3 * Implementation of GiNaC's products of expressions. */
6 * GiNaC Copyright (C) 1999-2002 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
30 #include "operators.h"
37 GINAC_IMPLEMENT_REGISTERED_CLASS(mul, expairseq)
40 // default ctor, dtor, copy ctor, assignment operator and helpers
45 tinfo_key = TINFO_mul;
57 mul::mul(const ex & lh, const ex & rh)
59 tinfo_key = TINFO_mul;
61 construct_from_2_ex(lh,rh);
62 GINAC_ASSERT(is_canonical());
65 mul::mul(const exvector & v)
67 tinfo_key = TINFO_mul;
69 construct_from_exvector(v);
70 GINAC_ASSERT(is_canonical());
73 mul::mul(const epvector & v)
75 tinfo_key = TINFO_mul;
77 construct_from_epvector(v);
78 GINAC_ASSERT(is_canonical());
81 mul::mul(const epvector & v, const ex & oc)
83 tinfo_key = TINFO_mul;
85 construct_from_epvector(v);
86 GINAC_ASSERT(is_canonical());
89 mul::mul(epvector * vp, const ex & oc)
91 tinfo_key = TINFO_mul;
94 construct_from_epvector(*vp);
96 GINAC_ASSERT(is_canonical());
99 mul::mul(const ex & lh, const ex & mh, const ex & rh)
101 tinfo_key = TINFO_mul;
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());
116 DEFAULT_ARCHIVING(mul)
119 // functions overriding virtual functions from base classes
123 void mul::print(const print_context & c, unsigned level) const
125 if (is_a<print_tree>(c)) {
127 inherited::print(c, level);
129 } else if (is_a<print_csrc>(c)) {
131 if (precedence() <= level)
134 if (!overall_coeff.is_equal(_ex1)) {
135 overall_coeff.print(c, precedence());
139 // Print arguments, separated by "*" or "/"
140 epvector::const_iterator it = seq.begin(), itend = seq.end();
141 while (it != itend) {
143 // If the first argument is a negative integer power, it gets printed as "1.0/<expr>"
144 if (it == seq.begin() && ex_to<numeric>(it->coeff).is_integer() && it->coeff.info(info_flags::negative)) {
145 if (is_a<print_csrc_cl_N>(c))
151 // If the exponent is 1 or -1, it is left out
152 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
153 it->rest.print(c, precedence());
155 // Outer parens around ex needed for broken gcc-2.95 parser:
156 (ex(power(it->rest, abs(ex_to<numeric>(it->coeff))))).print(c, level);
159 // Separator is "/" for negative integer powers, "*" otherwise
162 if (ex_to<numeric>(it->coeff).is_integer() && it->coeff.info(info_flags::negative))
169 if (precedence() <= level)
172 } else if (is_a<print_python_repr>(c)) {
173 c.s << class_name() << '(';
175 for (unsigned i=1; i<nops(); ++i) {
182 if (precedence() <= level) {
183 if (is_a<print_latex>(c))
191 // First print the overall numeric coefficient
192 const numeric &coeff = ex_to<numeric>(overall_coeff);
193 if (coeff.csgn() == -1)
195 if (!coeff.is_equal(_num1) &&
196 !coeff.is_equal(_num_1)) {
197 if (coeff.is_rational()) {
198 if (coeff.is_negative())
203 if (coeff.csgn() == -1)
204 (-coeff).print(c, precedence());
206 coeff.print(c, precedence());
208 if (is_a<print_latex>(c))
214 // Then proceed with the remaining factors
215 epvector::const_iterator it = seq.begin(), itend = seq.end();
216 while (it != itend) {
218 if (is_a<print_latex>(c))
225 recombine_pair_to_ex(*it).print(c, precedence());
229 if (precedence() <= level) {
230 if (is_a<print_latex>(c))
238 bool mul::info(unsigned inf) const
241 case info_flags::polynomial:
242 case info_flags::integer_polynomial:
243 case info_flags::cinteger_polynomial:
244 case info_flags::rational_polynomial:
245 case info_flags::crational_polynomial:
246 case info_flags::rational_function: {
247 epvector::const_iterator i = seq.begin(), end = seq.end();
249 if (!(recombine_pair_to_ex(*i).info(inf)))
253 return overall_coeff.info(inf);
255 case info_flags::algebraic: {
256 epvector::const_iterator i = seq.begin(), end = seq.end();
258 if ((recombine_pair_to_ex(*i).info(inf)))
265 return inherited::info(inf);
268 int mul::degree(const ex & s) const
270 // Sum up degrees of factors
272 epvector::const_iterator i = seq.begin(), end = seq.end();
274 if (ex_to<numeric>(i->coeff).is_integer())
275 deg_sum += i->rest.degree(s) * ex_to<numeric>(i->coeff).to_int();
281 int mul::ldegree(const ex & s) const
283 // Sum up degrees of factors
285 epvector::const_iterator i = seq.begin(), end = seq.end();
287 if (ex_to<numeric>(i->coeff).is_integer())
288 deg_sum += i->rest.ldegree(s) * ex_to<numeric>(i->coeff).to_int();
294 ex mul::coeff(const ex & s, int n) const
297 coeffseq.reserve(seq.size()+1);
300 // product of individual coeffs
301 // if a non-zero power of s is found, the resulting product will be 0
302 epvector::const_iterator i = seq.begin(), end = seq.end();
304 coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
307 coeffseq.push_back(overall_coeff);
308 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
311 epvector::const_iterator i = seq.begin(), end = seq.end();
312 bool coeff_found = false;
314 ex t = recombine_pair_to_ex(*i);
315 ex c = t.coeff(s, n);
317 coeffseq.push_back(c);
320 coeffseq.push_back(t);
325 coeffseq.push_back(overall_coeff);
326 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
332 /** Perform automatic term rewriting rules in this class. In the following
333 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
334 * stand for such expressions that contain a plain number.
336 * - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
340 * @param level cut-off in recursive evaluation */
341 ex mul::eval(int level) const
343 epvector *evaled_seqp = evalchildren(level);
345 // do more evaluation later
346 return (new mul(evaled_seqp,overall_coeff))->
347 setflag(status_flags::dynallocated);
350 #ifdef DO_GINAC_ASSERT
351 epvector::const_iterator i = seq.begin(), end = seq.end();
353 GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
354 (!(ex_to<numeric>(i->coeff).is_integer())));
355 GINAC_ASSERT(!(i->is_canonical_numeric()));
356 if (is_exactly_a<numeric>(recombine_pair_to_ex(*i)))
357 print(print_tree(std::cerr));
358 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
360 expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
361 GINAC_ASSERT(p.rest.is_equal(i->rest));
362 GINAC_ASSERT(p.coeff.is_equal(i->coeff));
366 #endif // def DO_GINAC_ASSERT
368 if (flags & status_flags::evaluated) {
369 GINAC_ASSERT(seq.size()>0);
370 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
374 int seq_size = seq.size();
375 if (overall_coeff.is_zero()) {
378 } else if (seq_size==0) {
380 return overall_coeff;
381 } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
383 return recombine_pair_to_ex(*(seq.begin()));
384 } else if ((seq_size==1) &&
385 is_exactly_a<add>((*seq.begin()).rest) &&
386 ex_to<numeric>((*seq.begin()).coeff).is_equal(_num1)) {
387 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
388 const add & addref = ex_to<add>((*seq.begin()).rest);
389 epvector *distrseq = new epvector();
390 distrseq->reserve(addref.seq.size());
391 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
393 distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
396 return (new add(distrseq,
397 ex_to<numeric>(addref.overall_coeff).
398 mul_dyn(ex_to<numeric>(overall_coeff))))
399 ->setflag(status_flags::dynallocated | status_flags::evaluated);
404 ex mul::evalf(int level) const
407 return mul(seq,overall_coeff);
409 if (level==-max_recursion_level)
410 throw(std::runtime_error("max recursion level reached"));
412 epvector *s = new epvector();
413 s->reserve(seq.size());
416 epvector::const_iterator i = seq.begin(), end = seq.end();
418 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
422 return mul(s, overall_coeff.evalf(level));
425 ex mul::evalm(void) const
428 if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
429 && is_a<matrix>(seq[0].rest))
430 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
432 // Evaluate children first, look whether there are any matrices at all
433 // (there can be either no matrices or one matrix; if there were more
434 // than one matrix, it would be a non-commutative product)
435 epvector *s = new epvector;
436 s->reserve(seq.size());
438 bool have_matrix = false;
439 epvector::iterator the_matrix;
441 epvector::const_iterator i = seq.begin(), end = seq.end();
443 const ex &m = recombine_pair_to_ex(*i).evalm();
444 s->push_back(split_ex_to_pair(m));
445 if (is_a<matrix>(m)) {
447 the_matrix = s->end() - 1;
454 // The product contained a matrix. We will multiply all other factors
456 matrix m = ex_to<matrix>(the_matrix->rest);
457 s->erase(the_matrix);
458 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
459 return m.mul_scalar(scalar);
462 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
465 ex mul::simplify_ncmul(const exvector & v) const
468 return inherited::simplify_ncmul(v);
470 // Find first noncommutative element and call its simplify_ncmul()
471 epvector::const_iterator i = seq.begin(), end = seq.end();
473 if (i->rest.return_type() == return_types::noncommutative)
474 return i->rest.simplify_ncmul(v);
477 return inherited::simplify_ncmul(v);
482 /** Implementation of ex::diff() for a product. It applies the product rule.
484 ex mul::derivative(const symbol & s) const
486 unsigned num = seq.size();
490 // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
491 epvector mulseq = seq;
492 epvector::const_iterator i = seq.begin(), end = seq.end();
493 epvector::iterator i2 = mulseq.begin();
495 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
498 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
502 return (new add(addseq))->setflag(status_flags::dynallocated);
505 int mul::compare_same_type(const basic & other) const
507 return inherited::compare_same_type(other);
510 unsigned mul::return_type(void) const
513 // mul without factors: should not happen, but commutes
514 return return_types::commutative;
517 bool all_commutative = true;
518 epvector::const_iterator noncommutative_element; // point to first found nc element
520 epvector::const_iterator i = seq.begin(), end = seq.end();
522 unsigned rt = i->rest.return_type();
523 if (rt == return_types::noncommutative_composite)
524 return rt; // one ncc -> mul also ncc
525 if ((rt == return_types::noncommutative) && (all_commutative)) {
526 // first nc element found, remember position
527 noncommutative_element = i;
528 all_commutative = false;
530 if ((rt == return_types::noncommutative) && (!all_commutative)) {
531 // another nc element found, compare type_infos
532 if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
533 // diffent types -> mul is ncc
534 return return_types::noncommutative_composite;
539 // all factors checked
540 return all_commutative ? return_types::commutative : return_types::noncommutative;
543 unsigned mul::return_type_tinfo(void) const
546 return tinfo_key; // mul without factors: should not happen
548 // return type_info of first noncommutative element
549 epvector::const_iterator i = seq.begin(), end = seq.end();
551 if (i->rest.return_type() == return_types::noncommutative)
552 return i->rest.return_type_tinfo();
555 // no noncommutative element found, should not happen
559 ex mul::thisexpairseq(const epvector & v, const ex & oc) const
561 return (new mul(v, oc))->setflag(status_flags::dynallocated);
564 ex mul::thisexpairseq(epvector * vp, const ex & oc) const
566 return (new mul(vp, oc))->setflag(status_flags::dynallocated);
569 expair mul::split_ex_to_pair(const ex & e) const
571 if (is_exactly_a<power>(e)) {
572 const power & powerref = ex_to<power>(e);
573 if (is_exactly_a<numeric>(powerref.exponent))
574 return expair(powerref.basis,powerref.exponent);
576 return expair(e,_ex1);
579 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
582 // to avoid duplication of power simplification rules,
583 // we create a temporary power object
584 // otherwise it would be hard to correctly evaluate
585 // expression like (4^(1/3))^(3/2)
586 if (c.is_equal(_ex1))
587 return split_ex_to_pair(e);
589 return split_ex_to_pair(power(e,c));
592 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
595 // to avoid duplication of power simplification rules,
596 // we create a temporary power object
597 // otherwise it would be hard to correctly evaluate
598 // expression like (4^(1/3))^(3/2)
599 if (c.is_equal(_ex1))
602 return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
605 ex mul::recombine_pair_to_ex(const expair & p) const
607 if (ex_to<numeric>(p.coeff).is_equal(_num1))
610 return (new power(p.rest,p.coeff))->setflag(status_flags::dynallocated);
613 bool mul::expair_needs_further_processing(epp it)
615 if (is_exactly_a<mul>(it->rest) &&
616 ex_to<numeric>(it->coeff).is_integer()) {
617 // combined pair is product with integer power -> expand it
618 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
621 if (is_exactly_a<numeric>(it->rest)) {
622 expair ep = split_ex_to_pair(recombine_pair_to_ex(*it));
623 if (!ep.is_equal(*it)) {
624 // combined pair is a numeric power which can be simplified
628 if (it->coeff.is_equal(_ex1)) {
629 // combined pair has coeff 1 and must be moved to the end
636 ex mul::default_overall_coeff(void) const
641 void mul::combine_overall_coeff(const ex & c)
643 GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
644 GINAC_ASSERT(is_exactly_a<numeric>(c));
645 overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
648 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
650 GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
651 GINAC_ASSERT(is_exactly_a<numeric>(c1));
652 GINAC_ASSERT(is_exactly_a<numeric>(c2));
653 overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
656 bool mul::can_make_flat(const expair & p) const
658 GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
659 // this assertion will probably fail somewhere
660 // it would require a more careful make_flat, obeying the power laws
661 // probably should return true only if p.coeff is integer
662 return ex_to<numeric>(p.coeff).is_equal(_num1);
665 ex mul::expand(unsigned options) const
667 // First, expand the children
668 epvector * expanded_seqp = expandchildren(options);
669 const epvector & expanded_seq = (expanded_seqp == NULL) ? seq : *expanded_seqp;
671 // Now, look for all the factors that are sums and multiply each one out
672 // with the next one that is found while collecting the factors which are
674 int number_of_adds = 0;
675 ex last_expanded = _ex1;
677 non_adds.reserve(expanded_seq.size());
678 epvector::const_iterator cit = expanded_seq.begin(), last = expanded_seq.end();
679 while (cit != last) {
680 if (is_exactly_a<add>(cit->rest) &&
681 (cit->coeff.is_equal(_ex1))) {
683 if (is_exactly_a<add>(last_expanded)) {
685 // Expand a product of two sums, simple and robust version.
686 const add & add1 = ex_to<add>(last_expanded);
687 const add & add2 = ex_to<add>(cit->rest);
688 const int n1 = add1.nops();
689 const int n2 = add2.nops();
692 distrseq.reserve(n2);
693 for (int i1=0; i1<n1; ++i1) {
695 // cache the first operand (for efficiency):
696 const ex op1 = add1.op(i1);
697 for (int i2=0; i2<n2; ++i2)
698 distrseq.push_back(op1 * add2.op(i2));
699 tmp_accu += (new add(distrseq))->
700 setflag(status_flags::dynallocated);
702 last_expanded = tmp_accu;
704 // Expand a product of two sums, aggressive version.
705 // Caring for the overall coefficients in separate loops can
706 // sometimes give a performance gain of up to 15%!
708 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
709 // add2 is for the inner loop and should be the bigger of the two sums
710 // in the presence of asymptotically good sorting:
711 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
712 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
713 const epvector::const_iterator add1begin = add1.seq.begin();
714 const epvector::const_iterator add1end = add1.seq.end();
715 const epvector::const_iterator add2begin = add2.seq.begin();
716 const epvector::const_iterator add2end = add2.seq.end();
718 distrseq.reserve(add1.seq.size()+add2.seq.size());
719 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
720 if (!add1.overall_coeff.is_zero()) {
721 if (add1.overall_coeff.is_equal(_ex1))
722 distrseq.insert(distrseq.end(),add2begin,add2end);
724 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
725 distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
727 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
728 if (!add2.overall_coeff.is_zero()) {
729 if (add2.overall_coeff.is_equal(_ex1))
730 distrseq.insert(distrseq.end(),add1begin,add1end);
732 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
733 distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
735 // Compute the new overall coefficient and put it together:
736 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
737 // Multiply explicitly all non-numeric terms of add1 and add2:
738 for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
739 // We really have to combine terms here in order to compactify
740 // the result. Otherwise it would become waayy tooo bigg.
743 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
744 // Don't push_back expairs which might have a rest that evaluates to a numeric,
745 // since that would violate an invariant of expairseq:
746 const ex rest = (new mul(i1->rest, i2->rest))->setflag(status_flags::dynallocated);
747 if (is_exactly_a<numeric>(rest))
748 oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
750 distrseq.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
752 tmp_accu += (new add(distrseq, oc))->setflag(status_flags::dynallocated);
754 last_expanded = tmp_accu;
757 non_adds.push_back(split_ex_to_pair(last_expanded));
758 last_expanded = cit->rest;
761 non_adds.push_back(*cit);
766 delete expanded_seqp;
768 // Now the only remaining thing to do is to multiply the factors which
769 // were not sums into the "last_expanded" sum
770 if (is_exactly_a<add>(last_expanded)) {
771 const add & finaladd = ex_to<add>(last_expanded);
773 int n = finaladd.nops();
775 for (int i=0; i<n; ++i) {
776 epvector factors = non_adds;
777 factors.push_back(split_ex_to_pair(finaladd.op(i)));
778 distrseq.push_back((new mul(factors, overall_coeff))->
779 setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
781 return ((new add(distrseq))->
782 setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
784 non_adds.push_back(split_ex_to_pair(last_expanded));
785 return (new mul(non_adds, overall_coeff))->
786 setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
791 // new virtual functions which can be overridden by derived classes
797 // non-virtual functions in this class
801 /** Member-wise expand the expairs representing this sequence. This must be
802 * overridden from expairseq::expandchildren() and done iteratively in order
803 * to allow for early cancallations and thus safe memory.
806 * @return pointer to epvector containing expanded representation or zero
807 * pointer, if sequence is unchanged. */
808 epvector * mul::expandchildren(unsigned options) const
810 const epvector::const_iterator last = seq.end();
811 epvector::const_iterator cit = seq.begin();
813 const ex & factor = recombine_pair_to_ex(*cit);
814 const ex & expanded_factor = factor.expand(options);
815 if (!are_ex_trivially_equal(factor,expanded_factor)) {
817 // something changed, copy seq, eval and return it
818 epvector *s = new epvector;
819 s->reserve(seq.size());
821 // copy parts of seq which are known not to have changed
822 epvector::const_iterator cit2 = seq.begin();
827 // copy first changed element
828 s->push_back(split_ex_to_pair(expanded_factor));
832 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
840 return 0; // nothing has changed