3 * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
6 * GiNaC Copyright (C) 1999-2008 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()
43 #include "relational.h"
48 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
49 print_func<print_dflt>(&power::do_print_dflt).
50 print_func<print_latex>(&power::do_print_latex).
51 print_func<print_csrc>(&power::do_print_csrc).
52 print_func<print_python>(&power::do_print_python).
53 print_func<print_python_repr>(&power::do_print_python_repr).
54 print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))
56 typedef std::vector<int> intvector;
59 // default constructor
74 void power::read_archive(const archive_node &n, lst &sym_lst)
76 inherited::read_archive(n, sym_lst);
77 n.find_ex("basis", basis, sym_lst);
78 n.find_ex("exponent", exponent, sym_lst);
81 void power::archive(archive_node &n) const
83 inherited::archive(n);
84 n.add_ex("basis", basis);
85 n.add_ex("exponent", exponent);
89 // functions overriding virtual functions from base classes
94 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
96 // Ordinary output of powers using '^' or '**'
97 if (precedence() <= level)
98 c.s << openbrace << '(';
99 basis.print(c, precedence());
102 exponent.print(c, precedence());
104 if (precedence() <= level)
105 c.s << ')' << closebrace;
108 void power::do_print_dflt(const print_dflt & c, unsigned level) const
110 if (exponent.is_equal(_ex1_2)) {
112 // Square roots are printed in a special way
118 print_power(c, "^", "", "", level);
121 void power::do_print_latex(const print_latex & c, unsigned level) const
123 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
125 // Powers with negative numeric exponents are printed as fractions
127 power(basis, -exponent).eval().print(c);
130 } else if (exponent.is_equal(_ex1_2)) {
132 // Square roots are printed in a special way
138 print_power(c, "^", "{", "}", level);
141 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
143 // Optimal output of integer powers of symbols to aid compiler CSE.
144 // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
145 // to learn why such a parenthesation is really necessary.
148 } else if (exp == 2) {
152 } else if (exp & 1) {
155 print_sym_pow(c, x, exp-1);
158 print_sym_pow(c, x, exp >> 1);
160 print_sym_pow(c, x, exp >> 1);
165 void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const
167 if (exponent.is_equal(_ex_1)) {
180 void power::do_print_csrc(const print_csrc & c, unsigned level) const
182 // Integer powers of symbols are printed in a special, optimized way
183 if (exponent.info(info_flags::integer)
184 && (is_a<symbol>(basis) || is_a<constant>(basis))) {
185 int exp = ex_to<numeric>(exponent).to_int();
192 print_sym_pow(c, ex_to<symbol>(basis), exp);
195 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
196 } else if (exponent.is_equal(_ex_1)) {
201 // Otherwise, use the pow() function
211 void power::do_print_python(const print_python & c, unsigned level) const
213 print_power(c, "**", "", "", level);
216 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
218 c.s << class_name() << '(';
225 bool power::info(unsigned inf) const
228 case info_flags::polynomial:
229 case info_flags::integer_polynomial:
230 case info_flags::cinteger_polynomial:
231 case info_flags::rational_polynomial:
232 case info_flags::crational_polynomial:
233 return exponent.info(info_flags::nonnegint) &&
235 case info_flags::rational_function:
236 return exponent.info(info_flags::integer) &&
238 case info_flags::algebraic:
239 return !exponent.info(info_flags::integer) ||
241 case info_flags::expanded:
242 return (flags & status_flags::expanded);
243 case info_flags::has_indices: {
244 if (flags & status_flags::has_indices)
246 else if (flags & status_flags::has_no_indices)
248 else if (basis.info(info_flags::has_indices)) {
249 setflag(status_flags::has_indices);
250 clearflag(status_flags::has_no_indices);
253 clearflag(status_flags::has_indices);
254 setflag(status_flags::has_no_indices);
259 return inherited::info(inf);
262 size_t power::nops() const
267 ex power::op(size_t i) const
271 return i==0 ? basis : exponent;
274 ex power::map(map_function & f) const
276 const ex &mapped_basis = f(basis);
277 const ex &mapped_exponent = f(exponent);
279 if (!are_ex_trivially_equal(basis, mapped_basis)
280 || !are_ex_trivially_equal(exponent, mapped_exponent))
281 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
286 bool power::is_polynomial(const ex & var) const
288 if (exponent.has(var))
290 if (!exponent.info(info_flags::nonnegint))
292 return basis.is_polynomial(var);
295 int power::degree(const ex & s) const
297 if (is_equal(ex_to<basic>(s)))
299 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
300 if (basis.is_equal(s))
301 return ex_to<numeric>(exponent).to_int();
303 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
304 } else if (basis.has(s))
305 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
310 int power::ldegree(const ex & s) const
312 if (is_equal(ex_to<basic>(s)))
314 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
315 if (basis.is_equal(s))
316 return ex_to<numeric>(exponent).to_int();
318 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
319 } else if (basis.has(s))
320 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
325 ex power::coeff(const ex & s, int n) const
327 if (is_equal(ex_to<basic>(s)))
328 return n==1 ? _ex1 : _ex0;
329 else if (!basis.is_equal(s)) {
330 // basis not equal to s
337 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
339 int int_exp = ex_to<numeric>(exponent).to_int();
345 // non-integer exponents are treated as zero
354 /** Perform automatic term rewriting rules in this class. In the following
355 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
356 * stand for such expressions that contain a plain number.
357 * - ^(x,0) -> 1 (also handles ^(0,0))
359 * - ^(0,c) -> 0 or exception (depending on the real part of c)
361 * - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
362 * - ^(^(x,c1),c2) -> ^(x,c1*c2) if x is positive and c1 is real.
363 * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
364 * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
365 * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
366 * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
368 * @param level cut-off in recursive evaluation */
369 ex power::eval(int level) const
371 if ((level==1) && (flags & status_flags::evaluated))
373 else if (level == -max_recursion_level)
374 throw(std::runtime_error("max recursion level reached"));
376 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
377 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
379 bool basis_is_numerical = false;
380 bool exponent_is_numerical = false;
381 const numeric *num_basis;
382 const numeric *num_exponent;
384 if (is_exactly_a<numeric>(ebasis)) {
385 basis_is_numerical = true;
386 num_basis = &ex_to<numeric>(ebasis);
388 if (is_exactly_a<numeric>(eexponent)) {
389 exponent_is_numerical = true;
390 num_exponent = &ex_to<numeric>(eexponent);
393 // ^(x,0) -> 1 (0^0 also handled here)
394 if (eexponent.is_zero()) {
395 if (ebasis.is_zero())
396 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
402 if (eexponent.is_equal(_ex1))
405 // ^(0,c1) -> 0 or exception (depending on real value of c1)
406 if (ebasis.is_zero() && exponent_is_numerical) {
407 if ((num_exponent->real()).is_zero())
408 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
409 else if ((num_exponent->real()).is_negative())
410 throw (pole_error("power::eval(): division by zero",1));
416 if (ebasis.is_equal(_ex1))
419 // power of a function calculated by separate rules defined for this function
420 if (is_exactly_a<function>(ebasis))
421 return ex_to<function>(ebasis).power(eexponent);
423 // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
424 if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
425 return power(ebasis.op(0), ebasis.op(1) * eexponent);
427 if (exponent_is_numerical) {
429 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
430 // except if c1,c2 are rational, but c1^c2 is not)
431 if (basis_is_numerical) {
432 const bool basis_is_crational = num_basis->is_crational();
433 const bool exponent_is_crational = num_exponent->is_crational();
434 if (!basis_is_crational || !exponent_is_crational) {
435 // return a plain float
436 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
437 status_flags::evaluated |
438 status_flags::expanded);
441 const numeric res = num_basis->power(*num_exponent);
442 if (res.is_crational()) {
445 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
447 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
448 if (basis_is_crational && exponent_is_crational
449 && num_exponent->is_real()
450 && !num_exponent->is_integer()) {
451 const numeric n = num_exponent->numer();
452 const numeric m = num_exponent->denom();
454 numeric q = iquo(n, m, r);
455 if (r.is_negative()) {
459 if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
460 if (num_basis->is_rational() && !num_basis->is_integer()) {
461 // try it for numerator and denominator separately, in order to
462 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
463 const numeric bnum = num_basis->numer();
464 const numeric bden = num_basis->denom();
465 const numeric res_bnum = bnum.power(*num_exponent);
466 const numeric res_bden = bden.power(*num_exponent);
467 if (res_bnum.is_integer())
468 return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
469 if (res_bden.is_integer())
470 return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
474 // assemble resulting product, but allowing for a re-evaluation,
475 // because otherwise we'll end up with something like
476 // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
477 // instead of 7/16*7^(1/3).
478 ex prod = power(*num_basis,r.div(m));
479 return prod*power(*num_basis,q);
484 // ^(^(x,c1),c2) -> ^(x,c1*c2)
485 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
486 // case c1==1 should not happen, see below!)
487 if (is_exactly_a<power>(ebasis)) {
488 const power & sub_power = ex_to<power>(ebasis);
489 const ex & sub_basis = sub_power.basis;
490 const ex & sub_exponent = sub_power.exponent;
491 if (is_exactly_a<numeric>(sub_exponent)) {
492 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
493 GINAC_ASSERT(num_sub_exponent!=numeric(1));
494 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()) {
495 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
500 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
501 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
502 return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
505 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
506 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
507 numeric icont = ebasis.integer_content();
508 const numeric lead_coeff =
509 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
511 const bool canonicalizable = lead_coeff.is_integer();
512 const bool unit_normal = lead_coeff.is_pos_integer();
513 if (canonicalizable && (! unit_normal))
514 icont = icont.mul(*_num_1_p);
516 if (canonicalizable && (icont != *_num1_p)) {
517 const add& addref = ex_to<add>(ebasis);
518 add* addp = new add(addref);
519 addp->setflag(status_flags::dynallocated);
520 addp->clearflag(status_flags::hash_calculated);
521 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
522 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
523 i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
525 const numeric c = icont.power(*num_exponent);
526 if (likely(c != *_num1_p))
527 return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
529 return power(*addp, *num_exponent);
533 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
534 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
535 if (is_exactly_a<mul>(ebasis)) {
536 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
537 const mul & mulref = ex_to<mul>(ebasis);
538 if (!mulref.overall_coeff.is_equal(_ex1)) {
539 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
540 if (num_coeff.is_real()) {
541 if (num_coeff.is_positive()) {
542 mul *mulp = new mul(mulref);
543 mulp->overall_coeff = _ex1;
544 mulp->clearflag(status_flags::evaluated);
545 mulp->clearflag(status_flags::hash_calculated);
546 return (new mul(power(*mulp,exponent),
547 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
549 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
550 if (!num_coeff.is_equal(*_num_1_p)) {
551 mul *mulp = new mul(mulref);
552 mulp->overall_coeff = _ex_1;
553 mulp->clearflag(status_flags::evaluated);
554 mulp->clearflag(status_flags::hash_calculated);
555 return (new mul(power(*mulp,exponent),
556 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
563 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
564 if (num_exponent->is_pos_integer() &&
565 ebasis.return_type() != return_types::commutative &&
566 !is_a<matrix>(ebasis)) {
567 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
571 if (are_ex_trivially_equal(ebasis,basis) &&
572 are_ex_trivially_equal(eexponent,exponent)) {
575 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
576 status_flags::evaluated);
579 ex power::evalf(int level) const
586 eexponent = exponent;
587 } else if (level == -max_recursion_level) {
588 throw(std::runtime_error("max recursion level reached"));
590 ebasis = basis.evalf(level-1);
591 if (!is_exactly_a<numeric>(exponent))
592 eexponent = exponent.evalf(level-1);
594 eexponent = exponent;
597 return power(ebasis,eexponent);
600 ex power::evalm() const
602 const ex ebasis = basis.evalm();
603 const ex eexponent = exponent.evalm();
604 if (is_a<matrix>(ebasis)) {
605 if (is_exactly_a<numeric>(eexponent)) {
606 return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
609 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
612 bool power::has(const ex & other, unsigned options) const
614 if (!(options & has_options::algebraic))
615 return basic::has(other, options);
616 if (!is_a<power>(other))
617 return basic::has(other, options);
618 if (!exponent.info(info_flags::integer)
619 || !other.op(1).info(info_flags::integer))
620 return basic::has(other, options);
621 if (exponent.info(info_flags::posint)
622 && other.op(1).info(info_flags::posint)
623 && ex_to<numeric>(exponent).to_int()
624 > ex_to<numeric>(other.op(1)).to_int()
625 && basis.match(other.op(0)))
627 if (exponent.info(info_flags::negint)
628 && other.op(1).info(info_flags::negint)
629 && ex_to<numeric>(exponent).to_int()
630 < ex_to<numeric>(other.op(1)).to_int()
631 && basis.match(other.op(0)))
633 return basic::has(other, options);
637 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
639 ex power::subs(const exmap & m, unsigned options) const
641 const ex &subsed_basis = basis.subs(m, options);
642 const ex &subsed_exponent = exponent.subs(m, options);
644 if (!are_ex_trivially_equal(basis, subsed_basis)
645 || !are_ex_trivially_equal(exponent, subsed_exponent))
646 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
648 if (!(options & subs_options::algebraic))
649 return subs_one_level(m, options);
651 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
652 int nummatches = std::numeric_limits<int>::max();
654 if (tryfactsubs(*this, it->first, nummatches, repls)) {
655 ex anum = it->second.subs(repls, subs_options::no_pattern);
656 ex aden = it->first.subs(repls, subs_options::no_pattern);
657 ex result = (*this)*power(anum/aden, nummatches);
658 return (ex_to<basic>(result)).subs_one_level(m, options);
662 return subs_one_level(m, options);
665 ex power::eval_ncmul(const exvector & v) const
667 return inherited::eval_ncmul(v);
670 ex power::conjugate() const
672 ex newbasis = basis.conjugate();
673 ex newexponent = exponent.conjugate();
674 if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
677 return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
680 ex power::real_part() const
682 if (exponent.info(info_flags::integer)) {
683 ex basis_real = basis.real_part();
684 if (basis_real == basis)
686 realsymbol a("a"),b("b");
688 if (exponent.info(info_flags::posint))
689 result = power(a+I*b,exponent);
691 result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
692 result = result.expand();
693 result = result.real_part();
694 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
698 ex a = basis.real_part();
699 ex b = basis.imag_part();
700 ex c = exponent.real_part();
701 ex d = exponent.imag_part();
702 return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
705 ex power::imag_part() const
707 if (exponent.info(info_flags::integer)) {
708 ex basis_real = basis.real_part();
709 if (basis_real == basis)
711 realsymbol a("a"),b("b");
713 if (exponent.info(info_flags::posint))
714 result = power(a+I*b,exponent);
716 result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
717 result = result.expand();
718 result = result.imag_part();
719 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
723 ex a=basis.real_part();
724 ex b=basis.imag_part();
725 ex c=exponent.real_part();
726 ex d=exponent.imag_part();
728 power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
735 /** Implementation of ex::diff() for a power.
737 ex power::derivative(const symbol & s) const
739 if (is_a<numeric>(exponent)) {
740 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
743 newseq.push_back(expair(basis, exponent - _ex1));
744 newseq.push_back(expair(basis.diff(s), _ex1));
745 return mul(newseq, exponent);
747 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
749 add(mul(exponent.diff(s), log(basis)),
750 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
754 int power::compare_same_type(const basic & other) const
756 GINAC_ASSERT(is_exactly_a<power>(other));
757 const power &o = static_cast<const power &>(other);
759 int cmpval = basis.compare(o.basis);
763 return exponent.compare(o.exponent);
766 unsigned power::return_type() const
768 return basis.return_type();
771 return_type_t power::return_type_tinfo() const
773 return basis.return_type_tinfo();
776 ex power::expand(unsigned options) const
778 if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
779 // A special case worth optimizing.
780 setflag(status_flags::expanded);
784 const ex expanded_basis = basis.expand(options);
785 const ex expanded_exponent = exponent.expand(options);
787 // x^(a+b) -> x^a * x^b
788 if (is_exactly_a<add>(expanded_exponent)) {
789 const add &a = ex_to<add>(expanded_exponent);
791 distrseq.reserve(a.seq.size() + 1);
792 epvector::const_iterator last = a.seq.end();
793 epvector::const_iterator cit = a.seq.begin();
795 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
799 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
800 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
801 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
802 int int_exponent = num_exponent.to_int();
803 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
804 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
806 distrseq.push_back(power(expanded_basis, a.overall_coeff));
808 distrseq.push_back(power(expanded_basis, a.overall_coeff));
810 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
811 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
812 return r.expand(options);
815 if (!is_exactly_a<numeric>(expanded_exponent) ||
816 !ex_to<numeric>(expanded_exponent).is_integer()) {
817 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
820 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
824 // integer numeric exponent
825 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
826 int int_exponent = num_exponent.to_int();
829 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
830 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
832 // (x*y)^n -> x^n * y^n
833 if (is_exactly_a<mul>(expanded_basis))
834 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
836 // cannot expand further
837 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
840 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
844 // new virtual functions which can be overridden by derived classes
850 // non-virtual functions in this class
853 /** expand a^n where a is an add and n is a positive integer.
854 * @see power::expand */
855 ex power::expand_add(const add & a, int n, unsigned options) const
858 return expand_add_2(a, options);
860 const size_t m = a.nops();
862 // The number of terms will be the number of combinatorial compositions,
863 // i.e. the number of unordered arrangements of m nonnegative integers
864 // which sum up to n. It is frequently written as C_n(m) and directly
865 // related with binomial coefficients:
866 result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
868 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
869 intvector upper_limit(m-1);
871 for (size_t l=0; l<m-1; ++l) {
880 for (std::size_t l = 0; l < m - 1; ++l) {
881 const ex & b = a.op(l);
882 GINAC_ASSERT(!is_exactly_a<add>(b));
883 GINAC_ASSERT(!is_exactly_a<power>(b) ||
884 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
885 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
886 !is_exactly_a<add>(ex_to<power>(b).basis) ||
887 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
888 !is_exactly_a<power>(ex_to<power>(b).basis));
889 if (is_exactly_a<mul>(b))
890 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
892 term.push_back(power(b,k[l]));
895 const ex & b = a.op(m - 1);
896 GINAC_ASSERT(!is_exactly_a<add>(b));
897 GINAC_ASSERT(!is_exactly_a<power>(b) ||
898 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
899 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
900 !is_exactly_a<add>(ex_to<power>(b).basis) ||
901 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
902 !is_exactly_a<power>(ex_to<power>(b).basis));
903 if (is_exactly_a<mul>(b))
904 term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
906 term.push_back(power(b,n-k_cum[m-2]));
908 numeric f = binomial(numeric(n),numeric(k[0]));
909 for (std::size_t l = 1; l < m - 1; ++l)
910 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
914 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
918 std::size_t l = m - 2;
919 while ((++k[l]) > upper_limit[l]) {
931 // recalc k_cum[] and upper_limit[]
932 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
934 for (size_t i=l+1; i<m-1; ++i)
935 k_cum[i] = k_cum[i-1]+k[i];
937 for (size_t i=l+1; i<m-1; ++i)
938 upper_limit[i] = n-k_cum[i-1];
941 return (new add(result))->setflag(status_flags::dynallocated |
942 status_flags::expanded);
946 /** Special case of power::expand_add. Expands a^2 where a is an add.
947 * @see power::expand_add */
948 ex power::expand_add_2(const add & a, unsigned options) const
951 size_t a_nops = a.nops();
952 sum.reserve((a_nops*(a_nops+1))/2);
953 epvector::const_iterator last = a.seq.end();
955 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
956 // first part: ignore overall_coeff and expand other terms
957 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
958 const ex & r = cit0->rest;
959 const ex & c = cit0->coeff;
961 GINAC_ASSERT(!is_exactly_a<add>(r));
962 GINAC_ASSERT(!is_exactly_a<power>(r) ||
963 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
964 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
965 !is_exactly_a<add>(ex_to<power>(r).basis) ||
966 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
967 !is_exactly_a<power>(ex_to<power>(r).basis));
969 if (c.is_equal(_ex1)) {
970 if (is_exactly_a<mul>(r)) {
971 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
974 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
978 if (is_exactly_a<mul>(r)) {
979 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
980 ex_to<numeric>(c).power_dyn(*_num2_p)));
982 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
983 ex_to<numeric>(c).power_dyn(*_num2_p)));
987 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
988 const ex & r1 = cit1->rest;
989 const ex & c1 = cit1->coeff;
990 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
991 _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
995 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
997 // second part: add terms coming from overall_factor (if != 0)
998 if (!a.overall_coeff.is_zero()) {
999 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1001 sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1004 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1007 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1009 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1012 /** Expand factors of m in m^n where m is a mul and n is an integer.
1013 * @see power::expand */
1014 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1016 GINAC_ASSERT(n.is_integer());
1022 // do not bother to rename indices if there are no any.
1023 if ((!(options & expand_options::expand_rename_idx))
1024 && m.info(info_flags::has_indices))
1025 options |= expand_options::expand_rename_idx;
1026 // Leave it to multiplication since dummy indices have to be renamed
1027 if ((options & expand_options::expand_rename_idx) &&
1028 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1030 exvector va = get_all_dummy_indices(m);
1031 sort(va.begin(), va.end(), ex_is_less());
1033 for (int i=1; i < n.to_int(); i++)
1034 result *= rename_dummy_indices_uniquely(va, m);
1039 distrseq.reserve(m.seq.size());
1040 bool need_reexpand = false;
1042 epvector::const_iterator last = m.seq.end();
1043 epvector::const_iterator cit = m.seq.begin();
1045 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1046 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1047 // this happens when e.g. (a+b)^(1/2) gets squared and
1048 // the resulting product needs to be reexpanded
1049 need_reexpand = true;
1051 distrseq.push_back(p);
1055 const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1057 return ex(result).expand(options);
1059 return result.setflag(status_flags::expanded);
1063 GINAC_BIND_UNARCHIVER(power);
1065 } // namespace GiNaC