3 * Implementation of GiNaC's indexed expressions. */
6 * GiNaC Copyright (C) 1999-2006 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"
46 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(indexed, exprseq,
47 print_func<print_context>(&indexed::do_print).
48 print_func<print_latex>(&indexed::do_print_latex).
49 print_func<print_tree>(&indexed::do_print_tree))
52 // default constructor
55 indexed::indexed() : symtree(not_symmetric())
57 tinfo_key = &indexed::tinfo_static;
64 indexed::indexed(const ex & b) : inherited(b), symtree(not_symmetric())
66 tinfo_key = &indexed::tinfo_static;
70 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(not_symmetric())
72 tinfo_key = &indexed::tinfo_static;
76 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(not_symmetric())
78 tinfo_key = &indexed::tinfo_static;
82 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(not_symmetric())
84 tinfo_key = &indexed::tinfo_static;
88 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())
90 tinfo_key = &indexed::tinfo_static;
94 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
96 tinfo_key = &indexed::tinfo_static;
100 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
102 tinfo_key = &indexed::tinfo_static;
106 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)
108 tinfo_key = &indexed::tinfo_static;
112 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(not_symmetric())
114 seq.insert(seq.end(), v.begin(), v.end());
115 tinfo_key = &indexed::tinfo_static;
119 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
121 seq.insert(seq.end(), v.begin(), v.end());
122 tinfo_key = &indexed::tinfo_static;
126 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
128 tinfo_key = &indexed::tinfo_static;
131 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
133 tinfo_key = &indexed::tinfo_static;
136 indexed::indexed(const symmetry & symm, std::auto_ptr<exvector> vp) : inherited(vp), symtree(symm)
138 tinfo_key = &indexed::tinfo_static;
145 indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
147 if (!n.find_ex("symmetry", symtree, sym_lst)) {
148 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
150 n.find_unsigned("symmetry", symm);
159 symtree = not_symmetric();
162 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
166 void indexed::archive(archive_node &n) const
168 inherited::archive(n);
169 n.add_ex("symmetry", symtree);
172 DEFAULT_UNARCHIVE(indexed)
175 // functions overriding virtual functions from base classes
178 void indexed::printindices(const print_context & c, unsigned level) const
180 if (seq.size() > 1) {
182 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
184 if (is_a<print_latex>(c)) {
186 // TeX output: group by variance
188 bool covariant = true;
190 while (it != itend) {
191 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
192 if (first || cur_covariant != covariant) { // Variance changed
193 // The empty {} prevents indices from ending up on top of each other
196 covariant = cur_covariant;
212 while (it != itend) {
220 void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
222 if (precedence() <= level)
223 c.s << openbrace << '(';
225 seq[0].print(c, precedence());
227 printindices(c, level);
228 if (precedence() <= level)
229 c.s << ')' << closebrace;
232 void indexed::do_print(const print_context & c, unsigned level) const
234 print_indexed(c, "", "", level);
237 void indexed::do_print_latex(const print_latex & c, unsigned level) const
239 print_indexed(c, "{", "}", level);
242 void indexed::do_print_tree(const print_tree & c, unsigned level) const
244 c.s << std::string(level, ' ') << class_name() << " @" << this
245 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
246 << ", " << seq.size()-1 << " indices"
247 << ", symmetry=" << symtree << std::endl;
248 seq[0].print(c, level + c.delta_indent);
249 printindices(c, level + c.delta_indent);
252 bool indexed::info(unsigned inf) const
254 if (inf == info_flags::indexed) return true;
255 if (inf == info_flags::has_indices) return seq.size() > 1;
256 return inherited::info(inf);
259 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
260 bool operator() (const ex & e, unsigned inf) const {
261 return !(ex_to<idx>(e).get_value().info(inf));
265 bool indexed::all_index_values_are(unsigned inf) const
267 // No indices? Then no property can be fulfilled
272 return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
275 int indexed::compare_same_type(const basic & other) const
277 GINAC_ASSERT(is_a<indexed>(other));
278 return inherited::compare_same_type(other);
281 ex indexed::eval(int level) const
283 // First evaluate children, then we will end up here again
285 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
287 const ex &base = seq[0];
289 // If the base object is 0, the whole object is 0
293 // If the base object is a product, pull out the numeric factor
294 if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
296 ex f = ex_to<numeric>(base.op(base.nops() - 1));
298 return f * thiscontainer(v);
301 if(this->tinfo()==&indexed::tinfo_static && seq.size()==1)
304 // Canonicalize indices according to the symmetry properties
305 if (seq.size() > 2) {
307 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
308 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
309 if (sig != INT_MAX) {
310 // Something has changed while sorting indices, more evaluations later
313 return ex(sig) * thiscontainer(v);
317 // Let the class of the base object perform additional evaluations
318 return ex_to<basic>(base).eval_indexed(*this);
321 ex indexed::real_part() const
323 if(op(0).info(info_flags::real))
325 return real_part_function(*this).hold();
328 ex indexed::imag_part() const
330 if(op(0).info(info_flags::real))
332 return imag_part_function(*this).hold();
335 ex indexed::thiscontainer(const exvector & v) const
337 return indexed(ex_to<symmetry>(symtree), v);
340 ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
342 return indexed(ex_to<symmetry>(symtree), vp);
345 unsigned indexed::return_type() const
347 if(is_a<matrix>(op(0)))
348 return return_types::commutative;
350 return op(0).return_type();
353 ex indexed::expand(unsigned options) const
355 GINAC_ASSERT(seq.size() > 0);
357 if (options & expand_options::expand_indexed) {
358 ex newbase = seq[0].expand(options);
359 if (is_exactly_a<add>(newbase)) {
361 for (size_t i=0; i<newbase.nops(); i++) {
363 s[0] = newbase.op(i);
364 sum += thiscontainer(s).expand(options);
368 if (!are_ex_trivially_equal(newbase, seq[0])) {
371 return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
374 return inherited::expand(options);
378 // virtual functions which can be overridden by derived classes
384 // non-virtual functions in this class
387 /** Check whether all indices are of class idx and validate the symmetry
388 * tree. This function is used internally to make sure that all constructed
389 * indexed objects really carry indices and not some other classes. */
390 void indexed::validate() const
392 GINAC_ASSERT(seq.size() > 0);
393 exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
394 while (it != itend) {
396 throw(std::invalid_argument("indices of indexed object must be of type idx"));
400 if (!symtree.is_zero()) {
401 if (!is_exactly_a<symmetry>(symtree))
402 throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
403 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
407 /** Implementation of ex::diff() for an indexed object always returns 0.
410 ex indexed::derivative(const symbol & s) const
419 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
420 bool operator() (const ex &lh, const ex &rh) const
426 // Replacing the dimension might cause an error (e.g. with
427 // index classes that only work in a fixed number of dimensions)
428 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
435 /** Check whether two sorted index vectors are consistent (i.e. equal). */
436 static bool indices_consistent(const exvector & v1, const exvector & v2)
438 // Number of indices must be the same
439 if (v1.size() != v2.size())
442 return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
445 exvector indexed::get_indices() const
447 GINAC_ASSERT(seq.size() >= 1);
448 return exvector(seq.begin() + 1, seq.end());
451 exvector indexed::get_dummy_indices() const
453 exvector free_indices, dummy_indices;
454 find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
455 return dummy_indices;
458 exvector indexed::get_dummy_indices(const indexed & other) const
460 exvector indices = get_free_indices();
461 exvector other_indices = other.get_free_indices();
462 indices.insert(indices.end(), other_indices.begin(), other_indices.end());
463 exvector dummy_indices;
464 find_dummy_indices(indices, dummy_indices);
465 return dummy_indices;
468 bool indexed::has_dummy_index_for(const ex & i) const
470 exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
471 while (it != itend) {
472 if (is_dummy_pair(*it, i))
479 exvector indexed::get_free_indices() const
481 exvector free_indices, dummy_indices;
482 find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
486 exvector add::get_free_indices() const
488 exvector free_indices;
489 for (size_t i=0; i<nops(); i++) {
491 free_indices = op(i).get_free_indices();
493 exvector free_indices_of_term = op(i).get_free_indices();
494 if (!indices_consistent(free_indices, free_indices_of_term))
495 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
501 exvector mul::get_free_indices() const
503 // Concatenate free indices of all factors
505 for (size_t i=0; i<nops(); i++) {
506 exvector free_indices_of_factor = op(i).get_free_indices();
507 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
510 // And remove the dummy indices
511 exvector free_indices, dummy_indices;
512 find_free_and_dummy(un, free_indices, dummy_indices);
516 exvector ncmul::get_free_indices() const
518 // Concatenate free indices of all factors
520 for (size_t i=0; i<nops(); i++) {
521 exvector free_indices_of_factor = op(i).get_free_indices();
522 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
525 // And remove the dummy indices
526 exvector free_indices, dummy_indices;
527 find_free_and_dummy(un, free_indices, dummy_indices);
531 struct is_summation_idx : public std::unary_function<ex, bool> {
532 bool operator()(const ex & e)
534 return is_dummy_pair(e, e);
538 exvector integral::get_free_indices() const
540 if (a.get_free_indices().size() || b.get_free_indices().size())
541 throw (std::runtime_error("integral::get_free_indices: boundary values should not have free indices"));
542 return f.get_free_indices();
545 template<class T> size_t number_of_type(const exvector&v)
548 for(exvector::const_iterator i=v.begin(); i!=v.end(); ++i)
549 if(is_exactly_a<T>(*i))
554 /** Rename dummy indices in an expression.
556 * @param e Expression to work on
557 * @param local_dummy_indices The set of dummy indices that appear in the
559 * @param global_dummy_indices The set of dummy indices that have appeared
560 * before and which we would like to use in "e", too. This gets updated
562 template<class T> static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
564 size_t global_size = number_of_type<T>(global_dummy_indices),
565 local_size = number_of_type<T>(local_dummy_indices);
567 // Any local dummy indices at all?
571 if (global_size < local_size) {
573 // More local indices than we encountered before, add the new ones
575 size_t old_global_size = global_size;
576 int remaining = local_size - global_size;
577 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
578 while (it != itend && remaining > 0) {
579 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()) {
580 global_dummy_indices.push_back(*it);
587 // If this is the first set of local indices, do nothing
588 if (old_global_size == 0)
591 GINAC_ASSERT(local_size <= global_size);
593 // Construct vectors of index symbols
594 exvector local_syms, global_syms;
595 local_syms.reserve(local_size);
596 global_syms.reserve(local_size);
597 for (size_t i=0; local_syms.size()!=local_size; i++)
598 if(is_exactly_a<T>(local_dummy_indices[i]))
599 local_syms.push_back(local_dummy_indices[i].op(0));
600 shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
601 for (size_t i=0; global_syms.size()!=local_size; i++) // don't use more global symbols than necessary
602 if(is_exactly_a<T>(global_dummy_indices[i]))
603 global_syms.push_back(global_dummy_indices[i].op(0));
604 shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
606 // Remove common indices
607 exvector local_uniq, global_uniq;
608 set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
609 set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
611 // Replace remaining non-common local index symbols by global ones
612 if (local_uniq.empty())
615 while (global_uniq.size() > local_uniq.size())
616 global_uniq.pop_back();
617 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
621 /** Given a set of indices, extract those of class varidx. */
622 static void find_variant_indices(const exvector & v, exvector & variant_indices)
624 exvector::const_iterator it1, itend;
625 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
626 if (is_exactly_a<varidx>(*it1))
627 variant_indices.push_back(*it1);
631 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
634 * @param e Object to work on
635 * @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
636 * @param moved_indices The set of indices that have been repositioned (will be changed by this function)
637 * @return true if 'e' was changed */
638 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
640 bool something_changed = false;
642 // If a dummy index is encountered for the first time in the
643 // product, pull it up, otherwise, pull it down
644 exvector::const_iterator it2, it2start, it2end;
645 for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
646 if (!is_exactly_a<varidx>(*it2))
649 exvector::iterator vit, vitend;
650 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
651 if (it2->op(0).is_equal(vit->op(0))) {
652 if (ex_to<varidx>(*it2).is_covariant()) {
654 *it2 == ex_to<varidx>(*it2).toggle_variance(),
655 ex_to<varidx>(*it2).toggle_variance() == *it2
656 ), subs_options::no_pattern);
657 something_changed = true;
658 it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
659 it2start = ex_to<indexed>(e).seq.begin();
660 it2end = ex_to<indexed>(e).seq.end();
662 moved_indices.push_back(*vit);
663 variant_dummy_indices.erase(vit);
668 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
669 if (it2->op(0).is_equal(vit->op(0))) {
670 if (ex_to<varidx>(*it2).is_contravariant()) {
671 e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance(), subs_options::no_pattern);
672 something_changed = true;
673 it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
674 it2start = ex_to<indexed>(e).seq.begin();
675 it2end = ex_to<indexed>(e).seq.end();
684 return something_changed;
687 /* Ordering that only compares the base expressions of indexed objects. */
688 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
689 bool operator() (const ex &lh, const ex &rh) const
691 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
695 /* An auxiliary function used by simplify_indexed() and expand_dummy_sum()
696 * It returns an exvector of factors from the supplied product */
697 static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
699 // Remember whether the product was commutative or noncommutative
700 // (because we chop it into factors and need to reassemble later)
701 non_commutative = is_exactly_a<ncmul>(e);
703 // Collect factors in an exvector, store squares twice
704 v.reserve(e.nops() * 2);
706 if (is_exactly_a<power>(e)) {
707 // We only get called for simple squares, split a^2 -> a*a
708 GINAC_ASSERT(e.op(1).is_equal(_ex2));
709 v.push_back(e.op(0));
710 v.push_back(e.op(0));
712 for (size_t i=0; i<e.nops(); i++) {
714 if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
715 v.push_back(f.op(0));
716 v.push_back(f.op(0));
717 } else if (is_exactly_a<ncmul>(f)) {
718 // Noncommutative factor found, split it as well
719 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
720 for (size_t j=0; j<f.nops(); j++)
721 v.push_back(f.op(j));
728 template<class T> ex idx_symmetrization(const ex& r,const exvector& local_dummy_indices)
729 { exvector dummy_syms;
730 dummy_syms.reserve(r.nops());
731 for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
732 if(is_exactly_a<T>(*it))
733 dummy_syms.push_back(it->op(0));
734 if(dummy_syms.size() < 2)
736 ex q=symmetrize(r, dummy_syms);
740 // Forward declaration needed in absence of friend injection, C.f. [namespace.memdef]:
741 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp);
743 /** Simplify product of indexed expressions (commutative, noncommutative and
744 * simple squares), return list of free indices. */
745 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
747 // Collect factors in an exvector
750 // Remember whether the product was commutative or noncommutative
751 // (because we chop it into factors and need to reassemble later)
752 bool non_commutative;
753 product_to_exvector(e, v, non_commutative);
755 // Perform contractions
756 bool something_changed = false;
757 GINAC_ASSERT(v.size() > 1);
758 exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
759 for (it1 = v.begin(); it1 != next_to_last; it1++) {
762 if (!is_a<indexed>(*it1))
765 bool first_noncommutative = (it1->return_type() != return_types::commutative);
767 // Indexed factor found, get free indices and look for contraction
769 exvector free1, dummy1;
770 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
772 exvector::iterator it2;
773 for (it2 = it1 + 1; it2 != itend; it2++) {
775 if (!is_a<indexed>(*it2))
778 bool second_noncommutative = (it2->return_type() != return_types::commutative);
780 // Find free indices of second factor and merge them with free
781 // indices of first factor
783 find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
784 un.insert(un.end(), free1.begin(), free1.end());
786 // Check whether the two factors share dummy indices
787 exvector free, dummy;
788 find_free_and_dummy(un, free, dummy);
789 size_t num_dummies = dummy.size();
790 if (num_dummies == 0)
793 // At least one dummy index, is it a defined scalar product?
794 bool contracted = false;
795 if (free.empty() && it1->nops()==2 && it2->nops()==2) {
797 ex dim = minimal_dim(
798 ex_to<idx>(it1->op(1)).get_dim(),
799 ex_to<idx>(it2->op(1)).get_dim()
802 // User-defined scalar product?
803 if (sp.is_defined(*it1, *it2, dim)) {
805 // Yes, substitute it
806 *it1 = sp.evaluate(*it1, *it2, dim);
808 goto contraction_done;
812 // Try to contract the first one with the second one
813 contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
816 // That didn't work; maybe the second object knows how to
817 // contract itself with the first one
818 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
822 if (first_noncommutative || second_noncommutative
823 || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
824 || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
825 || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
827 // One of the factors became a sum or product:
828 // re-expand expression and run again
829 // Non-commutative products are always re-expanded to give
830 // eval_ncmul() the chance to re-order and canonicalize
832 ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
833 return simplify_indexed(r, free_indices, dummy_indices, sp);
836 // Both objects may have new indices now or they might
837 // even not be indexed objects any more, so we have to
839 something_changed = true;
845 // Find free indices (concatenate them all and call find_free_and_dummy())
846 // and all dummy indices that appear
847 exvector un, individual_dummy_indices;
848 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
849 exvector free_indices_of_factor;
850 if (is_a<indexed>(*it1)) {
851 exvector dummy_indices_of_factor;
852 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);
853 individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
855 free_indices_of_factor = it1->get_free_indices();
856 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
858 exvector local_dummy_indices;
859 find_free_and_dummy(un, free_indices, local_dummy_indices);
860 local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
862 // Filter out the dummy indices with variance
863 exvector variant_dummy_indices;
864 find_variant_indices(local_dummy_indices, variant_dummy_indices);
866 // Any indices with variance present at all?
867 if (!variant_dummy_indices.empty()) {
869 // Yes, bring the product into a canonical order that only depends on
870 // the base expressions of indexed objects
871 if (!non_commutative)
872 std::sort(v.begin(), v.end(), ex_base_is_less());
874 exvector moved_indices;
876 // Iterate over all indexed objects in the product
877 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
878 if (!is_a<indexed>(*it1))
881 if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
882 something_changed = true;
887 if (something_changed)
888 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
892 // The result should be symmetric with respect to exchange of dummy
893 // indices, so if the symmetrization vanishes, the whole expression is
894 // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
895 ex q = idx_symmetrization<idx>(r, local_dummy_indices);
897 free_indices.clear();
900 q = idx_symmetrization<varidx>(q, local_dummy_indices);
902 free_indices.clear();
905 q = idx_symmetrization<spinidx>(q, local_dummy_indices);
907 free_indices.clear();
911 // Dummy index renaming
912 r = rename_dummy_indices<idx>(r, dummy_indices, local_dummy_indices);
913 r = rename_dummy_indices<varidx>(r, dummy_indices, local_dummy_indices);
914 r = rename_dummy_indices<spinidx>(r, dummy_indices, local_dummy_indices);
916 // Product of indexed object with a scalar?
917 if (is_exactly_a<mul>(r) && r.nops() == 2
918 && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
919 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
924 /** This structure stores the original and symmetrized versions of terms
925 * obtained during the simplification of sums. */
928 terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
930 ex orig; /**< original term */
931 ex symm; /**< symmtrized term */
934 class terminfo_is_less {
936 bool operator() (const terminfo & ti1, const terminfo & ti2) const
938 return (ti1.symm.compare(ti2.symm) < 0);
942 /** This structure stores the individual symmetrized terms obtained during
943 * the simplification of sums. */
946 symminfo() : num(0) {}
948 symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
950 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
951 coeff = symmterm_.op(symmterm_.nops()-1);
952 symmterm = symmterm_ / coeff;
955 symmterm = symmterm_;
959 ex symmterm; /**< symmetrized term */
960 ex coeff; /**< coefficient of symmetrized term */
961 ex orig; /**< original term */
962 size_t num; /**< how many symmetrized terms resulted from the original term */
965 class symminfo_is_less_by_symmterm {
967 bool operator() (const symminfo & si1, const symminfo & si2) const
969 return (si1.symmterm.compare(si2.symmterm) < 0);
973 class symminfo_is_less_by_orig {
975 bool operator() (const symminfo & si1, const symminfo & si2) const
977 return (si1.orig.compare(si2.orig) < 0);
981 bool hasindex(const ex &x, const ex &sym)
983 if(is_a<idx>(x) && x.op(0)==sym)
986 for(size_t i=0; i<x.nops(); ++i)
987 if(hasindex(x.op(i), sym))
992 /** Simplify indexed expression, return list of free indices. */
993 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
995 // Expand the expression
996 ex e_expanded = e.expand();
998 // Simplification of single indexed object: just find the free indices
999 // and perform dummy index renaming/repositioning
1000 if (is_a<indexed>(e_expanded)) {
1002 // Find the dummy indices
1003 const indexed &i = ex_to<indexed>(e_expanded);
1004 exvector local_dummy_indices;
1005 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
1007 // Filter out the dummy indices with variance
1008 exvector variant_dummy_indices;
1009 find_variant_indices(local_dummy_indices, variant_dummy_indices);
1011 // Any indices with variance present at all?
1012 if (!variant_dummy_indices.empty()) {
1014 // Yes, reposition them
1015 exvector moved_indices;
1016 reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
1019 // Rename the dummy indices
1020 e_expanded = rename_dummy_indices<idx>(e_expanded, dummy_indices, local_dummy_indices);
1021 e_expanded = rename_dummy_indices<varidx>(e_expanded, dummy_indices, local_dummy_indices);
1022 e_expanded = rename_dummy_indices<spinidx>(e_expanded, dummy_indices, local_dummy_indices);
1026 // Simplification of sum = sum of simplifications, check consistency of
1027 // free indices in each term
1028 if (is_exactly_a<add>(e_expanded)) {
1031 free_indices.clear();
1033 for (size_t i=0; i<e_expanded.nops(); i++) {
1034 exvector free_indices_of_term;
1035 ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
1036 if (!term.is_zero()) {
1038 free_indices = free_indices_of_term;
1042 if (!indices_consistent(free_indices, free_indices_of_term)) {
1043 std::ostringstream s;
1044 s << "simplify_indexed: inconsistent indices in sum: ";
1045 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
1046 throw (std::runtime_error(s.str()));
1048 if (is_a<indexed>(sum) && is_a<indexed>(term))
1049 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
1056 // If the sum turns out to be zero, we are finished
1057 if (sum.is_zero()) {
1058 free_indices.clear();
1062 // More than one term and more than one dummy index?
1063 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
1064 if (num_terms_orig < 2 || dummy_indices.size() < 2)
1067 // Chop the sum into terms and symmetrize each one over the dummy
1069 std::vector<terminfo> terms;
1070 for (size_t i=0; i<sum.nops(); i++) {
1071 const ex & term = sum.op(i);
1072 exvector dummy_indices_of_term;
1073 dummy_indices_of_term.reserve(dummy_indices.size());
1074 for(exvector::iterator i=dummy_indices.begin(); i!=dummy_indices.end(); ++i)
1075 if(hasindex(term,i->op(0)))
1076 dummy_indices_of_term.push_back(*i);
1077 ex term_symm = idx_symmetrization<idx>(term, dummy_indices_of_term);
1078 term_symm = idx_symmetrization<varidx>(term_symm, dummy_indices_of_term);
1079 term_symm = idx_symmetrization<spinidx>(term_symm, dummy_indices_of_term);
1080 if (term_symm.is_zero())
1082 terms.push_back(terminfo(term, term_symm));
1085 // Sort by symmetrized terms
1086 std::sort(terms.begin(), terms.end(), terminfo_is_less());
1088 // Combine equal symmetrized terms
1089 std::vector<terminfo> terms_pass2;
1090 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1092 std::vector<terminfo>::const_iterator j = i + 1;
1093 while (j != terms.end() && j->symm == i->symm) {
1097 terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1101 // If there is only one term left, we are finished
1102 if (terms_pass2.size() == 1)
1103 return terms_pass2[0].orig;
1105 // Chop the symmetrized terms into subterms
1106 std::vector<symminfo> sy;
1107 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1108 if (is_exactly_a<add>(i->symm)) {
1109 size_t num = i->symm.nops();
1110 for (size_t j=0; j<num; j++)
1111 sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1113 sy.push_back(symminfo(i->symm, i->orig, 1));
1116 // Sort by symmetrized subterms
1117 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1119 // Combine equal symmetrized subterms
1120 std::vector<symminfo> sy_pass2;
1122 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1124 // Combine equal terms
1125 std::vector<symminfo>::const_iterator j = i + 1;
1126 if (j != sy.end() && j->symmterm == i->symmterm) {
1128 // More than one term, collect the coefficients
1129 ex coeff = i->coeff;
1130 while (j != sy.end() && j->symmterm == i->symmterm) {
1135 // Add combined term to result
1136 if (!coeff.is_zero())
1137 result.push_back(coeff * i->symmterm);
1141 // Single term, store for second pass
1142 sy_pass2.push_back(*i);
1148 // Were there any remaining terms that didn't get combined?
1149 if (sy_pass2.size() > 0) {
1151 // Yes, sort by their original terms
1152 std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1154 for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1156 // How many symmetrized terms of this original term are left?
1158 std::vector<symminfo>::const_iterator j = i + 1;
1159 while (j != sy_pass2.end() && j->orig == i->orig) {
1164 if (num == i->num) {
1166 // All terms left, then add the original term to the result
1167 result.push_back(i->orig);
1171 // Some terms were combined with others, add up the remaining symmetrized terms
1172 std::vector<symminfo>::const_iterator k;
1173 for (k=i; k!=j; k++)
1174 result.push_back(k->coeff * k->symmterm);
1181 // Add all resulting terms
1182 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1183 if (sum_symm.is_zero())
1184 free_indices.clear();
1188 // Simplification of products
1189 if (is_exactly_a<mul>(e_expanded)
1190 || is_exactly_a<ncmul>(e_expanded)
1191 || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1192 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1194 // Cannot do anything
1195 free_indices.clear();
1199 /** Simplify/canonicalize expression containing indexed objects. This
1200 * performs contraction of dummy indices where possible and checks whether
1201 * the free indices in sums are consistent.
1203 * @param options Simplification options (currently unused)
1204 * @return simplified expression */
1205 ex ex::simplify_indexed(unsigned options) const
1207 exvector free_indices, dummy_indices;
1209 return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1212 /** Simplify/canonicalize expression containing indexed objects. This
1213 * performs contraction of dummy indices where possible, checks whether
1214 * the free indices in sums are consistent, and automatically replaces
1215 * scalar products by known values if desired.
1217 * @param sp Scalar products to be replaced automatically
1218 * @param options Simplification options (currently unused)
1219 * @return simplified expression */
1220 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1222 exvector free_indices, dummy_indices;
1223 return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1226 /** Symmetrize expression over its free indices. */
1227 ex ex::symmetrize() const
1229 return GiNaC::symmetrize(*this, get_free_indices());
1232 /** Antisymmetrize expression over its free indices. */
1233 ex ex::antisymmetrize() const
1235 return GiNaC::antisymmetrize(*this, get_free_indices());
1238 /** Symmetrize expression by cyclic permutation over its free indices. */
1239 ex ex::symmetrize_cyclic() const
1241 return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1248 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1250 // If indexed, extract base objects
1251 ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1252 ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1254 // Enforce canonical order in pair
1255 if (s1.compare(s2) > 0) {
1264 bool spmapkey::operator==(const spmapkey &other) const
1266 if (!v1.is_equal(other.v1))
1268 if (!v2.is_equal(other.v2))
1270 if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1273 return dim.is_equal(other.dim);
1276 bool spmapkey::operator<(const spmapkey &other) const
1278 int cmp = v1.compare(other.v1);
1281 cmp = v2.compare(other.v2);
1285 // Objects are equal, now check dimensions
1286 if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1289 return dim.compare(other.dim) < 0;
1292 void spmapkey::debugprint() const
1294 std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1297 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1299 spm[spmapkey(v1, v2)] = sp;
1302 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1304 spm[spmapkey(v1, v2, dim)] = sp;
1307 void scalar_products::add_vectors(const lst & l, const ex & dim)
1309 // Add all possible pairs of products
1310 for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1311 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1312 add(*it1, *it2, *it1 * *it2);
1315 void scalar_products::clear()
1320 /** Check whether scalar product pair is defined. */
1321 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1323 return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1326 /** Return value of defined scalar product pair. */
1327 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1329 return spm.find(spmapkey(v1, v2, dim))->second;
1332 void scalar_products::debugprint() const
1334 std::cerr << "map size=" << spm.size() << std::endl;
1335 spmap::const_iterator i = spm.begin(), end = spm.end();
1337 const spmapkey & k = i->first;
1338 std::cerr << "item key=";
1340 std::cerr << ", value=" << i->second << std::endl;
1345 exvector get_all_dummy_indices_safely(const ex & e)
1347 if (is_a<indexed>(e))
1348 return ex_to<indexed>(e).get_dummy_indices();
1349 else if (is_a<power>(e) && e.op(1)==2) {
1350 return e.op(0).get_free_indices();
1352 else if (is_a<mul>(e) || is_a<ncmul>(e)) {
1354 exvector free_indices;
1355 for (int i=0; i<e.nops(); ++i) {
1356 exvector dummies_of_factor = get_all_dummy_indices_safely(e.op(i));
1357 dummies.insert(dummies.end(), dummies_of_factor.begin(),
1358 dummies_of_factor.end());
1359 exvector free_of_factor = e.op(i).get_free_indices();
1360 free_indices.insert(free_indices.begin(), free_of_factor.begin(),
1361 free_of_factor.end());
1363 exvector free_out, dummy_out;
1364 find_free_and_dummy(free_indices.begin(), free_indices.end(), free_out,
1366 dummies.insert(dummies.end(), dummy_out.begin(), dummy_out.end());
1369 else if(is_a<add>(e)) {
1371 for(int i=0; i<e.nops(); ++i) {
1372 exvector dummies_of_term = get_all_dummy_indices_safely(e.op(i));
1373 sort(dummies_of_term.begin(), dummies_of_term.end());
1375 set_union(result.begin(), result.end(), dummies_of_term.begin(),
1376 dummies_of_term.end(), std::back_inserter<exvector>(new_vec),
1378 result.swap(new_vec);
1385 /** Returns all dummy indices from the exvector */
1386 exvector get_all_dummy_indices(const ex & e)
1390 product_to_exvector(e, p, nc);
1391 exvector::const_iterator ip = p.begin(), ipend = p.end();
1393 while (ip != ipend) {
1394 if (is_a<indexed>(*ip)) {
1395 v1 = ex_to<indexed>(*ip).get_dummy_indices();
1396 v.insert(v.end(), v1.begin(), v1.end());
1397 exvector::const_iterator ip1 = ip+1;
1398 while (ip1 != ipend) {
1399 if (is_a<indexed>(*ip1)) {
1400 v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
1401 v.insert(v.end(), v1.begin(), v1.end());
1411 lst rename_dummy_indices_uniquely(const exvector & va, const exvector & vb)
1413 exvector common_indices;
1414 set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
1415 if (common_indices.empty()) {
1416 return lst(lst(), lst());
1418 exvector new_indices, old_indices;
1419 old_indices.reserve(2*common_indices.size());
1420 new_indices.reserve(2*common_indices.size());
1421 exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
1422 while (ip != ipend) {
1423 ex newsym=(new symbol)->setflag(status_flags::dynallocated);
1425 if(is_exactly_a<spinidx>(*ip))
1426 newidx = (new spinidx(newsym, ex_to<spinidx>(*ip).get_dim(),
1427 ex_to<spinidx>(*ip).is_covariant(),
1428 ex_to<spinidx>(*ip).is_dotted()))
1429 -> setflag(status_flags::dynallocated);
1430 else if (is_exactly_a<varidx>(*ip))
1431 newidx = (new varidx(newsym, ex_to<varidx>(*ip).get_dim(),
1432 ex_to<varidx>(*ip).is_covariant()))
1433 -> setflag(status_flags::dynallocated);
1435 newidx = (new idx(newsym, ex_to<idx>(*ip).get_dim()))
1436 -> setflag(status_flags::dynallocated);
1437 old_indices.push_back(*ip);
1438 new_indices.push_back(newidx);
1439 if(is_a<varidx>(*ip)) {
1440 old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
1441 new_indices.push_back(ex_to<varidx>(newidx).toggle_variance());
1445 return lst(lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end()));
1449 ex rename_dummy_indices_uniquely(const exvector & va, const exvector & vb, const ex & b)
1451 lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1452 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);
1455 ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
1457 exvector va = get_all_dummy_indices_safely(a);
1458 if (va.size() > 0) {
1459 exvector vb = get_all_dummy_indices_safely(b);
1460 if (vb.size() > 0) {
1461 sort(va.begin(), va.end(), ex_is_less());
1462 sort(vb.begin(), vb.end(), ex_is_less());
1463 lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1464 if (indices_subs.op(0).nops() > 0)
1465 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);
1471 ex rename_dummy_indices_uniquely(exvector & va, const ex & b, bool modify_va)
1473 if (va.size() > 0) {
1474 exvector vb = get_all_dummy_indices_safely(b);
1475 if (vb.size() > 0) {
1476 sort(vb.begin(), vb.end(), ex_is_less());
1477 lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1478 if (indices_subs.op(0).nops() > 0) {
1480 for (lst::const_iterator i = ex_to<lst>(indices_subs.op(1)).begin(); i != ex_to<lst>(indices_subs.op(1)).end(); ++i)
1482 exvector uncommon_indices;
1483 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());
1484 exvector::const_iterator ip = uncommon_indices.begin(), ipend = uncommon_indices.end();
1485 while (ip != ipend) {
1489 sort(va.begin(), va.end(), ex_is_less());
1491 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);
1498 ex expand_dummy_sum(const ex & e, bool subs_idx)
1500 ex e_expanded = e.expand();
1501 pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
1502 if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
1503 return e_expanded.map(fcn);
1504 } else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded) || is_a<indexed>(e_expanded)) {
1506 if (is_a<indexed>(e_expanded))
1507 v = ex_to<indexed>(e_expanded).get_dummy_indices();
1509 v = get_all_dummy_indices(e_expanded);
1510 ex result = e_expanded;
1511 for(exvector::const_iterator it=v.begin(); it!=v.end(); ++it) {
1513 if (ex_to<idx>(nu).get_dim().info(info_flags::nonnegint)) {
1514 int idim = ex_to<numeric>(ex_to<idx>(nu).get_dim()).to_int();
1516 for (int i=0; i < idim; i++) {
1517 if (subs_idx && is_a<varidx>(nu)) {
1518 ex other = ex_to<varidx>(nu).toggle_variance();
1519 en += result.subs(lst(
1521 other == idx(i, idim)
1524 en += result.subs( nu.op(0) == i );
1536 } // namespace GiNaC