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)
76 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
78 debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
79 n.find_ex("basis", basis, sym_lst);
80 n.find_ex("exponent", exponent, sym_lst);
83 void power::archive(archive_node &n) const
85 inherited::archive(n);
86 n.add_ex("basis", basis);
87 n.add_ex("exponent", exponent);
90 DEFAULT_UNARCHIVE(power)
93 // functions overriding virtual functions from base classes
98 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
100 // Optimal output of integer powers of symbols to aid compiler CSE.
101 // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
102 // to learn why such a parenthisation is really necessary.
105 } else if (exp == 2) {
109 } else if (exp & 1) {
112 print_sym_pow(c, x, exp-1);
115 print_sym_pow(c, x, exp >> 1);
117 print_sym_pow(c, x, exp >> 1);
122 void power::print(const print_context & c, unsigned level) const
124 debugmsg("power print", LOGLEVEL_PRINT);
126 if (is_a<print_tree>(c)) {
128 inherited::print(c, level);
130 } else if (is_a<print_csrc>(c)) {
132 // Integer powers of symbols are printed in a special, optimized way
133 if (exponent.info(info_flags::integer)
134 && (is_exactly_a<symbol>(basis) || is_exactly_a<constant>(basis))) {
135 int exp = ex_to<numeric>(exponent).to_int();
140 if (is_a<print_csrc_cl_N>(c))
145 print_sym_pow(c, ex_to<symbol>(basis), exp);
148 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
149 } else if (exponent.compare(_num_1()) == 0) {
150 if (is_a<print_csrc_cl_N>(c))
157 // Otherwise, use the pow() or expt() (CLN) functions
159 if (is_a<print_csrc_cl_N>(c))
171 if (exponent.is_equal(_ex1_2())) {
172 if (is_a<print_latex>(c))
177 if (is_a<print_latex>(c))
182 if (precedence() <= level) {
183 if (is_a<print_latex>(c))
188 basis.print(c, precedence());
190 if (is_a<print_latex>(c))
192 exponent.print(c, precedence());
193 if (is_a<print_latex>(c))
195 if (precedence() <= level) {
196 if (is_a<print_latex>(c))
205 bool power::info(unsigned inf) const
208 case info_flags::polynomial:
209 case info_flags::integer_polynomial:
210 case info_flags::cinteger_polynomial:
211 case info_flags::rational_polynomial:
212 case info_flags::crational_polynomial:
213 return exponent.info(info_flags::nonnegint);
214 case info_flags::rational_function:
215 return exponent.info(info_flags::integer);
216 case info_flags::algebraic:
217 return (!exponent.info(info_flags::integer) ||
220 return inherited::info(inf);
223 unsigned power::nops() const
228 ex & power::let_op(int i)
233 return i==0 ? basis : exponent;
236 ex power::map(map_function & f) const
238 return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
241 int power::degree(const ex & s) const
243 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to<numeric>(exponent).is_integer()) {
244 if (basis.is_equal(s))
245 return ex_to<numeric>(exponent).to_int();
247 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
252 int power::ldegree(const ex & s) const
254 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to<numeric>(exponent).is_integer()) {
255 if (basis.is_equal(s))
256 return ex_to<numeric>(exponent).to_int();
258 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
263 ex power::coeff(const ex & s, int n) const
265 if (!basis.is_equal(s)) {
266 // basis not equal to s
273 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to<numeric>(exponent).is_integer()) {
275 int int_exp = ex_to<numeric>(exponent).to_int();
281 // non-integer exponents are treated as zero
290 /** Perform automatic term rewriting rules in this class. In the following
291 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
292 * stand for such expressions that contain a plain number.
293 * - ^(x,0) -> 1 (also handles ^(0,0))
295 * - ^(0,c) -> 0 or exception (depending on the real part of c)
297 * - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
298 * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
299 * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
300 * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
301 * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
303 * @param level cut-off in recursive evaluation */
304 ex power::eval(int level) const
306 debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
308 if ((level==1) && (flags & status_flags::evaluated))
310 else if (level == -max_recursion_level)
311 throw(std::runtime_error("max recursion level reached"));
313 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
314 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
316 bool basis_is_numerical = false;
317 bool exponent_is_numerical = false;
318 const numeric *num_basis;
319 const numeric *num_exponent;
321 if (is_exactly_of_type(*ebasis.bp,numeric)) {
322 basis_is_numerical = true;
323 num_basis = static_cast<const numeric *>(ebasis.bp);
325 if (is_exactly_of_type(*eexponent.bp,numeric)) {
326 exponent_is_numerical = true;
327 num_exponent = static_cast<const numeric *>(eexponent.bp);
330 // ^(x,0) -> 1 (0^0 also handled here)
331 if (eexponent.is_zero()) {
332 if (ebasis.is_zero())
333 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
339 if (eexponent.is_equal(_ex1()))
342 // ^(0,c1) -> 0 or exception (depending on real value of c1)
343 if (ebasis.is_zero() && exponent_is_numerical) {
344 if ((num_exponent->real()).is_zero())
345 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
346 else if ((num_exponent->real()).is_negative())
347 throw (pole_error("power::eval(): division by zero",1));
353 if (ebasis.is_equal(_ex1()))
356 if (exponent_is_numerical) {
358 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
359 // except if c1,c2 are rational, but c1^c2 is not)
360 if (basis_is_numerical) {
361 const bool basis_is_crational = num_basis->is_crational();
362 const bool exponent_is_crational = num_exponent->is_crational();
363 if (!basis_is_crational || !exponent_is_crational) {
364 // return a plain float
365 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
366 status_flags::evaluated |
367 status_flags::expanded);
370 const numeric res = num_basis->power(*num_exponent);
371 if (res.is_crational()) {
374 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
376 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
377 if (basis_is_crational && exponent_is_crational
378 && num_exponent->is_real()
379 && !num_exponent->is_integer()) {
380 const numeric n = num_exponent->numer();
381 const numeric m = num_exponent->denom();
383 numeric q = iquo(n, m, r);
384 if (r.is_negative()) {
388 if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
389 if (num_basis->is_rational() && !num_basis->is_integer()) {
390 // try it for numerator and denominator separately, in order to
391 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
392 const numeric bnum = num_basis->numer();
393 const numeric bden = num_basis->denom();
394 const numeric res_bnum = bnum.power(*num_exponent);
395 const numeric res_bden = bden.power(*num_exponent);
396 if (res_bnum.is_integer())
397 return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
398 if (res_bden.is_integer())
399 return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
403 // assemble resulting product, but allowing for a re-evaluation,
404 // because otherwise we'll end up with something like
405 // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
406 // instead of 7/16*7^(1/3).
407 ex prod = power(*num_basis,r.div(m));
408 return prod*power(*num_basis,q);
413 // ^(^(x,c1),c2) -> ^(x,c1*c2)
414 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
415 // case c1==1 should not happen, see below!)
416 if (is_ex_exactly_of_type(ebasis,power)) {
417 const power & sub_power = ex_to<power>(ebasis);
418 const ex & sub_basis = sub_power.basis;
419 const ex & sub_exponent = sub_power.exponent;
420 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
421 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
422 GINAC_ASSERT(num_sub_exponent!=numeric(1));
423 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1()).is_negative())
424 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
428 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
429 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
430 return expand_mul(ex_to<mul>(ebasis), *num_exponent);
433 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
434 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
435 if (is_ex_exactly_of_type(ebasis,mul)) {
436 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
437 const mul & mulref = ex_to<mul>(ebasis);
438 if (!mulref.overall_coeff.is_equal(_ex1())) {
439 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
440 if (num_coeff.is_real()) {
441 if (num_coeff.is_positive()) {
442 mul *mulp = new mul(mulref);
443 mulp->overall_coeff = _ex1();
444 mulp->clearflag(status_flags::evaluated);
445 mulp->clearflag(status_flags::hash_calculated);
446 return (new mul(power(*mulp,exponent),
447 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
449 GINAC_ASSERT(num_coeff.compare(_num0())<0);
450 if (num_coeff.compare(_num_1())!=0) {
451 mul *mulp = new mul(mulref);
452 mulp->overall_coeff = _ex_1();
453 mulp->clearflag(status_flags::evaluated);
454 mulp->clearflag(status_flags::hash_calculated);
455 return (new mul(power(*mulp,exponent),
456 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
463 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
464 if (num_exponent->is_pos_integer() &&
465 ebasis.return_type() != return_types::commutative &&
466 !is_ex_of_type(ebasis,matrix)) {
467 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
471 if (are_ex_trivially_equal(ebasis,basis) &&
472 are_ex_trivially_equal(eexponent,exponent)) {
475 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
476 status_flags::evaluated);
479 ex power::evalf(int level) const
481 debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
488 eexponent = exponent;
489 } else if (level == -max_recursion_level) {
490 throw(std::runtime_error("max recursion level reached"));
492 ebasis = basis.evalf(level-1);
493 if (!is_ex_exactly_of_type(eexponent,numeric))
494 eexponent = exponent.evalf(level-1);
496 eexponent = exponent;
499 return power(ebasis,eexponent);
502 ex power::evalm(void) const
504 const ex ebasis = basis.evalm();
505 const ex eexponent = exponent.evalm();
506 if (is_ex_of_type(ebasis,matrix)) {
507 if (is_ex_of_type(eexponent,numeric)) {
508 return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
511 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
514 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
516 const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
517 const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
519 if (are_ex_trivially_equal(basis, subsed_basis)
520 && are_ex_trivially_equal(exponent, subsed_exponent))
521 return basic::subs(ls, lr, no_pattern);
523 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
526 ex power::simplify_ncmul(const exvector & v) const
528 return inherited::simplify_ncmul(v);
533 /** Implementation of ex::diff() for a power.
535 ex power::derivative(const symbol & s) const
537 if (exponent.info(info_flags::real)) {
538 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
541 newseq.push_back(expair(basis, exponent - _ex1()));
542 newseq.push_back(expair(basis.diff(s), _ex1()));
543 return mul(newseq, exponent);
545 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
547 add(mul(exponent.diff(s), log(basis)),
548 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
552 int power::compare_same_type(const basic & other) const
554 GINAC_ASSERT(is_exactly_of_type(other, power));
555 const power &o = static_cast<const power &>(other);
557 int cmpval = basis.compare(o.basis);
561 return exponent.compare(o.exponent);
564 unsigned power::return_type(void) const
566 return basis.return_type();
569 unsigned power::return_type_tinfo(void) const
571 return basis.return_type_tinfo();
574 ex power::expand(unsigned options) const
576 if (options == 0 && (flags & status_flags::expanded))
579 const ex expanded_basis = basis.expand(options);
580 const ex expanded_exponent = exponent.expand(options);
582 // x^(a+b) -> x^a * x^b
583 if (is_ex_exactly_of_type(expanded_exponent, add)) {
584 const add &a = ex_to<add>(expanded_exponent);
586 distrseq.reserve(a.seq.size() + 1);
587 epvector::const_iterator last = a.seq.end();
588 epvector::const_iterator cit = a.seq.begin();
590 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
594 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
595 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
596 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
597 int int_exponent = num_exponent.to_int();
598 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
599 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
601 distrseq.push_back(power(expanded_basis, a.overall_coeff));
603 distrseq.push_back(power(expanded_basis, a.overall_coeff));
605 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
606 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
610 if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
611 !ex_to<numeric>(expanded_exponent).is_integer()) {
612 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
615 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
619 // integer numeric exponent
620 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
621 int int_exponent = num_exponent.to_int();
624 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
625 return expand_add(ex_to<add>(expanded_basis), int_exponent);
627 // (x*y)^n -> x^n * y^n
628 if (is_ex_exactly_of_type(expanded_basis,mul))
629 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
631 // cannot expand further
632 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
635 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
639 // new virtual functions which can be overridden by derived classes
645 // non-virtual functions in this class
648 /** expand a^n where a is an add and n is an integer.
649 * @see power::expand */
650 ex power::expand_add(const add & a, int n) const
653 return expand_add_2(a);
657 sum.reserve((n+1)*(m-1));
659 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
660 intvector upper_limit(m-1);
663 for (int l=0; l<m-1; l++) {
672 for (l=0; l<m-1; l++) {
673 const ex & b = a.op(l);
674 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
675 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
676 !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
677 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
678 !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
679 !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
680 !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
681 if (is_ex_exactly_of_type(b,mul))
682 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
684 term.push_back(power(b,k[l]));
687 const ex & b = a.op(l);
688 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
689 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
690 !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
691 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
692 !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
693 !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
694 !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
695 if (is_ex_exactly_of_type(b,mul))
696 term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
698 term.push_back(power(b,n-k_cum[m-2]));
700 numeric f = binomial(numeric(n),numeric(k[0]));
701 for (l=1; l<m-1; l++)
702 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
706 // TODO: Can we optimize this? Alex seemed to think so...
707 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
711 while ((l>=0) && ((++k[l])>upper_limit[l])) {
717 // recalc k_cum[] and upper_limit[]
721 k_cum[l] = k_cum[l-1]+k[l];
723 for (int i=l+1; i<m-1; i++)
724 k_cum[i] = k_cum[i-1]+k[i];
726 for (int i=l+1; i<m-1; i++)
727 upper_limit[i] = n-k_cum[i-1];
729 return (new add(sum))->setflag(status_flags::dynallocated |
730 status_flags::expanded );
734 /** Special case of power::expand_add. Expands a^2 where a is an add.
735 * @see power::expand_add */
736 ex power::expand_add_2(const add & a) const
739 unsigned a_nops = a.nops();
740 sum.reserve((a_nops*(a_nops+1))/2);
741 epvector::const_iterator last = a.seq.end();
743 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
744 // first part: ignore overall_coeff and expand other terms
745 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
746 const ex & r = cit0->rest;
747 const ex & c = cit0->coeff;
749 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
750 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
751 !is_ex_exactly_of_type(ex_to<power>(r).exponent,numeric) ||
752 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
753 !is_ex_exactly_of_type(ex_to<power>(r).basis,add) ||
754 !is_ex_exactly_of_type(ex_to<power>(r).basis,mul) ||
755 !is_ex_exactly_of_type(ex_to<power>(r).basis,power));
757 if (are_ex_trivially_equal(c,_ex1())) {
758 if (is_ex_exactly_of_type(r,mul)) {
759 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
762 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
766 if (is_ex_exactly_of_type(r,mul)) {
767 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
768 ex_to<numeric>(c).power_dyn(_num2())));
770 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
771 ex_to<numeric>(c).power_dyn(_num2())));
775 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
776 const ex & r1 = cit1->rest;
777 const ex & c1 = cit1->coeff;
778 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
779 _num2().mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
783 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
785 // second part: add terms coming from overall_factor (if != 0)
786 if (!a.overall_coeff.is_zero()) {
787 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
789 sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2())));
792 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2()),_ex1()));
795 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
797 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
800 /** Expand factors of m in m^n where m is a mul and n is and integer
801 * @see power::expand */
802 ex power::expand_mul(const mul & m, const numeric & n) const
808 distrseq.reserve(m.seq.size());
809 epvector::const_iterator last = m.seq.end();
810 epvector::const_iterator cit = m.seq.begin();
812 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
813 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
815 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
816 // since n is an integer
817 distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
821 return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
826 ex sqrt(const ex & a)
828 return power(a,_ex1_2());