3 * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
6 * GiNaC Copyright (C) 1999-2001 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
28 #include "expairseq.h"
34 #include "inifcns.h" // for log() in power::derivative()
44 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
46 typedef std::vector<int> intvector;
49 // default ctor, dtor, copy ctor assignment operator and helpers
52 power::power() : inherited(TINFO_power)
54 debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
57 void power::copy(const power & other)
59 inherited::copy(other);
61 exponent = other.exponent;
64 DEFAULT_DESTROY(power)
70 power::power(const ex & lh, const ex & rh) : inherited(TINFO_power), basis(lh), exponent(rh)
72 debugmsg("power ctor from ex,ex",LOGLEVEL_CONSTRUCT);
75 /** Ctor from an ex and a bare numeric. This is somewhat more efficient than
76 * the normal ctor from two ex whenever it can be used. */
77 power::power(const ex & lh, const numeric & rh) : inherited(TINFO_power), basis(lh), exponent(rh)
79 debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
86 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
88 debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
89 n.find_ex("basis", basis, sym_lst);
90 n.find_ex("exponent", exponent, sym_lst);
93 void power::archive(archive_node &n) const
95 inherited::archive(n);
96 n.add_ex("basis", basis);
97 n.add_ex("exponent", exponent);
100 DEFAULT_UNARCHIVE(power)
103 // functions overriding virtual functions from bases classes
108 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
110 // Optimal output of integer powers of symbols to aid compiler CSE.
111 // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
112 // to learn why such a hack is really necessary.
115 } else if (exp == 2) {
119 } else if (exp & 1) {
122 print_sym_pow(c, x, exp-1);
125 print_sym_pow(c, x, exp >> 1);
127 print_sym_pow(c, x, exp >> 1);
132 void power::print(const print_context & c, unsigned level) const
134 debugmsg("power print", LOGLEVEL_PRINT);
136 if (is_a<print_tree>(c)) {
138 inherited::print(c, level);
140 } else if (is_a<print_csrc>(c)) {
142 // Integer powers of symbols are printed in a special, optimized way
143 if (exponent.info(info_flags::integer)
144 && (is_exactly_a<symbol>(basis) || is_exactly_a<constant>(basis))) {
145 int exp = ex_to<numeric>(exponent).to_int();
150 if (is_a<print_csrc_cl_N>(c))
155 print_sym_pow(c, ex_to<symbol>(basis), exp);
158 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
159 } else if (exponent.compare(_num_1()) == 0) {
160 if (is_a<print_csrc_cl_N>(c))
167 // Otherwise, use the pow() or expt() (CLN) functions
169 if (is_a<print_csrc_cl_N>(c))
181 if (exponent.is_equal(_ex1_2())) {
182 if (is_a<print_latex>(c))
187 if (is_a<print_latex>(c))
192 if (precedence() <= level) {
193 if (is_a<print_latex>(c))
198 basis.print(c, precedence());
200 if (is_a<print_latex>(c))
202 exponent.print(c, precedence());
203 if (is_a<print_latex>(c))
205 if (precedence() <= level) {
206 if (is_a<print_latex>(c))
215 bool power::info(unsigned inf) const
218 case info_flags::polynomial:
219 case info_flags::integer_polynomial:
220 case info_flags::cinteger_polynomial:
221 case info_flags::rational_polynomial:
222 case info_flags::crational_polynomial:
223 return exponent.info(info_flags::nonnegint);
224 case info_flags::rational_function:
225 return exponent.info(info_flags::integer);
226 case info_flags::algebraic:
227 return (!exponent.info(info_flags::integer) ||
230 return inherited::info(inf);
233 unsigned power::nops() const
238 ex & power::let_op(int i)
243 return i==0 ? basis : exponent;
246 ex power::map(map_function & f) const
248 return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
251 int power::degree(const ex & s) const
253 if (is_exactly_of_type(*exponent.bp,numeric)) {
254 if (basis.is_equal(s)) {
255 if (ex_to<numeric>(exponent).is_integer())
256 return ex_to<numeric>(exponent).to_int();
260 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
265 int power::ldegree(const ex & s) const
267 if (is_exactly_of_type(*exponent.bp,numeric)) {
268 if (basis.is_equal(s)) {
269 if (ex_to<numeric>(exponent).is_integer())
270 return ex_to<numeric>(exponent).to_int();
274 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
279 ex power::coeff(const ex & s, int n) const
281 if (!basis.is_equal(s)) {
282 // basis not equal to s
289 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to<numeric>(exponent).is_integer()) {
291 int int_exp = ex_to<numeric>(exponent).to_int();
297 // non-integer exponents are treated as zero
306 ex power::eval(int level) const
308 // simplifications: ^(x,0) -> 1 (0^0 handled here)
310 // ^(0,c1) -> 0 or exception (depending on real value of c1)
312 // ^(c1,c2) -> *(c1^n,c1^(c2-n)) (c1, c2 numeric(), 0<(c2-n)<1 except if c1,c2 are rational, but c1^c2 is not)
313 // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
314 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
315 // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
316 // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
318 debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
320 if ((level==1) && (flags & status_flags::evaluated))
322 else if (level == -max_recursion_level)
323 throw(std::runtime_error("max recursion level reached"));
325 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
326 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
328 bool basis_is_numerical = false;
329 bool exponent_is_numerical = false;
331 numeric * num_exponent;
333 if (is_exactly_of_type(*ebasis.bp,numeric)) {
334 basis_is_numerical = true;
335 num_basis = static_cast<numeric *>(ebasis.bp);
337 if (is_exactly_of_type(*eexponent.bp,numeric)) {
338 exponent_is_numerical = true;
339 num_exponent = static_cast<numeric *>(eexponent.bp);
342 // ^(x,0) -> 1 (0^0 also handled here)
343 if (eexponent.is_zero()) {
344 if (ebasis.is_zero())
345 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
351 if (eexponent.is_equal(_ex1()))
354 // ^(0,c1) -> 0 or exception (depending on real value of c1)
355 if (ebasis.is_zero() && exponent_is_numerical) {
356 if ((num_exponent->real()).is_zero())
357 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
358 else if ((num_exponent->real()).is_negative())
359 throw (pole_error("power::eval(): division by zero",1));
365 if (ebasis.is_equal(_ex1()))
368 if (exponent_is_numerical) {
370 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
371 // except if c1,c2 are rational, but c1^c2 is not)
372 if (basis_is_numerical) {
373 bool basis_is_crational = num_basis->is_crational();
374 bool exponent_is_crational = num_exponent->is_crational();
375 numeric res = num_basis->power(*num_exponent);
377 if ((!basis_is_crational || !exponent_is_crational)
378 || res.is_crational()) {
381 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
383 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
384 if (basis_is_crational && exponent_is_crational
385 && num_exponent->is_real()
386 && !num_exponent->is_integer()) {
387 numeric n = num_exponent->numer();
388 numeric m = num_exponent->denom();
390 numeric q = iquo(n, m, r);
391 if (r.is_negative()) {
395 if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
399 res.push_back(expair(ebasis,r.div(m)));
400 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
405 // ^(^(x,c1),c2) -> ^(x,c1*c2)
406 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
407 // case c1==1 should not happen, see below!)
408 if (is_ex_exactly_of_type(ebasis,power)) {
409 const power & sub_power = ex_to<power>(ebasis);
410 const ex & sub_basis = sub_power.basis;
411 const ex & sub_exponent = sub_power.exponent;
412 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
413 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
414 GINAC_ASSERT(num_sub_exponent!=numeric(1));
415 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1()).is_negative())
416 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
420 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
421 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
422 return expand_mul(ex_to<mul>(ebasis), *num_exponent);
425 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
426 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
427 if (is_ex_exactly_of_type(ebasis,mul)) {
428 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
429 const mul & mulref = ex_to<mul>(ebasis);
430 if (!mulref.overall_coeff.is_equal(_ex1())) {
431 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
432 if (num_coeff.is_real()) {
433 if (num_coeff.is_positive()) {
434 mul * mulp = new mul(mulref);
435 mulp->overall_coeff = _ex1();
436 mulp->clearflag(status_flags::evaluated);
437 mulp->clearflag(status_flags::hash_calculated);
438 return (new mul(power(*mulp,exponent),
439 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
441 GINAC_ASSERT(num_coeff.compare(_num0())<0);
442 if (num_coeff.compare(_num_1())!=0) {
443 mul * mulp = new mul(mulref);
444 mulp->overall_coeff = _ex_1();
445 mulp->clearflag(status_flags::evaluated);
446 mulp->clearflag(status_flags::hash_calculated);
447 return (new mul(power(*mulp,exponent),
448 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
455 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
456 if (num_exponent->is_pos_integer() &&
457 ebasis.return_type() != return_types::commutative &&
458 !is_ex_of_type(ebasis,matrix)) {
459 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
463 if (are_ex_trivially_equal(ebasis,basis) &&
464 are_ex_trivially_equal(eexponent,exponent)) {
467 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
468 status_flags::evaluated);
471 ex power::evalf(int level) const
473 debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
480 eexponent = exponent;
481 } else if (level == -max_recursion_level) {
482 throw(std::runtime_error("max recursion level reached"));
484 ebasis = basis.evalf(level-1);
485 if (!is_ex_exactly_of_type(eexponent,numeric))
486 eexponent = exponent.evalf(level-1);
488 eexponent = exponent;
491 return power(ebasis,eexponent);
494 ex power::evalm(void) const
496 ex ebasis = basis.evalm();
497 ex eexponent = exponent.evalm();
498 if (is_ex_of_type(ebasis,matrix)) {
499 if (is_ex_of_type(eexponent,numeric)) {
500 return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
503 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
506 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
508 const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
509 const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
511 if (are_ex_trivially_equal(basis, subsed_basis)
512 && are_ex_trivially_equal(exponent, subsed_exponent))
513 return basic::subs(ls, lr, no_pattern);
515 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
518 ex power::simplify_ncmul(const exvector & v) const
520 return inherited::simplify_ncmul(v);
525 /** Implementation of ex::diff() for a power.
527 ex power::derivative(const symbol & s) const
529 if (exponent.info(info_flags::real)) {
530 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
533 newseq.push_back(expair(basis, exponent - _ex1()));
534 newseq.push_back(expair(basis.diff(s), _ex1()));
535 return mul(newseq, exponent);
537 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
539 add(mul(exponent.diff(s), log(basis)),
540 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
544 int power::compare_same_type(const basic & other) const
546 GINAC_ASSERT(is_exactly_of_type(other, power));
547 const power &o = static_cast<const power &>(other);
549 int cmpval = basis.compare(o.basis);
553 return exponent.compare(o.exponent);
556 unsigned power::return_type(void) const
558 return basis.return_type();
561 unsigned power::return_type_tinfo(void) const
563 return basis.return_type_tinfo();
566 ex power::expand(unsigned options) const
568 if (flags & status_flags::expanded)
571 ex expanded_basis = basis.expand(options);
572 ex expanded_exponent = exponent.expand(options);
574 // x^(a+b) -> x^a * x^b
575 if (is_ex_exactly_of_type(expanded_exponent, add)) {
576 const add &a = ex_to<add>(expanded_exponent);
578 distrseq.reserve(a.seq.size() + 1);
579 epvector::const_iterator last = a.seq.end();
580 epvector::const_iterator cit = a.seq.begin();
582 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
586 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
587 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
588 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
589 int int_exponent = num_exponent.to_int();
590 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
591 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
593 distrseq.push_back(power(expanded_basis, a.overall_coeff));
595 distrseq.push_back(power(expanded_basis, a.overall_coeff));
597 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
598 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
602 if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
603 !ex_to<numeric>(expanded_exponent).is_integer()) {
604 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
607 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
611 // integer numeric exponent
612 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
613 int int_exponent = num_exponent.to_int();
616 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
617 return expand_add(ex_to<add>(expanded_basis), int_exponent);
619 // (x*y)^n -> x^n * y^n
620 if (is_ex_exactly_of_type(expanded_basis,mul))
621 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
623 // cannot expand further
624 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
627 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
631 // new virtual functions which can be overridden by derived classes
637 // non-virtual functions in this class
640 /** expand a^n where a is an add and n is an integer.
641 * @see power::expand */
642 ex power::expand_add(const add & a, int n) const
645 return expand_add_2(a);
649 sum.reserve((n+1)*(m-1));
651 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
652 intvector upper_limit(m-1);
655 for (int l=0; l<m-1; l++) {
664 for (l=0; l<m-1; l++) {
665 const ex & b = a.op(l);
666 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
667 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
668 !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
669 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
670 !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
671 !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
672 !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
673 if (is_ex_exactly_of_type(b,mul))
674 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
676 term.push_back(power(b,k[l]));
679 const ex & b = a.op(l);
680 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
681 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
682 !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
683 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
684 !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
685 !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
686 !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
687 if (is_ex_exactly_of_type(b,mul))
688 term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
690 term.push_back(power(b,n-k_cum[m-2]));
692 numeric f = binomial(numeric(n),numeric(k[0]));
693 for (l=1; l<m-1; l++)
694 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
699 cout << "begin term" << endl;
700 for (int i=0; i<m-1; i++) {
701 cout << "k[" << i << "]=" << k[i] << endl;
702 cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
703 cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
705 for_each(term.begin(), term.end(), ostream_iterator<ex>(cout, "\n"));
706 cout << "end term" << endl;
709 // TODO: optimize this
710 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
714 while ((l>=0)&&((++k[l])>upper_limit[l])) {
720 // recalc k_cum[] and upper_limit[]
724 k_cum[l] = k_cum[l-1]+k[l];
726 for (int i=l+1; i<m-1; i++)
727 k_cum[i] = k_cum[i-1]+k[i];
729 for (int i=l+1; i<m-1; i++)
730 upper_limit[i] = n-k_cum[i-1];
732 return (new add(sum))->setflag(status_flags::dynallocated |
733 status_flags::expanded );
737 /** Special case of power::expand_add. Expands a^2 where a is an add.
738 * @see power::expand_add */
739 ex power::expand_add_2(const add & a) const
742 unsigned a_nops = a.nops();
743 sum.reserve((a_nops*(a_nops+1))/2);
744 epvector::const_iterator last = a.seq.end();
746 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
747 // first part: ignore overall_coeff and expand other terms
748 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
749 const ex & r = (*cit0).rest;
750 const ex & c = (*cit0).coeff;
752 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
753 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
754 !is_ex_exactly_of_type(ex_to<power>(r).exponent,numeric) ||
755 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
756 !is_ex_exactly_of_type(ex_to<power>(r).basis,add) ||
757 !is_ex_exactly_of_type(ex_to<power>(r).basis,mul) ||
758 !is_ex_exactly_of_type(ex_to<power>(r).basis,power));
760 if (are_ex_trivially_equal(c,_ex1())) {
761 if (is_ex_exactly_of_type(r,mul)) {
762 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
765 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
769 if (is_ex_exactly_of_type(r,mul)) {
770 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
771 ex_to<numeric>(c).power_dyn(_num2())));
773 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
774 ex_to<numeric>(c).power_dyn(_num2())));
778 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
779 const ex & r1 = (*cit1).rest;
780 const ex & c1 = (*cit1).coeff;
781 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
782 _num2().mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
786 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
788 // second part: add terms coming from overall_factor (if != 0)
789 if (!a.overall_coeff.is_zero()) {
790 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
791 sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to<numeric>(a.overall_coeff).mul_dyn(_num2())));
793 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2()),_ex1()));
796 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
798 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
801 /** Expand factors of m in m^n where m is a mul and n is and integer
802 * @see power::expand */
803 ex power::expand_mul(const mul & m, const numeric & n) const
809 distrseq.reserve(m.seq.size());
810 epvector::const_iterator last = m.seq.end();
811 epvector::const_iterator cit = m.seq.begin();
813 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
814 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
816 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
817 // since n is an integer
818 distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
822 return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
826 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
827 unsigned options) const
829 ex rest_power = ex(power(basis,exponent.add(_num_1()))).
830 expand(options | expand_options::internal_do_not_expand_power_operands);
832 return ex(mul(rest_power,basis),0).
833 expand(options | expand_options::internal_do_not_expand_mul_operands);
839 ex sqrt(const ex & a)
841 return power(a,_ex1_2());