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"
41 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
43 typedef std::vector<int> intvector;
46 // default ctor, dtor, copy ctor assignment operator and helpers
51 power::power() : basic(TINFO_power)
53 debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
58 void power::copy(const power & other)
60 inherited::copy(other);
62 exponent = other.exponent;
65 void power::destroy(bool call_parent)
67 if (call_parent) inherited::destroy(call_parent);
76 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
78 debugmsg("power ctor from ex,ex",LOGLEVEL_CONSTRUCT);
79 GINAC_ASSERT(basis.return_type()==return_types::commutative);
82 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
84 debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
85 GINAC_ASSERT(basis.return_type()==return_types::commutative);
92 /** Construct object from archive_node. */
93 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
95 debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
96 n.find_ex("basis", basis, sym_lst);
97 n.find_ex("exponent", exponent, sym_lst);
100 /** Unarchive the object. */
101 ex power::unarchive(const archive_node &n, const lst &sym_lst)
103 return (new power(n, sym_lst))->setflag(status_flags::dynallocated);
106 /** Archive the object. */
107 void power::archive(archive_node &n) const
109 inherited::archive(n);
110 n.add_ex("basis", basis);
111 n.add_ex("exponent", exponent);
115 // functions overriding virtual functions from bases classes
120 void power::print(std::ostream & os, unsigned upper_precedence) const
122 debugmsg("power print",LOGLEVEL_PRINT);
123 if (exponent.is_equal(_ex1_2())) {
124 os << "sqrt(" << basis << ")";
126 if (precedence<=upper_precedence) os << "(";
127 basis.print(os,precedence);
129 exponent.print(os,precedence);
130 if (precedence<=upper_precedence) os << ")";
134 void power::printraw(std::ostream & os) const
136 debugmsg("power printraw",LOGLEVEL_PRINT);
138 os << class_name() << "(";
141 exponent.printraw(os);
142 os << ",hash=" << hashvalue << ",flags=" << flags << ")";
145 void power::printtree(std::ostream & os, unsigned indent) const
147 debugmsg("power printtree",LOGLEVEL_PRINT);
149 os << std::string(indent,' ') << class_name()
150 << ", hash=" << hashvalue
151 << " (0x" << std::hex << hashvalue << std::dec << ")"
152 << ", flags=" << flags << std::endl;
153 basis.printtree(os, indent+delta_indent);
154 exponent.printtree(os, indent+delta_indent);
157 static void print_sym_pow(std::ostream & os, unsigned type, const symbol &x, int exp)
159 // Optimal output of integer powers of symbols to aid compiler CSE.
160 // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
161 // to learn why such a hack is really necessary.
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"));
354 if (eexponent.is_equal(_ex1()))
357 // ^(0,c1) -> 0 or exception (depending on real value of c1)
358 if (ebasis.is_zero() && exponent_is_numerical) {
359 if ((num_exponent->real()).is_zero())
360 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
361 else if ((num_exponent->real()).is_negative())
362 throw (pole_error("power::eval(): division by zero",1));
368 if (ebasis.is_equal(_ex1()))
371 if (basis_is_numerical && exponent_is_numerical) {
372 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
373 // except if c1,c2 are rational, but c1^c2 is not)
374 bool basis_is_crational = num_basis->is_crational();
375 bool exponent_is_crational = num_exponent->is_crational();
376 numeric res = num_basis->power(*num_exponent);
378 if ((!basis_is_crational || !exponent_is_crational)
379 || res.is_crational()) {
382 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
383 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
384 if (basis_is_crational && exponent_is_crational
385 && num_exponent->is_real()
386 && !num_exponent->is_integer()) {
387 numeric n = num_exponent->numer();
388 numeric m = num_exponent->denom();
390 numeric q = iquo(n, m, r);
391 if (r.is_negative()) {
395 if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
399 res.push_back(expair(ebasis,r.div(m)));
400 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
405 // ^(^(x,c1),c2) -> ^(x,c1*c2)
406 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
407 // case c1==1 should not happen, see below!)
408 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
409 const power & sub_power = ex_to_power(ebasis);
410 const ex & sub_basis = sub_power.basis;
411 const ex & sub_exponent = sub_power.exponent;
412 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
413 const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
414 GINAC_ASSERT(num_sub_exponent!=numeric(1));
415 if (num_exponent->is_integer() || abs(num_sub_exponent)<1)
416 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()) {
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);
602 // (x*y)^n -> x^n * y^n
603 if (is_ex_exactly_of_type(expanded_basis,mul))
604 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
606 // cannot expand further
607 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
610 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
614 // new virtual functions which can be overridden by derived classes
620 // non-virtual functions in this class
623 /** expand a^n where a is an add and n is an integer.
624 * @see power::expand */
625 ex power::expand_add(const add & a, int n) const
628 return expand_add_2(a);
632 sum.reserve((n+1)*(m-1));
634 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
635 intvector upper_limit(m-1);
638 for (int l=0; l<m-1; l++) {
647 for (l=0; l<m-1; 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(k[l])));
659 term.push_back(power(b,k[l]));
662 const ex & b = a.op(l);
663 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
664 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
665 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
666 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
667 !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
668 !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
669 !is_ex_exactly_of_type(ex_to_power(b).basis,power));
670 if (is_ex_exactly_of_type(b,mul))
671 term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
673 term.push_back(power(b,n-k_cum[m-2]));
675 numeric f = binomial(numeric(n),numeric(k[0]));
676 for (l=1; l<m-1; l++)
677 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
682 cout << "begin term" << endl;
683 for (int i=0; i<m-1; i++) {
684 cout << "k[" << i << "]=" << k[i] << endl;
685 cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
686 cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
688 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
689 cout << *cit << endl;
691 cout << "end term" << endl;
694 // TODO: optimize this
695 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
699 while ((l>=0)&&((++k[l])>upper_limit[l])) {
705 // recalc k_cum[] and upper_limit[]
709 k_cum[l] = k_cum[l-1]+k[l];
711 for (int i=l+1; i<m-1; i++)
712 k_cum[i] = k_cum[i-1]+k[i];
714 for (int i=l+1; i<m-1; i++)
715 upper_limit[i] = n-k_cum[i-1];
717 return (new add(sum))->setflag(status_flags::dynallocated |
718 status_flags::expanded );
722 /** Special case of power::expand_add. Expands a^2 where a is an add.
723 * @see power::expand_add */
724 ex power::expand_add_2(const add & a) const
727 unsigned a_nops = a.nops();
728 sum.reserve((a_nops*(a_nops+1))/2);
729 epvector::const_iterator last = a.seq.end();
731 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
732 // first part: ignore overall_coeff and expand other terms
733 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
734 const ex & r = (*cit0).rest;
735 const ex & c = (*cit0).coeff;
737 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
738 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
739 !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
740 !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
741 !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
742 !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
743 !is_ex_exactly_of_type(ex_to_power(r).basis,power));
745 if (are_ex_trivially_equal(c,_ex1())) {
746 if (is_ex_exactly_of_type(r,mul)) {
747 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
750 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
754 if (is_ex_exactly_of_type(r,mul)) {
755 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
756 ex_to_numeric(c).power_dyn(_num2())));
758 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
759 ex_to_numeric(c).power_dyn(_num2())));
763 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
764 const ex & r1 = (*cit1).rest;
765 const ex & c1 = (*cit1).coeff;
766 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
767 _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
771 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
773 // second part: add terms coming from overall_factor (if != 0)
774 if (!a.overall_coeff.is_zero()) {
775 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
776 sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
778 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
781 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
783 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
786 /** Expand factors of m in m^n where m is a mul and n is and integer
787 * @see power::expand */
788 ex power::expand_mul(const mul & m, const numeric & n) const
794 distrseq.reserve(m.seq.size());
795 epvector::const_iterator last = m.seq.end();
796 epvector::const_iterator cit = m.seq.begin();
798 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
799 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
801 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
802 // since n is an integer
803 distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
807 return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
811 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
812 unsigned options) const
819 const add & addref=static_cast<const add &>(*basis.bp);
823 ex first_operands=add(splitseq);
824 ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
826 int n=exponent.to_int();
827 for (int k=0; k<=n; k++) {
828 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
829 * power(last_operand,numeric(n-k)));
831 return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
836 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
837 unsigned options) const
839 ex rest_power = ex(power(basis,exponent.add(_num_1()))).
840 expand(options | expand_options::internal_do_not_expand_power_operands);
842 return ex(mul(rest_power,basis),0).
843 expand(options | expand_options::internal_do_not_expand_mul_operands);
848 // static member variables
853 unsigned power::precedence = 60;
857 ex sqrt(const ex & a)
859 return power(a,_ex1_2());