3 * Implementation of GiNaC's indexed expressions. */
6 * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
8 * This program is free software; you can redistribute it and/or modify
9 * it under the terms of the GNU General Public License as published by
10 * the Free Software Foundation; either version 2 of the License, or
11 * (at your option) any later version.
13 * This program is distributed in the hope that it will be useful,
14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 * GNU General Public License for more details.
18 * You should have received a copy of the GNU General Public License
19 * along with this program; if not, write to the Free Software
20 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
33 #include "relational.h"
35 #include "operators.h"
45 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(indexed, exprseq,
46 print_func<print_context>(&indexed::do_print).
47 print_func<print_latex>(&indexed::do_print_latex).
48 print_func<print_tree>(&indexed::do_print_tree))
51 // default constructor
54 indexed::indexed() : symtree(not_symmetric())
56 tinfo_key = TINFO_indexed;
63 indexed::indexed(const ex & b) : inherited(b), symtree(not_symmetric())
65 tinfo_key = TINFO_indexed;
69 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(not_symmetric())
71 tinfo_key = TINFO_indexed;
75 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(not_symmetric())
77 tinfo_key = TINFO_indexed;
81 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(not_symmetric())
83 tinfo_key = TINFO_indexed;
87 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(not_symmetric())
89 tinfo_key = TINFO_indexed;
93 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
95 tinfo_key = TINFO_indexed;
99 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
101 tinfo_key = TINFO_indexed;
105 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(symm)
107 tinfo_key = TINFO_indexed;
111 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(not_symmetric())
113 seq.insert(seq.end(), v.begin(), v.end());
114 tinfo_key = TINFO_indexed;
118 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
120 seq.insert(seq.end(), v.begin(), v.end());
121 tinfo_key = TINFO_indexed;
125 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
127 tinfo_key = TINFO_indexed;
130 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
132 tinfo_key = TINFO_indexed;
135 indexed::indexed(const symmetry & symm, std::auto_ptr<exvector> vp) : inherited(vp), symtree(symm)
137 tinfo_key = TINFO_indexed;
144 indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
146 if (!n.find_ex("symmetry", symtree, sym_lst)) {
147 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
149 n.find_unsigned("symmetry", symm);
158 symtree = not_symmetric();
161 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
165 void indexed::archive(archive_node &n) const
167 inherited::archive(n);
168 n.add_ex("symmetry", symtree);
171 DEFAULT_UNARCHIVE(indexed)
174 // functions overriding virtual functions from base classes
177 void indexed::printindices(const print_context & c, unsigned level) const
179 if (seq.size() > 1) {
181 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
183 if (is_a<print_latex>(c)) {
185 // TeX output: group by variance
187 bool covariant = true;
189 while (it != itend) {
190 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
191 if (first || cur_covariant != covariant) { // Variance changed
192 // The empty {} prevents indices from ending up on top of each other
195 covariant = cur_covariant;
211 while (it != itend) {
219 void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
221 if (precedence() <= level)
222 c.s << openbrace << '(';
224 seq[0].print(c, precedence());
226 printindices(c, level);
227 if (precedence() <= level)
228 c.s << ')' << closebrace;
231 void indexed::do_print(const print_context & c, unsigned level) const
233 print_indexed(c, "", "", level);
236 void indexed::do_print_latex(const print_latex & c, unsigned level) const
238 print_indexed(c, "{", "}", level);
241 void indexed::do_print_tree(const print_tree & c, unsigned level) const
243 c.s << std::string(level, ' ') << class_name() << " @" << this
244 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
245 << ", " << seq.size()-1 << " indices"
246 << ", symmetry=" << symtree << std::endl;
247 seq[0].print(c, level + c.delta_indent);
248 printindices(c, level + c.delta_indent);
251 bool indexed::info(unsigned inf) const
253 if (inf == info_flags::indexed) return true;
254 if (inf == info_flags::has_indices) return seq.size() > 1;
255 return inherited::info(inf);
258 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
259 bool operator() (const ex & e, unsigned inf) const {
260 return !(ex_to<idx>(e).get_value().info(inf));
264 bool indexed::all_index_values_are(unsigned inf) const
266 // No indices? Then no property can be fulfilled
271 return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
274 int indexed::compare_same_type(const basic & other) const
276 GINAC_ASSERT(is_a<indexed>(other));
277 return inherited::compare_same_type(other);
280 ex indexed::eval(int level) const
282 // First evaluate children, then we will end up here again
284 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
286 const ex &base = seq[0];
288 // If the base object is 0, the whole object is 0
292 // If the base object is a product, pull out the numeric factor
293 if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
295 ex f = ex_to<numeric>(base.op(base.nops() - 1));
297 return f * thiscontainer(v);
300 if(this->tinfo()==TINFO_indexed && seq.size()==1)
303 // Canonicalize indices according to the symmetry properties
304 if (seq.size() > 2) {
306 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
307 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
308 if (sig != INT_MAX) {
309 // Something has changed while sorting indices, more evaluations later
312 return ex(sig) * thiscontainer(v);
316 // Let the class of the base object perform additional evaluations
317 return ex_to<basic>(base).eval_indexed(*this);
320 ex indexed::thiscontainer(const exvector & v) const
322 return indexed(ex_to<symmetry>(symtree), v);
325 ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
327 return indexed(ex_to<symmetry>(symtree), vp);
330 unsigned indexed::return_type() const
332 if(is_a<matrix>(op(0)))
333 return return_types::commutative;
335 return op(0).return_type();
338 ex indexed::expand(unsigned options) const
340 GINAC_ASSERT(seq.size() > 0);
342 if (options & expand_options::expand_indexed) {
343 ex newbase = seq[0].expand(options);
344 if (is_exactly_a<add>(newbase)) {
346 for (size_t i=0; i<newbase.nops(); i++) {
348 s[0] = newbase.op(i);
349 sum += thiscontainer(s).expand(options);
353 if (!are_ex_trivially_equal(newbase, seq[0])) {
356 return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
359 return inherited::expand(options);
363 // virtual functions which can be overridden by derived classes
369 // non-virtual functions in this class
372 /** Check whether all indices are of class idx and validate the symmetry
373 * tree. This function is used internally to make sure that all constructed
374 * indexed objects really carry indices and not some other classes. */
375 void indexed::validate() const
377 GINAC_ASSERT(seq.size() > 0);
378 exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
379 while (it != itend) {
381 throw(std::invalid_argument("indices of indexed object must be of type idx"));
385 if (!symtree.is_zero()) {
386 if (!is_exactly_a<symmetry>(symtree))
387 throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
388 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
392 /** Implementation of ex::diff() for an indexed object always returns 0.
395 ex indexed::derivative(const symbol & s) const
404 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
405 bool operator() (const ex &lh, const ex &rh) const
411 // Replacing the dimension might cause an error (e.g. with
412 // index classes that only work in a fixed number of dimensions)
413 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
420 /** Check whether two sorted index vectors are consistent (i.e. equal). */
421 static bool indices_consistent(const exvector & v1, const exvector & v2)
423 // Number of indices must be the same
424 if (v1.size() != v2.size())
427 return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
430 exvector indexed::get_indices() const
432 GINAC_ASSERT(seq.size() >= 1);
433 return exvector(seq.begin() + 1, seq.end());
436 exvector indexed::get_dummy_indices() const
438 exvector free_indices, dummy_indices;
439 find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
440 return dummy_indices;
443 exvector indexed::get_dummy_indices(const indexed & other) const
445 exvector indices = get_free_indices();
446 exvector other_indices = other.get_free_indices();
447 indices.insert(indices.end(), other_indices.begin(), other_indices.end());
448 exvector dummy_indices;
449 find_dummy_indices(indices, dummy_indices);
450 return dummy_indices;
453 bool indexed::has_dummy_index_for(const ex & i) const
455 exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
456 while (it != itend) {
457 if (is_dummy_pair(*it, i))
464 exvector indexed::get_free_indices() const
466 exvector free_indices, dummy_indices;
467 find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
471 exvector add::get_free_indices() const
473 exvector free_indices;
474 for (size_t i=0; i<nops(); i++) {
476 free_indices = op(i).get_free_indices();
478 exvector free_indices_of_term = op(i).get_free_indices();
479 if (!indices_consistent(free_indices, free_indices_of_term))
480 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
486 exvector mul::get_free_indices() const
488 // Concatenate free indices of all factors
490 for (size_t i=0; i<nops(); i++) {
491 exvector free_indices_of_factor = op(i).get_free_indices();
492 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
495 // And remove the dummy indices
496 exvector free_indices, dummy_indices;
497 find_free_and_dummy(un, free_indices, dummy_indices);
501 exvector ncmul::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 struct is_summation_idx : public std::unary_function<ex, bool> {
517 bool operator()(const ex & e)
519 return is_dummy_pair(e, e);
523 exvector power::get_free_indices() const
525 // Get free indices of basis
526 exvector basis_indices = basis.get_free_indices();
528 if (exponent.info(info_flags::even)) {
529 // If the exponent is an even number, then any "free" index that
530 // forms a dummy pair with itself is actually a summation index
531 exvector really_free;
532 std::remove_copy_if(basis_indices.begin(), basis_indices.end(),
533 std::back_inserter(really_free), is_summation_idx());
536 return basis_indices;
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 // If a dummy index is encountered for the first time in the
644 // product, pull it up, otherwise, pull it down
645 exvector::const_iterator it2, it2start, it2end;
646 for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
647 if (!is_exactly_a<varidx>(*it2))
650 exvector::iterator vit, vitend;
651 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
652 if (it2->op(0).is_equal(vit->op(0))) {
653 if (ex_to<varidx>(*it2).is_covariant()) {
655 *it2 == ex_to<varidx>(*it2).toggle_variance(),
656 ex_to<varidx>(*it2).toggle_variance() == *it2
657 ), subs_options::no_pattern);
658 something_changed = true;
659 it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
660 it2start = ex_to<indexed>(e).seq.begin();
661 it2end = ex_to<indexed>(e).seq.end();
663 moved_indices.push_back(*vit);
664 variant_dummy_indices.erase(vit);
669 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
670 if (it2->op(0).is_equal(vit->op(0))) {
671 if (ex_to<varidx>(*it2).is_contravariant()) {
672 e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance(), subs_options::no_pattern);
673 something_changed = true;
674 it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
675 it2start = ex_to<indexed>(e).seq.begin();
676 it2end = ex_to<indexed>(e).seq.end();
685 return something_changed;
688 /* Ordering that only compares the base expressions of indexed objects. */
689 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
690 bool operator() (const ex &lh, const ex &rh) const
692 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
696 /* An auxiliary function used by simplify_indexed() and expand_dummy_sum()
697 * It returns an exvector of factors from the supplied product */
698 static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
700 // Remember whether the product was commutative or noncommutative
701 // (because we chop it into factors and need to reassemble later)
702 non_commutative = is_exactly_a<ncmul>(e);
704 // Collect factors in an exvector, store squares twice
705 v.reserve(e.nops() * 2);
707 if (is_exactly_a<power>(e)) {
708 // We only get called for simple squares, split a^2 -> a*a
709 GINAC_ASSERT(e.op(1).is_equal(_ex2));
710 v.push_back(e.op(0));
711 v.push_back(e.op(0));
713 for (size_t i=0; i<e.nops(); i++) {
715 if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
716 v.push_back(f.op(0));
717 v.push_back(f.op(0));
718 } else if (is_exactly_a<ncmul>(f)) {
719 // Noncommutative factor found, split it as well
720 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
721 for (size_t j=0; j<f.nops(); j++)
722 v.push_back(f.op(j));
729 template<class T> ex idx_symmetrization(const ex& r,const exvector& local_dummy_indices)
730 { exvector dummy_syms;
731 dummy_syms.reserve(r.nops());
732 for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
733 if(is_exactly_a<T>(*it))
734 dummy_syms.push_back(it->op(0));
735 if(dummy_syms.size() < 2)
737 ex q=symmetrize(r, dummy_syms);
741 // Forward declaration needed in absence of friend injection, C.f. [namespace.memdef]:
742 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp);
744 /** Simplify product of indexed expressions (commutative, noncommutative and
745 * simple squares), return list of free indices. */
746 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
748 // Collect factors in an exvector
751 // Remember whether the product was commutative or noncommutative
752 // (because we chop it into factors and need to reassemble later)
753 bool non_commutative;
754 product_to_exvector(e, v, non_commutative);
756 // Perform contractions
757 bool something_changed = false;
758 GINAC_ASSERT(v.size() > 1);
759 exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
760 for (it1 = v.begin(); it1 != next_to_last; it1++) {
763 if (!is_a<indexed>(*it1))
766 bool first_noncommutative = (it1->return_type() != return_types::commutative);
768 // Indexed factor found, get free indices and look for contraction
770 exvector free1, dummy1;
771 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
773 exvector::iterator it2;
774 for (it2 = it1 + 1; it2 != itend; it2++) {
776 if (!is_a<indexed>(*it2))
779 bool second_noncommutative = (it2->return_type() != return_types::commutative);
781 // Find free indices of second factor and merge them with free
782 // indices of first factor
784 find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
785 un.insert(un.end(), free1.begin(), free1.end());
787 // Check whether the two factors share dummy indices
788 exvector free, dummy;
789 find_free_and_dummy(un, free, dummy);
790 size_t num_dummies = dummy.size();
791 if (num_dummies == 0)
794 // At least one dummy index, is it a defined scalar product?
795 bool contracted = false;
799 // Find minimal dimension of all indices of both factors
800 exvector::const_iterator dit = ex_to<indexed>(*it1).seq.begin() + 1, ditend = ex_to<indexed>(*it1).seq.end();
801 ex dim = ex_to<idx>(*dit).get_dim();
803 for (; dit != ditend; ++dit) {
804 dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
806 dit = ex_to<indexed>(*it2).seq.begin() + 1;
807 ditend = ex_to<indexed>(*it2).seq.end();
808 for (; dit != ditend; ++dit) {
809 dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
812 // User-defined scalar product?
813 if (sp.is_defined(*it1, *it2, dim)) {
815 // Yes, substitute it
816 *it1 = sp.evaluate(*it1, *it2, dim);
818 goto contraction_done;
820 } catch (const std::runtime_error&) {}
823 // Try to contract the first one with the second one
824 contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
827 // That didn't work; maybe the second object knows how to
828 // contract itself with the first one
829 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
833 if (first_noncommutative || second_noncommutative
834 || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
835 || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
836 || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
838 // One of the factors became a sum or product:
839 // re-expand expression and run again
840 // Non-commutative products are always re-expanded to give
841 // eval_ncmul() the chance to re-order and canonicalize
843 ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
844 return simplify_indexed(r, free_indices, dummy_indices, sp);
847 // Both objects may have new indices now or they might
848 // even not be indexed objects any more, so we have to
850 something_changed = true;
856 // Find free indices (concatenate them all and call find_free_and_dummy())
857 // and all dummy indices that appear
858 exvector un, individual_dummy_indices;
859 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
860 exvector free_indices_of_factor;
861 if (is_a<indexed>(*it1)) {
862 exvector dummy_indices_of_factor;
863 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);
864 individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
866 free_indices_of_factor = it1->get_free_indices();
867 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
869 exvector local_dummy_indices;
870 find_free_and_dummy(un, free_indices, local_dummy_indices);
871 local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
873 // Filter out the dummy indices with variance
874 exvector variant_dummy_indices;
875 find_variant_indices(local_dummy_indices, variant_dummy_indices);
877 // Any indices with variance present at all?
878 if (!variant_dummy_indices.empty()) {
880 // Yes, bring the product into a canonical order that only depends on
881 // the base expressions of indexed objects
882 if (!non_commutative)
883 std::sort(v.begin(), v.end(), ex_base_is_less());
885 exvector moved_indices;
887 // Iterate over all indexed objects in the product
888 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
889 if (!is_a<indexed>(*it1))
892 if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
893 something_changed = true;
898 if (something_changed)
899 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
903 // The result should be symmetric with respect to exchange of dummy
904 // indices, so if the symmetrization vanishes, the whole expression is
905 // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
906 ex q = idx_symmetrization<idx>(r, local_dummy_indices);
908 free_indices.clear();
911 q = idx_symmetrization<varidx>(q, local_dummy_indices);
913 free_indices.clear();
916 q = idx_symmetrization<spinidx>(q, local_dummy_indices);
918 free_indices.clear();
922 // Dummy index renaming
923 r = rename_dummy_indices<idx>(r, dummy_indices, local_dummy_indices);
924 r = rename_dummy_indices<varidx>(r, dummy_indices, local_dummy_indices);
925 r = rename_dummy_indices<spinidx>(r, dummy_indices, local_dummy_indices);
927 // Product of indexed object with a scalar?
928 if (is_exactly_a<mul>(r) && r.nops() == 2
929 && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
930 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
935 /** This structure stores the original and symmetrized versions of terms
936 * obtained during the simplification of sums. */
939 terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
941 ex orig; /**< original term */
942 ex symm; /**< symmtrized term */
945 class terminfo_is_less {
947 bool operator() (const terminfo & ti1, const terminfo & ti2) const
949 return (ti1.symm.compare(ti2.symm) < 0);
953 /** This structure stores the individual symmetrized terms obtained during
954 * the simplification of sums. */
957 symminfo() : num(0) {}
959 symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
961 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
962 coeff = symmterm_.op(symmterm_.nops()-1);
963 symmterm = symmterm_ / coeff;
966 symmterm = symmterm_;
970 ex symmterm; /**< symmetrized term */
971 ex coeff; /**< coefficient of symmetrized term */
972 ex orig; /**< original term */
973 size_t num; /**< how many symmetrized terms resulted from the original term */
976 class symminfo_is_less_by_symmterm {
978 bool operator() (const symminfo & si1, const symminfo & si2) const
980 return (si1.symmterm.compare(si2.symmterm) < 0);
984 class symminfo_is_less_by_orig {
986 bool operator() (const symminfo & si1, const symminfo & si2) const
988 return (si1.orig.compare(si2.orig) < 0);
992 bool hasindex(const ex &x, const ex &sym)
994 if(is_a<idx>(x) && x.op(0)==sym)
997 for(size_t i=0; i<x.nops(); ++i)
998 if(hasindex(x.op(i), sym))
1003 /** Simplify indexed expression, return list of free indices. */
1004 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
1006 // Expand the expression
1007 ex e_expanded = e.expand();
1009 // Simplification of single indexed object: just find the free indices
1010 // and perform dummy index renaming/repositioning
1011 if (is_a<indexed>(e_expanded)) {
1013 // Find the dummy indices
1014 const indexed &i = ex_to<indexed>(e_expanded);
1015 exvector local_dummy_indices;
1016 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
1018 // Filter out the dummy indices with variance
1019 exvector variant_dummy_indices;
1020 find_variant_indices(local_dummy_indices, variant_dummy_indices);
1022 // Any indices with variance present at all?
1023 if (!variant_dummy_indices.empty()) {
1025 // Yes, reposition them
1026 exvector moved_indices;
1027 reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
1030 // Rename the dummy indices
1031 e_expanded = rename_dummy_indices<idx>(e_expanded, dummy_indices, local_dummy_indices);
1032 e_expanded = rename_dummy_indices<varidx>(e_expanded, dummy_indices, local_dummy_indices);
1033 e_expanded = rename_dummy_indices<spinidx>(e_expanded, dummy_indices, local_dummy_indices);
1037 // Simplification of sum = sum of simplifications, check consistency of
1038 // free indices in each term
1039 if (is_exactly_a<add>(e_expanded)) {
1042 free_indices.clear();
1044 for (size_t i=0; i<e_expanded.nops(); i++) {
1045 exvector free_indices_of_term;
1046 ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
1047 if (!term.is_zero()) {
1049 free_indices = free_indices_of_term;
1053 if (!indices_consistent(free_indices, free_indices_of_term)) {
1054 std::ostringstream s;
1055 s << "simplify_indexed: inconsistent indices in sum: ";
1056 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
1057 throw (std::runtime_error(s.str()));
1059 if (is_a<indexed>(sum) && is_a<indexed>(term))
1060 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
1067 // If the sum turns out to be zero, we are finished
1068 if (sum.is_zero()) {
1069 free_indices.clear();
1073 // More than one term and more than one dummy index?
1074 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
1075 if (num_terms_orig < 2 || dummy_indices.size() < 2)
1078 // Chop the sum into terms and symmetrize each one over the dummy
1080 std::vector<terminfo> terms;
1081 for (size_t i=0; i<sum.nops(); i++) {
1082 const ex & term = sum.op(i);
1083 exvector dummy_indices_of_term;
1084 dummy_indices_of_term.reserve(dummy_indices.size());
1085 for(exvector::iterator i=dummy_indices.begin(); i!=dummy_indices.end(); ++i)
1086 if(hasindex(term,i->op(0)))
1087 dummy_indices_of_term.push_back(*i);
1088 ex term_symm = idx_symmetrization<idx>(term, dummy_indices_of_term);
1089 term_symm = idx_symmetrization<varidx>(term_symm, dummy_indices_of_term);
1090 term_symm = idx_symmetrization<spinidx>(term_symm, dummy_indices_of_term);
1091 if (term_symm.is_zero())
1093 terms.push_back(terminfo(term, term_symm));
1096 // Sort by symmetrized terms
1097 std::sort(terms.begin(), terms.end(), terminfo_is_less());
1099 // Combine equal symmetrized terms
1100 std::vector<terminfo> terms_pass2;
1101 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1103 std::vector<terminfo>::const_iterator j = i + 1;
1104 while (j != terms.end() && j->symm == i->symm) {
1108 terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1112 // If there is only one term left, we are finished
1113 if (terms_pass2.size() == 1)
1114 return terms_pass2[0].orig;
1116 // Chop the symmetrized terms into subterms
1117 std::vector<symminfo> sy;
1118 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1119 if (is_exactly_a<add>(i->symm)) {
1120 size_t num = i->symm.nops();
1121 for (size_t j=0; j<num; j++)
1122 sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1124 sy.push_back(symminfo(i->symm, i->orig, 1));
1127 // Sort by symmetrized subterms
1128 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1130 // Combine equal symmetrized subterms
1131 std::vector<symminfo> sy_pass2;
1133 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1135 // Combine equal terms
1136 std::vector<symminfo>::const_iterator j = i + 1;
1137 if (j != sy.end() && j->symmterm == i->symmterm) {
1139 // More than one term, collect the coefficients
1140 ex coeff = i->coeff;
1141 while (j != sy.end() && j->symmterm == i->symmterm) {
1146 // Add combined term to result
1147 if (!coeff.is_zero())
1148 result.push_back(coeff * i->symmterm);
1152 // Single term, store for second pass
1153 sy_pass2.push_back(*i);
1159 // Were there any remaining terms that didn't get combined?
1160 if (sy_pass2.size() > 0) {
1162 // Yes, sort by their original terms
1163 std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1165 for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1167 // How many symmetrized terms of this original term are left?
1169 std::vector<symminfo>::const_iterator j = i + 1;
1170 while (j != sy_pass2.end() && j->orig == i->orig) {
1175 if (num == i->num) {
1177 // All terms left, then add the original term to the result
1178 result.push_back(i->orig);
1182 // Some terms were combined with others, add up the remaining symmetrized terms
1183 std::vector<symminfo>::const_iterator k;
1184 for (k=i; k!=j; k++)
1185 result.push_back(k->coeff * k->symmterm);
1192 // Add all resulting terms
1193 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1194 if (sum_symm.is_zero())
1195 free_indices.clear();
1199 // Simplification of products
1200 if (is_exactly_a<mul>(e_expanded)
1201 || is_exactly_a<ncmul>(e_expanded)
1202 || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1203 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1205 // Cannot do anything
1206 free_indices.clear();
1210 /** Simplify/canonicalize expression containing indexed objects. This
1211 * performs contraction of dummy indices where possible and checks whether
1212 * the free indices in sums are consistent.
1214 * @param options Simplification options (currently unused)
1215 * @return simplified expression */
1216 ex ex::simplify_indexed(unsigned options) const
1218 exvector free_indices, dummy_indices;
1220 return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1223 /** Simplify/canonicalize expression containing indexed objects. This
1224 * performs contraction of dummy indices where possible, checks whether
1225 * the free indices in sums are consistent, and automatically replaces
1226 * scalar products by known values if desired.
1228 * @param sp Scalar products to be replaced automatically
1229 * @param options Simplification options (currently unused)
1230 * @return simplified expression */
1231 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1233 exvector free_indices, dummy_indices;
1234 return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1237 /** Symmetrize expression over its free indices. */
1238 ex ex::symmetrize() const
1240 return GiNaC::symmetrize(*this, get_free_indices());
1243 /** Antisymmetrize expression over its free indices. */
1244 ex ex::antisymmetrize() const
1246 return GiNaC::antisymmetrize(*this, get_free_indices());
1249 /** Symmetrize expression by cyclic permutation over its free indices. */
1250 ex ex::symmetrize_cyclic() const
1252 return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1259 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1261 // If indexed, extract base objects
1262 ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1263 ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1265 // Enforce canonical order in pair
1266 if (s1.compare(s2) > 0) {
1275 bool spmapkey::operator==(const spmapkey &other) const
1277 if (!v1.is_equal(other.v1))
1279 if (!v2.is_equal(other.v2))
1281 if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1284 return dim.is_equal(other.dim);
1287 bool spmapkey::operator<(const spmapkey &other) const
1289 int cmp = v1.compare(other.v1);
1292 cmp = v2.compare(other.v2);
1296 // Objects are equal, now check dimensions
1297 if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1300 return dim.compare(other.dim) < 0;
1303 void spmapkey::debugprint() const
1305 std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1308 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1310 spm[spmapkey(v1, v2)] = sp;
1313 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1315 spm[spmapkey(v1, v2, dim)] = sp;
1318 void scalar_products::add_vectors(const lst & l, const ex & dim)
1320 // Add all possible pairs of products
1321 for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1322 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1323 add(*it1, *it2, *it1 * *it2);
1326 void scalar_products::clear()
1331 /** Check whether scalar product pair is defined. */
1332 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1334 return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1337 /** Return value of defined scalar product pair. */
1338 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1340 return spm.find(spmapkey(v1, v2, dim))->second;
1343 void scalar_products::debugprint() const
1345 std::cerr << "map size=" << spm.size() << std::endl;
1346 spmap::const_iterator i = spm.begin(), end = spm.end();
1348 const spmapkey & k = i->first;
1349 std::cerr << "item key=";
1351 std::cerr << ", value=" << i->second << std::endl;
1356 /** Returns all dummy indices from the exvector */
1357 exvector get_all_dummy_indices(const ex & e)
1361 product_to_exvector(e, p, nc);
1362 exvector::const_iterator ip = p.begin(), ipend = p.end();
1364 while (ip != ipend) {
1365 if (is_a<indexed>(*ip)) {
1366 v1 = ex_to<indexed>(*ip).get_dummy_indices();
1367 v.insert(v.end(), v1.begin(), v1.end());
1368 exvector::const_iterator ip1 = ip+1;
1369 while (ip1 != ipend) {
1370 if (is_a<indexed>(*ip1)) {
1371 v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
1372 v.insert(v.end(), v1.begin(), v1.end());
1382 lst rename_dummy_indices_uniquely(const exvector & va, const exvector & vb)
1384 exvector common_indices;
1385 set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
1386 if (common_indices.empty()) {
1387 return lst(lst(), lst());
1389 exvector new_indices, old_indices;
1390 old_indices.reserve(2*common_indices.size());
1391 new_indices.reserve(2*common_indices.size());
1392 exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
1393 while (ip != ipend) {
1394 ex newsym=(new symbol)->setflag(status_flags::dynallocated);
1396 if(is_exactly_a<spinidx>(*ip))
1397 newidx = (new spinidx(newsym, ex_to<spinidx>(*ip).get_dim(),
1398 ex_to<spinidx>(*ip).is_covariant(),
1399 ex_to<spinidx>(*ip).is_dotted()))
1400 -> setflag(status_flags::dynallocated);
1401 else if (is_exactly_a<varidx>(*ip))
1402 newidx = (new varidx(newsym, ex_to<varidx>(*ip).get_dim(),
1403 ex_to<varidx>(*ip).is_covariant()))
1404 -> setflag(status_flags::dynallocated);
1406 newidx = (new idx(newsym, ex_to<idx>(*ip).get_dim()))
1407 -> setflag(status_flags::dynallocated);
1408 old_indices.push_back(*ip);
1409 new_indices.push_back(newidx);
1410 if(is_a<varidx>(*ip)) {
1411 old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
1412 new_indices.push_back(ex_to<varidx>(newidx).toggle_variance());
1416 return lst(lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end()));
1420 ex rename_dummy_indices_uniquely(const exvector & va, const exvector & vb, const ex & b)
1422 lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1423 return (indices_subs.op(0).nops()>0 ? b.subs((lst)indices_subs.op(0), (lst)indices_subs.op(1), subs_options::no_pattern) : b);
1426 ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
1428 exvector va = get_all_dummy_indices(a);
1429 if (va.size() > 0) {
1430 exvector vb = get_all_dummy_indices(b);
1431 if (vb.size() > 0) {
1432 sort(va.begin(), va.end(), ex_is_less());
1433 sort(vb.begin(), vb.end(), ex_is_less());
1434 lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1435 if (indices_subs.op(0).nops() > 0)
1436 return b.subs((lst)indices_subs.op(0), (lst)indices_subs.op(1), subs_options::no_pattern);
1442 ex rename_dummy_indices_uniquely(exvector & va, const ex & b, bool modify_va)
1444 if (va.size() > 0) {
1445 exvector vb = get_all_dummy_indices(b);
1446 if (vb.size() > 0) {
1447 sort(vb.begin(), vb.end(), ex_is_less());
1448 lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1449 if (indices_subs.op(0).nops() > 0) {
1451 for (lst::const_iterator i = ex_to<lst>(indices_subs.op(1)).begin(); i != ex_to<lst>(indices_subs.op(1)).end(); ++i)
1453 exvector uncommon_indices;
1454 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());
1455 exvector::const_iterator ip = uncommon_indices.begin(), ipend = uncommon_indices.end();
1456 while (ip != ipend) {
1460 sort(va.begin(), va.end(), ex_is_less());
1462 return b.subs((lst)indices_subs.op(0), (lst)indices_subs.op(1), subs_options::no_pattern);
1469 ex expand_dummy_sum(const ex & e, bool subs_idx)
1471 ex e_expanded = e.expand();
1472 pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
1473 if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
1474 return e_expanded.map(fcn);
1475 } else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded)) {
1476 exvector v = get_all_dummy_indices(e_expanded);
1477 exvector::const_iterator it = v.begin(), itend = v.end();
1478 while (it != itend) {
1479 varidx nu = ex_to<varidx>(*it);
1480 if (nu.is_dim_numeric()) {
1482 for (int i=0; i < ex_to<numeric>(nu.get_dim()).to_int(); i++) {
1483 if (is_a<varidx>(nu) && !subs_idx) {
1484 en += e_expanded.subs(lst(nu == varidx(i, nu.get_dim(), true), nu.toggle_variance() == varidx(i, nu.get_dim())));
1486 en += e_expanded.subs(lst(nu == idx(i, nu.get_dim()), nu.toggle_variance() == idx(i, nu.get_dim())));
1489 return expand_dummy_sum(en, subs_idx);
1494 } else if (is_a<indexed>(e_expanded)) {
1495 exvector v = ex_to<indexed>(e_expanded).get_dummy_indices();
1496 exvector::const_iterator it = v.begin(), itend = v.end();
1497 while (it != itend) {
1498 varidx nu = ex_to<varidx>(*it);
1499 if (nu.is_dim_numeric()) {
1501 for (int i=0; i < ex_to<numeric>(nu.get_dim()).to_int(); i++) {
1502 if (is_a<varidx>(nu) && !subs_idx) {
1503 en += e_expanded.subs(lst(nu == varidx(i, nu.get_dim(), true), nu.toggle_variance() == varidx(i, nu.get_dim())));
1505 en += e_expanded.subs(lst(nu == idx(i, nu.get_dim()), nu.toggle_variance() == idx(i, nu.get_dim())));
1508 return expand_dummy_sum(en, subs_idx);
1518 } // namespace GiNaC