                       unsigned options)
110 {
111         const ex arg_pt = arg.subs(rel);
112         if (arg_pt.info(info_flags::numeric)
113             && ex_to<numeric>(arg_pt).real().is_zero()
114             && !(options & series_options::suppress_branchcut))
115                 throw (std::domain_error("csgn_series(): on imaginary axis"));
116         
117         epvector seq;
118         seq.push_back(expair(csgn(arg_pt), _ex0));
119         return pseries(rel,seq);
120 }
121
122 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
123                         evalf_func(csgn_evalf).
124                         series_func(csgn_series));
125
126
127 //////////
128 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
129 // This function is closely related to the unwinding number K, sometimes found
130 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
131 //////////
132
133 static ex eta_evalf(const ex &x, const ex &y)
134 {
135         // It seems like we basically have to replicate the eval function here,
136         // since the expression might not be fully evaluated yet.
137         if (x.info(info_flags::positive) || y.info(info_flags::positive))
138                 return _ex0;
139
140         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
141                 const numeric nx = ex_to<numeric>(x);
142                 const numeric ny = ex_to<numeric>(y);
143                 const numeric nxy = ex_to<numeric>(x*y);
144                 int cut = 0;
145                 if (nx.is_real() && nx.is_negative())
146                         cut -= 4;
147                 if (ny.is_real() && ny.is_negative())
148                         cut -= 4;
149                 if (nxy.is_real() && nxy.is_negative())
150                         cut += 4;
151                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
152                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
153         }
154
155         return eta(x,y).hold();
156 }
157
158 static ex eta_eval(const ex &x, const ex &y)
159 {
160         // trivial:  eta(x,c) -> 0  if c is real and positive
161         if (x.info(info_flags::positive) || y.info(info_flags::positive))
162                 return _ex0;
163
164         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
165                 // don't call eta_evalf here because it would call Pi.evalf()!
166                 const numeric nx = ex_to<numeric>(x);
167                 const numeric ny = ex_to<numeric>(y);
168                 const numeric nxy = ex_to<numeric>(x*y);
169                 int cut = 0;
170                 if (nx.is_real() && nx.is_negative())
171                         cut -= 4;
172                 if (ny.is_real() && ny.is_negative())
173                         cut -= 4;
174                 if (nxy.is_real() && nxy.is_negative())
175                         cut += 4;
176                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
177                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
178         }
179         
180         return eta(x,y).hold();
181 }
182
183 static ex eta_series(const ex & x, const ex & y,
184                      const relational & rel,
185                      int order,
186                      unsigned options)
187 {
188         const ex x_pt = x.subs(rel);
189         const ex y_pt = y.subs(rel);
190         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
191             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
192             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
193                         throw (std::domain_error("eta_series(): on discontinuity"));
194         epvector seq;
195         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
196         return pseries(rel,seq);
197 }
198
199 REGISTER_FUNCTION(eta, eval_func(eta_eval).
200                        evalf_func(eta_evalf).
201                        series_func(eta_series).
202                        latex_name("\\eta").
203                        set_symmetry(sy_symm(0, 1)));
204
205
206 //////////
207 // dilogarithm
208 //////////
209
210 static ex Li2_evalf(const ex & x)
211 {
212         if (is_exactly_a<numeric>(x))
213                 return Li2(ex_to<numeric>(x));
214         
215         return Li2(x).hold();
216 }
217
218 static ex Li2_eval(const ex & x)
219 {
220         if (x.info(info_flags::numeric)) {
221                 // Li2(0) -> 0
222                 if (x.is_zero())
223                         return _ex0;
224                 // Li2(1) -> Pi^2/6
225                 if (x.is_equal(_ex1))
226                         return power(Pi,_ex2)/_ex6;
227                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
228                 if (x.is_equal(_ex1_2))
229                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
230                 // Li2(-1) -> -Pi^2/12
231                 if (x.is_equal(_ex_1))
232                         return -power(Pi,_ex2)/_ex12;
233                 // Li2(I) -> -Pi^2/48+Catalan*I
234                 if (x.is_equal(I))
235                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
236                 // Li2(-I) -> -Pi^2/48-Catalan*I
237                 if (x.is_equal(-I))
238                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
239                 // Li2(float)
240                 if (!x.info(info_flags::crational))
241                         return Li2(ex_to<numeric>(x));
242         }
243         
244         return Li2(x).hold();
245 }
246
247 static ex Li2_deriv(const ex & x, unsigned deriv_param)
248 {
249         GINAC_ASSERT(deriv_param==0);
250         
251         // d/dx Li2(x) -> -log(1-x)/x
252         return -log(_ex1-x)/x;
253 }
254
255 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
256 {
257         const ex x_pt = x.subs(rel);
258         if (x_pt.info(info_flags::numeric)) {
259                 // First special case: x==0 (derivatives have poles)
260                 if (x_pt.is_zero()) {
261                         // method:
262                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot 
263                         // simply substitute x==0.  The limit, however, exists: it is 1.
264                         // We also know all higher derivatives' limits:
265                         // (d/dx)^n Li2(x) == n!/n^2.
266                         // So the primitive series expansion is
267                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
268                         // and so on.
269                         // We first construct such a primitive series expansion manually in
270                         // a dummy symbol s and then insert the argument's series expansion
271                         // for s.  Reexpanding the resulting series returns the desired
272                         // result.
273                         const symbol s;
274                         ex ser;
275                         // manually construct the primitive expansion
276                         for (int i=1; i<order; ++i)
277                                 ser += pow(s,i) / pow(numeric(i), _num2);
278                         // substitute the argument's series expansion
279                         ser = ser.subs(s==x.series(rel, order));
280                         // maybe that was terminating, so add a proper order term
281                         epvector nseq;
282                         nseq.push_back(expair(Order(_ex1), order));
283                         ser += pseries(rel, nseq);
284                         // reexpanding it will collapse the series again
285                         return ser.series(rel, order);
286                         // NB: Of course, this still does not allow us to compute anything
287                         // like sin(Li2(x)).series(x==0,2), since then this code here is
288                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
289                         // substitution x==0.  Probably limits *are* needed for the general
290                         // cases.  In case L'Hospital's rule is implemented for limits and
291                         // basic::series() takes care of this, this whole block is probably
292                         // obsolete!
293                 }
294                 // second special case: x==1 (branch point)
295                 if (x_pt.is_equal(_ex1)) {
296                         // method:
297                         // construct series manually in a dummy symbol s
298                         const symbol s;
299                         ex ser = zeta(_ex2);
300                         // manually construct the primitive expansion
301                         for (int i=1; i<order; ++i)
302                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
303                         // substitute the argument's series expansion
304                         ser = ser.subs(s==x.series(rel, order));
305                         // maybe that was terminating, so add a proper order term
306                         epvector nseq;
307                         nseq.push_back(expair(Order(_ex1), order));
308                         ser += pseries(rel, nseq);
309                         // reexpanding it will collapse the series again
310                         return ser.series(rel, order);
311                 }
312                 // third special case: x real, >=1 (branch cut)
313                 if (!(options & series_options::suppress_branchcut) &&
314                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
315                         // method:
316                         // This is the branch cut: assemble the primitive series manually
317                         // and then add the corresponding complex step function.
318                         const symbol &s = ex_to<symbol>(rel.lhs());
319                         const ex point = rel.rhs();
320                         const symbol foo;
321                         epvector seq;
322                         // zeroth order term:
323                         seq.push_back(expair(Li2(x_pt), _ex0));
324                         // compute the intermediate terms:
325                         ex replarg = series(Li2(x), s==foo, order);
326                         for (unsigned i=1; i<replarg.nops()-1; ++i)
327                                 seq.push_back(expair((replarg.op(i)/power(s-foo,i)).series(foo==point,1,options).op(0).subs(foo==s),i));
328                         // append an order term:
329                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
330                         return pseries(rel, seq);
331                 }
332         }
333         // all other cases should be safe, by now:
334         throw do_taylor();  // caught by function::series()
335 }
336
337 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
338                        evalf_func(Li2_evalf).
339                        derivative_func(Li2_deriv).
340                        series_func(Li2_series).
341                        latex_name("\\mbox{Li}_2"));
342
343 //////////
344 // trilogarithm
345 //////////
346
347 static ex Li3_eval(const ex & x)
348 {
349         if (x.is_zero())
350                 return x;
351         return Li3(x).hold();
352 }
353
354 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
355                        latex_name("\\mbox{Li}_3"));
356
357 //////////
358 // factorial
359 //////////
360
361 static ex factorial_evalf(const ex & x)
362 {
363         return factorial(x).hold();
364 }
365
366 static ex factorial_eval(const ex & x)
367 {
368         if (is_ex_exactly_of_type(x, numeric))
369                 return factorial(ex_to<numeric>(x));
370         else
371                 return factorial(x).hold();
372 }
373
374 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
375                              evalf_func(factorial_evalf));
376
377 //////////
378 // binomial
379 //////////
380
381 static ex binomial_evalf(const ex & x, const ex & y)
382 {
383         return binomial(x, y).hold();
384 }
385
386 static ex binomial_eval(const ex & x, const ex &y)
387 {
388         if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
389                 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
390         else
391                 return binomial(x, y).hold();
392 }
393
394 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
395                             evalf_func(binomial_evalf));
396
397 //////////
398 // Order term function (for truncated power series)
399 //////////
400
401 static ex Order_eval(const ex & x)
402 {
403         if (is_ex_exactly_of_type(x, numeric)) {
404                 // O(c) -> O(1) or 0
405                 if (!x.is_zero())
406                         return Order(_ex1).hold();
407                 else
408                         return _ex0;
409         } else if (is_ex_exactly_of_type(x, mul)) {
410                 const mul &m = ex_to<mul>(x);
411                 // O(c*expr) -> O(expr)
412                 if (is_ex_exactly_of_type(m.op(m.nops() - 1), numeric))
413                         return Order(x / m.op(m.nops() - 1)).hold();
414         }
415         return Order(x).hold();
416 }
417
418 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
419 {
420         // Just wrap the function into a pseries object
421         epvector new_seq;
422         GINAC_ASSERT(is_a<symbol>(r.lhs()));
423         const symbol &s = ex_to<symbol>(r.lhs());
424         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
425         return pseries(r, new_seq);
426 }
427
428 // Differentiation is handled in function::derivative because of its special requirements
429
430 REGISTER_FUNCTION(Order, eval_func(Order_eval).
431                          series_func(Order_series).
432                          latex_name("\\mathcal{O}"));
433
434 //////////
435 // Solve linear system
436 //////////
437
438 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
439 {
440         // solve a system of linear equations
441         if (eqns.info(info_flags::relation_equal)) {
442                 if (!symbols.info(info_flags::symbol))
443                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
444                 const ex sol = lsolve(lst(eqns),lst(symbols));
445                 
446                 GINAC_ASSERT(sol.nops()==1);
447                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
448                 
449                 return sol.op(0).op(1); // return rhs of first solution
450         }
451         
452         // syntax checks
453         if (!eqns.info(info_flags::list)) {
454                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
455         }
456         for (unsigned i=0; i<eqns.nops(); i++) {
457                 if (!eqns.op(i).info(info_flags::relation_equal)) {
458                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
459                 }
460         }
461         if (!symbols.info(info_flags::list)) {
462                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
463         }
464         for (unsigned i=0; i<symbols.nops(); i++) {
465                 if (!symbols.op(i).info(info_flags::symbol)) {
466                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
467                 }
468         }
469         
470         // build matrix from equation system
471         matrix sys(eqns.nops(),symbols.nops());
472         matrix rhs(eqns.nops(),1);
473         matrix vars(symbols.nops(),1);
474         
475         for (unsigned r=0; r<eqns.nops(); r++) {
476                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
477                 ex linpart = eq;
478                 for (unsigned c=0; c<symbols.nops(); c++) {
479                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
480                         linpart -= co*symbols.op(c);
481                         sys(r,c) = co;
482                 }
483                 linpart = linpart.expand();
484                 rhs(r,0) = -linpart;
485         }
486         
487         // test if system is linear and fill vars matrix
488         for (unsigned i=0; i<symbols.nops(); i++) {
489                 vars(i,0) = symbols.op(i);
490                 if (sys.has(symbols.op(i)))
491                         throw(std::logic_error("lsolve: system is not linear"));
492                 if (rhs.has(symbols.op(i)))
493                         throw(std::logic_error("lsolve: system is not linear"));
494         }
495         
496         matrix solution;
497         try {
498                 solution = sys.solve(vars,rhs,options);
499         } catch (const std::runtime_error & e) {
500                 // Probably singular matrix or otherwise overdetermined system:
501                 // It is consistent to return an empty list
502                 return lst();
503         }
504         GINAC_ASSERT(solution.cols()==1);
505         GINAC_ASSERT(solution.rows()==symbols.nops());
506         
507         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
508         lst sollist;
509         for (unsigned i=0; i<symbols.nops(); i++)
510                 sollist.append(symbols.op(i)==solution(i,0));
511         
512         return sollist;
513 }
514
515 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
516  * for static lib (so ginsh will see them). */
517 unsigned force_include_tgamma = function_index_tgamma;
518 unsigned force_include_zeta1 = function_index_zeta1;
519
520 } // namespace GiNaC