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 exponent.print(c, precedence());
201 if (precedence() <= level) {
202 if (is_of_type(c, print_latex))
211 bool power::info(unsigned inf) const
214 case info_flags::polynomial:
215 case info_flags::integer_polynomial:
216 case info_flags::cinteger_polynomial:
217 case info_flags::rational_polynomial:
218 case info_flags::crational_polynomial:
219 return exponent.info(info_flags::nonnegint);
220 case info_flags::rational_function:
221 return exponent.info(info_flags::integer);
222 case info_flags::algebraic:
223 return (!exponent.info(info_flags::integer) ||
226 return inherited::info(inf);
229 unsigned power::nops() const
234 ex & power::let_op(int i)
239 return i==0 ? basis : exponent;
242 int power::degree(const ex & s) const
244 if (is_exactly_of_type(*exponent.bp,numeric)) {
245 if (basis.is_equal(s)) {
246 if (ex_to_numeric(exponent).is_integer())
247 return ex_to_numeric(exponent).to_int();
251 return basis.degree(s) * ex_to_numeric(exponent).to_int();
256 int power::ldegree(const ex & s) const
258 if (is_exactly_of_type(*exponent.bp,numeric)) {
259 if (basis.is_equal(s)) {
260 if (ex_to_numeric(exponent).is_integer())
261 return ex_to_numeric(exponent).to_int();
265 return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
270 ex power::coeff(const ex & s, int n) const
272 if (!basis.is_equal(s)) {
273 // basis not equal to s
280 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
282 int int_exp = ex_to_numeric(exponent).to_int();
288 // non-integer exponents are treated as zero
297 ex power::eval(int level) const
299 // simplifications: ^(x,0) -> 1 (0^0 handled here)
301 // ^(0,c1) -> 0 or exception (depending on real value of c1)
303 // ^(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)
304 // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
305 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
306 // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
307 // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
309 debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
311 if ((level==1) && (flags & status_flags::evaluated))
313 else if (level == -max_recursion_level)
314 throw(std::runtime_error("max recursion level reached"));
316 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
317 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
319 bool basis_is_numerical = 0;
320 bool exponent_is_numerical = 0;
322 numeric * num_exponent;
324 if (is_exactly_of_type(*ebasis.bp,numeric)) {
325 basis_is_numerical = 1;
326 num_basis = static_cast<numeric *>(ebasis.bp);
328 if (is_exactly_of_type(*eexponent.bp,numeric)) {
329 exponent_is_numerical = 1;
330 num_exponent = static_cast<numeric *>(eexponent.bp);
333 // ^(x,0) -> 1 (0^0 also handled here)
334 if (eexponent.is_zero()) {
335 if (ebasis.is_zero())
336 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
342 if (eexponent.is_equal(_ex1()))
345 // ^(0,c1) -> 0 or exception (depending on real value of c1)
346 if (ebasis.is_zero() && exponent_is_numerical) {
347 if ((num_exponent->real()).is_zero())
348 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
349 else if ((num_exponent->real()).is_negative())
350 throw (pole_error("power::eval(): division by zero",1));
356 if (ebasis.is_equal(_ex1()))
359 if (basis_is_numerical && exponent_is_numerical) {
360 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
361 // except if c1,c2 are rational, but c1^c2 is not)
362 bool basis_is_crational = num_basis->is_crational();
363 bool exponent_is_crational = num_exponent->is_crational();
364 numeric res = num_basis->power(*num_exponent);
366 if ((!basis_is_crational || !exponent_is_crational)
367 || res.is_crational()) {
370 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
371 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
372 if (basis_is_crational && exponent_is_crational
373 && num_exponent->is_real()
374 && !num_exponent->is_integer()) {
375 numeric n = num_exponent->numer();
376 numeric m = num_exponent->denom();
378 numeric q = iquo(n, m, r);
379 if (r.is_negative()) {
383 if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
387 res.push_back(expair(ebasis,r.div(m)));
388 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
393 // ^(^(x,c1),c2) -> ^(x,c1*c2)
394 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
395 // case c1==1 should not happen, see below!)
396 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
397 const power & sub_power = ex_to_power(ebasis);
398 const ex & sub_basis = sub_power.basis;
399 const ex & sub_exponent = sub_power.exponent;
400 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
401 const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
402 GINAC_ASSERT(num_sub_exponent!=numeric(1));
403 if (num_exponent->is_integer() || abs(num_sub_exponent)<1)
404 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
408 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
409 if (exponent_is_numerical && num_exponent->is_integer() &&
410 is_ex_exactly_of_type(ebasis,mul)) {
411 return expand_mul(ex_to_mul(ebasis), *num_exponent);
414 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
415 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
416 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
417 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
418 const mul & mulref = ex_to_mul(ebasis);
419 if (!mulref.overall_coeff.is_equal(_ex1())) {
420 const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
421 if (num_coeff.is_real()) {
422 if (num_coeff.is_positive()) {
423 mul * mulp = new mul(mulref);
424 mulp->overall_coeff = _ex1();
425 mulp->clearflag(status_flags::evaluated);
426 mulp->clearflag(status_flags::hash_calculated);
427 return (new mul(power(*mulp,exponent),
428 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
430 GINAC_ASSERT(num_coeff.compare(_num0())<0);
431 if (num_coeff.compare(_num_1())!=0) {
432 mul * mulp = new mul(mulref);
433 mulp->overall_coeff = _ex_1();
434 mulp->clearflag(status_flags::evaluated);
435 mulp->clearflag(status_flags::hash_calculated);
436 return (new mul(power(*mulp,exponent),
437 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
444 if (are_ex_trivially_equal(ebasis,basis) &&
445 are_ex_trivially_equal(eexponent,exponent)) {
448 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
449 status_flags::evaluated);
452 ex power::evalf(int level) const
454 debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
461 eexponent = exponent;
462 } else if (level == -max_recursion_level) {
463 throw(std::runtime_error("max recursion level reached"));
465 ebasis = basis.evalf(level-1);
466 if (!is_ex_exactly_of_type(eexponent,numeric))
467 eexponent = exponent.evalf(level-1);
469 eexponent = exponent;
472 return power(ebasis,eexponent);
475 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
477 const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
478 const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
480 if (are_ex_trivially_equal(basis, subsed_basis)
481 && are_ex_trivially_equal(exponent, subsed_exponent))
482 return basic::subs(ls, lr, no_pattern);
484 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
487 ex power::simplify_ncmul(const exvector & v) const
489 return inherited::simplify_ncmul(v);
494 /** Implementation of ex::diff() for a power.
496 ex power::derivative(const symbol & s) const
498 if (exponent.info(info_flags::real)) {
499 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
502 newseq.push_back(expair(basis, exponent - _ex1()));
503 newseq.push_back(expair(basis.diff(s), _ex1()));
504 return mul(newseq, exponent);
506 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
508 add(mul(exponent.diff(s), log(basis)),
509 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
513 int power::compare_same_type(const basic & other) const
515 GINAC_ASSERT(is_exactly_of_type(other, power));
516 const power & o=static_cast<const power &>(const_cast<basic &>(other));
519 cmpval=basis.compare(o.basis);
521 return exponent.compare(o.exponent);
526 unsigned power::return_type(void) const
528 return basis.return_type();
531 unsigned power::return_type_tinfo(void) const
533 return basis.return_type_tinfo();
536 ex power::expand(unsigned options) const
538 if (flags & status_flags::expanded)
541 ex expanded_basis = basis.expand(options);
542 ex expanded_exponent = exponent.expand(options);
544 // x^(a+b) -> x^a * x^b
545 if (is_ex_exactly_of_type(expanded_exponent, add)) {
546 const add &a = ex_to_add(expanded_exponent);
548 distrseq.reserve(a.seq.size() + 1);
549 epvector::const_iterator last = a.seq.end();
550 epvector::const_iterator cit = a.seq.begin();
552 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
556 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
557 if (ex_to_numeric(a.overall_coeff).is_integer()) {
558 const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
559 int int_exponent = num_exponent.to_int();
560 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
561 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
563 distrseq.push_back(power(expanded_basis, a.overall_coeff));
565 distrseq.push_back(power(expanded_basis, a.overall_coeff));
567 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
568 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
572 if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
573 !ex_to_numeric(expanded_exponent).is_integer()) {
574 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
577 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
581 // integer numeric exponent
582 const numeric & num_exponent = ex_to_numeric(expanded_exponent);
583 int int_exponent = num_exponent.to_int();
586 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
587 return expand_add(ex_to_add(expanded_basis), int_exponent);
589 // (x*y)^n -> x^n * y^n
590 if (is_ex_exactly_of_type(expanded_basis,mul))
591 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
593 // cannot expand further
594 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
597 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
601 // new virtual functions which can be overridden by derived classes
607 // non-virtual functions in this class
610 /** expand a^n where a is an add and n is an integer.
611 * @see power::expand */
612 ex power::expand_add(const add & a, int n) const
615 return expand_add_2(a);
619 sum.reserve((n+1)*(m-1));
621 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
622 intvector upper_limit(m-1);
625 for (int l=0; l<m-1; l++) {
634 for (l=0; l<m-1; l++) {
635 const ex & b = a.op(l);
636 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
637 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
638 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
639 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
640 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
641 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
642 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
643 if (is_ex_exactly_of_type(b,mul))
644 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
646 term.push_back(power(b,k[l]));
649 const ex & b = a.op(l);
650 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
651 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
652 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
653 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
654 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
655 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
656 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
657 if (is_ex_exactly_of_type(b,mul))
658 term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
660 term.push_back(power(b,n-k_cum[m-2]));
662 numeric f = binomial(numeric(n),numeric(k[0]));
663 for (l=1; l<m-1; l++)
664 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
669 cout << "begin term" << endl;
670 for (int i=0; i<m-1; i++) {
671 cout << "k[" << i << "]=" << k[i] << endl;
672 cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
673 cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
675 for_each(term.begin(), term.end(), ostream_iterator<ex>(cout, "\n"));
676 cout << "end term" << endl;
679 // TODO: optimize this
680 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
684 while ((l>=0)&&((++k[l])>upper_limit[l])) {
690 // recalc k_cum[] and upper_limit[]
694 k_cum[l] = k_cum[l-1]+k[l];
696 for (int i=l+1; i<m-1; i++)
697 k_cum[i] = k_cum[i-1]+k[i];
699 for (int i=l+1; i<m-1; i++)
700 upper_limit[i] = n-k_cum[i-1];
702 return (new add(sum))->setflag(status_flags::dynallocated |
703 status_flags::expanded );
707 /** Special case of power::expand_add. Expands a^2 where a is an add.
708 * @see power::expand_add */
709 ex power::expand_add_2(const add & a) const
712 unsigned a_nops = a.nops();
713 sum.reserve((a_nops*(a_nops+1))/2);
714 epvector::const_iterator last = a.seq.end();
716 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
717 // first part: ignore overall_coeff and expand other terms
718 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
719 const ex & r = (*cit0).rest;
720 const ex & c = (*cit0).coeff;
722 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
723 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
724 !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
725 !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
726 !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
727 !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
728 !is_ex_exactly_of_type(ex_to_power(r).basis,power));
730 if (are_ex_trivially_equal(c,_ex1())) {
731 if (is_ex_exactly_of_type(r,mul)) {
732 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
735 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
739 if (is_ex_exactly_of_type(r,mul)) {
740 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
741 ex_to_numeric(c).power_dyn(_num2())));
743 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
744 ex_to_numeric(c).power_dyn(_num2())));
748 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
749 const ex & r1 = (*cit1).rest;
750 const ex & c1 = (*cit1).coeff;
751 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
752 _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
756 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
758 // second part: add terms coming from overall_factor (if != 0)
759 if (!a.overall_coeff.is_zero()) {
760 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
761 sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
763 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
766 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
768 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
771 /** Expand factors of m in m^n where m is a mul and n is and integer
772 * @see power::expand */
773 ex power::expand_mul(const mul & m, const numeric & n) const
779 distrseq.reserve(m.seq.size());
780 epvector::const_iterator last = m.seq.end();
781 epvector::const_iterator cit = m.seq.begin();
783 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
784 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
786 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
787 // since n is an integer
788 distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
792 return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
796 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
797 unsigned options) const
804 const add & addref=static_cast<const add &>(*basis.bp);
808 ex first_operands=add(splitseq);
809 ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
811 int n=exponent.to_int();
812 for (int k=0; k<=n; k++) {
813 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
814 * power(last_operand,numeric(n-k)));
816 return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
821 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
822 unsigned options) const
824 ex rest_power = ex(power(basis,exponent.add(_num_1()))).
825 expand(options | expand_options::internal_do_not_expand_power_operands);
827 return ex(mul(rest_power,basis),0).
828 expand(options | expand_options::internal_do_not_expand_mul_operands);
834 ex sqrt(const ex & a)
836 return power(a,_ex1_2());