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())) {
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 exponent.print(c, precedence);
199 if (precedence <= level) {
200 if (is_of_type(c, print_latex))
209 bool power::info(unsigned inf) const
212 case info_flags::polynomial:
213 case info_flags::integer_polynomial:
214 case info_flags::cinteger_polynomial:
215 case info_flags::rational_polynomial:
216 case info_flags::crational_polynomial:
217 return exponent.info(info_flags::nonnegint);
218 case info_flags::rational_function:
219 return exponent.info(info_flags::integer);
220 case info_flags::algebraic:
221 return (!exponent.info(info_flags::integer) ||
224 return inherited::info(inf);
227 unsigned power::nops() const
232 ex & power::let_op(int i)
237 return i==0 ? basis : exponent;
240 int power::degree(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.degree(s) * ex_to_numeric(exponent).to_int();
254 int power::ldegree(const ex & s) const
256 if (is_exactly_of_type(*exponent.bp,numeric)) {
257 if (basis.is_equal(s)) {
258 if (ex_to_numeric(exponent).is_integer())
259 return ex_to_numeric(exponent).to_int();
263 return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
268 ex power::coeff(const ex & s, int n) const
270 if (!basis.is_equal(s)) {
271 // basis not equal to s
278 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
280 int int_exp = ex_to_numeric(exponent).to_int();
286 // non-integer exponents are treated as zero
295 ex power::eval(int level) const
297 // simplifications: ^(x,0) -> 1 (0^0 handled here)
299 // ^(0,c1) -> 0 or exception (depending on real value of c1)
301 // ^(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)
302 // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
303 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
304 // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
305 // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
307 debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
309 if ((level==1) && (flags & status_flags::evaluated))
311 else if (level == -max_recursion_level)
312 throw(std::runtime_error("max recursion level reached"));
314 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
315 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
317 bool basis_is_numerical = 0;
318 bool exponent_is_numerical = 0;
320 numeric * num_exponent;
322 if (is_exactly_of_type(*ebasis.bp,numeric)) {
323 basis_is_numerical = 1;
324 num_basis = static_cast<numeric *>(ebasis.bp);
326 if (is_exactly_of_type(*eexponent.bp,numeric)) {
327 exponent_is_numerical = 1;
328 num_exponent = static_cast<numeric *>(eexponent.bp);
331 // ^(x,0) -> 1 (0^0 also handled here)
332 if (eexponent.is_zero()) {
333 if (ebasis.is_zero())
334 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
340 if (eexponent.is_equal(_ex1()))
343 // ^(0,c1) -> 0 or exception (depending on real value of c1)
344 if (ebasis.is_zero() && exponent_is_numerical) {
345 if ((num_exponent->real()).is_zero())
346 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
347 else if ((num_exponent->real()).is_negative())
348 throw (pole_error("power::eval(): division by zero",1));
354 if (ebasis.is_equal(_ex1()))
357 if (basis_is_numerical && exponent_is_numerical) {
358 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
359 // except if c1,c2 are rational, but c1^c2 is not)
360 bool basis_is_crational = num_basis->is_crational();
361 bool exponent_is_crational = num_exponent->is_crational();
362 numeric res = num_basis->power(*num_exponent);
364 if ((!basis_is_crational || !exponent_is_crational)
365 || res.is_crational()) {
368 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
369 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
370 if (basis_is_crational && exponent_is_crational
371 && num_exponent->is_real()
372 && !num_exponent->is_integer()) {
373 numeric n = num_exponent->numer();
374 numeric m = num_exponent->denom();
376 numeric q = iquo(n, m, r);
377 if (r.is_negative()) {
381 if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
385 res.push_back(expair(ebasis,r.div(m)));
386 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
391 // ^(^(x,c1),c2) -> ^(x,c1*c2)
392 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
393 // case c1==1 should not happen, see below!)
394 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
395 const power & sub_power = ex_to_power(ebasis);
396 const ex & sub_basis = sub_power.basis;
397 const ex & sub_exponent = sub_power.exponent;
398 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
399 const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
400 GINAC_ASSERT(num_sub_exponent!=numeric(1));
401 if (num_exponent->is_integer() || abs(num_sub_exponent)<1)
402 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
406 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
407 if (exponent_is_numerical && num_exponent->is_integer() &&
408 is_ex_exactly_of_type(ebasis,mul)) {
409 return expand_mul(ex_to_mul(ebasis), *num_exponent);
412 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
413 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
414 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
415 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
416 const mul & mulref = ex_to_mul(ebasis);
417 if (!mulref.overall_coeff.is_equal(_ex1())) {
418 const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
419 if (num_coeff.is_real()) {
420 if (num_coeff.is_positive()) {
421 mul * mulp = new mul(mulref);
422 mulp->overall_coeff = _ex1();
423 mulp->clearflag(status_flags::evaluated);
424 mulp->clearflag(status_flags::hash_calculated);
425 return (new mul(power(*mulp,exponent),
426 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
428 GINAC_ASSERT(num_coeff.compare(_num0())<0);
429 if (num_coeff.compare(_num_1())!=0) {
430 mul * mulp = new mul(mulref);
431 mulp->overall_coeff = _ex_1();
432 mulp->clearflag(status_flags::evaluated);
433 mulp->clearflag(status_flags::hash_calculated);
434 return (new mul(power(*mulp,exponent),
435 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
442 if (are_ex_trivially_equal(ebasis,basis) &&
443 are_ex_trivially_equal(eexponent,exponent)) {
446 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
447 status_flags::evaluated);
450 ex power::evalf(int level) const
452 debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
459 eexponent = exponent;
460 } else if (level == -max_recursion_level) {
461 throw(std::runtime_error("max recursion level reached"));
463 ebasis = basis.evalf(level-1);
464 if (!is_ex_exactly_of_type(eexponent,numeric))
465 eexponent = exponent.evalf(level-1);
467 eexponent = exponent;
470 return power(ebasis,eexponent);
473 ex power::subs(const lst & ls, const lst & lr) const
475 const ex & subsed_basis=basis.subs(ls,lr);
476 const ex & subsed_exponent=exponent.subs(ls,lr);
478 if (are_ex_trivially_equal(basis,subsed_basis)&&
479 are_ex_trivially_equal(exponent,subsed_exponent)) {
480 return inherited::subs(ls, lr);
483 return power(subsed_basis, subsed_exponent);
486 ex power::simplify_ncmul(const exvector & v) const
488 return inherited::simplify_ncmul(v);
493 /** Implementation of ex::diff() for a power.
495 ex power::derivative(const symbol & s) const
497 if (exponent.info(info_flags::real)) {
498 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
501 newseq.push_back(expair(basis, exponent - _ex1()));
502 newseq.push_back(expair(basis.diff(s), _ex1()));
503 return mul(newseq, exponent);
505 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
507 add(mul(exponent.diff(s), log(basis)),
508 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
512 int power::compare_same_type(const basic & other) const
514 GINAC_ASSERT(is_exactly_of_type(other, power));
515 const power & o=static_cast<const power &>(const_cast<basic &>(other));
518 cmpval=basis.compare(o.basis);
520 return exponent.compare(o.exponent);
525 unsigned power::return_type(void) const
527 return basis.return_type();
530 unsigned power::return_type_tinfo(void) const
532 return basis.return_type_tinfo();
535 ex power::expand(unsigned options) const
537 if (flags & status_flags::expanded)
540 ex expanded_basis = basis.expand(options);
541 ex expanded_exponent = exponent.expand(options);
543 // x^(a+b) -> x^a * x^b
544 if (is_ex_exactly_of_type(expanded_exponent, add)) {
545 const add &a = ex_to_add(expanded_exponent);
547 distrseq.reserve(a.seq.size() + 1);
548 epvector::const_iterator last = a.seq.end();
549 epvector::const_iterator cit = a.seq.begin();
551 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
555 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
556 if (ex_to_numeric(a.overall_coeff).is_integer()) {
557 const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
558 int int_exponent = num_exponent.to_int();
559 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
560 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
562 distrseq.push_back(power(expanded_basis, a.overall_coeff));
564 distrseq.push_back(power(expanded_basis, a.overall_coeff));
566 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
567 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
571 if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
572 !ex_to_numeric(expanded_exponent).is_integer()) {
573 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
576 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
580 // integer numeric exponent
581 const numeric & num_exponent = ex_to_numeric(expanded_exponent);
582 int int_exponent = num_exponent.to_int();
585 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
586 return expand_add(ex_to_add(expanded_basis), int_exponent);
588 // (x*y)^n -> x^n * y^n
589 if (is_ex_exactly_of_type(expanded_basis,mul))
590 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
592 // cannot expand further
593 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
596 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
600 // new virtual functions which can be overridden by derived classes
606 // non-virtual functions in this class
609 /** expand a^n where a is an add and n is an integer.
610 * @see power::expand */
611 ex power::expand_add(const add & a, int n) const
614 return expand_add_2(a);
618 sum.reserve((n+1)*(m-1));
620 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
621 intvector upper_limit(m-1);
624 for (int l=0; l<m-1; l++) {
633 for (l=0; l<m-1; 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(k[l])));
645 term.push_back(power(b,k[l]));
648 const ex & b = a.op(l);
649 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
650 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
651 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
652 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
653 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
654 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
655 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
656 if (is_ex_exactly_of_type(b,mul))
657 term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
659 term.push_back(power(b,n-k_cum[m-2]));
661 numeric f = binomial(numeric(n),numeric(k[0]));
662 for (l=1; l<m-1; l++)
663 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
668 cout << "begin term" << endl;
669 for (int i=0; i<m-1; i++) {
670 cout << "k[" << i << "]=" << k[i] << endl;
671 cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
672 cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
674 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
675 cout << *cit << endl;
677 cout << "end term" << endl;
680 // TODO: optimize this
681 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
685 while ((l>=0)&&((++k[l])>upper_limit[l])) {
691 // recalc k_cum[] and upper_limit[]
695 k_cum[l] = k_cum[l-1]+k[l];
697 for (int i=l+1; i<m-1; i++)
698 k_cum[i] = k_cum[i-1]+k[i];
700 for (int i=l+1; i<m-1; i++)
701 upper_limit[i] = n-k_cum[i-1];
703 return (new add(sum))->setflag(status_flags::dynallocated |
704 status_flags::expanded );
708 /** Special case of power::expand_add. Expands a^2 where a is an add.
709 * @see power::expand_add */
710 ex power::expand_add_2(const add & a) const
713 unsigned a_nops = a.nops();
714 sum.reserve((a_nops*(a_nops+1))/2);
715 epvector::const_iterator last = a.seq.end();
717 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
718 // first part: ignore overall_coeff and expand other terms
719 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
720 const ex & r = (*cit0).rest;
721 const ex & c = (*cit0).coeff;
723 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
724 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
725 !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
726 !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
727 !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
728 !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
729 !is_ex_exactly_of_type(ex_to_power(r).basis,power));
731 if (are_ex_trivially_equal(c,_ex1())) {
732 if (is_ex_exactly_of_type(r,mul)) {
733 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
736 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
740 if (is_ex_exactly_of_type(r,mul)) {
741 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
742 ex_to_numeric(c).power_dyn(_num2())));
744 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
745 ex_to_numeric(c).power_dyn(_num2())));
749 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
750 const ex & r1 = (*cit1).rest;
751 const ex & c1 = (*cit1).coeff;
752 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
753 _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
757 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
759 // second part: add terms coming from overall_factor (if != 0)
760 if (!a.overall_coeff.is_zero()) {
761 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
762 sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
764 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
767 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
769 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
772 /** Expand factors of m in m^n where m is a mul and n is and integer
773 * @see power::expand */
774 ex power::expand_mul(const mul & m, const numeric & n) const
780 distrseq.reserve(m.seq.size());
781 epvector::const_iterator last = m.seq.end();
782 epvector::const_iterator cit = m.seq.begin();
784 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
785 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
787 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
788 // since n is an integer
789 distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
793 return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
797 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
798 unsigned options) const
805 const add & addref=static_cast<const add &>(*basis.bp);
809 ex first_operands=add(splitseq);
810 ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
812 int n=exponent.to_int();
813 for (int k=0; k<=n; k++) {
814 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
815 * power(last_operand,numeric(n-k)));
817 return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
822 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
823 unsigned options) const
825 ex rest_power = ex(power(basis,exponent.add(_num_1()))).
826 expand(options | expand_options::internal_do_not_expand_power_operands);
828 return ex(mul(rest_power,basis),0).
829 expand(options | expand_options::internal_do_not_expand_mul_operands);
834 // static member variables
839 unsigned power::precedence = 60;
843 ex sqrt(const ex & a)
845 return power(a,_ex1_2());