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 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(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(ostream & os) const
166 debugmsg("power printraw",LOGLEVEL_PRINT);
171 exponent.printraw(os);
172 os << ",hash=" << hashvalue << ",flags=" << flags << ")";
175 void power::printtree(ostream & os, unsigned indent) const
177 debugmsg("power printtree",LOGLEVEL_PRINT);
179 os << string(indent,' ') << "power: "
180 << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
181 << ", flags=" << flags << endl;
182 basis.printtree(os,indent+delta_indent);
183 exponent.printtree(os,indent+delta_indent);
186 static void print_sym_pow(ostream & os, unsigned type, const symbol &x, int exp)
188 // Optimal output of integer powers of symbols to aid compiler CSE
190 x.printcsrc(os, type, 0);
191 } else if (exp == 2) {
192 x.printcsrc(os, type, 0);
194 x.printcsrc(os, type, 0);
195 } else if (exp & 1) {
198 print_sym_pow(os, type, x, exp-1);
201 print_sym_pow(os, type, x, exp >> 1);
203 print_sym_pow(os, type, x, exp >> 1);
208 void power::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const
210 debugmsg("power print csrc", LOGLEVEL_PRINT);
212 // Integer powers of symbols are printed in a special, optimized way
213 if (exponent.info(info_flags::integer) &&
214 (is_ex_exactly_of_type(basis, symbol) ||
215 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
253 if (inf==info_flags::polynomial ||
254 inf==info_flags::integer_polynomial ||
255 inf==info_flags::cinteger_polynomial ||
256 inf==info_flags::rational_polynomial ||
257 inf==info_flags::crational_polynomial) {
258 return exponent.info(info_flags::nonnegint);
259 } else if (inf==info_flags::rational_function) {
260 return exponent.info(info_flags::integer);
262 return inherited::info(inf);
266 unsigned power::nops() const
271 ex & power::let_op(int i)
276 return i==0 ? basis : exponent;
279 int power::degree(const symbol & s) const
281 if (is_exactly_of_type(*exponent.bp,numeric)) {
282 if ((*basis.bp).compare(s)==0)
283 return ex_to_numeric(exponent).to_int();
285 return basis.degree(s) * ex_to_numeric(exponent).to_int();
290 int power::ldegree(const symbol & s) const
292 if (is_exactly_of_type(*exponent.bp,numeric)) {
293 if ((*basis.bp).compare(s)==0)
294 return ex_to_numeric(exponent).to_int();
296 return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
301 ex power::coeff(const symbol & s, int n) const
303 if ((*basis.bp).compare(s)!=0) {
304 // basis not equal to s
310 } else if (is_exactly_of_type(*exponent.bp,numeric)&&
311 (static_cast<const numeric &>(*exponent.bp).compare(numeric(n))==0)) {
318 ex power::eval(int level) const
320 // simplifications: ^(x,0) -> 1 (0^0 handled here)
322 // ^(0,c1) -> 0 or exception (depending on real value of c1)
324 // ^(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)
325 // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
326 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
327 // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
328 // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
330 debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
332 if ((level==1) && (flags & status_flags::evaluated))
334 else if (level == -max_recursion_level)
335 throw(std::runtime_error("max recursion level reached"));
337 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
338 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
340 bool basis_is_numerical = 0;
341 bool exponent_is_numerical = 0;
343 numeric * num_exponent;
345 if (is_exactly_of_type(*ebasis.bp,numeric)) {
346 basis_is_numerical = 1;
347 num_basis = static_cast<numeric *>(ebasis.bp);
349 if (is_exactly_of_type(*eexponent.bp,numeric)) {
350 exponent_is_numerical = 1;
351 num_exponent = static_cast<numeric *>(eexponent.bp);
354 // ^(x,0) -> 1 (0^0 also handled here)
355 if (eexponent.is_zero())
356 if (ebasis.is_zero())
357 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
362 if (eexponent.is_equal(_ex1()))
365 // ^(0,c1) -> 0 or exception (depending on real value of c1)
366 if (ebasis.is_zero() && exponent_is_numerical) {
367 if ((num_exponent->real()).is_zero())
368 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
369 else if ((num_exponent->real()).is_negative())
370 throw (std::overflow_error("power::eval(): division by zero"));
376 if (ebasis.is_equal(_ex1()))
379 if (basis_is_numerical && exponent_is_numerical) {
380 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
381 // except if c1,c2 are rational, but c1^c2 is not)
382 bool basis_is_crational = num_basis->is_crational();
383 bool exponent_is_crational = num_exponent->is_crational();
384 numeric res = (*num_basis).power(*num_exponent);
386 if ((!basis_is_crational || !exponent_is_crational)
387 || res.is_crational()) {
390 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
391 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
392 if (basis_is_crational && exponent_is_crational
393 && num_exponent->is_real()
394 && !num_exponent->is_integer()) {
395 numeric n = num_exponent->numer();
396 numeric m = num_exponent->denom();
398 numeric q = iquo(n, m, r);
399 if (r.is_negative()) {
403 if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
407 res.push_back(expair(ebasis,r.div(m)));
408 return (new mul(res,ex(num_basis->power(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
413 // ^(^(x,c1),c2) -> ^(x,c1*c2)
414 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
415 // case c1==1 should not happen, see below!)
416 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
417 const power & sub_power = ex_to_power(ebasis);
418 const ex & sub_basis = sub_power.basis;
419 const ex & sub_exponent = sub_power.exponent;
420 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
421 const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
422 GINAC_ASSERT(num_sub_exponent!=numeric(1));
423 if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
424 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
429 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
430 if (exponent_is_numerical && num_exponent->is_integer() &&
431 is_ex_exactly_of_type(ebasis,mul)) {
432 return expand_mul(ex_to_mul(ebasis), *num_exponent);
435 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
436 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
437 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
438 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
439 const mul & mulref=ex_to_mul(ebasis);
440 if (!mulref.overall_coeff.is_equal(_ex1())) {
441 const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff);
442 if (num_coeff.is_real()) {
443 if (num_coeff.is_positive()>0) {
444 mul * mulp=new mul(mulref);
445 mulp->overall_coeff=_ex1();
446 mulp->clearflag(status_flags::evaluated);
447 mulp->clearflag(status_flags::hash_calculated);
448 return (new mul(power(*mulp,exponent),
449 power(num_coeff,*num_exponent)))->
450 setflag(status_flags::dynallocated);
452 GINAC_ASSERT(num_coeff.compare(_num0())<0);
453 if (num_coeff.compare(_num_1())!=0) {
454 mul * mulp=new mul(mulref);
455 mulp->overall_coeff=_ex_1();
456 mulp->clearflag(status_flags::evaluated);
457 mulp->clearflag(status_flags::hash_calculated);
458 return (new mul(power(*mulp,exponent),
459 power(abs(num_coeff),*num_exponent)))->
460 setflag(status_flags::dynallocated);
467 if (are_ex_trivially_equal(ebasis,basis) &&
468 are_ex_trivially_equal(eexponent,exponent)) {
471 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
472 status_flags::evaluated);
475 ex power::evalf(int level) const
477 debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
484 eexponent = exponent;
485 } else if (level == -max_recursion_level) {
486 throw(std::runtime_error("max recursion level reached"));
488 ebasis = basis.evalf(level-1);
489 if (!is_ex_exactly_of_type(eexponent,numeric))
490 eexponent = exponent.evalf(level-1);
492 eexponent = exponent;
495 return power(ebasis,eexponent);
498 ex power::subs(const lst & ls, const lst & lr) const
500 const ex & subsed_basis=basis.subs(ls,lr);
501 const ex & subsed_exponent=exponent.subs(ls,lr);
503 if (are_ex_trivially_equal(basis,subsed_basis)&&
504 are_ex_trivially_equal(exponent,subsed_exponent)) {
508 return power(subsed_basis, subsed_exponent);
511 ex power::simplify_ncmul(const exvector & v) const
513 return inherited::simplify_ncmul(v);
518 /** Implementation of ex::diff() for a power.
520 ex power::derivative(const symbol & s) const
522 if (exponent.info(info_flags::real)) {
523 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
524 return mul(mul(exponent, power(basis, exponent - _ex1())), basis.diff(s));
526 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
527 return mul(power(basis, exponent),
528 add(mul(exponent.diff(s), log(basis)),
529 mul(mul(exponent, basis.diff(s)), power(basis, -1))));
533 int power::compare_same_type(const basic & other) const
535 GINAC_ASSERT(is_exactly_of_type(other, power));
536 const power & o=static_cast<const power &>(const_cast<basic &>(other));
539 cmpval=basis.compare(o.basis);
541 return exponent.compare(o.exponent);
546 unsigned power::return_type(void) const
548 return basis.return_type();
551 unsigned power::return_type_tinfo(void) const
553 return basis.return_type_tinfo();
556 ex power::expand(unsigned options) const
558 if (flags & status_flags::expanded)
561 ex expanded_basis = basis.expand(options);
563 if (!is_ex_exactly_of_type(exponent,numeric) ||
564 !ex_to_numeric(exponent).is_integer()) {
565 if (are_ex_trivially_equal(basis,expanded_basis)) {
568 return (new power(expanded_basis,exponent))->
569 setflag(status_flags::dynallocated |
570 status_flags::expanded);
574 // integer numeric exponent
575 const numeric & num_exponent = ex_to_numeric(exponent);
576 int int_exponent = num_exponent.to_int();
578 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
579 return expand_add(ex_to_add(expanded_basis), int_exponent);
582 if (is_ex_exactly_of_type(expanded_basis,mul)) {
583 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
586 // cannot expand further
587 if (are_ex_trivially_equal(basis,expanded_basis)) {
590 return (new power(expanded_basis,exponent))->
591 setflag(status_flags::dynallocated |
592 status_flags::expanded);
597 // new virtual functions which can be overridden by derived classes
603 // non-virtual functions in this class
606 /** expand a^n where a is an add and n is an integer.
607 * @see power::expand */
608 ex power::expand_add(const add & a, int n) const
611 return expand_add_2(a);
615 sum.reserve((n+1)*(m-1));
617 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
618 intvector upper_limit(m-1);
621 for (int l=0; l<m-1; l++) {
630 for (l=0; l<m-1; l++) {
631 const ex & b = a.op(l);
632 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
633 GINAC_ASSERT(!is_ex_exactly_of_type(b,power)||
634 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
635 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer()||
636 !is_ex_exactly_of_type(ex_to_power(b).basis,add)||
637 !is_ex_exactly_of_type(ex_to_power(b).basis,mul)||
638 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
639 if (is_ex_exactly_of_type(b,mul)) {
640 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
642 term.push_back(power(b,k[l]));
646 const ex & b = a.op(l);
647 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
648 GINAC_ASSERT(!is_ex_exactly_of_type(b,power)||
649 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
650 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer()||
651 !is_ex_exactly_of_type(ex_to_power(b).basis,add)||
652 !is_ex_exactly_of_type(ex_to_power(b).basis,mul)||
653 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
654 if (is_ex_exactly_of_type(b,mul)) {
655 term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
657 term.push_back(power(b,n-k_cum[m-2]));
660 numeric f = binomial(numeric(n),numeric(k[0]));
661 for (l=1; l<m-1; l++) {
662 f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
667 cout << "begin term" << endl;
668 for (int i=0; i<m-1; i++) {
669 cout << "k[" << i << "]=" << k[i] << endl;
670 cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
671 cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
673 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
674 cout << *cit << endl;
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];
700 for (int i=l+1; i<m-1; i++) {
701 upper_limit[i]=n-k_cum[i-1];
704 return (new add(sum))->setflag(status_flags::dynallocated |
705 status_flags::expanded );
709 /** Special case of power::expand_add. Expands a^2 where a is an add.
710 * @see power::expand_add */
711 ex power::expand_add_2(const add & a) const
714 unsigned a_nops=a.nops();
715 sum.reserve((a_nops*(a_nops+1))/2);
716 epvector::const_iterator last=a.seq.end();
718 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
719 // first part: ignore overall_coeff and expand other terms
720 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
721 const ex & r=(*cit0).rest;
722 const ex & c=(*cit0).coeff;
724 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
725 GINAC_ASSERT(!is_ex_exactly_of_type(r,power)||
726 !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric)||
727 !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer()||
728 !is_ex_exactly_of_type(ex_to_power(r).basis,add)||
729 !is_ex_exactly_of_type(ex_to_power(r).basis,mul)||
730 !is_ex_exactly_of_type(ex_to_power(r).basis,power));
732 if (are_ex_trivially_equal(c,_ex1())) {
733 if (is_ex_exactly_of_type(r,mul)) {
734 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),_ex1()));
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_equal(_ex0())) {
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 |
770 status_flags::expanded );
773 /** Expand factors of m in m^n where m is a mul and n is and integer
774 * @see power::expand */
775 ex power::expand_mul(const mul & m, const numeric & n) const
777 if (n.is_equal(_num0()))
781 distrseq.reserve(m.seq.size());
782 epvector::const_iterator last = m.seq.end();
783 epvector::const_iterator cit = m.seq.begin();
785 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
786 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
788 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
789 // since n is an integer
790 distrseq.push_back(expair((*cit).rest,
791 ex_to_numeric((*cit).coeff).mul(n)));
795 return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))
796 ->setflag(status_flags::dynallocated);
800 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
801 unsigned options) const
808 const add & addref=static_cast<const add &>(*basis.bp);
812 ex first_operands=add(splitseq);
813 ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
815 int n=exponent.to_int();
816 for (int k=0; k<=n; k++) {
817 distrseq.push_back(binomial(n,k)*power(first_operands,numeric(k))*
818 power(last_operand,numeric(n-k)));
820 return ex((new add(distrseq))->setflag(status_flags::expanded |
821 status_flags::dynallocated )).
827 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
828 unsigned options) const
830 ex rest_power=ex(power(basis,exponent.add(_num_1()))).
831 expand(options | expand_options::internal_do_not_expand_power_operands);
833 return ex(mul(rest_power,basis),0).
834 expand(options | expand_options::internal_do_not_expand_mul_operands);
839 // static member variables
844 unsigned power::precedence = 60;
850 const power some_power;
851 const type_info & typeid_power=typeid(some_power);
855 ex sqrt(const ex & a)
857 return power(a,_ex1_2());
860 #ifndef NO_NAMESPACE_GINAC
862 #endif // ndef NO_NAMESPACE_GINAC