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 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
76 debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
77 GINAC_ASSERT(basis.return_type()==return_types::commutative);
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())) {
184 if (precedence <= level)
186 basis.print(c, precedence);
188 exponent.print(c, precedence);
189 if (precedence <= level)
195 bool power::info(unsigned inf) const
198 case info_flags::polynomial:
199 case info_flags::integer_polynomial:
200 case info_flags::cinteger_polynomial:
201 case info_flags::rational_polynomial:
202 case info_flags::crational_polynomial:
203 return exponent.info(info_flags::nonnegint);
204 case info_flags::rational_function:
205 return exponent.info(info_flags::integer);
206 case info_flags::algebraic:
207 return (!exponent.info(info_flags::integer) ||
210 return inherited::info(inf);
213 unsigned power::nops() const
218 ex & power::let_op(int i)
223 return i==0 ? basis : exponent;
226 int power::degree(const ex & s) const
228 if (is_exactly_of_type(*exponent.bp,numeric)) {
229 if (basis.is_equal(s)) {
230 if (ex_to_numeric(exponent).is_integer())
231 return ex_to_numeric(exponent).to_int();
235 return basis.degree(s) * ex_to_numeric(exponent).to_int();
240 int power::ldegree(const ex & s) const
242 if (is_exactly_of_type(*exponent.bp,numeric)) {
243 if (basis.is_equal(s)) {
244 if (ex_to_numeric(exponent).is_integer())
245 return ex_to_numeric(exponent).to_int();
249 return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
254 ex power::coeff(const ex & s, int n) const
256 if (!basis.is_equal(s)) {
257 // basis not equal to s
264 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
266 int int_exp = ex_to_numeric(exponent).to_int();
272 // non-integer exponents are treated as zero
281 ex power::eval(int level) const
283 // simplifications: ^(x,0) -> 1 (0^0 handled here)
285 // ^(0,c1) -> 0 or exception (depending on real value of c1)
287 // ^(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)
288 // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
289 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
290 // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
291 // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
293 debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
295 if ((level==1) && (flags & status_flags::evaluated))
297 else if (level == -max_recursion_level)
298 throw(std::runtime_error("max recursion level reached"));
300 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
301 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
303 bool basis_is_numerical = 0;
304 bool exponent_is_numerical = 0;
306 numeric * num_exponent;
308 if (is_exactly_of_type(*ebasis.bp,numeric)) {
309 basis_is_numerical = 1;
310 num_basis = static_cast<numeric *>(ebasis.bp);
312 if (is_exactly_of_type(*eexponent.bp,numeric)) {
313 exponent_is_numerical = 1;
314 num_exponent = static_cast<numeric *>(eexponent.bp);
317 // ^(x,0) -> 1 (0^0 also handled here)
318 if (eexponent.is_zero()) {
319 if (ebasis.is_zero())
320 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
326 if (eexponent.is_equal(_ex1()))
329 // ^(0,c1) -> 0 or exception (depending on real value of c1)
330 if (ebasis.is_zero() && exponent_is_numerical) {
331 if ((num_exponent->real()).is_zero())
332 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
333 else if ((num_exponent->real()).is_negative())
334 throw (pole_error("power::eval(): division by zero",1));
340 if (ebasis.is_equal(_ex1()))
343 if (basis_is_numerical && exponent_is_numerical) {
344 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
345 // except if c1,c2 are rational, but c1^c2 is not)
346 bool basis_is_crational = num_basis->is_crational();
347 bool exponent_is_crational = num_exponent->is_crational();
348 numeric res = num_basis->power(*num_exponent);
350 if ((!basis_is_crational || !exponent_is_crational)
351 || res.is_crational()) {
354 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
355 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
356 if (basis_is_crational && exponent_is_crational
357 && num_exponent->is_real()
358 && !num_exponent->is_integer()) {
359 numeric n = num_exponent->numer();
360 numeric m = num_exponent->denom();
362 numeric q = iquo(n, m, r);
363 if (r.is_negative()) {
367 if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
371 res.push_back(expair(ebasis,r.div(m)));
372 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
377 // ^(^(x,c1),c2) -> ^(x,c1*c2)
378 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
379 // case c1==1 should not happen, see below!)
380 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
381 const power & sub_power = ex_to_power(ebasis);
382 const ex & sub_basis = sub_power.basis;
383 const ex & sub_exponent = sub_power.exponent;
384 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
385 const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
386 GINAC_ASSERT(num_sub_exponent!=numeric(1));
387 if (num_exponent->is_integer() || abs(num_sub_exponent)<1)
388 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
392 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
393 if (exponent_is_numerical && num_exponent->is_integer() &&
394 is_ex_exactly_of_type(ebasis,mul)) {
395 return expand_mul(ex_to_mul(ebasis), *num_exponent);
398 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
399 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
400 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
401 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
402 const mul & mulref = ex_to_mul(ebasis);
403 if (!mulref.overall_coeff.is_equal(_ex1())) {
404 const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
405 if (num_coeff.is_real()) {
406 if (num_coeff.is_positive()) {
407 mul * mulp = new mul(mulref);
408 mulp->overall_coeff = _ex1();
409 mulp->clearflag(status_flags::evaluated);
410 mulp->clearflag(status_flags::hash_calculated);
411 return (new mul(power(*mulp,exponent),
412 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
414 GINAC_ASSERT(num_coeff.compare(_num0())<0);
415 if (num_coeff.compare(_num_1())!=0) {
416 mul * mulp = new mul(mulref);
417 mulp->overall_coeff = _ex_1();
418 mulp->clearflag(status_flags::evaluated);
419 mulp->clearflag(status_flags::hash_calculated);
420 return (new mul(power(*mulp,exponent),
421 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
428 if (are_ex_trivially_equal(ebasis,basis) &&
429 are_ex_trivially_equal(eexponent,exponent)) {
432 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
433 status_flags::evaluated);
436 ex power::evalf(int level) const
438 debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
445 eexponent = exponent;
446 } else if (level == -max_recursion_level) {
447 throw(std::runtime_error("max recursion level reached"));
449 ebasis = basis.evalf(level-1);
450 if (!is_ex_exactly_of_type(eexponent,numeric))
451 eexponent = exponent.evalf(level-1);
453 eexponent = exponent;
456 return power(ebasis,eexponent);
459 ex power::subs(const lst & ls, const lst & lr) const
461 const ex & subsed_basis=basis.subs(ls,lr);
462 const ex & subsed_exponent=exponent.subs(ls,lr);
464 if (are_ex_trivially_equal(basis,subsed_basis)&&
465 are_ex_trivially_equal(exponent,subsed_exponent)) {
466 return inherited::subs(ls, lr);
469 return power(subsed_basis, subsed_exponent);
472 ex power::simplify_ncmul(const exvector & v) const
474 return inherited::simplify_ncmul(v);
479 /** Implementation of ex::diff() for a power.
481 ex power::derivative(const symbol & s) const
483 if (exponent.info(info_flags::real)) {
484 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
487 newseq.push_back(expair(basis, exponent - _ex1()));
488 newseq.push_back(expair(basis.diff(s), _ex1()));
489 return mul(newseq, exponent);
491 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
493 add(mul(exponent.diff(s), log(basis)),
494 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
498 int power::compare_same_type(const basic & other) const
500 GINAC_ASSERT(is_exactly_of_type(other, power));
501 const power & o=static_cast<const power &>(const_cast<basic &>(other));
504 cmpval=basis.compare(o.basis);
506 return exponent.compare(o.exponent);
511 unsigned power::return_type(void) const
513 return basis.return_type();
516 unsigned power::return_type_tinfo(void) const
518 return basis.return_type_tinfo();
521 ex power::expand(unsigned options) const
523 if (flags & status_flags::expanded)
526 ex expanded_basis = basis.expand(options);
527 ex expanded_exponent = exponent.expand(options);
529 // x^(a+b) -> x^a * x^b
530 if (is_ex_exactly_of_type(expanded_exponent, add)) {
531 const add &a = ex_to_add(expanded_exponent);
533 distrseq.reserve(a.seq.size() + 1);
534 epvector::const_iterator last = a.seq.end();
535 epvector::const_iterator cit = a.seq.begin();
537 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
541 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
542 if (ex_to_numeric(a.overall_coeff).is_integer()) {
543 const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
544 int int_exponent = num_exponent.to_int();
545 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
546 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
548 distrseq.push_back(power(expanded_basis, a.overall_coeff));
550 distrseq.push_back(power(expanded_basis, a.overall_coeff));
552 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
553 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
557 if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
558 !ex_to_numeric(expanded_exponent).is_integer()) {
559 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
562 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
566 // integer numeric exponent
567 const numeric & num_exponent = ex_to_numeric(expanded_exponent);
568 int int_exponent = num_exponent.to_int();
571 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
572 return expand_add(ex_to_add(expanded_basis), int_exponent);
574 // (x*y)^n -> x^n * y^n
575 if (is_ex_exactly_of_type(expanded_basis,mul))
576 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
578 // cannot expand further
579 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
582 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
586 // new virtual functions which can be overridden by derived classes
592 // non-virtual functions in this class
595 /** expand a^n where a is an add and n is an integer.
596 * @see power::expand */
597 ex power::expand_add(const add & a, int n) const
600 return expand_add_2(a);
604 sum.reserve((n+1)*(m-1));
606 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
607 intvector upper_limit(m-1);
610 for (int l=0; l<m-1; l++) {
619 for (l=0; l<m-1; l++) {
620 const ex & b = a.op(l);
621 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
622 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
623 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
624 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
625 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
626 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
627 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
628 if (is_ex_exactly_of_type(b,mul))
629 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
631 term.push_back(power(b,k[l]));
634 const ex & b = a.op(l);
635 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
636 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
637 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
638 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
639 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
640 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
641 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
642 if (is_ex_exactly_of_type(b,mul))
643 term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
645 term.push_back(power(b,n-k_cum[m-2]));
647 numeric f = binomial(numeric(n),numeric(k[0]));
648 for (l=1; l<m-1; l++)
649 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
654 cout << "begin term" << endl;
655 for (int i=0; i<m-1; i++) {
656 cout << "k[" << i << "]=" << k[i] << endl;
657 cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
658 cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
660 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
661 cout << *cit << endl;
663 cout << "end term" << endl;
666 // TODO: optimize this
667 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
671 while ((l>=0)&&((++k[l])>upper_limit[l])) {
677 // recalc k_cum[] and upper_limit[]
681 k_cum[l] = k_cum[l-1]+k[l];
683 for (int i=l+1; i<m-1; i++)
684 k_cum[i] = k_cum[i-1]+k[i];
686 for (int i=l+1; i<m-1; i++)
687 upper_limit[i] = n-k_cum[i-1];
689 return (new add(sum))->setflag(status_flags::dynallocated |
690 status_flags::expanded );
694 /** Special case of power::expand_add. Expands a^2 where a is an add.
695 * @see power::expand_add */
696 ex power::expand_add_2(const add & a) const
699 unsigned a_nops = a.nops();
700 sum.reserve((a_nops*(a_nops+1))/2);
701 epvector::const_iterator last = a.seq.end();
703 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
704 // first part: ignore overall_coeff and expand other terms
705 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
706 const ex & r = (*cit0).rest;
707 const ex & c = (*cit0).coeff;
709 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
710 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
711 !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
712 !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
713 !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
714 !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
715 !is_ex_exactly_of_type(ex_to_power(r).basis,power));
717 if (are_ex_trivially_equal(c,_ex1())) {
718 if (is_ex_exactly_of_type(r,mul)) {
719 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
722 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
726 if (is_ex_exactly_of_type(r,mul)) {
727 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
728 ex_to_numeric(c).power_dyn(_num2())));
730 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
731 ex_to_numeric(c).power_dyn(_num2())));
735 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
736 const ex & r1 = (*cit1).rest;
737 const ex & c1 = (*cit1).coeff;
738 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
739 _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
743 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
745 // second part: add terms coming from overall_factor (if != 0)
746 if (!a.overall_coeff.is_zero()) {
747 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
748 sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
750 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
753 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
755 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
758 /** Expand factors of m in m^n where m is a mul and n is and integer
759 * @see power::expand */
760 ex power::expand_mul(const mul & m, const numeric & n) const
766 distrseq.reserve(m.seq.size());
767 epvector::const_iterator last = m.seq.end();
768 epvector::const_iterator cit = m.seq.begin();
770 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
771 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
773 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
774 // since n is an integer
775 distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
779 return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
783 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
784 unsigned options) const
791 const add & addref=static_cast<const add &>(*basis.bp);
795 ex first_operands=add(splitseq);
796 ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
798 int n=exponent.to_int();
799 for (int k=0; k<=n; k++) {
800 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
801 * power(last_operand,numeric(n-k)));
803 return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
808 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
809 unsigned options) const
811 ex rest_power = ex(power(basis,exponent.add(_num_1()))).
812 expand(options | expand_options::internal_do_not_expand_power_operands);
814 return ex(mul(rest_power,basis),0).
815 expand(options | expand_options::internal_do_not_expand_mul_operands);
820 // static member variables
825 unsigned power::precedence = 60;
829 ex sqrt(const ex & a)
831 return power(a,_ex1_2());