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"
42 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
44 typedef std::vector<int> intvector;
47 // default ctor, dtor, copy ctor assignment operator and helpers
50 power::power() : basic(TINFO_power)
52 debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
55 void power::copy(const power & other)
57 inherited::copy(other);
59 exponent = other.exponent;
62 DEFAULT_DESTROY(power)
68 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
70 debugmsg("power ctor from ex,ex",LOGLEVEL_CONSTRUCT);
73 /** Ctor from an ex and a bare numeric. This is somewhat more efficient than
74 * the normal ctor from two ex whenever it can be used. */
75 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
77 debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
84 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
86 debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
87 n.find_ex("basis", basis, sym_lst);
88 n.find_ex("exponent", exponent, sym_lst);
91 void power::archive(archive_node &n) const
93 inherited::archive(n);
94 n.add_ex("basis", basis);
95 n.add_ex("exponent", exponent);
98 DEFAULT_UNARCHIVE(power)
101 // functions overriding virtual functions from bases classes
106 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
108 // Optimal output of integer powers of symbols to aid compiler CSE.
109 // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
110 // to learn why such a hack is really necessary.
113 } else if (exp == 2) {
117 } else if (exp & 1) {
120 print_sym_pow(c, x, exp-1);
123 print_sym_pow(c, x, exp >> 1);
125 print_sym_pow(c, x, exp >> 1);
130 void power::print(const print_context & c, unsigned level) const
132 debugmsg("power print", LOGLEVEL_PRINT);
134 if (is_of_type(c, print_tree)) {
136 inherited::print(c, level);
138 } else if (is_of_type(c, print_csrc)) {
140 // Integer powers of symbols are printed in a special, optimized way
141 if (exponent.info(info_flags::integer)
142 && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
143 int exp = ex_to_numeric(exponent).to_int();
148 if (is_of_type(c, print_csrc_cl_N))
153 print_sym_pow(c, ex_to_symbol(basis), exp);
156 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
157 } else if (exponent.compare(_num_1()) == 0) {
158 if (is_of_type(c, print_csrc_cl_N))
165 // Otherwise, use the pow() or expt() (CLN) functions
167 if (is_of_type(c, print_csrc_cl_N))
179 if (exponent.is_equal(_ex1_2())) {
180 if (is_of_type(c, print_latex))
185 if (is_of_type(c, print_latex))
190 if (precedence() <= level) {
191 if (is_of_type(c, print_latex))
196 basis.print(c, precedence());
198 if (is_of_type(c, print_latex))
200 exponent.print(c, precedence());
201 if (is_of_type(c, print_latex))
203 if (precedence() <= level) {
204 if (is_of_type(c, print_latex))
213 bool power::info(unsigned inf) const
216 case info_flags::polynomial:
217 case info_flags::integer_polynomial:
218 case info_flags::cinteger_polynomial:
219 case info_flags::rational_polynomial:
220 case info_flags::crational_polynomial:
221 return exponent.info(info_flags::nonnegint);
222 case info_flags::rational_function:
223 return exponent.info(info_flags::integer);
224 case info_flags::algebraic:
225 return (!exponent.info(info_flags::integer) ||
228 return inherited::info(inf);
231 unsigned power::nops() const
236 ex & power::let_op(int i)
241 return i==0 ? basis : exponent;
244 ex power::map(map_func f) const
246 return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
249 int power::degree(const ex & s) const
251 if (is_exactly_of_type(*exponent.bp,numeric)) {
252 if (basis.is_equal(s)) {
253 if (ex_to_numeric(exponent).is_integer())
254 return ex_to_numeric(exponent).to_int();
258 return basis.degree(s) * ex_to_numeric(exponent).to_int();
263 int power::ldegree(const ex & s) const
265 if (is_exactly_of_type(*exponent.bp,numeric)) {
266 if (basis.is_equal(s)) {
267 if (ex_to_numeric(exponent).is_integer())
268 return ex_to_numeric(exponent).to_int();
272 return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
277 ex power::coeff(const ex & s, int n) const
279 if (!basis.is_equal(s)) {
280 // basis not equal to s
287 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
289 int int_exp = ex_to_numeric(exponent).to_int();
295 // non-integer exponents are treated as zero
304 ex power::eval(int level) const
306 // simplifications: ^(x,0) -> 1 (0^0 handled here)
308 // ^(0,c1) -> 0 or exception (depending on real value of c1)
310 // ^(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)
311 // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
312 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
313 // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
314 // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
316 debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
318 if ((level==1) && (flags & status_flags::evaluated))
320 else if (level == -max_recursion_level)
321 throw(std::runtime_error("max recursion level reached"));
323 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
324 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
326 bool basis_is_numerical = false;
327 bool exponent_is_numerical = false;
329 numeric * num_exponent;
331 if (is_exactly_of_type(*ebasis.bp,numeric)) {
332 basis_is_numerical = true;
333 num_basis = static_cast<numeric *>(ebasis.bp);
335 if (is_exactly_of_type(*eexponent.bp,numeric)) {
336 exponent_is_numerical = true;
337 num_exponent = static_cast<numeric *>(eexponent.bp);
340 // ^(x,0) -> 1 (0^0 also handled here)
341 if (eexponent.is_zero()) {
342 if (ebasis.is_zero())
343 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
349 if (eexponent.is_equal(_ex1()))
352 // ^(0,c1) -> 0 or exception (depending on real value of c1)
353 if (ebasis.is_zero() && exponent_is_numerical) {
354 if ((num_exponent->real()).is_zero())
355 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
356 else if ((num_exponent->real()).is_negative())
357 throw (pole_error("power::eval(): division by zero",1));
363 if (ebasis.is_equal(_ex1()))
366 if (exponent_is_numerical) {
368 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
369 // except if c1,c2 are rational, but c1^c2 is not)
370 if (basis_is_numerical) {
371 bool basis_is_crational = num_basis->is_crational();
372 bool exponent_is_crational = num_exponent->is_crational();
373 numeric res = num_basis->power(*num_exponent);
375 if ((!basis_is_crational || !exponent_is_crational)
376 || res.is_crational()) {
379 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
381 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
382 if (basis_is_crational && exponent_is_crational
383 && num_exponent->is_real()
384 && !num_exponent->is_integer()) {
385 numeric n = num_exponent->numer();
386 numeric m = num_exponent->denom();
388 numeric q = iquo(n, m, r);
389 if (r.is_negative()) {
393 if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
397 res.push_back(expair(ebasis,r.div(m)));
398 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
403 // ^(^(x,c1),c2) -> ^(x,c1*c2)
404 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
405 // case c1==1 should not happen, see below!)
406 if (is_ex_exactly_of_type(ebasis,power)) {
407 const power & sub_power = ex_to_power(ebasis);
408 const ex & sub_basis = sub_power.basis;
409 const ex & sub_exponent = sub_power.exponent;
410 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
411 const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
412 GINAC_ASSERT(num_sub_exponent!=numeric(1));
413 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1()).is_negative())
414 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
418 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
419 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
420 return expand_mul(ex_to_mul(ebasis), *num_exponent);
423 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
424 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
425 if (is_ex_exactly_of_type(ebasis,mul)) {
426 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
427 const mul & mulref = ex_to_mul(ebasis);
428 if (!mulref.overall_coeff.is_equal(_ex1())) {
429 const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
430 if (num_coeff.is_real()) {
431 if (num_coeff.is_positive()) {
432 mul * mulp = new mul(mulref);
433 mulp->overall_coeff = _ex1();
434 mulp->clearflag(status_flags::evaluated);
435 mulp->clearflag(status_flags::hash_calculated);
436 return (new mul(power(*mulp,exponent),
437 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
439 GINAC_ASSERT(num_coeff.compare(_num0())<0);
440 if (num_coeff.compare(_num_1())!=0) {
441 mul * mulp = new mul(mulref);
442 mulp->overall_coeff = _ex_1();
443 mulp->clearflag(status_flags::evaluated);
444 mulp->clearflag(status_flags::hash_calculated);
445 return (new mul(power(*mulp,exponent),
446 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
453 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer)
454 if (num_exponent->is_pos_integer() && ebasis.return_type() != return_types::commutative) {
455 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
459 if (are_ex_trivially_equal(ebasis,basis) &&
460 are_ex_trivially_equal(eexponent,exponent)) {
463 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
464 status_flags::evaluated);
467 ex power::evalf(int level) const
469 debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
476 eexponent = exponent;
477 } else if (level == -max_recursion_level) {
478 throw(std::runtime_error("max recursion level reached"));
480 ebasis = basis.evalf(level-1);
481 if (!is_ex_exactly_of_type(eexponent,numeric))
482 eexponent = exponent.evalf(level-1);
484 eexponent = exponent;
487 return power(ebasis,eexponent);
490 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
492 const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
493 const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
495 if (are_ex_trivially_equal(basis, subsed_basis)
496 && are_ex_trivially_equal(exponent, subsed_exponent))
497 return basic::subs(ls, lr, no_pattern);
499 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
502 ex power::simplify_ncmul(const exvector & v) const
504 return inherited::simplify_ncmul(v);
509 /** Implementation of ex::diff() for a power.
511 ex power::derivative(const symbol & s) const
513 if (exponent.info(info_flags::real)) {
514 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
517 newseq.push_back(expair(basis, exponent - _ex1()));
518 newseq.push_back(expair(basis.diff(s), _ex1()));
519 return mul(newseq, exponent);
521 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
523 add(mul(exponent.diff(s), log(basis)),
524 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
528 int power::compare_same_type(const basic & other) const
530 GINAC_ASSERT(is_exactly_of_type(other, power));
531 const power & o=static_cast<const power &>(const_cast<basic &>(other));
534 cmpval=basis.compare(o.basis);
536 return exponent.compare(o.exponent);
541 unsigned power::return_type(void) const
543 return basis.return_type();
546 unsigned power::return_type_tinfo(void) const
548 return basis.return_type_tinfo();
551 ex power::expand(unsigned options) const
553 if (flags & status_flags::expanded)
556 ex expanded_basis = basis.expand(options);
557 ex expanded_exponent = exponent.expand(options);
559 // x^(a+b) -> x^a * x^b
560 if (is_ex_exactly_of_type(expanded_exponent, add)) {
561 const add &a = ex_to_add(expanded_exponent);
563 distrseq.reserve(a.seq.size() + 1);
564 epvector::const_iterator last = a.seq.end();
565 epvector::const_iterator cit = a.seq.begin();
567 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
571 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
572 if (ex_to_numeric(a.overall_coeff).is_integer()) {
573 const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
574 int int_exponent = num_exponent.to_int();
575 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
576 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
578 distrseq.push_back(power(expanded_basis, a.overall_coeff));
580 distrseq.push_back(power(expanded_basis, a.overall_coeff));
582 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
583 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
587 if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
588 !ex_to_numeric(expanded_exponent).is_integer()) {
589 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
592 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
596 // integer numeric exponent
597 const numeric & num_exponent = ex_to_numeric(expanded_exponent);
598 int int_exponent = num_exponent.to_int();
601 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
602 return expand_add(ex_to_add(expanded_basis), int_exponent);
604 // (x*y)^n -> x^n * y^n
605 if (is_ex_exactly_of_type(expanded_basis,mul))
606 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
608 // cannot expand further
609 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
612 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
616 // new virtual functions which can be overridden by derived classes
622 // non-virtual functions in this class
625 /** expand a^n where a is an add and n is an integer.
626 * @see power::expand */
627 ex power::expand_add(const add & a, int n) const
630 return expand_add_2(a);
634 sum.reserve((n+1)*(m-1));
636 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
637 intvector upper_limit(m-1);
640 for (int l=0; l<m-1; l++) {
649 for (l=0; l<m-1; l++) {
650 const ex & b = a.op(l);
651 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
652 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
653 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
654 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
655 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
656 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
657 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
658 if (is_ex_exactly_of_type(b,mul))
659 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
661 term.push_back(power(b,k[l]));
664 const ex & b = a.op(l);
665 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
666 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
667 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
668 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
669 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
670 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
671 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
672 if (is_ex_exactly_of_type(b,mul))
673 term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
675 term.push_back(power(b,n-k_cum[m-2]));
677 numeric f = binomial(numeric(n),numeric(k[0]));
678 for (l=1; l<m-1; l++)
679 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
684 cout << "begin term" << endl;
685 for (int i=0; i<m-1; i++) {
686 cout << "k[" << i << "]=" << k[i] << endl;
687 cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
688 cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
690 for_each(term.begin(), term.end(), ostream_iterator<ex>(cout, "\n"));
691 cout << "end term" << endl;
694 // TODO: optimize this
695 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
699 while ((l>=0)&&((++k[l])>upper_limit[l])) {
705 // recalc k_cum[] and upper_limit[]
709 k_cum[l] = k_cum[l-1]+k[l];
711 for (int i=l+1; i<m-1; i++)
712 k_cum[i] = k_cum[i-1]+k[i];
714 for (int i=l+1; i<m-1; i++)
715 upper_limit[i] = n-k_cum[i-1];
717 return (new add(sum))->setflag(status_flags::dynallocated |
718 status_flags::expanded );
722 /** Special case of power::expand_add. Expands a^2 where a is an add.
723 * @see power::expand_add */
724 ex power::expand_add_2(const add & a) const
727 unsigned a_nops = a.nops();
728 sum.reserve((a_nops*(a_nops+1))/2);
729 epvector::const_iterator last = a.seq.end();
731 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
732 // first part: ignore overall_coeff and expand other terms
733 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
734 const ex & r = (*cit0).rest;
735 const ex & c = (*cit0).coeff;
737 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
738 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
739 !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
740 !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
741 !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
742 !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
743 !is_ex_exactly_of_type(ex_to_power(r).basis,power));
745 if (are_ex_trivially_equal(c,_ex1())) {
746 if (is_ex_exactly_of_type(r,mul)) {
747 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
750 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
754 if (is_ex_exactly_of_type(r,mul)) {
755 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
756 ex_to_numeric(c).power_dyn(_num2())));
758 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
759 ex_to_numeric(c).power_dyn(_num2())));
763 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
764 const ex & r1 = (*cit1).rest;
765 const ex & c1 = (*cit1).coeff;
766 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
767 _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
771 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
773 // second part: add terms coming from overall_factor (if != 0)
774 if (!a.overall_coeff.is_zero()) {
775 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
776 sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
778 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
781 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
783 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
786 /** Expand factors of m in m^n where m is a mul and n is and integer
787 * @see power::expand */
788 ex power::expand_mul(const mul & m, const numeric & n) const
794 distrseq.reserve(m.seq.size());
795 epvector::const_iterator last = m.seq.end();
796 epvector::const_iterator cit = m.seq.begin();
798 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
799 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
801 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
802 // since n is an integer
803 distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
807 return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
811 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
812 unsigned options) const
814 ex rest_power = ex(power(basis,exponent.add(_num_1()))).
815 expand(options | expand_options::internal_do_not_expand_power_operands);
817 return ex(mul(rest_power,basis),0).
818 expand(options | expand_options::internal_do_not_expand_mul_operands);
824 ex sqrt(const ex & a)
826 return power(a,_ex1_2());