ab46ae0e91c8a589a8599ca3300de76352636eed
[ginac.git] / ginac / mul.cpp
1 /** @file mul.cpp
2  *
3  *  Implementation of GiNaC's products of expressions. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
7  *
8  *  This program is free software; you can redistribute it and/or modify
9  *  it under the terms of the GNU General Public License as published by
10  *  the Free Software Foundation; either version 2 of the License, or
11  *  (at your option) any later version.
12  *
13  *  This program is distributed in the hope that it will be useful,
14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
16  *  GNU General Public License for more details.
17  *
18  *  You should have received a copy of the GNU General Public License
19  *  along with this program; if not, write to the Free Software
20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
21  */
22
23 #include <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 = TINFO_mul;
55 }
56
57 //////////
58 // other constructors
59 //////////
60
61 // public
62
63 mul::mul(const ex & lh, const ex & rh)
64 {
65         tinfo_key = TINFO_mul;
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 = TINFO_mul;
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 = TINFO_mul;
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)
88 {
89         tinfo_key = TINFO_mul;
90         overall_coeff = oc;
91         construct_from_epvector(v);
92         GINAC_ASSERT(is_canonical());
93 }
94
95 mul::mul(std::auto_ptr<epvector> vp, const ex & oc)
96 {
97         tinfo_key = TINFO_mul;
98         GINAC_ASSERT(vp.get()!=0);
99         overall_coeff = oc;
100         construct_from_epvector(*vp);
101         GINAC_ASSERT(is_canonical());
102 }
103
104 mul::mul(const ex & lh, const ex & mh, const ex & rh)
105 {
106         tinfo_key = TINFO_mul;
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                 overall_coeff.print(c, precedence());
222                 c.s << "*";
223         }
224
225         // Print arguments, separated by "*" or "/"
226         epvector::const_iterator it = seq.begin(), itend = seq.end();
227         while (it != itend) {
228
229                 // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
230                 bool needclosingparenthesis = false;
231                 if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
232                         if (is_a<print_csrc_cl_N>(c)) {
233                                 c.s << "recip(";
234                                 needclosingparenthesis = true;
235                         } else
236                                 c.s << "1.0/";
237                 }
238
239                 // If the exponent is 1 or -1, it is left out
240                 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
241                         it->rest.print(c, precedence());
242                 else if (it->coeff.info(info_flags::negint))
243                         // Outer parens around ex needed for broken GCC parser:
244                         (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
245                 else
246                         // Outer parens around ex needed for broken GCC parser:
247                         (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
248
249                 if (needclosingparenthesis)
250                         c.s << ")";
251
252                 // Separator is "/" for negative integer powers, "*" otherwise
253                 ++it;
254                 if (it != itend) {
255                         if (it->coeff.info(info_flags::negint))
256                                 c.s << "/";
257                         else
258                                 c.s << "*";
259                 }
260         }
261
262         if (precedence() <= level)
263                 c.s << ")";
264 }
265
266 void mul::do_print_python_repr(const print_python_repr & c, unsigned level) const
267 {
268         c.s << class_name() << '(';
269         op(0).print(c);
270         for (size_t i=1; i<nops(); ++i) {
271                 c.s << ',';
272                 op(i).print(c);
273         }
274         c.s << ')';
275 }
276
277 bool mul::info(unsigned inf) const
278 {
279         switch (inf) {
280                 case info_flags::polynomial:
281                 case info_flags::integer_polynomial:
282                 case info_flags::cinteger_polynomial:
283                 case info_flags::rational_polynomial:
284                 case info_flags::crational_polynomial:
285                 case info_flags::rational_function: {
286                         epvector::const_iterator i = seq.begin(), end = seq.end();
287                         while (i != end) {
288                                 if (!(recombine_pair_to_ex(*i).info(inf)))
289                                         return false;
290                                 ++i;
291                         }
292                         return overall_coeff.info(inf);
293                 }
294                 case info_flags::algebraic: {
295                         epvector::const_iterator i = seq.begin(), end = seq.end();
296                         while (i != end) {
297                                 if ((recombine_pair_to_ex(*i).info(inf)))
298                                         return true;
299                                 ++i;
300                         }
301                         return false;
302                 }
303         }
304         return inherited::info(inf);
305 }
306
307 int mul::degree(const ex & s) const
308 {
309         // Sum up degrees of factors
310         int deg_sum = 0;
311         epvector::const_iterator i = seq.begin(), end = seq.end();
312         while (i != end) {
313                 if (ex_to<numeric>(i->coeff).is_integer())
314                         deg_sum += i->rest.degree(s) * ex_to<numeric>(i->coeff).to_int();
315                 ++i;
316         }
317         return deg_sum;
318 }
319
320 int mul::ldegree(const ex & s) const
321 {
322         // Sum up degrees of factors
323         int deg_sum = 0;
324         epvector::const_iterator i = seq.begin(), end = seq.end();
325         while (i != end) {
326                 if (ex_to<numeric>(i->coeff).is_integer())
327                         deg_sum += i->rest.ldegree(s) * ex_to<numeric>(i->coeff).to_int();
328                 ++i;
329         }
330         return deg_sum;
331 }
332
333 ex mul::coeff(const ex & s, int n) const
334 {
335         exvector coeffseq;
336         coeffseq.reserve(seq.size()+1);
337         
338         if (n==0) {
339                 // product of individual coeffs
340                 // if a non-zero power of s is found, the resulting product will be 0
341                 epvector::const_iterator i = seq.begin(), end = seq.end();
342                 while (i != end) {
343                         coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
344                         ++i;
345                 }
346                 coeffseq.push_back(overall_coeff);
347                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
348         }
349         
350         epvector::const_iterator i = seq.begin(), end = seq.end();
351         bool coeff_found = false;
352         while (i != end) {
353                 ex t = recombine_pair_to_ex(*i);
354                 ex c = t.coeff(s, n);
355                 if (!c.is_zero()) {
356                         coeffseq.push_back(c);
357                         coeff_found = 1;
358                 } else {
359                         coeffseq.push_back(t);
360                 }
361                 ++i;
362         }
363         if (coeff_found) {
364                 coeffseq.push_back(overall_coeff);
365                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
366         }
367         
368         return _ex0;
369 }
370
371 /** Perform automatic term rewriting rules in this class.  In the following
372  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
373  *  stand for such expressions that contain a plain number.
374  *  - *(...,x;0) -> 0
375  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
376  *  - *(x;1) -> x
377  *  - *(;c) -> c
378  *
379  *  @param level cut-off in recursive evaluation */
380 ex mul::eval(int level) const
381 {
382         std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
383         if (evaled_seqp.get()) {
384                 // do more evaluation later
385                 return (new mul(evaled_seqp, overall_coeff))->
386                            setflag(status_flags::dynallocated);
387         }
388         
389 #ifdef DO_GINAC_ASSERT
390         epvector::const_iterator i = seq.begin(), end = seq.end();
391         while (i != end) {
392                 GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
393                              (!(ex_to<numeric>(i->coeff).is_integer())));
394                 GINAC_ASSERT(!(i->is_canonical_numeric()));
395                 if (is_exactly_a<numeric>(recombine_pair_to_ex(*i)))
396                     print(print_tree(std::cerr));
397                 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
398                 /* for paranoia */
399                 expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
400                 GINAC_ASSERT(p.rest.is_equal(i->rest));
401                 GINAC_ASSERT(p.coeff.is_equal(i->coeff));
402                 /* end paranoia */
403                 ++i;
404         }
405 #endif // def DO_GINAC_ASSERT
406         
407         if (flags & status_flags::evaluated) {
408                 GINAC_ASSERT(seq.size()>0);
409                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
410                 return *this;
411         }
412         
413         int seq_size = seq.size();
414         if (overall_coeff.is_zero()) {
415                 // *(...,x;0) -> 0
416                 return _ex0;
417         } else if (seq_size==0) {
418                 // *(;c) -> c
419                 return overall_coeff;
420         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
421                 // *(x;1) -> x
422                 return recombine_pair_to_ex(*(seq.begin()));
423         } else if ((seq_size==1) &&
424                    is_exactly_a<add>((*seq.begin()).rest) &&
425                    ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
426                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
427                 const add & addref = ex_to<add>((*seq.begin()).rest);
428                 std::auto_ptr<epvector> distrseq(new epvector);
429                 distrseq->reserve(addref.seq.size());
430                 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
431                 while (i != end) {
432                         distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
433                         ++i;
434                 }
435                 return (new add(distrseq,
436                                 ex_to<numeric>(addref.overall_coeff).
437                                 mul_dyn(ex_to<numeric>(overall_coeff))))
438                       ->setflag(status_flags::dynallocated | status_flags::evaluated);
439         }
440         return this->hold();
441 }
442
443 ex mul::evalf(int level) const
444 {
445         if (level==1)
446                 return mul(seq,overall_coeff);
447         
448         if (level==-max_recursion_level)
449                 throw(std::runtime_error("max recursion level reached"));
450         
451         std::auto_ptr<epvector> s(new epvector);
452         s->reserve(seq.size());
453
454         --level;
455         epvector::const_iterator i = seq.begin(), end = seq.end();
456         while (i != end) {
457                 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
458                                                            i->coeff));
459                 ++i;
460         }
461         return mul(s, overall_coeff.evalf(level));
462 }
463
464 ex mul::evalm() const
465 {
466         // numeric*matrix
467         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
468          && is_a<matrix>(seq[0].rest))
469                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
470
471         // Evaluate children first, look whether there are any matrices at all
472         // (there can be either no matrices or one matrix; if there were more
473         // than one matrix, it would be a non-commutative product)
474         std::auto_ptr<epvector> s(new epvector);
475         s->reserve(seq.size());
476
477         bool have_matrix = false;
478         epvector::iterator the_matrix;
479
480         epvector::const_iterator i = seq.begin(), end = seq.end();
481         while (i != end) {
482                 const ex &m = recombine_pair_to_ex(*i).evalm();
483                 s->push_back(split_ex_to_pair(m));
484                 if (is_a<matrix>(m)) {
485                         have_matrix = true;
486                         the_matrix = s->end() - 1;
487                 }
488                 ++i;
489         }
490
491         if (have_matrix) {
492
493                 // The product contained a matrix. We will multiply all other factors
494                 // into that matrix.
495                 matrix m = ex_to<matrix>(the_matrix->rest);
496                 s->erase(the_matrix);
497                 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
498                 return m.mul_scalar(scalar);
499
500         } else
501                 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
502 }
503
504 ex mul::eval_ncmul(const exvector & v) const
505 {
506         if (seq.empty())
507                 return inherited::eval_ncmul(v);
508
509         // Find first noncommutative element and call its eval_ncmul()
510         epvector::const_iterator i = seq.begin(), end = seq.end();
511         while (i != end) {
512                 if (i->rest.return_type() == return_types::noncommutative)
513                         return i->rest.eval_ncmul(v);
514                 ++i;
515         }
516         return inherited::eval_ncmul(v);
517 }
518
519 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, lst & repls)
520 {       
521         ex origbase;
522         int origexponent;
523         int origexpsign;
524
525         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
526                 origbase = origfactor.op(0);
527                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
528                 origexponent = expon > 0 ? expon : -expon;
529                 origexpsign = expon > 0 ? 1 : -1;
530         } else {
531                 origbase = origfactor;
532                 origexponent = 1;
533                 origexpsign = 1;
534         }
535
536         ex patternbase;
537         int patternexponent;
538         int patternexpsign;
539
540         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
541                 patternbase = patternfactor.op(0);
542                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
543                 patternexponent = expon > 0 ? expon : -expon;
544                 patternexpsign = expon > 0 ? 1 : -1;
545         } else {
546                 patternbase = patternfactor;
547                 patternexponent = 1;
548                 patternexpsign = 1;
549         }
550
551         lst saverepls = repls;
552         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
553                 return false;
554         repls = saverepls;
555
556         int newnummatches = origexponent / patternexponent;
557         if (newnummatches < nummatches)
558                 nummatches = newnummatches;
559         return true;
560 }
561
562 /** Checks wheter e matches to the pattern pat and the (possibly to be updated
563   * list of replacements repls. This matching is in the sense of algebraic
564   * substitutions. Matching starts with pat.op(factor) of the pattern because
565   * the factors before this one have already been matched. The (possibly
566   * updated) number of matches is in nummatches. subsed[i] is true for factors
567   * that already have been replaced by previous substitutions and matched[i]
568   * is true for factors that have been matched by the current match.
569   */
570 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, lst &repls,
571                 int factor, int &nummatches, const std::vector<bool> &subsed,
572                 std::vector<bool> &matched)
573 {
574         if (factor == pat.nops())
575                 return true;
576
577         for (size_t i=0; i<e.nops(); ++i) {
578                 if(subsed[i] || matched[i])
579                         continue;
580                 lst newrepls = repls;
581                 int newnummatches = nummatches;
582                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
583                         matched[i] = true;
584                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
585                                         newnummatches, subsed, matched)) {
586                                 repls = newrepls;
587                                 nummatches = newnummatches;
588                                 return true;
589                         }
590                         else
591                                 matched[i] = false;
592                 }
593         }
594
595         return false;
596 }
597
598 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
599 {       
600         std::vector<bool> subsed(seq.size(), false);
601         exvector subsresult(seq.size());
602
603         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
604
605                 if (is_exactly_a<mul>(it->first)) {
606 retry1:
607                         int nummatches = std::numeric_limits<int>::max();
608                         std::vector<bool> currsubsed(seq.size(), false);
609                         bool succeed = true;
610                         lst repls;
611                         
612                         if(!algebraic_match_mul_with_mul(*this, it->first, repls, 0, nummatches, subsed, currsubsed))
613                                 continue;
614
615                         bool foundfirstsubsedfactor = false;
616                         for (size_t j=0; j<subsed.size(); j++) {
617                                 if (currsubsed[j]) {
618                                         if (foundfirstsubsedfactor)
619                                                 subsresult[j] = op(j);
620                                         else {
621                                                 foundfirstsubsedfactor = true;
622                                                 subsresult[j] = op(j) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches);
623                                         }
624                                         subsed[j] = true;
625                                 }
626                         }
627                         goto retry1;
628
629                 } else {
630 retry2:
631                         int nummatches = std::numeric_limits<int>::max();
632                         lst repls;
633
634                         for (size_t j=0; j<this->nops(); j++) {
635                                 if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)) {
636                                         subsed[j] = true;
637                                         subsresult[j] = op(j) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches);
638                                         goto retry2;
639                                 }
640                         }
641                 }
642         }
643
644         bool subsfound = false;
645         for (size_t i=0; i<subsed.size(); i++) {
646                 if (subsed[i]) {
647                         subsfound = true;
648                         break;
649                 }
650         }
651         if (!subsfound)
652                 return subs_one_level(m, options | subs_options::algebraic);
653
654         exvector ev; ev.reserve(nops());
655         for (size_t i=0; i<nops(); i++) {
656                 if (subsed[i])
657                         ev.push_back(subsresult[i]);
658                 else
659                         ev.push_back(op(i));
660         }
661
662         return (new mul(ev))->setflag(status_flags::dynallocated);
663 }
664
665 // protected
666
667 /** Implementation of ex::diff() for a product.  It applies the product rule.
668  *  @see ex::diff */
669 ex mul::derivative(const symbol & s) const
670 {
671         size_t num = seq.size();
672         exvector addseq;
673         addseq.reserve(num);
674         
675         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
676         epvector mulseq = seq;
677         epvector::const_iterator i = seq.begin(), end = seq.end();
678         epvector::iterator i2 = mulseq.begin();
679         while (i != end) {
680                 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
681                                              i->rest.diff(s));
682                 ep.swap(*i2);
683                 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
684                 ep.swap(*i2);
685                 ++i; ++i2;
686         }
687         return (new add(addseq))->setflag(status_flags::dynallocated);
688 }
689
690 int mul::compare_same_type(const basic & other) const
691 {
692         return inherited::compare_same_type(other);
693 }
694
695 unsigned mul::return_type() const
696 {
697         if (seq.empty()) {
698                 // mul without factors: should not happen, but commutates
699                 return return_types::commutative;
700         }
701         
702         bool all_commutative = true;
703         epvector::const_iterator noncommutative_element; // point to first found nc element
704         
705         epvector::const_iterator i = seq.begin(), end = seq.end();
706         while (i != end) {
707                 unsigned rt = i->rest.return_type();
708                 if (rt == return_types::noncommutative_composite)
709                         return rt; // one ncc -> mul also ncc
710                 if ((rt == return_types::noncommutative) && (all_commutative)) {
711                         // first nc element found, remember position
712                         noncommutative_element = i;
713                         all_commutative = false;
714                 }
715                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
716                         // another nc element found, compare type_infos
717                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
718                                 // diffent types -> mul is ncc
719                                 return return_types::noncommutative_composite;
720                         }
721                 }
722                 ++i;
723         }
724         // all factors checked
725         return all_commutative ? return_types::commutative : return_types::noncommutative;
726 }
727    
728 unsigned mul::return_type_tinfo() const
729 {
730         if (seq.empty())
731                 return tinfo_key;  // mul without factors: should not happen
732         
733         // return type_info of first noncommutative element
734         epvector::const_iterator i = seq.begin(), end = seq.end();
735         while (i != end) {
736                 if (i->rest.return_type() == return_types::noncommutative)
737                         return i->rest.return_type_tinfo();
738                 ++i;
739         }
740         // no noncommutative element found, should not happen
741         return tinfo_key;
742 }
743
744 ex mul::thisexpairseq(const epvector & v, const ex & oc) const
745 {
746         return (new mul(v, oc))->setflag(status_flags::dynallocated);
747 }
748
749 ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc) const
750 {
751         return (new mul(vp, oc))->setflag(status_flags::dynallocated);
752 }
753
754 expair mul::split_ex_to_pair(const ex & e) const
755 {
756         if (is_exactly_a<power>(e)) {
757                 const power & powerref = ex_to<power>(e);
758                 if (is_exactly_a<numeric>(powerref.exponent))
759                         return expair(powerref.basis,powerref.exponent);
760         }
761         return expair(e,_ex1);
762 }
763         
764 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
765                                           const ex & c) const
766 {
767         // to avoid duplication of power simplification rules,
768         // we create a temporary power object
769         // otherwise it would be hard to correctly evaluate
770         // expression like (4^(1/3))^(3/2)
771         if (c.is_equal(_ex1))
772                 return split_ex_to_pair(e);
773
774         return split_ex_to_pair(power(e,c));
775 }
776         
777 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
778                                             const ex & c) const
779 {
780         // to avoid duplication of power simplification rules,
781         // we create a temporary power object
782         // otherwise it would be hard to correctly evaluate
783         // expression like (4^(1/3))^(3/2)
784         if (c.is_equal(_ex1))
785                 return p;
786
787         return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
788 }
789         
790 ex mul::recombine_pair_to_ex(const expair & p) const
791 {
792         if (ex_to<numeric>(p.coeff).is_equal(*_num1_p)) 
793                 return p.rest;
794         else
795                 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
796 }
797
798 bool mul::expair_needs_further_processing(epp it)
799 {
800         if (is_exactly_a<mul>(it->rest) &&
801                 ex_to<numeric>(it->coeff).is_integer()) {
802                 // combined pair is product with integer power -> expand it
803                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
804                 return true;
805         }
806         if (is_exactly_a<numeric>(it->rest)) {
807                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
808                 if (!ep.is_equal(*it)) {
809                         // combined pair is a numeric power which can be simplified
810                         *it = ep;
811                         return true;
812                 }
813                 if (it->coeff.is_equal(_ex1)) {
814                         // combined pair has coeff 1 and must be moved to the end
815                         return true;
816                 }
817         }
818         return false;
819 }       
820
821 ex mul::default_overall_coeff() const
822 {
823         return _ex1;
824 }
825
826 void mul::combine_overall_coeff(const ex & c)
827 {
828         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
829         GINAC_ASSERT(is_exactly_a<numeric>(c));
830         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
831 }
832
833 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
834 {
835         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
836         GINAC_ASSERT(is_exactly_a<numeric>(c1));
837         GINAC_ASSERT(is_exactly_a<numeric>(c2));
838         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
839 }
840
841 bool mul::can_make_flat(const expair & p) const
842 {
843         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
844         // this assertion will probably fail somewhere
845         // it would require a more careful make_flat, obeying the power laws
846         // probably should return true only if p.coeff is integer
847         return ex_to<numeric>(p.coeff).is_equal(*_num1_p);
848 }
849
850 bool mul::can_be_further_expanded(const ex & e)
851 {
852         if (is_exactly_a<mul>(e)) {
853                 for (epvector::const_iterator cit = ex_to<mul>(e).seq.begin(); cit != ex_to<mul>(e).seq.end(); ++cit) {
854                         if (is_exactly_a<add>(cit->rest) && cit->coeff.info(info_flags::posint))
855                                 return true;
856                 }
857         } else if (is_exactly_a<power>(e)) {
858                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
859                         return true;
860         }
861         return false;
862 }
863
864 ex mul::expand(unsigned options) const
865 {
866         // First, expand the children
867         std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
868         const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
869
870         // Now, look for all the factors that are sums and multiply each one out
871         // with the next one that is found while collecting the factors which are
872         // not sums
873         ex last_expanded = _ex1;
874
875         epvector non_adds;
876         non_adds.reserve(expanded_seq.size());
877
878         for (epvector::const_iterator cit = expanded_seq.begin(); cit != expanded_seq.end(); ++cit) {
879                 if (is_exactly_a<add>(cit->rest) &&
880                         (cit->coeff.is_equal(_ex1))) {
881                         if (is_exactly_a<add>(last_expanded)) {
882
883                                 // Expand a product of two sums, aggressive version.
884                                 // Caring for the overall coefficients in separate loops can
885                                 // sometimes give a performance gain of up to 15%!
886
887                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
888                                 // add2 is for the inner loop and should be the bigger of the two sums
889                                 // in the presence of asymptotically good sorting:
890                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
891                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
892                                 const epvector::const_iterator add1begin = add1.seq.begin();
893                                 const epvector::const_iterator add1end   = add1.seq.end();
894                                 const epvector::const_iterator add2begin = add2.seq.begin();
895                                 const epvector::const_iterator add2end   = add2.seq.end();
896                                 epvector distrseq;
897                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
898
899                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
900                                 if (!add1.overall_coeff.is_zero()) {
901                                         if (add1.overall_coeff.is_equal(_ex1))
902                                                 distrseq.insert(distrseq.end(),add2begin,add2end);
903                                         else
904                                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
905                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
906                                 }
907
908                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
909                                 if (!add2.overall_coeff.is_zero()) {
910                                         if (add2.overall_coeff.is_equal(_ex1))
911                                                 distrseq.insert(distrseq.end(),add1begin,add1end);
912                                         else
913                                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
914                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
915                                 }
916
917                                 // Compute the new overall coefficient and put it together:
918                                 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
919
920                                 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
921
922                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i) {
923                                         add_indices = get_all_dummy_indices(i->rest);
924                                         add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
925                                 }
926                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i) {
927                                         add_indices = get_all_dummy_indices(i->rest);
928                                         add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
929                                 }
930
931                                 sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
932                                 sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
933                                 lst dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
934
935                                 // Multiply explicitly all non-numeric terms of add1 and add2:
936                                 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
937                                         // We really have to combine terms here in order to compactify
938                                         // the result.  Otherwise it would become waayy tooo bigg.
939                                         numeric oc;
940                                         distrseq.clear();
941                                         ex i2_new = (dummy_subs.op(0).nops()>0? 
942                                                                  i2->rest.subs((lst)dummy_subs.op(0), (lst)dummy_subs.op(1), subs_options::no_pattern) : i2->rest);
943                                         for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
944                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
945                                                 // since that would violate an invariant of expairseq:
946                                                 const ex rest = (new mul(i1->rest, i2_new))->setflag(status_flags::dynallocated);
947                                                 if (is_exactly_a<numeric>(rest)) {
948                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
949                                                 } else {
950                                                         distrseq.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
951                                                 }
952                                         }
953                                         tmp_accu += (new add(distrseq, oc))->setflag(status_flags::dynallocated);
954                                 }
955                                 last_expanded = tmp_accu;
956
957                         } else {
958                                 if (!last_expanded.is_equal(_ex1))
959                                         non_adds.push_back(split_ex_to_pair(last_expanded));
960                                 last_expanded = cit->rest;
961                         }
962
963                 } else {
964                         non_adds.push_back(*cit);
965                 }
966         }
967
968         // Now the only remaining thing to do is to multiply the factors which
969         // were not sums into the "last_expanded" sum
970         if (is_exactly_a<add>(last_expanded)) {
971                 size_t n = last_expanded.nops();
972                 exvector distrseq;
973                 distrseq.reserve(n);
974                 exvector va = get_all_dummy_indices(mul(non_adds));
975                 sort(va.begin(), va.end(), ex_is_less());
976
977                 for (size_t i=0; i<n; ++i) {
978                         epvector factors = non_adds;
979                         factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
980                         ex term = (new mul(factors, overall_coeff))->setflag(status_flags::dynallocated);
981                         if (can_be_further_expanded(term)) {
982                                 distrseq.push_back(term.expand());
983                         } else {
984                                 if (options == 0)
985                                         ex_to<basic>(term).setflag(status_flags::expanded);
986                                 distrseq.push_back(term);
987                         }
988                 }
989
990                 return ((new add(distrseq))->
991                         setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
992         }
993
994         non_adds.push_back(split_ex_to_pair(last_expanded));
995         ex result = (new mul(non_adds, overall_coeff))->setflag(status_flags::dynallocated);
996         if (can_be_further_expanded(result)) {
997                 return result.expand();
998         } else {
999                 if (options == 0)
1000                         ex_to<basic>(result).setflag(status_flags::expanded);
1001                 return result;
1002         }
1003 }
1004
1005   
1006 //////////
1007 // new virtual functions which can be overridden by derived classes
1008 //////////
1009
1010 // none
1011
1012 //////////
1013 // non-virtual functions in this class
1014 //////////
1015
1016
1017 /** Member-wise expand the expairs representing this sequence.  This must be
1018  *  overridden from expairseq::expandchildren() and done iteratively in order
1019  *  to allow for early cancallations and thus safe memory.
1020  *
1021  *  @see mul::expand()
1022  *  @return pointer to epvector containing expanded representation or zero
1023  *  pointer, if sequence is unchanged. */
1024 std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
1025 {
1026         const epvector::const_iterator last = seq.end();
1027         epvector::const_iterator cit = seq.begin();
1028         while (cit!=last) {
1029                 const ex & factor = recombine_pair_to_ex(*cit);
1030                 const ex & expanded_factor = factor.expand(options);
1031                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1032                         
1033                         // something changed, copy seq, eval and return it
1034                         std::auto_ptr<epvector> s(new epvector);
1035                         s->reserve(seq.size());
1036                         
1037                         // copy parts of seq which are known not to have changed
1038                         epvector::const_iterator cit2 = seq.begin();
1039                         while (cit2!=cit) {
1040                                 s->push_back(*cit2);
1041                                 ++cit2;
1042                         }
1043
1044                         // copy first changed element
1045                         s->push_back(split_ex_to_pair(expanded_factor));
1046                         ++cit2;
1047
1048                         // copy rest
1049                         while (cit2!=last) {
1050                                 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1051                                 ++cit2;
1052                         }
1053                         return s;
1054                 }
1055                 ++cit;
1056         }
1057         
1058         return std::auto_ptr<epvector>(0); // nothing has changed
1059 }
1060
1061 } // namespace GiNaC