trancendental function branch-point sanity
[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 "ncmul.h"
33 #include "numeric.h"
34 #include "power.h"
35 #include "relational.h"
36 #include "pseries.h"
37 #include "symbol.h"
38 #include "utils.h"
39
40 #ifndef NO_NAMESPACE_GINAC
41 namespace GiNaC {
42 #endif // ndef NO_NAMESPACE_GINAC
43
44 //////////
45 // absolute value
46 //////////
47
48 static ex abs_evalf(const ex & arg)
49 {
50         BEGIN_TYPECHECK
51                 TYPECHECK(arg,numeric)
52         END_TYPECHECK(abs(arg))
53         
54         return abs(ex_to_numeric(arg));
55 }
56
57 static ex abs_eval(const ex & arg)
58 {
59         if (is_ex_exactly_of_type(arg, numeric))
60                 return abs(ex_to_numeric(arg));
61         else
62                 return abs(arg).hold();
63 }
64
65 REGISTER_FUNCTION(abs, eval_func(abs_eval).
66                        evalf_func(abs_evalf));
67
68
69 //////////
70 // Complex sign
71 //////////
72
73 static ex csgn_evalf(const ex & arg)
74 {
75         BEGIN_TYPECHECK
76                 TYPECHECK(arg,numeric)
77         END_TYPECHECK(csgn(arg))
78         
79         return csgn(ex_to_numeric(arg));
80 }
81
82 static ex csgn_eval(const ex & arg)
83 {
84         if (is_ex_exactly_of_type(arg, numeric))
85                 return csgn(ex_to_numeric(arg));
86         
87         else if (is_ex_exactly_of_type(arg, mul)) {
88                 numeric oc = ex_to_numeric(arg.op(arg.nops()-1));
89                 if (oc.is_real()) {
90                         if (oc > 0)
91                                 // csgn(42*x) -> csgn(x)
92                                 return csgn(arg/oc).hold();
93                         else
94                                 // csgn(-42*x) -> -csgn(x)
95                                 return -csgn(arg/oc).hold();
96                 }
97                 if (oc.real().is_zero()) {
98                         if (oc.imag() > 0)
99                                 // csgn(42*I*x) -> csgn(I*x)
100                                 return csgn(I*arg/oc).hold();
101                         else
102                                 // csgn(-42*I*x) -> -csgn(I*x)
103                                 return -csgn(I*arg/oc).hold();
104                 }
105         }
106    
107         return csgn(arg).hold();
108 }
109
110 static ex csgn_series(const ex & arg,
111                       const relational & rel,
112                       int order,
113                       unsigned options)
114 {
115         const ex arg_pt = arg.subs(rel);
116         if (arg_pt.info(info_flags::numeric)
117             && ex_to_numeric(arg_pt).real().is_zero()
118             && !(options & series_options::suppress_branchcut))
119                 throw (std::domain_error("csgn_series(): on imaginary axis"));
120         
121         epvector seq;
122         seq.push_back(expair(csgn(arg_pt), _ex0()));
123         return pseries(rel,seq);
124 }
125
126 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
127                         evalf_func(csgn_evalf).
128                         series_func(csgn_series));
129
130
131 //////////
132 // Eta function: log(x*y) == log(x) + log(y) + eta(x,y).
133 //////////
134
135 static ex eta_evalf(const ex & x, const ex & y)
136 {
137         BEGIN_TYPECHECK
138                 TYPECHECK(x,numeric)
139                 TYPECHECK(y,numeric)
140         END_TYPECHECK(eta(x,y))
141                 
142         numeric xim = imag(ex_to_numeric(x));
143         numeric yim = imag(ex_to_numeric(y));
144         numeric xyim = imag(ex_to_numeric(x*y));
145         return evalf(I/4*Pi)*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
146 }
147
148 static ex eta_eval(const ex & x, const ex & y)
149 {
150         if (is_ex_exactly_of_type(x, numeric) &&
151                 is_ex_exactly_of_type(y, numeric)) {
152                 // don't call eta_evalf here because it would call Pi.evalf()!
153                 numeric xim = imag(ex_to_numeric(x));
154                 numeric yim = imag(ex_to_numeric(y));
155                 numeric xyim = imag(ex_to_numeric(x*y));
156                 return (I/4)*Pi*((csgn(-xim)+1)*(csgn(-yim)+1)*(csgn(xyim)+1)-(csgn(xim)+1)*(csgn(yim)+1)*(csgn(-xyim)+1));
157         }
158         
159         return eta(x,y).hold();
160 }
161
162 static ex eta_series(const ex & arg1,
163                      const ex & arg2,
164                      const relational & rel,
165                      int order,
166                      unsigned options)
167 {
168         const ex arg1_pt = arg1.subs(rel);
169         const ex arg2_pt = arg2.subs(rel);
170         if (ex_to_numeric(arg1_pt).imag().is_zero() ||
171                 ex_to_numeric(arg2_pt).imag().is_zero() ||
172                 ex_to_numeric(arg1_pt*arg2_pt).imag().is_zero()) {
173                 throw (std::domain_error("eta_series(): on discontinuity"));
174         }
175         epvector seq;
176         seq.push_back(expair(eta(arg1_pt,arg2_pt), _ex0()));
177         return pseries(rel,seq);
178 }
179
180 REGISTER_FUNCTION(eta, eval_func(eta_eval).
181                        evalf_func(eta_evalf).
182                        series_func(eta_series));
183
184
185 //////////
186 // dilogarithm
187 //////////
188
189 static ex Li2_evalf(const ex & x)
190 {
191         BEGIN_TYPECHECK
192                 TYPECHECK(x,numeric)
193         END_TYPECHECK(Li2(x))
194         
195         return Li2(ex_to_numeric(x));  // -> numeric Li2(numeric)
196 }
197
198 static ex Li2_eval(const ex & x)
199 {
200         if (x.info(info_flags::numeric)) {
201                 // Li2(0) -> 0
202                 if (x.is_zero())
203                         return _ex0();
204                 // Li2(1) -> Pi^2/6
205                 if (x.is_equal(_ex1()))
206                         return power(Pi,_ex2())/_ex6();
207                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
208                 if (x.is_equal(_ex1_2()))
209                         return power(Pi,_ex2())/_ex12() + power(log(_ex2()),_ex2())*_ex_1_2();
210                 // Li2(-1) -> -Pi^2/12
211                 if (x.is_equal(_ex_1()))
212                         return -power(Pi,_ex2())/_ex12();
213                 // Li2(I) -> -Pi^2/48+Catalan*I
214                 if (x.is_equal(I))
215                         return power(Pi,_ex2())/_ex_48() + Catalan*I;
216                 // Li2(-I) -> -Pi^2/48-Catalan*I
217                 if (x.is_equal(-I))
218                         return power(Pi,_ex2())/_ex_48() - Catalan*I;
219                 // Li2(float)
220                 if (!x.info(info_flags::crational))
221                         return Li2_evalf(x);
222         }
223         
224         return Li2(x).hold();
225 }
226
227 static ex Li2_deriv(const ex & x, unsigned deriv_param)
228 {
229         GINAC_ASSERT(deriv_param==0);
230         
231         // d/dx Li2(x) -> -log(1-x)/x
232         return -log(1-x)/x;
233 }
234
235 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
236 {
237         const ex x_pt = x.subs(rel);
238         if (x_pt.info(info_flags::numeric)) {
239                 // First special case: x==0 (derivatives have poles)
240                 if (x_pt.is_zero()) {
241                         // method:
242                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot 
243                         // simply substitute x==0.  The limit, however, exists: it is 1.
244                         // We also know all higher derivatives' limits:
245                         // (d/dx)^n Li2(x) == n!/n^2.
246                         // So the primitive series expansion is
247                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
248                         // and so on.
249                         // We first construct such a primitive series expansion manually in
250                         // a dummy symbol s and then insert the argument's series expansion
251                         // for s.  Reexpanding the resulting series returns the desired
252                         // result.
253                         const symbol s;
254                         ex ser;
255                         // manually construct the primitive expansion
256                         for (int i=1; i<order; ++i)
257                                 ser += pow(s,i) / pow(numeric(i), _num2());
258                         // substitute the argument's series expansion
259                         ser = ser.subs(s==x.series(rel, order));
260                         // maybe that was terminating, so add a proper order term
261                         epvector nseq;
262                         nseq.push_back(expair(Order(_ex1()), order));
263                         ser += pseries(rel, nseq);
264                         // reexpanding it will collapse the series again
265                         return ser.series(rel, order);
266                         // NB: Of course, this still does not allow us to compute anything
267                         // like sin(Li2(x)).series(x==0,2), since then this code here is
268                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
269                         // substitution x==0.  Probably limits *are* needed for the general
270                         // cases.  In case L'Hospital's rule is implemented for limits and
271                         // basic::series() takes care of this, this whole block is probably
272                         // obsolete!
273                 }
274                 // second special case: x==1 (branch point)
275                 if (x_pt == _ex1()) {
276                         // method:
277                         // construct series manually in a dummy symbol s
278                         const symbol s;
279                         ex ser = zeta(2);
280                         // manually construct the primitive expansion
281                         for (int i=1; i<order; ++i)
282                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
283                         // substitute the argument's series expansion
284                         ser = ser.subs(s==x.series(rel, order));
285                         // maybe that was terminating, so add a proper order term
286                         epvector nseq;
287                         nseq.push_back(expair(Order(_ex1()), order));
288                         ser += pseries(rel, nseq);
289                         // reexpanding it will collapse the series again
290                         return ser.series(rel, order);
291                 }
292                 // third special case: x real, >=1 (branch cut)
293                 if (!(options & series_options::suppress_branchcut) &&
294                         ex_to_numeric(x_pt).is_real() && ex_to_numeric(x_pt)>1) {
295                         // method:
296                         // This is the branch cut: assemble the primitive series manually
297                         // and then add the corresponding complex step function.
298                         const symbol *s = static_cast<symbol *>(rel.lhs().bp);
299                         const ex point = rel.rhs();
300                         const symbol foo;
301                         epvector seq;
302                         // zeroth order term:
303                         seq.push_back(expair(Li2(x_pt), _ex0()));
304                         // compute the intermediate terms:
305                         ex replarg = series(Li2(x), *s==foo, order);
306                         for (unsigned i=1; i<replarg.nops()-1; ++i)
307                                 seq.push_back(expair((replarg.op(i)/power(*s-foo,i)).series(foo==point,1,options).op(0).subs(foo==*s),i));
308                         // append an order term:
309                         seq.push_back(expair(Order(_ex1()), replarg.nops()-1));
310                         return pseries(rel, seq);
311                 }
312         }
313         // all other cases should be safe, by now:
314         throw do_taylor();  // caught by function::series()
315 }
316
317 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
318                        evalf_func(Li2_evalf).
319                        derivative_func(Li2_deriv).
320                        series_func(Li2_series));
321
322 //////////
323 // trilogarithm
324 //////////
325
326 static ex Li3_eval(const ex & x)
327 {
328         if (x.is_zero())
329                 return x;
330         return Li3(x).hold();
331 }
332
333 REGISTER_FUNCTION(Li3, eval_func(Li3_eval));
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
411 //////////
412 // Inert partial differentiation operator
413 //////////
414
415 static ex Derivative_eval(const ex & f, const ex & l)
416 {
417         if (!is_ex_exactly_of_type(f, function)) {
418                 throw(std::invalid_argument("Derivative(): 1st argument must be a function"));
419         }
420         if (!is_ex_exactly_of_type(l, lst)) {
421                 throw(std::invalid_argument("Derivative(): 2nd argument must be a list"));
422         }
423         return Derivative(f, l).hold();
424 }
425
426 REGISTER_FUNCTION(Derivative, eval_func(Derivative_eval));
427
428 //////////
429 // Solve linear system
430 //////////
431
432 ex lsolve(const ex &eqns, const ex &symbols)
433 {
434         // solve a system of linear equations
435         if (eqns.info(info_flags::relation_equal)) {
436                 if (!symbols.info(info_flags::symbol))
437                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
438                 ex sol=lsolve(lst(eqns),lst(symbols));
439                 
440                 GINAC_ASSERT(sol.nops()==1);
441                 GINAC_ASSERT(is_ex_exactly_of_type(sol.op(0),relational));
442                 
443                 return sol.op(0).op(1); // return rhs of first solution
444         }
445         
446         // syntax checks
447         if (!eqns.info(info_flags::list)) {
448                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
449         }
450         for (unsigned i=0; i<eqns.nops(); i++) {
451                 if (!eqns.op(i).info(info_flags::relation_equal)) {
452                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
453                 }
454         }
455         if (!symbols.info(info_flags::list)) {
456                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
457         }
458         for (unsigned i=0; i<symbols.nops(); i++) {
459                 if (!symbols.op(i).info(info_flags::symbol)) {
460                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
461                 }
462         }
463         
464         // build matrix from equation system
465         matrix sys(eqns.nops(),symbols.nops());
466         matrix rhs(eqns.nops(),1);
467         matrix vars(symbols.nops(),1);
468         
469         for (unsigned r=0; r<eqns.nops(); r++) {
470                 ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
471                 ex linpart = eq;
472                 for (unsigned c=0; c<symbols.nops(); c++) {
473                         ex co = eq.coeff(ex_to_symbol(symbols.op(c)),1);
474                         linpart -= co*symbols.op(c);
475                         sys.set(r,c,co);
476                 }
477                 linpart = linpart.expand();
478                 rhs.set(r,0,-linpart);
479         }
480         
481         // test if system is linear and fill vars matrix
482         for (unsigned i=0; i<symbols.nops(); i++) {
483                 vars.set(i,0,symbols.op(i));
484                 if (sys.has(symbols.op(i)))
485                         throw(std::logic_error("lsolve: system is not linear"));
486                 if (rhs.has(symbols.op(i)))
487                         throw(std::logic_error("lsolve: system is not linear"));
488         }
489         
490         matrix solution;
491         try {
492                 solution = sys.solve(vars,rhs);
493         } catch (const std::runtime_error & e) {
494                 // Probably singular matrix or otherwise overdetermined system:
495                 // It is consistent to return an empty list
496                 return lst();
497         }    
498         GINAC_ASSERT(solution.cols()==1);
499         GINAC_ASSERT(solution.rows()==symbols.nops());
500         
501         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
502         lst sollist;
503         for (unsigned i=0; i<symbols.nops(); i++)
504                 sollist.append(symbols.op(i)==solution(i,0));
505         
506         return sollist;
507 }
508
509 /** non-commutative power. */
510 ex ncpower(const ex &basis, unsigned exponent)
511 {
512         if (exponent==0) {
513                 return _ex1();
514         }
515
516         exvector v;
517         v.reserve(exponent);
518         for (unsigned i=0; i<exponent; ++i) {
519                 v.push_back(basis);
520         }
521
522         return ncmul(v,1);
523 }
524
525 /** Force inclusion of functions from initcns_gamma and inifcns_zeta
526  *  for static lib (so ginsh will see them). */
527 unsigned force_include_tgamma = function_index_tgamma;
528 unsigned force_include_zeta1 = function_index_zeta1;
529
530 #ifndef NO_NAMESPACE_GINAC
531 } // namespace GiNaC
532 #endif // ndef NO_NAMESPACE_GINAC