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"
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 void power::copy(const power & other)
62 inherited::copy(other);
64 exponent=other.exponent;
67 void power::destroy(bool call_parent)
69 if (call_parent) inherited::destroy(call_parent);
78 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
80 debugmsg("power constructor from ex,ex",LOGLEVEL_CONSTRUCT);
81 GINAC_ASSERT(basis.return_type()==return_types::commutative);
84 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
86 debugmsg("power constructor from ex,numeric",LOGLEVEL_CONSTRUCT);
87 GINAC_ASSERT(basis.return_type()==return_types::commutative);
94 /** Construct object from archive_node. */
95 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
97 debugmsg("power constructor from archive_node", LOGLEVEL_CONSTRUCT);
98 n.find_ex("basis", basis, sym_lst);
99 n.find_ex("exponent", exponent, sym_lst);
102 /** Unarchive the object. */
103 ex power::unarchive(const archive_node &n, const lst &sym_lst)
105 return (new power(n, sym_lst))->setflag(status_flags::dynallocated);
108 /** Archive the object. */
109 void power::archive(archive_node &n) const
111 inherited::archive(n);
112 n.add_ex("basis", basis);
113 n.add_ex("exponent", exponent);
117 // functions overriding virtual functions from bases classes
122 void power::print(std::ostream & os, unsigned upper_precedence) const
124 debugmsg("power print",LOGLEVEL_PRINT);
125 if (exponent.is_equal(_ex1_2())) {
126 os << "sqrt(" << basis << ")";
128 if (precedence<=upper_precedence) os << "(";
129 basis.print(os,precedence);
131 exponent.print(os,precedence);
132 if (precedence<=upper_precedence) os << ")";
136 void power::printraw(std::ostream & os) const
138 debugmsg("power printraw",LOGLEVEL_PRINT);
143 exponent.printraw(os);
144 os << ",hash=" << hashvalue << ",flags=" << flags << ")";
147 void power::printtree(std::ostream & os, unsigned indent) const
149 debugmsg("power printtree",LOGLEVEL_PRINT);
151 os << std::string(indent,' ') << "power: "
152 << "hash=" << hashvalue
153 << " (0x" << std::hex << hashvalue << std::dec << ")"
154 << ", flags=" << flags << std::endl;
155 basis.printtree(os, indent+delta_indent);
156 exponent.printtree(os, indent+delta_indent);
159 static void print_sym_pow(std::ostream & os, unsigned type, const symbol &x, int exp)
161 // Optimal output of integer powers of symbols to aid compiler CSE
163 x.printcsrc(os, type, 0);
164 } else if (exp == 2) {
165 x.printcsrc(os, type, 0);
167 x.printcsrc(os, type, 0);
168 } else if (exp & 1) {
171 print_sym_pow(os, type, x, exp-1);
174 print_sym_pow(os, type, x, exp >> 1);
176 print_sym_pow(os, type, x, exp >> 1);
181 void power::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
183 debugmsg("power print csrc", LOGLEVEL_PRINT);
185 // Integer powers of symbols are printed in a special, optimized way
186 if (exponent.info(info_flags::integer)
187 && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
188 int exp = ex_to_numeric(exponent).to_int();
193 if (type == csrc_types::ctype_cl_N)
198 print_sym_pow(os, type, static_cast<const symbol &>(*basis.bp), exp);
201 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
202 } else if (exponent.compare(_num_1()) == 0) {
203 if (type == csrc_types::ctype_cl_N)
207 basis.bp->printcsrc(os, type, 0);
210 // Otherwise, use the pow() or expt() (CLN) functions
212 if (type == csrc_types::ctype_cl_N)
216 basis.bp->printcsrc(os, type, 0);
218 exponent.bp->printcsrc(os, type, 0);
223 bool power::info(unsigned inf) const
226 case info_flags::polynomial:
227 case info_flags::integer_polynomial:
228 case info_flags::cinteger_polynomial:
229 case info_flags::rational_polynomial:
230 case info_flags::crational_polynomial:
231 return exponent.info(info_flags::nonnegint);
232 case info_flags::rational_function:
233 return exponent.info(info_flags::integer);
234 case info_flags::algebraic:
235 return (!exponent.info(info_flags::integer) ||
238 return inherited::info(inf);
241 unsigned power::nops() const
246 ex & power::let_op(int i)
251 return i==0 ? basis : exponent;
254 int power::degree(const symbol & s) const
256 if (is_exactly_of_type(*exponent.bp,numeric)) {
257 if ((*basis.bp).compare(s)==0) {
258 if (ex_to_numeric(exponent).is_integer())
259 return ex_to_numeric(exponent).to_int();
263 return basis.degree(s) * ex_to_numeric(exponent).to_int();
268 int power::ldegree(const symbol & s) const
270 if (is_exactly_of_type(*exponent.bp,numeric)) {
271 if ((*basis.bp).compare(s)==0) {
272 if (ex_to_numeric(exponent).is_integer())
273 return ex_to_numeric(exponent).to_int();
277 return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
282 ex power::coeff(const symbol & s, int n) const
284 if ((*basis.bp).compare(s)!=0) {
285 // basis not equal to s
292 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
294 int int_exp = ex_to_numeric(exponent).to_int();
300 // non-integer exponents are treated as zero
309 ex power::eval(int level) const
311 // simplifications: ^(x,0) -> 1 (0^0 handled here)
313 // ^(0,c1) -> 0 or exception (depending on real value of c1)
315 // ^(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)
316 // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
317 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
318 // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
319 // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
321 debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
323 if ((level==1) && (flags & status_flags::evaluated))
325 else if (level == -max_recursion_level)
326 throw(std::runtime_error("max recursion level reached"));
328 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
329 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
331 bool basis_is_numerical = 0;
332 bool exponent_is_numerical = 0;
334 numeric * num_exponent;
336 if (is_exactly_of_type(*ebasis.bp,numeric)) {
337 basis_is_numerical = 1;
338 num_basis = static_cast<numeric *>(ebasis.bp);
340 if (is_exactly_of_type(*eexponent.bp,numeric)) {
341 exponent_is_numerical = 1;
342 num_exponent = static_cast<numeric *>(eexponent.bp);
345 // ^(x,0) -> 1 (0^0 also handled here)
346 if (eexponent.is_zero())
347 if (ebasis.is_zero())
348 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
353 if (eexponent.is_equal(_ex1()))
356 // ^(0,c1) -> 0 or exception (depending on real value of c1)
357 if (ebasis.is_zero() && exponent_is_numerical) {
358 if ((num_exponent->real()).is_zero())
359 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
360 else if ((num_exponent->real()).is_negative())
361 throw (pole_error("power::eval(): division by zero",1));
367 if (ebasis.is_equal(_ex1()))
370 if (basis_is_numerical && exponent_is_numerical) {
371 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
372 // except if c1,c2 are rational, but c1^c2 is not)
373 bool basis_is_crational = num_basis->is_crational();
374 bool exponent_is_crational = num_exponent->is_crational();
375 numeric res = (*num_basis).power(*num_exponent);
377 if ((!basis_is_crational || !exponent_is_crational)
378 || res.is_crational()) {
381 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
382 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
383 if (basis_is_crational && exponent_is_crational
384 && num_exponent->is_real()
385 && !num_exponent->is_integer()) {
386 numeric n = num_exponent->numer();
387 numeric m = num_exponent->denom();
389 numeric q = iquo(n, m, r);
390 if (r.is_negative()) {
394 if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
398 res.push_back(expair(ebasis,r.div(m)));
399 return (new mul(res,ex(num_basis->power(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
404 // ^(^(x,c1),c2) -> ^(x,c1*c2)
405 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
406 // case c1==1 should not happen, see below!)
407 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
408 const power & sub_power = ex_to_power(ebasis);
409 const ex & sub_basis = sub_power.basis;
410 const ex & sub_exponent = sub_power.exponent;
411 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
412 const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
413 GINAC_ASSERT(num_sub_exponent!=numeric(1));
414 if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
415 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
420 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
421 if (exponent_is_numerical && num_exponent->is_integer() &&
422 is_ex_exactly_of_type(ebasis,mul)) {
423 return expand_mul(ex_to_mul(ebasis), *num_exponent);
426 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
427 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
428 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
429 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
430 const mul & mulref=ex_to_mul(ebasis);
431 if (!mulref.overall_coeff.is_equal(_ex1())) {
432 const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff);
433 if (num_coeff.is_real()) {
434 if (num_coeff.is_positive()>0) {
435 mul * mulp=new mul(mulref);
436 mulp->overall_coeff=_ex1();
437 mulp->clearflag(status_flags::evaluated);
438 mulp->clearflag(status_flags::hash_calculated);
439 return (new mul(power(*mulp,exponent),
440 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
442 GINAC_ASSERT(num_coeff.compare(_num0())<0);
443 if (num_coeff.compare(_num_1())!=0) {
444 mul * mulp=new mul(mulref);
445 mulp->overall_coeff=_ex_1();
446 mulp->clearflag(status_flags::evaluated);
447 mulp->clearflag(status_flags::hash_calculated);
448 return (new mul(power(*mulp,exponent),
449 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
456 if (are_ex_trivially_equal(ebasis,basis) &&
457 are_ex_trivially_equal(eexponent,exponent)) {
460 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
461 status_flags::evaluated);
464 ex power::evalf(int level) const
466 debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
473 eexponent = exponent;
474 } else if (level == -max_recursion_level) {
475 throw(std::runtime_error("max recursion level reached"));
477 ebasis = basis.evalf(level-1);
478 if (!is_ex_exactly_of_type(eexponent,numeric))
479 eexponent = exponent.evalf(level-1);
481 eexponent = exponent;
484 return power(ebasis,eexponent);
487 ex power::subs(const lst & ls, const lst & lr) const
489 const ex & subsed_basis=basis.subs(ls,lr);
490 const ex & subsed_exponent=exponent.subs(ls,lr);
492 if (are_ex_trivially_equal(basis,subsed_basis)&&
493 are_ex_trivially_equal(exponent,subsed_exponent)) {
497 return power(subsed_basis, subsed_exponent);
500 ex power::simplify_ncmul(const exvector & v) const
502 return inherited::simplify_ncmul(v);
507 /** Implementation of ex::diff() for a power.
509 ex power::derivative(const symbol & s) const
511 if (exponent.info(info_flags::real)) {
512 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
515 newseq.push_back(expair(basis, exponent - _ex1()));
516 newseq.push_back(expair(basis.diff(s), _ex1()));
517 return mul(newseq, exponent);
519 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
521 add(mul(exponent.diff(s), log(basis)),
522 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
526 int power::compare_same_type(const basic & other) const
528 GINAC_ASSERT(is_exactly_of_type(other, power));
529 const power & o=static_cast<const power &>(const_cast<basic &>(other));
532 cmpval=basis.compare(o.basis);
534 return exponent.compare(o.exponent);
539 unsigned power::return_type(void) const
541 return basis.return_type();
544 unsigned power::return_type_tinfo(void) const
546 return basis.return_type_tinfo();
549 ex power::expand(unsigned options) const
551 if (flags & status_flags::expanded)
554 ex expanded_basis = basis.expand(options);
555 ex expanded_exponent = exponent.expand(options);
557 // x^(a+b) -> x^a * x^b
558 if (is_ex_exactly_of_type(expanded_exponent, add)) {
559 const add &a = ex_to_add(expanded_exponent);
561 distrseq.reserve(a.seq.size() + 1);
562 epvector::const_iterator last = a.seq.end();
563 epvector::const_iterator cit = a.seq.begin();
565 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
569 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
570 if (ex_to_numeric(a.overall_coeff).is_integer()) {
571 const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
572 int int_exponent = num_exponent.to_int();
573 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
574 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
576 distrseq.push_back(power(expanded_basis, a.overall_coeff));
578 distrseq.push_back(power(expanded_basis, a.overall_coeff));
580 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
581 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
585 if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
586 !ex_to_numeric(expanded_exponent).is_integer()) {
587 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
590 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
594 // integer numeric exponent
595 const numeric & num_exponent = ex_to_numeric(expanded_exponent);
596 int int_exponent = num_exponent.to_int();
599 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
600 return expand_add(ex_to_add(expanded_basis), int_exponent);
603 // (x*y)^n -> x^n * y^n
604 if (is_ex_exactly_of_type(expanded_basis,mul)) {
605 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
608 // cannot expand further
609 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
612 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
617 // new virtual functions which can be overridden by derived classes
623 // non-virtual functions in this class
626 /** expand a^n where a is an add and n is an integer.
627 * @see power::expand */
628 ex power::expand_add(const add & a, int n) const
631 return expand_add_2(a);
635 sum.reserve((n+1)*(m-1));
637 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
638 intvector upper_limit(m-1);
641 for (int l=0; l<m-1; l++) {
650 for (l=0; l<m-1; l++) {
651 const ex & b = a.op(l);
652 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
653 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
654 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
655 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
656 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
657 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
658 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
659 if (is_ex_exactly_of_type(b,mul)) {
660 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
662 term.push_back(power(b,k[l]));
666 const ex & b = a.op(l);
667 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
668 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
669 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
670 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
671 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
672 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
673 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
674 if (is_ex_exactly_of_type(b,mul)) {
675 term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
677 term.push_back(power(b,n-k_cum[m-2]));
680 numeric f = binomial(numeric(n),numeric(k[0]));
681 for (l=1; l<m-1; l++) {
682 f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
687 cout << "begin term" << endl;
688 for (int i=0; i<m-1; i++) {
689 cout << "k[" << i << "]=" << k[i] << endl;
690 cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
691 cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
693 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
694 cout << *cit << endl;
696 cout << "end term" << endl;
699 // TODO: optimize this
700 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
704 while ((l>=0)&&((++k[l])>upper_limit[l])) {
710 // recalc k_cum[] and upper_limit[]
714 k_cum[l]=k_cum[l-1]+k[l];
716 for (int i=l+1; i<m-1; i++) {
717 k_cum[i]=k_cum[i-1]+k[i];
720 for (int i=l+1; i<m-1; i++) {
721 upper_limit[i]=n-k_cum[i-1];
724 return (new add(sum))->setflag(status_flags::dynallocated |
725 status_flags::expanded );
729 /** Special case of power::expand_add. Expands a^2 where a is an add.
730 * @see power::expand_add */
731 ex power::expand_add_2(const add & a) const
734 unsigned a_nops=a.nops();
735 sum.reserve((a_nops*(a_nops+1))/2);
736 epvector::const_iterator last=a.seq.end();
738 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
739 // first part: ignore overall_coeff and expand other terms
740 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
741 const ex & r=(*cit0).rest;
742 const ex & c=(*cit0).coeff;
744 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
745 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
746 !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
747 !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
748 !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
749 !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
750 !is_ex_exactly_of_type(ex_to_power(r).basis,power));
752 if (are_ex_trivially_equal(c,_ex1())) {
753 if (is_ex_exactly_of_type(r,mul)) {
754 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
757 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
761 if (is_ex_exactly_of_type(r,mul)) {
762 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
763 ex_to_numeric(c).power_dyn(_num2())));
765 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
766 ex_to_numeric(c).power_dyn(_num2())));
770 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
771 const ex & r1=(*cit1).rest;
772 const ex & c1=(*cit1).coeff;
773 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
774 _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
778 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
780 // second part: add terms coming from overall_factor (if != 0)
781 if (!a.overall_coeff.is_equal(_ex0())) {
782 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
783 sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
785 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
788 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
790 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
793 /** Expand factors of m in m^n where m is a mul and n is and integer
794 * @see power::expand */
795 ex power::expand_mul(const mul & m, const numeric & n) const
797 if (n.is_equal(_num0()))
801 distrseq.reserve(m.seq.size());
802 epvector::const_iterator last = m.seq.end();
803 epvector::const_iterator cit = m.seq.begin();
805 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
806 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
808 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
809 // since n is an integer
810 distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
814 return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
818 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
819 unsigned options) const
826 const add & addref=static_cast<const add &>(*basis.bp);
830 ex first_operands=add(splitseq);
831 ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
833 int n=exponent.to_int();
834 for (int k=0; k<=n; k++) {
835 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
836 * power(last_operand,numeric(n-k)));
838 return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
843 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
844 unsigned options) const
846 ex rest_power = ex(power(basis,exponent.add(_num_1()))).
847 expand(options | expand_options::internal_do_not_expand_power_operands);
849 return ex(mul(rest_power,basis),0).
850 expand(options | expand_options::internal_do_not_expand_mul_operands);
855 // static member variables
860 unsigned power::precedence = 60;
864 ex sqrt(const ex & a)
866 return power(a,_ex1_2());
869 #ifndef NO_NAMESPACE_GINAC
871 #endif // ndef NO_NAMESPACE_GINAC