3 * Implementation of GiNaC's indexed expressions. */
6 * GiNaC Copyright (C) 1999-2015 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
29 #include "relational.h"
31 #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())
64 indexed::indexed(const ex & b) : inherited{b}, symtree(not_symmetric())
69 indexed::indexed(const ex & b, const ex & i1) : inherited{b, i1}, symtree(not_symmetric())
74 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited{b, i1, i2}, symtree(not_symmetric())
79 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited{b, i1, i2, i3}, symtree(not_symmetric())
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())
89 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited{b, i1, i2}, symtree(symm)
94 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited{b, i1, i2, i3}, symtree(symm)
99 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)
104 indexed::indexed(const ex & b, const exvector & v) : inherited{b}, symtree(not_symmetric())
106 seq.insert(seq.end(), v.begin(), v.end());
110 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited{b}, symtree(symm)
112 seq.insert(seq.end(), v.begin(), v.end());
116 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
120 indexed::indexed(const symmetry & symm, const exvector & v) : inherited(v), symtree(symm)
124 indexed::indexed(const symmetry & symm, exvector && v) : inherited(std::move(v)), symtree(symm)
132 void indexed::read_archive(const archive_node &n, lst &sym_lst)
134 inherited::read_archive(n, sym_lst);
135 if (!n.find_ex("symmetry", symtree, sym_lst)) {
136 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
138 n.find_unsigned("symmetry", symm);
147 symtree = not_symmetric();
150 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
153 GINAC_BIND_UNARCHIVER(indexed);
155 void indexed::archive(archive_node &n) const
157 inherited::archive(n);
158 n.add_ex("symmetry", symtree);
162 // functions overriding virtual functions from base classes
165 void indexed::printindices(const print_context & c, unsigned level) const
167 if (seq.size() > 1) {
169 auto it = seq.begin() + 1, itend = seq.end();
171 if (is_a<print_latex>(c)) {
173 // TeX output: group by variance
175 bool covariant = true;
177 while (it != itend) {
178 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
179 if (first || cur_covariant != covariant) { // Variance changed
180 // The empty {} prevents indices from ending up on top of each other
183 covariant = cur_covariant;
199 while (it != itend) {
207 void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
209 if (precedence() <= level)
210 c.s << openbrace << '(';
212 seq[0].print(c, precedence());
214 printindices(c, level);
215 if (precedence() <= level)
216 c.s << ')' << closebrace;
219 void indexed::do_print(const print_context & c, unsigned level) const
221 print_indexed(c, "", "", level);
224 void indexed::do_print_latex(const print_latex & c, unsigned level) const
226 print_indexed(c, "{", "}", level);
229 void indexed::do_print_tree(const print_tree & c, unsigned level) const
231 c.s << std::string(level, ' ') << class_name() << " @" << this
232 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
233 << ", " << seq.size()-1 << " indices"
234 << ", symmetry=" << symtree << std::endl;
235 seq[0].print(c, level + c.delta_indent);
236 printindices(c, level + c.delta_indent);
239 bool indexed::info(unsigned inf) const
241 if (inf == info_flags::indexed) return true;
242 if (inf == info_flags::has_indices) return seq.size() > 1;
243 return inherited::info(inf);
246 bool indexed::all_index_values_are(unsigned inf) const
248 // No indices? Then no property can be fulfilled
253 return find_if(seq.begin() + 1, seq.end(),
254 [inf](const ex & e) { return !(ex_to<idx>(e).get_value().info(inf)); }) == seq.end();
257 int indexed::compare_same_type(const basic & other) const
259 GINAC_ASSERT(is_a<indexed>(other));
260 return inherited::compare_same_type(other);
263 ex indexed::eval(int level) const
265 // First evaluate children, then we will end up here again
267 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
269 const ex &base = seq[0];
271 // If the base object is 0, the whole object is 0
275 // If the base object is a product, pull out the numeric factor
276 if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
278 ex f = ex_to<numeric>(base.op(base.nops() - 1));
280 return f * thiscontainer(v);
283 if((typeid(*this) == typeid(indexed)) && seq.size()==1)
286 // Canonicalize indices according to the symmetry properties
287 if (seq.size() > 2) {
289 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
290 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
291 if (sig != std::numeric_limits<int>::max()) {
292 // Something has changed while sorting indices, more evaluations later
295 return ex(sig) * thiscontainer(v);
299 // Let the class of the base object perform additional evaluations
300 return ex_to<basic>(base).eval_indexed(*this);
303 ex indexed::real_part() const
305 if(op(0).info(info_flags::real))
307 return real_part_function(*this).hold();
310 ex indexed::imag_part() const
312 if(op(0).info(info_flags::real))
314 return imag_part_function(*this).hold();
317 ex indexed::thiscontainer(const exvector & v) const
319 return indexed(ex_to<symmetry>(symtree), v);
322 ex indexed::thiscontainer(exvector && v) const
324 return indexed(ex_to<symmetry>(symtree), std::move(v));
327 unsigned indexed::return_type() const
329 if(is_a<matrix>(op(0)))
330 return return_types::commutative;
332 return op(0).return_type();
335 ex indexed::expand(unsigned options) const
337 GINAC_ASSERT(seq.size() > 0);
339 if (options & expand_options::expand_indexed) {
340 ex newbase = seq[0].expand(options);
341 if (is_exactly_a<add>(newbase)) {
343 for (size_t i=0; i<newbase.nops(); i++) {
345 s[0] = newbase.op(i);
346 sum += thiscontainer(s).expand(options);
350 if (!are_ex_trivially_equal(newbase, seq[0])) {
353 return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
356 return inherited::expand(options);
360 // virtual functions which can be overridden by derived classes
366 // non-virtual functions in this class
369 /** Check whether all indices are of class idx and validate the symmetry
370 * tree. This function is used internally to make sure that all constructed
371 * indexed objects really carry indices and not some other classes. */
372 void indexed::validate() const
374 GINAC_ASSERT(seq.size() > 0);
375 auto it = seq.begin() + 1, itend = seq.end();
376 while (it != itend) {
378 throw(std::invalid_argument("indices of indexed object must be of type idx"));
382 if (!symtree.is_zero()) {
383 if (!is_exactly_a<symmetry>(symtree))
384 throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
385 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
389 /** Implementation of ex::diff() for an indexed object always returns 0.
392 ex indexed::derivative(const symbol & s) const
401 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
402 bool operator() (const ex &lh, const ex &rh) const
408 // Replacing the dimension might cause an error (e.g. with
409 // index classes that only work in a fixed number of dimensions)
410 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
417 /** Check whether two sorted index vectors are consistent (i.e. equal). */
418 static bool indices_consistent(const exvector & v1, const exvector & v2)
420 // Number of indices must be the same
421 if (v1.size() != v2.size())
424 return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
427 exvector indexed::get_indices() const
429 GINAC_ASSERT(seq.size() >= 1);
430 return exvector(seq.begin() + 1, seq.end());
433 exvector indexed::get_dummy_indices() const
435 exvector free_indices, dummy_indices;
436 find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
437 return dummy_indices;
440 exvector indexed::get_dummy_indices(const indexed & other) const
442 exvector indices = get_free_indices();
443 exvector other_indices = other.get_free_indices();
444 indices.insert(indices.end(), other_indices.begin(), other_indices.end());
445 exvector dummy_indices;
446 find_dummy_indices(indices, dummy_indices);
447 return dummy_indices;
450 bool indexed::has_dummy_index_for(const ex & i) const
452 auto it = seq.begin() + 1, itend = seq.end();
453 while (it != itend) {
454 if (is_dummy_pair(*it, i))
461 exvector indexed::get_free_indices() const
463 exvector free_indices, dummy_indices;
464 find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
468 exvector add::get_free_indices() const
470 exvector free_indices;
471 for (size_t i=0; i<nops(); i++) {
473 free_indices = op(i).get_free_indices();
475 exvector free_indices_of_term = op(i).get_free_indices();
476 if (!indices_consistent(free_indices, free_indices_of_term))
477 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
483 exvector mul::get_free_indices() const
485 // Concatenate free indices of all factors
487 for (size_t i=0; i<nops(); i++) {
488 exvector free_indices_of_factor = op(i).get_free_indices();
489 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
492 // And remove the dummy indices
493 exvector free_indices, dummy_indices;
494 find_free_and_dummy(un, free_indices, dummy_indices);
498 exvector ncmul::get_free_indices() const
500 // Concatenate free indices of all factors
502 for (size_t i=0; i<nops(); i++) {
503 exvector free_indices_of_factor = op(i).get_free_indices();
504 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
507 // And remove the dummy indices
508 exvector free_indices, dummy_indices;
509 find_free_and_dummy(un, free_indices, dummy_indices);
513 struct is_summation_idx : public std::unary_function<ex, bool> {
514 bool operator()(const ex & e)
516 return is_dummy_pair(e, e);
520 exvector integral::get_free_indices() const
522 if (a.get_free_indices().size() || b.get_free_indices().size())
523 throw (std::runtime_error("integral::get_free_indices: boundary values should not have free indices"));
524 return f.get_free_indices();
527 template<class T> size_t number_of_type(const exvector&v)
531 if (is_exactly_a<T>(it))
536 /** Rename dummy indices in an expression.
538 * @param e Expression to work on
539 * @param local_dummy_indices The set of dummy indices that appear in the
541 * @param global_dummy_indices The set of dummy indices that have appeared
542 * before and which we would like to use in "e", too. This gets updated
544 template<class T> static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
546 size_t global_size = number_of_type<T>(global_dummy_indices),
547 local_size = number_of_type<T>(local_dummy_indices);
549 // Any local dummy indices at all?
553 if (global_size < local_size) {
555 // More local indices than we encountered before, add the new ones
557 size_t old_global_size = global_size;
558 int remaining = local_size - global_size;
559 auto it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
560 while (it != itend && remaining > 0) {
561 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()) {
562 global_dummy_indices.push_back(*it);
569 // If this is the first set of local indices, do nothing
570 if (old_global_size == 0)
573 GINAC_ASSERT(local_size <= global_size);
575 // Construct vectors of index symbols
576 exvector local_syms, global_syms;
577 local_syms.reserve(local_size);
578 global_syms.reserve(local_size);
579 for (size_t i=0; local_syms.size()!=local_size; i++)
580 if(is_exactly_a<T>(local_dummy_indices[i]))
581 local_syms.push_back(local_dummy_indices[i].op(0));
582 shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
583 for (size_t i=0; global_syms.size()!=local_size; i++) // don't use more global symbols than necessary
584 if(is_exactly_a<T>(global_dummy_indices[i]))
585 global_syms.push_back(global_dummy_indices[i].op(0));
586 shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
588 // Remove common indices
589 exvector local_uniq, global_uniq;
590 set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
591 set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
593 // Replace remaining non-common local index symbols by global ones
594 if (local_uniq.empty())
597 while (global_uniq.size() > local_uniq.size())
598 global_uniq.pop_back();
599 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
603 /** Given a set of indices, extract those of class varidx. */
604 static void find_variant_indices(const exvector & v, exvector & variant_indices)
606 exvector::const_iterator it1, itend;
607 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
608 if (is_exactly_a<varidx>(*it1))
609 variant_indices.push_back(*it1);
613 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
616 * @param e Object to work on
617 * @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
618 * @param moved_indices The set of indices that have been repositioned (will be changed by this function)
619 * @return true if 'e' was changed */
620 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
622 bool something_changed = false;
624 // Find dummy symbols that occur twice in the same indexed object.
625 exvector local_var_dummies;
626 local_var_dummies.reserve(e.nops()/2);
627 for (size_t i=1; i<e.nops(); ++i) {
628 if (!is_a<varidx>(e.op(i)))
630 for (size_t j=i+1; j<e.nops(); ++j) {
631 if (is_dummy_pair(e.op(i), e.op(j))) {
632 local_var_dummies.push_back(e.op(i));
633 for (auto k = variant_dummy_indices.begin(); k!=variant_dummy_indices.end(); ++k) {
634 if (e.op(i).op(0) == k->op(0)) {
635 variant_dummy_indices.erase(k);
644 // In the case where a dummy symbol occurs twice in the same indexed object
645 // we try all possibilities of raising/lowering and keep the least one in
646 // the sense of ex_is_less.
648 size_t numpossibs = 1 << local_var_dummies.size();
649 for (size_t i=0; i<numpossibs; ++i) {
651 for (size_t j=0; j<local_var_dummies.size(); ++j) {
654 ex curr_idx = local_var_dummies[j];
655 ex curr_toggle = ex_to<varidx>(curr_idx).toggle_variance();
656 m[curr_idx] = curr_toggle;
657 m[curr_toggle] = curr_idx;
659 try_e = e.subs(m, subs_options::no_pattern);
661 if(ex_is_less()(try_e, optimal_e))
663 something_changed = true;
668 if (!is_a<indexed>(e))
671 exvector seq = ex_to<indexed>(e).seq;
673 // If a dummy index is encountered for the first time in the
674 // product, pull it up, otherwise, pull it down
675 for (auto it2 = seq.begin()+1, it2end = seq.end(); it2 != it2end; ++it2) {
676 if (!is_exactly_a<varidx>(*it2))
679 exvector::iterator vit, vitend;
680 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
681 if (it2->op(0).is_equal(vit->op(0))) {
682 if (ex_to<varidx>(*it2).is_covariant()) {
684 * N.B. we don't want to use
687 * *it2 == ex_to<varidx>(*it2).toggle_variance(),
688 * ex_to<varidx>(*it2).toggle_variance() == *it2
689 * }, subs_options::no_pattern);
691 * since this can trigger non-trivial repositioning of indices,
692 * e.g. due to non-trivial symmetry properties of e, thus
693 * invalidating iterators
695 *it2 = ex_to<varidx>(*it2).toggle_variance();
696 something_changed = true;
698 moved_indices.push_back(*vit);
699 variant_dummy_indices.erase(vit);
704 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
705 if (it2->op(0).is_equal(vit->op(0))) {
706 if (ex_to<varidx>(*it2).is_contravariant()) {
707 *it2 = ex_to<varidx>(*it2).toggle_variance();
708 something_changed = true;
717 if (something_changed)
718 e = ex_to<indexed>(e).thiscontainer(seq);
720 return something_changed;
723 /* Ordering that only compares the base expressions of indexed objects. */
724 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
725 bool operator() (const ex &lh, const ex &rh) const
727 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
731 /* An auxiliary function used by simplify_indexed() and expand_dummy_sum()
732 * It returns an exvector of factors from the supplied product */
733 static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
735 // Remember whether the product was commutative or noncommutative
736 // (because we chop it into factors and need to reassemble later)
737 non_commutative = is_exactly_a<ncmul>(e);
739 // Collect factors in an exvector, store squares twice
740 v.reserve(e.nops() * 2);
742 if (is_exactly_a<power>(e)) {
743 // We only get called for simple squares, split a^2 -> a*a
744 GINAC_ASSERT(e.op(1).is_equal(_ex2));
745 v.push_back(e.op(0));
746 v.push_back(e.op(0));
748 for (size_t i=0; i<e.nops(); i++) {
750 if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
751 v.push_back(f.op(0));
752 v.push_back(f.op(0));
753 } else if (is_exactly_a<ncmul>(f)) {
754 // Noncommutative factor found, split it as well
755 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
756 for (size_t j=0; j<f.nops(); j++)
757 v.push_back(f.op(j));
764 template<class T> ex idx_symmetrization(const ex& r,const exvector& local_dummy_indices)
765 { exvector dummy_syms;
766 dummy_syms.reserve(r.nops());
767 for (auto & it : local_dummy_indices)
768 if(is_exactly_a<T>(it))
769 dummy_syms.push_back(it.op(0));
770 if(dummy_syms.size() < 2)
772 ex q=symmetrize(r, dummy_syms);
776 // Forward declaration needed in absence of friend injection, C.f. [namespace.memdef]:
777 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp);
779 /** Simplify product of indexed expressions (commutative, noncommutative and
780 * simple squares), return list of free indices. */
781 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
783 // Collect factors in an exvector
786 // Remember whether the product was commutative or noncommutative
787 // (because we chop it into factors and need to reassemble later)
788 bool non_commutative;
789 product_to_exvector(e, v, non_commutative);
791 // Perform contractions
792 bool something_changed = false;
793 bool has_nonsymmetric = false;
794 GINAC_ASSERT(v.size() > 1);
795 exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
796 for (it1 = v.begin(); it1 != next_to_last; it1++) {
799 if (!is_a<indexed>(*it1))
802 bool first_noncommutative = (it1->return_type() != return_types::commutative);
803 bool first_nonsymmetric = ex_to<symmetry>(ex_to<indexed>(*it1).get_symmetry()).has_nonsymmetric();
805 // Indexed factor found, get free indices and look for contraction
807 exvector free1, dummy1;
808 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
810 exvector::iterator it2;
811 for (it2 = it1 + 1; it2 != itend; it2++) {
813 if (!is_a<indexed>(*it2))
816 bool second_noncommutative = (it2->return_type() != return_types::commutative);
818 // Find free indices of second factor and merge them with free
819 // indices of first factor
821 find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
822 un.insert(un.end(), free1.begin(), free1.end());
824 // Check whether the two factors share dummy indices
825 exvector free, dummy;
826 find_free_and_dummy(un, free, dummy);
827 size_t num_dummies = dummy.size();
828 if (num_dummies == 0)
831 // At least one dummy index, is it a defined scalar product?
832 bool contracted = false;
833 if (free.empty() && it1->nops()==2 && it2->nops()==2) {
835 ex dim = minimal_dim(
836 ex_to<idx>(it1->op(1)).get_dim(),
837 ex_to<idx>(it2->op(1)).get_dim()
840 // User-defined scalar product?
841 if (sp.is_defined(*it1, *it2, dim)) {
843 // Yes, substitute it
844 *it1 = sp.evaluate(*it1, *it2, dim);
846 goto contraction_done;
850 // Try to contract the first one with the second one
851 contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
854 // That didn't work; maybe the second object knows how to
855 // contract itself with the first one
856 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
860 if (first_noncommutative || second_noncommutative
861 || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
862 || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
863 || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
865 // One of the factors became a sum or product:
866 // re-expand expression and run again
867 // Non-commutative products are always re-expanded to give
868 // eval_ncmul() the chance to re-order and canonicalize
870 bool is_a_product = (is_exactly_a<mul>(*it1) || is_exactly_a<ncmul>(*it1)) &&
871 (is_exactly_a<mul>(*it2) || is_exactly_a<ncmul>(*it2));
872 ex r = (non_commutative ? ex(ncmul(std::move(v))) : ex(mul(std::move(v))));
874 // If new expression is a product we can call this function again,
875 // otherwise we need to pass argument to simplify_indexed() to be expanded
877 return simplify_indexed_product(r, free_indices, dummy_indices, sp);
879 return simplify_indexed(r, free_indices, dummy_indices, sp);
882 // Both objects may have new indices now or they might
883 // even not be indexed objects any more, so we have to
885 something_changed = true;
888 else if (!has_nonsymmetric &&
889 (first_nonsymmetric ||
890 ex_to<symmetry>(ex_to<indexed>(*it2).get_symmetry()).has_nonsymmetric())) {
891 has_nonsymmetric = true;
896 // Find free indices (concatenate them all and call find_free_and_dummy())
897 // and all dummy indices that appear
898 exvector un, individual_dummy_indices;
899 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
900 exvector free_indices_of_factor;
901 if (is_a<indexed>(*it1)) {
902 exvector dummy_indices_of_factor;
903 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);
904 individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
906 free_indices_of_factor = it1->get_free_indices();
907 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
909 exvector local_dummy_indices;
910 find_free_and_dummy(un, free_indices, local_dummy_indices);
911 local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
913 // Filter out the dummy indices with variance
914 exvector variant_dummy_indices;
915 find_variant_indices(local_dummy_indices, variant_dummy_indices);
917 // Any indices with variance present at all?
918 if (!variant_dummy_indices.empty()) {
920 // Yes, bring the product into a canonical order that only depends on
921 // the base expressions of indexed objects
922 if (!non_commutative)
923 std::sort(v.begin(), v.end(), ex_base_is_less());
925 exvector moved_indices;
927 // Iterate over all indexed objects in the product
928 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
929 if (!is_a<indexed>(*it1))
932 if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
933 something_changed = true;
938 if (something_changed)
939 r = non_commutative ? ex(ncmul(std::move(v))) : ex(mul(std::move(v)));
943 // The result should be symmetric with respect to exchange of dummy
944 // indices, so if the symmetrization vanishes, the whole expression is
945 // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
946 if (has_nonsymmetric) {
947 ex q = idx_symmetrization<idx>(r, local_dummy_indices);
949 free_indices.clear();
952 q = idx_symmetrization<varidx>(q, local_dummy_indices);
954 free_indices.clear();
957 q = idx_symmetrization<spinidx>(q, local_dummy_indices);
959 free_indices.clear();
964 // Dummy index renaming
965 r = rename_dummy_indices<idx>(r, dummy_indices, local_dummy_indices);
966 r = rename_dummy_indices<varidx>(r, dummy_indices, local_dummy_indices);
967 r = rename_dummy_indices<spinidx>(r, dummy_indices, local_dummy_indices);
969 // Product of indexed object with a scalar?
970 if (is_exactly_a<mul>(r) && r.nops() == 2
971 && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
972 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
977 /** This structure stores the original and symmetrized versions of terms
978 * obtained during the simplification of sums. */
981 terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
983 ex orig; /**< original term */
984 ex symm; /**< symmetrized term */
987 class terminfo_is_less {
989 bool operator() (const terminfo & ti1, const terminfo & ti2) const
991 return (ti1.symm.compare(ti2.symm) < 0);
995 /** This structure stores the individual symmetrized terms obtained during
996 * the simplification of sums. */
999 symminfo() : num(0) {}
1001 symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
1003 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
1004 coeff = symmterm_.op(symmterm_.nops()-1);
1005 symmterm = symmterm_ / coeff;
1008 symmterm = symmterm_;
1012 ex symmterm; /**< symmetrized term */
1013 ex coeff; /**< coefficient of symmetrized term */
1014 ex orig; /**< original term */
1015 size_t num; /**< how many symmetrized terms resulted from the original term */
1018 class symminfo_is_less_by_symmterm {
1020 bool operator() (const symminfo & si1, const symminfo & si2) const
1022 return (si1.symmterm.compare(si2.symmterm) < 0);
1026 class symminfo_is_less_by_orig {
1028 bool operator() (const symminfo & si1, const symminfo & si2) const
1030 return (si1.orig.compare(si2.orig) < 0);
1034 bool hasindex(const ex &x, const ex &sym)
1036 if(is_a<idx>(x) && x.op(0)==sym)
1039 for(size_t i=0; i<x.nops(); ++i)
1040 if(hasindex(x.op(i), sym))
1045 /** Simplify indexed expression, return list of free indices. */
1046 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
1048 // Expand the expression
1049 ex e_expanded = e.expand();
1051 // Simplification of single indexed object: just find the free indices
1052 // and perform dummy index renaming/repositioning
1053 if (is_a<indexed>(e_expanded)) {
1055 // Find the dummy indices
1056 const indexed &i = ex_to<indexed>(e_expanded);
1057 exvector local_dummy_indices;
1058 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
1060 // Filter out the dummy indices with variance
1061 exvector variant_dummy_indices;
1062 find_variant_indices(local_dummy_indices, variant_dummy_indices);
1064 // Any indices with variance present at all?
1065 if (!variant_dummy_indices.empty()) {
1067 // Yes, reposition them
1068 exvector moved_indices;
1069 reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
1072 // Rename the dummy indices
1073 e_expanded = rename_dummy_indices<idx>(e_expanded, dummy_indices, local_dummy_indices);
1074 e_expanded = rename_dummy_indices<varidx>(e_expanded, dummy_indices, local_dummy_indices);
1075 e_expanded = rename_dummy_indices<spinidx>(e_expanded, dummy_indices, local_dummy_indices);
1079 // Simplification of sum = sum of simplifications, check consistency of
1080 // free indices in each term
1081 if (is_exactly_a<add>(e_expanded)) {
1084 free_indices.clear();
1086 for (size_t i=0; i<e_expanded.nops(); i++) {
1087 exvector free_indices_of_term;
1088 ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
1089 if (!term.is_zero()) {
1091 free_indices = free_indices_of_term;
1095 if (!indices_consistent(free_indices, free_indices_of_term)) {
1096 std::ostringstream s;
1097 s << "simplify_indexed: inconsistent indices in sum: ";
1098 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
1099 throw (std::runtime_error(s.str()));
1101 if (is_a<indexed>(sum) && is_a<indexed>(term))
1102 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
1109 // If the sum turns out to be zero, we are finished
1110 if (sum.is_zero()) {
1111 free_indices.clear();
1115 // More than one term and more than one dummy index?
1116 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
1117 if (num_terms_orig < 2 || dummy_indices.size() < 2)
1120 // Chop the sum into terms and symmetrize each one over the dummy
1122 std::vector<terminfo> terms;
1123 for (size_t i=0; i<sum.nops(); i++) {
1124 const ex & term = sum.op(i);
1125 exvector dummy_indices_of_term;
1126 dummy_indices_of_term.reserve(dummy_indices.size());
1127 for (auto & i : dummy_indices)
1128 if (hasindex(term,i.op(0)))
1129 dummy_indices_of_term.push_back(i);
1130 ex term_symm = idx_symmetrization<idx>(term, dummy_indices_of_term);
1131 term_symm = idx_symmetrization<varidx>(term_symm, dummy_indices_of_term);
1132 term_symm = idx_symmetrization<spinidx>(term_symm, dummy_indices_of_term);
1133 if (term_symm.is_zero())
1135 terms.push_back(terminfo(term, term_symm));
1138 // Sort by symmetrized terms
1139 std::sort(terms.begin(), terms.end(), terminfo_is_less());
1141 // Combine equal symmetrized terms
1142 std::vector<terminfo> terms_pass2;
1143 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1146 while (j != terms.end() && j->symm == i->symm) {
1150 terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1154 // If there is only one term left, we are finished
1155 if (terms_pass2.size() == 1)
1156 return terms_pass2[0].orig;
1158 // Chop the symmetrized terms into subterms
1159 std::vector<symminfo> sy;
1160 for (auto & i : terms_pass2) {
1161 if (is_exactly_a<add>(i.symm)) {
1162 size_t num = i.symm.nops();
1163 for (size_t j=0; j<num; j++)
1164 sy.push_back(symminfo(i.symm.op(j), i.orig, num));
1166 sy.push_back(symminfo(i.symm, i.orig, 1));
1169 // Sort by symmetrized subterms
1170 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1172 // Combine equal symmetrized subterms
1173 std::vector<symminfo> sy_pass2;
1175 for (auto i=sy.begin(); i!=sy.end(); ) {
1177 // Combine equal terms
1179 if (j != sy.end() && j->symmterm == i->symmterm) {
1181 // More than one term, collect the coefficients
1182 ex coeff = i->coeff;
1183 while (j != sy.end() && j->symmterm == i->symmterm) {
1188 // Add combined term to result
1189 if (!coeff.is_zero())
1190 result.push_back(coeff * i->symmterm);
1194 // Single term, store for second pass
1195 sy_pass2.push_back(*i);
1201 // Were there any remaining terms that didn't get combined?
1202 if (sy_pass2.size() > 0) {
1204 // Yes, sort by their original terms
1205 std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1207 for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1209 // How many symmetrized terms of this original term are left?
1212 while (j != sy_pass2.end() && j->orig == i->orig) {
1217 if (num == i->num) {
1219 // All terms left, then add the original term to the result
1220 result.push_back(i->orig);
1224 // Some terms were combined with others, add up the remaining symmetrized terms
1225 std::vector<symminfo>::const_iterator k;
1226 for (k=i; k!=j; k++)
1227 result.push_back(k->coeff * k->symmterm);
1234 // Add all resulting terms
1235 ex sum_symm = dynallocate<add>(result);
1236 if (sum_symm.is_zero())
1237 free_indices.clear();
1241 // Simplification of products
1242 if (is_exactly_a<mul>(e_expanded)
1243 || is_exactly_a<ncmul>(e_expanded)
1244 || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1245 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1247 // Cannot do anything
1248 free_indices.clear();
1252 /** Simplify/canonicalize expression containing indexed objects. This
1253 * performs contraction of dummy indices where possible and checks whether
1254 * the free indices in sums are consistent.
1256 * @param options Simplification options (currently unused)
1257 * @return simplified expression */
1258 ex ex::simplify_indexed(unsigned options) const
1260 exvector free_indices, dummy_indices;
1262 return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1265 /** Simplify/canonicalize expression containing indexed objects. This
1266 * performs contraction of dummy indices where possible, checks whether
1267 * the free indices in sums are consistent, and automatically replaces
1268 * scalar products by known values if desired.
1270 * @param sp Scalar products to be replaced automatically
1271 * @param options Simplification options (currently unused)
1272 * @return simplified expression */
1273 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1275 exvector free_indices, dummy_indices;
1276 return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1279 /** Symmetrize expression over its free indices. */
1280 ex ex::symmetrize() const
1282 return GiNaC::symmetrize(*this, get_free_indices());
1285 /** Antisymmetrize expression over its free indices. */
1286 ex ex::antisymmetrize() const
1288 return GiNaC::antisymmetrize(*this, get_free_indices());
1291 /** Symmetrize expression by cyclic permutation over its free indices. */
1292 ex ex::symmetrize_cyclic() const
1294 return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1301 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1303 // If indexed, extract base objects
1304 ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1305 ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1307 // Enforce canonical order in pair
1308 if (s1.compare(s2) > 0) {
1317 bool spmapkey::operator==(const spmapkey &other) const
1319 if (!v1.is_equal(other.v1))
1321 if (!v2.is_equal(other.v2))
1323 if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1326 return dim.is_equal(other.dim);
1329 bool spmapkey::operator<(const spmapkey &other) const
1331 int cmp = v1.compare(other.v1);
1334 cmp = v2.compare(other.v2);
1338 // Objects are equal, now check dimensions
1339 if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1342 return dim.compare(other.dim) < 0;
1345 void spmapkey::debugprint() const
1347 std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1350 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1352 spm[spmapkey(v1, v2)] = sp;
1355 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1357 spm[spmapkey(v1, v2, dim)] = sp;
1360 void scalar_products::add_vectors(const lst & l, const ex & dim)
1362 // Add all possible pairs of products
1363 for (auto & it1 : l)
1364 for (auto & it2 : l)
1365 add(it1, it2, it1 * it2);
1368 void scalar_products::clear()
1373 /** Check whether scalar product pair is defined. */
1374 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1376 return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1379 /** Return value of defined scalar product pair. */
1380 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1382 return spm.find(spmapkey(v1, v2, dim))->second;
1385 void scalar_products::debugprint() const
1387 std::cerr << "map size=" << spm.size() << std::endl;
1388 for (auto & it : spm) {
1389 const spmapkey & k = it.first;
1390 std::cerr << "item key=";
1392 std::cerr << ", value=" << it.second << std::endl;
1396 exvector get_all_dummy_indices_safely(const ex & e)
1398 if (is_a<indexed>(e))
1399 return ex_to<indexed>(e).get_dummy_indices();
1400 else if (is_a<power>(e) && e.op(1)==2) {
1401 return e.op(0).get_free_indices();
1403 else if (is_a<mul>(e) || is_a<ncmul>(e)) {
1405 exvector free_indices;
1406 for (std::size_t i = 0; i < e.nops(); ++i) {
1407 exvector dummies_of_factor = get_all_dummy_indices_safely(e.op(i));
1408 dummies.insert(dummies.end(), dummies_of_factor.begin(),
1409 dummies_of_factor.end());
1410 exvector free_of_factor = e.op(i).get_free_indices();
1411 free_indices.insert(free_indices.begin(), free_of_factor.begin(),
1412 free_of_factor.end());
1414 exvector free_out, dummy_out;
1415 find_free_and_dummy(free_indices.begin(), free_indices.end(), free_out,
1417 dummies.insert(dummies.end(), dummy_out.begin(), dummy_out.end());
1420 else if(is_a<add>(e)) {
1422 for(std::size_t i = 0; i < e.nops(); ++i) {
1423 exvector dummies_of_term = get_all_dummy_indices_safely(e.op(i));
1424 sort(dummies_of_term.begin(), dummies_of_term.end());
1426 set_union(result.begin(), result.end(), dummies_of_term.begin(),
1427 dummies_of_term.end(), std::back_inserter<exvector>(new_vec),
1429 result.swap(new_vec);
1436 /** Returns all dummy indices from the exvector */
1437 exvector get_all_dummy_indices(const ex & e)
1441 product_to_exvector(e, p, nc);
1442 auto ip = p.begin(), ipend = p.end();
1444 while (ip != ipend) {
1445 if (is_a<indexed>(*ip)) {
1446 v1 = ex_to<indexed>(*ip).get_dummy_indices();
1447 v.insert(v.end(), v1.begin(), v1.end());
1449 while (ip1 != ipend) {
1450 if (is_a<indexed>(*ip1)) {
1451 v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
1452 v.insert(v.end(), v1.begin(), v1.end());
1462 lst rename_dummy_indices_uniquely(const exvector & va, const exvector & vb)
1464 exvector common_indices;
1465 set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
1466 if (common_indices.empty()) {
1467 return lst{lst{}, lst{}};
1469 exvector new_indices, old_indices;
1470 old_indices.reserve(2*common_indices.size());
1471 new_indices.reserve(2*common_indices.size());
1472 exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
1473 while (ip != ipend) {
1474 ex newsym = dynallocate<symbol>();
1476 if(is_exactly_a<spinidx>(*ip))
1477 newidx = dynallocate<spinidx>(newsym, ex_to<spinidx>(*ip).get_dim(),
1478 ex_to<spinidx>(*ip).is_covariant(),
1479 ex_to<spinidx>(*ip).is_dotted());
1480 else if (is_exactly_a<varidx>(*ip))
1481 newidx = dynallocate<varidx>(newsym, ex_to<varidx>(*ip).get_dim(),
1482 ex_to<varidx>(*ip).is_covariant());
1484 newidx = dynallocate<idx>(newsym, ex_to<idx>(*ip).get_dim());
1485 old_indices.push_back(*ip);
1486 new_indices.push_back(newidx);
1487 if(is_a<varidx>(*ip)) {
1488 old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
1489 new_indices.push_back(ex_to<varidx>(newidx).toggle_variance());
1493 return lst{lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end())};
1497 ex rename_dummy_indices_uniquely(const exvector & va, const exvector & vb, const ex & b)
1499 lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1500 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);
1503 ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
1505 exvector va = get_all_dummy_indices_safely(a);
1506 if (va.size() > 0) {
1507 exvector vb = get_all_dummy_indices_safely(b);
1508 if (vb.size() > 0) {
1509 sort(va.begin(), va.end(), ex_is_less());
1510 sort(vb.begin(), vb.end(), ex_is_less());
1511 lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1512 if (indices_subs.op(0).nops() > 0)
1513 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);
1519 ex rename_dummy_indices_uniquely(exvector & va, const ex & b, bool modify_va)
1521 if (va.size() > 0) {
1522 exvector vb = get_all_dummy_indices_safely(b);
1523 if (vb.size() > 0) {
1524 sort(vb.begin(), vb.end(), ex_is_less());
1525 lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1526 if (indices_subs.op(0).nops() > 0) {
1528 for (auto & i : ex_to<lst>(indices_subs.op(1)))
1530 exvector uncommon_indices;
1531 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());
1532 for (auto & ip : uncommon_indices)
1534 sort(va.begin(), va.end(), ex_is_less());
1536 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);
1543 ex expand_dummy_sum(const ex & e, bool subs_idx)
1545 ex e_expanded = e.expand();
1546 pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
1547 if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
1548 return e_expanded.map(fcn);
1549 } else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded) || is_a<indexed>(e_expanded)) {
1551 if (is_a<indexed>(e_expanded))
1552 v = ex_to<indexed>(e_expanded).get_dummy_indices();
1554 v = get_all_dummy_indices(e_expanded);
1555 ex result = e_expanded;
1556 for (const auto & nu : v) {
1557 if (ex_to<idx>(nu).get_dim().info(info_flags::nonnegint)) {
1558 int idim = ex_to<numeric>(ex_to<idx>(nu).get_dim()).to_int();
1560 for (int i=0; i < idim; i++) {
1561 if (subs_idx && is_a<varidx>(nu)) {
1562 ex other = ex_to<varidx>(nu).toggle_variance();
1563 en += result.subs(lst{
1565 other == idx(i, idim)
1568 en += result.subs( nu.op(0) == i );
1580 } // namespace GiNaC