3 * Implementation of GiNaC's products of expressions. */
6 * GiNaC Copyright (C) 1999-2011 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
26 #include "operators.h"
42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mul, expairseq,
43 print_func<print_context>(&mul::do_print).
44 print_func<print_latex>(&mul::do_print_latex).
45 print_func<print_csrc>(&mul::do_print_csrc).
46 print_func<print_tree>(&mul::do_print_tree).
47 print_func<print_python_repr>(&mul::do_print_python_repr))
51 // default constructor
64 mul::mul(const ex & lh, const ex & rh)
67 construct_from_2_ex(lh,rh);
68 GINAC_ASSERT(is_canonical());
71 mul::mul(const exvector & v)
74 construct_from_exvector(v);
75 GINAC_ASSERT(is_canonical());
78 mul::mul(const epvector & v)
81 construct_from_epvector(v);
82 GINAC_ASSERT(is_canonical());
85 mul::mul(const epvector & v, const ex & oc, bool do_index_renaming)
88 construct_from_epvector(v, do_index_renaming);
89 GINAC_ASSERT(is_canonical());
92 mul::mul(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming)
94 GINAC_ASSERT(vp.get()!=0);
96 construct_from_epvector(*vp, do_index_renaming);
97 GINAC_ASSERT(is_canonical());
100 mul::mul(const ex & lh, const ex & mh, const ex & rh)
104 factors.push_back(lh);
105 factors.push_back(mh);
106 factors.push_back(rh);
107 overall_coeff = _ex1;
108 construct_from_exvector(factors);
109 GINAC_ASSERT(is_canonical());
117 // functions overriding virtual functions from base classes
120 void mul::print_overall_coeff(const print_context & c, const char *mul_sym) const
122 const numeric &coeff = ex_to<numeric>(overall_coeff);
123 if (coeff.csgn() == -1)
125 if (!coeff.is_equal(*_num1_p) &&
126 !coeff.is_equal(*_num_1_p)) {
127 if (coeff.is_rational()) {
128 if (coeff.is_negative())
133 if (coeff.csgn() == -1)
134 (-coeff).print(c, precedence());
136 coeff.print(c, precedence());
142 void mul::do_print(const print_context & c, unsigned level) const
144 if (precedence() <= level)
147 print_overall_coeff(c, "*");
149 epvector::const_iterator it = seq.begin(), itend = seq.end();
151 while (it != itend) {
156 recombine_pair_to_ex(*it).print(c, precedence());
160 if (precedence() <= level)
164 void mul::do_print_latex(const print_latex & c, unsigned level) const
166 if (precedence() <= level)
169 print_overall_coeff(c, " ");
171 // Separate factors into those with negative numeric exponent
173 epvector::const_iterator it = seq.begin(), itend = seq.end();
174 exvector neg_powers, others;
175 while (it != itend) {
176 GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
177 if (ex_to<numeric>(it->coeff).is_negative())
178 neg_powers.push_back(recombine_pair_to_ex(expair(it->rest, -(it->coeff))));
180 others.push_back(recombine_pair_to_ex(*it));
184 if (!neg_powers.empty()) {
186 // Factors with negative exponent are printed as a fraction
188 mul(others).eval().print(c);
190 mul(neg_powers).eval().print(c);
195 // All other factors are printed in the ordinary way
196 exvector::const_iterator vit = others.begin(), vitend = others.end();
197 while (vit != vitend) {
199 vit->print(c, precedence());
204 if (precedence() <= level)
208 void mul::do_print_csrc(const print_csrc & c, unsigned level) const
210 if (precedence() <= level)
213 if (!overall_coeff.is_equal(_ex1)) {
214 if (overall_coeff.is_equal(_ex_1))
217 overall_coeff.print(c, precedence());
222 // Print arguments, separated by "*" or "/"
223 epvector::const_iterator it = seq.begin(), itend = seq.end();
224 while (it != itend) {
226 // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
227 bool needclosingparenthesis = false;
228 if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
229 if (is_a<print_csrc_cl_N>(c)) {
231 needclosingparenthesis = true;
236 // If the exponent is 1 or -1, it is left out
237 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
238 it->rest.print(c, precedence());
239 else if (it->coeff.info(info_flags::negint))
240 // Outer parens around ex needed for broken GCC parser:
241 (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
243 // Outer parens around ex needed for broken GCC parser:
244 (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
246 if (needclosingparenthesis)
249 // Separator is "/" for negative integer powers, "*" otherwise
252 if (it->coeff.info(info_flags::negint))
259 if (precedence() <= level)
263 void mul::do_print_python_repr(const print_python_repr & c, unsigned level) const
265 c.s << class_name() << '(';
267 for (size_t i=1; i<nops(); ++i) {
274 bool mul::info(unsigned inf) const
277 case info_flags::polynomial:
278 case info_flags::integer_polynomial:
279 case info_flags::cinteger_polynomial:
280 case info_flags::rational_polynomial:
281 case info_flags::real:
282 case info_flags::rational:
283 case info_flags::integer:
284 case info_flags::crational:
285 case info_flags::cinteger:
286 case info_flags::positive:
287 case info_flags::nonnegative:
288 case info_flags::posint:
289 case info_flags::nonnegint:
290 case info_flags::even:
291 case info_flags::crational_polynomial:
292 case info_flags::rational_function: {
293 epvector::const_iterator i = seq.begin(), end = seq.end();
295 if (!(recombine_pair_to_ex(*i).info(inf)))
299 if (overall_coeff.is_equal(*_num1_p) && inf == info_flags::even)
301 return overall_coeff.info(inf);
303 case info_flags::algebraic: {
304 epvector::const_iterator i = seq.begin(), end = seq.end();
306 if ((recombine_pair_to_ex(*i).info(inf)))
312 case info_flags::negative: {
314 epvector::const_iterator i = seq.begin(), end = seq.end();
316 const ex& factor = recombine_pair_to_ex(*i++);
317 if (factor.info(info_flags::positive))
319 else if (factor.info(info_flags::negative))
324 if (overall_coeff.info(info_flags::negative))
328 case info_flags::negint: {
330 epvector::const_iterator i = seq.begin(), end = seq.end();
332 const ex& factor = recombine_pair_to_ex(*i++);
333 if (factor.info(info_flags::posint))
335 else if (factor.info(info_flags::negint))
340 if (overall_coeff.info(info_flags::negint))
342 else if (!overall_coeff.info(info_flags::posint))
347 return inherited::info(inf);
350 bool mul::is_polynomial(const ex & var) const
352 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
353 if (!i->rest.is_polynomial(var) ||
354 (i->rest.has(var) && !i->coeff.info(info_flags::integer))) {
361 int mul::degree(const ex & s) const
363 // Sum up degrees of factors
365 epvector::const_iterator i = seq.begin(), end = seq.end();
367 if (ex_to<numeric>(i->coeff).is_integer())
368 deg_sum += recombine_pair_to_ex(*i).degree(s);
371 throw std::runtime_error("mul::degree() undefined degree because of non-integer exponent");
378 int mul::ldegree(const ex & s) const
380 // Sum up degrees of factors
382 epvector::const_iterator i = seq.begin(), end = seq.end();
384 if (ex_to<numeric>(i->coeff).is_integer())
385 deg_sum += recombine_pair_to_ex(*i).ldegree(s);
388 throw std::runtime_error("mul::ldegree() undefined degree because of non-integer exponent");
395 ex mul::coeff(const ex & s, int n) const
398 coeffseq.reserve(seq.size()+1);
401 // product of individual coeffs
402 // if a non-zero power of s is found, the resulting product will be 0
403 epvector::const_iterator i = seq.begin(), end = seq.end();
405 coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
408 coeffseq.push_back(overall_coeff);
409 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
412 epvector::const_iterator i = seq.begin(), end = seq.end();
413 bool coeff_found = false;
415 ex t = recombine_pair_to_ex(*i);
416 ex c = t.coeff(s, n);
418 coeffseq.push_back(c);
421 coeffseq.push_back(t);
426 coeffseq.push_back(overall_coeff);
427 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
433 /** Perform automatic term rewriting rules in this class. In the following
434 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
435 * stand for such expressions that contain a plain number.
437 * - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
441 * @param level cut-off in recursive evaluation */
442 ex mul::eval(int level) const
444 std::auto_ptr<epvector> evaled_seqp = evalchildren(level);
445 if (evaled_seqp.get()) {
446 // do more evaluation later
447 return (new mul(evaled_seqp, overall_coeff))->
448 setflag(status_flags::dynallocated);
451 if (flags & status_flags::evaluated) {
452 GINAC_ASSERT(seq.size()>0);
453 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
457 size_t seq_size = seq.size();
458 if (overall_coeff.is_zero()) {
461 } else if (seq_size==0) {
463 return overall_coeff;
464 } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
466 return recombine_pair_to_ex(*(seq.begin()));
467 } else if ((seq_size==1) &&
468 is_exactly_a<add>((*seq.begin()).rest) &&
469 ex_to<numeric>((*seq.begin()).coeff).is_equal(*_num1_p)) {
470 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
471 const add & addref = ex_to<add>((*seq.begin()).rest);
472 std::auto_ptr<epvector> distrseq(new epvector);
473 distrseq->reserve(addref.seq.size());
474 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
476 distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
479 return (new add(distrseq,
480 ex_to<numeric>(addref.overall_coeff).
481 mul_dyn(ex_to<numeric>(overall_coeff)))
482 )->setflag(status_flags::dynallocated | status_flags::evaluated);
483 } else if ((seq_size >= 2) && (! (flags & status_flags::expanded))) {
484 // Strip the content and the unit part from each term. Thus
485 // things like (-x+a)*(3*x-3*a) automagically turn into - 3*(x-a)^2
487 epvector::const_iterator last = seq.end();
488 epvector::const_iterator i = seq.begin();
489 epvector::const_iterator j = seq.begin();
490 std::auto_ptr<epvector> s(new epvector);
491 numeric oc = *_num1_p;
492 bool something_changed = false;
494 if (likely(! (is_a<add>(i->rest) && i->coeff.is_equal(_ex1)))) {
495 // power::eval has such a rule, no need to handle powers here
500 // XXX: What is the best way to check if the polynomial is a primitive?
501 numeric c = i->rest.integer_content();
502 const numeric lead_coeff =
503 ex_to<numeric>(ex_to<add>(i->rest).seq.begin()->coeff).div(c);
504 const bool canonicalizable = lead_coeff.is_integer();
506 // XXX: The main variable is chosen in a random way, so this code
507 // does NOT transform the term into the canonical form (thus, in some
508 // very unlucky event it can even loop forever). Hopefully the main
509 // variable will be the same for all terms in *this
510 const bool unit_normal = lead_coeff.is_pos_integer();
511 if (likely((c == *_num1_p) && ((! canonicalizable) || unit_normal))) {
516 if (! something_changed) {
517 s->reserve(seq_size);
518 something_changed = true;
521 while ((j!=i) && (j!=last)) {
527 c = c.mul(*_num_1_p);
531 // divide add by the number in place to save at least 2 .eval() calls
532 const add& addref = ex_to<add>(i->rest);
533 add* primitive = new add(addref);
534 primitive->setflag(status_flags::dynallocated);
535 primitive->clearflag(status_flags::hash_calculated);
536 primitive->overall_coeff = ex_to<numeric>(primitive->overall_coeff).div_dyn(c);
537 for (epvector::iterator ai = primitive->seq.begin(); ai != primitive->seq.end(); ++ai)
538 ai->coeff = ex_to<numeric>(ai->coeff).div_dyn(c);
540 s->push_back(expair(*primitive, _ex1));
545 if (something_changed) {
550 return (new mul(s, ex_to<numeric>(overall_coeff).mul_dyn(oc))
551 )->setflag(status_flags::dynallocated);
558 ex mul::evalf(int level) const
561 return mul(seq,overall_coeff);
563 if (level==-max_recursion_level)
564 throw(std::runtime_error("max recursion level reached"));
566 std::auto_ptr<epvector> s(new epvector);
567 s->reserve(seq.size());
570 epvector::const_iterator i = seq.begin(), end = seq.end();
572 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
576 return mul(s, overall_coeff.evalf(level));
579 void mul::find_real_imag(ex & rp, ex & ip) const
581 rp = overall_coeff.real_part();
582 ip = overall_coeff.imag_part();
583 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
584 ex factor = recombine_pair_to_ex(*i);
585 ex new_rp = factor.real_part();
586 ex new_ip = factor.imag_part();
587 if(new_ip.is_zero()) {
591 ex temp = rp*new_rp - ip*new_ip;
592 ip = ip*new_rp + rp*new_ip;
600 ex mul::real_part() const
603 find_real_imag(rp, ip);
607 ex mul::imag_part() const
610 find_real_imag(rp, ip);
614 ex mul::evalm() const
617 if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
618 && is_a<matrix>(seq[0].rest))
619 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
621 // Evaluate children first, look whether there are any matrices at all
622 // (there can be either no matrices or one matrix; if there were more
623 // than one matrix, it would be a non-commutative product)
624 std::auto_ptr<epvector> s(new epvector);
625 s->reserve(seq.size());
627 bool have_matrix = false;
628 epvector::iterator the_matrix;
630 epvector::const_iterator i = seq.begin(), end = seq.end();
632 const ex &m = recombine_pair_to_ex(*i).evalm();
633 s->push_back(split_ex_to_pair(m));
634 if (is_a<matrix>(m)) {
636 the_matrix = s->end() - 1;
643 // The product contained a matrix. We will multiply all other factors
645 matrix m = ex_to<matrix>(the_matrix->rest);
646 s->erase(the_matrix);
647 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
648 return m.mul_scalar(scalar);
651 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
654 ex mul::eval_ncmul(const exvector & v) const
657 return inherited::eval_ncmul(v);
659 // Find first noncommutative element and call its eval_ncmul()
660 epvector::const_iterator i = seq.begin(), end = seq.end();
662 if (i->rest.return_type() == return_types::noncommutative)
663 return i->rest.eval_ncmul(v);
666 return inherited::eval_ncmul(v);
669 bool tryfactsubs(const ex & origfactor, const ex & patternfactor, int & nummatches, exmap& repls)
675 if (is_exactly_a<power>(origfactor) && origfactor.op(1).info(info_flags::integer)) {
676 origbase = origfactor.op(0);
677 int expon = ex_to<numeric>(origfactor.op(1)).to_int();
678 origexponent = expon > 0 ? expon : -expon;
679 origexpsign = expon > 0 ? 1 : -1;
681 origbase = origfactor;
690 if (is_exactly_a<power>(patternfactor) && patternfactor.op(1).info(info_flags::integer)) {
691 patternbase = patternfactor.op(0);
692 int expon = ex_to<numeric>(patternfactor.op(1)).to_int();
693 patternexponent = expon > 0 ? expon : -expon;
694 patternexpsign = expon > 0 ? 1 : -1;
696 patternbase = patternfactor;
701 exmap saverepls = repls;
702 if (origexponent < patternexponent || origexpsign != patternexpsign || !origbase.match(patternbase,saverepls))
706 int newnummatches = origexponent / patternexponent;
707 if (newnummatches < nummatches)
708 nummatches = newnummatches;
712 /** Checks wheter e matches to the pattern pat and the (possibly to be updated)
713 * list of replacements repls. This matching is in the sense of algebraic
714 * substitutions. Matching starts with pat.op(factor) of the pattern because
715 * the factors before this one have already been matched. The (possibly
716 * updated) number of matches is in nummatches. subsed[i] is true for factors
717 * that already have been replaced by previous substitutions and matched[i]
718 * is true for factors that have been matched by the current match.
720 bool algebraic_match_mul_with_mul(const mul &e, const ex &pat, exmap& repls,
721 int factor, int &nummatches, const std::vector<bool> &subsed,
722 std::vector<bool> &matched)
724 GINAC_ASSERT(subsed.size() == e.nops());
725 GINAC_ASSERT(matched.size() == e.nops());
727 if (factor == (int)pat.nops())
730 for (size_t i=0; i<e.nops(); ++i) {
731 if(subsed[i] || matched[i])
733 exmap newrepls = repls;
734 int newnummatches = nummatches;
735 if (tryfactsubs(e.op(i), pat.op(factor), newnummatches, newrepls)) {
737 if (algebraic_match_mul_with_mul(e, pat, newrepls, factor+1,
738 newnummatches, subsed, matched)) {
740 nummatches = newnummatches;
751 bool mul::has(const ex & pattern, unsigned options) const
753 if(!(options&has_options::algebraic))
754 return basic::has(pattern,options);
755 if(is_a<mul>(pattern)) {
757 int nummatches = std::numeric_limits<int>::max();
758 std::vector<bool> subsed(nops(), false);
759 std::vector<bool> matched(nops(), false);
760 if(algebraic_match_mul_with_mul(*this, pattern, repls, 0, nummatches,
764 return basic::has(pattern, options);
767 ex mul::algebraic_subs_mul(const exmap & m, unsigned options) const
769 std::vector<bool> subsed(nops(), false);
773 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
775 if (is_exactly_a<mul>(it->first)) {
777 int nummatches = std::numeric_limits<int>::max();
778 std::vector<bool> currsubsed(nops(), false);
781 if(!algebraic_match_mul_with_mul(*this, it->first, repls, 0, nummatches, subsed, currsubsed))
784 for (size_t j=0; j<subsed.size(); j++)
788 = it->first.subs(repls, subs_options::no_pattern);
789 divide_by *= power(subsed_pattern, nummatches);
791 = it->second.subs(repls, subs_options::no_pattern);
792 multiply_by *= power(subsed_result, nummatches);
797 for (size_t j=0; j<this->nops(); j++) {
798 int nummatches = std::numeric_limits<int>::max();
800 if (!subsed[j] && tryfactsubs(op(j), it->first, nummatches, repls)){
803 = it->first.subs(repls, subs_options::no_pattern);
804 divide_by *= power(subsed_pattern, nummatches);
806 = it->second.subs(repls, subs_options::no_pattern);
807 multiply_by *= power(subsed_result, nummatches);
813 bool subsfound = false;
814 for (size_t i=0; i<subsed.size(); i++) {
821 return subs_one_level(m, options | subs_options::algebraic);
823 return ((*this)/divide_by)*multiply_by;
826 ex mul::conjugate() const
828 // The base class' method is wrong here because we have to be careful at
829 // branch cuts. power::conjugate takes care of that already, so use it.
830 epvector *newepv = 0;
831 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
833 newepv->push_back(split_ex_to_pair(recombine_pair_to_ex(*i).conjugate()));
836 ex x = recombine_pair_to_ex(*i);
837 ex c = x.conjugate();
841 newepv = new epvector;
842 newepv->reserve(seq.size());
843 for (epvector::const_iterator j=seq.begin(); j!=i; ++j) {
844 newepv->push_back(*j);
846 newepv->push_back(split_ex_to_pair(c));
848 ex x = overall_coeff.conjugate();
849 if (!newepv && are_ex_trivially_equal(x, overall_coeff)) {
852 ex result = thisexpairseq(newepv ? *newepv : seq, x);
860 /** Implementation of ex::diff() for a product. It applies the product rule.
862 ex mul::derivative(const symbol & s) const
864 size_t num = seq.size();
868 // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
869 epvector mulseq = seq;
870 epvector::const_iterator i = seq.begin(), end = seq.end();
871 epvector::iterator i2 = mulseq.begin();
873 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
876 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
880 return (new add(addseq))->setflag(status_flags::dynallocated);
883 int mul::compare_same_type(const basic & other) const
885 return inherited::compare_same_type(other);
888 unsigned mul::return_type() const
891 // mul without factors: should not happen, but commutates
892 return return_types::commutative;
895 bool all_commutative = true;
896 epvector::const_iterator noncommutative_element; // point to first found nc element
898 epvector::const_iterator i = seq.begin(), end = seq.end();
900 unsigned rt = i->rest.return_type();
901 if (rt == return_types::noncommutative_composite)
902 return rt; // one ncc -> mul also ncc
903 if ((rt == return_types::noncommutative) && (all_commutative)) {
904 // first nc element found, remember position
905 noncommutative_element = i;
906 all_commutative = false;
908 if ((rt == return_types::noncommutative) && (!all_commutative)) {
909 // another nc element found, compare type_infos
910 if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
911 // different types -> mul is ncc
912 return return_types::noncommutative_composite;
917 // all factors checked
918 return all_commutative ? return_types::commutative : return_types::noncommutative;
921 return_type_t mul::return_type_tinfo() const
924 return make_return_type_t<mul>(); // mul without factors: should not happen
926 // return type_info of first noncommutative element
927 epvector::const_iterator i = seq.begin(), end = seq.end();
929 if (i->rest.return_type() == return_types::noncommutative)
930 return i->rest.return_type_tinfo();
933 // no noncommutative element found, should not happen
934 return make_return_type_t<mul>();
937 ex mul::thisexpairseq(const epvector & v, const ex & oc, bool do_index_renaming) const
939 return (new mul(v, oc, do_index_renaming))->setflag(status_flags::dynallocated);
942 ex mul::thisexpairseq(std::auto_ptr<epvector> vp, const ex & oc, bool do_index_renaming) const
944 return (new mul(vp, oc, do_index_renaming))->setflag(status_flags::dynallocated);
947 expair mul::split_ex_to_pair(const ex & e) const
949 if (is_exactly_a<power>(e)) {
950 const power & powerref = ex_to<power>(e);
951 if (is_exactly_a<numeric>(powerref.exponent))
952 return expair(powerref.basis,powerref.exponent);
954 return expair(e,_ex1);
957 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
960 // to avoid duplication of power simplification rules,
961 // we create a temporary power object
962 // otherwise it would be hard to correctly evaluate
963 // expression like (4^(1/3))^(3/2)
964 if (c.is_equal(_ex1))
965 return split_ex_to_pair(e);
967 return split_ex_to_pair(power(e,c));
970 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
973 // to avoid duplication of power simplification rules,
974 // we create a temporary power object
975 // otherwise it would be hard to correctly evaluate
976 // expression like (4^(1/3))^(3/2)
977 if (c.is_equal(_ex1))
980 return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
983 ex mul::recombine_pair_to_ex(const expair & p) const
985 if (ex_to<numeric>(p.coeff).is_equal(*_num1_p))
988 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
991 bool mul::expair_needs_further_processing(epp it)
993 if (is_exactly_a<mul>(it->rest) &&
994 ex_to<numeric>(it->coeff).is_integer()) {
995 // combined pair is product with integer power -> expand it
996 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
999 if (is_exactly_a<numeric>(it->rest)) {
1000 if (it->coeff.is_equal(_ex1)) {
1001 // pair has coeff 1 and must be moved to the end
1004 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
1005 if (!ep.is_equal(*it)) {
1006 // combined pair is a numeric power which can be simplified
1014 ex mul::default_overall_coeff() const
1019 void mul::combine_overall_coeff(const ex & c)
1021 GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1022 GINAC_ASSERT(is_exactly_a<numeric>(c));
1023 overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
1026 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
1028 GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
1029 GINAC_ASSERT(is_exactly_a<numeric>(c1));
1030 GINAC_ASSERT(is_exactly_a<numeric>(c2));
1031 overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
1034 bool mul::can_make_flat(const expair & p) const
1036 GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
1037 // this assertion will probably fail somewhere
1038 // it would require a more careful make_flat, obeying the power laws
1039 // probably should return true only if p.coeff is integer
1040 return ex_to<numeric>(p.coeff).is_equal(*_num1_p);
1043 bool mul::can_be_further_expanded(const ex & e)
1045 if (is_exactly_a<mul>(e)) {
1046 for (epvector::const_iterator cit = ex_to<mul>(e).seq.begin(); cit != ex_to<mul>(e).seq.end(); ++cit) {
1047 if (is_exactly_a<add>(cit->rest) && cit->coeff.info(info_flags::posint))
1050 } else if (is_exactly_a<power>(e)) {
1051 if (is_exactly_a<add>(e.op(0)) && e.op(1).info(info_flags::posint))
1057 ex mul::expand(unsigned options) const
1060 // trivial case: expanding the monomial (~ 30% of all calls)
1061 epvector::const_iterator i = seq.begin(), seq_end = seq.end();
1062 while ((i != seq.end()) && is_a<symbol>(i->rest) && i->coeff.info(info_flags::integer))
1065 setflag(status_flags::expanded);
1070 // do not rename indices if the object has no indices at all
1071 if ((!(options & expand_options::expand_rename_idx)) &&
1072 this->info(info_flags::has_indices))
1073 options |= expand_options::expand_rename_idx;
1075 const bool skip_idx_rename = !(options & expand_options::expand_rename_idx);
1077 // First, expand the children
1078 std::auto_ptr<epvector> expanded_seqp = expandchildren(options);
1079 const epvector & expanded_seq = (expanded_seqp.get() ? *expanded_seqp : seq);
1081 // Now, look for all the factors that are sums and multiply each one out
1082 // with the next one that is found while collecting the factors which are
1084 ex last_expanded = _ex1;
1087 non_adds.reserve(expanded_seq.size());
1089 for (epvector::const_iterator cit = expanded_seq.begin(); cit != expanded_seq.end(); ++cit) {
1090 if (is_exactly_a<add>(cit->rest) &&
1091 (cit->coeff.is_equal(_ex1))) {
1092 if (is_exactly_a<add>(last_expanded)) {
1094 // Expand a product of two sums, aggressive version.
1095 // Caring for the overall coefficients in separate loops can
1096 // sometimes give a performance gain of up to 15%!
1098 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
1099 // add2 is for the inner loop and should be the bigger of the two sums
1100 // in the presence of asymptotically good sorting:
1101 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
1102 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
1103 const epvector::const_iterator add1begin = add1.seq.begin();
1104 const epvector::const_iterator add1end = add1.seq.end();
1105 const epvector::const_iterator add2begin = add2.seq.begin();
1106 const epvector::const_iterator add2end = add2.seq.end();
1108 distrseq.reserve(add1.seq.size()+add2.seq.size());
1110 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
1111 if (!add1.overall_coeff.is_zero()) {
1112 if (add1.overall_coeff.is_equal(_ex1))
1113 distrseq.insert(distrseq.end(),add2begin,add2end);
1115 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
1116 distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
1119 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
1120 if (!add2.overall_coeff.is_zero()) {
1121 if (add2.overall_coeff.is_equal(_ex1))
1122 distrseq.insert(distrseq.end(),add1begin,add1end);
1124 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
1125 distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
1128 // Compute the new overall coefficient and put it together:
1129 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
1131 exvector add1_dummy_indices, add2_dummy_indices, add_indices;
1134 if (!skip_idx_rename) {
1135 for (epvector::const_iterator i=add1begin; i!=add1end; ++i) {
1136 add_indices = get_all_dummy_indices_safely(i->rest);
1137 add1_dummy_indices.insert(add1_dummy_indices.end(), add_indices.begin(), add_indices.end());
1139 for (epvector::const_iterator i=add2begin; i!=add2end; ++i) {
1140 add_indices = get_all_dummy_indices_safely(i->rest);
1141 add2_dummy_indices.insert(add2_dummy_indices.end(), add_indices.begin(), add_indices.end());
1144 sort(add1_dummy_indices.begin(), add1_dummy_indices.end(), ex_is_less());
1145 sort(add2_dummy_indices.begin(), add2_dummy_indices.end(), ex_is_less());
1146 dummy_subs = rename_dummy_indices_uniquely(add1_dummy_indices, add2_dummy_indices);
1149 // Multiply explicitly all non-numeric terms of add1 and add2:
1150 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
1151 // We really have to combine terms here in order to compactify
1152 // the result. Otherwise it would become waayy tooo bigg.
1153 numeric oc(*_num0_p);
1155 distrseq2.reserve(add1.seq.size());
1156 const ex i2_new = (skip_idx_rename || (dummy_subs.op(0).nops() == 0) ?
1158 i2->rest.subs(ex_to<lst>(dummy_subs.op(0)),
1159 ex_to<lst>(dummy_subs.op(1)), subs_options::no_pattern));
1160 for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
1161 // Don't push_back expairs which might have a rest that evaluates to a numeric,
1162 // since that would violate an invariant of expairseq:
1163 const ex rest = (new mul(i1->rest, i2_new))->setflag(status_flags::dynallocated);
1164 if (is_exactly_a<numeric>(rest)) {
1165 oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
1167 distrseq2.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
1170 tmp_accu += (new add(distrseq2, oc))->setflag(status_flags::dynallocated);
1172 last_expanded = tmp_accu;
1174 if (!last_expanded.is_equal(_ex1))
1175 non_adds.push_back(split_ex_to_pair(last_expanded));
1176 last_expanded = cit->rest;
1180 non_adds.push_back(*cit);
1184 // Now the only remaining thing to do is to multiply the factors which
1185 // were not sums into the "last_expanded" sum
1186 if (is_exactly_a<add>(last_expanded)) {
1187 size_t n = last_expanded.nops();
1189 distrseq.reserve(n);
1191 if (! skip_idx_rename) {
1192 va = get_all_dummy_indices_safely(mul(non_adds));
1193 sort(va.begin(), va.end(), ex_is_less());
1196 for (size_t i=0; i<n; ++i) {
1197 epvector factors = non_adds;
1198 if (skip_idx_rename)
1199 factors.push_back(split_ex_to_pair(last_expanded.op(i)));
1201 factors.push_back(split_ex_to_pair(rename_dummy_indices_uniquely(va, last_expanded.op(i))));
1202 ex term = (new mul(factors, overall_coeff))->setflag(status_flags::dynallocated);
1203 if (can_be_further_expanded(term)) {
1204 distrseq.push_back(term.expand());
1207 ex_to<basic>(term).setflag(status_flags::expanded);
1208 distrseq.push_back(term);
1212 return ((new add(distrseq))->
1213 setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
1216 non_adds.push_back(split_ex_to_pair(last_expanded));
1217 ex result = (new mul(non_adds, overall_coeff))->setflag(status_flags::dynallocated);
1218 if (can_be_further_expanded(result)) {
1219 return result.expand();
1222 ex_to<basic>(result).setflag(status_flags::expanded);
1229 // new virtual functions which can be overridden by derived classes
1235 // non-virtual functions in this class
1239 /** Member-wise expand the expairs representing this sequence. This must be
1240 * overridden from expairseq::expandchildren() and done iteratively in order
1241 * to allow for early cancallations and thus safe memory.
1243 * @see mul::expand()
1244 * @return pointer to epvector containing expanded representation or zero
1245 * pointer, if sequence is unchanged. */
1246 std::auto_ptr<epvector> mul::expandchildren(unsigned options) const
1248 const epvector::const_iterator last = seq.end();
1249 epvector::const_iterator cit = seq.begin();
1251 const ex & factor = recombine_pair_to_ex(*cit);
1252 const ex & expanded_factor = factor.expand(options);
1253 if (!are_ex_trivially_equal(factor,expanded_factor)) {
1255 // something changed, copy seq, eval and return it
1256 std::auto_ptr<epvector> s(new epvector);
1257 s->reserve(seq.size());
1259 // copy parts of seq which are known not to have changed
1260 epvector::const_iterator cit2 = seq.begin();
1262 s->push_back(*cit2);
1266 // copy first changed element
1267 s->push_back(split_ex_to_pair(expanded_factor));
1271 while (cit2!=last) {
1272 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
1280 return std::auto_ptr<epvector>(0); // nothing has changed
1283 GINAC_BIND_UNARCHIVER(mul);
1285 } // namespace GiNaC