fixed comments
[ginac.git] / ginac / inifcns.cpp
1 /** @file inifcns.cpp
2  *
3  *  Implementation of GiNaC's initially known functions. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2001 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 "utils.h"
37
38 namespace GiNaC {
39
40 //////////
41 // absolute value
42 //////////
43
44 static ex abs_evalf(const ex & arg)
45 {
46         BEGIN_TYPECHECK
47                 TYPECHECK(arg,numeric)
48         END_TYPECHECK(abs(arg))
49         
50         return abs(ex_to<numeric>(arg));
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         BEGIN_TYPECHECK
72                 TYPECHECK(arg,numeric)
73         END_TYPECHECK(csgn(arg))
74         
75         return csgn(ex_to<numeric>(arg));
76 }
77
78 static ex csgn_eval(const ex & arg)
79 {
80         if (is_ex_exactly_of_type(arg, numeric))
81                 return csgn(ex_to<numeric>(arg));
82         
83         else if (is_ex_of_type(arg, mul) &&
84                  is_ex_of_type(arg.op(arg.nops()-1),numeric)) {
85                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
86                 if (oc.is_real()) {
87                         if (oc > 0)
88                                 // csgn(42*x) -> csgn(x)
89                                 return csgn(arg/oc).hold();
90                         else
91                                 // csgn(-42*x) -> -csgn(x)
92                                 return -csgn(arg/oc).hold();
93                 }
94                 if (oc.real().is_zero()) {
95                         if (oc.imag() > 0)
96                                 // csgn(42*I*x) -> csgn(I*x)
97                                 return csgn(I*arg/oc).hold();
98                         else
99                                 // csgn(-42*I*x) -> -csgn(I*x)
100                                 return -csgn(I*arg/oc).hold();
101                 }
102         }
103         
104         return csgn(arg).hold();
105 }
106
107 static ex csgn_series(const ex & arg,
108                       const relational & rel,
109                       int order,
110                       unsigned options)
111 {
112         const ex arg_pt = arg.subs(rel);
113         if (arg_pt.info(info_flags::numeric)
114             && ex_to<numeric>(arg_pt).real().is_zero()
115             && !(options & series_options::suppress_branchcut))
116                 throw (std::domain_error("csgn_series(): on imaginary axis"));
117         
118         epvector seq;
119         seq.push_back(expair(csgn(arg_pt), _ex0()));
120         return pseries(rel,seq);
121 }
122
123 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
124                         evalf_func(csgn_evalf).
125                         series_func(csgn_series));
126
127
128 //////////
129 // Eta function: log(x*y) == log(x) + log(y) + eta(x,y).
130 //////////
131
132 static ex eta_evalf(const ex & x, const ex & y)
133 {
134         BEGIN_TYPECHECK
135                 TYPECHECK(x,numeric)
136                 TYPECHECK(y,numeric)
137         END_TYPECHECK(eta(x,y))
138                 
139         numeric xim = imag(ex_to<numeric>(x));
140         numeric yim = imag(ex_to<numeric>(y));
141         numeric xyim = imag(ex_to<numeric>(x*y));
142         return evalf(I/4*Pi)*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
143 }
144
145 static ex eta_eval(const ex & x, const ex & y)
146 {
147         if (is_ex_exactly_of_type(x, numeric) &&
148                 is_ex_exactly_of_type(y, numeric)) {
149                 // don't call eta_evalf here because it would call Pi.evalf()!
150                 numeric xim = imag(ex_to<numeric>(x));
151                 numeric yim = imag(ex_to<numeric>(y));
152                 numeric xyim = imag(ex_to<numeric>(x*y));
153                 return (I/4)*Pi*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
154         }
155         
156         return eta(x,y).hold();
157 }
158
159 static ex eta_series(const ex & arg1,
160                      const ex & arg2,
161                      const relational & rel,
162                      int order,
163                      unsigned options)
164 {
165         const ex arg1_pt = arg1.subs(rel);
166         const ex arg2_pt = arg2.subs(rel);
167         if (ex_to<numeric>(arg1_pt).imag().is_zero() ||
168                 ex_to<numeric>(arg2_pt).imag().is_zero() ||
169                 ex_to<numeric>(arg1_pt*arg2_pt).imag().is_zero()) {
170                 throw (std::domain_error("eta_series(): on discontinuity"));
171         }
172         epvector seq;
173         seq.push_back(expair(eta(arg1_pt,arg2_pt), _ex0()));
174         return pseries(rel,seq);
175 }
176
177 REGISTER_FUNCTION(eta, eval_func(eta_eval).
178                        evalf_func(eta_evalf).
179                        series_func(eta_series).
180                        latex_name("\\eta"));
181
182
183 //////////
184 // dilogarithm
185 //////////
186
187 static ex Li2_evalf(const ex & x)
188 {
189         BEGIN_TYPECHECK
190                 TYPECHECK(x,numeric)
191         END_TYPECHECK(Li2(x))
192         
193         return Li2(ex_to<numeric>(x));  // -> numeric Li2(numeric)
194 }
195
196 static ex Li2_eval(const ex & x)
197 {
198         if (x.info(info_flags::numeric)) {
199                 // Li2(0) -> 0
200                 if (x.is_zero())
201                         return _ex0();
202                 // Li2(1) -> Pi^2/6
203                 if (x.is_equal(_ex1()))
204                         return power(Pi,_ex2())/_ex6();
205                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
206                 if (x.is_equal(_ex1_2()))
207                         return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
208                 // Li2(-1) -> -Pi^2/12
209                 if (x.is_equal(_ex_1()))
210                         return -power(Pi,_ex2())/_ex12();
211                 // Li2(I) -> -Pi^2/48+Catalan*I
212                 if (x.is_equal(I))
213                         return power(Pi,_ex2())/_ex_48() + Catalan*I;
214                 // Li2(-I) -> -Pi^2/48-Catalan*I
215                 if (x.is_equal(-I))
216                         return power(Pi,_ex2())/_ex_48() - Catalan*I;
217                 // Li2(float)
218                 if (!x.info(info_flags::crational))
219                         return Li2_evalf(x);
220         }
221         
222         return Li2(x).hold();
223 }
224
225 static ex Li2_deriv(const ex & x, unsigned deriv_param)
226 {
227         GINAC_ASSERT(deriv_param==0);
228         
229         // d/dx Li2(x) -> -log(1-x)/x
230         return -log(1-x)/x;
231 }
232
233 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
234 {
235         const ex x_pt = x.subs(rel);
236         if (x_pt.info(info_flags::numeric)) {
237                 // First special case: x==0 (derivatives have poles)
238                 if (x_pt.is_zero()) {
239                         // method:
240                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot 
241                         // simply substitute x==0.  The limit, however, exists: it is 1.
242                         // We also know all higher derivatives' limits:
243                         // (d/dx)^n Li2(x) == n!/n^2.
244                         // So the primitive series expansion is
245                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
246                         // and so on.
247                         // We first construct such a primitive series expansion manually in
248                         // a dummy symbol s and then insert the argument's series expansion
249                         // for s.  Reexpanding the resulting series returns the desired
250                         // result.
251                         const symbol s;
252                         ex ser;
253                         // manually construct the primitive expansion
254                         for (int i=1; i<order; ++i)
255                                 ser += pow(s,i) / pow(numeric(i), _num2());
256                         // substitute the argument's series expansion
257                         ser = ser.subs(s==x.series(rel, order));
258                         // maybe that was terminating, so add a proper order term
259                         epvector nseq;
260                         nseq.push_back(expair(Order(_ex1()), order));
261                         ser += pseries(rel, nseq);
262                         // reexpanding it will collapse the series again
263                         return ser.series(rel, order);
264                         // NB: Of course, this still does not allow us to compute anything
265                         // like sin(Li2(x)).series(x==0,2), since then this code here is
266                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
267                         // substitution x==0.  Probably limits *are* needed for the general
268                         // cases.  In case L'Hospital's rule is implemented for limits and
269                         // basic::series() takes care of this, this whole block is probably
270                         // obsolete!
271                 }
272                 // second special case: x==1 (branch point)
273                 if (x_pt == _ex1()) {
274                         // method:
275                         // construct series manually in a dummy symbol s
276                         const symbol s;
277                         ex ser = zeta(2);
278                         // manually construct the primitive expansion
279                         for (int i=1; i<order; ++i)
280                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
281                         // substitute the argument's series expansion
282                         ser = ser.subs(s==x.series(rel, order));
283                         // maybe that was terminating, so add a proper order term
284                         epvector nseq;
285                         nseq.push_back(expair(Order(_ex1()), order));
286                         ser += pseries(rel, nseq);
287                         // reexpanding it will collapse the series again
288                         return ser.series(rel, order);
289                 }
290                 // third special case: x real, >=1 (branch cut)
291                 if (!(options & series_options::suppress_branchcut) &&
292                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
293                         // method:
294                         // This is the branch cut: assemble the primitive series manually
295                         // and then add the corresponding complex step function.
296                         const symbol *s = static_cast<symbol *>(rel.lhs().bp);
297                         const ex point = rel.rhs();
298                         const symbol foo;
299                         epvector seq;
300                         // zeroth order term:
301                         seq.push_back(expair(Li2(x_pt), _ex0()));
302                         // compute the intermediate terms:
303                         ex replarg = series(Li2(x), *s==foo, order);
304                         for (unsigned i=1; i<replarg.nops()-1; ++i)
305                                 seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,options).op(0).subs(foo==*s),i));
306                         // append an order term:
307                         seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
308                         return pseries(rel, seq);
309                 }
310         }
311         // all other cases should be safe, by now:
312         throw do_taylor();  // caught by function::series()
313 }
314
315 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
316                        evalf_func(Li2_evalf).
317                        derivative_func(Li2_deriv).
318                        series_func(Li2_series).
319                        latex_name("\\mbox{Li}_2"));
320
321 //////////
322 // trilogarithm
323 //////////
324
325 static ex Li3_eval(const ex & x)
326 {
327         if (x.is_zero())
328                 return x;
329         return Li3(x).hold();
330 }
331
332 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
333                        latex_name("\\mbox{Li}_3"));
334
335 //////////
336 // factorial
337 //////////
338
339 static ex factorial_evalf(const ex & x)
340 {
341         return factorial(x).hold();
342 }
343
344 static ex factorial_eval(const ex & x)
345 {
346         if (is_ex_exactly_of_type(x, numeric))
347                 return factorial(ex_to<numeric>(x));
348         else
349                 return factorial(x).hold();
350 }
351
352 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
353                              evalf_func(factorial_evalf));
354
355 //////////
356 // binomial
357 //////////
358
359 static ex binomial_evalf(const ex & x, const ex & y)
360 {
361         return binomial(x, y).hold();
362 }
363
364 static ex binomial_eval(const ex & x, const ex &y)
365 {
366         if (is_ex_exactly_of_type(x, numeric) && is_ex_exactly_of_type(y, numeric))
367                 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
368         else
369                 return binomial(x, y).hold();
370 }
371
372 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
373                             evalf_func(binomial_evalf));
374
375 //////////
376 // Order term function (for truncated power series)
377 //////////
378
379 static ex Order_eval(const ex & x)
380 {
381         if (is_ex_exactly_of_type(x, numeric)) {
382                 // O(c) -> O(1) or 0
383                 if (!x.is_zero())
384                         return Order(_ex1()).hold();
385                 else
386                         return _ex0();
387         } else if (is_ex_exactly_of_type(x, mul)) {
388                 mul *m = static_cast<mul *>(x.bp);
389                 // O(c*expr) -> O(expr)
390                 if (is_ex_exactly_of_type(m->op(m->nops() - 1), numeric))
391                         return Order(x / m->op(m->nops() - 1)).hold();
392         }
393         return Order(x).hold();
394 }
395
396 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
397 {
398         // Just wrap the function into a pseries object
399         epvector new_seq;
400         GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
401         const symbol *s = static_cast<symbol *>(r.lhs().bp);
402         new_seq.push_back(expair(Order(_ex1()), numeric(std::min(x.ldegree(*s), order))));
403         return pseries(r, new_seq);
404 }
405
406 // Differentiation is handled in function::derivative because of its special requirements
407
408 REGISTER_FUNCTION(Order, eval_func(Order_eval).
409                          series_func(Order_series).
410                          latex_name("\\mathcal{O}"));
411
412 //////////
413 // Inert partial differentiation operator
414 //////////
415
416 static ex Derivative_eval(const ex & f, const ex & l)
417 {
418         if (!is_ex_exactly_of_type(f, function)) {
419                 throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
420         }
421         if (!is_ex_exactly_of_type(l, lst)) {
422                 throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
423         }
424         return Derivative(f, l).hold();
425 }
426
427 REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
428
429 //////////
430 // Solve linear system
431 //////////
432
433 ex lsolve(const ex &eqns, const ex &symbols)
434 {
435         // solve a system of linear equations
436         if (eqns.info(info_flags::relation_equal)) {
437                 if (!symbols.info(info_flags::symbol))
438                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
439                 ex sol=lsolve(lst(eqns),lst(symbols));
440                 
441                 GINAC_ASSERT(sol.nops()==1);
442                 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
443                 
444                 return sol.op(0).op(1); // return rhs of first solution
445         }
446         
447         // syntax checks
448         if (!eqns.info(info_flags::list)) {
449                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
450         }
451         for (unsigned i=0; i<eqns.nops(); i++) {
452                 if (!eqns.op(i).info(info_flags::relation_equal)) {
453                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
454                 }
455         }
456         if (!symbols.info(info_flags::list)) {
457                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
458         }
459         for (unsigned i=0; i<symbols.nops(); i++) {
460                 if (!symbols.op(i).info(info_flags::symbol)) {
461                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
462                 }
463         }
464         
465         // build matrix from equation system
466         matrix sys(eqns.nops(),symbols.nops());
467         matrix rhs(eqns.nops(),1);
468         matrix vars(symbols.nops(),1);
469         
470         for (unsigned r=0; r<eqns.nops(); r++) {
471                 ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
472                 ex linpart = eq;
473                 for (unsigned c=0; c<symbols.nops(); c++) {
474                         ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
475                         linpart -= co*symbols.op(c);
476                         sys(r,c) = co;
477                 }
478                 linpart = linpart.expand();
479                 rhs(r,0) = -linpart;
480         }
481         
482         // test if system is linear and fill vars matrix
483         for (unsigned i=0; i<symbols.nops(); i++) {
484                 vars(i,0) = symbols.op(i);
485                 if (sys.has(symbols.op(i)))
486                         throw(std::logic_error("lsolve: system is not linear"));
487                 if (rhs.has(symbols.op(i)))
488                         throw(std::logic_error("lsolve: system is not linear"));
489         }
490         
491         matrix solution;
492         try {
493                 solution = sys.solve(vars,rhs);
494         } catch (const std::runtime_error & e) {
495                 // Probably singular matrix or otherwise overdetermined system:
496                 // It is consistent to return an empty list
497                 return lst();
498         }    
499         GINAC_ASSERT(solution.cols()==1);
500         GINAC_ASSERT(solution.rows()==symbols.nops());
501         
502         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
503         lst sollist;
504         for (unsigned i=0; i<symbols.nops(); i++)
505                 sollist.append(symbols.op(i)==solution(i,0));
506         
507         return sollist;
508 }
509
510 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
511  * for static lib (so ginsh will see them). */
512 unsigned force_include_tgamma = function_index_tgamma;
513 unsigned force_include_zeta1 = function_index_zeta1;
514
515 } // namespace GiNaC