3 * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
6 * GiNaC Copyright (C) 1999-2014 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
24 #include "expairseq.h"
30 #include "operators.h"
31 #include "inifcns.h" // for log() in power::derivative()
38 #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::positive:
244 return basis.info(info_flags::positive) && exponent.info(info_flags::real);
245 case info_flags::nonnegative:
246 return basis.info(info_flags::real) && exponent.info(info_flags::integer) && exponent.info(info_flags::even);
247 case info_flags::has_indices: {
248 if (flags & status_flags::has_indices)
250 else if (flags & status_flags::has_no_indices)
252 else if (basis.info(info_flags::has_indices)) {
253 setflag(status_flags::has_indices);
254 clearflag(status_flags::has_no_indices);
257 clearflag(status_flags::has_indices);
258 setflag(status_flags::has_no_indices);
263 return inherited::info(inf);
266 size_t power::nops() const
271 ex power::op(size_t i) const
275 return i==0 ? basis : exponent;
278 ex power::map(map_function & f) const
280 const ex &mapped_basis = f(basis);
281 const ex &mapped_exponent = f(exponent);
283 if (!are_ex_trivially_equal(basis, mapped_basis)
284 || !are_ex_trivially_equal(exponent, mapped_exponent))
285 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
290 bool power::is_polynomial(const ex & var) const
292 if (basis.is_polynomial(var)) {
294 // basis is non-constant polynomial in var
295 return exponent.info(info_flags::nonnegint);
297 // basis is constant in var
298 return !exponent.has(var);
300 // basis is a non-polynomial function of var
304 int power::degree(const ex & s) const
306 if (is_equal(ex_to<basic>(s)))
308 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
309 if (basis.is_equal(s))
310 return ex_to<numeric>(exponent).to_int();
312 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
313 } else if (basis.has(s))
314 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
319 int power::ldegree(const ex & s) const
321 if (is_equal(ex_to<basic>(s)))
323 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
324 if (basis.is_equal(s))
325 return ex_to<numeric>(exponent).to_int();
327 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
328 } else if (basis.has(s))
329 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
334 ex power::coeff(const ex & s, int n) const
336 if (is_equal(ex_to<basic>(s)))
337 return n==1 ? _ex1 : _ex0;
338 else if (!basis.is_equal(s)) {
339 // basis not equal to s
346 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
348 int int_exp = ex_to<numeric>(exponent).to_int();
354 // non-integer exponents are treated as zero
363 /** Perform automatic term rewriting rules in this class. In the following
364 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
365 * stand for such expressions that contain a plain number.
366 * - ^(x,0) -> 1 (also handles ^(0,0))
368 * - ^(0,c) -> 0 or exception (depending on the real part of c)
370 * - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
371 * - ^(^(x,c1),c2) -> ^(x,c1*c2) if x is positive and c1 is real.
372 * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0), case c1=1 should not happen, see below!)
373 * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
374 * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
375 * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
377 * @param level cut-off in recursive evaluation */
378 ex power::eval(int level) const
380 if ((level==1) && (flags & status_flags::evaluated))
382 else if (level == -max_recursion_level)
383 throw(std::runtime_error("max recursion level reached"));
385 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
386 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
388 const numeric *num_basis = NULL;
389 const numeric *num_exponent = NULL;
391 if (is_exactly_a<numeric>(ebasis)) {
392 num_basis = &ex_to<numeric>(ebasis);
394 if (is_exactly_a<numeric>(eexponent)) {
395 num_exponent = &ex_to<numeric>(eexponent);
398 // ^(x,0) -> 1 (0^0 also handled here)
399 if (eexponent.is_zero()) {
400 if (ebasis.is_zero())
401 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
407 if (eexponent.is_equal(_ex1))
410 // ^(0,c1) -> 0 or exception (depending on real value of c1)
411 if ( ebasis.is_zero() && num_exponent ) {
412 if ((num_exponent->real()).is_zero())
413 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
414 else if ((num_exponent->real()).is_negative())
415 throw (pole_error("power::eval(): division by zero",1));
421 if (ebasis.is_equal(_ex1))
424 // power of a function calculated by separate rules defined for this function
425 if (is_exactly_a<function>(ebasis))
426 return ex_to<function>(ebasis).power(eexponent);
428 // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
429 if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
430 return power(ebasis.op(0), ebasis.op(1) * eexponent);
432 if ( num_exponent ) {
434 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
435 // except if c1,c2 are rational, but c1^c2 is not)
437 const bool basis_is_crational = num_basis->is_crational();
438 const bool exponent_is_crational = num_exponent->is_crational();
439 if (!basis_is_crational || !exponent_is_crational) {
440 // return a plain float
441 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
442 status_flags::evaluated |
443 status_flags::expanded);
446 const numeric res = num_basis->power(*num_exponent);
447 if (res.is_crational()) {
450 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
452 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
453 if (basis_is_crational && exponent_is_crational
454 && num_exponent->is_real()
455 && !num_exponent->is_integer()) {
456 const numeric n = num_exponent->numer();
457 const numeric m = num_exponent->denom();
459 numeric q = iquo(n, m, r);
460 if (r.is_negative()) {
464 if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
465 if (num_basis->is_rational() && !num_basis->is_integer()) {
466 // try it for numerator and denominator separately, in order to
467 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
468 const numeric bnum = num_basis->numer();
469 const numeric bden = num_basis->denom();
470 const numeric res_bnum = bnum.power(*num_exponent);
471 const numeric res_bden = bden.power(*num_exponent);
472 if (res_bnum.is_integer())
473 return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
474 if (res_bden.is_integer())
475 return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
479 // assemble resulting product, but allowing for a re-evaluation,
480 // because otherwise we'll end up with something like
481 // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
482 // instead of 7/16*7^(1/3).
483 ex prod = power(*num_basis,r.div(m));
484 return prod*power(*num_basis,q);
489 // ^(^(x,c1),c2) -> ^(x,c1*c2)
490 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
491 // case c1==1 should not happen, see below!)
492 if (is_exactly_a<power>(ebasis)) {
493 const power & sub_power = ex_to<power>(ebasis);
494 const ex & sub_basis = sub_power.basis;
495 const ex & sub_exponent = sub_power.exponent;
496 if (is_exactly_a<numeric>(sub_exponent)) {
497 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
498 GINAC_ASSERT(num_sub_exponent!=numeric(1));
499 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()
500 || (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
501 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
506 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
507 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
508 return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
511 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
512 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
513 numeric icont = ebasis.integer_content();
514 const numeric lead_coeff =
515 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
517 const bool canonicalizable = lead_coeff.is_integer();
518 const bool unit_normal = lead_coeff.is_pos_integer();
519 if (canonicalizable && (! unit_normal))
520 icont = icont.mul(*_num_1_p);
522 if (canonicalizable && (icont != *_num1_p)) {
523 const add& addref = ex_to<add>(ebasis);
524 add* addp = new add(addref);
525 addp->setflag(status_flags::dynallocated);
526 addp->clearflag(status_flags::hash_calculated);
527 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
528 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
529 i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
531 const numeric c = icont.power(*num_exponent);
532 if (likely(c != *_num1_p))
533 return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
535 return power(*addp, *num_exponent);
539 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
540 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
541 if (is_exactly_a<mul>(ebasis)) {
542 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
543 const mul & mulref = ex_to<mul>(ebasis);
544 if (!mulref.overall_coeff.is_equal(_ex1)) {
545 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
546 if (num_coeff.is_real()) {
547 if (num_coeff.is_positive()) {
548 mul *mulp = new mul(mulref);
549 mulp->overall_coeff = _ex1;
550 mulp->setflag(status_flags::dynallocated);
551 mulp->clearflag(status_flags::evaluated);
552 mulp->clearflag(status_flags::hash_calculated);
553 return (new mul(power(*mulp,exponent),
554 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
556 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
557 if (!num_coeff.is_equal(*_num_1_p)) {
558 mul *mulp = new mul(mulref);
559 mulp->overall_coeff = _ex_1;
560 mulp->setflag(status_flags::dynallocated);
561 mulp->clearflag(status_flags::evaluated);
562 mulp->clearflag(status_flags::hash_calculated);
563 return (new mul(power(*mulp,exponent),
564 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
571 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
572 if (num_exponent->is_pos_integer() &&
573 ebasis.return_type() != return_types::commutative &&
574 !is_a<matrix>(ebasis)) {
575 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
579 if (are_ex_trivially_equal(ebasis,basis) &&
580 are_ex_trivially_equal(eexponent,exponent)) {
583 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
584 status_flags::evaluated);
587 ex power::evalf(int level) const
594 eexponent = exponent;
595 } else if (level == -max_recursion_level) {
596 throw(std::runtime_error("max recursion level reached"));
598 ebasis = basis.evalf(level-1);
599 if (!is_exactly_a<numeric>(exponent))
600 eexponent = exponent.evalf(level-1);
602 eexponent = exponent;
605 return power(ebasis,eexponent);
608 ex power::evalm() const
610 const ex ebasis = basis.evalm();
611 const ex eexponent = exponent.evalm();
612 if (is_a<matrix>(ebasis)) {
613 if (is_exactly_a<numeric>(eexponent)) {
614 return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
617 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
620 bool power::has(const ex & other, unsigned options) const
622 if (!(options & has_options::algebraic))
623 return basic::has(other, options);
624 if (!is_a<power>(other))
625 return basic::has(other, options);
626 if (!exponent.info(info_flags::integer)
627 || !other.op(1).info(info_flags::integer))
628 return basic::has(other, options);
629 if (exponent.info(info_flags::posint)
630 && other.op(1).info(info_flags::posint)
631 && ex_to<numeric>(exponent).to_int()
632 > ex_to<numeric>(other.op(1)).to_int()
633 && basis.match(other.op(0)))
635 if (exponent.info(info_flags::negint)
636 && other.op(1).info(info_flags::negint)
637 && ex_to<numeric>(exponent).to_int()
638 < ex_to<numeric>(other.op(1)).to_int()
639 && basis.match(other.op(0)))
641 return basic::has(other, options);
645 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
647 ex power::subs(const exmap & m, unsigned options) const
649 const ex &subsed_basis = basis.subs(m, options);
650 const ex &subsed_exponent = exponent.subs(m, options);
652 if (!are_ex_trivially_equal(basis, subsed_basis)
653 || !are_ex_trivially_equal(exponent, subsed_exponent))
654 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
656 if (!(options & subs_options::algebraic))
657 return subs_one_level(m, options);
659 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
660 int nummatches = std::numeric_limits<int>::max();
662 if (tryfactsubs(*this, it->first, nummatches, repls)) {
663 ex anum = it->second.subs(repls, subs_options::no_pattern);
664 ex aden = it->first.subs(repls, subs_options::no_pattern);
665 ex result = (*this)*power(anum/aden, nummatches);
666 return (ex_to<basic>(result)).subs_one_level(m, options);
670 return subs_one_level(m, options);
673 ex power::eval_ncmul(const exvector & v) const
675 return inherited::eval_ncmul(v);
678 ex power::conjugate() const
680 // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
681 // branch cut which runs along the negative real axis.
682 if (basis.info(info_flags::positive)) {
683 ex newexponent = exponent.conjugate();
684 if (are_ex_trivially_equal(exponent, newexponent)) {
687 return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
689 if (exponent.info(info_flags::integer)) {
690 ex newbasis = basis.conjugate();
691 if (are_ex_trivially_equal(basis, newbasis)) {
694 return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
696 return conjugate_function(*this).hold();
699 ex power::real_part() const
701 if (exponent.info(info_flags::integer)) {
702 ex basis_real = basis.real_part();
703 if (basis_real == basis)
705 realsymbol a("a"),b("b");
707 if (exponent.info(info_flags::posint))
708 result = power(a+I*b,exponent);
710 result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
711 result = result.expand();
712 result = result.real_part();
713 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
717 ex a = basis.real_part();
718 ex b = basis.imag_part();
719 ex c = exponent.real_part();
720 ex d = exponent.imag_part();
721 return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
724 ex power::imag_part() const
726 if (exponent.info(info_flags::integer)) {
727 ex basis_real = basis.real_part();
728 if (basis_real == basis)
730 realsymbol a("a"),b("b");
732 if (exponent.info(info_flags::posint))
733 result = power(a+I*b,exponent);
735 result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
736 result = result.expand();
737 result = result.imag_part();
738 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
742 ex a=basis.real_part();
743 ex b=basis.imag_part();
744 ex c=exponent.real_part();
745 ex d=exponent.imag_part();
746 return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
753 /** Implementation of ex::diff() for a power.
755 ex power::derivative(const symbol & s) const
757 if (is_a<numeric>(exponent)) {
758 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
761 newseq.push_back(expair(basis, exponent - _ex1));
762 newseq.push_back(expair(basis.diff(s), _ex1));
763 return mul(newseq, exponent);
765 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
767 add(mul(exponent.diff(s), log(basis)),
768 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
772 int power::compare_same_type(const basic & other) const
774 GINAC_ASSERT(is_exactly_a<power>(other));
775 const power &o = static_cast<const power &>(other);
777 int cmpval = basis.compare(o.basis);
781 return exponent.compare(o.exponent);
784 unsigned power::return_type() const
786 return basis.return_type();
789 return_type_t power::return_type_tinfo() const
791 return basis.return_type_tinfo();
794 ex power::expand(unsigned options) const
796 if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
797 // A special case worth optimizing.
798 setflag(status_flags::expanded);
802 const ex expanded_basis = basis.expand(options);
803 const ex expanded_exponent = exponent.expand(options);
805 // x^(a+b) -> x^a * x^b
806 if (is_exactly_a<add>(expanded_exponent)) {
807 const add &a = ex_to<add>(expanded_exponent);
809 distrseq.reserve(a.seq.size() + 1);
810 epvector::const_iterator last = a.seq.end();
811 epvector::const_iterator cit = a.seq.begin();
813 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
817 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
818 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
819 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
820 int int_exponent = num_exponent.to_int();
821 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
822 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
824 distrseq.push_back(power(expanded_basis, a.overall_coeff));
826 distrseq.push_back(power(expanded_basis, a.overall_coeff));
828 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
829 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
830 return r.expand(options);
833 if (!is_exactly_a<numeric>(expanded_exponent) ||
834 !ex_to<numeric>(expanded_exponent).is_integer()) {
835 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
838 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
842 // integer numeric exponent
843 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
844 int int_exponent = num_exponent.to_int();
847 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
848 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
850 // (x*y)^n -> x^n * y^n
851 if (is_exactly_a<mul>(expanded_basis))
852 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
854 // cannot expand further
855 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
858 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
862 // new virtual functions which can be overridden by derived classes
868 // non-virtual functions in this class
871 /** expand a^n where a is an add and n is a positive integer.
872 * @see power::expand */
873 ex power::expand_add(const add & a, int n, unsigned options) const
876 return expand_add_2(a, options);
878 const size_t m = a.nops();
880 // The number of terms will be the number of combinatorial compositions,
881 // i.e. the number of unordered arrangements of m nonnegative integers
882 // which sum up to n. It is frequently written as C_n(m) and directly
883 // related with binomial coefficients:
884 result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
886 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
887 intvector upper_limit(m-1);
889 for (size_t l=0; l<m-1; ++l) {
898 for (std::size_t l = 0; l < m - 1; ++l) {
899 const ex & b = a.op(l);
900 GINAC_ASSERT(!is_exactly_a<add>(b));
901 GINAC_ASSERT(!is_exactly_a<power>(b) ||
902 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
903 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
904 !is_exactly_a<add>(ex_to<power>(b).basis) ||
905 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
906 !is_exactly_a<power>(ex_to<power>(b).basis));
907 if (is_exactly_a<mul>(b))
908 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
910 term.push_back(power(b,k[l]));
913 const ex & b = a.op(m - 1);
914 GINAC_ASSERT(!is_exactly_a<add>(b));
915 GINAC_ASSERT(!is_exactly_a<power>(b) ||
916 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
917 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
918 !is_exactly_a<add>(ex_to<power>(b).basis) ||
919 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
920 !is_exactly_a<power>(ex_to<power>(b).basis));
921 if (is_exactly_a<mul>(b))
922 term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
924 term.push_back(power(b,n-k_cum[m-2]));
926 numeric f = binomial(numeric(n),numeric(k[0]));
927 for (std::size_t l = 1; l < m - 1; ++l)
928 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
932 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
936 std::size_t l = m - 2;
937 while ((++k[l]) > upper_limit[l]) {
949 // recalc k_cum[] and upper_limit[]
950 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
952 for (size_t i=l+1; i<m-1; ++i)
953 k_cum[i] = k_cum[i-1]+k[i];
955 for (size_t i=l+1; i<m-1; ++i)
956 upper_limit[i] = n-k_cum[i-1];
959 return (new add(result))->setflag(status_flags::dynallocated |
960 status_flags::expanded);
964 /** Special case of power::expand_add. Expands a^2 where a is an add.
965 * @see power::expand_add */
966 ex power::expand_add_2(const add & a, unsigned options) const
969 size_t a_nops = a.nops();
970 sum.reserve((a_nops*(a_nops+1))/2);
971 epvector::const_iterator last = a.seq.end();
973 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
974 // first part: ignore overall_coeff and expand other terms
975 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
976 const ex & r = cit0->rest;
977 const ex & c = cit0->coeff;
979 GINAC_ASSERT(!is_exactly_a<add>(r));
980 GINAC_ASSERT(!is_exactly_a<power>(r) ||
981 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
982 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
983 !is_exactly_a<add>(ex_to<power>(r).basis) ||
984 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
985 !is_exactly_a<power>(ex_to<power>(r).basis));
987 if (c.is_equal(_ex1)) {
988 if (is_exactly_a<mul>(r)) {
989 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
992 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
996 if (is_exactly_a<mul>(r)) {
997 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
998 ex_to<numeric>(c).power_dyn(*_num2_p)));
1000 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1001 ex_to<numeric>(c).power_dyn(*_num2_p)));
1005 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1006 const ex & r1 = cit1->rest;
1007 const ex & c1 = cit1->coeff;
1008 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
1009 _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1013 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
1015 // second part: add terms coming from overall_factor (if != 0)
1016 if (!a.overall_coeff.is_zero()) {
1017 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1019 sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1022 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1025 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1027 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1030 /** Expand factors of m in m^n where m is a mul and n is an integer.
1031 * @see power::expand */
1032 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1034 GINAC_ASSERT(n.is_integer());
1040 // do not bother to rename indices if there are no any.
1041 if ((!(options & expand_options::expand_rename_idx))
1042 && m.info(info_flags::has_indices))
1043 options |= expand_options::expand_rename_idx;
1044 // Leave it to multiplication since dummy indices have to be renamed
1045 if ((options & expand_options::expand_rename_idx) &&
1046 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1048 exvector va = get_all_dummy_indices(m);
1049 sort(va.begin(), va.end(), ex_is_less());
1051 for (int i=1; i < n.to_int(); i++)
1052 result *= rename_dummy_indices_uniquely(va, m);
1057 distrseq.reserve(m.seq.size());
1058 bool need_reexpand = false;
1060 epvector::const_iterator last = m.seq.end();
1061 epvector::const_iterator cit = m.seq.begin();
1063 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1064 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1065 // this happens when e.g. (a+b)^(1/2) gets squared and
1066 // the resulting product needs to be reexpanded
1067 need_reexpand = true;
1069 distrseq.push_back(p);
1073 const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1075 return ex(result).expand(options);
1077 return result.setflag(status_flags::expanded);
1081 GINAC_BIND_UNARCHIVER(power);
1083 } // namespace GiNaC