3 * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
6 * GiNaC Copyright (C) 1999-2000 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"
39 #ifndef NO_NAMESPACE_GINAC
41 #endif // ndef NO_NAMESPACE_GINAC
43 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
45 typedef std::vector<int> intvector;
48 // default constructor, destructor, copy constructor assignment operator and helpers
53 power::power() : basic(TINFO_power)
55 debugmsg("power default constructor",LOGLEVEL_CONSTRUCT);
60 debugmsg("power destructor",LOGLEVEL_DESTRUCT);
64 power::power(const power & other)
66 debugmsg("power copy constructor",LOGLEVEL_CONSTRUCT);
70 const power & power::operator=(const power & other)
72 debugmsg("power operator=",LOGLEVEL_ASSIGNMENT);
82 void power::copy(const power & other)
84 inherited::copy(other);
86 exponent=other.exponent;
89 void power::destroy(bool call_parent)
91 if (call_parent) inherited::destroy(call_parent);
100 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
102 debugmsg("power constructor from ex,ex",LOGLEVEL_CONSTRUCT);
103 GINAC_ASSERT(basis.return_type()==return_types::commutative);
106 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
108 debugmsg("power constructor from ex,numeric",LOGLEVEL_CONSTRUCT);
109 GINAC_ASSERT(basis.return_type()==return_types::commutative);
116 /** Construct object from archive_node. */
117 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
119 debugmsg("power constructor from archive_node", LOGLEVEL_CONSTRUCT);
120 n.find_ex("basis", basis, sym_lst);
121 n.find_ex("exponent", exponent, sym_lst);
124 /** Unarchive the object. */
125 ex power::unarchive(const archive_node &n, const lst &sym_lst)
127 return (new power(n, sym_lst))->setflag(status_flags::dynallocated);
130 /** Archive the object. */
131 void power::archive(archive_node &n) const
133 inherited::archive(n);
134 n.add_ex("basis", basis);
135 n.add_ex("exponent", exponent);
139 // functions overriding virtual functions from bases classes
144 basic * power::duplicate() const
146 debugmsg("power duplicate",LOGLEVEL_DUPLICATE);
147 return new power(*this);
150 void power::print(std::ostream & os, unsigned upper_precedence) const
152 debugmsg("power print",LOGLEVEL_PRINT);
153 if (exponent.is_equal(_ex1_2())) {
154 os << "sqrt(" << basis << ")";
156 if (precedence<=upper_precedence) os << "(";
157 basis.print(os,precedence);
159 exponent.print(os,precedence);
160 if (precedence<=upper_precedence) os << ")";
164 void power::printraw(std::ostream & os) const
166 debugmsg("power printraw",LOGLEVEL_PRINT);
171 exponent.printraw(os);
172 os << ",hash=" << hashvalue << ",flags=" << flags << ")";
175 void power::printtree(std::ostream & os, unsigned indent) const
177 debugmsg("power printtree",LOGLEVEL_PRINT);
179 os << std::string(indent,' ') << "power: "
180 << "hash=" << hashvalue
181 << " (0x" << std::hex << hashvalue << std::dec << ")"
182 << ", flags=" << flags << std::endl;
183 basis.printtree(os, indent+delta_indent);
184 exponent.printtree(os, indent+delta_indent);
187 static void print_sym_pow(std::ostream & os, unsigned type, const symbol &x, int exp)
189 // Optimal output of integer powers of symbols to aid compiler CSE
191 x.printcsrc(os, type, 0);
192 } else if (exp == 2) {
193 x.printcsrc(os, type, 0);
195 x.printcsrc(os, type, 0);
196 } else if (exp & 1) {
199 print_sym_pow(os, type, x, exp-1);
202 print_sym_pow(os, type, x, exp >> 1);
204 print_sym_pow(os, type, x, exp >> 1);
209 void power::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
211 debugmsg("power print csrc", LOGLEVEL_PRINT);
213 // Integer powers of symbols are printed in a special, optimized way
214 if (exponent.info(info_flags::integer)
215 && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
216 int exp = ex_to_numeric(exponent).to_int();
221 if (type == csrc_types::ctype_cl_N)
226 print_sym_pow(os, type, static_cast<const symbol &>(*basis.bp), exp);
229 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
230 } else if (exponent.compare(_num_1()) == 0) {
231 if (type == csrc_types::ctype_cl_N)
235 basis.bp->printcsrc(os, type, 0);
238 // Otherwise, use the pow() or expt() (CLN) functions
240 if (type == csrc_types::ctype_cl_N)
244 basis.bp->printcsrc(os, type, 0);
246 exponent.bp->printcsrc(os, type, 0);
251 bool power::info(unsigned inf) const
254 case info_flags::polynomial:
255 case info_flags::integer_polynomial:
256 case info_flags::cinteger_polynomial:
257 case info_flags::rational_polynomial:
258 case info_flags::crational_polynomial:
259 return exponent.info(info_flags::nonnegint);
260 case info_flags::rational_function:
261 return exponent.info(info_flags::integer);
262 case info_flags::algebraic:
263 return (!exponent.info(info_flags::integer) ||
266 return inherited::info(inf);
269 unsigned power::nops() const
274 ex & power::let_op(int i)
279 return i==0 ? basis : exponent;
282 int power::degree(const symbol & s) const
284 if (is_exactly_of_type(*exponent.bp,numeric)) {
285 if ((*basis.bp).compare(s)==0)
286 return ex_to_numeric(exponent).to_int();
288 return basis.degree(s) * ex_to_numeric(exponent).to_int();
293 int power::ldegree(const symbol & s) const
295 if (is_exactly_of_type(*exponent.bp,numeric)) {
296 if ((*basis.bp).compare(s)==0)
297 return ex_to_numeric(exponent).to_int();
299 return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
304 ex power::coeff(const symbol & s, int n) const
306 if ((*basis.bp).compare(s)!=0) {
307 // basis not equal to s
313 } else if (is_exactly_of_type(*exponent.bp,numeric)&&
314 (static_cast<const numeric &>(*exponent.bp).compare(numeric(n))==0)) {
321 ex power::eval(int level) const
323 // simplifications: ^(x,0) -> 1 (0^0 handled here)
325 // ^(0,c1) -> 0 or exception (depending on real value of c1)
327 // ^(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)
328 // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
329 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
330 // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
331 // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
333 debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
335 if ((level==1) && (flags & status_flags::evaluated))
337 else if (level == -max_recursion_level)
338 throw(std::runtime_error("max recursion level reached"));
340 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
341 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
343 bool basis_is_numerical = 0;
344 bool exponent_is_numerical = 0;
346 numeric * num_exponent;
348 if (is_exactly_of_type(*ebasis.bp,numeric)) {
349 basis_is_numerical = 1;
350 num_basis = static_cast<numeric *>(ebasis.bp);
352 if (is_exactly_of_type(*eexponent.bp,numeric)) {
353 exponent_is_numerical = 1;
354 num_exponent = static_cast<numeric *>(eexponent.bp);
357 // ^(x,0) -> 1 (0^0 also handled here)
358 if (eexponent.is_zero())
359 if (ebasis.is_zero())
360 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
365 if (eexponent.is_equal(_ex1()))
368 // ^(0,c1) -> 0 or exception (depending on real value of c1)
369 if (ebasis.is_zero() && exponent_is_numerical) {
370 if ((num_exponent->real()).is_zero())
371 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
372 else if ((num_exponent->real()).is_negative())
373 throw (pole_error("power::eval(): division by zero",1));
379 if (ebasis.is_equal(_ex1()))
382 if (basis_is_numerical && exponent_is_numerical) {
383 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
384 // except if c1,c2 are rational, but c1^c2 is not)
385 bool basis_is_crational = num_basis->is_crational();
386 bool exponent_is_crational = num_exponent->is_crational();
387 numeric res = (*num_basis).power(*num_exponent);
389 if ((!basis_is_crational || !exponent_is_crational)
390 || res.is_crational()) {
393 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
394 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
395 if (basis_is_crational && exponent_is_crational
396 && num_exponent->is_real()
397 && !num_exponent->is_integer()) {
398 numeric n = num_exponent->numer();
399 numeric m = num_exponent->denom();
401 numeric q = iquo(n, m, r);
402 if (r.is_negative()) {
406 if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
410 res.push_back(expair(ebasis,r.div(m)));
411 return (new mul(res,ex(num_basis->power(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
416 // ^(^(x,c1),c2) -> ^(x,c1*c2)
417 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
418 // case c1==1 should not happen, see below!)
419 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
420 const power & sub_power = ex_to_power(ebasis);
421 const ex & sub_basis = sub_power.basis;
422 const ex & sub_exponent = sub_power.exponent;
423 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
424 const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
425 GINAC_ASSERT(num_sub_exponent!=numeric(1));
426 if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
427 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
432 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
433 if (exponent_is_numerical && num_exponent->is_integer() &&
434 is_ex_exactly_of_type(ebasis,mul)) {
435 return expand_mul(ex_to_mul(ebasis), *num_exponent);
438 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
439 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
440 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
441 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
442 const mul & mulref=ex_to_mul(ebasis);
443 if (!mulref.overall_coeff.is_equal(_ex1())) {
444 const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff);
445 if (num_coeff.is_real()) {
446 if (num_coeff.is_positive()>0) {
447 mul * mulp=new mul(mulref);
448 mulp->overall_coeff=_ex1();
449 mulp->clearflag(status_flags::evaluated);
450 mulp->clearflag(status_flags::hash_calculated);
451 return (new mul(power(*mulp,exponent),
452 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
454 GINAC_ASSERT(num_coeff.compare(_num0())<0);
455 if (num_coeff.compare(_num_1())!=0) {
456 mul * mulp=new mul(mulref);
457 mulp->overall_coeff=_ex_1();
458 mulp->clearflag(status_flags::evaluated);
459 mulp->clearflag(status_flags::hash_calculated);
460 return (new mul(power(*mulp,exponent),
461 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
468 if (are_ex_trivially_equal(ebasis,basis) &&
469 are_ex_trivially_equal(eexponent,exponent)) {
472 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
473 status_flags::evaluated);
476 ex power::evalf(int level) const
478 debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
485 eexponent = exponent;
486 } else if (level == -max_recursion_level) {
487 throw(std::runtime_error("max recursion level reached"));
489 ebasis = basis.evalf(level-1);
490 if (!is_ex_exactly_of_type(eexponent,numeric))
491 eexponent = exponent.evalf(level-1);
493 eexponent = exponent;
496 return power(ebasis,eexponent);
499 ex power::subs(const lst & ls, const lst & lr) const
501 const ex & subsed_basis=basis.subs(ls,lr);
502 const ex & subsed_exponent=exponent.subs(ls,lr);
504 if (are_ex_trivially_equal(basis,subsed_basis)&&
505 are_ex_trivially_equal(exponent,subsed_exponent)) {
509 return power(subsed_basis, subsed_exponent);
512 ex power::simplify_ncmul(const exvector & v) const
514 return inherited::simplify_ncmul(v);
519 /** Implementation of ex::diff() for a power.
521 ex power::derivative(const symbol & s) const
523 if (exponent.info(info_flags::real)) {
524 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
527 newseq.push_back(expair(basis, exponent - _ex1()));
528 newseq.push_back(expair(basis.diff(s), _ex1()));
529 return mul(newseq, exponent);
531 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
532 return mul(power(basis, exponent),
533 add(mul(exponent.diff(s), log(basis)),
534 mul(mul(exponent, basis.diff(s)), power(basis, -1))));
538 int power::compare_same_type(const basic & other) const
540 GINAC_ASSERT(is_exactly_of_type(other, power));
541 const power & o=static_cast<const power &>(const_cast<basic &>(other));
544 cmpval=basis.compare(o.basis);
546 return exponent.compare(o.exponent);
551 unsigned power::return_type(void) const
553 return basis.return_type();
556 unsigned power::return_type_tinfo(void) const
558 return basis.return_type_tinfo();
561 ex power::expand(unsigned options) const
563 if (flags & status_flags::expanded)
566 ex expanded_basis = basis.expand(options);
568 if (!is_ex_exactly_of_type(exponent,numeric) ||
569 !ex_to_numeric(exponent).is_integer()) {
570 if (are_ex_trivially_equal(basis,expanded_basis)) {
573 return (new power(expanded_basis,exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
577 // integer numeric exponent
578 const numeric & num_exponent = ex_to_numeric(exponent);
579 int int_exponent = num_exponent.to_int();
581 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
582 return expand_add(ex_to_add(expanded_basis), int_exponent);
585 if (is_ex_exactly_of_type(expanded_basis,mul)) {
586 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
589 // cannot expand further
590 if (are_ex_trivially_equal(basis,expanded_basis)) {
593 return (new power(expanded_basis,exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
598 // new virtual functions which can be overridden by derived classes
604 // non-virtual functions in this class
607 /** expand a^n where a is an add and n is an integer.
608 * @see power::expand */
609 ex power::expand_add(const add & a, int n) const
612 return expand_add_2(a);
616 sum.reserve((n+1)*(m-1));
618 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
619 intvector upper_limit(m-1);
622 for (int l=0; l<m-1; l++) {
631 for (l=0; l<m-1; l++) {
632 const ex & b = a.op(l);
633 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
634 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
635 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
636 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
637 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
638 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
639 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
640 if (is_ex_exactly_of_type(b,mul)) {
641 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
643 term.push_back(power(b,k[l]));
647 const ex & b = a.op(l);
648 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
649 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
650 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
651 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
652 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
653 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
654 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
655 if (is_ex_exactly_of_type(b,mul)) {
656 term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
658 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=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];
701 for (int i=l+1; i<m-1; i++) {
702 upper_limit[i]=n-k_cum[i-1];
705 return (new add(sum))->setflag(status_flags::dynallocated |
706 status_flags::expanded );
710 /** Special case of power::expand_add. Expands a^2 where a is an add.
711 * @see power::expand_add */
712 ex power::expand_add_2(const add & a) const
715 unsigned a_nops=a.nops();
716 sum.reserve((a_nops*(a_nops+1))/2);
717 epvector::const_iterator last=a.seq.end();
719 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
720 // first part: ignore overall_coeff and expand other terms
721 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
722 const ex & r=(*cit0).rest;
723 const ex & c=(*cit0).coeff;
725 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
726 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
727 !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
728 !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
729 !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
730 !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
731 !is_ex_exactly_of_type(ex_to_power(r).basis,power));
733 if (are_ex_trivially_equal(c,_ex1())) {
734 if (is_ex_exactly_of_type(r,mul)) {
735 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
738 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
742 if (is_ex_exactly_of_type(r,mul)) {
743 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
744 ex_to_numeric(c).power_dyn(_num2())));
746 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
747 ex_to_numeric(c).power_dyn(_num2())));
751 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
752 const ex & r1=(*cit1).rest;
753 const ex & c1=(*cit1).coeff;
754 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
755 _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
759 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
761 // second part: add terms coming from overall_factor (if != 0)
762 if (!a.overall_coeff.is_equal(_ex0())) {
763 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
764 sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
766 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
769 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
771 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
774 /** Expand factors of m in m^n where m is a mul and n is and integer
775 * @see power::expand */
776 ex power::expand_mul(const mul & m, const numeric & n) const
778 if (n.is_equal(_num0()))
782 distrseq.reserve(m.seq.size());
783 epvector::const_iterator last = m.seq.end();
784 epvector::const_iterator cit = m.seq.begin();
786 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
787 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
789 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
790 // since n is an integer
791 distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
795 return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
799 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
800 unsigned options) const
807 const add & addref=static_cast<const add &>(*basis.bp);
811 ex first_operands=add(splitseq);
812 ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
814 int n=exponent.to_int();
815 for (int k=0; k<=n; k++) {
816 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
817 * power(last_operand,numeric(n-k)));
819 return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
824 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
825 unsigned options) const
827 ex rest_power = ex(power(basis,exponent.add(_num_1()))).
828 expand(options | expand_options::internal_do_not_expand_power_operands);
830 return ex(mul(rest_power,basis),0).
831 expand(options | expand_options::internal_do_not_expand_mul_operands);
836 // static member variables
841 unsigned power::precedence = 60;
847 const power some_power;
848 const type_info & typeid_power=typeid(some_power);
852 ex sqrt(const ex & a)
854 return power(a,_ex1_2());
857 #ifndef NO_NAMESPACE_GINAC
859 #endif // ndef NO_NAMESPACE_GINAC