d693373cae62c6cd622495d05722aa9ce7065d99
[ginac.git] / ginac / indexed.cpp
1 /** @file indexed.cpp
2  *
3  *  Implementation of GiNaC's indexed expressions. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
7  *
8  *  This program is free software; you can redistribute it and/or modify
9  *  it under the terms of the GNU General Public License as published by
10  *  the Free Software Foundation; either version 2 of the License, or
11  *  (at your option) any later version.
12  *
13  *  This program is distributed in the hope that it will be useful,
14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
16  *  GNU General Public License for more details.
17  *
18  *  You should have received a copy of the GNU General Public License
19  *  along with this program; if not, write to the Free Software
20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
21  */
22
23 #include <iostream>
24 #include <sstream>
25 #include <stdexcept>
26
27 #include "indexed.h"
28 #include "idx.h"
29 #include "add.h"
30 #include "mul.h"
31 #include "ncmul.h"
32 #include "power.h"
33 #include "relational.h"
34 #include "symmetry.h"
35 #include "operators.h"
36 #include "lst.h"
37 #include "archive.h"
38 #include "utils.h"
39
40 namespace GiNaC {
41
42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(indexed, exprseq,
43   print_func<print_context>(&indexed::do_print).
44   print_func<print_latex>(&indexed::do_print_latex).
45   print_func<print_tree>(&indexed::do_print_tree))
46
47 //////////
48 // default constructor
49 //////////
50
51 indexed::indexed() : symtree(not_symmetric())
52 {
53         tinfo_key = TINFO_indexed;
54 }
55
56 //////////
57 // other constructors
58 //////////
59
60 indexed::indexed(const ex & b) : inherited(b), symtree(not_symmetric())
61 {
62         tinfo_key = TINFO_indexed;
63         validate();
64 }
65
66 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(not_symmetric())
67 {
68         tinfo_key = TINFO_indexed;
69         validate();
70 }
71
72 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(not_symmetric())
73 {
74         tinfo_key = TINFO_indexed;
75         validate();
76 }
77
78 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(not_symmetric())
79 {
80         tinfo_key = TINFO_indexed;
81         validate();
82 }
83
84 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(not_symmetric())
85 {
86         tinfo_key = TINFO_indexed;
87         validate();
88 }
89
90 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
91 {
92         tinfo_key = TINFO_indexed;
93         validate();
94 }
95
96 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
97 {
98         tinfo_key = TINFO_indexed;
99         validate();
100 }
101
102 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(symm)
103 {
104         tinfo_key = TINFO_indexed;
105         validate();
106 }
107
108 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(not_symmetric())
109 {
110         seq.insert(seq.end(), v.begin(), v.end());
111         tinfo_key = TINFO_indexed;
112         validate();
113 }
114
115 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
116 {
117         seq.insert(seq.end(), v.begin(), v.end());
118         tinfo_key = TINFO_indexed;
119         validate();
120 }
121
122 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
123 {
124         tinfo_key = TINFO_indexed;
125 }
126
127 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
128 {
129         tinfo_key = TINFO_indexed;
130 }
131
132 indexed::indexed(const symmetry & symm, std::auto_ptr<exvector> vp) : inherited(vp), symtree(symm)
133 {
134         tinfo_key = TINFO_indexed;
135 }
136
137 //////////
138 // archiving
139 //////////
140
141 indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
142 {
143         if (!n.find_ex("symmetry", symtree, sym_lst)) {
144                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
145                 unsigned symm = 0;
146                 n.find_unsigned("symmetry", symm);
147                 switch (symm) {
148                         case 1:
149                                 symtree = sy_symm();
150                                 break;
151                         case 2:
152                                 symtree = sy_anti();
153                                 break;
154                         default:
155                                 symtree = not_symmetric();
156                                 break;
157                 }
158                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
159         }
160 }
161
162 void indexed::archive(archive_node &n) const
163 {
164         inherited::archive(n);
165         n.add_ex("symmetry", symtree);
166 }
167
168 DEFAULT_UNARCHIVE(indexed)
169
170 //////////
171 // functions overriding virtual functions from base classes
172 //////////
173
174 void indexed::printindices(const print_context & c, unsigned level) const
175 {
176         if (seq.size() > 1) {
177
178                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
179
180                 if (is_a<print_latex>(c)) {
181
182                         // TeX output: group by variance
183                         bool first = true;
184                         bool covariant = true;
185
186                         while (it != itend) {
187                                 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
188                                 if (first || cur_covariant != covariant) { // Variance changed
189                                         // The empty {} prevents indices from ending up on top of each other
190                                         if (!first)
191                                                 c.s << "}{}";
192                                         covariant = cur_covariant;
193                                         if (covariant)
194                                                 c.s << "_{";
195                                         else
196                                                 c.s << "^{";
197                                 }
198                                 it->print(c, level);
199                                 c.s << " ";
200                                 first = false;
201                                 it++;
202                         }
203                         c.s << "}";
204
205                 } else {
206
207                         // Ordinary output
208                         while (it != itend) {
209                                 it->print(c, level);
210                                 it++;
211                         }
212                 }
213         }
214 }
215
216 void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
217 {
218         if (precedence() <= level)
219                 c.s << openbrace << '(';
220         c.s << openbrace;
221         seq[0].print(c, precedence());
222         c.s << closebrace;
223         printindices(c, level);
224         if (precedence() <= level)
225                 c.s << ')' << closebrace;
226 }
227
228 void indexed::do_print(const print_context & c, unsigned level) const
229 {
230         print_indexed(c, "", "", level);
231 }
232
233 void indexed::do_print_latex(const print_latex & c, unsigned level) const
234 {
235         print_indexed(c, "{", "}", level);
236 }
237
238 void indexed::do_print_tree(const print_tree & c, unsigned level) const
239 {
240         c.s << std::string(level, ' ') << class_name() << " @" << this
241             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
242             << ", " << seq.size()-1 << " indices"
243             << ", symmetry=" << symtree << std::endl;
244         seq[0].print(c, level + c.delta_indent);
245         printindices(c, level + c.delta_indent);
246 }
247
248 bool indexed::info(unsigned inf) const
249 {
250         if (inf == info_flags::indexed) return true;
251         if (inf == info_flags::has_indices) return seq.size() > 1;
252         return inherited::info(inf);
253 }
254
255 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
256         bool operator() (const ex & e, unsigned inf) const {
257                 return !(ex_to<idx>(e).get_value().info(inf));
258         }
259 };
260
261 bool indexed::all_index_values_are(unsigned inf) const
262 {
263         // No indices? Then no property can be fulfilled
264         if (seq.size() < 2)
265                 return false;
266
267         // Check all indices
268         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
269 }
270
271 int indexed::compare_same_type(const basic & other) const
272 {
273         GINAC_ASSERT(is_a<indexed>(other));
274         return inherited::compare_same_type(other);
275 }
276
277 ex indexed::eval(int level) const
278 {
279         // First evaluate children, then we will end up here again
280         if (level > 1)
281                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
282
283         const ex &base = seq[0];
284
285         // If the base object is 0, the whole object is 0
286         if (base.is_zero())
287                 return _ex0;
288
289         // If the base object is a product, pull out the numeric factor
290         if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
291                 exvector v(seq);
292                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
293                 v[0] = seq[0] / f;
294                 return f * thiscontainer(v);
295         }
296
297         // Canonicalize indices according to the symmetry properties
298         if (seq.size() > 2) {
299                 exvector v = seq;
300                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
301                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
302                 if (sig != INT_MAX) {
303                         // Something has changed while sorting indices, more evaluations later
304                         if (sig == 0)
305                                 return _ex0;
306                         return ex(sig) * thiscontainer(v);
307                 }
308         }
309
310         // Let the class of the base object perform additional evaluations
311         return ex_to<basic>(base).eval_indexed(*this);
312 }
313
314 ex indexed::thiscontainer(const exvector & v) const
315 {
316         return indexed(ex_to<symmetry>(symtree), v);
317 }
318
319 ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
320 {
321         return indexed(ex_to<symmetry>(symtree), vp);
322 }
323
324 ex indexed::expand(unsigned options) const
325 {
326         GINAC_ASSERT(seq.size() > 0);
327
328         if (options & expand_options::expand_indexed) {
329                 ex newbase = seq[0].expand(options);
330                 if (is_exactly_a<add>(newbase)) {
331                         ex sum = _ex0;
332                         for (size_t i=0; i<newbase.nops(); i++) {
333                                 exvector s = seq;
334                                 s[0] = newbase.op(i);
335                                 sum += thiscontainer(s).expand(options);
336                         }
337                         return sum;
338                 }
339                 if (!are_ex_trivially_equal(newbase, seq[0])) {
340                         exvector s = seq;
341                         s[0] = newbase;
342                         return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
343                 }
344         }
345         return inherited::expand(options);
346 }
347
348 //////////
349 // virtual functions which can be overridden by derived classes
350 //////////
351
352 // none
353
354 //////////
355 // non-virtual functions in this class
356 //////////
357
358 /** Check whether all indices are of class idx and validate the symmetry
359  *  tree. This function is used internally to make sure that all constructed
360  *  indexed objects really carry indices and not some other classes. */
361 void indexed::validate() const
362 {
363         GINAC_ASSERT(seq.size() > 0);
364         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
365         while (it != itend) {
366                 if (!is_a<idx>(*it))
367                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
368                 it++;
369         }
370
371         if (!symtree.is_zero()) {
372                 if (!is_exactly_a<symmetry>(symtree))
373                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
374                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
375         }
376 }
377
378 /** Implementation of ex::diff() for an indexed object always returns 0.
379  *
380  *  @see ex::diff */
381 ex indexed::derivative(const symbol & s) const
382 {
383         return _ex0;
384 }
385
386 //////////
387 // global functions
388 //////////
389
390 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
391         bool operator() (const ex &lh, const ex &rh) const
392         {
393                 if (lh.is_equal(rh))
394                         return true;
395                 else
396                         try {
397                                 // Replacing the dimension might cause an error (e.g. with
398                                 // index classes that only work in a fixed number of dimensions)
399                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
400                         } catch (...) {
401                                 return false;
402                         }
403         }
404 };
405
406 /** Check whether two sorted index vectors are consistent (i.e. equal). */
407 static bool indices_consistent(const exvector & v1, const exvector & v2)
408 {
409         // Number of indices must be the same
410         if (v1.size() != v2.size())
411                 return false;
412
413         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
414 }
415
416 exvector indexed::get_indices() const
417 {
418         GINAC_ASSERT(seq.size() >= 1);
419         return exvector(seq.begin() + 1, seq.end());
420 }
421
422 exvector indexed::get_dummy_indices() const
423 {
424         exvector free_indices, dummy_indices;
425         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
426         return dummy_indices;
427 }
428
429 exvector indexed::get_dummy_indices(const indexed & other) const
430 {
431         exvector indices = get_free_indices();
432         exvector other_indices = other.get_free_indices();
433         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
434         exvector dummy_indices;
435         find_dummy_indices(indices, dummy_indices);
436         return dummy_indices;
437 }
438
439 bool indexed::has_dummy_index_for(const ex & i) const
440 {
441         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
442         while (it != itend) {
443                 if (is_dummy_pair(*it, i))
444                         return true;
445                 it++;
446         }
447         return false;
448 }
449
450 exvector indexed::get_free_indices() const
451 {
452         exvector free_indices, dummy_indices;
453         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
454         return free_indices;
455 }
456
457 exvector add::get_free_indices() const
458 {
459         exvector free_indices;
460         for (size_t i=0; i<nops(); i++) {
461                 if (i == 0)
462                         free_indices = op(i).get_free_indices();
463                 else {
464                         exvector free_indices_of_term = op(i).get_free_indices();
465                         if (!indices_consistent(free_indices, free_indices_of_term))
466                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
467                 }
468         }
469         return free_indices;
470 }
471
472 exvector mul::get_free_indices() const
473 {
474         // Concatenate free indices of all factors
475         exvector un;
476         for (size_t i=0; i<nops(); i++) {
477                 exvector free_indices_of_factor = op(i).get_free_indices();
478                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
479         }
480
481         // And remove the dummy indices
482         exvector free_indices, dummy_indices;
483         find_free_and_dummy(un, free_indices, dummy_indices);
484         return free_indices;
485 }
486
487 exvector ncmul::get_free_indices() const
488 {
489         // Concatenate free indices of all factors
490         exvector un;
491         for (size_t i=0; i<nops(); i++) {
492                 exvector free_indices_of_factor = op(i).get_free_indices();
493                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
494         }
495
496         // And remove the dummy indices
497         exvector free_indices, dummy_indices;
498         find_free_and_dummy(un, free_indices, dummy_indices);
499         return free_indices;
500 }
501
502 struct is_summation_idx : public std::unary_function<ex, bool> {
503         bool operator()(const ex & e)
504         {
505                 return is_dummy_pair(e, e);
506         }
507 };
508
509 exvector power::get_free_indices() const
510 {
511         // Get free indices of basis
512         exvector basis_indices = basis.get_free_indices();
513
514         if (exponent.info(info_flags::even)) {
515                 // If the exponent is an even number, then any "free" index that
516                 // forms a dummy pair with itself is actually a summation index
517                 exvector really_free;
518                 std::remove_copy_if(basis_indices.begin(), basis_indices.end(),
519                                     std::back_inserter(really_free), is_summation_idx());
520                 return really_free;
521         } else
522                 return basis_indices;
523 }
524
525 /** Rename dummy indices in an expression.
526  *
527  *  @param e Expression to work on
528  *  @param local_dummy_indices The set of dummy indices that appear in the
529  *    expression "e"
530  *  @param global_dummy_indices The set of dummy indices that have appeared
531  *    before and which we would like to use in "e", too. This gets updated
532  *    by the function */
533 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
534 {
535         size_t global_size = global_dummy_indices.size(),
536                local_size = local_dummy_indices.size();
537
538         // Any local dummy indices at all?
539         if (local_size == 0)
540                 return e;
541
542         if (global_size < local_size) {
543
544                 // More local indices than we encountered before, add the new ones
545                 // to the global set
546                 size_t old_global_size = global_size;
547                 int remaining = local_size - global_size;
548                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
549                 while (it != itend && remaining > 0) {
550                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(op0_is_equal(), *it)) == global_dummy_indices.end()) {
551                                 global_dummy_indices.push_back(*it);
552                                 global_size++;
553                                 remaining--;
554                         }
555                         it++;
556                 }
557
558                 // If this is the first set of local indices, do nothing
559                 if (old_global_size == 0)
560                         return e;
561         }
562         GINAC_ASSERT(local_size <= global_size);
563
564         // Construct vectors of index symbols
565         exvector local_syms, global_syms;
566         local_syms.reserve(local_size);
567         global_syms.reserve(local_size);
568         for (size_t i=0; i<local_size; i++)
569                 local_syms.push_back(local_dummy_indices[i].op(0));
570         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
571         for (size_t i=0; i<local_size; i++) // don't use more global symbols than necessary
572                 global_syms.push_back(global_dummy_indices[i].op(0));
573         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
574
575         // Remove common indices
576         exvector local_uniq, global_uniq;
577         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
578         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
579
580         // Replace remaining non-common local index symbols by global ones
581         if (local_uniq.empty())
582                 return e;
583         else {
584                 while (global_uniq.size() > local_uniq.size())
585                         global_uniq.pop_back();
586                 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
587         }
588 }
589
590 /** Given a set of indices, extract those of class varidx. */
591 static void find_variant_indices(const exvector & v, exvector & variant_indices)
592 {
593         exvector::const_iterator it1, itend;
594         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
595                 if (is_exactly_a<varidx>(*it1))
596                         variant_indices.push_back(*it1);
597         }
598 }
599
600 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
601  *  variance.
602  *
603  *  @param e Object to work on
604  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
605  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
606  *  @return true if 'e' was changed */
607 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
608 {
609         bool something_changed = false;
610
611         // If a dummy index is encountered for the first time in the
612         // product, pull it up, otherwise, pull it down
613         exvector::const_iterator it2, it2start, it2end;
614         for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
615                 if (!is_exactly_a<varidx>(*it2))
616                         continue;
617
618                 exvector::iterator vit, vitend;
619                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
620                         if (it2->op(0).is_equal(vit->op(0))) {
621                                 if (ex_to<varidx>(*it2).is_covariant()) {
622                                         e = e.subs(lst(
623                                                 *it2 == ex_to<varidx>(*it2).toggle_variance(),
624                                                 ex_to<varidx>(*it2).toggle_variance() == *it2
625                                         ), subs_options::no_pattern);
626                                         something_changed = true;
627                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
628                                         it2start = ex_to<indexed>(e).seq.begin();
629                                         it2end = ex_to<indexed>(e).seq.end();
630                                 }
631                                 moved_indices.push_back(*vit);
632                                 variant_dummy_indices.erase(vit);
633                                 goto next_index;
634                         }
635                 }
636
637                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
638                         if (it2->op(0).is_equal(vit->op(0))) {
639                                 if (ex_to<varidx>(*it2).is_contravariant()) {
640                                         e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance(), subs_options::no_pattern);
641                                         something_changed = true;
642                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
643                                         it2start = ex_to<indexed>(e).seq.begin();
644                                         it2end = ex_to<indexed>(e).seq.end();
645                                 }
646                                 goto next_index;
647                         }
648                 }
649
650 next_index: ;
651         }
652
653         return something_changed;
654 }
655
656 /* Ordering that only compares the base expressions of indexed objects. */
657 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
658         bool operator() (const ex &lh, const ex &rh) const
659         {
660                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
661         }
662 };
663
664 /** Simplify product of indexed expressions (commutative, noncommutative and
665  *  simple squares), return list of free indices. */
666 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
667 {
668         // Remember whether the product was commutative or noncommutative
669         // (because we chop it into factors and need to reassemble later)
670         bool non_commutative = is_exactly_a<ncmul>(e);
671
672         // Collect factors in an exvector, store squares twice
673         exvector v;
674         v.reserve(e.nops() * 2);
675
676         if (is_exactly_a<power>(e)) {
677                 // We only get called for simple squares, split a^2 -> a*a
678                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
679                 v.push_back(e.op(0));
680                 v.push_back(e.op(0));
681         } else {
682                 for (size_t i=0; i<e.nops(); i++) {
683                         ex f = e.op(i);
684                         if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
685                                 v.push_back(f.op(0));
686                     v.push_back(f.op(0));
687                         } else if (is_exactly_a<ncmul>(f)) {
688                                 // Noncommutative factor found, split it as well
689                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
690                                 for (size_t j=0; j<f.nops(); j++)
691                                         v.push_back(f.op(j));
692                         } else
693                                 v.push_back(f);
694                 }
695         }
696
697         // Perform contractions
698         bool something_changed = false;
699         GINAC_ASSERT(v.size() > 1);
700         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
701         for (it1 = v.begin(); it1 != next_to_last; it1++) {
702
703 try_again:
704                 if (!is_a<indexed>(*it1))
705                         continue;
706
707                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
708
709                 // Indexed factor found, get free indices and look for contraction
710                 // candidates
711                 exvector free1, dummy1;
712                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
713
714                 exvector::iterator it2;
715                 for (it2 = it1 + 1; it2 != itend; it2++) {
716
717                         if (!is_a<indexed>(*it2))
718                                 continue;
719
720                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
721
722                         // Find free indices of second factor and merge them with free
723                         // indices of first factor
724                         exvector un;
725                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
726                         un.insert(un.end(), free1.begin(), free1.end());
727
728                         // Check whether the two factors share dummy indices
729                         exvector free, dummy;
730                         find_free_and_dummy(un, free, dummy);
731                         size_t num_dummies = dummy.size();
732                         if (num_dummies == 0)
733                                 continue;
734
735                         // At least one dummy index, is it a defined scalar product?
736                         bool contracted = false;
737                         if (free.empty()) {
738
739                                 // Find minimal dimension of all indices of both factors
740                                 exvector::const_iterator dit = ex_to<indexed>(*it1).seq.begin() + 1, ditend = ex_to<indexed>(*it1).seq.end();
741                                 ex dim = ex_to<idx>(*dit).get_dim();
742                                 ++dit;
743                                 for (; dit != ditend; ++dit) {
744                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
745                                 }
746                                 dit = ex_to<indexed>(*it2).seq.begin() + 1;
747                                 ditend = ex_to<indexed>(*it2).seq.end();
748                                 for (; dit != ditend; ++dit) {
749                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
750                                 }
751
752                                 // User-defined scalar product?
753                                 if (sp.is_defined(*it1, *it2, dim)) {
754
755                                         // Yes, substitute it
756                                         *it1 = sp.evaluate(*it1, *it2, dim);
757                                         *it2 = _ex1;
758                                         goto contraction_done;
759                                 }
760                         }
761
762                         // Try to contract the first one with the second one
763                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
764                         if (!contracted) {
765
766                                 // That didn't work; maybe the second object knows how to
767                                 // contract itself with the first one
768                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
769                         }
770                         if (contracted) {
771 contraction_done:
772                                 if (first_noncommutative || second_noncommutative
773                                  || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
774                                  || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
775                                  || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
776
777                                         // One of the factors became a sum or product:
778                                         // re-expand expression and run again
779                                         // Non-commutative products are always re-expanded to give
780                                         // eval_ncmul() the chance to re-order and canonicalize
781                                         // the product
782                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
783                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
784                                 }
785
786                                 // Both objects may have new indices now or they might
787                                 // even not be indexed objects any more, so we have to
788                                 // start over
789                                 something_changed = true;
790                                 goto try_again;
791                         }
792                 }
793         }
794
795         // Find free indices (concatenate them all and call find_free_and_dummy())
796         // and all dummy indices that appear
797         exvector un, individual_dummy_indices;
798         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
799                 exvector free_indices_of_factor;
800                 if (is_a<indexed>(*it1)) {
801                         exvector dummy_indices_of_factor;
802                         find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free_indices_of_factor, dummy_indices_of_factor);
803                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
804                 } else
805                         free_indices_of_factor = it1->get_free_indices();
806                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
807         }
808         exvector local_dummy_indices;
809         find_free_and_dummy(un, free_indices, local_dummy_indices);
810         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
811
812         // Filter out the dummy indices with variance
813         exvector variant_dummy_indices;
814         find_variant_indices(local_dummy_indices, variant_dummy_indices);
815
816         // Any indices with variance present at all?
817         if (!variant_dummy_indices.empty()) {
818
819                 // Yes, bring the product into a canonical order that only depends on
820                 // the base expressions of indexed objects
821                 if (!non_commutative)
822                         std::sort(v.begin(), v.end(), ex_base_is_less());
823
824                 exvector moved_indices;
825
826                 // Iterate over all indexed objects in the product
827                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
828                         if (!is_a<indexed>(*it1))
829                                 continue;
830
831                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
832                                 something_changed = true;
833                 }
834         }
835
836         ex r;
837         if (something_changed)
838                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
839         else
840                 r = e;
841
842         // The result should be symmetric with respect to exchange of dummy
843         // indices, so if the symmetrization vanishes, the whole expression is
844         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
845         if (local_dummy_indices.size() >= 2) {
846                 exvector dummy_syms;
847                 dummy_syms.reserve(local_dummy_indices.size());
848                 for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
849                         dummy_syms.push_back(it->op(0));
850                 if (symmetrize(r, dummy_syms).is_zero()) {
851                         free_indices.clear();
852                         return _ex0;
853                 }
854         }
855
856         // Dummy index renaming
857         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
858
859         // Product of indexed object with a scalar?
860         if (is_exactly_a<mul>(r) && r.nops() == 2
861          && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
862                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
863         else
864                 return r;
865 }
866
867 /** This structure stores the original and symmetrized versions of terms
868  *  obtained during the simplification of sums. */
869 class terminfo {
870 public:
871         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
872
873         ex orig; /**< original term */
874         ex symm; /**< symmtrized term */
875 };
876
877 class terminfo_is_less {
878 public:
879         bool operator() (const terminfo & ti1, const terminfo & ti2) const
880         {
881                 return (ti1.symm.compare(ti2.symm) < 0);
882         }
883 };
884
885 /** This structure stores the individual symmetrized terms obtained during
886  *  the simplification of sums. */
887 class symminfo {
888 public:
889         symminfo() : num(0) {}
890
891         symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
892         {
893                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
894                         coeff = symmterm_.op(symmterm_.nops()-1);
895                         symmterm = symmterm_ / coeff;
896                 } else {
897                         coeff = 1;
898                         symmterm = symmterm_;
899                 }
900         }
901
902         ex symmterm;  /**< symmetrized term */
903         ex coeff;     /**< coefficient of symmetrized term */
904         ex orig;      /**< original term */
905         size_t num; /**< how many symmetrized terms resulted from the original term */
906 };
907
908 class symminfo_is_less_by_symmterm {
909 public:
910         bool operator() (const symminfo & si1, const symminfo & si2) const
911         {
912                 return (si1.symmterm.compare(si2.symmterm) < 0);
913         }
914 };
915
916 class symminfo_is_less_by_orig {
917 public:
918         bool operator() (const symminfo & si1, const symminfo & si2) const
919         {
920                 return (si1.orig.compare(si2.orig) < 0);
921         }
922 };
923
924 /** Simplify indexed expression, return list of free indices. */
925 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
926 {
927         // Expand the expression
928         ex e_expanded = e.expand();
929
930         // Simplification of single indexed object: just find the free indices
931         // and perform dummy index renaming/repositioning
932         if (is_a<indexed>(e_expanded)) {
933
934                 // Find the dummy indices
935                 const indexed &i = ex_to<indexed>(e_expanded);
936                 exvector local_dummy_indices;
937                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
938
939                 // Filter out the dummy indices with variance
940                 exvector variant_dummy_indices;
941                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
942
943                 // Any indices with variance present at all?
944                 if (!variant_dummy_indices.empty()) {
945
946                         // Yes, reposition them
947                         exvector moved_indices;
948                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
949                 }
950
951                 // Rename the dummy indices
952                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
953         }
954
955         // Simplification of sum = sum of simplifications, check consistency of
956         // free indices in each term
957         if (is_exactly_a<add>(e_expanded)) {
958                 bool first = true;
959                 ex sum;
960                 free_indices.clear();
961
962                 for (size_t i=0; i<e_expanded.nops(); i++) {
963                         exvector free_indices_of_term;
964                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
965                         if (!term.is_zero()) {
966                                 if (first) {
967                                         free_indices = free_indices_of_term;
968                                         sum = term;
969                                         first = false;
970                                 } else {
971                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
972                                                 std::ostringstream s;
973                                                 s << "simplify_indexed: inconsistent indices in sum: ";
974                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
975                                                 throw (std::runtime_error(s.str()));
976                                         }
977                                         if (is_a<indexed>(sum) && is_a<indexed>(term))
978                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
979                                         else
980                                                 sum += term;
981                                 }
982                         }
983                 }
984
985                 // If the sum turns out to be zero, we are finished
986                 if (sum.is_zero()) {
987                         free_indices.clear();
988                         return sum;
989                 }
990
991                 // More than one term and more than one dummy index?
992                 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
993                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
994                         return sum;
995
996                 // Yes, construct vector of all dummy index symbols
997                 exvector dummy_syms;
998                 dummy_syms.reserve(dummy_indices.size());
999                 for (exvector::const_iterator it = dummy_indices.begin(); it != dummy_indices.end(); ++it)
1000                         dummy_syms.push_back(it->op(0));
1001
1002                 // Chop the sum into terms and symmetrize each one over the dummy
1003                 // indices
1004                 std::vector<terminfo> terms;
1005                 for (size_t i=0; i<sum.nops(); i++) {
1006                         const ex & term = sum.op(i);
1007                         ex term_symm = symmetrize(term, dummy_syms);
1008                         if (term_symm.is_zero())
1009                                 continue;
1010                         terms.push_back(terminfo(term, term_symm));
1011                 }
1012
1013                 // Sort by symmetrized terms
1014                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
1015
1016                 // Combine equal symmetrized terms
1017                 std::vector<terminfo> terms_pass2;
1018                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1019                         size_t num = 1;
1020                         std::vector<terminfo>::const_iterator j = i + 1;
1021                         while (j != terms.end() && j->symm == i->symm) {
1022                                 num++;
1023                                 j++;
1024                         }
1025                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1026                         i = j;
1027                 }
1028
1029                 // If there is only one term left, we are finished
1030                 if (terms_pass2.size() == 1)
1031                         return terms_pass2[0].orig;
1032
1033                 // Chop the symmetrized terms into subterms
1034                 std::vector<symminfo> sy;
1035                 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1036                         if (is_exactly_a<add>(i->symm)) {
1037                                 size_t num = i->symm.nops();
1038                                 for (size_t j=0; j<num; j++)
1039                                         sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1040                         } else
1041                                 sy.push_back(symminfo(i->symm, i->orig, 1));
1042                 }
1043
1044                 // Sort by symmetrized subterms
1045                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1046
1047                 // Combine equal symmetrized subterms
1048                 std::vector<symminfo> sy_pass2;
1049                 exvector result;
1050                 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1051
1052                         // Combine equal terms
1053                         std::vector<symminfo>::const_iterator j = i + 1;
1054                         if (j != sy.end() && j->symmterm == i->symmterm) {
1055
1056                                 // More than one term, collect the coefficients
1057                                 ex coeff = i->coeff;
1058                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1059                                         coeff += j->coeff;
1060                                         j++;
1061                                 }
1062
1063                                 // Add combined term to result
1064                                 if (!coeff.is_zero())
1065                                         result.push_back(coeff * i->symmterm);
1066
1067                         } else {
1068
1069                                 // Single term, store for second pass
1070                                 sy_pass2.push_back(*i);
1071                         }
1072
1073                         i = j;
1074                 }
1075
1076                 // Were there any remaining terms that didn't get combined?
1077                 if (sy_pass2.size() > 0) {
1078
1079                         // Yes, sort by their original terms
1080                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1081
1082                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1083
1084                                 // How many symmetrized terms of this original term are left?
1085                                 size_t num = 1;
1086                                 std::vector<symminfo>::const_iterator j = i + 1;
1087                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1088                                         num++;
1089                                         j++;
1090                                 }
1091
1092                                 if (num == i->num) {
1093
1094                                         // All terms left, then add the original term to the result
1095                                         result.push_back(i->orig);
1096
1097                                 } else {
1098
1099                                         // Some terms were combined with others, add up the remaining symmetrized terms
1100                                         std::vector<symminfo>::const_iterator k;
1101                                         for (k=i; k!=j; k++)
1102                                                 result.push_back(k->coeff * k->symmterm);
1103                                 }
1104
1105                                 i = j;
1106                         }
1107                 }
1108
1109                 // Add all resulting terms
1110                 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1111                 if (sum_symm.is_zero())
1112                         free_indices.clear();
1113                 return sum_symm;
1114         }
1115
1116         // Simplification of products
1117         if (is_exactly_a<mul>(e_expanded)
1118          || is_exactly_a<ncmul>(e_expanded)
1119          || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1120                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1121
1122         // Cannot do anything
1123         free_indices.clear();
1124         return e_expanded;
1125 }
1126
1127 /** Simplify/canonicalize expression containing indexed objects. This
1128  *  performs contraction of dummy indices where possible and checks whether
1129  *  the free indices in sums are consistent.
1130  *
1131  *  @param options Simplification options (currently unused)
1132  *  @return simplified expression */
1133 ex ex::simplify_indexed(unsigned options) const
1134 {
1135         exvector free_indices, dummy_indices;
1136         scalar_products sp;
1137         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1138 }
1139
1140 /** Simplify/canonicalize expression containing indexed objects. This
1141  *  performs contraction of dummy indices where possible, checks whether
1142  *  the free indices in sums are consistent, and automatically replaces
1143  *  scalar products by known values if desired.
1144  *
1145  *  @param sp Scalar products to be replaced automatically
1146  *  @param options Simplification options (currently unused)
1147  *  @return simplified expression */
1148 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1149 {
1150         exvector free_indices, dummy_indices;
1151         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1152 }
1153
1154 /** Symmetrize expression over its free indices. */
1155 ex ex::symmetrize() const
1156 {
1157         return GiNaC::symmetrize(*this, get_free_indices());
1158 }
1159
1160 /** Antisymmetrize expression over its free indices. */
1161 ex ex::antisymmetrize() const
1162 {
1163         return GiNaC::antisymmetrize(*this, get_free_indices());
1164 }
1165
1166 /** Symmetrize expression by cyclic permutation over its free indices. */
1167 ex ex::symmetrize_cyclic() const
1168 {
1169         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1170 }
1171
1172 //////////
1173 // helper classes
1174 //////////
1175
1176 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1177 {
1178         // If indexed, extract base objects
1179         ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1180         ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1181
1182         // Enforce canonical order in pair
1183         if (s1.compare(s2) > 0) {
1184                 v1 = s2;
1185                 v2 = s1;
1186         } else {
1187                 v1 = s1;
1188                 v2 = s2;
1189         }
1190 }
1191
1192 bool spmapkey::operator==(const spmapkey &other) const
1193 {
1194         if (!v1.is_equal(other.v1))
1195                 return false;
1196         if (!v2.is_equal(other.v2))
1197                 return false;
1198         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1199                 return true;
1200         else
1201                 return dim.is_equal(other.dim);
1202 }
1203
1204 bool spmapkey::operator<(const spmapkey &other) const
1205 {
1206         int cmp = v1.compare(other.v1);
1207         if (cmp)
1208                 return cmp < 0;
1209         cmp = v2.compare(other.v2);
1210         if (cmp)
1211                 return cmp < 0;
1212
1213         // Objects are equal, now check dimensions
1214         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1215                 return false;
1216         else
1217                 return dim.compare(other.dim) < 0;
1218 }
1219
1220 void spmapkey::debugprint() const
1221 {
1222         std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1223 }
1224
1225 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1226 {
1227         spm[spmapkey(v1, v2)] = sp;
1228 }
1229
1230 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1231 {
1232         spm[spmapkey(v1, v2, dim)] = sp;
1233 }
1234
1235 void scalar_products::add_vectors(const lst & l, const ex & dim)
1236 {
1237         // Add all possible pairs of products
1238         for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1239                 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1240                         add(*it1, *it2, *it1 * *it2);
1241 }
1242
1243 void scalar_products::clear()
1244 {
1245         spm.clear();
1246 }
1247
1248 /** Check whether scalar product pair is defined. */
1249 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1250 {
1251         return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1252 }
1253
1254 /** Return value of defined scalar product pair. */
1255 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1256 {
1257         return spm.find(spmapkey(v1, v2, dim))->second;
1258 }
1259
1260 void scalar_products::debugprint() const
1261 {
1262         std::cerr << "map size=" << spm.size() << std::endl;
1263         spmap::const_iterator i = spm.begin(), end = spm.end();
1264         while (i != end) {
1265                 const spmapkey & k = i->first;
1266                 std::cerr << "item key=";
1267                 k.debugprint();
1268                 std::cerr << ", value=" << i->second << std::endl;
1269                 ++i;
1270         }
1271 }
1272
1273 } // namespace GiNaC