3 * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
6 * GiNaC Copyright (C) 1999-2003 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
29 #include "expairseq.h"
35 #include "operators.h"
36 #include "inifcns.h" // for log() in power::derivative()
47 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
49 typedef std::vector<int> intvector;
52 // default constructor
55 power::power() : inherited(TINFO_power) { }
67 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
69 n.find_ex("basis", basis, sym_lst);
70 n.find_ex("exponent", exponent, sym_lst);
73 void power::archive(archive_node &n) const
75 inherited::archive(n);
76 n.add_ex("basis", basis);
77 n.add_ex("exponent", exponent);
80 DEFAULT_UNARCHIVE(power)
83 // functions overriding virtual functions from base classes
88 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
90 // Optimal output of integer powers of symbols to aid compiler CSE.
91 // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
92 // to learn why such a parenthesation is really necessary.
95 } else if (exp == 2) {
102 print_sym_pow(c, x, exp-1);
105 print_sym_pow(c, x, exp >> 1);
107 print_sym_pow(c, x, exp >> 1);
112 void power::print(const print_context & c, unsigned level) const
114 if (is_a<print_tree>(c)) {
116 inherited::print(c, level);
118 } else if (is_a<print_csrc>(c)) {
120 // Integer powers of symbols are printed in a special, optimized way
121 if (exponent.info(info_flags::integer)
122 && (is_a<symbol>(basis) || is_a<constant>(basis))) {
123 int exp = ex_to<numeric>(exponent).to_int();
128 if (is_a<print_csrc_cl_N>(c))
133 print_sym_pow(c, ex_to<symbol>(basis), exp);
136 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
137 } else if (exponent.is_equal(_ex_1)) {
138 if (is_a<print_csrc_cl_N>(c))
145 // Otherwise, use the pow() or expt() (CLN) functions
147 if (is_a<print_csrc_cl_N>(c))
157 } else if (is_a<print_python_repr>(c)) {
159 c.s << class_name() << '(';
167 bool is_tex = is_a<print_latex>(c);
169 if (is_tex && is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
171 // Powers with negative numeric exponents are printed as fractions in TeX
173 power(basis, -exponent).eval().print(c);
176 } else if (exponent.is_equal(_ex1_2)) {
178 // Square roots are printed in a special way
179 c.s << (is_tex ? "\\sqrt{" : "sqrt(");
181 c.s << (is_tex ? '}' : ')');
185 // Ordinary output of powers using '^' or '**'
186 if (precedence() <= level)
187 c.s << (is_tex ? "{(" : "(");
188 basis.print(c, precedence());
189 if (is_a<print_python>(c))
195 exponent.print(c, precedence());
198 if (precedence() <= level)
199 c.s << (is_tex ? ")}" : ")");
204 bool power::info(unsigned inf) const
207 case info_flags::polynomial:
208 case info_flags::integer_polynomial:
209 case info_flags::cinteger_polynomial:
210 case info_flags::rational_polynomial:
211 case info_flags::crational_polynomial:
212 return exponent.info(info_flags::nonnegint);
213 case info_flags::rational_function:
214 return exponent.info(info_flags::integer);
215 case info_flags::algebraic:
216 return (!exponent.info(info_flags::integer) ||
219 return inherited::info(inf);
222 size_t power::nops() const
227 ex power::op(size_t i) const
231 return i==0 ? basis : exponent;
234 ex power::map(map_function & f) const
236 return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
239 int power::degree(const ex & s) const
241 if (is_equal(ex_to<basic>(s)))
243 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
244 if (basis.is_equal(s))
245 return ex_to<numeric>(exponent).to_int();
247 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
248 } else if (basis.has(s))
249 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
254 int power::ldegree(const ex & s) const
256 if (is_equal(ex_to<basic>(s)))
258 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
259 if (basis.is_equal(s))
260 return ex_to<numeric>(exponent).to_int();
262 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
263 } else if (basis.has(s))
264 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
269 ex power::coeff(const ex & s, int n) const
271 if (is_equal(ex_to<basic>(s)))
272 return n==1 ? _ex1 : _ex0;
273 else if (!basis.is_equal(s)) {
274 // basis not equal to s
281 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
283 int int_exp = ex_to<numeric>(exponent).to_int();
289 // non-integer exponents are treated as zero
298 /** Perform automatic term rewriting rules in this class. In the following
299 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
300 * stand for such expressions that contain a plain number.
301 * - ^(x,0) -> 1 (also handles ^(0,0))
303 * - ^(0,c) -> 0 or exception (depending on the real part of c)
305 * - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
306 * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
307 * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
308 * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
309 * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
311 * @param level cut-off in recursive evaluation */
312 ex power::eval(int level) const
314 if ((level==1) && (flags & status_flags::evaluated))
316 else if (level == -max_recursion_level)
317 throw(std::runtime_error("max recursion level reached"));
319 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
320 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
322 bool basis_is_numerical = false;
323 bool exponent_is_numerical = false;
324 const numeric *num_basis;
325 const numeric *num_exponent;
327 if (is_exactly_a<numeric>(ebasis)) {
328 basis_is_numerical = true;
329 num_basis = &ex_to<numeric>(ebasis);
331 if (is_exactly_a<numeric>(eexponent)) {
332 exponent_is_numerical = true;
333 num_exponent = &ex_to<numeric>(eexponent);
336 // ^(x,0) -> 1 (0^0 also handled here)
337 if (eexponent.is_zero()) {
338 if (ebasis.is_zero())
339 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
345 if (eexponent.is_equal(_ex1))
348 // ^(0,c1) -> 0 or exception (depending on real value of c1)
349 if (ebasis.is_zero() && exponent_is_numerical) {
350 if ((num_exponent->real()).is_zero())
351 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
352 else if ((num_exponent->real()).is_negative())
353 throw (pole_error("power::eval(): division by zero",1));
359 if (ebasis.is_equal(_ex1))
362 if (exponent_is_numerical) {
364 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
365 // except if c1,c2 are rational, but c1^c2 is not)
366 if (basis_is_numerical) {
367 const bool basis_is_crational = num_basis->is_crational();
368 const bool exponent_is_crational = num_exponent->is_crational();
369 if (!basis_is_crational || !exponent_is_crational) {
370 // return a plain float
371 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
372 status_flags::evaluated |
373 status_flags::expanded);
376 const numeric res = num_basis->power(*num_exponent);
377 if (res.is_crational()) {
380 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
382 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
383 if (basis_is_crational && exponent_is_crational
384 && num_exponent->is_real()
385 && !num_exponent->is_integer()) {
386 const numeric n = num_exponent->numer();
387 const 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
395 if (num_basis->is_rational() && !num_basis->is_integer()) {
396 // try it for numerator and denominator separately, in order to
397 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
398 const numeric bnum = num_basis->numer();
399 const numeric bden = num_basis->denom();
400 const numeric res_bnum = bnum.power(*num_exponent);
401 const numeric res_bden = bden.power(*num_exponent);
402 if (res_bnum.is_integer())
403 return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
404 if (res_bden.is_integer())
405 return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
409 // assemble resulting product, but allowing for a re-evaluation,
410 // because otherwise we'll end up with something like
411 // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
412 // instead of 7/16*7^(1/3).
413 ex prod = power(*num_basis,r.div(m));
414 return prod*power(*num_basis,q);
419 // ^(^(x,c1),c2) -> ^(x,c1*c2)
420 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
421 // case c1==1 should not happen, see below!)
422 if (is_exactly_a<power>(ebasis)) {
423 const power & sub_power = ex_to<power>(ebasis);
424 const ex & sub_basis = sub_power.basis;
425 const ex & sub_exponent = sub_power.exponent;
426 if (is_exactly_a<numeric>(sub_exponent)) {
427 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
428 GINAC_ASSERT(num_sub_exponent!=numeric(1));
429 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1).is_negative())
430 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
434 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
435 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
436 return expand_mul(ex_to<mul>(ebasis), *num_exponent);
439 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
440 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
441 if (is_exactly_a<mul>(ebasis)) {
442 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
443 const mul & mulref = ex_to<mul>(ebasis);
444 if (!mulref.overall_coeff.is_equal(_ex1)) {
445 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
446 if (num_coeff.is_real()) {
447 if (num_coeff.is_positive()) {
448 mul *mulp = new mul(mulref);
449 mulp->overall_coeff = _ex1;
450 mulp->clearflag(status_flags::evaluated);
451 mulp->clearflag(status_flags::hash_calculated);
452 return (new mul(power(*mulp,exponent),
453 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
455 GINAC_ASSERT(num_coeff.compare(_num0)<0);
456 if (!num_coeff.is_equal(_num_1)) {
457 mul *mulp = new mul(mulref);
458 mulp->overall_coeff = _ex_1;
459 mulp->clearflag(status_flags::evaluated);
460 mulp->clearflag(status_flags::hash_calculated);
461 return (new mul(power(*mulp,exponent),
462 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
469 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
470 if (num_exponent->is_pos_integer() &&
471 ebasis.return_type() != return_types::commutative &&
472 !is_a<matrix>(ebasis)) {
473 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
477 if (are_ex_trivially_equal(ebasis,basis) &&
478 are_ex_trivially_equal(eexponent,exponent)) {
481 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
482 status_flags::evaluated);
485 ex power::evalf(int level) const
492 eexponent = exponent;
493 } else if (level == -max_recursion_level) {
494 throw(std::runtime_error("max recursion level reached"));
496 ebasis = basis.evalf(level-1);
497 if (!is_exactly_a<numeric>(exponent))
498 eexponent = exponent.evalf(level-1);
500 eexponent = exponent;
503 return power(ebasis,eexponent);
506 ex power::evalm() const
508 const ex ebasis = basis.evalm();
509 const ex eexponent = exponent.evalm();
510 if (is_a<matrix>(ebasis)) {
511 if (is_exactly_a<numeric>(eexponent)) {
512 return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
515 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
519 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
521 ex power::subs(const exmap & m, unsigned options) const
523 const ex &subsed_basis = basis.subs(m, options);
524 const ex &subsed_exponent = exponent.subs(m, options);
526 if (!are_ex_trivially_equal(basis, subsed_basis)
527 || !are_ex_trivially_equal(exponent, subsed_exponent))
528 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
530 if (!(options & subs_options::algebraic))
531 return subs_one_level(m, options);
533 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
534 int nummatches = std::numeric_limits<int>::max();
536 if (tryfactsubs(*this, it->first, nummatches, repls))
537 return (ex_to<basic>((*this) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches))).subs_one_level(m, options);
540 return subs_one_level(m, options);
543 ex power::eval_ncmul(const exvector & v) const
545 return inherited::eval_ncmul(v);
550 /** Implementation of ex::diff() for a power.
552 ex power::derivative(const symbol & s) const
554 if (exponent.info(info_flags::real)) {
555 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
558 newseq.push_back(expair(basis, exponent - _ex1));
559 newseq.push_back(expair(basis.diff(s), _ex1));
560 return mul(newseq, exponent);
562 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
564 add(mul(exponent.diff(s), log(basis)),
565 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
569 int power::compare_same_type(const basic & other) const
571 GINAC_ASSERT(is_exactly_a<power>(other));
572 const power &o = static_cast<const power &>(other);
574 int cmpval = basis.compare(o.basis);
578 return exponent.compare(o.exponent);
581 unsigned power::return_type() const
583 return basis.return_type();
586 unsigned power::return_type_tinfo() const
588 return basis.return_type_tinfo();
591 ex power::expand(unsigned options) const
593 if (options == 0 && (flags & status_flags::expanded))
596 const ex expanded_basis = basis.expand(options);
597 const ex expanded_exponent = exponent.expand(options);
599 // x^(a+b) -> x^a * x^b
600 if (is_exactly_a<add>(expanded_exponent)) {
601 const add &a = ex_to<add>(expanded_exponent);
603 distrseq.reserve(a.seq.size() + 1);
604 epvector::const_iterator last = a.seq.end();
605 epvector::const_iterator cit = a.seq.begin();
607 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
611 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
612 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
613 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
614 int int_exponent = num_exponent.to_int();
615 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
616 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
618 distrseq.push_back(power(expanded_basis, a.overall_coeff));
620 distrseq.push_back(power(expanded_basis, a.overall_coeff));
622 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
623 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
627 if (!is_exactly_a<numeric>(expanded_exponent) ||
628 !ex_to<numeric>(expanded_exponent).is_integer()) {
629 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
632 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
636 // integer numeric exponent
637 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
638 int int_exponent = num_exponent.to_int();
641 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
642 return expand_add(ex_to<add>(expanded_basis), int_exponent);
644 // (x*y)^n -> x^n * y^n
645 if (is_exactly_a<mul>(expanded_basis))
646 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
648 // cannot expand further
649 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
652 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
656 // new virtual functions which can be overridden by derived classes
662 // non-virtual functions in this class
665 /** expand a^n where a is an add and n is a positive integer.
666 * @see power::expand */
667 ex power::expand_add(const add & a, int n) const
670 return expand_add_2(a);
672 const size_t m = a.nops();
674 // The number of terms will be the number of combinatorial compositions,
675 // i.e. the number of unordered arrangement of m nonnegative integers
676 // which sum up to n. It is frequently written as C_n(m) and directly
677 // related with binomial coefficients:
678 result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
680 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
681 intvector upper_limit(m-1);
684 for (size_t l=0; l<m-1; ++l) {
693 for (l=0; l<m-1; ++l) {
694 const ex & b = a.op(l);
695 GINAC_ASSERT(!is_exactly_a<add>(b));
696 GINAC_ASSERT(!is_exactly_a<power>(b) ||
697 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
698 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
699 !is_exactly_a<add>(ex_to<power>(b).basis) ||
700 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
701 !is_exactly_a<power>(ex_to<power>(b).basis));
702 if (is_exactly_a<mul>(b))
703 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
705 term.push_back(power(b,k[l]));
708 const ex & b = a.op(l);
709 GINAC_ASSERT(!is_exactly_a<add>(b));
710 GINAC_ASSERT(!is_exactly_a<power>(b) ||
711 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
712 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
713 !is_exactly_a<add>(ex_to<power>(b).basis) ||
714 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
715 !is_exactly_a<power>(ex_to<power>(b).basis));
716 if (is_exactly_a<mul>(b))
717 term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
719 term.push_back(power(b,n-k_cum[m-2]));
721 numeric f = binomial(numeric(n),numeric(k[0]));
722 for (l=1; l<m-1; ++l)
723 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
727 result.push_back((new mul(term))->setflag(status_flags::dynallocated));
731 while ((l>=0) && ((++k[l])>upper_limit[l])) {
737 // recalc k_cum[] and upper_limit[]
738 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
740 for (size_t i=l+1; i<m-1; ++i)
741 k_cum[i] = k_cum[i-1]+k[i];
743 for (size_t i=l+1; i<m-1; ++i)
744 upper_limit[i] = n-k_cum[i-1];
747 return (new add(result))->setflag(status_flags::dynallocated |
748 status_flags::expanded);
752 /** Special case of power::expand_add. Expands a^2 where a is an add.
753 * @see power::expand_add */
754 ex power::expand_add_2(const add & a) const
757 size_t a_nops = a.nops();
758 sum.reserve((a_nops*(a_nops+1))/2);
759 epvector::const_iterator last = a.seq.end();
761 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
762 // first part: ignore overall_coeff and expand other terms
763 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
764 const ex & r = cit0->rest;
765 const ex & c = cit0->coeff;
767 GINAC_ASSERT(!is_exactly_a<add>(r));
768 GINAC_ASSERT(!is_exactly_a<power>(r) ||
769 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
770 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
771 !is_exactly_a<add>(ex_to<power>(r).basis) ||
772 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
773 !is_exactly_a<power>(ex_to<power>(r).basis));
775 if (c.is_equal(_ex1)) {
776 if (is_exactly_a<mul>(r)) {
777 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
780 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
784 if (is_exactly_a<mul>(r)) {
785 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r),_num2),
786 ex_to<numeric>(c).power_dyn(_num2)));
788 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
789 ex_to<numeric>(c).power_dyn(_num2)));
793 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
794 const ex & r1 = cit1->rest;
795 const ex & c1 = cit1->coeff;
796 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
797 _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
801 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
803 // second part: add terms coming from overall_factor (if != 0)
804 if (!a.overall_coeff.is_zero()) {
805 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
807 sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
810 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
813 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
815 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
818 /** Expand factors of m in m^n where m is a mul and n is and integer.
819 * @see power::expand */
820 ex power::expand_mul(const mul & m, const numeric & n) const
822 GINAC_ASSERT(n.is_integer());
828 distrseq.reserve(m.seq.size());
829 epvector::const_iterator last = m.seq.end();
830 epvector::const_iterator cit = m.seq.begin();
832 if (is_exactly_a<numeric>(cit->rest)) {
833 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
835 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
836 // since n is an integer
837 distrseq.push_back(expair(cit->rest, ex_to<numeric>(cit->coeff).mul(n)));
841 return (new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);