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"
33 #include "relational.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);
71 GINAC_ASSERT(basis.return_type()==return_types::commutative);
74 /** Ctor from an ex and a bare numeric. This is somewhat more efficient than
75 * the normal ctor from two ex whenever it can be used. */
76 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
78 debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
79 GINAC_ASSERT(basis.return_type()==return_types::commutative);
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_of_type(c, print_tree)) {
138 inherited::print(c, level);
140 } else if (is_of_type(c, print_csrc)) {
142 // Integer powers of symbols are printed in a special, optimized way
143 if (exponent.info(info_flags::integer)
144 && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
145 int exp = ex_to_numeric(exponent).to_int();
150 if (is_of_type(c, print_csrc_cl_N))
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_of_type(c, print_csrc_cl_N))
167 // Otherwise, use the pow() or expt() (CLN) functions
169 if (is_of_type(c, print_csrc_cl_N))
181 if (exponent.is_equal(_ex1_2())) {
182 if (is_of_type(c, print_latex))
187 if (is_of_type(c, print_latex))
192 if (precedence() <= level) {
193 if (is_of_type(c, print_latex))
198 basis.print(c, precedence());
200 if (is_of_type(c, print_latex))
202 exponent.print(c, precedence());
203 if (is_of_type(c, print_latex))
205 if (precedence() <= level) {
206 if (is_of_type(c, print_latex))
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 int power::degree(const ex & s) const
248 if (is_exactly_of_type(*exponent.bp,numeric)) {
249 if (basis.is_equal(s)) {
250 if (ex_to_numeric(exponent).is_integer())
251 return ex_to_numeric(exponent).to_int();
255 return basis.degree(s) * ex_to_numeric(exponent).to_int();
260 int power::ldegree(const ex & s) const
262 if (is_exactly_of_type(*exponent.bp,numeric)) {
263 if (basis.is_equal(s)) {
264 if (ex_to_numeric(exponent).is_integer())
265 return ex_to_numeric(exponent).to_int();
269 return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
274 ex power::coeff(const ex & s, int n) const
276 if (!basis.is_equal(s)) {
277 // basis not equal to s
284 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
286 int int_exp = ex_to_numeric(exponent).to_int();
292 // non-integer exponents are treated as zero
301 ex power::eval(int level) const
303 // simplifications: ^(x,0) -> 1 (0^0 handled here)
305 // ^(0,c1) -> 0 or exception (depending on real value of c1)
307 // ^(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)
308 // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
309 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
310 // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
311 // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
313 debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
315 if ((level==1) && (flags & status_flags::evaluated))
317 else if (level == -max_recursion_level)
318 throw(std::runtime_error("max recursion level reached"));
320 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
321 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
323 bool basis_is_numerical = 0;
324 bool exponent_is_numerical = 0;
326 numeric * num_exponent;
328 if (is_exactly_of_type(*ebasis.bp,numeric)) {
329 basis_is_numerical = 1;
330 num_basis = static_cast<numeric *>(ebasis.bp);
332 if (is_exactly_of_type(*eexponent.bp,numeric)) {
333 exponent_is_numerical = 1;
334 num_exponent = static_cast<numeric *>(eexponent.bp);
337 // ^(x,0) -> 1 (0^0 also handled here)
338 if (eexponent.is_zero()) {
339 if (ebasis.is_zero())
340 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
346 if (eexponent.is_equal(_ex1()))
349 // ^(0,c1) -> 0 or exception (depending on real value of c1)
350 if (ebasis.is_zero() && exponent_is_numerical) {
351 if ((num_exponent->real()).is_zero())
352 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
353 else if ((num_exponent->real()).is_negative())
354 throw (pole_error("power::eval(): division by zero",1));
360 if (ebasis.is_equal(_ex1()))
363 if (basis_is_numerical && exponent_is_numerical) {
364 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
365 // except if c1,c2 are rational, but c1^c2 is not)
366 bool basis_is_crational = num_basis->is_crational();
367 bool exponent_is_crational = num_exponent->is_crational();
368 numeric res = num_basis->power(*num_exponent);
370 if ((!basis_is_crational || !exponent_is_crational)
371 || res.is_crational()) {
374 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
375 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
376 if (basis_is_crational && exponent_is_crational
377 && num_exponent->is_real()
378 && !num_exponent->is_integer()) {
379 numeric n = num_exponent->numer();
380 numeric m = num_exponent->denom();
382 numeric q = iquo(n, m, r);
383 if (r.is_negative()) {
387 if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
391 res.push_back(expair(ebasis,r.div(m)));
392 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
397 // ^(^(x,c1),c2) -> ^(x,c1*c2)
398 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
399 // case c1==1 should not happen, see below!)
400 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
401 const power & sub_power = ex_to_power(ebasis);
402 const ex & sub_basis = sub_power.basis;
403 const ex & sub_exponent = sub_power.exponent;
404 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
405 const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
406 GINAC_ASSERT(num_sub_exponent!=numeric(1));
407 if (num_exponent->is_integer() || abs(num_sub_exponent)<1)
408 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
412 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
413 if (exponent_is_numerical && num_exponent->is_integer() &&
414 is_ex_exactly_of_type(ebasis,mul)) {
415 return expand_mul(ex_to_mul(ebasis), *num_exponent);
418 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
419 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
420 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
421 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
422 const mul & mulref = ex_to_mul(ebasis);
423 if (!mulref.overall_coeff.is_equal(_ex1())) {
424 const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
425 if (num_coeff.is_real()) {
426 if (num_coeff.is_positive()) {
427 mul * mulp = new mul(mulref);
428 mulp->overall_coeff = _ex1();
429 mulp->clearflag(status_flags::evaluated);
430 mulp->clearflag(status_flags::hash_calculated);
431 return (new mul(power(*mulp,exponent),
432 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
434 GINAC_ASSERT(num_coeff.compare(_num0())<0);
435 if (num_coeff.compare(_num_1())!=0) {
436 mul * mulp = new mul(mulref);
437 mulp->overall_coeff = _ex_1();
438 mulp->clearflag(status_flags::evaluated);
439 mulp->clearflag(status_flags::hash_calculated);
440 return (new mul(power(*mulp,exponent),
441 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
448 if (are_ex_trivially_equal(ebasis,basis) &&
449 are_ex_trivially_equal(eexponent,exponent)) {
452 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
453 status_flags::evaluated);
456 ex power::evalf(int level) const
458 debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
465 eexponent = exponent;
466 } else if (level == -max_recursion_level) {
467 throw(std::runtime_error("max recursion level reached"));
469 ebasis = basis.evalf(level-1);
470 if (!is_ex_exactly_of_type(eexponent,numeric))
471 eexponent = exponent.evalf(level-1);
473 eexponent = exponent;
476 return power(ebasis,eexponent);
479 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
481 const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
482 const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
484 if (are_ex_trivially_equal(basis, subsed_basis)
485 && are_ex_trivially_equal(exponent, subsed_exponent))
486 return basic::subs(ls, lr, no_pattern);
488 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
491 ex power::simplify_ncmul(const exvector & v) const
493 return inherited::simplify_ncmul(v);
498 /** Implementation of ex::diff() for a power.
500 ex power::derivative(const symbol & s) const
502 if (exponent.info(info_flags::real)) {
503 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
506 newseq.push_back(expair(basis, exponent - _ex1()));
507 newseq.push_back(expair(basis.diff(s), _ex1()));
508 return mul(newseq, exponent);
510 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
512 add(mul(exponent.diff(s), log(basis)),
513 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
517 int power::compare_same_type(const basic & other) const
519 GINAC_ASSERT(is_exactly_of_type(other, power));
520 const power & o=static_cast<const power &>(const_cast<basic &>(other));
523 cmpval=basis.compare(o.basis);
525 return exponent.compare(o.exponent);
530 unsigned power::return_type(void) const
532 return basis.return_type();
535 unsigned power::return_type_tinfo(void) const
537 return basis.return_type_tinfo();
540 ex power::expand(unsigned options) const
542 if (flags & status_flags::expanded)
545 ex expanded_basis = basis.expand(options);
546 ex expanded_exponent = exponent.expand(options);
548 // x^(a+b) -> x^a * x^b
549 if (is_ex_exactly_of_type(expanded_exponent, add)) {
550 const add &a = ex_to_add(expanded_exponent);
552 distrseq.reserve(a.seq.size() + 1);
553 epvector::const_iterator last = a.seq.end();
554 epvector::const_iterator cit = a.seq.begin();
556 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
560 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
561 if (ex_to_numeric(a.overall_coeff).is_integer()) {
562 const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
563 int int_exponent = num_exponent.to_int();
564 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
565 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
567 distrseq.push_back(power(expanded_basis, a.overall_coeff));
569 distrseq.push_back(power(expanded_basis, a.overall_coeff));
571 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
572 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
576 if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
577 !ex_to_numeric(expanded_exponent).is_integer()) {
578 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
581 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
585 // integer numeric exponent
586 const numeric & num_exponent = ex_to_numeric(expanded_exponent);
587 int int_exponent = num_exponent.to_int();
590 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
591 return expand_add(ex_to_add(expanded_basis), int_exponent);
593 // (x*y)^n -> x^n * y^n
594 if (is_ex_exactly_of_type(expanded_basis,mul))
595 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
597 // cannot expand further
598 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
601 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
605 // new virtual functions which can be overridden by derived classes
611 // non-virtual functions in this class
614 /** expand a^n where a is an add and n is an integer.
615 * @see power::expand */
616 ex power::expand_add(const add & a, int n) const
619 return expand_add_2(a);
623 sum.reserve((n+1)*(m-1));
625 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
626 intvector upper_limit(m-1);
629 for (int l=0; l<m-1; l++) {
638 for (l=0; l<m-1; l++) {
639 const ex & b = a.op(l);
640 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
641 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
642 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
643 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
644 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
645 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
646 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
647 if (is_ex_exactly_of_type(b,mul))
648 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
650 term.push_back(power(b,k[l]));
653 const ex & b = a.op(l);
654 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
655 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
656 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
657 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
658 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
659 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
660 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
661 if (is_ex_exactly_of_type(b,mul))
662 term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
664 term.push_back(power(b,n-k_cum[m-2]));
666 numeric f = binomial(numeric(n),numeric(k[0]));
667 for (l=1; l<m-1; l++)
668 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
673 cout << "begin term" << endl;
674 for (int i=0; i<m-1; i++) {
675 cout << "k[" << i << "]=" << k[i] << endl;
676 cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
677 cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
679 for_each(term.begin(), term.end(), ostream_iterator<ex>(cout, "\n"));
680 cout << "end term" << endl;
683 // TODO: optimize this
684 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
688 while ((l>=0)&&((++k[l])>upper_limit[l])) {
694 // recalc k_cum[] and upper_limit[]
698 k_cum[l] = k_cum[l-1]+k[l];
700 for (int i=l+1; i<m-1; i++)
701 k_cum[i] = k_cum[i-1]+k[i];
703 for (int i=l+1; i<m-1; i++)
704 upper_limit[i] = n-k_cum[i-1];
706 return (new add(sum))->setflag(status_flags::dynallocated |
707 status_flags::expanded );
711 /** Special case of power::expand_add. Expands a^2 where a is an add.
712 * @see power::expand_add */
713 ex power::expand_add_2(const add & a) const
716 unsigned a_nops = a.nops();
717 sum.reserve((a_nops*(a_nops+1))/2);
718 epvector::const_iterator last = a.seq.end();
720 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
721 // first part: ignore overall_coeff and expand other terms
722 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
723 const ex & r = (*cit0).rest;
724 const ex & c = (*cit0).coeff;
726 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
727 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
728 !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
729 !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
730 !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
731 !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
732 !is_ex_exactly_of_type(ex_to_power(r).basis,power));
734 if (are_ex_trivially_equal(c,_ex1())) {
735 if (is_ex_exactly_of_type(r,mul)) {
736 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
739 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
743 if (is_ex_exactly_of_type(r,mul)) {
744 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
745 ex_to_numeric(c).power_dyn(_num2())));
747 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
748 ex_to_numeric(c).power_dyn(_num2())));
752 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
753 const ex & r1 = (*cit1).rest;
754 const ex & c1 = (*cit1).coeff;
755 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
756 _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
760 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
762 // second part: add terms coming from overall_factor (if != 0)
763 if (!a.overall_coeff.is_zero()) {
764 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
765 sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
767 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
770 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
772 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
775 /** Expand factors of m in m^n where m is a mul and n is and integer
776 * @see power::expand */
777 ex power::expand_mul(const mul & m, const numeric & n) const
783 distrseq.reserve(m.seq.size());
784 epvector::const_iterator last = m.seq.end();
785 epvector::const_iterator cit = m.seq.begin();
787 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
788 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
790 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
791 // since n is an integer
792 distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
796 return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
800 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
801 unsigned options) const
803 ex rest_power = ex(power(basis,exponent.add(_num_1()))).
804 expand(options | expand_options::internal_do_not_expand_power_operands);
806 return ex(mul(rest_power,basis),0).
807 expand(options | expand_options::internal_do_not_expand_mul_operands);
813 ex sqrt(const ex & a)
815 return power(a,_ex1_2());