3 * Implementation of GiNaC's indexed expressions. */
6 * GiNaC Copyright (C) 1999-2008 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
34 #include "relational.h"
36 #include "operators.h"
47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(indexed, exprseq,
48 print_func<print_context>(&indexed::do_print).
49 print_func<print_latex>(&indexed::do_print_latex).
50 print_func<print_tree>(&indexed::do_print_tree))
53 // default constructor
56 indexed::indexed() : symtree(not_symmetric())
58 tinfo_key = &indexed::tinfo_static;
65 indexed::indexed(const ex & b) : inherited(b), symtree(not_symmetric())
67 tinfo_key = &indexed::tinfo_static;
71 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(not_symmetric())
73 tinfo_key = &indexed::tinfo_static;
77 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(not_symmetric())
79 tinfo_key = &indexed::tinfo_static;
83 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(not_symmetric())
85 tinfo_key = &indexed::tinfo_static;
89 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())
91 tinfo_key = &indexed::tinfo_static;
95 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
97 tinfo_key = &indexed::tinfo_static;
101 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
103 tinfo_key = &indexed::tinfo_static;
107 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)
109 tinfo_key = &indexed::tinfo_static;
113 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(not_symmetric())
115 seq.insert(seq.end(), v.begin(), v.end());
116 tinfo_key = &indexed::tinfo_static;
120 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
122 seq.insert(seq.end(), v.begin(), v.end());
123 tinfo_key = &indexed::tinfo_static;
127 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
129 tinfo_key = &indexed::tinfo_static;
132 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
134 tinfo_key = &indexed::tinfo_static;
137 indexed::indexed(const symmetry & symm, std::auto_ptr<exvector> vp) : inherited(vp), symtree(symm)
139 tinfo_key = &indexed::tinfo_static;
146 indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
148 if (!n.find_ex("symmetry", symtree, sym_lst)) {
149 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
151 n.find_unsigned("symmetry", symm);
160 symtree = not_symmetric();
163 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
167 void indexed::archive(archive_node &n) const
169 inherited::archive(n);
170 n.add_ex("symmetry", symtree);
173 DEFAULT_UNARCHIVE(indexed)
176 // functions overriding virtual functions from base classes
179 void indexed::printindices(const print_context & c, unsigned level) const
181 if (seq.size() > 1) {
183 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
185 if (is_a<print_latex>(c)) {
187 // TeX output: group by variance
189 bool covariant = true;
191 while (it != itend) {
192 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
193 if (first || cur_covariant != covariant) { // Variance changed
194 // The empty {} prevents indices from ending up on top of each other
197 covariant = cur_covariant;
213 while (it != itend) {
221 void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
223 if (precedence() <= level)
224 c.s << openbrace << '(';
226 seq[0].print(c, precedence());
228 printindices(c, level);
229 if (precedence() <= level)
230 c.s << ')' << closebrace;
233 void indexed::do_print(const print_context & c, unsigned level) const
235 print_indexed(c, "", "", level);
238 void indexed::do_print_latex(const print_latex & c, unsigned level) const
240 print_indexed(c, "{", "}", level);
243 void indexed::do_print_tree(const print_tree & c, unsigned level) const
245 c.s << std::string(level, ' ') << class_name() << " @" << this
246 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
247 << ", " << seq.size()-1 << " indices"
248 << ", symmetry=" << symtree << std::endl;
249 seq[0].print(c, level + c.delta_indent);
250 printindices(c, level + c.delta_indent);
253 bool indexed::info(unsigned inf) const
255 if (inf == info_flags::indexed) return true;
256 if (inf == info_flags::has_indices) return seq.size() > 1;
257 return inherited::info(inf);
260 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
261 bool operator() (const ex & e, unsigned inf) const {
262 return !(ex_to<idx>(e).get_value().info(inf));
266 bool indexed::all_index_values_are(unsigned inf) const
268 // No indices? Then no property can be fulfilled
273 return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
276 int indexed::compare_same_type(const basic & other) const
278 GINAC_ASSERT(is_a<indexed>(other));
279 return inherited::compare_same_type(other);
282 ex indexed::eval(int level) const
284 // First evaluate children, then we will end up here again
286 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
288 const ex &base = seq[0];
290 // If the base object is 0, the whole object is 0
294 // If the base object is a product, pull out the numeric factor
295 if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
297 ex f = ex_to<numeric>(base.op(base.nops() - 1));
299 return f * thiscontainer(v);
302 if(this->tinfo()==&indexed::tinfo_static && seq.size()==1)
305 // Canonicalize indices according to the symmetry properties
306 if (seq.size() > 2) {
308 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
309 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
310 if (sig != std::numeric_limits<int>::max()) {
311 // Something has changed while sorting indices, more evaluations later
314 return ex(sig) * thiscontainer(v);
318 // Let the class of the base object perform additional evaluations
319 return ex_to<basic>(base).eval_indexed(*this);
322 ex indexed::real_part() const
324 if(op(0).info(info_flags::real))
326 return real_part_function(*this).hold();
329 ex indexed::imag_part() const
331 if(op(0).info(info_flags::real))
333 return imag_part_function(*this).hold();
336 ex indexed::thiscontainer(const exvector & v) const
338 return indexed(ex_to<symmetry>(symtree), v);
341 ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
343 return indexed(ex_to<symmetry>(symtree), vp);
346 unsigned indexed::return_type() const
348 if(is_a<matrix>(op(0)))
349 return return_types::commutative;
351 return op(0).return_type();
354 ex indexed::expand(unsigned options) const
356 GINAC_ASSERT(seq.size() > 0);
358 if (options & expand_options::expand_indexed) {
359 ex newbase = seq[0].expand(options);
360 if (is_exactly_a<add>(newbase)) {
362 for (size_t i=0; i<newbase.nops(); i++) {
364 s[0] = newbase.op(i);
365 sum += thiscontainer(s).expand(options);
369 if (!are_ex_trivially_equal(newbase, seq[0])) {
372 return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
375 return inherited::expand(options);
379 // virtual functions which can be overridden by derived classes
385 // non-virtual functions in this class
388 /** Check whether all indices are of class idx and validate the symmetry
389 * tree. This function is used internally to make sure that all constructed
390 * indexed objects really carry indices and not some other classes. */
391 void indexed::validate() const
393 GINAC_ASSERT(seq.size() > 0);
394 exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
395 while (it != itend) {
397 throw(std::invalid_argument("indices of indexed object must be of type idx"));
401 if (!symtree.is_zero()) {
402 if (!is_exactly_a<symmetry>(symtree))
403 throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
404 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
408 /** Implementation of ex::diff() for an indexed object always returns 0.
411 ex indexed::derivative(const symbol & s) const
420 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
421 bool operator() (const ex &lh, const ex &rh) const
427 // Replacing the dimension might cause an error (e.g. with
428 // index classes that only work in a fixed number of dimensions)
429 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
436 /** Check whether two sorted index vectors are consistent (i.e. equal). */
437 static bool indices_consistent(const exvector & v1, const exvector & v2)
439 // Number of indices must be the same
440 if (v1.size() != v2.size())
443 return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
446 exvector indexed::get_indices() const
448 GINAC_ASSERT(seq.size() >= 1);
449 return exvector(seq.begin() + 1, seq.end());
452 exvector indexed::get_dummy_indices() const
454 exvector free_indices, dummy_indices;
455 find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
456 return dummy_indices;
459 exvector indexed::get_dummy_indices(const indexed & other) const
461 exvector indices = get_free_indices();
462 exvector other_indices = other.get_free_indices();
463 indices.insert(indices.end(), other_indices.begin(), other_indices.end());
464 exvector dummy_indices;
465 find_dummy_indices(indices, dummy_indices);
466 return dummy_indices;
469 bool indexed::has_dummy_index_for(const ex & i) const
471 exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
472 while (it != itend) {
473 if (is_dummy_pair(*it, i))
480 exvector indexed::get_free_indices() const
482 exvector free_indices, dummy_indices;
483 find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
487 exvector add::get_free_indices() const
489 exvector free_indices;
490 for (size_t i=0; i<nops(); i++) {
492 free_indices = op(i).get_free_indices();
494 exvector free_indices_of_term = op(i).get_free_indices();
495 if (!indices_consistent(free_indices, free_indices_of_term))
496 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
502 exvector mul::get_free_indices() const
504 // Concatenate free indices of all factors
506 for (size_t i=0; i<nops(); i++) {
507 exvector free_indices_of_factor = op(i).get_free_indices();
508 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
511 // And remove the dummy indices
512 exvector free_indices, dummy_indices;
513 find_free_and_dummy(un, free_indices, dummy_indices);
517 exvector ncmul::get_free_indices() const
519 // Concatenate free indices of all factors
521 for (size_t i=0; i<nops(); i++) {
522 exvector free_indices_of_factor = op(i).get_free_indices();
523 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
526 // And remove the dummy indices
527 exvector free_indices, dummy_indices;
528 find_free_and_dummy(un, free_indices, dummy_indices);
532 struct is_summation_idx : public std::unary_function<ex, bool> {
533 bool operator()(const ex & e)
535 return is_dummy_pair(e, e);
539 exvector integral::get_free_indices() const
541 if (a.get_free_indices().size() || b.get_free_indices().size())
542 throw (std::runtime_error("integral::get_free_indices: boundary values should not have free indices"));
543 return f.get_free_indices();
546 template<class T> size_t number_of_type(const exvector&v)
549 for(exvector::const_iterator i=v.begin(); i!=v.end(); ++i)
550 if(is_exactly_a<T>(*i))
555 /** Rename dummy indices in an expression.
557 * @param e Expression to work on
558 * @param local_dummy_indices The set of dummy indices that appear in the
560 * @param global_dummy_indices The set of dummy indices that have appeared
561 * before and which we would like to use in "e", too. This gets updated
563 template<class T> static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
565 size_t global_size = number_of_type<T>(global_dummy_indices),
566 local_size = number_of_type<T>(local_dummy_indices);
568 // Any local dummy indices at all?
572 if (global_size < local_size) {
574 // More local indices than we encountered before, add the new ones
576 size_t old_global_size = global_size;
577 int remaining = local_size - global_size;
578 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
579 while (it != itend && remaining > 0) {
580 if (is_exactly_a<T>(*it) && find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(idx_is_equal_ignore_dim(), *it)) == global_dummy_indices.end()) {
581 global_dummy_indices.push_back(*it);
588 // If this is the first set of local indices, do nothing
589 if (old_global_size == 0)
592 GINAC_ASSERT(local_size <= global_size);
594 // Construct vectors of index symbols
595 exvector local_syms, global_syms;
596 local_syms.reserve(local_size);
597 global_syms.reserve(local_size);
598 for (size_t i=0; local_syms.size()!=local_size; i++)
599 if(is_exactly_a<T>(local_dummy_indices[i]))
600 local_syms.push_back(local_dummy_indices[i].op(0));
601 shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
602 for (size_t i=0; global_syms.size()!=local_size; i++) // don't use more global symbols than necessary
603 if(is_exactly_a<T>(global_dummy_indices[i]))
604 global_syms.push_back(global_dummy_indices[i].op(0));
605 shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
607 // Remove common indices
608 exvector local_uniq, global_uniq;
609 set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
610 set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
612 // Replace remaining non-common local index symbols by global ones
613 if (local_uniq.empty())
616 while (global_uniq.size() > local_uniq.size())
617 global_uniq.pop_back();
618 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
622 /** Given a set of indices, extract those of class varidx. */
623 static void find_variant_indices(const exvector & v, exvector & variant_indices)
625 exvector::const_iterator it1, itend;
626 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
627 if (is_exactly_a<varidx>(*it1))
628 variant_indices.push_back(*it1);
632 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
635 * @param e Object to work on
636 * @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
637 * @param moved_indices The set of indices that have been repositioned (will be changed by this function)
638 * @return true if 'e' was changed */
639 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
641 bool something_changed = false;
643 // Find dummy symbols that occur twice in the same indexed object.
644 exvector local_var_dummies;
645 local_var_dummies.reserve(e.nops()/2);
646 for (size_t i=1; i<e.nops(); ++i) {
647 if (!is_a<varidx>(e.op(i)))
649 for (size_t j=i+1; j<e.nops(); ++j) {
650 if (is_dummy_pair(e.op(i), e.op(j))) {
651 local_var_dummies.push_back(e.op(i));
652 for (exvector::iterator k = variant_dummy_indices.begin();
653 k!=variant_dummy_indices.end(); ++k) {
654 if (e.op(i).op(0) == k->op(0)) {
655 variant_dummy_indices.erase(k);
664 // In the case where a dummy symbol occurs twice in the same indexed object
665 // we try all posibilities of raising/lowering and keep the least one in
666 // the sense of ex_is_less.
668 size_t numpossibs = 1 << local_var_dummies.size();
669 for (size_t i=0; i<numpossibs; ++i) {
671 for (size_t j=0; j<local_var_dummies.size(); ++j) {
674 ex curr_idx = local_var_dummies[j];
675 ex curr_toggle = ex_to<varidx>(curr_idx).toggle_variance();
676 m[curr_idx] = curr_toggle;
677 m[curr_toggle] = curr_idx;
679 try_e = e.subs(m, subs_options::no_pattern);
681 if(ex_is_less()(try_e, optimal_e))
683 something_changed = true;
688 if (!is_a<indexed>(e))
691 exvector seq = ex_to<indexed>(e).seq;
693 // If a dummy index is encountered for the first time in the
694 // product, pull it up, otherwise, pull it down
695 for (exvector::iterator it2 = seq.begin()+1, it2end = seq.end();
696 it2 != it2end; ++it2) {
697 if (!is_exactly_a<varidx>(*it2))
700 exvector::iterator vit, vitend;
701 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
702 if (it2->op(0).is_equal(vit->op(0))) {
703 if (ex_to<varidx>(*it2).is_covariant()) {
705 * N.B. we don't want to use
708 * *it2 == ex_to<varidx>(*it2).toggle_variance(),
709 * ex_to<varidx>(*it2).toggle_variance() == *it2
710 * ), subs_options::no_pattern);
712 * since this can trigger non-trivial repositioning of indices,
713 * e.g. due to non-trivial symmetry properties of e, thus
714 * invalidating iterators
716 *it2 = ex_to<varidx>(*it2).toggle_variance();
717 something_changed = true;
719 moved_indices.push_back(*vit);
720 variant_dummy_indices.erase(vit);
725 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
726 if (it2->op(0).is_equal(vit->op(0))) {
727 if (ex_to<varidx>(*it2).is_contravariant()) {
728 *it2 = ex_to<varidx>(*it2).toggle_variance();
729 something_changed = true;
738 if (something_changed)
739 e = ex_to<indexed>(e).thiscontainer(seq);
741 return something_changed;
744 /* Ordering that only compares the base expressions of indexed objects. */
745 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
746 bool operator() (const ex &lh, const ex &rh) const
748 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
752 /* An auxiliary function used by simplify_indexed() and expand_dummy_sum()
753 * It returns an exvector of factors from the supplied product */
754 static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
756 // Remember whether the product was commutative or noncommutative
757 // (because we chop it into factors and need to reassemble later)
758 non_commutative = is_exactly_a<ncmul>(e);
760 // Collect factors in an exvector, store squares twice
761 v.reserve(e.nops() * 2);
763 if (is_exactly_a<power>(e)) {
764 // We only get called for simple squares, split a^2 -> a*a
765 GINAC_ASSERT(e.op(1).is_equal(_ex2));
766 v.push_back(e.op(0));
767 v.push_back(e.op(0));
769 for (size_t i=0; i<e.nops(); i++) {
771 if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
772 v.push_back(f.op(0));
773 v.push_back(f.op(0));
774 } else if (is_exactly_a<ncmul>(f)) {
775 // Noncommutative factor found, split it as well
776 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
777 for (size_t j=0; j<f.nops(); j++)
778 v.push_back(f.op(j));
785 template<class T> ex idx_symmetrization(const ex& r,const exvector& local_dummy_indices)
786 { exvector dummy_syms;
787 dummy_syms.reserve(r.nops());
788 for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
789 if(is_exactly_a<T>(*it))
790 dummy_syms.push_back(it->op(0));
791 if(dummy_syms.size() < 2)
793 ex q=symmetrize(r, dummy_syms);
797 // Forward declaration needed in absence of friend injection, C.f. [namespace.memdef]:
798 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp);
800 /** Simplify product of indexed expressions (commutative, noncommutative and
801 * simple squares), return list of free indices. */
802 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
804 // Collect factors in an exvector
807 // Remember whether the product was commutative or noncommutative
808 // (because we chop it into factors and need to reassemble later)
809 bool non_commutative;
810 product_to_exvector(e, v, non_commutative);
812 // Perform contractions
813 bool something_changed = false;
814 GINAC_ASSERT(v.size() > 1);
815 exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
816 for (it1 = v.begin(); it1 != next_to_last; it1++) {
819 if (!is_a<indexed>(*it1))
822 bool first_noncommutative = (it1->return_type() != return_types::commutative);
824 // Indexed factor found, get free indices and look for contraction
826 exvector free1, dummy1;
827 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
829 exvector::iterator it2;
830 for (it2 = it1 + 1; it2 != itend; it2++) {
832 if (!is_a<indexed>(*it2))
835 bool second_noncommutative = (it2->return_type() != return_types::commutative);
837 // Find free indices of second factor and merge them with free
838 // indices of first factor
840 find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
841 un.insert(un.end(), free1.begin(), free1.end());
843 // Check whether the two factors share dummy indices
844 exvector free, dummy;
845 find_free_and_dummy(un, free, dummy);
846 size_t num_dummies = dummy.size();
847 if (num_dummies == 0)
850 // At least one dummy index, is it a defined scalar product?
851 bool contracted = false;
852 if (free.empty() && it1->nops()==2 && it2->nops()==2) {
854 ex dim = minimal_dim(
855 ex_to<idx>(it1->op(1)).get_dim(),
856 ex_to<idx>(it2->op(1)).get_dim()
859 // User-defined scalar product?
860 if (sp.is_defined(*it1, *it2, dim)) {
862 // Yes, substitute it
863 *it1 = sp.evaluate(*it1, *it2, dim);
865 goto contraction_done;
869 // Try to contract the first one with the second one
870 contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
873 // That didn't work; maybe the second object knows how to
874 // contract itself with the first one
875 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
879 if (first_noncommutative || second_noncommutative
880 || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
881 || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
882 || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
884 // One of the factors became a sum or product:
885 // re-expand expression and run again
886 // Non-commutative products are always re-expanded to give
887 // eval_ncmul() the chance to re-order and canonicalize
889 ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
890 return simplify_indexed(r, free_indices, dummy_indices, sp);
893 // Both objects may have new indices now or they might
894 // even not be indexed objects any more, so we have to
896 something_changed = true;
902 // Find free indices (concatenate them all and call find_free_and_dummy())
903 // and all dummy indices that appear
904 exvector un, individual_dummy_indices;
905 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
906 exvector free_indices_of_factor;
907 if (is_a<indexed>(*it1)) {
908 exvector dummy_indices_of_factor;
909 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);
910 individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
912 free_indices_of_factor = it1->get_free_indices();
913 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
915 exvector local_dummy_indices;
916 find_free_and_dummy(un, free_indices, local_dummy_indices);
917 local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
919 // Filter out the dummy indices with variance
920 exvector variant_dummy_indices;
921 find_variant_indices(local_dummy_indices, variant_dummy_indices);
923 // Any indices with variance present at all?
924 if (!variant_dummy_indices.empty()) {
926 // Yes, bring the product into a canonical order that only depends on
927 // the base expressions of indexed objects
928 if (!non_commutative)
929 std::sort(v.begin(), v.end(), ex_base_is_less());
931 exvector moved_indices;
933 // Iterate over all indexed objects in the product
934 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
935 if (!is_a<indexed>(*it1))
938 if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
939 something_changed = true;
944 if (something_changed)
945 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
949 // The result should be symmetric with respect to exchange of dummy
950 // indices, so if the symmetrization vanishes, the whole expression is
951 // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
952 ex q = idx_symmetrization<idx>(r, local_dummy_indices);
954 free_indices.clear();
957 q = idx_symmetrization<varidx>(q, local_dummy_indices);
959 free_indices.clear();
962 q = idx_symmetrization<spinidx>(q, local_dummy_indices);
964 free_indices.clear();
968 // Dummy index renaming
969 r = rename_dummy_indices<idx>(r, dummy_indices, local_dummy_indices);
970 r = rename_dummy_indices<varidx>(r, dummy_indices, local_dummy_indices);
971 r = rename_dummy_indices<spinidx>(r, dummy_indices, local_dummy_indices);
973 // Product of indexed object with a scalar?
974 if (is_exactly_a<mul>(r) && r.nops() == 2
975 && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
976 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
981 /** This structure stores the original and symmetrized versions of terms
982 * obtained during the simplification of sums. */
985 terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
987 ex orig; /**< original term */
988 ex symm; /**< symmtrized term */
991 class terminfo_is_less {
993 bool operator() (const terminfo & ti1, const terminfo & ti2) const
995 return (ti1.symm.compare(ti2.symm) < 0);
999 /** This structure stores the individual symmetrized terms obtained during
1000 * the simplification of sums. */
1003 symminfo() : num(0) {}
1005 symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
1007 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
1008 coeff = symmterm_.op(symmterm_.nops()-1);
1009 symmterm = symmterm_ / coeff;
1012 symmterm = symmterm_;
1016 ex symmterm; /**< symmetrized term */
1017 ex coeff; /**< coefficient of symmetrized term */
1018 ex orig; /**< original term */
1019 size_t num; /**< how many symmetrized terms resulted from the original term */
1022 class symminfo_is_less_by_symmterm {
1024 bool operator() (const symminfo & si1, const symminfo & si2) const
1026 return (si1.symmterm.compare(si2.symmterm) < 0);
1030 class symminfo_is_less_by_orig {
1032 bool operator() (const symminfo & si1, const symminfo & si2) const
1034 return (si1.orig.compare(si2.orig) < 0);
1038 bool hasindex(const ex &x, const ex &sym)
1040 if(is_a<idx>(x) && x.op(0)==sym)
1043 for(size_t i=0; i<x.nops(); ++i)
1044 if(hasindex(x.op(i), sym))
1049 /** Simplify indexed expression, return list of free indices. */
1050 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
1052 // Expand the expression
1053 ex e_expanded = e.expand();
1055 // Simplification of single indexed object: just find the free indices
1056 // and perform dummy index renaming/repositioning
1057 if (is_a<indexed>(e_expanded)) {
1059 // Find the dummy indices
1060 const indexed &i = ex_to<indexed>(e_expanded);
1061 exvector local_dummy_indices;
1062 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
1064 // Filter out the dummy indices with variance
1065 exvector variant_dummy_indices;
1066 find_variant_indices(local_dummy_indices, variant_dummy_indices);
1068 // Any indices with variance present at all?
1069 if (!variant_dummy_indices.empty()) {
1071 // Yes, reposition them
1072 exvector moved_indices;
1073 reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
1076 // Rename the dummy indices
1077 e_expanded = rename_dummy_indices<idx>(e_expanded, dummy_indices, local_dummy_indices);
1078 e_expanded = rename_dummy_indices<varidx>(e_expanded, dummy_indices, local_dummy_indices);
1079 e_expanded = rename_dummy_indices<spinidx>(e_expanded, dummy_indices, local_dummy_indices);
1083 // Simplification of sum = sum of simplifications, check consistency of
1084 // free indices in each term
1085 if (is_exactly_a<add>(e_expanded)) {
1088 free_indices.clear();
1090 for (size_t i=0; i<e_expanded.nops(); i++) {
1091 exvector free_indices_of_term;
1092 ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
1093 if (!term.is_zero()) {
1095 free_indices = free_indices_of_term;
1099 if (!indices_consistent(free_indices, free_indices_of_term)) {
1100 std::ostringstream s;
1101 s << "simplify_indexed: inconsistent indices in sum: ";
1102 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
1103 throw (std::runtime_error(s.str()));
1105 if (is_a<indexed>(sum) && is_a<indexed>(term))
1106 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
1113 // If the sum turns out to be zero, we are finished
1114 if (sum.is_zero()) {
1115 free_indices.clear();
1119 // More than one term and more than one dummy index?
1120 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
1121 if (num_terms_orig < 2 || dummy_indices.size() < 2)
1124 // Chop the sum into terms and symmetrize each one over the dummy
1126 std::vector<terminfo> terms;
1127 for (size_t i=0; i<sum.nops(); i++) {
1128 const ex & term = sum.op(i);
1129 exvector dummy_indices_of_term;
1130 dummy_indices_of_term.reserve(dummy_indices.size());
1131 for(exvector::iterator i=dummy_indices.begin(); i!=dummy_indices.end(); ++i)
1132 if(hasindex(term,i->op(0)))
1133 dummy_indices_of_term.push_back(*i);
1134 ex term_symm = idx_symmetrization<idx>(term, dummy_indices_of_term);
1135 term_symm = idx_symmetrization<varidx>(term_symm, dummy_indices_of_term);
1136 term_symm = idx_symmetrization<spinidx>(term_symm, dummy_indices_of_term);
1137 if (term_symm.is_zero())
1139 terms.push_back(terminfo(term, term_symm));
1142 // Sort by symmetrized terms
1143 std::sort(terms.begin(), terms.end(), terminfo_is_less());
1145 // Combine equal symmetrized terms
1146 std::vector<terminfo> terms_pass2;
1147 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1149 std::vector<terminfo>::const_iterator j = i + 1;
1150 while (j != terms.end() && j->symm == i->symm) {
1154 terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1158 // If there is only one term left, we are finished
1159 if (terms_pass2.size() == 1)
1160 return terms_pass2[0].orig;
1162 // Chop the symmetrized terms into subterms
1163 std::vector<symminfo> sy;
1164 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1165 if (is_exactly_a<add>(i->symm)) {
1166 size_t num = i->symm.nops();
1167 for (size_t j=0; j<num; j++)
1168 sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1170 sy.push_back(symminfo(i->symm, i->orig, 1));
1173 // Sort by symmetrized subterms
1174 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1176 // Combine equal symmetrized subterms
1177 std::vector<symminfo> sy_pass2;
1179 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1181 // Combine equal terms
1182 std::vector<symminfo>::const_iterator j = i + 1;
1183 if (j != sy.end() && j->symmterm == i->symmterm) {
1185 // More than one term, collect the coefficients
1186 ex coeff = i->coeff;
1187 while (j != sy.end() && j->symmterm == i->symmterm) {
1192 // Add combined term to result
1193 if (!coeff.is_zero())
1194 result.push_back(coeff * i->symmterm);
1198 // Single term, store for second pass
1199 sy_pass2.push_back(*i);
1205 // Were there any remaining terms that didn't get combined?
1206 if (sy_pass2.size() > 0) {
1208 // Yes, sort by their original terms
1209 std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1211 for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1213 // How many symmetrized terms of this original term are left?
1215 std::vector<symminfo>::const_iterator j = i + 1;
1216 while (j != sy_pass2.end() && j->orig == i->orig) {
1221 if (num == i->num) {
1223 // All terms left, then add the original term to the result
1224 result.push_back(i->orig);
1228 // Some terms were combined with others, add up the remaining symmetrized terms
1229 std::vector<symminfo>::const_iterator k;
1230 for (k=i; k!=j; k++)
1231 result.push_back(k->coeff * k->symmterm);
1238 // Add all resulting terms
1239 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1240 if (sum_symm.is_zero())
1241 free_indices.clear();
1245 // Simplification of products
1246 if (is_exactly_a<mul>(e_expanded)
1247 || is_exactly_a<ncmul>(e_expanded)
1248 || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1249 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1251 // Cannot do anything
1252 free_indices.clear();
1256 /** Simplify/canonicalize expression containing indexed objects. This
1257 * performs contraction of dummy indices where possible and checks whether
1258 * the free indices in sums are consistent.
1260 * @param options Simplification options (currently unused)
1261 * @return simplified expression */
1262 ex ex::simplify_indexed(unsigned options) const
1264 exvector free_indices, dummy_indices;
1266 return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1269 /** Simplify/canonicalize expression containing indexed objects. This
1270 * performs contraction of dummy indices where possible, checks whether
1271 * the free indices in sums are consistent, and automatically replaces
1272 * scalar products by known values if desired.
1274 * @param sp Scalar products to be replaced automatically
1275 * @param options Simplification options (currently unused)
1276 * @return simplified expression */
1277 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1279 exvector free_indices, dummy_indices;
1280 return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1283 /** Symmetrize expression over its free indices. */
1284 ex ex::symmetrize() const
1286 return GiNaC::symmetrize(*this, get_free_indices());
1289 /** Antisymmetrize expression over its free indices. */
1290 ex ex::antisymmetrize() const
1292 return GiNaC::antisymmetrize(*this, get_free_indices());
1295 /** Symmetrize expression by cyclic permutation over its free indices. */
1296 ex ex::symmetrize_cyclic() const
1298 return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1305 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1307 // If indexed, extract base objects
1308 ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1309 ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1311 // Enforce canonical order in pair
1312 if (s1.compare(s2) > 0) {
1321 bool spmapkey::operator==(const spmapkey &other) const
1323 if (!v1.is_equal(other.v1))
1325 if (!v2.is_equal(other.v2))
1327 if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1330 return dim.is_equal(other.dim);
1333 bool spmapkey::operator<(const spmapkey &other) const
1335 int cmp = v1.compare(other.v1);
1338 cmp = v2.compare(other.v2);
1342 // Objects are equal, now check dimensions
1343 if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1346 return dim.compare(other.dim) < 0;
1349 void spmapkey::debugprint() const
1351 std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1354 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1356 spm[spmapkey(v1, v2)] = sp;
1359 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1361 spm[spmapkey(v1, v2, dim)] = sp;
1364 void scalar_products::add_vectors(const lst & l, const ex & dim)
1366 // Add all possible pairs of products
1367 for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1368 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1369 add(*it1, *it2, *it1 * *it2);
1372 void scalar_products::clear()
1377 /** Check whether scalar product pair is defined. */
1378 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1380 return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1383 /** Return value of defined scalar product pair. */
1384 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1386 return spm.find(spmapkey(v1, v2, dim))->second;
1389 void scalar_products::debugprint() const
1391 std::cerr << "map size=" << spm.size() << std::endl;
1392 spmap::const_iterator i = spm.begin(), end = spm.end();
1394 const spmapkey & k = i->first;
1395 std::cerr << "item key=";
1397 std::cerr << ", value=" << i->second << std::endl;
1402 exvector get_all_dummy_indices_safely(const ex & e)
1404 if (is_a<indexed>(e))
1405 return ex_to<indexed>(e).get_dummy_indices();
1406 else if (is_a<power>(e) && e.op(1)==2) {
1407 return e.op(0).get_free_indices();
1409 else if (is_a<mul>(e) || is_a<ncmul>(e)) {
1411 exvector free_indices;
1412 for (std::size_t i = 0; i < e.nops(); ++i) {
1413 exvector dummies_of_factor = get_all_dummy_indices_safely(e.op(i));
1414 dummies.insert(dummies.end(), dummies_of_factor.begin(),
1415 dummies_of_factor.end());
1416 exvector free_of_factor = e.op(i).get_free_indices();
1417 free_indices.insert(free_indices.begin(), free_of_factor.begin(),
1418 free_of_factor.end());
1420 exvector free_out, dummy_out;
1421 find_free_and_dummy(free_indices.begin(), free_indices.end(), free_out,
1423 dummies.insert(dummies.end(), dummy_out.begin(), dummy_out.end());
1426 else if(is_a<add>(e)) {
1428 for(std::size_t i = 0; i < e.nops(); ++i) {
1429 exvector dummies_of_term = get_all_dummy_indices_safely(e.op(i));
1430 sort(dummies_of_term.begin(), dummies_of_term.end());
1432 set_union(result.begin(), result.end(), dummies_of_term.begin(),
1433 dummies_of_term.end(), std::back_inserter<exvector>(new_vec),
1435 result.swap(new_vec);
1442 /** Returns all dummy indices from the exvector */
1443 exvector get_all_dummy_indices(const ex & e)
1447 product_to_exvector(e, p, nc);
1448 exvector::const_iterator ip = p.begin(), ipend = p.end();
1450 while (ip != ipend) {
1451 if (is_a<indexed>(*ip)) {
1452 v1 = ex_to<indexed>(*ip).get_dummy_indices();
1453 v.insert(v.end(), v1.begin(), v1.end());
1454 exvector::const_iterator ip1 = ip+1;
1455 while (ip1 != ipend) {
1456 if (is_a<indexed>(*ip1)) {
1457 v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
1458 v.insert(v.end(), v1.begin(), v1.end());
1468 lst rename_dummy_indices_uniquely(const exvector & va, const exvector & vb)
1470 exvector common_indices;
1471 set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
1472 if (common_indices.empty()) {
1473 return lst(lst(), lst());
1475 exvector new_indices, old_indices;
1476 old_indices.reserve(2*common_indices.size());
1477 new_indices.reserve(2*common_indices.size());
1478 exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
1479 while (ip != ipend) {
1480 ex newsym=(new symbol)->setflag(status_flags::dynallocated);
1482 if(is_exactly_a<spinidx>(*ip))
1483 newidx = (new spinidx(newsym, ex_to<spinidx>(*ip).get_dim(),
1484 ex_to<spinidx>(*ip).is_covariant(),
1485 ex_to<spinidx>(*ip).is_dotted()))
1486 -> setflag(status_flags::dynallocated);
1487 else if (is_exactly_a<varidx>(*ip))
1488 newidx = (new varidx(newsym, ex_to<varidx>(*ip).get_dim(),
1489 ex_to<varidx>(*ip).is_covariant()))
1490 -> setflag(status_flags::dynallocated);
1492 newidx = (new idx(newsym, ex_to<idx>(*ip).get_dim()))
1493 -> setflag(status_flags::dynallocated);
1494 old_indices.push_back(*ip);
1495 new_indices.push_back(newidx);
1496 if(is_a<varidx>(*ip)) {
1497 old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
1498 new_indices.push_back(ex_to<varidx>(newidx).toggle_variance());
1502 return lst(lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end()));
1506 ex rename_dummy_indices_uniquely(const exvector & va, const exvector & vb, const ex & b)
1508 lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1509 return (indices_subs.op(0).nops()>0 ? b.subs(ex_to<lst>(indices_subs.op(0)), ex_to<lst>(indices_subs.op(1)), subs_options::no_pattern|subs_options::no_index_renaming) : b);
1512 ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
1514 exvector va = get_all_dummy_indices_safely(a);
1515 if (va.size() > 0) {
1516 exvector vb = get_all_dummy_indices_safely(b);
1517 if (vb.size() > 0) {
1518 sort(va.begin(), va.end(), ex_is_less());
1519 sort(vb.begin(), vb.end(), ex_is_less());
1520 lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1521 if (indices_subs.op(0).nops() > 0)
1522 return b.subs(ex_to<lst>(indices_subs.op(0)), ex_to<lst>(indices_subs.op(1)), subs_options::no_pattern|subs_options::no_index_renaming);
1528 ex rename_dummy_indices_uniquely(exvector & va, const ex & b, bool modify_va)
1530 if (va.size() > 0) {
1531 exvector vb = get_all_dummy_indices_safely(b);
1532 if (vb.size() > 0) {
1533 sort(vb.begin(), vb.end(), ex_is_less());
1534 lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1535 if (indices_subs.op(0).nops() > 0) {
1537 for (lst::const_iterator i = ex_to<lst>(indices_subs.op(1)).begin(); i != ex_to<lst>(indices_subs.op(1)).end(); ++i)
1539 exvector uncommon_indices;
1540 set_difference(vb.begin(), vb.end(), indices_subs.op(0).begin(), indices_subs.op(0).end(), std::back_insert_iterator<exvector>(uncommon_indices), ex_is_less());
1541 exvector::const_iterator ip = uncommon_indices.begin(), ipend = uncommon_indices.end();
1542 while (ip != ipend) {
1546 sort(va.begin(), va.end(), ex_is_less());
1548 return b.subs(ex_to<lst>(indices_subs.op(0)), ex_to<lst>(indices_subs.op(1)), subs_options::no_pattern|subs_options::no_index_renaming);
1555 ex expand_dummy_sum(const ex & e, bool subs_idx)
1557 ex e_expanded = e.expand();
1558 pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
1559 if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
1560 return e_expanded.map(fcn);
1561 } else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded) || is_a<indexed>(e_expanded)) {
1563 if (is_a<indexed>(e_expanded))
1564 v = ex_to<indexed>(e_expanded).get_dummy_indices();
1566 v = get_all_dummy_indices(e_expanded);
1567 ex result = e_expanded;
1568 for(exvector::const_iterator it=v.begin(); it!=v.end(); ++it) {
1570 if (ex_to<idx>(nu).get_dim().info(info_flags::nonnegint)) {
1571 int idim = ex_to<numeric>(ex_to<idx>(nu).get_dim()).to_int();
1573 for (int i=0; i < idim; i++) {
1574 if (subs_idx && is_a<varidx>(nu)) {
1575 ex other = ex_to<varidx>(nu).toggle_variance();
1576 en += result.subs(lst(
1578 other == idx(i, idim)
1581 en += result.subs( nu.op(0) == i );
1593 } // namespace GiNaC