3 * Implementation of GiNaC's indexed expressions. */
6 * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
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.
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.
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
33 #include "relational.h"
35 #include "operators.h"
45 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(indexed, exprseq,
46 print_func<print_context>(&indexed::do_print).
47 print_func<print_latex>(&indexed::do_print_latex).
48 print_func<print_tree>(&indexed::do_print_tree))
51 // default constructor
54 indexed::indexed() : symtree(not_symmetric())
56 tinfo_key = TINFO_indexed;
63 indexed::indexed(const ex & b) : inherited(b), symtree(not_symmetric())
65 tinfo_key = TINFO_indexed;
69 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(not_symmetric())
71 tinfo_key = TINFO_indexed;
75 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(not_symmetric())
77 tinfo_key = TINFO_indexed;
81 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(not_symmetric())
83 tinfo_key = TINFO_indexed;
87 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())
89 tinfo_key = TINFO_indexed;
93 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
95 tinfo_key = TINFO_indexed;
99 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
101 tinfo_key = TINFO_indexed;
105 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)
107 tinfo_key = TINFO_indexed;
111 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(not_symmetric())
113 seq.insert(seq.end(), v.begin(), v.end());
114 tinfo_key = TINFO_indexed;
118 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
120 seq.insert(seq.end(), v.begin(), v.end());
121 tinfo_key = TINFO_indexed;
125 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
127 tinfo_key = TINFO_indexed;
130 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
132 tinfo_key = TINFO_indexed;
135 indexed::indexed(const symmetry & symm, std::auto_ptr<exvector> vp) : inherited(vp), symtree(symm)
137 tinfo_key = TINFO_indexed;
144 indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
146 if (!n.find_ex("symmetry", symtree, sym_lst)) {
147 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
149 n.find_unsigned("symmetry", symm);
158 symtree = not_symmetric();
161 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
165 void indexed::archive(archive_node &n) const
167 inherited::archive(n);
168 n.add_ex("symmetry", symtree);
171 DEFAULT_UNARCHIVE(indexed)
174 // functions overriding virtual functions from base classes
177 void indexed::printindices(const print_context & c, unsigned level) const
179 if (seq.size() > 1) {
181 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
183 if (is_a<print_latex>(c)) {
185 // TeX output: group by variance
187 bool covariant = true;
189 while (it != itend) {
190 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
191 if (first || cur_covariant != covariant) { // Variance changed
192 // The empty {} prevents indices from ending up on top of each other
195 covariant = cur_covariant;
211 while (it != itend) {
219 void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
221 if (precedence() <= level)
222 c.s << openbrace << '(';
224 seq[0].print(c, precedence());
226 printindices(c, level);
227 if (precedence() <= level)
228 c.s << ')' << closebrace;
231 void indexed::do_print(const print_context & c, unsigned level) const
233 print_indexed(c, "", "", level);
236 void indexed::do_print_latex(const print_latex & c, unsigned level) const
238 print_indexed(c, "{", "}", level);
241 void indexed::do_print_tree(const print_tree & c, unsigned level) const
243 c.s << std::string(level, ' ') << class_name() << " @" << this
244 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
245 << ", " << seq.size()-1 << " indices"
246 << ", symmetry=" << symtree << std::endl;
247 seq[0].print(c, level + c.delta_indent);
248 printindices(c, level + c.delta_indent);
251 bool indexed::info(unsigned inf) const
253 if (inf == info_flags::indexed) return true;
254 if (inf == info_flags::has_indices) return seq.size() > 1;
255 return inherited::info(inf);
258 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
259 bool operator() (const ex & e, unsigned inf) const {
260 return !(ex_to<idx>(e).get_value().info(inf));
264 bool indexed::all_index_values_are(unsigned inf) const
266 // No indices? Then no property can be fulfilled
271 return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
274 int indexed::compare_same_type(const basic & other) const
276 GINAC_ASSERT(is_a<indexed>(other));
277 return inherited::compare_same_type(other);
280 ex indexed::eval(int level) const
282 // First evaluate children, then we will end up here again
284 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
286 const ex &base = seq[0];
288 // If the base object is 0, the whole object is 0
292 // If the base object is a product, pull out the numeric factor
293 if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
295 ex f = ex_to<numeric>(base.op(base.nops() - 1));
297 return f * thiscontainer(v);
300 // Canonicalize indices according to the symmetry properties
301 if (seq.size() > 2) {
303 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
304 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
305 if (sig != INT_MAX) {
306 // Something has changed while sorting indices, more evaluations later
309 return ex(sig) * thiscontainer(v);
313 // Let the class of the base object perform additional evaluations
314 return ex_to<basic>(base).eval_indexed(*this);
317 ex indexed::thiscontainer(const exvector & v) const
319 return indexed(ex_to<symmetry>(symtree), v);
322 ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
324 return indexed(ex_to<symmetry>(symtree), vp);
327 ex indexed::expand(unsigned options) const
329 GINAC_ASSERT(seq.size() > 0);
331 if (options & expand_options::expand_indexed) {
332 ex newbase = seq[0].expand(options);
333 if (is_exactly_a<add>(newbase)) {
335 for (size_t i=0; i<newbase.nops(); i++) {
337 s[0] = newbase.op(i);
338 sum += thiscontainer(s).expand(options);
342 if (!are_ex_trivially_equal(newbase, seq[0])) {
345 return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
348 return inherited::expand(options);
352 // virtual functions which can be overridden by derived classes
358 // non-virtual functions in this class
361 /** Check whether all indices are of class idx and validate the symmetry
362 * tree. This function is used internally to make sure that all constructed
363 * indexed objects really carry indices and not some other classes. */
364 void indexed::validate() const
366 GINAC_ASSERT(seq.size() > 0);
367 exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
368 while (it != itend) {
370 throw(std::invalid_argument("indices of indexed object must be of type idx"));
374 if (!symtree.is_zero()) {
375 if (!is_exactly_a<symmetry>(symtree))
376 throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
377 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
381 /** Implementation of ex::diff() for an indexed object always returns 0.
384 ex indexed::derivative(const symbol & s) const
393 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
394 bool operator() (const ex &lh, const ex &rh) const
400 // Replacing the dimension might cause an error (e.g. with
401 // index classes that only work in a fixed number of dimensions)
402 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
409 /** Check whether two sorted index vectors are consistent (i.e. equal). */
410 static bool indices_consistent(const exvector & v1, const exvector & v2)
412 // Number of indices must be the same
413 if (v1.size() != v2.size())
416 return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
419 exvector indexed::get_indices() const
421 GINAC_ASSERT(seq.size() >= 1);
422 return exvector(seq.begin() + 1, seq.end());
425 exvector indexed::get_dummy_indices() const
427 exvector free_indices, dummy_indices;
428 find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
429 return dummy_indices;
432 exvector indexed::get_dummy_indices(const indexed & other) const
434 exvector indices = get_free_indices();
435 exvector other_indices = other.get_free_indices();
436 indices.insert(indices.end(), other_indices.begin(), other_indices.end());
437 exvector dummy_indices;
438 find_dummy_indices(indices, dummy_indices);
439 return dummy_indices;
442 bool indexed::has_dummy_index_for(const ex & i) const
444 exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
445 while (it != itend) {
446 if (is_dummy_pair(*it, i))
453 exvector indexed::get_free_indices() const
455 exvector free_indices, dummy_indices;
456 find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
460 exvector add::get_free_indices() const
462 exvector free_indices;
463 for (size_t i=0; i<nops(); i++) {
465 free_indices = op(i).get_free_indices();
467 exvector free_indices_of_term = op(i).get_free_indices();
468 if (!indices_consistent(free_indices, free_indices_of_term))
469 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
475 exvector mul::get_free_indices() const
477 // Concatenate free indices of all factors
479 for (size_t i=0; i<nops(); i++) {
480 exvector free_indices_of_factor = op(i).get_free_indices();
481 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
484 // And remove the dummy indices
485 exvector free_indices, dummy_indices;
486 find_free_and_dummy(un, free_indices, dummy_indices);
490 exvector ncmul::get_free_indices() const
492 // Concatenate free indices of all factors
494 for (size_t i=0; i<nops(); i++) {
495 exvector free_indices_of_factor = op(i).get_free_indices();
496 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
499 // And remove the dummy indices
500 exvector free_indices, dummy_indices;
501 find_free_and_dummy(un, free_indices, dummy_indices);
505 struct is_summation_idx : public std::unary_function<ex, bool> {
506 bool operator()(const ex & e)
508 return is_dummy_pair(e, e);
512 exvector power::get_free_indices() const
514 // Get free indices of basis
515 exvector basis_indices = basis.get_free_indices();
517 if (exponent.info(info_flags::even)) {
518 // If the exponent is an even number, then any "free" index that
519 // forms a dummy pair with itself is actually a summation index
520 exvector really_free;
521 std::remove_copy_if(basis_indices.begin(), basis_indices.end(),
522 std::back_inserter(really_free), is_summation_idx());
525 return basis_indices;
528 exvector integral::get_free_indices() const
530 if (a.get_free_indices().size() || b.get_free_indices().size())
531 throw (std::runtime_error("integral::get_free_indices: boundary values should not have free indices"));
532 return f.get_free_indices();
535 /** Rename dummy indices in an expression.
537 * @param e Expression to work on
538 * @param local_dummy_indices The set of dummy indices that appear in the
540 * @param global_dummy_indices The set of dummy indices that have appeared
541 * before and which we would like to use in "e", too. This gets updated
543 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
545 size_t global_size = global_dummy_indices.size(),
546 local_size = local_dummy_indices.size();
548 // Any local dummy indices at all?
552 if (global_size < local_size) {
554 // More local indices than we encountered before, add the new ones
556 size_t old_global_size = global_size;
557 int remaining = local_size - global_size;
558 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
559 while (it != itend && remaining > 0) {
560 if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(op0_is_equal(), *it)) == global_dummy_indices.end()) {
561 global_dummy_indices.push_back(*it);
568 // If this is the first set of local indices, do nothing
569 if (old_global_size == 0)
572 GINAC_ASSERT(local_size <= global_size);
574 // Construct vectors of index symbols
575 exvector local_syms, global_syms;
576 local_syms.reserve(local_size);
577 global_syms.reserve(local_size);
578 for (size_t i=0; i<local_size; i++)
579 local_syms.push_back(local_dummy_indices[i].op(0));
580 shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
581 for (size_t i=0; i<local_size; i++) // don't use more global symbols than necessary
582 global_syms.push_back(global_dummy_indices[i].op(0));
583 shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
585 // Remove common indices
586 exvector local_uniq, global_uniq;
587 set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
588 set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
590 // Replace remaining non-common local index symbols by global ones
591 if (local_uniq.empty())
594 while (global_uniq.size() > local_uniq.size())
595 global_uniq.pop_back();
596 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
600 /** Given a set of indices, extract those of class varidx. */
601 static void find_variant_indices(const exvector & v, exvector & variant_indices)
603 exvector::const_iterator it1, itend;
604 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
605 if (is_exactly_a<varidx>(*it1))
606 variant_indices.push_back(*it1);
610 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
613 * @param e Object to work on
614 * @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
615 * @param moved_indices The set of indices that have been repositioned (will be changed by this function)
616 * @return true if 'e' was changed */
617 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
619 bool something_changed = false;
621 // If a dummy index is encountered for the first time in the
622 // product, pull it up, otherwise, pull it down
623 exvector::const_iterator it2, it2start, it2end;
624 for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
625 if (!is_exactly_a<varidx>(*it2))
628 exvector::iterator vit, vitend;
629 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
630 if (it2->op(0).is_equal(vit->op(0))) {
631 if (ex_to<varidx>(*it2).is_covariant()) {
633 *it2 == ex_to<varidx>(*it2).toggle_variance(),
634 ex_to<varidx>(*it2).toggle_variance() == *it2
635 ), subs_options::no_pattern);
636 something_changed = true;
637 it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
638 it2start = ex_to<indexed>(e).seq.begin();
639 it2end = ex_to<indexed>(e).seq.end();
641 moved_indices.push_back(*vit);
642 variant_dummy_indices.erase(vit);
647 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
648 if (it2->op(0).is_equal(vit->op(0))) {
649 if (ex_to<varidx>(*it2).is_contravariant()) {
650 e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance(), subs_options::no_pattern);
651 something_changed = true;
652 it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
653 it2start = ex_to<indexed>(e).seq.begin();
654 it2end = ex_to<indexed>(e).seq.end();
663 return something_changed;
666 /* Ordering that only compares the base expressions of indexed objects. */
667 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
668 bool operator() (const ex &lh, const ex &rh) const
670 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
674 /* An auxiliary function used by simplify_indexed() and expand_dummy_sum()
675 * It returns an exvector of factors from the supplied product */
676 static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
678 // Remember whether the product was commutative or noncommutative
679 // (because we chop it into factors and need to reassemble later)
680 non_commutative = is_exactly_a<ncmul>(e);
682 // Collect factors in an exvector, store squares twice
683 v.reserve(e.nops() * 2);
685 if (is_exactly_a<power>(e)) {
686 // We only get called for simple squares, split a^2 -> a*a
687 GINAC_ASSERT(e.op(1).is_equal(_ex2));
688 v.push_back(e.op(0));
689 v.push_back(e.op(0));
691 for (size_t i=0; i<e.nops(); i++) {
693 if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
694 v.push_back(f.op(0));
695 v.push_back(f.op(0));
696 } else if (is_exactly_a<ncmul>(f)) {
697 // Noncommutative factor found, split it as well
698 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
699 for (size_t j=0; j<f.nops(); j++)
700 v.push_back(f.op(j));
707 /** Simplify product of indexed expressions (commutative, noncommutative and
708 * simple squares), return list of free indices. */
709 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
711 // Collect factors in an exvector
714 // Remember whether the product was commutative or noncommutative
715 // (because we chop it into factors and need to reassemble later)
716 bool non_commutative;
717 product_to_exvector(e, v, non_commutative);
719 // Perform contractions
720 bool something_changed = false;
721 GINAC_ASSERT(v.size() > 1);
722 exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
723 for (it1 = v.begin(); it1 != next_to_last; it1++) {
726 if (!is_a<indexed>(*it1))
729 bool first_noncommutative = (it1->return_type() != return_types::commutative);
731 // Indexed factor found, get free indices and look for contraction
733 exvector free1, dummy1;
734 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
736 exvector::iterator it2;
737 for (it2 = it1 + 1; it2 != itend; it2++) {
739 if (!is_a<indexed>(*it2))
742 bool second_noncommutative = (it2->return_type() != return_types::commutative);
744 // Find free indices of second factor and merge them with free
745 // indices of first factor
747 find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
748 un.insert(un.end(), free1.begin(), free1.end());
750 // Check whether the two factors share dummy indices
751 exvector free, dummy;
752 find_free_and_dummy(un, free, dummy);
753 size_t num_dummies = dummy.size();
754 if (num_dummies == 0)
757 // At least one dummy index, is it a defined scalar product?
758 bool contracted = false;
761 // Find minimal dimension of all indices of both factors
762 exvector::const_iterator dit = ex_to<indexed>(*it1).seq.begin() + 1, ditend = ex_to<indexed>(*it1).seq.end();
763 ex dim = ex_to<idx>(*dit).get_dim();
765 for (; dit != ditend; ++dit) {
766 dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
768 dit = ex_to<indexed>(*it2).seq.begin() + 1;
769 ditend = ex_to<indexed>(*it2).seq.end();
770 for (; dit != ditend; ++dit) {
771 dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
774 // User-defined scalar product?
775 if (sp.is_defined(*it1, *it2, dim)) {
777 // Yes, substitute it
778 *it1 = sp.evaluate(*it1, *it2, dim);
780 goto contraction_done;
784 // Try to contract the first one with the second one
785 contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
788 // That didn't work; maybe the second object knows how to
789 // contract itself with the first one
790 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
794 if (first_noncommutative || second_noncommutative
795 || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
796 || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
797 || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
799 // One of the factors became a sum or product:
800 // re-expand expression and run again
801 // Non-commutative products are always re-expanded to give
802 // eval_ncmul() the chance to re-order and canonicalize
804 ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
805 return simplify_indexed(r, free_indices, dummy_indices, sp);
808 // Both objects may have new indices now or they might
809 // even not be indexed objects any more, so we have to
811 something_changed = true;
817 // Find free indices (concatenate them all and call find_free_and_dummy())
818 // and all dummy indices that appear
819 exvector un, individual_dummy_indices;
820 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
821 exvector free_indices_of_factor;
822 if (is_a<indexed>(*it1)) {
823 exvector dummy_indices_of_factor;
824 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);
825 individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
827 free_indices_of_factor = it1->get_free_indices();
828 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
830 exvector local_dummy_indices;
831 find_free_and_dummy(un, free_indices, local_dummy_indices);
832 local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
834 // Filter out the dummy indices with variance
835 exvector variant_dummy_indices;
836 find_variant_indices(local_dummy_indices, variant_dummy_indices);
838 // Any indices with variance present at all?
839 if (!variant_dummy_indices.empty()) {
841 // Yes, bring the product into a canonical order that only depends on
842 // the base expressions of indexed objects
843 if (!non_commutative)
844 std::sort(v.begin(), v.end(), ex_base_is_less());
846 exvector moved_indices;
848 // Iterate over all indexed objects in the product
849 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
850 if (!is_a<indexed>(*it1))
853 if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
854 something_changed = true;
859 if (something_changed)
860 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
864 // The result should be symmetric with respect to exchange of dummy
865 // indices, so if the symmetrization vanishes, the whole expression is
866 // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
867 if (local_dummy_indices.size() >= 2) {
869 dummy_syms.reserve(local_dummy_indices.size());
870 for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
871 dummy_syms.push_back(it->op(0));
872 if (symmetrize(r, dummy_syms).is_zero()) {
873 free_indices.clear();
878 // Dummy index renaming
879 r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
881 // Product of indexed object with a scalar?
882 if (is_exactly_a<mul>(r) && r.nops() == 2
883 && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
884 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
889 /** This structure stores the original and symmetrized versions of terms
890 * obtained during the simplification of sums. */
893 terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
895 ex orig; /**< original term */
896 ex symm; /**< symmtrized term */
899 class terminfo_is_less {
901 bool operator() (const terminfo & ti1, const terminfo & ti2) const
903 return (ti1.symm.compare(ti2.symm) < 0);
907 /** This structure stores the individual symmetrized terms obtained during
908 * the simplification of sums. */
911 symminfo() : num(0) {}
913 symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
915 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
916 coeff = symmterm_.op(symmterm_.nops()-1);
917 symmterm = symmterm_ / coeff;
920 symmterm = symmterm_;
924 ex symmterm; /**< symmetrized term */
925 ex coeff; /**< coefficient of symmetrized term */
926 ex orig; /**< original term */
927 size_t num; /**< how many symmetrized terms resulted from the original term */
930 class symminfo_is_less_by_symmterm {
932 bool operator() (const symminfo & si1, const symminfo & si2) const
934 return (si1.symmterm.compare(si2.symmterm) < 0);
938 class symminfo_is_less_by_orig {
940 bool operator() (const symminfo & si1, const symminfo & si2) const
942 return (si1.orig.compare(si2.orig) < 0);
946 /** Simplify indexed expression, return list of free indices. */
947 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
949 // Expand the expression
950 ex e_expanded = e.expand();
952 // Simplification of single indexed object: just find the free indices
953 // and perform dummy index renaming/repositioning
954 if (is_a<indexed>(e_expanded)) {
956 // Find the dummy indices
957 const indexed &i = ex_to<indexed>(e_expanded);
958 exvector local_dummy_indices;
959 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
961 // Filter out the dummy indices with variance
962 exvector variant_dummy_indices;
963 find_variant_indices(local_dummy_indices, variant_dummy_indices);
965 // Any indices with variance present at all?
966 if (!variant_dummy_indices.empty()) {
968 // Yes, reposition them
969 exvector moved_indices;
970 reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
973 // Rename the dummy indices
974 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
977 // Simplification of sum = sum of simplifications, check consistency of
978 // free indices in each term
979 if (is_exactly_a<add>(e_expanded)) {
982 free_indices.clear();
984 for (size_t i=0; i<e_expanded.nops(); i++) {
985 exvector free_indices_of_term;
986 ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
987 if (!term.is_zero()) {
989 free_indices = free_indices_of_term;
993 if (!indices_consistent(free_indices, free_indices_of_term)) {
994 std::ostringstream s;
995 s << "simplify_indexed: inconsistent indices in sum: ";
996 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
997 throw (std::runtime_error(s.str()));
999 if (is_a<indexed>(sum) && is_a<indexed>(term))
1000 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
1007 // If the sum turns out to be zero, we are finished
1008 if (sum.is_zero()) {
1009 free_indices.clear();
1013 // More than one term and more than one dummy index?
1014 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
1015 if (num_terms_orig < 2 || dummy_indices.size() < 2)
1018 // Yes, construct vector of all dummy index symbols
1019 exvector dummy_syms;
1020 dummy_syms.reserve(dummy_indices.size());
1021 for (exvector::const_iterator it = dummy_indices.begin(); it != dummy_indices.end(); ++it)
1022 dummy_syms.push_back(it->op(0));
1024 // Chop the sum into terms and symmetrize each one over the dummy
1026 std::vector<terminfo> terms;
1027 for (size_t i=0; i<sum.nops(); i++) {
1028 const ex & term = sum.op(i);
1029 ex term_symm = symmetrize(term, dummy_syms);
1030 if (term_symm.is_zero())
1032 terms.push_back(terminfo(term, term_symm));
1035 // Sort by symmetrized terms
1036 std::sort(terms.begin(), terms.end(), terminfo_is_less());
1038 // Combine equal symmetrized terms
1039 std::vector<terminfo> terms_pass2;
1040 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1042 std::vector<terminfo>::const_iterator j = i + 1;
1043 while (j != terms.end() && j->symm == i->symm) {
1047 terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1051 // If there is only one term left, we are finished
1052 if (terms_pass2.size() == 1)
1053 return terms_pass2[0].orig;
1055 // Chop the symmetrized terms into subterms
1056 std::vector<symminfo> sy;
1057 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1058 if (is_exactly_a<add>(i->symm)) {
1059 size_t num = i->symm.nops();
1060 for (size_t j=0; j<num; j++)
1061 sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1063 sy.push_back(symminfo(i->symm, i->orig, 1));
1066 // Sort by symmetrized subterms
1067 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1069 // Combine equal symmetrized subterms
1070 std::vector<symminfo> sy_pass2;
1072 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1074 // Combine equal terms
1075 std::vector<symminfo>::const_iterator j = i + 1;
1076 if (j != sy.end() && j->symmterm == i->symmterm) {
1078 // More than one term, collect the coefficients
1079 ex coeff = i->coeff;
1080 while (j != sy.end() && j->symmterm == i->symmterm) {
1085 // Add combined term to result
1086 if (!coeff.is_zero())
1087 result.push_back(coeff * i->symmterm);
1091 // Single term, store for second pass
1092 sy_pass2.push_back(*i);
1098 // Were there any remaining terms that didn't get combined?
1099 if (sy_pass2.size() > 0) {
1101 // Yes, sort by their original terms
1102 std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1104 for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1106 // How many symmetrized terms of this original term are left?
1108 std::vector<symminfo>::const_iterator j = i + 1;
1109 while (j != sy_pass2.end() && j->orig == i->orig) {
1114 if (num == i->num) {
1116 // All terms left, then add the original term to the result
1117 result.push_back(i->orig);
1121 // Some terms were combined with others, add up the remaining symmetrized terms
1122 std::vector<symminfo>::const_iterator k;
1123 for (k=i; k!=j; k++)
1124 result.push_back(k->coeff * k->symmterm);
1131 // Add all resulting terms
1132 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1133 if (sum_symm.is_zero())
1134 free_indices.clear();
1138 // Simplification of products
1139 if (is_exactly_a<mul>(e_expanded)
1140 || is_exactly_a<ncmul>(e_expanded)
1141 || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1142 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1144 // Cannot do anything
1145 free_indices.clear();
1149 /** Simplify/canonicalize expression containing indexed objects. This
1150 * performs contraction of dummy indices where possible and checks whether
1151 * the free indices in sums are consistent.
1153 * @param options Simplification options (currently unused)
1154 * @return simplified expression */
1155 ex ex::simplify_indexed(unsigned options) const
1157 exvector free_indices, dummy_indices;
1159 return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1162 /** Simplify/canonicalize expression containing indexed objects. This
1163 * performs contraction of dummy indices where possible, checks whether
1164 * the free indices in sums are consistent, and automatically replaces
1165 * scalar products by known values if desired.
1167 * @param sp Scalar products to be replaced automatically
1168 * @param options Simplification options (currently unused)
1169 * @return simplified expression */
1170 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1172 exvector free_indices, dummy_indices;
1173 return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1176 /** Symmetrize expression over its free indices. */
1177 ex ex::symmetrize() const
1179 return GiNaC::symmetrize(*this, get_free_indices());
1182 /** Antisymmetrize expression over its free indices. */
1183 ex ex::antisymmetrize() const
1185 return GiNaC::antisymmetrize(*this, get_free_indices());
1188 /** Symmetrize expression by cyclic permutation over its free indices. */
1189 ex ex::symmetrize_cyclic() const
1191 return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1198 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1200 // If indexed, extract base objects
1201 ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1202 ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1204 // Enforce canonical order in pair
1205 if (s1.compare(s2) > 0) {
1214 bool spmapkey::operator==(const spmapkey &other) const
1216 if (!v1.is_equal(other.v1))
1218 if (!v2.is_equal(other.v2))
1220 if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1223 return dim.is_equal(other.dim);
1226 bool spmapkey::operator<(const spmapkey &other) const
1228 int cmp = v1.compare(other.v1);
1231 cmp = v2.compare(other.v2);
1235 // Objects are equal, now check dimensions
1236 if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1239 return dim.compare(other.dim) < 0;
1242 void spmapkey::debugprint() const
1244 std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1247 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1249 spm[spmapkey(v1, v2)] = sp;
1252 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1254 spm[spmapkey(v1, v2, dim)] = sp;
1257 void scalar_products::add_vectors(const lst & l, const ex & dim)
1259 // Add all possible pairs of products
1260 for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1261 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1262 add(*it1, *it2, *it1 * *it2);
1265 void scalar_products::clear()
1270 /** Check whether scalar product pair is defined. */
1271 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1273 return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1276 /** Return value of defined scalar product pair. */
1277 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1279 return spm.find(spmapkey(v1, v2, dim))->second;
1282 void scalar_products::debugprint() const
1284 std::cerr << "map size=" << spm.size() << std::endl;
1285 spmap::const_iterator i = spm.begin(), end = spm.end();
1287 const spmapkey & k = i->first;
1288 std::cerr << "item key=";
1290 std::cerr << ", value=" << i->second << std::endl;
1295 /** Returns all dummy indices from the exvector */
1296 exvector get_all_dummy_indices(const ex & e)
1300 product_to_exvector(e, p, nc);
1301 exvector::const_iterator ip = p.begin(), ipend = p.end();
1303 while (ip != ipend) {
1304 if (is_a<indexed>(*ip)) {
1305 v1 = ex_to<indexed>(*ip).get_dummy_indices();
1306 v.insert(v.end(), v1.begin(), v1.end());
1307 exvector::const_iterator ip1 = ip+1;
1308 while (ip1 != ipend) {
1309 if (is_a<indexed>(*ip1)) {
1310 v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
1311 v.insert(v.end(), v1.begin(), v1.end());
1321 ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
1323 exvector va = get_all_dummy_indices(a), vb = get_all_dummy_indices(b), common_indices;
1324 set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
1325 if (common_indices.empty()) {
1328 exvector new_indices, old_indices;
1329 old_indices.reserve(2*common_indices.size());
1330 new_indices.reserve(2*common_indices.size());
1331 exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
1332 while (ip != ipend) {
1333 if (is_a<varidx>(*ip)) {
1334 varidx mu((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(*ip).get_dim(), ex_to<varidx>(*ip).is_covariant());
1335 old_indices.push_back(*ip);
1336 new_indices.push_back(mu);
1337 old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
1338 new_indices.push_back(mu.toggle_variance());
1340 old_indices.push_back(*ip);
1341 new_indices.push_back(idx((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(*ip).get_dim()));
1345 return b.subs(lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end()), subs_options::no_pattern);
1349 ex expand_dummy_sum(const ex & e, bool subs_idx)
1351 ex e_expanded = e.expand();
1352 pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
1353 if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
1354 return e_expanded.map(fcn);
1355 } else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded)) {
1356 exvector v = get_all_dummy_indices(e_expanded);
1357 exvector::const_iterator it = v.begin(), itend = v.end();
1358 while (it != itend) {
1359 varidx nu = ex_to<varidx>(*it);
1360 if (nu.is_dim_numeric()) {
1362 for (int i=0; i < ex_to<numeric>(nu.get_dim()).to_int(); i++) {
1363 if (is_a<varidx>(nu) && !subs_idx) {
1364 en += e_expanded.subs(lst(nu == varidx(i, nu.get_dim(), true), nu.toggle_variance() == varidx(i, nu.get_dim())));
1366 en += e_expanded.subs(lst(nu == idx(i, nu.get_dim()), nu.toggle_variance() == idx(i, nu.get_dim())));
1369 return expand_dummy_sum(en, subs_idx);
1374 } else if (is_a<indexed>(e_expanded)) {
1375 exvector v = ex_to<indexed>(e_expanded).get_dummy_indices();
1376 exvector::const_iterator it = v.begin(), itend = v.end();
1377 while (it != itend) {
1378 varidx nu = ex_to<varidx>(*it);
1379 if (nu.is_dim_numeric()) {
1381 for (int i=0; i < ex_to<numeric>(nu.get_dim()).to_int(); i++) {
1382 if (is_a<varidx>(nu) && !subs_idx) {
1383 en += e_expanded.subs(lst(nu == varidx(i, nu.get_dim(), true), nu.toggle_variance() == varidx(i, nu.get_dim())));
1385 en += e_expanded.subs(lst(nu == idx(i, nu.get_dim()), nu.toggle_variance() == idx(i, nu.get_dim())));
1388 return expand_dummy_sum(en, subs_idx);
1398 } // namespace GiNaC