dd96a70fafdb15114985a1dce12d6d4aeba3569b
[ginac.git] / ginac / mul.cpp
1 /** @file mul.cpp
2  *
3  *  Implementation of GiNaC's products of expressions. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2007 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 <iostream>
24 #include <vector>
25 #include <stdexcept>
26 #include <limits>
27
28 #include "mul.h"
29 #include "add.h"
30 #include "power.h"
31 #include "operators.h"
32 #include "matrix.h"
33 #include "indexed.h"
34 #include "lst.h"
35 #include "archive.h"
36 #include "utils.h"
37
38 namespace GiNaC {
39
40 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mul, expairseq,
41   print_func<print_context>(&mul::do_print).
42   print_func<print_latex>(&mul::do_print_latex).
43   print_func<print_csrc>(&mul::do_print_csrc).
44   print_func<print_tree>(&mul::do_print_tree).
45   print_func<print_python_repr>(&mul::do_print_python_repr))
46
47
48 //////////
49 // default constructor
50 //////////
51
52 mul::mul()
53 {
54         tinfo_key = &mul::tinfo_static;
55 }
56
57 //////////
58 // other constructors
59 //////////
60
61 // public
62
63 mul::mul(const ex & lh, const ex & rh)
64 {
65         tinfo_key = &mul::tinfo_static;
66         overall_coeff = _ex1;
67         construct_from_2_ex(lh,rh);
68         GINAC_ASSERT(is_canonical());
69 }
70
71 mul::mul(const exvector & v)
72 {
73         tinfo_key = &mul::tinfo_static;
74         overall_coeff = _ex1;
75         construct_from_exvector(v);
76         GINAC_ASSERT(is_canonical());
77 }
78
79 mul::mul(const epvector & v)
80 {
81         tinfo_key = &mul::tinfo_static;
82         overall_coeff = _ex1;
83         construct_from_epvector(v);
84         GINAC_ASSERT(is_canonical());
85 }
86
87 mul::mul(const epvector & v, const ex & oc, bool do_index_renaming)
88 {
89         tinfo_key = &mul::tinfo_static;
90         overall_coeff = oc;
91         construct_from_epvector(v, do_index_renaming);
92         GINAC_ASSERT(is_canonical());
93 }
94
95 mul::mul(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming)
96 {
97         tinfo_key = &mul::tinfo_static;
98         GINAC_ASSERT(vp.get()!=0);
99         overall_coeff = oc;
100         construct_from_epvector(*vp, do_index_renaming);
101         GINAC_ASSERT(is_canonical());
102 }
103
104 mul::mul(const ex & lh, const ex & mh, const ex & rh)
105 {
106         tinfo_key = &mul::tinfo_static;
107         exvector factors;
108         factors.reserve(3);
109         factors.push_back(lh);
110         factors.push_back(mh);
111         factors.push_back(rh);
112         overall_coeff = _ex1;
113         construct_from_exvector(factors);
114         GINAC_ASSERT(is_canonical());
115 }
116
117 //////////
118 // archiving
119 //////////
120
121 DEFAULT_ARCHIVING(mul)
122
123 //////////
124 // functions overriding virtual functions from base classes
125 //////////
126
127 void mul::print_overall_coeff(const print_context & c, const char *mul_sym) const
128 {
129         const numeric &coeff = ex_to<numeric>(overall_coeff);
130         if (coeff.csgn() == -1)
131                 c.s << '-';
132         if (!coeff.is_equal(*_num1_p) &&
133                 !coeff.is_equal(*_num_1_p)) {
134                 if (coeff.is_rational()) {
135                         if (coeff.is_negative())
136                                 (-coeff).print(c);
137                         else
138                                 coeff.print(c);
139                 } else {
140                         if (coeff.csgn() == -1)
141                                 (-coeff).print(c, precedence());
142                         else
143                                 coeff.print(c, precedence());
144                 }
145                 c.s << mul_sym;
146         }
147 }
148
149 void mul::do_print(const print_context & c, unsigned level) const
150 {
151         if (precedence() <= level)
152                 c.s << '(';
153
154         print_overall_coeff(c, "*");
155
156         epvector::const_iterator it = seq.begin(), itend = seq.end();
157         bool first = true;
158         while (it != itend) {
159                 if (!first)
160                         c.s << '*';
161                 else
162                         first = false;
163                 recombine_pair_to_ex(*it).print(c, precedence());
164                 ++it;
165         }
166
167         if (precedence() <= level)
168                 c.s << ')';
169 }
170
171 void mul::do_print_latex(const print_latex & c, unsigned level) const
172 {
173         if (precedence() <= level)
174                 c.s << "{(";
175
176         print_overall_coeff(c, " ");
177
178         // Separate factors into those with negative numeric exponent
179         // and all others
180         epvector::const_iterator it = seq.begin(), itend = seq.end();
181         exvector neg_powers, others;
182         while (it != itend) {
183                 GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
184                 if (ex_to<numeric>(it->coeff).is_negative())
185                         neg_powers.push_back(recombine_pair_to_ex(expair(it->rest, -(it->coeff))));
186                 else
187                         others.push_back(recombine_pair_to_ex(*it));
188                 ++it;
189         }
190
191         if (!neg_powers.empty()) {
192
193                 // Factors with negative exponent are printed as a fraction
194                 c.s << "\\frac{";
195                 mul(others).eval().print(c);
196                 c.s << "}{";
197                 mul(neg_powers).eval().print(c);
198                 c.s << "}";
199
200         } else {
201
202                 // All other factors are printed in the ordinary way
203                 exvector::const_iterator vit = others.begin(), vitend = others.end();
204                 while (vit != vitend) {
205                         c.s << ' ';
206                         vit->print(c, precedence());
207                         ++vit;
208                 }
209         }
210
211         if (precedence() <= level)
212                 c.s << ")}";
213 }
214
215 void mul::do_print_csrc(const print_csrc & c, unsigned level) const
216 {
217         if (precedence() <= level)
218                 c.s << "(";
219
220         if (!overall_coeff.is_equal(_ex1)) {
221                 if (overall_coeff.is_equal(_ex_1))
222                         c.s << "-";
223                 else {
224                         overall_coeff.print(c, precedence());
225                         c.s << "*";
226                 }
227         }
228
229         // Print arguments, separated by "*" or "/"
230         epvector::const_iterator it = seq.begin(), itend = seq.end();
231         while (it != itend) {
232
233                 // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
234                 bool needclosingparenthesis = false;
235                 if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
236                         if (is_a<print_csrc_cl_N>(c)) {
237                                 c.s << "recip(";
238                                 needclosingparenthesis = true;
239                         } else
240                                 c.s << "1.0/";
241                 }
242
243                 // If the exponent is 1 or -1, it is left out
244                 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
245                         it->rest.print(c, precedence());
246                 else if (it->coeff.info(info_flags::negint))
247                         // Outer parens around ex needed for broken GCC parser:
248                         (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
249                 else
250                         // Outer parens around ex needed for broken GCC parser:
251                         (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
252
253                 if (needclosingparenthesis)
254                         c.s << ")";
255
256                 // Separator is "/" for negative integer powers, "*" otherwise
257                 ++it;
258                 if (it != itend) {
259                         if (it->coeff.info(info_flags::negint))
260                                 c.s << "/";
261                         else
262                                 c.s << "*";
263                 }
264         }
265
266         if (precedence() <= level)
267                 c.s << ")";
268 }
269
270 void mul::do_print_python_repr(const print_python_repr & c, unsigned level) const
271 {
272         c.s << class_name() << '(';
273         op(0).print(c);
274         for (size_t i=1; i<nops(); ++i) {
275                 c.s << ',';
276                 op(i).print(c);
277         }
278         c.s << ')';
279 }
280
281 bool mul::info(unsigned inf) const
282 {
283         switch (inf) {
284                 case info_flags::polynomial:
285                 case info_flags::integer_polynomial:
286                 case info_flags::cinteger_polynomial:
287                 case info_flags::rational_polynomial:
288                 case info_flags::crational_polynomial:
289                 case info_flags::rational_function: {
290                         epvector::const_iterator i = seq.begin(), end = seq.end();
291                         while (i != end) {
292                                 if (!(recombine_pair_to_ex(*i).info(inf)))
293                                         return false;
294                                 ++i;
295                         }
296                         return overall_coeff.info(inf);
297                 }
298                 case info_flags::algebraic: {
299                         epvector::const_iterator i = seq.begin(), end = seq.end();
300                         while (i != end) {
301                                 if ((recombine_pair_to_ex(*i).info(inf)))
302                                         return true;
303                                 ++i;
304                         }
305                         return false;
306                 }
307         }
308         return inherited::info(inf);
309 }
310
311 int mul::degree(const ex & s) const
312 {
313         // Sum up degrees of factors
314         int deg_sum = 0;
315         epvector::const_iterator i = seq.begin(), end = seq.end();
316         while (i != end) {
317                 if (ex_to<numeric>(i->coeff).is_integer())
318                         deg_sum += i->rest.degree(s) * ex_to<numeric>(i->coeff).to_int();
319                 ++i;
320         }
321         return deg_sum;
322 }
323
324 int mul::ldegree(const ex & s) const
325 {
326         // Sum up degrees of factors
327         int deg_sum = 0;
328         epvector::const_iterator i = seq.begin(), end = seq.end();
329         while (i != end) {
330                 if (ex_to<numeric>(i->coeff).is_integer())
331                         deg_sum += i->rest.ldegree(s) * ex_to<numeric>(i->coeff).to_int();
332                 ++i;
333         }
334         return deg_sum;
335 }
336
337 ex mul::coeff(const ex & s, int n) const
338 {
339         exvector coeffseq;
340         coeffseq.reserve(seq.size()+1);
341         
342         if (n==0) {
343                 // product of individual coeffs
344                 // if a non-zero power of s is found, the resulting product will be 0
345                 epvector::const_iterator i = seq.begin(), end = seq.end();
346                 while (i != end) {
347                         coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
348                         ++i;
349                 }
350                 coeffseq.push_back(overall_coeff);
351                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
352         }
353         
354         epvector::const_iterator i = seq.begin(), end = seq.end();
355         bool coeff_found = false;
356         while (i != end) {
357                 ex t = recombine_pair_to_ex(*i);
358                 ex c = t.coeff(s, n);
359                 if (!c.is_zero()) {
360                         coeffseq.push_back(c);
361                         coeff_found = 1;
362                 } else {
363                         coeffseq.push_back(t);
364                 }
365                 ++i;
366         }
367         if (coeff_found) {
368                 coeffseq.push_back(overall_coeff);
369                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
370         }
371         
372         return _ex0;
373 }
374
375 /** Perform automatic term rewriting rules in this class.  In the following
376  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
377  *  stand for such expressions that contain a plain number.
378  *  - *(...,x;0) -> 0
379  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
380  *  - *(x;1) -> x
381  *  - *(;c) -> c
382  *
383  *  @param level cut-off in recursive evaluation */
384 ex mul::eval(int level) const
385 {
386         std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
387         if (evaled_seqp.get()) {
388                 // do more evaluation later
389                 return (new mul(evaled_seqp, overall_coeff))->
390                            setflag(status_flags::dynallocated);
391         }
392         
393 #ifdef DO_GINAC_ASSERT
394         epvector::const_iterator i = seq.begin(), end = seq.end();
395         while (i != end) {
396                 GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
397                              (!(ex_to<numeric>(i->coeff).is_integer())));
398                 GINAC_ASSERT(!(i->is_canonical_numeric()));
399                 if (is_exactly_a<numeric>(recombine_pair_to_ex(*i)))
400                     print(print_tree(std::cerr));
401                 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
402                 /* for paranoia */
403                 expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
404                 GINAC_ASSERT(p.rest.is_equal(i->rest));
405                 GINAC_ASSERT(p.coeff.is_equal(i->coeff));
406                 /* end paranoia */
407                 ++i;
408         }
409 #endif // def DO_GINAC_ASSERT
410         
411         if (flags & status_flags::evaluated) {
412                 GINAC_ASSERT(seq.size()>0);
413                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
414                 return *this;
415         }
416         
417         int seq_size = seq.size();
418         if (overall_coeff.is_zero()) {
419                 // *(...,x;0) -> 0
420                 return _ex0;
421         } else if (seq_size==0) {
422                 // *(;c) -> c
423                 return overall_coeff;
424         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
425                 // *(x;1) -> x
426                 return recombine_pair_to_ex(*(seq.begin()));
427         } else if ((seq_size==1) &&
428                    is_exactly_a<add>((*seq.begin()).rest) &&
429                    ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
430                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
431                 const add & addref = ex_to<add>((*seq.begin()).rest);
432                 std::auto_ptr<epvector> distrseq(new epvector);
433                 distrseq->reserve(addref.seq.size());
434                 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
435                 while (i != end) {
436                         distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
437                         ++i;
438                 }
439                 return (new add(distrseq,
440                                 ex_to<numeric>(addref.overall_coeff).
441                                 mul_dyn(ex_to<numeric>(overall_coeff))))
442                       ->setflag(status_flags::dynallocated | status_flags::evaluated);
443         }
444         return this->hold();
445 }
446
447 ex mul::evalf(int level) const
448 {
449         if (level==1)
450                 return mul(seq,overall_coeff);
451         
452         if (level==-max_recursion_level)
453                 throw(std::runtime_error("max recursion level reached"));
454         
455         std::auto_ptr<epvector> s(new epvector);
456         s->reserve(seq.size());
457
458         --level;
459         epvector::const_iterator i = seq.begin(), end = seq.end();
460         while (i != end) {
461                 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
462                                                            i->coeff));
463                 ++i;
464         }
465         return mul(s, overall_coeff.evalf(level));
466 }
467
468 void mul::find_real_imag(ex & rp, ex & ip) const
469 {
470         rp = overall_coeff.real_part();
471         ip = overall_coeff.imag_part();
472         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
473                 ex factor = recombine_pair_to_ex(*i);
474                 ex new_rp = factor.real_part();
475                 ex new_ip = factor.imag_part();
476                 if(new_ip.is_zero()) {
477                         rp *= new_rp;
478                         ip *= new_rp;
479                 } else {
480                         ex temp = rp*new_rp - ip*new_ip;
481                         ip = ip*new_rp + rp*new_ip;
482                         rp = temp;
483                 }
484         }
485         rp = rp.expand();
486         ip = ip.expand();
487 }
488
489 ex mul::real_part() const
490 {
491         ex rp, ip;
492         find_real_imag(rp, ip);
493         return rp;
494 }
495
496 ex mul::imag_part() const
497 {
498         ex rp, ip;
499         find_real_imag(rp, ip);
500         return ip;
501 }
502
503 ex mul::evalm() const
504 {
505         // numeric*matrix
506         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
507          && is_a<matrix>(seq[0].rest))
508                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
509
510         // Evaluate children first, look whether there are any matrices at all
511         // (there can be either no matrices or one matrix; if there were more
512         // than one matrix, it would be a non-commutative product)
513         std::auto_ptr<epvector> s(new epvector);
514         s->reserve(seq.size());
515
516         bool have_matrix = false;
517         epvector::iterator the_matrix;
518
519         epvector::const_iterator i = seq.begin(), end = seq.end();
520         while (i != end) {
521                 const ex &m = recombine_pair_to_ex(*i).evalm();
522                 s->push_back(split_ex_to_pair(m));
523                 if (is_a<matrix>(m)) {
524                         have_matrix = true;
525                         the_matrix = s->end() - 1;
526                 }
527                 ++i;
528         }
529
530         if (have_matrix) {
531
532                 // The product contained a matrix. We will multiply all other factors
533                 // into that matrix.
534                 matrix m = ex_to<matrix>(the_matrix->rest);
535                 s->erase(the_matrix);
536                 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
537                 return m.mul_scalar(scalar);
538
539         } else
540                 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
541 }
542
543 ex mul::eval_ncmul(const exvector & v) const
544 {
545         if (seq.empty())
546                 return inherited::eval_ncmul(v);
547
548         // Find first noncommutative element and call its eval_ncmul()
549         epvector::const_iterator i = seq.begin(), end = seq.end();
550         while (i != end) {
551                 if (i->rest.return_type() == return_types::noncommutative)
552                         return i->rest.eval_ncmul(v);
553                 ++i;
554         }
555         return inherited::eval_ncmul(v);
556 }
557
558 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, lst & repls)
559 {       
560         ex origbase;
561         int origexponent;
562         int origexpsign;
563
564         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
565                 origbase = origfactor.op(0);
566                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
567                 origexponent = expon > 0 ? expon : -expon;
568                 origexpsign = expon > 0 ? 1 : -1;
569         } else {
570                 origbase = origfactor;
571                 origexponent = 1;
572                 origexpsign = 1;
573         }
574
575         ex patternbase;
576         int patternexponent;
577         int patternexpsign;
578
579         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
580                 patternbase = patternfactor.op(0);
581                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
582                 patternexponent = expon > 0 ? expon : -expon;
583                 patternexpsign = expon > 0 ? 1 : -1;
584         } else {
585                 patternbase = patternfactor;
586                 patternexponent = 1;
587                 patternexpsign = 1;
588         }
589
590         lst saverepls = repls;
591         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
592                 return false;
593         repls = saverepls;
594
595         int newnummatches = origexponent / patternexponent;
596         if (newnummatches < nummatches)
597                 nummatches = newnummatches;
598         return true;
599 }
600
601 /** Checks wheter e matches to the pattern pat and the (possibly to be updated)
602   * list of replacements repls. This matching is in the sense of algebraic
603   * substitutions. Matching starts with pat.op(factor) of the pattern because
604   * the factors before this one have already been matched. The (possibly
605   * updated) number of matches is in nummatches. subsed[i] is true for factors
606   * that already have been replaced by previous substitutions and matched[i]
607   * is true for factors that have been matched by the current match.
608   */
609 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, lst &repls,
610                 int factor, int &nummatches, const std::vector<bool> &subsed,
611                 std::vector<bool> &matched)
612 {
613         if (factor == pat.nops())
614                 return true;
615
616         for (size_t i=0; i<e.nops(); ++i) {
617                 if(subsed[i] || matched[i])
618                         continue;
619                 lst newrepls = repls;
620                 int newnummatches = nummatches;
621                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
622                         matched[i] = true;
623                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
624                                         newnummatches, subsed, matched)) {
625                                 repls = newrepls;
626                                 nummatches = newnummatches;
627                                 return true;
628                         }
629                         else
630                                 matched[i] = false;
631                 }
632         }
633
634         return false;
635 }
636
637 bool mul::has(const ex & pattern, unsigned options) const
638 {
639         if(!(options&has_options::algebraic))
640                 return basic::has(pattern,options);
641         if(is_a<mul>(pattern)) {
642                 lst repls;
643                 int nummatches = std::numeric_limits<int>::max();
644                 std::vector<bool> subsed(seq.size(), false);
645                 std::vector<bool> matched(seq.size(), false);
646                 if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
647                                 subsed, matched))
648                         return true;
649         }
650         return basic::has(pattern, options);
651 }
652
653 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
654 {       
655         std::vector<bool> subsed(seq.size(), false);
656         exvector subsresult(seq.size());
657         ex divide_by = 1;
658         ex multiply_by = 1;
659
660         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
661
662                 if (is_exactly_a<mul>(it->first)) {
663 retry1:
664                         int nummatches = std::numeric_limits<int>::max();
665                         std::vector<bool> currsubsed(seq.size(), false);
666                         lst repls;
667                         
668                         if(!algebraic_match_mul_with_mul(*this, it->first, repls, 0, nummatches, subsed, currsubsed))
669                                 continue;
670
671                         for (size_t j=0; j<subsed.size(); j++)
672                                 if (currsubsed[j])
673                                         subsed[j] = true;
674                         ex subsed_pattern
675                                 = it->first.subs(ex(repls), subs_options::no_pattern);
676                         divide_by *= power(subsed_pattern, nummatches);
677                         ex subsed_result
678                                 = it->second.subs(ex(repls), subs_options::no_pattern);
679                         multiply_by *= power(subsed_result, nummatches);
680                         goto retry1;
681
682                 } else {
683
684                         for (size_t j=0; j<this->nops(); j++) {
685                                 int nummatches = std::numeric_limits<int>::max();
686                                 lst repls;
687                                 if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)){
688                                         subsed[j] = true;
689                                         ex subsed_pattern
690                                                 = it->first.subs(ex(repls), subs_options::no_pattern);
691                                         divide_by *= power(subsed_pattern, nummatches);
692                                         ex subsed_result
693                                                 = it->second.subs(ex(repls), subs_options::no_pattern);
694                                         multiply_by *= power(subsed_result, nummatches);
695                                 }
696                         }
697                 }
698         }
699
700         bool subsfound = false;
701         for (size_t i=0; i<subsed.size(); i++) {
702                 if (subsed[i]) {
703                         subsfound = true;
704                         break;
705                 }
706         }
707         if (!subsfound)
708                 return subs_one_level(m, options | subs_options::algebraic);
709
710         return ((*this)/divide_by)*multiply_by;
711 }
712
713 // protected
714
715 /** Implementation of ex::diff() for a product.  It applies the product rule.
716  *  @see ex::diff */
717 ex mul::derivative(const symbol & s) const
718 {
719         size_t num = seq.size();
720         exvector addseq;
721         addseq.reserve(num);
722         
723         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
724         epvector mulseq = seq;
725         epvector::const_iterator i = seq.begin(), end = seq.end();
726         epvector::iterator i2 = mulseq.begin();
727         while (i != end) {
728                 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
729                                              i->rest.diff(s));
730                 ep.swap(*i2);
731                 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
732                 ep.swap(*i2);
733                 ++i; ++i2;
734         }
735         return (new add(addseq))->setflag(status_flags::dynallocated);
736 }
737
738 int mul::compare_same_type(const basic & other) const
739 {
740         return inherited::compare_same_type(other);
741 }
742
743 unsigned mul::return_type() const
744 {
745         if (seq.empty()) {
746                 // mul without factors: should not happen, but commutates
747                 return return_types::commutative;
748         }
749         
750         bool all_commutative = true;
751         epvector::const_iterator noncommutative_element; // point to first found nc element
752         
753         epvector::const_iterator i = seq.begin(), end = seq.end();
754         while (i != end) {
755                 unsigned rt = i->rest.return_type();
756                 if (rt == return_types::noncommutative_composite)
757                         return rt; // one ncc -> mul also ncc
758                 if ((rt == return_types::noncommutative) && (all_commutative)) {
759                         // first nc element found, remember position
760                         noncommutative_element = i;
761                         all_commutative = false;
762                 }
763                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
764                         // another nc element found, compare type_infos
765                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
766                                         // different types -> mul is ncc
767                                         return return_types::noncommutative_composite;
768                         }
769                 }
770                 ++i;
771         }
772         // all factors checked
773         return all_commutative ? return_types::commutative : return_types::noncommutative;
774 }
775    
776 tinfo_t mul::return_type_tinfo() const
777 {
778         if (seq.empty())
779                 return this;  // mul without factors: should not happen
780         
781         // return type_info of first noncommutative element
782         epvector::const_iterator i = seq.begin(), end = seq.end();
783         while (i != end) {
784                 if (i->rest.return_type() == return_types::noncommutative)
785                         return i->rest.return_type_tinfo();
786                 ++i;
787         }
788         // no noncommutative element found, should not happen
789         return this;
790 }
791
792 ex mul::thisexpairseq(const epvector & v, const ex & oc, bool do_index_renaming) const
793 {
794         return (new mul(v, oc, do_index_renaming))->setflag(status_flags::dynallocated);
795 }
796
797 ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming) const
798 {
799         return (new mul(vp, oc, do_index_renaming))->setflag(status_flags::dynallocated);
800 }
801
802 expair mul::split_ex_to_pair(const ex & e) const
803 {
804         if (is_exactly_a<power>(e)) {
805                 const power & powerref = ex_to<power>(e);
806                 if (is_exactly_a<numeric>(powerref.exponent))
807                         return expair(powerref.basis,powerref.exponent);
808         }
809         return expair(e,_ex1);
810 }
811         
812 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
813                                           const ex & c) const
814 {
815         // to avoid duplication of power simplification rules,
816         // we create a temporary power object
817         // otherwise it would be hard to correctly evaluate
818         // expression like (4^(1/3))^(3/2)
819         if (c.is_equal(_ex1))
820                 return split_ex_to_pair(e);
821
822         return split_ex_to_pair(power(e,c));
823 }
824         
825 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
826                                             const ex & c) const
827 {
828         // to avoid duplication of power simplification rules,
829         // we create a temporary power object
830         // otherwise it would be hard to correctly evaluate
831         // expression like (4^(1/3))^(3/2)
832         if (c.is_equal(_ex1))
833                 return p;
834
835         return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
836 }
837         
838 ex mul::recombine_pair_to_ex(const expair & p) const
839 {
840         if (ex_to<numeric>(p.coeff).is_equal(*_num1_p)) 
841                 return p.rest;
842         else
843                 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
844 }
845
846 bool mul::expair_needs_further_processing(epp it)
847 {
848         if (is_exactly_a<mul>(it->rest) &&
849                 ex_to<numeric>(it->coeff).is_integer()) {
850                 // combined pair is product with integer power -> expand it
851                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
852                 return true;
853         }
854         if (is_exactly_a<numeric>(it->rest)) {
855                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
856                 if (!ep.is_equal(*it)) {
857                         // combined pair is a numeric power which can be simplified
858                         *it = ep;
859                         return true;
860                 }
861                 if (it->coeff.is_equal(_ex1)) {
862                         // combined pair has coeff 1 and must be moved to the end
863                         return true;
864                 }
865         }
866         return false;
867 }       
868
869 ex mul::default_overall_coeff() const
870 {
871         return _ex1;
872 }
873
874 void mul::combine_overall_coeff(const ex & c)
875 {
876         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
877         GINAC_ASSERT(is_exactly_a<numeric>(c));
878         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
879 }
880
881 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
882 {
883         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
884         GINAC_ASSERT(is_exactly_a<numeric>(c1));
885         GINAC_ASSERT(is_exactly_a<numeric>(c2));
886         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
887 }
888
889 bool mul::can_make_flat(const expair & p) const
890 {
891         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
892         // this assertion will probably fail somewhere
893         // it would require a more careful make_flat, obeying the power laws
894         // probably should return true only if p.coeff is integer
895         return ex_to<numeric>(p.coeff).is_equal(*_num1_p);
896 }
897
898 bool mul::can_be_further_expanded(const ex & e)
899 {
900         if (is_exactly_a<mul>(e)) {
901                 for (epvector::const_iterator cit = ex_to<mul>(e).seq.begin(); cit != ex_to<mul>(e).seq.end(); ++cit) {
902                         if (is_exactly_a<add>(cit->rest) && cit->coeff.info(info_flags::posint))
903                                 return true;
904                 }
905         } else if (is_exactly_a<power>(e)) {
906                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
907                         return true;
908         }
909         return false;
910 }
911
912 ex mul::expand(unsigned options) const
913 {
914         // First, expand the children
915         std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
916         const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
917
918         // Now, look for all the factors that are sums and multiply each one out
919         // with the next one that is found while collecting the factors which are
920         // not sums
921         ex last_expanded = _ex1;
922
923         epvector non_adds;
924         non_adds.reserve(expanded_seq.size());
925
926         for (epvector::const_iterator cit = expanded_seq.begin(); cit != expanded_seq.end(); ++cit) {
927                 if (is_exactly_a<add>(cit->rest) &&
928                         (cit->coeff.is_equal(_ex1))) {
929                         if (is_exactly_a<add>(last_expanded)) {
930
931                                 // Expand a product of two sums, aggressive version.
932                                 // Caring for the overall coefficients in separate loops can
933                                 // sometimes give a performance gain of up to 15%!
934
935                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
936                                 // add2 is for the inner loop and should be the bigger of the two sums
937                                 // in the presence of asymptotically good sorting:
938                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
939                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
940                                 const epvector::const_iterator add1begin = add1.seq.begin();
941                                 const epvector::const_iterator add1end   = add1.seq.end();
942                                 const epvector::const_iterator add2begin = add2.seq.begin();
943                                 const epvector::const_iterator add2end   = add2.seq.end();
944                                 epvector distrseq;
945                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
946
947                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
948                                 if (!add1.overall_coeff.is_zero()) {
949                                         if (add1.overall_coeff.is_equal(_ex1))
950                                                 distrseq.insert(distrseq.end(),add2begin,add2end);
951                                         else
952                                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
953                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
954                                 }
955
956                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
957                                 if (!add2.overall_coeff.is_zero()) {
958                                         if (add2.overall_coeff.is_equal(_ex1))
959                                                 distrseq.insert(distrseq.end(),add1begin,add1end);
960                                         else
961                                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
962                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
963                                 }
964
965                                 // Compute the new overall coefficient and put it together:
966                                 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
967
968                                 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
969
970                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i) {
971                                         add_indices = get_all_dummy_indices_safely(i->rest);
972                                         add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
973                                 }
974                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i) {
975                                         add_indices = get_all_dummy_indices_safely(i->rest);
976                                         add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
977                                 }
978
979                                 sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
980                                 sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
981                                 lst dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
982
983                                 // Multiply explicitly all non-numeric terms of add1 and add2:
984                                 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
985                                         // We really have to combine terms here in order to compactify
986                                         // the result.  Otherwise it would become waayy tooo bigg.
987                                         numeric oc;
988                                         distrseq.clear();
989                                         ex i2_new = (dummy_subs.op(0).nops()>0? 
990                                                                  i2->rest.subs((lst)dummy_subs.op(0), (lst)dummy_subs.op(1), subs_options::no_pattern) : i2->rest);
991                                         for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
992                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
993                                                 // since that would violate an invariant of expairseq:
994                                                 const ex rest = (new mul(i1->rest, i2_new))->setflag(status_flags::dynallocated);
995                                                 if (is_exactly_a<numeric>(rest)) {
996                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
997                                                 } else {
998                                                         distrseq.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
999                                                 }
1000                                         }
1001                                         tmp_accu += (new add(distrseq, oc))->setflag(status_flags::dynallocated);
1002                                 }
1003                                 last_expanded = tmp_accu;
1004
1005                         } else {
1006                                 if (!last_expanded.is_equal(_ex1))
1007                                         non_adds.push_back(split_ex_to_pair(last_expanded));
1008                                 last_expanded = cit->rest;
1009                         }
1010
1011                 } else {
1012                         non_adds.push_back(*cit);
1013                 }
1014         }
1015
1016         // Now the only remaining thing to do is to multiply the factors which
1017         // were not sums into the "last_expanded" sum
1018         if (is_exactly_a<add>(last_expanded)) {
1019                 size_t n = last_expanded.nops();
1020                 exvector distrseq;
1021                 distrseq.reserve(n);
1022                 exvector va = get_all_dummy_indices_safely(mul(non_adds));
1023                 sort(va.begin(), va.end(), ex_is_less());
1024
1025                 for (size_t i=0; i<n; ++i) {
1026                         epvector factors = non_adds;
1027                         factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
1028                         ex term = (new mul(factors, overall_coeff))->setflag(status_flags::dynallocated);
1029                         if (can_be_further_expanded(term)) {
1030                                 distrseq.push_back(term.expand());
1031                         } else {
1032                                 if (options == 0)
1033                                         ex_to<basic>(term).setflag(status_flags::expanded);
1034                                 distrseq.push_back(term);
1035                         }
1036                 }
1037
1038                 return ((new add(distrseq))->
1039                         setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
1040         }
1041
1042         non_adds.push_back(split_ex_to_pair(last_expanded));
1043         ex result = (new mul(non_adds, overall_coeff))->setflag(status_flags::dynallocated);
1044         if (can_be_further_expanded(result)) {
1045                 return result.expand();
1046         } else {
1047                 if (options == 0)
1048                         ex_to<basic>(result).setflag(status_flags::expanded);
1049                 return result;
1050         }
1051 }
1052
1053   
1054 //////////
1055 // new virtual functions which can be overridden by derived classes
1056 //////////
1057
1058 // none
1059
1060 //////////
1061 // non-virtual functions in this class
1062 //////////
1063
1064
1065 /** Member-wise expand the expairs representing this sequence.  This must be
1066  *  overridden from expairseq::expandchildren() and done iteratively in order
1067  *  to allow for early cancallations and thus safe memory.
1068  *
1069  *  @see mul::expand()
1070  *  @return pointer to epvector containing expanded representation or zero
1071  *  pointer, if sequence is unchanged. */
1072 std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
1073 {
1074         const epvector::const_iterator last = seq.end();
1075         epvector::const_iterator cit = seq.begin();
1076         while (cit!=last) {
1077                 const ex & factor = recombine_pair_to_ex(*cit);
1078                 const ex & expanded_factor = factor.expand(options);
1079                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1080                         
1081                         // something changed, copy seq, eval and return it
1082                         std::auto_ptr<epvector> s(new epvector);
1083                         s->reserve(seq.size());
1084                         
1085                         // copy parts of seq which are known not to have changed
1086                         epvector::const_iterator cit2 = seq.begin();
1087                         while (cit2!=cit) {
1088                                 s->push_back(*cit2);
1089                                 ++cit2;
1090                         }
1091
1092                         // copy first changed element
1093                         s->push_back(split_ex_to_pair(expanded_factor));
1094                         ++cit2;
1095
1096                         // copy rest
1097                         while (cit2!=last) {
1098                                 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1099                                 ++cit2;
1100                         }
1101                         return s;
1102                 }
1103                 ++cit;
1104         }
1105         
1106         return std::auto_ptr<epvector>(0); // nothing has changed
1107 }
1108
1109 } // namespace GiNaC