3 * Implementation of GiNaC's products of expressions. */
6 * GiNaC Copyright (C) 1999-2003 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
36 GINAC_IMPLEMENT_REGISTERED_CLASS(mul, expairseq)
39 // default ctor, dtor, copy ctor, assignment operator and helpers
44 tinfo_key = TINFO_mul;
56 mul::mul(const ex & lh, const ex & rh)
58 tinfo_key = TINFO_mul;
60 construct_from_2_ex(lh,rh);
61 GINAC_ASSERT(is_canonical());
64 mul::mul(const exvector & v)
66 tinfo_key = TINFO_mul;
68 construct_from_exvector(v);
69 GINAC_ASSERT(is_canonical());
72 mul::mul(const epvector & v)
74 tinfo_key = TINFO_mul;
76 construct_from_epvector(v);
77 GINAC_ASSERT(is_canonical());
80 mul::mul(const epvector & v, const ex & oc)
82 tinfo_key = TINFO_mul;
84 construct_from_epvector(v);
85 GINAC_ASSERT(is_canonical());
88 mul::mul(epvector * vp, const ex & oc)
90 tinfo_key = TINFO_mul;
93 construct_from_epvector(*vp);
95 GINAC_ASSERT(is_canonical());
98 mul::mul(const ex & lh, const ex & mh, const ex & rh)
100 tinfo_key = TINFO_mul;
103 factors.push_back(lh);
104 factors.push_back(mh);
105 factors.push_back(rh);
106 overall_coeff = _ex1;
107 construct_from_exvector(factors);
108 GINAC_ASSERT(is_canonical());
115 DEFAULT_ARCHIVING(mul)
118 // 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 bool needclosingparenthesis = false;
145 if (it == seq.begin() && it->coeff.info(info_flags::negint)) {
146 if (is_a<print_csrc_cl_N>(c)) {
148 needclosingparenthesis = true;
153 // If the exponent is 1 or -1, it is left out
154 if (it->coeff.is_equal(_ex1) || it->coeff.is_equal(_ex_1))
155 it->rest.print(c, precedence());
156 else if (it->coeff.info(info_flags::negint))
157 // Outer parens around ex needed for broken gcc-2.95 parser:
158 (ex(power(it->rest, -ex_to<numeric>(it->coeff)))).print(c, level);
160 // Outer parens around ex needed for broken gcc-2.95 parser:
161 (ex(power(it->rest, ex_to<numeric>(it->coeff)))).print(c, level);
163 if (needclosingparenthesis)
166 // Separator is "/" for negative integer powers, "*" otherwise
169 if (it->coeff.info(info_flags::negint))
176 if (precedence() <= level)
179 } else if (is_a<print_python_repr>(c)) {
180 c.s << class_name() << '(';
182 for (unsigned i=1; i<nops(); ++i) {
189 if (precedence() <= level) {
190 if (is_a<print_latex>(c))
196 // First print the overall numeric coefficient
197 numeric coeff = ex_to<numeric>(overall_coeff);
198 if (coeff.csgn() == -1)
200 if (!coeff.is_equal(_num1) &&
201 !coeff.is_equal(_num_1)) {
202 if (coeff.is_rational()) {
203 if (coeff.is_negative())
208 if (coeff.csgn() == -1)
209 (-coeff).print(c, precedence());
211 coeff.print(c, precedence());
213 if (is_a<print_latex>(c))
219 // Then proceed with the remaining factors
220 epvector::const_iterator it = seq.begin(), itend = seq.end();
221 if (is_a<print_latex>(c)) {
223 // Separate factors into those with negative numeric exponent
225 exvector neg_powers, others;
226 while (it != itend) {
227 GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
228 if (ex_to<numeric>(it->coeff).is_negative())
229 neg_powers.push_back(recombine_pair_to_ex(expair(it->rest, -(it->coeff))));
231 others.push_back(recombine_pair_to_ex(*it));
235 if (!neg_powers.empty()) {
237 // Factors with negative exponent are printed as a fraction
239 mul(others).eval().print(c);
241 mul(neg_powers).eval().print(c);
246 // All other factors are printed in the ordinary way
247 exvector::const_iterator vit = others.begin(), vitend = others.end();
248 while (vit != vitend) {
250 vit->print(c, precedence());
258 while (it != itend) {
263 recombine_pair_to_ex(*it).print(c, precedence());
268 if (precedence() <= level) {
269 if (is_a<print_latex>(c))
277 bool mul::info(unsigned inf) const
280 case info_flags::polynomial:
281 case info_flags::integer_polynomial:
282 case info_flags::cinteger_polynomial:
283 case info_flags::rational_polynomial:
284 case info_flags::crational_polynomial:
285 case info_flags::rational_function: {
286 epvector::const_iterator i = seq.begin(), end = seq.end();
288 if (!(recombine_pair_to_ex(*i).info(inf)))
292 return overall_coeff.info(inf);
294 case info_flags::algebraic: {
295 epvector::const_iterator i = seq.begin(), end = seq.end();
297 if ((recombine_pair_to_ex(*i).info(inf)))
304 return inherited::info(inf);
307 int mul::degree(const ex & s) const
309 // Sum up degrees of factors
311 epvector::const_iterator i = seq.begin(), end = seq.end();
313 if (ex_to<numeric>(i->coeff).is_integer())
314 deg_sum += i->rest.degree(s) * ex_to<numeric>(i->coeff).to_int();
320 int mul::ldegree(const ex & s) const
322 // Sum up degrees of factors
324 epvector::const_iterator i = seq.begin(), end = seq.end();
326 if (ex_to<numeric>(i->coeff).is_integer())
327 deg_sum += i->rest.ldegree(s) * ex_to<numeric>(i->coeff).to_int();
333 ex mul::coeff(const ex & s, int n) const
336 coeffseq.reserve(seq.size()+1);
339 // product of individual coeffs
340 // if a non-zero power of s is found, the resulting product will be 0
341 epvector::const_iterator i = seq.begin(), end = seq.end();
343 coeffseq.push_back(recombine_pair_to_ex(*i).coeff(s,n));
346 coeffseq.push_back(overall_coeff);
347 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
350 epvector::const_iterator i = seq.begin(), end = seq.end();
351 bool coeff_found = false;
353 ex t = recombine_pair_to_ex(*i);
354 ex c = t.coeff(s, n);
356 coeffseq.push_back(c);
359 coeffseq.push_back(t);
364 coeffseq.push_back(overall_coeff);
365 return (new mul(coeffseq))->setflag(status_flags::dynallocated);
371 /** Perform automatic term rewriting rules in this class. In the following
372 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
373 * stand for such expressions that contain a plain number.
375 * - *(+(x1,x2,...);c) -> *(+(*(x1,c),*(x2,c),...))
379 * @param level cut-off in recursive evaluation */
380 ex mul::eval(int level) const
382 epvector *evaled_seqp = evalchildren(level);
384 // do more evaluation later
385 return (new mul(evaled_seqp,overall_coeff))->
386 setflag(status_flags::dynallocated);
389 #ifdef DO_GINAC_ASSERT
390 epvector::const_iterator i = seq.begin(), end = seq.end();
392 GINAC_ASSERT((!is_exactly_a<mul>(i->rest)) ||
393 (!(ex_to<numeric>(i->coeff).is_integer())));
394 GINAC_ASSERT(!(i->is_canonical_numeric()));
395 if (is_ex_exactly_of_type(recombine_pair_to_ex(*i), numeric))
396 print(print_tree(std::cerr));
397 GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*i)));
399 expair p = split_ex_to_pair(recombine_pair_to_ex(*i));
400 GINAC_ASSERT(p.rest.is_equal(i->rest));
401 GINAC_ASSERT(p.coeff.is_equal(i->coeff));
405 #endif // def DO_GINAC_ASSERT
407 if (flags & status_flags::evaluated) {
408 GINAC_ASSERT(seq.size()>0);
409 GINAC_ASSERT(seq.size()>1 || !overall_coeff.is_equal(_ex1));
413 int seq_size = seq.size();
414 if (overall_coeff.is_zero()) {
417 } else if (seq_size==0) {
419 return overall_coeff;
420 } else if (seq_size==1 && overall_coeff.is_equal(_ex1)) {
422 return recombine_pair_to_ex(*(seq.begin()));
423 } else if ((seq_size==1) &&
424 is_ex_exactly_of_type((*seq.begin()).rest,add) &&
425 ex_to<numeric>((*seq.begin()).coeff).is_equal(_num1)) {
426 // *(+(x,y,...);c) -> +(*(x,c),*(y,c),...) (c numeric(), no powers of +())
427 const add & addref = ex_to<add>((*seq.begin()).rest);
428 epvector *distrseq = new epvector();
429 distrseq->reserve(addref.seq.size());
430 epvector::const_iterator i = addref.seq.begin(), end = addref.seq.end();
432 distrseq->push_back(addref.combine_pair_with_coeff_to_pair(*i, overall_coeff));
435 return (new add(distrseq,
436 ex_to<numeric>(addref.overall_coeff).
437 mul_dyn(ex_to<numeric>(overall_coeff))))
438 ->setflag(status_flags::dynallocated | status_flags::evaluated);
443 ex mul::evalf(int level) const
446 return mul(seq,overall_coeff);
448 if (level==-max_recursion_level)
449 throw(std::runtime_error("max recursion level reached"));
451 epvector *s = new epvector();
452 s->reserve(seq.size());
455 epvector::const_iterator i = seq.begin(), end = seq.end();
457 s->push_back(combine_ex_with_coeff_to_pair(i->rest.evalf(level),
461 return mul(s, overall_coeff.evalf(level));
464 ex mul::evalm(void) const
467 if (seq.size() == 1 && seq[0].coeff.is_equal(_ex1)
468 && is_ex_of_type(seq[0].rest, matrix))
469 return ex_to<matrix>(seq[0].rest).mul(ex_to<numeric>(overall_coeff));
471 // Evaluate children first, look whether there are any matrices at all
472 // (there can be either no matrices or one matrix; if there were more
473 // than one matrix, it would be a non-commutative product)
474 epvector *s = new epvector;
475 s->reserve(seq.size());
477 bool have_matrix = false;
478 epvector::iterator the_matrix;
480 epvector::const_iterator i = seq.begin(), end = seq.end();
482 const ex &m = recombine_pair_to_ex(*i).evalm();
483 s->push_back(split_ex_to_pair(m));
484 if (is_ex_of_type(m, matrix)) {
486 the_matrix = s->end() - 1;
493 // The product contained a matrix. We will multiply all other factors
495 matrix m = ex_to<matrix>(the_matrix->rest);
496 s->erase(the_matrix);
497 ex scalar = (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
498 return m.mul_scalar(scalar);
501 return (new mul(s, overall_coeff))->setflag(status_flags::dynallocated);
504 ex mul::simplify_ncmul(const exvector & v) const
507 return inherited::simplify_ncmul(v);
509 // Find first noncommutative element and call its simplify_ncmul()
510 epvector::const_iterator i = seq.begin(), end = seq.end();
512 if (i->rest.return_type() == return_types::noncommutative)
513 return i->rest.simplify_ncmul(v);
516 return inherited::simplify_ncmul(v);
521 /** Implementation of ex::diff() for a product. It applies the product rule.
523 ex mul::derivative(const symbol & s) const
525 unsigned num = seq.size();
529 // D(a*b*c) = D(a)*b*c + a*D(b)*c + a*b*D(c)
530 epvector mulseq = seq;
531 epvector::const_iterator i = seq.begin(), end = seq.end();
532 epvector::iterator i2 = mulseq.begin();
534 expair ep = split_ex_to_pair(power(i->rest, i->coeff - _ex1) *
537 addseq.push_back((new mul(mulseq, overall_coeff * i->coeff))->setflag(status_flags::dynallocated));
541 return (new add(addseq))->setflag(status_flags::dynallocated);
544 int mul::compare_same_type(const basic & other) const
546 return inherited::compare_same_type(other);
549 bool mul::is_equal_same_type(const basic & other) const
551 return inherited::is_equal_same_type(other);
554 unsigned mul::return_type(void) const
557 // mul without factors: should not happen, but commutes
558 return return_types::commutative;
561 bool all_commutative = true;
562 epvector::const_iterator noncommutative_element; // point to first found nc element
564 epvector::const_iterator i = seq.begin(), end = seq.end();
566 unsigned rt = i->rest.return_type();
567 if (rt == return_types::noncommutative_composite)
568 return rt; // one ncc -> mul also ncc
569 if ((rt == return_types::noncommutative) && (all_commutative)) {
570 // first nc element found, remember position
571 noncommutative_element = i;
572 all_commutative = false;
574 if ((rt == return_types::noncommutative) && (!all_commutative)) {
575 // another nc element found, compare type_infos
576 if (noncommutative_element->rest.return_type_tinfo() != i->rest.return_type_tinfo()) {
577 // diffent types -> mul is ncc
578 return return_types::noncommutative_composite;
583 // all factors checked
584 return all_commutative ? return_types::commutative : return_types::noncommutative;
587 unsigned mul::return_type_tinfo(void) const
590 return tinfo_key; // mul without factors: should not happen
592 // return type_info of first noncommutative element
593 epvector::const_iterator i = seq.begin(), end = seq.end();
595 if (i->rest.return_type() == return_types::noncommutative)
596 return i->rest.return_type_tinfo();
599 // no noncommutative element found, should not happen
603 ex mul::thisexpairseq(const epvector & v, const ex & oc) const
605 return (new mul(v, oc))->setflag(status_flags::dynallocated);
608 ex mul::thisexpairseq(epvector * vp, const ex & oc) const
610 return (new mul(vp, oc))->setflag(status_flags::dynallocated);
613 expair mul::split_ex_to_pair(const ex & e) const
615 if (is_ex_exactly_of_type(e,power)) {
616 const power & powerref = ex_to<power>(e);
617 if (is_ex_exactly_of_type(powerref.exponent,numeric))
618 return expair(powerref.basis,powerref.exponent);
620 return expair(e,_ex1);
623 expair mul::combine_ex_with_coeff_to_pair(const ex & e,
626 // to avoid duplication of power simplification rules,
627 // we create a temporary power object
628 // otherwise it would be hard to correctly simplify
629 // expression like (4^(1/3))^(3/2)
630 if (are_ex_trivially_equal(c,_ex1))
631 return split_ex_to_pair(e);
633 return split_ex_to_pair(power(e,c));
636 expair mul::combine_pair_with_coeff_to_pair(const expair & p,
639 // to avoid duplication of power simplification rules,
640 // we create a temporary power object
641 // otherwise it would be hard to correctly simplify
642 // expression like (4^(1/3))^(3/2)
643 if (are_ex_trivially_equal(c,_ex1))
646 return split_ex_to_pair(power(recombine_pair_to_ex(p),c));
649 ex mul::recombine_pair_to_ex(const expair & p) const
651 if (ex_to<numeric>(p.coeff).is_equal(_num1))
654 return power(p.rest,p.coeff);
657 bool mul::expair_needs_further_processing(epp it)
659 if (is_ex_exactly_of_type((*it).rest,mul) &&
660 ex_to<numeric>((*it).coeff).is_integer()) {
661 // combined pair is product with integer power -> expand it
662 *it = split_ex_to_pair(recombine_pair_to_ex(*it));
665 if (is_ex_exactly_of_type((*it).rest,numeric)) {
666 expair ep=split_ex_to_pair(recombine_pair_to_ex(*it));
667 if (!ep.is_equal(*it)) {
668 // combined pair is a numeric power which can be simplified
672 if (ex_to<numeric>((*it).coeff).is_equal(_num1)) {
673 // combined pair has coeff 1 and must be moved to the end
680 ex mul::default_overall_coeff(void) const
685 void mul::combine_overall_coeff(const ex & c)
687 GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
688 GINAC_ASSERT(is_exactly_a<numeric>(c));
689 overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c));
692 void mul::combine_overall_coeff(const ex & c1, const ex & c2)
694 GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
695 GINAC_ASSERT(is_exactly_a<numeric>(c1));
696 GINAC_ASSERT(is_exactly_a<numeric>(c2));
697 overall_coeff = ex_to<numeric>(overall_coeff).mul_dyn(ex_to<numeric>(c1).power(ex_to<numeric>(c2)));
700 bool mul::can_make_flat(const expair & p) const
702 GINAC_ASSERT(is_exactly_a<numeric>(p.coeff));
703 // this assertion will probably fail somewhere
704 // it would require a more careful make_flat, obeying the power laws
705 // probably should return true only if p.coeff is integer
706 return ex_to<numeric>(p.coeff).is_equal(_num1);
709 ex mul::expand(unsigned options) const
711 // First, expand the children
712 epvector * expanded_seqp = expandchildren(options);
713 const epvector & expanded_seq = (expanded_seqp == NULL) ? seq : *expanded_seqp;
715 // Now, look for all the factors that are sums and multiply each one out
716 // with the next one that is found while collecting the factors which are
718 int number_of_adds = 0;
719 ex last_expanded = _ex1;
721 non_adds.reserve(expanded_seq.size());
722 epvector::const_iterator cit = expanded_seq.begin(), last = expanded_seq.end();
723 while (cit != last) {
724 if (is_ex_exactly_of_type(cit->rest, add) &&
725 (cit->coeff.is_equal(_ex1))) {
727 if (is_ex_exactly_of_type(last_expanded, add)) {
729 // Expand a product of two sums, simple and robust version.
730 const add & add1 = ex_to<add>(last_expanded);
731 const add & add2 = ex_to<add>(cit->rest);
732 const int n1 = add1.nops();
733 const int n2 = add2.nops();
736 distrseq.reserve(n2);
737 for (int i1=0; i1<n1; ++i1) {
739 // cache the first operand (for efficiency):
740 const ex op1 = add1.op(i1);
741 for (int i2=0; i2<n2; ++i2)
742 distrseq.push_back(op1 * add2.op(i2));
743 tmp_accu += (new add(distrseq))->
744 setflag(status_flags::dynallocated);
746 last_expanded = tmp_accu;
748 // Expand a product of two sums, aggressive version.
749 // Caring for the overall coefficients in separate loops can
750 // sometimes give a performance gain of up to 15%!
752 const int sizedifference = ex_to<add>(last_expanded).seq.size()-ex_to<add>(cit->rest).seq.size();
753 // add2 is for the inner loop and should be the bigger of the two sums
754 // in the presence of asymptotically good sorting:
755 const add& add1 = (sizedifference<0 ? ex_to<add>(last_expanded) : ex_to<add>(cit->rest));
756 const add& add2 = (sizedifference<0 ? ex_to<add>(cit->rest) : ex_to<add>(last_expanded));
757 const epvector::const_iterator add1begin = add1.seq.begin();
758 const epvector::const_iterator add1end = add1.seq.end();
759 const epvector::const_iterator add2begin = add2.seq.begin();
760 const epvector::const_iterator add2end = add2.seq.end();
762 distrseq.reserve(add1.seq.size()+add2.seq.size());
763 // Multiply add2 with the overall coefficient of add1 and append it to distrseq:
764 if (!add1.overall_coeff.is_zero()) {
765 if (add1.overall_coeff.is_equal(_ex1))
766 distrseq.insert(distrseq.end(),add2begin,add2end);
768 for (epvector::const_iterator i=add2begin; i!=add2end; ++i)
769 distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add1.overall_coeff))));
771 // Multiply add1 with the overall coefficient of add2 and append it to distrseq:
772 if (!add2.overall_coeff.is_zero()) {
773 if (add2.overall_coeff.is_equal(_ex1))
774 distrseq.insert(distrseq.end(),add1begin,add1end);
776 for (epvector::const_iterator i=add1begin; i!=add1end; ++i)
777 distrseq.push_back(expair(i->rest, ex_to<numeric>(i->coeff).mul_dyn(ex_to<numeric>(add2.overall_coeff))));
779 // Compute the new overall coefficient and put it together:
780 ex tmp_accu = (new add(distrseq, add1.overall_coeff*add2.overall_coeff))->setflag(status_flags::dynallocated);
781 // Multiply explicitly all non-numeric terms of add1 and add2:
782 for (epvector::const_iterator i1=add1begin; i1!=add1end; ++i1) {
783 // We really have to combine terms here in order to compactify
784 // the result. Otherwise it would become waayy tooo bigg.
787 for (epvector::const_iterator i2=add2begin; i2!=add2end; ++i2) {
788 // Don't push_back expairs which might have a rest that evaluates to a numeric,
789 // since that would violate an invariant of expairseq:
790 const ex rest = (new mul(i1->rest, i2->rest))->setflag(status_flags::dynallocated);
791 if (is_ex_exactly_of_type(rest, numeric))
792 oc += ex_to<numeric>(rest).mul(ex_to<numeric>(i1->coeff).mul(ex_to<numeric>(i2->coeff)));
794 distrseq.push_back(expair(rest, ex_to<numeric>(i1->coeff).mul_dyn(ex_to<numeric>(i2->coeff))));
796 tmp_accu += (new add(distrseq, oc))->setflag(status_flags::dynallocated);
798 last_expanded = tmp_accu;
801 non_adds.push_back(split_ex_to_pair(last_expanded));
802 last_expanded = cit->rest;
805 non_adds.push_back(*cit);
810 delete expanded_seqp;
812 // Now the only remaining thing to do is to multiply the factors which
813 // were not sums into the "last_expanded" sum
814 if (is_ex_exactly_of_type(last_expanded, add)) {
815 const add & finaladd = ex_to<add>(last_expanded);
817 int n = finaladd.nops();
819 for (int i=0; i<n; ++i) {
820 epvector factors = non_adds;
821 factors.push_back(split_ex_to_pair(finaladd.op(i)));
822 distrseq.push_back((new mul(factors, overall_coeff))->
823 setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
825 return ((new add(distrseq))->
826 setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0)));
828 non_adds.push_back(split_ex_to_pair(last_expanded));
829 return (new mul(non_adds, overall_coeff))->
830 setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
835 // new virtual functions which can be overridden by derived classes
841 // non-virtual functions in this class
845 /** Member-wise expand the expairs representing this sequence. This must be
846 * overridden from expairseq::expandchildren() and done iteratively in order
847 * to allow for early cancallations and thus safe memory.
850 * @return pointer to epvector containing expanded representation or zero
851 * pointer, if sequence is unchanged. */
852 epvector * mul::expandchildren(unsigned options) const
854 const epvector::const_iterator last = seq.end();
855 epvector::const_iterator cit = seq.begin();
857 const ex & factor = recombine_pair_to_ex(*cit);
858 const ex & expanded_factor = factor.expand(options);
859 if (!are_ex_trivially_equal(factor,expanded_factor)) {
861 // something changed, copy seq, eval and return it
862 epvector *s = new epvector;
863 s->reserve(seq.size());
865 // copy parts of seq which are known not to have changed
866 epvector::const_iterator cit2 = seq.begin();
871 // copy first changed element
872 s->push_back(split_ex_to_pair(expanded_factor));
876 s->push_back(split_ex_to_pair(recombine_pair_to_ex(*cit2).expand(options)));
884 return 0; // nothing has changed