Add support for Texinfo-5.0.
[ginac.git] / ginac / mul.cpp
1 /** @file mul.cpp
2  *
3  *  Implementation of GiNaC's products of expressions. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2011 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 "mul.h"
24 #include "add.h"
25 #include "power.h"
26 #include "operators.h"
27 #include "matrix.h"
28 #include "indexed.h"
29 #include "lst.h"
30 #include "archive.h"
31 #include "utils.h"
32 #include "symbol.h"
33 #include "compiler.h"
34
35 #include <iostream>
36 #include <limits>
37 #include <stdexcept>
38 #include <vector>
39
40 namespace GiNaC {
41
42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mul, expairseq,
43   print_func<print_context>(&mul::do_print).
44   print_func<print_latex>(&mul::do_print_latex).
45   print_func<print_csrc>(&mul::do_print_csrc).
46   print_func<print_tree>(&mul::do_print_tree).
47   print_func<print_python_repr>(&mul::do_print_python_repr))
48
49
50 //////////
51 // default constructor
52 //////////
53
54 mul::mul()
55 {
56 }
57
58 //////////
59 // other constructors
60 //////////
61
62 // public
63
64 mul::mul(const ex & lh, const ex & rh)
65 {
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         overall_coeff = _ex1;
74         construct_from_exvector(v);
75         GINAC_ASSERT(is_canonical());
76 }
77
78 mul::mul(const epvector & v)
79 {
80         overall_coeff = _ex1;
81         construct_from_epvector(v);
82         GINAC_ASSERT(is_canonical());
83 }
84
85 mul::mul(const epvector & v, const ex & oc, bool do_index_renaming)
86 {
87         overall_coeff = oc;
88         construct_from_epvector(v, do_index_renaming);
89         GINAC_ASSERT(is_canonical());
90 }
91
92 mul::mul(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming)
93 {
94         GINAC_ASSERT(vp.get()!=0);
95         overall_coeff = oc;
96         construct_from_epvector(*vp, do_index_renaming);
97         GINAC_ASSERT(is_canonical());
98 }
99
100 mul::mul(const ex & lh, const ex & mh, const ex & rh)
101 {
102         exvector factors;
103         factors.reserve(3);
104         factors.push_back(lh);
105         factors.push_back(mh);
106         factors.push_back(rh);
107         overall_coeff = _ex1;
108         construct_from_exvector(factors);
109         GINAC_ASSERT(is_canonical());
110 }
111
112 //////////
113 // archiving
114 //////////
115
116 //////////
117 // functions overriding virtual functions from base classes
118 //////////
119
120 void mul::print_overall_coeff(const print_context & c, const char *mul_sym) const
121 {
122         const numeric &coeff = ex_to<numeric>(overall_coeff);
123         if (coeff.csgn() == -1)
124                 c.s << '-';
125         if (!coeff.is_equal(*_num1_p) &&
126                 !coeff.is_equal(*_num_1_p)) {
127                 if (coeff.is_rational()) {
128                         if (coeff.is_negative())
129                                 (-coeff).print(c);
130                         else
131                                 coeff.print(c);
132                 } else {
133                         if (coeff.csgn() == -1)
134                                 (-coeff).print(c, precedence());
135                         else
136                                 coeff.print(c, precedence());
137                 }
138                 c.s << mul_sym;
139         }
140 }
141
142 void mul::do_print(const print_context & c, unsigned level) const
143 {
144         if (precedence() <= level)
145                 c.s << '(';
146
147         print_overall_coeff(c, "*");
148
149         epvector::const_iterator it = seq.begin(), itend = seq.end();
150         bool first = true;
151         while (it != itend) {
152                 if (!first)
153                         c.s << '*';
154                 else
155                         first = false;
156                 recombine_pair_to_ex(*it).print(c, precedence());
157                 ++it;
158         }
159
160         if (precedence() <= level)
161                 c.s << ')';
162 }
163
164 void mul::do_print_latex(const print_latex & c, unsigned level) const
165 {
166         if (precedence() <= level)
167                 c.s << "{(";
168
169         print_overall_coeff(c, " ");
170
171         // Separate factors into those with negative numeric exponent
172         // and all others
173         epvector::const_iterator it = seq.begin(), itend = seq.end();
174         exvector neg_powers, others;
175         while (it != itend) {
176                 GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
177                 if (ex_to<numeric>(it->coeff).is_negative())
178                         neg_powers.push_back(recombine_pair_to_ex(expair(it->rest, -(it->coeff))));
179                 else
180                         others.push_back(recombine_pair_to_ex(*it));
181                 ++it;
182         }
183
184         if (!neg_powers.empty()) {
185
186                 // Factors with negative exponent are printed as a fraction
187                 c.s << "\\frac{";
188                 mul(others).eval().print(c);
189                 c.s << "}{";
190                 mul(neg_powers).eval().print(c);
191                 c.s << "}";
192
193         } else {
194
195                 // All other factors are printed in the ordinary way
196                 exvector::const_iterator vit = others.begin(), vitend = others.end();
197                 while (vit != vitend) {
198                         c.s << ' ';
199                         vit->print(c, precedence());
200                         ++vit;
201                 }
202         }
203
204         if (precedence() <= level)
205                 c.s << ")}";
206 }
207
208 void mul::do_print_csrc(const print_csrc & c, unsigned level) const
209 {
210         if (precedence() <= level)
211                 c.s << "(";
212
213         if (!overall_coeff.is_equal(_ex1)) {
214                 if (overall_coeff.is_equal(_ex_1))
215                         c.s << "-";
216                 else {
217                         overall_coeff.print(c, precedence());
218                         c.s << "*";
219                 }
220         }
221
222         // Print arguments, separated by "*" or "/"
223         epvector::const_iterator it = seq.begin(), itend = seq.end();
224         while (it != itend) {
225
226                 // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
227                 bool needclosingparenthesis = false;
228                 if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
229                         if (is_a<print_csrc_cl_N>(c)) {
230                                 c.s << "recip(";
231                                 needclosingparenthesis = true;
232                         } else
233                                 c.s << "1.0/";
234                 }
235
236                 // If the exponent is 1 or -1, it is left out
237                 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
238                         it->rest.print(c, precedence());
239                 else if (it->coeff.info(info_flags::negint))
240                         // Outer parens around ex needed for broken GCC parser:
241                         (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
242                 else
243                         // Outer parens around ex needed for broken GCC parser:
244                         (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
245
246                 if (needclosingparenthesis)
247                         c.s << ")";
248
249                 // Separator is "/" for negative integer powers, "*" otherwise
250                 ++it;
251                 if (it != itend) {
252                         if (it->coeff.info(info_flags::negint))
253                                 c.s << "/";
254                         else
255                                 c.s << "*";
256                 }
257         }
258
259         if (precedence() <= level)
260                 c.s << ")";
261 }
262
263 void mul::do_print_python_repr(const print_python_repr & c, unsigned level) const
264 {
265         c.s << class_name() << '(';
266         op(0).print(c);
267         for (size_t i=1; i<nops(); ++i) {
268                 c.s << ',';
269                 op(i).print(c);
270         }
271         c.s << ')';
272 }
273
274 bool mul::info(unsigned inf) const
275 {
276         switch (inf) {
277                 case info_flags::polynomial:
278                 case info_flags::integer_polynomial:
279                 case info_flags::cinteger_polynomial:
280                 case info_flags::rational_polynomial:
281                 case info_flags::real:
282                 case info_flags::rational:
283                 case info_flags::integer:
284                 case info_flags::crational:
285                 case info_flags::cinteger:
286                 case info_flags::positive:
287                 case info_flags::nonnegative:
288                 case info_flags::posint:
289                 case info_flags::nonnegint:
290                 case info_flags::even:
291                 case info_flags::crational_polynomial:
292                 case info_flags::rational_function: {
293                         epvector::const_iterator i = seq.begin(), end = seq.end();
294                         while (i != end) {
295                                 if (!(recombine_pair_to_ex(*i).info(inf)))
296                                         return false;
297                                 ++i;
298                         }
299                         if (overall_coeff.is_equal(*_num1_p) && inf == info_flags::even)
300                                 return true;
301                         return overall_coeff.info(inf);
302                 }
303                 case info_flags::algebraic: {
304                         epvector::const_iterator i = seq.begin(), end = seq.end();
305                         while (i != end) {
306                                 if ((recombine_pair_to_ex(*i).info(inf)))
307                                         return true;
308                                 ++i;
309                         }
310                         return false;
311                 }
312                 case info_flags::negative: {
313                         bool neg = false;
314                         epvector::const_iterator i = seq.begin(), end = seq.end();
315                         while (i != end) {
316                                 const ex& factor = recombine_pair_to_ex(*i++);
317                                 if (factor.info(info_flags::positive))
318                                         continue;
319                                 else if (factor.info(info_flags::negative))
320                                         neg = !neg;
321                                 else
322                                         return false;
323                         }
324                         if (overall_coeff.info(info_flags::negative))
325                                 neg = !neg;
326                         return neg;
327                 }
328                 case info_flags::negint: {
329                         bool neg = false;
330                         epvector::const_iterator i = seq.begin(), end = seq.end();
331                         while (i != end) {
332                                 const ex& factor = recombine_pair_to_ex(*i++);
333                                 if (factor.info(info_flags::posint))
334                                         continue;
335                                 else if (factor.info(info_flags::negint))
336                                         neg = !neg;
337                                 else
338                                         return false;
339                         }
340                         if (overall_coeff.info(info_flags::negint))
341                                 neg = !neg;
342                         else if (!overall_coeff.info(info_flags::posint))
343                                 return false;
344                         return neg;
345                 }
346         }
347         return inherited::info(inf);
348 }
349
350 bool mul::is_polynomial(const ex & var) const
351 {
352         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
353                 if (!i->rest.is_polynomial(var) ||
354                     (i->rest.has(var) && !i->coeff.info(info_flags::integer))) {
355                         return false;
356                 }
357         }
358         return true;
359 }
360
361 int mul::degree(const ex & s) const
362 {
363         // Sum up degrees of factors
364         int deg_sum = 0;
365         epvector::const_iterator i = seq.begin(), end = seq.end();
366         while (i != end) {
367                 if (ex_to<numeric>(i->coeff).is_integer())
368                         deg_sum += recombine_pair_to_ex(*i).degree(s);
369                 else {
370                         if (i->rest.has(s))
371                                 throw std::runtime_error("mul::degree() undefined degree because of non-integer exponent");
372                 }
373                 ++i;
374         }
375         return deg_sum;
376 }
377
378 int mul::ldegree(const ex & s) const
379 {
380         // Sum up degrees of factors
381         int deg_sum = 0;
382         epvector::const_iterator i = seq.begin(), end = seq.end();
383         while (i != end) {
384                 if (ex_to<numeric>(i->coeff).is_integer())
385                         deg_sum += recombine_pair_to_ex(*i).ldegree(s);
386                 else {
387                         if (i->rest.has(s))
388                                 throw std::runtime_error("mul::ldegree() undefined degree because of non-integer exponent");
389                 }
390                 ++i;
391         }
392         return deg_sum;
393 }
394
395 ex mul::coeff(const ex & s, int n) const
396 {
397         exvector coeffseq;
398         coeffseq.reserve(seq.size()+1);
399         
400         if (n==0) {
401                 // product of individual coeffs
402                 // if a non-zero power of s is found, the resulting product will be 0
403                 epvector::const_iterator i = seq.begin(), end = seq.end();
404                 while (i != end) {
405                         coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
406                         ++i;
407                 }
408                 coeffseq.push_back(overall_coeff);
409                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
410         }
411         
412         epvector::const_iterator i = seq.begin(), end = seq.end();
413         bool coeff_found = false;
414         while (i != end) {
415                 ex t = recombine_pair_to_ex(*i);
416                 ex c = t.coeff(s, n);
417                 if (!c.is_zero()) {
418                         coeffseq.push_back(c);
419                         coeff_found = 1;
420                 } else {
421                         coeffseq.push_back(t);
422                 }
423                 ++i;
424         }
425         if (coeff_found) {
426                 coeffseq.push_back(overall_coeff);
427                 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
428         }
429         
430         return _ex0;
431 }
432
433 /** Perform automatic term rewriting rules in this class.  In the following
434  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
435  *  stand for such expressions that contain a plain number.
436  *  - *(...,x;0) -> 0
437  *  - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
438  *  - *(x;1) -> x
439  *  - *(;c) -> c
440  *
441  *  @param level cut-off in recursive evaluation */
442 ex mul::eval(int level) const
443 {
444         std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
445         if (evaled_seqp.get()) {
446                 // do more evaluation later
447                 return (new mul(evaled_seqp, overall_coeff))->
448                            setflag(status_flags::dynallocated);
449         }
450         
451         if (flags & status_flags::evaluated) {
452                 GINAC_ASSERT(seq.size()>0);
453                 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
454                 return *this;
455         }
456         
457         size_t seq_size = seq.size();
458         if (overall_coeff.is_zero()) {
459                 // *(...,x;0) -> 0
460                 return _ex0;
461         } else if (seq_size==0) {
462                 // *(;c) -> c
463                 return overall_coeff;
464         } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
465                 // *(x;1) -> x
466                 return recombine_pair_to_ex(*(seq.begin()));
467         } else if ((seq_size==1) &&
468                    is_exactly_a<add>((*seq.begin()).rest) &&
469                    ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
470                 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
471                 const add & addref = ex_to<add>((*seq.begin()).rest);
472                 std::auto_ptr<epvector> distrseq(new epvector);
473                 distrseq->reserve(addref.seq.size());
474                 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
475                 while (i != end) {
476                         distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
477                         ++i;
478                 }
479                 return (new add(distrseq,
480                                 ex_to<numeric>(addref.overall_coeff).
481                                 mul_dyn(ex_to<numeric>(overall_coeff)))
482                        )->setflag(status_flags::dynallocated | status_flags::evaluated);
483         } else if ((seq_size >= 2) && (! (flags & status_flags::expanded))) {
484                 // Strip the content and the unit part from each term. Thus
485                 // things like (-x+a)*(3*x-3*a) automagically turn into - 3*(x-a)^2
486
487                 epvector::const_iterator last = seq.end();
488                 epvector::const_iterator i = seq.begin();
489                 epvector::const_iterator j = seq.begin();
490                 std::auto_ptr<epvector> s(new epvector);
491                 numeric oc = *_num1_p;
492                 bool something_changed = false;
493                 while (i!=last) {
494                         if (likely(! (is_a<add>(i->rest) && i->coeff.is_equal(_ex1)))) {
495                                 // power::eval has such a rule, no need to handle powers here
496                                 ++i;
497                                 continue;
498                         }
499
500                         // XXX: What is the best way to check if the polynomial is a primitive? 
501                         numeric c = i->rest.integer_content();
502                         const numeric lead_coeff =
503                                 ex_to<numeric>(ex_to<add>(i->rest).seq.begin()->coeff).div(c);
504                         const bool canonicalizable = lead_coeff.is_integer();
505
506                         // XXX: The main variable is chosen in a random way, so this code 
507                         // does NOT transform the term into the canonical form (thus, in some
508                         // very unlucky event it can even loop forever). Hopefully the main
509                         // variable will be the same for all terms in *this
510                         const bool unit_normal = lead_coeff.is_pos_integer();
511                         if (likely((c == *_num1_p) && ((! canonicalizable) || unit_normal))) {
512                                 ++i;
513                                 continue;
514                         }
515
516                         if (! something_changed) {
517                                 s->reserve(seq_size);
518                                 something_changed = true;
519                         }
520
521                         while ((j!=i) && (j!=last)) {
522                                 s->push_back(*j);
523                                 ++j;
524                         }
525
526                         if (! unit_normal)
527                                 c = c.mul(*_num_1_p);
528
529                         oc = oc.mul(c);
530
531                         // divide add by the number in place to save at least 2 .eval() calls
532                         const add& addref = ex_to<add>(i->rest);
533                         add* primitive = new add(addref);
534                         primitive->setflag(status_flags::dynallocated);
535                         primitive->clearflag(status_flags::hash_calculated);
536                         primitive->overall_coeff = ex_to<numeric>(primitive->overall_coeff).div_dyn(c);
537                         for (epvector::iterator ai = primitive->seq.begin(); ai != primitive->seq.end(); ++ai)
538                                 ai->coeff = ex_to<numeric>(ai->coeff).div_dyn(c);
539                         
540                         s->push_back(expair(*primitive, _ex1));
541
542                         ++i;
543                         ++j;
544                 }
545                 if (something_changed) {
546                         while (j!=last) {
547                                 s->push_back(*j);
548                                 ++j;
549                         }
550                         return (new mul(s, ex_to<numeric>(overall_coeff).mul_dyn(oc))
551                                )->setflag(status_flags::dynallocated);
552                 }
553         }
554
555         return this->hold();
556 }
557
558 ex mul::evalf(int level) const
559 {
560         if (level==1)
561                 return mul(seq,overall_coeff);
562         
563         if (level==-max_recursion_level)
564                 throw(std::runtime_error("max recursion level reached"));
565         
566         std::auto_ptr<epvector> s(new epvector);
567         s->reserve(seq.size());
568
569         --level;
570         epvector::const_iterator i = seq.begin(), end = seq.end();
571         while (i != end) {
572                 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
573                                                            i->coeff));
574                 ++i;
575         }
576         return mul(s, overall_coeff.evalf(level));
577 }
578
579 void mul::find_real_imag(ex & rp, ex & ip) const
580 {
581         rp = overall_coeff.real_part();
582         ip = overall_coeff.imag_part();
583         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
584                 ex factor = recombine_pair_to_ex(*i);
585                 ex new_rp = factor.real_part();
586                 ex new_ip = factor.imag_part();
587                 if(new_ip.is_zero()) {
588                         rp *= new_rp;
589                         ip *= new_rp;
590                 } else {
591                         ex temp = rp*new_rp - ip*new_ip;
592                         ip = ip*new_rp + rp*new_ip;
593                         rp = temp;
594                 }
595         }
596         rp = rp.expand();
597         ip = ip.expand();
598 }
599
600 ex mul::real_part() const
601 {
602         ex rp, ip;
603         find_real_imag(rp, ip);
604         return rp;
605 }
606
607 ex mul::imag_part() const
608 {
609         ex rp, ip;
610         find_real_imag(rp, ip);
611         return ip;
612 }
613
614 ex mul::evalm() const
615 {
616         // numeric*matrix
617         if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
618          && is_a<matrix>(seq[0].rest))
619                 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
620
621         // Evaluate children first, look whether there are any matrices at all
622         // (there can be either no matrices or one matrix; if there were more
623         // than one matrix, it would be a non-commutative product)
624         std::auto_ptr<epvector> s(new epvector);
625         s->reserve(seq.size());
626
627         bool have_matrix = false;
628         epvector::iterator the_matrix;
629
630         epvector::const_iterator i = seq.begin(), end = seq.end();
631         while (i != end) {
632                 const ex &m = recombine_pair_to_ex(*i).evalm();
633                 s->push_back(split_ex_to_pair(m));
634                 if (is_a<matrix>(m)) {
635                         have_matrix = true;
636                         the_matrix = s->end() - 1;
637                 }
638                 ++i;
639         }
640
641         if (have_matrix) {
642
643                 // The product contained a matrix. We will multiply all other factors
644                 // into that matrix.
645                 matrix m = ex_to<matrix>(the_matrix->rest);
646                 s->erase(the_matrix);
647                 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
648                 return m.mul_scalar(scalar);
649
650         } else
651                 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
652 }
653
654 ex mul::eval_ncmul(const exvector & v) const
655 {
656         if (seq.empty())
657                 return inherited::eval_ncmul(v);
658
659         // Find first noncommutative element and call its eval_ncmul()
660         epvector::const_iterator i = seq.begin(), end = seq.end();
661         while (i != end) {
662                 if (i->rest.return_type() == return_types::noncommutative)
663                         return i->rest.eval_ncmul(v);
664                 ++i;
665         }
666         return inherited::eval_ncmul(v);
667 }
668
669 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, exmap& repls)
670 {       
671         ex origbase;
672         int origexponent;
673         int origexpsign;
674
675         if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
676                 origbase = origfactor.op(0);
677                 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
678                 origexponent = expon > 0 ? expon : -expon;
679                 origexpsign = expon > 0 ? 1 : -1;
680         } else {
681                 origbase = origfactor;
682                 origexponent = 1;
683                 origexpsign = 1;
684         }
685
686         ex patternbase;
687         int patternexponent;
688         int patternexpsign;
689
690         if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
691                 patternbase = patternfactor.op(0);
692                 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
693                 patternexponent = expon > 0 ? expon : -expon;
694                 patternexpsign = expon > 0 ? 1 : -1;
695         } else {
696                 patternbase = patternfactor;
697                 patternexponent = 1;
698                 patternexpsign = 1;
699         }
700
701         exmap saverepls = repls;
702         if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
703                 return false;
704         repls = saverepls;
705
706         int newnummatches = origexponent / patternexponent;
707         if (newnummatches < nummatches)
708                 nummatches = newnummatches;
709         return true;
710 }
711
712 /** Checks wheter e matches to the pattern pat and the (possibly to be updated)
713   * list of replacements repls. This matching is in the sense of algebraic
714   * substitutions. Matching starts with pat.op(factor) of the pattern because
715   * the factors before this one have already been matched. The (possibly
716   * updated) number of matches is in nummatches. subsed[i] is true for factors
717   * that already have been replaced by previous substitutions and matched[i]
718   * is true for factors that have been matched by the current match.
719   */
720 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, exmap& repls,
721                                   int factor, int &nummatches, const std::vector<bool> &subsed,
722                                   std::vector<bool> &matched)
723 {
724         GINAC_ASSERT(subsed.size() == e.nops());
725         GINAC_ASSERT(matched.size() == e.nops());
726
727         if (factor == (int)pat.nops())
728                 return true;
729
730         for (size_t i=0; i<e.nops(); ++i) {
731                 if(subsed[i] || matched[i])
732                         continue;
733                 exmap newrepls = repls;
734                 int newnummatches = nummatches;
735                 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
736                         matched[i] = true;
737                         if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
738                                         newnummatches, subsed, matched)) {
739                                 repls = newrepls;
740                                 nummatches = newnummatches;
741                                 return true;
742                         }
743                         else
744                                 matched[i] = false;
745                 }
746         }
747
748         return false;
749 }
750
751 bool mul::has(const ex & pattern, unsigned options) const
752 {
753         if(!(options&has_options::algebraic))
754                 return basic::has(pattern,options);
755         if(is_a<mul>(pattern)) {
756                 exmap repls;
757                 int nummatches = std::numeric_limits<int>::max();
758                 std::vector<bool> subsed(nops(), false);
759                 std::vector<bool> matched(nops(), false);
760                 if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
761                                 subsed, matched))
762                         return true;
763         }
764         return basic::has(pattern, options);
765 }
766
767 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
768 {       
769         std::vector<bool> subsed(nops(), false);
770         ex divide_by = 1;
771         ex multiply_by = 1;
772
773         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
774
775                 if (is_exactly_a<mul>(it->first)) {
776 retry1:
777                         int nummatches = std::numeric_limits<int>::max();
778                         std::vector<bool> currsubsed(nops(), false);
779                         exmap repls;
780                         
781                         if(!algebraic_match_mul_with_mul(*this, it->first, repls, 0, nummatches, subsed, currsubsed))
782                                 continue;
783
784                         for (size_t j=0; j<subsed.size(); j++)
785                                 if (currsubsed[j])
786                                         subsed[j] = true;
787                         ex subsed_pattern
788                                 = it->first.subs(repls, subs_options::no_pattern);
789                         divide_by *= power(subsed_pattern, nummatches);
790                         ex subsed_result
791                                 = it->second.subs(repls, subs_options::no_pattern);
792                         multiply_by *= power(subsed_result, nummatches);
793                         goto retry1;
794
795                 } else {
796
797                         for (size_t j=0; j<this->nops(); j++) {
798                                 int nummatches = std::numeric_limits<int>::max();
799                                 exmap repls;
800                                 if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)){
801                                         subsed[j] = true;
802                                         ex subsed_pattern
803                                                 = it->first.subs(repls, subs_options::no_pattern);
804                                         divide_by *= power(subsed_pattern, nummatches);
805                                         ex subsed_result
806                                                 = it->second.subs(repls, subs_options::no_pattern);
807                                         multiply_by *= power(subsed_result, nummatches);
808                                 }
809                         }
810                 }
811         }
812
813         bool subsfound = false;
814         for (size_t i=0; i<subsed.size(); i++) {
815                 if (subsed[i]) {
816                         subsfound = true;
817                         break;
818                 }
819         }
820         if (!subsfound)
821                 return subs_one_level(m, options | subs_options::algebraic);
822
823         return ((*this)/divide_by)*multiply_by;
824 }
825
826 ex mul::conjugate() const
827 {
828         // The base class' method is wrong here because we have to be careful at
829         // branch cuts. power::conjugate takes care of that already, so use it.
830         epvector *newepv = 0;
831         for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
832                 if (newepv) {
833                         newepv->push_back(split_ex_to_pair(recombine_pair_to_ex(*i).conjugate()));
834                         continue;
835                 }
836                 ex x = recombine_pair_to_ex(*i);
837                 ex c = x.conjugate();
838                 if (c.is_equal(x)) {
839                         continue;
840                 }
841                 newepv = new epvector;
842                 newepv->reserve(seq.size());
843                 for (epvector::const_iterator j=seq.begin(); j!=i; ++j) {
844                         newepv->push_back(*j);
845                 }
846                 newepv->push_back(split_ex_to_pair(c));
847         }
848         ex x = overall_coeff.conjugate();
849         if (!newepv && are_ex_trivially_equal(x, overall_coeff)) {
850                 return *this;
851         }
852         ex result = thisexpairseq(newepv ? *newepv : seq, x);
853         delete newepv;
854         return result;
855 }
856
857
858 // protected
859
860 /** Implementation of ex::diff() for a product.  It applies the product rule.
861  *  @see ex::diff */
862 ex mul::derivative(const symbol & s) const
863 {
864         size_t num = seq.size();
865         exvector addseq;
866         addseq.reserve(num);
867         
868         // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
869         epvector mulseq = seq;
870         epvector::const_iterator i = seq.begin(), end = seq.end();
871         epvector::iterator i2 = mulseq.begin();
872         while (i != end) {
873                 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
874                                              i->rest.diff(s));
875                 ep.swap(*i2);
876                 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
877                 ep.swap(*i2);
878                 ++i; ++i2;
879         }
880         return (new add(addseq))->setflag(status_flags::dynallocated);
881 }
882
883 int mul::compare_same_type(const basic & other) const
884 {
885         return inherited::compare_same_type(other);
886 }
887
888 unsigned mul::return_type() const
889 {
890         if (seq.empty()) {
891                 // mul without factors: should not happen, but commutates
892                 return return_types::commutative;
893         }
894         
895         bool all_commutative = true;
896         epvector::const_iterator noncommutative_element; // point to first found nc element
897         
898         epvector::const_iterator i = seq.begin(), end = seq.end();
899         while (i != end) {
900                 unsigned rt = i->rest.return_type();
901                 if (rt == return_types::noncommutative_composite)
902                         return rt; // one ncc -> mul also ncc
903                 if ((rt == return_types::noncommutative) && (all_commutative)) {
904                         // first nc element found, remember position
905                         noncommutative_element = i;
906                         all_commutative = false;
907                 }
908                 if ((rt == return_types::noncommutative) && (!all_commutative)) {
909                         // another nc element found, compare type_infos
910                         if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
911                                         // different types -> mul is ncc
912                                         return return_types::noncommutative_composite;
913                         }
914                 }
915                 ++i;
916         }
917         // all factors checked
918         return all_commutative ? return_types::commutative : return_types::noncommutative;
919 }
920
921 return_type_t mul::return_type_tinfo() const
922 {
923         if (seq.empty())
924                 return make_return_type_t<mul>(); // mul without factors: should not happen
925         
926         // return type_info of first noncommutative element
927         epvector::const_iterator i = seq.begin(), end = seq.end();
928         while (i != end) {
929                 if (i->rest.return_type() == return_types::noncommutative)
930                         return i->rest.return_type_tinfo();
931                 ++i;
932         }
933         // no noncommutative element found, should not happen
934         return make_return_type_t<mul>();
935 }
936
937 ex mul::thisexpairseq(const epvector & v, const ex & oc, bool do_index_renaming) const
938 {
939         return (new mul(v, oc, do_index_renaming))->setflag(status_flags::dynallocated);
940 }
941
942 ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming) const
943 {
944         return (new mul(vp, oc, do_index_renaming))->setflag(status_flags::dynallocated);
945 }
946
947 expair mul::split_ex_to_pair(const ex & e) const
948 {
949         if (is_exactly_a<power>(e)) {
950                 const power & powerref = ex_to<power>(e);
951                 if (is_exactly_a<numeric>(powerref.exponent))
952                         return expair(powerref.basis,powerref.exponent);
953         }
954         return expair(e,_ex1);
955 }
956
957 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
958                                           const ex & c) const
959 {
960         // to avoid duplication of power simplification rules,
961         // we create a temporary power object
962         // otherwise it would be hard to correctly evaluate
963         // expression like (4^(1/3))^(3/2)
964         if (c.is_equal(_ex1))
965                 return split_ex_to_pair(e);
966
967         return split_ex_to_pair(power(e,c));
968 }
969
970 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
971                                             const ex & c) const
972 {
973         // to avoid duplication of power simplification rules,
974         // we create a temporary power object
975         // otherwise it would be hard to correctly evaluate
976         // expression like (4^(1/3))^(3/2)
977         if (c.is_equal(_ex1))
978                 return p;
979
980         return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
981 }
982
983 ex mul::recombine_pair_to_ex(const expair & p) const
984 {
985         if (ex_to<numeric>(p.coeff).is_equal(*_num1_p)) 
986                 return p.rest;
987         else
988                 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
989 }
990
991 bool mul::expair_needs_further_processing(epp it)
992 {
993         if (is_exactly_a<mul>(it->rest) &&
994             ex_to<numeric>(it->coeff).is_integer()) {
995                 // combined pair is product with integer power -> expand it
996                 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
997                 return true;
998         }
999         if (is_exactly_a<numeric>(it->rest)) {
1000                 if (it->coeff.is_equal(_ex1)) {
1001                         // pair has coeff 1 and must be moved to the end
1002                         return true;
1003                 }
1004                 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
1005                 if (!ep.is_equal(*it)) {
1006                         // combined pair is a numeric power which can be simplified
1007                         *it = ep;
1008                         return true;
1009                 }
1010         }
1011         return false;
1012 }       
1013
1014 ex mul::default_overall_coeff() const
1015 {
1016         return _ex1;
1017 }
1018
1019 void mul::combine_overall_coeff(const ex & c)
1020 {
1021         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1022         GINAC_ASSERT(is_exactly_a<numeric>(c));
1023         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
1024 }
1025
1026 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
1027 {
1028         GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1029         GINAC_ASSERT(is_exactly_a<numeric>(c1));
1030         GINAC_ASSERT(is_exactly_a<numeric>(c2));
1031         overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
1032 }
1033
1034 bool mul::can_make_flat(const expair & p) const
1035 {
1036         GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
1037         // this assertion will probably fail somewhere
1038         // it would require a more careful make_flat, obeying the power laws
1039         // probably should return true only if p.coeff is integer
1040         return ex_to<numeric>(p.coeff).is_equal(*_num1_p);
1041 }
1042
1043 bool mul::can_be_further_expanded(const ex & e)
1044 {
1045         if (is_exactly_a<mul>(e)) {
1046                 for (epvector::const_iterator cit = ex_to<mul>(e).seq.begin(); cit != ex_to<mul>(e).seq.end(); ++cit) {
1047                         if (is_exactly_a<add>(cit->rest) && cit->coeff.info(info_flags::posint))
1048                                 return true;
1049                 }
1050         } else if (is_exactly_a<power>(e)) {
1051                 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
1052                         return true;
1053         }
1054         return false;
1055 }
1056
1057 ex mul::expand(unsigned options) const
1058 {
1059         {
1060         // trivial case: expanding the monomial (~ 30% of all calls)
1061                 epvector::const_iterator i = seq.begin(), seq_end = seq.end();
1062                 while ((i != seq.end()) &&  is_a<symbol>(i->rest) && i->coeff.info(info_flags::integer))
1063                         ++i;
1064                 if (i == seq_end) {
1065                         setflag(status_flags::expanded);
1066                         return *this;
1067                 }
1068         }
1069
1070         // do not rename indices if the object has no indices at all
1071         if ((!(options & expand_options::expand_rename_idx)) && 
1072                         this->info(info_flags::has_indices))
1073                 options |= expand_options::expand_rename_idx;
1074
1075         const bool skip_idx_rename = !(options & expand_options::expand_rename_idx);
1076
1077         // First, expand the children
1078         std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
1079         const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
1080
1081         // Now, look for all the factors that are sums and multiply each one out
1082         // with the next one that is found while collecting the factors which are
1083         // not sums
1084         ex last_expanded = _ex1;
1085
1086         epvector non_adds;
1087         non_adds.reserve(expanded_seq.size());
1088
1089         for (epvector::const_iterator cit = expanded_seq.begin(); cit != expanded_seq.end(); ++cit) {
1090                 if (is_exactly_a<add>(cit->rest) &&
1091                         (cit->coeff.is_equal(_ex1))) {
1092                         if (is_exactly_a<add>(last_expanded)) {
1093
1094                                 // Expand a product of two sums, aggressive version.
1095                                 // Caring for the overall coefficients in separate loops can
1096                                 // sometimes give a performance gain of up to 15%!
1097
1098                                 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
1099                                 // add2 is for the inner loop and should be the bigger of the two sums
1100                                 // in the presence of asymptotically good sorting:
1101                                 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
1102                                 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
1103                                 const epvector::const_iterator add1begin = add1.seq.begin();
1104                                 const epvector::const_iterator add1end   = add1.seq.end();
1105                                 const epvector::const_iterator add2begin = add2.seq.begin();
1106                                 const epvector::const_iterator add2end   = add2.seq.end();
1107                                 epvector distrseq;
1108                                 distrseq.reserve(add1.seq.size()+add2.seq.size());
1109
1110                                 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
1111                                 if (!add1.overall_coeff.is_zero()) {
1112                                         if (add1.overall_coeff.is_equal(_ex1))
1113                                                 distrseq.insert(distrseq.end(),add2begin,add2end);
1114                                         else
1115                                                 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
1116                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
1117                                 }
1118
1119                                 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
1120                                 if (!add2.overall_coeff.is_zero()) {
1121                                         if (add2.overall_coeff.is_equal(_ex1))
1122                                                 distrseq.insert(distrseq.end(),add1begin,add1end);
1123                                         else
1124                                                 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
1125                                                         distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
1126                                 }
1127
1128                                 // Compute the new overall coefficient and put it together:
1129                                 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
1130
1131                                 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
1132                                 lst dummy_subs;
1133
1134                                 if (!skip_idx_rename) {
1135                                         for (epvector::const_iterator i=add1begin; i!=add1end; ++i) {
1136                                                 add_indices = get_all_dummy_indices_safely(i->rest);
1137                                                 add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
1138                                         }
1139                                         for (epvector::const_iterator i=add2begin; i!=add2end; ++i) {
1140                                                 add_indices = get_all_dummy_indices_safely(i->rest);
1141                                                 add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
1142                                         }
1143
1144                                         sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
1145                                         sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
1146                                         dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
1147                                 }
1148
1149                                 // Multiply explicitly all non-numeric terms of add1 and add2:
1150                                 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
1151                                         // We really have to combine terms here in order to compactify
1152                                         // the result.  Otherwise it would become waayy tooo bigg.
1153                                         numeric oc(*_num0_p);
1154                                         epvector distrseq2;
1155                                         distrseq2.reserve(add1.seq.size());
1156                                         const ex i2_new = (skip_idx_rename || (dummy_subs.op(0).nops() == 0) ?
1157                                                         i2->rest :
1158                                                         i2->rest.subs(ex_to<lst>(dummy_subs.op(0)), 
1159                                                                 ex_to<lst>(dummy_subs.op(1)), subs_options::no_pattern));
1160                                         for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
1161                                                 // Don't push_back expairs which might have a rest that evaluates to a numeric,
1162                                                 // since that would violate an invariant of expairseq:
1163                                                 const ex rest = (new mul(i1->rest, i2_new))->setflag(status_flags::dynallocated);
1164                                                 if (is_exactly_a<numeric>(rest)) {
1165                                                         oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
1166                                                 } else {
1167                                                         distrseq2.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
1168                                                 }
1169                                         }
1170                                         tmp_accu += (new add(distrseq2, oc))->setflag(status_flags::dynallocated);
1171                                 } 
1172                                 last_expanded = tmp_accu;
1173                         } else {
1174                                 if (!last_expanded.is_equal(_ex1))
1175                                         non_adds.push_back(split_ex_to_pair(last_expanded));
1176                                 last_expanded = cit->rest;
1177                         }
1178
1179                 } else {
1180                         non_adds.push_back(*cit);
1181                 }
1182         }
1183
1184         // Now the only remaining thing to do is to multiply the factors which
1185         // were not sums into the "last_expanded" sum
1186         if (is_exactly_a<add>(last_expanded)) {
1187                 size_t n = last_expanded.nops();
1188                 exvector distrseq;
1189                 distrseq.reserve(n);
1190                 exvector va;
1191                 if (! skip_idx_rename) {
1192                         va = get_all_dummy_indices_safely(mul(non_adds));
1193                         sort(va.begin(), va.end(), ex_is_less());
1194                 }
1195
1196                 for (size_t i=0; i<n; ++i) {
1197                         epvector factors = non_adds;
1198                         if (skip_idx_rename)
1199                                 factors.push_back(split_ex_to_pair(last_expanded.op(i)));
1200                         else
1201                                 factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
1202                         ex term = (new mul(factors, overall_coeff))->setflag(status_flags::dynallocated);
1203                         if (can_be_further_expanded(term)) {
1204                                 distrseq.push_back(term.expand());
1205                         } else {
1206                                 if (options == 0)
1207                                         ex_to<basic>(term).setflag(status_flags::expanded);
1208                                 distrseq.push_back(term);
1209                         }
1210                 }
1211
1212                 return ((new add(distrseq))->
1213                         setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
1214         }
1215
1216         non_adds.push_back(split_ex_to_pair(last_expanded));
1217         ex result = (new mul(non_adds, overall_coeff))->setflag(status_flags::dynallocated);
1218         if (can_be_further_expanded(result)) {
1219                 return result.expand();
1220         } else {
1221                 if (options == 0)
1222                         ex_to<basic>(result).setflag(status_flags::expanded);
1223                 return result;
1224         }
1225 }
1226
1227   
1228 //////////
1229 // new virtual functions which can be overridden by derived classes
1230 //////////
1231
1232 // none
1233
1234 //////////
1235 // non-virtual functions in this class
1236 //////////
1237
1238
1239 /** Member-wise expand the expairs representing this sequence.  This must be
1240  *  overridden from expairseq::expandchildren() and done iteratively in order
1241  *  to allow for early cancallations and thus safe memory.
1242  *
1243  *  @see mul::expand()
1244  *  @return pointer to epvector containing expanded representation or zero
1245  *  pointer, if sequence is unchanged. */
1246 std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
1247 {
1248         const epvector::const_iterator last = seq.end();
1249         epvector::const_iterator cit = seq.begin();
1250         while (cit!=last) {
1251                 const ex & factor = recombine_pair_to_ex(*cit);
1252                 const ex & expanded_factor = factor.expand(options);
1253                 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1254                         
1255                         // something changed, copy seq, eval and return it
1256                         std::auto_ptr<epvector> s(new epvector);
1257                         s->reserve(seq.size());
1258                         
1259                         // copy parts of seq which are known not to have changed
1260                         epvector::const_iterator cit2 = seq.begin();
1261                         while (cit2!=cit) {
1262                                 s->push_back(*cit2);
1263                                 ++cit2;
1264                         }
1265
1266                         // copy first changed element
1267                         s->push_back(split_ex_to_pair(expanded_factor));
1268                         ++cit2;
1269
1270                         // copy rest
1271                         while (cit2!=last) {
1272                                 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1273                                 ++cit2;
1274                         }
1275                         return s;
1276                 }
1277                 ++cit;
1278         }
1279         
1280         return std::auto_ptr<epvector>(0); // nothing has changed
1281 }
1282
1283 GINAC_BIND_UNARCHIVER(mul);
1284
1285 } // namespace GiNaC