3 * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
6 * GiNaC Copyright (C) 1999-2007 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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
29 #include "expairseq.h"
35 #include "operators.h"
36 #include "inifcns.h" // for log() in power::derivative()
46 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
47 print_func<print_dflt>(&power::do_print_dflt).
48 print_func<print_latex>(&power::do_print_latex).
49 print_func<print_csrc>(&power::do_print_csrc).
50 print_func<print_python>(&power::do_print_python).
51 print_func<print_python_repr>(&power::do_print_python_repr))
53 typedef std::vector<int> intvector;
56 // default constructor
59 power::power() : inherited(TINFO_power) { }
71 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
73 n.find_ex("basis", basis, sym_lst);
74 n.find_ex("exponent", exponent, sym_lst);
77 void power::archive(archive_node &n) const
79 inherited::archive(n);
80 n.add_ex("basis", basis);
81 n.add_ex("exponent", exponent);
84 DEFAULT_UNARCHIVE(power)
87 // functions overriding virtual functions from base classes
92 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
94 // Ordinary output of powers using '^' or '**'
95 if (precedence() <= level)
96 c.s << openbrace << '(';
97 basis.print(c, precedence());
100 exponent.print(c, precedence());
102 if (precedence() <= level)
103 c.s << ')' << closebrace;
106 void power::do_print_dflt(const print_dflt & c, unsigned level) const
108 if (exponent.is_equal(_ex1_2)) {
110 // Square roots are printed in a special way
116 print_power(c, "^", "", "", level);
119 void power::do_print_latex(const print_latex & c, unsigned level) const
121 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
123 // Powers with negative numeric exponents are printed as fractions
125 power(basis, -exponent).eval().print(c);
128 } else if (exponent.is_equal(_ex1_2)) {
130 // Square roots are printed in a special way
136 print_power(c, "^", "{", "}", level);
139 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
141 // Optimal output of integer powers of symbols to aid compiler CSE.
142 // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
143 // to learn why such a parenthesation is really necessary.
146 } else if (exp == 2) {
150 } else if (exp & 1) {
153 print_sym_pow(c, x, exp-1);
156 print_sym_pow(c, x, exp >> 1);
158 print_sym_pow(c, x, exp >> 1);
163 void power::do_print_csrc(const print_csrc & c, unsigned level) const
165 // Integer powers of symbols are printed in a special, optimized way
166 if (exponent.info(info_flags::integer)
167 && (is_a<symbol>(basis) || is_a<constant>(basis))) {
168 int exp = ex_to<numeric>(exponent).to_int();
173 if (is_a<print_csrc_cl_N>(c))
178 print_sym_pow(c, ex_to<symbol>(basis), exp);
181 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
182 } else if (exponent.is_equal(_ex_1)) {
183 if (is_a<print_csrc_cl_N>(c))
190 // Otherwise, use the pow() or expt() (CLN) functions
192 if (is_a<print_csrc_cl_N>(c))
203 void power::do_print_python(const print_python & c, unsigned level) const
205 print_power(c, "**", "", "", level);
208 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
210 c.s << class_name() << '(';
217 bool power::info(unsigned inf) const
220 case info_flags::polynomial:
221 case info_flags::integer_polynomial:
222 case info_flags::cinteger_polynomial:
223 case info_flags::rational_polynomial:
224 case info_flags::crational_polynomial:
225 return exponent.info(info_flags::nonnegint) &&
227 case info_flags::rational_function:
228 return exponent.info(info_flags::integer) &&
230 case info_flags::algebraic:
231 return !exponent.info(info_flags::integer) ||
234 return inherited::info(inf);
237 size_t power::nops() const
242 ex power::op(size_t i) const
246 return i==0 ? basis : exponent;
249 ex power::map(map_function & f) const
251 const ex &mapped_basis = f(basis);
252 const ex &mapped_exponent = f(exponent);
254 if (!are_ex_trivially_equal(basis, mapped_basis)
255 || !are_ex_trivially_equal(exponent, mapped_exponent))
256 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
261 int power::degree(const ex & s) const
263 if (is_equal(ex_to<basic>(s)))
265 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
266 if (basis.is_equal(s))
267 return ex_to<numeric>(exponent).to_int();
269 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
270 } else if (basis.has(s))
271 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
276 int power::ldegree(const ex & s) const
278 if (is_equal(ex_to<basic>(s)))
280 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
281 if (basis.is_equal(s))
282 return ex_to<numeric>(exponent).to_int();
284 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
285 } else if (basis.has(s))
286 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
291 ex power::coeff(const ex & s, int n) const
293 if (is_equal(ex_to<basic>(s)))
294 return n==1 ? _ex1 : _ex0;
295 else if (!basis.is_equal(s)) {
296 // basis not equal to s
303 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
305 int int_exp = ex_to<numeric>(exponent).to_int();
311 // non-integer exponents are treated as zero
320 /** Perform automatic term rewriting rules in this class. In the following
321 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
322 * stand for such expressions that contain a plain number.
323 * - ^(x,0) -> 1 (also handles ^(0,0))
325 * - ^(0,c) -> 0 or exception (depending on the real part of c)
327 * - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
328 * - ^(^(x,c1),c2) -> ^(x,c1*c2) if x is positive and c1 is real.
329 * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
330 * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
331 * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
332 * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
334 * @param level cut-off in recursive evaluation */
335 ex power::eval(int level) const
337 if ((level==1) && (flags & status_flags::evaluated))
339 else if (level == -max_recursion_level)
340 throw(std::runtime_error("max recursion level reached"));
342 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
343 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
345 bool basis_is_numerical = false;
346 bool exponent_is_numerical = false;
347 const numeric *num_basis;
348 const numeric *num_exponent;
350 if (is_exactly_a<numeric>(ebasis)) {
351 basis_is_numerical = true;
352 num_basis = &ex_to<numeric>(ebasis);
354 if (is_exactly_a<numeric>(eexponent)) {
355 exponent_is_numerical = true;
356 num_exponent = &ex_to<numeric>(eexponent);
359 // ^(x,0) -> 1 (0^0 also handled here)
360 if (eexponent.is_zero()) {
361 if (ebasis.is_zero())
362 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
368 if (eexponent.is_equal(_ex1))
371 // ^(0,c1) -> 0 or exception (depending on real value of c1)
372 if (ebasis.is_zero() && exponent_is_numerical) {
373 if ((num_exponent->real()).is_zero())
374 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
375 else if ((num_exponent->real()).is_negative())
376 throw (pole_error("power::eval(): division by zero",1));
382 if (ebasis.is_equal(_ex1))
385 // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
386 if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
387 return power(ebasis.op(0), ebasis.op(1) * eexponent);
389 if (exponent_is_numerical) {
391 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
392 // except if c1,c2 are rational, but c1^c2 is not)
393 if (basis_is_numerical) {
394 const bool basis_is_crational = num_basis->is_crational();
395 const bool exponent_is_crational = num_exponent->is_crational();
396 if (!basis_is_crational || !exponent_is_crational) {
397 // return a plain float
398 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
399 status_flags::evaluated |
400 status_flags::expanded);
403 const numeric res = num_basis->power(*num_exponent);
404 if (res.is_crational()) {
407 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
409 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
410 if (basis_is_crational && exponent_is_crational
411 && num_exponent->is_real()
412 && !num_exponent->is_integer()) {
413 const numeric n = num_exponent->numer();
414 const numeric m = num_exponent->denom();
416 numeric q = iquo(n, m, r);
417 if (r.is_negative()) {
421 if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
422 if (num_basis->is_rational() && !num_basis->is_integer()) {
423 // try it for numerator and denominator separately, in order to
424 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
425 const numeric bnum = num_basis->numer();
426 const numeric bden = num_basis->denom();
427 const numeric res_bnum = bnum.power(*num_exponent);
428 const numeric res_bden = bden.power(*num_exponent);
429 if (res_bnum.is_integer())
430 return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
431 if (res_bden.is_integer())
432 return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
436 // assemble resulting product, but allowing for a re-evaluation,
437 // because otherwise we'll end up with something like
438 // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
439 // instead of 7/16*7^(1/3).
440 ex prod = power(*num_basis,r.div(m));
441 return prod*power(*num_basis,q);
446 // ^(^(x,c1),c2) -> ^(x,c1*c2)
447 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
448 // case c1==1 should not happen, see below!)
449 if (is_exactly_a<power>(ebasis)) {
450 const power & sub_power = ex_to<power>(ebasis);
451 const ex & sub_basis = sub_power.basis;
452 const ex & sub_exponent = sub_power.exponent;
453 if (is_exactly_a<numeric>(sub_exponent)) {
454 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
455 GINAC_ASSERT(num_sub_exponent!=numeric(1));
456 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative())
457 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
461 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
462 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
463 return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
466 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
467 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
468 if (is_exactly_a<mul>(ebasis)) {
469 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
470 const mul & mulref = ex_to<mul>(ebasis);
471 if (!mulref.overall_coeff.is_equal(_ex1)) {
472 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
473 if (num_coeff.is_real()) {
474 if (num_coeff.is_positive()) {
475 mul *mulp = new mul(mulref);
476 mulp->overall_coeff = _ex1;
477 mulp->clearflag(status_flags::evaluated);
478 mulp->clearflag(status_flags::hash_calculated);
479 return (new mul(power(*mulp,exponent),
480 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
482 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
483 if (!num_coeff.is_equal(*_num_1_p)) {
484 mul *mulp = new mul(mulref);
485 mulp->overall_coeff = _ex_1;
486 mulp->clearflag(status_flags::evaluated);
487 mulp->clearflag(status_flags::hash_calculated);
488 return (new mul(power(*mulp,exponent),
489 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
496 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
497 if (num_exponent->is_pos_integer() &&
498 ebasis.return_type() != return_types::commutative &&
499 !is_a<matrix>(ebasis)) {
500 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
504 if (are_ex_trivially_equal(ebasis,basis) &&
505 are_ex_trivially_equal(eexponent,exponent)) {
508 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
509 status_flags::evaluated);
512 ex power::evalf(int level) const
519 eexponent = exponent;
520 } else if (level == -max_recursion_level) {
521 throw(std::runtime_error("max recursion level reached"));
523 ebasis = basis.evalf(level-1);
524 if (!is_exactly_a<numeric>(exponent))
525 eexponent = exponent.evalf(level-1);
527 eexponent = exponent;
530 return power(ebasis,eexponent);
533 ex power::evalm() const
535 const ex ebasis = basis.evalm();
536 const ex eexponent = exponent.evalm();
537 if (is_a<matrix>(ebasis)) {
538 if (is_exactly_a<numeric>(eexponent)) {
539 return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
542 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
546 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
548 ex power::subs(const exmap & m, unsigned options) const
550 const ex &subsed_basis = basis.subs(m, options);
551 const ex &subsed_exponent = exponent.subs(m, options);
553 if (!are_ex_trivially_equal(basis, subsed_basis)
554 || !are_ex_trivially_equal(exponent, subsed_exponent))
555 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
557 if (!(options & subs_options::algebraic))
558 return subs_one_level(m, options);
560 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
561 int nummatches = std::numeric_limits<int>::max();
563 if (tryfactsubs(*this, it->first, nummatches, repls))
564 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);
567 return subs_one_level(m, options);
570 ex power::eval_ncmul(const exvector & v) const
572 return inherited::eval_ncmul(v);
575 ex power::conjugate() const
577 ex newbasis = basis.conjugate();
578 ex newexponent = exponent.conjugate();
579 if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
582 return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
587 /** Implementation of ex::diff() for a power.
589 ex power::derivative(const symbol & s) const
591 if (is_a<numeric>(exponent)) {
592 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
595 newseq.push_back(expair(basis, exponent - _ex1));
596 newseq.push_back(expair(basis.diff(s), _ex1));
597 return mul(newseq, exponent);
599 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
601 add(mul(exponent.diff(s), log(basis)),
602 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
606 int power::compare_same_type(const basic & other) const
608 GINAC_ASSERT(is_exactly_a<power>(other));
609 const power &o = static_cast<const power &>(other);
611 int cmpval = basis.compare(o.basis);
615 return exponent.compare(o.exponent);
618 unsigned power::return_type() const
620 return basis.return_type();
623 unsigned power::return_type_tinfo() const
625 return basis.return_type_tinfo();
628 ex power::expand(unsigned options) const
630 if (options == 0 && (flags & status_flags::expanded))
633 const ex expanded_basis = basis.expand(options);
634 const ex expanded_exponent = exponent.expand(options);
636 // x^(a+b) -> x^a * x^b
637 if (is_exactly_a<add>(expanded_exponent)) {
638 const add &a = ex_to<add>(expanded_exponent);
640 distrseq.reserve(a.seq.size() + 1);
641 epvector::const_iterator last = a.seq.end();
642 epvector::const_iterator cit = a.seq.begin();
644 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
648 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
649 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
650 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
651 int int_exponent = num_exponent.to_int();
652 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
653 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
655 distrseq.push_back(power(expanded_basis, a.overall_coeff));
657 distrseq.push_back(power(expanded_basis, a.overall_coeff));
659 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
660 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
661 return r.expand(options);
664 if (!is_exactly_a<numeric>(expanded_exponent) ||
665 !ex_to<numeric>(expanded_exponent).is_integer()) {
666 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
669 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
673 // integer numeric exponent
674 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
675 int int_exponent = num_exponent.to_int();
678 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
679 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
681 // (x*y)^n -> x^n * y^n
682 if (is_exactly_a<mul>(expanded_basis))
683 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
685 // cannot expand further
686 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
689 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
693 // new virtual functions which can be overridden by derived classes
699 // non-virtual functions in this class
702 /** expand a^n where a is an add and n is a positive integer.
703 * @see power::expand */
704 ex power::expand_add(const add & a, int n, unsigned options) const
707 return expand_add_2(a, options);
709 const size_t m = a.nops();
711 // The number of terms will be the number of combinatorial compositions,
712 // i.e. the number of unordered arrangements of m nonnegative integers
713 // which sum up to n. It is frequently written as C_n(m) and directly
714 // related with binomial coefficients:
715 result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
717 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
718 intvector upper_limit(m-1);
721 for (size_t l=0; l<m-1; ++l) {
730 for (l=0; l<m-1; ++l) {
731 const ex & b = a.op(l);
732 GINAC_ASSERT(!is_exactly_a<add>(b));
733 GINAC_ASSERT(!is_exactly_a<power>(b) ||
734 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
735 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
736 !is_exactly_a<add>(ex_to<power>(b).basis) ||
737 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
738 !is_exactly_a<power>(ex_to<power>(b).basis));
739 if (is_exactly_a<mul>(b))
740 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
742 term.push_back(power(b,k[l]));
745 const ex & b = a.op(l);
746 GINAC_ASSERT(!is_exactly_a<add>(b));
747 GINAC_ASSERT(!is_exactly_a<power>(b) ||
748 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
749 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
750 !is_exactly_a<add>(ex_to<power>(b).basis) ||
751 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
752 !is_exactly_a<power>(ex_to<power>(b).basis));
753 if (is_exactly_a<mul>(b))
754 term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
756 term.push_back(power(b,n-k_cum[m-2]));
758 numeric f = binomial(numeric(n),numeric(k[0]));
759 for (l=1; l<m-1; ++l)
760 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
764 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
768 while ((l>=0) && ((++k[l])>upper_limit[l])) {
774 // recalc k_cum[] and upper_limit[]
775 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
777 for (size_t i=l+1; i<m-1; ++i)
778 k_cum[i] = k_cum[i-1]+k[i];
780 for (size_t i=l+1; i<m-1; ++i)
781 upper_limit[i] = n-k_cum[i-1];
784 return (new add(result))->setflag(status_flags::dynallocated |
785 status_flags::expanded);
789 /** Special case of power::expand_add. Expands a^2 where a is an add.
790 * @see power::expand_add */
791 ex power::expand_add_2(const add & a, unsigned options) const
794 size_t a_nops = a.nops();
795 sum.reserve((a_nops*(a_nops+1))/2);
796 epvector::const_iterator last = a.seq.end();
798 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
799 // first part: ignore overall_coeff and expand other terms
800 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
801 const ex & r = cit0->rest;
802 const ex & c = cit0->coeff;
804 GINAC_ASSERT(!is_exactly_a<add>(r));
805 GINAC_ASSERT(!is_exactly_a<power>(r) ||
806 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
807 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
808 !is_exactly_a<add>(ex_to<power>(r).basis) ||
809 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
810 !is_exactly_a<power>(ex_to<power>(r).basis));
812 if (c.is_equal(_ex1)) {
813 if (is_exactly_a<mul>(r)) {
814 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
817 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
821 if (is_exactly_a<mul>(r)) {
822 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
823 ex_to<numeric>(c).power_dyn(*_num2_p)));
825 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
826 ex_to<numeric>(c).power_dyn(*_num2_p)));
830 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
831 const ex & r1 = cit1->rest;
832 const ex & c1 = cit1->coeff;
833 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
834 _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
838 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
840 // second part: add terms coming from overall_factor (if != 0)
841 if (!a.overall_coeff.is_zero()) {
842 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
844 sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
847 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
850 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
852 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
855 /** Expand factors of m in m^n where m is a mul and n is and integer.
856 * @see power::expand */
857 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
859 GINAC_ASSERT(n.is_integer());
865 // Leave it to multiplication since dummy indices have to be renamed
866 if (get_all_dummy_indices(m).size() > 0 && n.is_positive()) {
868 for (int i=1; i < n.to_int(); i++)
869 result *= rename_dummy_indices_uniquely(m,m);
874 distrseq.reserve(m.seq.size());
875 bool need_reexpand = false;
877 epvector::const_iterator last = m.seq.end();
878 epvector::const_iterator cit = m.seq.begin();
880 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
881 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
882 // this happens when e.g. (a+b)^(1/2) gets squared and
883 // the resulting product needs to be reexpanded
884 need_reexpand = true;
886 distrseq.push_back(p);
890 const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
892 return ex(result).expand(options);
894 return result.setflag(status_flags::expanded);