3 * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
6 * GiNaC Copyright (C) 1999-2015 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::positive) && exponent.info(info_flags::real)) ||
247 (basis.info(info_flags::real) && exponent.info(info_flags::integer) && exponent.info(info_flags::even));
248 case info_flags::has_indices: {
249 if (flags & status_flags::has_indices)
251 else if (flags & status_flags::has_no_indices)
253 else if (basis.info(info_flags::has_indices)) {
254 setflag(status_flags::has_indices);
255 clearflag(status_flags::has_no_indices);
258 clearflag(status_flags::has_indices);
259 setflag(status_flags::has_no_indices);
264 return inherited::info(inf);
267 size_t power::nops() const
272 ex power::op(size_t i) const
276 return i==0 ? basis : exponent;
279 ex power::map(map_function & f) const
281 const ex &mapped_basis = f(basis);
282 const ex &mapped_exponent = f(exponent);
284 if (!are_ex_trivially_equal(basis, mapped_basis)
285 || !are_ex_trivially_equal(exponent, mapped_exponent))
286 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
291 bool power::is_polynomial(const ex & var) const
293 if (basis.is_polynomial(var)) {
295 // basis is non-constant polynomial in var
296 return exponent.info(info_flags::nonnegint);
298 // basis is constant in var
299 return !exponent.has(var);
301 // basis is a non-polynomial function of var
305 int power::degree(const ex & s) const
307 if (is_equal(ex_to<basic>(s)))
309 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
310 if (basis.is_equal(s))
311 return ex_to<numeric>(exponent).to_int();
313 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
314 } else if (basis.has(s))
315 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
320 int power::ldegree(const ex & s) const
322 if (is_equal(ex_to<basic>(s)))
324 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
325 if (basis.is_equal(s))
326 return ex_to<numeric>(exponent).to_int();
328 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
329 } else if (basis.has(s))
330 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
335 ex power::coeff(const ex & s, int n) const
337 if (is_equal(ex_to<basic>(s)))
338 return n==1 ? _ex1 : _ex0;
339 else if (!basis.is_equal(s)) {
340 // basis not equal to s
347 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
349 int int_exp = ex_to<numeric>(exponent).to_int();
355 // non-integer exponents are treated as zero
364 /** Perform automatic term rewriting rules in this class. In the following
365 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
366 * stand for such expressions that contain a plain number.
367 * - ^(x,0) -> 1 (also handles ^(0,0))
369 * - ^(0,c) -> 0 or exception (depending on the real part of c)
371 * - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
372 * - ^(^(x,c1),c2) -> ^(x,c1*c2) if x is positive and c1 is real.
373 * - ^(^(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!)
374 * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
375 * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
376 * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
378 * @param level cut-off in recursive evaluation */
379 ex power::eval(int level) const
381 if ((level==1) && (flags & status_flags::evaluated))
383 else if (level == -max_recursion_level)
384 throw(std::runtime_error("max recursion level reached"));
386 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
387 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
389 const numeric *num_basis = NULL;
390 const numeric *num_exponent = NULL;
392 if (is_exactly_a<numeric>(ebasis)) {
393 num_basis = &ex_to<numeric>(ebasis);
395 if (is_exactly_a<numeric>(eexponent)) {
396 num_exponent = &ex_to<numeric>(eexponent);
399 // ^(x,0) -> 1 (0^0 also handled here)
400 if (eexponent.is_zero()) {
401 if (ebasis.is_zero())
402 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
408 if (eexponent.is_equal(_ex1))
411 // ^(0,c1) -> 0 or exception (depending on real value of c1)
412 if ( ebasis.is_zero() && num_exponent ) {
413 if ((num_exponent->real()).is_zero())
414 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
415 else if ((num_exponent->real()).is_negative())
416 throw (pole_error("power::eval(): division by zero",1));
422 if (ebasis.is_equal(_ex1))
425 // power of a function calculated by separate rules defined for this function
426 if (is_exactly_a<function>(ebasis))
427 return ex_to<function>(ebasis).power(eexponent);
429 // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
430 if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
431 return power(ebasis.op(0), ebasis.op(1) * eexponent);
433 if ( num_exponent ) {
435 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
436 // except if c1,c2 are rational, but c1^c2 is not)
438 const bool basis_is_crational = num_basis->is_crational();
439 const bool exponent_is_crational = num_exponent->is_crational();
440 if (!basis_is_crational || !exponent_is_crational) {
441 // return a plain float
442 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
443 status_flags::evaluated |
444 status_flags::expanded);
447 const numeric res = num_basis->power(*num_exponent);
448 if (res.is_crational()) {
451 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
453 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
454 if (basis_is_crational && exponent_is_crational
455 && num_exponent->is_real()
456 && !num_exponent->is_integer()) {
457 const numeric n = num_exponent->numer();
458 const numeric m = num_exponent->denom();
460 numeric q = iquo(n, m, r);
461 if (r.is_negative()) {
465 if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
466 if (num_basis->is_rational() && !num_basis->is_integer()) {
467 // try it for numerator and denominator separately, in order to
468 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
469 const numeric bnum = num_basis->numer();
470 const numeric bden = num_basis->denom();
471 const numeric res_bnum = bnum.power(*num_exponent);
472 const numeric res_bden = bden.power(*num_exponent);
473 if (res_bnum.is_integer())
474 return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
475 if (res_bden.is_integer())
476 return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
480 // assemble resulting product, but allowing for a re-evaluation,
481 // because otherwise we'll end up with something like
482 // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
483 // instead of 7/16*7^(1/3).
484 ex prod = power(*num_basis,r.div(m));
485 return prod*power(*num_basis,q);
490 // ^(^(x,c1),c2) -> ^(x,c1*c2)
491 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
492 // case c1==1 should not happen, see below!)
493 if (is_exactly_a<power>(ebasis)) {
494 const power & sub_power = ex_to<power>(ebasis);
495 const ex & sub_basis = sub_power.basis;
496 const ex & sub_exponent = sub_power.exponent;
497 if (is_exactly_a<numeric>(sub_exponent)) {
498 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
499 GINAC_ASSERT(num_sub_exponent!=numeric(1));
500 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()
501 || (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
502 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
507 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
508 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
509 return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
512 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
513 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
514 numeric icont = ebasis.integer_content();
515 const numeric lead_coeff =
516 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
518 const bool canonicalizable = lead_coeff.is_integer();
519 const bool unit_normal = lead_coeff.is_pos_integer();
520 if (canonicalizable && (! unit_normal))
521 icont = icont.mul(*_num_1_p);
523 if (canonicalizable && (icont != *_num1_p)) {
524 const add& addref = ex_to<add>(ebasis);
525 add* addp = new add(addref);
526 addp->setflag(status_flags::dynallocated);
527 addp->clearflag(status_flags::hash_calculated);
528 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
529 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
530 i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
532 const numeric c = icont.power(*num_exponent);
533 if (likely(c != *_num1_p))
534 return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
536 return power(*addp, *num_exponent);
540 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
541 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
542 if (is_exactly_a<mul>(ebasis)) {
543 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
544 const mul & mulref = ex_to<mul>(ebasis);
545 if (!mulref.overall_coeff.is_equal(_ex1)) {
546 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
547 if (num_coeff.is_real()) {
548 if (num_coeff.is_positive()) {
549 mul *mulp = new mul(mulref);
550 mulp->overall_coeff = _ex1;
551 mulp->setflag(status_flags::dynallocated);
552 mulp->clearflag(status_flags::evaluated);
553 mulp->clearflag(status_flags::hash_calculated);
554 return (new mul(power(*mulp,exponent),
555 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
557 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
558 if (!num_coeff.is_equal(*_num_1_p)) {
559 mul *mulp = new mul(mulref);
560 mulp->overall_coeff = _ex_1;
561 mulp->setflag(status_flags::dynallocated);
562 mulp->clearflag(status_flags::evaluated);
563 mulp->clearflag(status_flags::hash_calculated);
564 return (new mul(power(*mulp,exponent),
565 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
572 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
573 if (num_exponent->is_pos_integer() &&
574 ebasis.return_type() != return_types::commutative &&
575 !is_a<matrix>(ebasis)) {
576 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
580 if (are_ex_trivially_equal(ebasis,basis) &&
581 are_ex_trivially_equal(eexponent,exponent)) {
584 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
585 status_flags::evaluated);
588 ex power::evalf(int level) const
595 eexponent = exponent;
596 } else if (level == -max_recursion_level) {
597 throw(std::runtime_error("max recursion level reached"));
599 ebasis = basis.evalf(level-1);
600 if (!is_exactly_a<numeric>(exponent))
601 eexponent = exponent.evalf(level-1);
603 eexponent = exponent;
606 return power(ebasis,eexponent);
609 ex power::evalm() const
611 const ex ebasis = basis.evalm();
612 const ex eexponent = exponent.evalm();
613 if (is_a<matrix>(ebasis)) {
614 if (is_exactly_a<numeric>(eexponent)) {
615 return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
618 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
621 bool power::has(const ex & other, unsigned options) const
623 if (!(options & has_options::algebraic))
624 return basic::has(other, options);
625 if (!is_a<power>(other))
626 return basic::has(other, options);
627 if (!exponent.info(info_flags::integer)
628 || !other.op(1).info(info_flags::integer))
629 return basic::has(other, options);
630 if (exponent.info(info_flags::posint)
631 && other.op(1).info(info_flags::posint)
632 && ex_to<numeric>(exponent).to_int()
633 > ex_to<numeric>(other.op(1)).to_int()
634 && basis.match(other.op(0)))
636 if (exponent.info(info_flags::negint)
637 && other.op(1).info(info_flags::negint)
638 && ex_to<numeric>(exponent).to_int()
639 < ex_to<numeric>(other.op(1)).to_int()
640 && basis.match(other.op(0)))
642 return basic::has(other, options);
646 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
648 ex power::subs(const exmap & m, unsigned options) const
650 const ex &subsed_basis = basis.subs(m, options);
651 const ex &subsed_exponent = exponent.subs(m, options);
653 if (!are_ex_trivially_equal(basis, subsed_basis)
654 || !are_ex_trivially_equal(exponent, subsed_exponent))
655 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
657 if (!(options & subs_options::algebraic))
658 return subs_one_level(m, options);
660 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
661 int nummatches = std::numeric_limits<int>::max();
663 if (tryfactsubs(*this, it->first, nummatches, repls)) {
664 ex anum = it->second.subs(repls, subs_options::no_pattern);
665 ex aden = it->first.subs(repls, subs_options::no_pattern);
666 ex result = (*this)*power(anum/aden, nummatches);
667 return (ex_to<basic>(result)).subs_one_level(m, options);
671 return subs_one_level(m, options);
674 ex power::eval_ncmul(const exvector & v) const
676 return inherited::eval_ncmul(v);
679 ex power::conjugate() const
681 // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
682 // branch cut which runs along the negative real axis.
683 if (basis.info(info_flags::positive)) {
684 ex newexponent = exponent.conjugate();
685 if (are_ex_trivially_equal(exponent, newexponent)) {
688 return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
690 if (exponent.info(info_flags::integer)) {
691 ex newbasis = basis.conjugate();
692 if (are_ex_trivially_equal(basis, newbasis)) {
695 return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
697 return conjugate_function(*this).hold();
700 ex power::real_part() const
702 if (exponent.info(info_flags::integer)) {
703 ex basis_real = basis.real_part();
704 if (basis_real == basis)
706 realsymbol a("a"),b("b");
708 if (exponent.info(info_flags::posint))
709 result = power(a+I*b,exponent);
711 result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
712 result = result.expand();
713 result = result.real_part();
714 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
718 ex a = basis.real_part();
719 ex b = basis.imag_part();
720 ex c = exponent.real_part();
721 ex d = exponent.imag_part();
722 return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
725 ex power::imag_part() const
727 if (exponent.info(info_flags::integer)) {
728 ex basis_real = basis.real_part();
729 if (basis_real == basis)
731 realsymbol a("a"),b("b");
733 if (exponent.info(info_flags::posint))
734 result = power(a+I*b,exponent);
736 result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
737 result = result.expand();
738 result = result.imag_part();
739 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
743 ex a=basis.real_part();
744 ex b=basis.imag_part();
745 ex c=exponent.real_part();
746 ex d=exponent.imag_part();
747 return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
754 /** Implementation of ex::diff() for a power.
756 ex power::derivative(const symbol & s) const
758 if (is_a<numeric>(exponent)) {
759 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
762 newseq.push_back(expair(basis, exponent - _ex1));
763 newseq.push_back(expair(basis.diff(s), _ex1));
764 return mul(newseq, exponent);
766 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
768 add(mul(exponent.diff(s), log(basis)),
769 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
773 int power::compare_same_type(const basic & other) const
775 GINAC_ASSERT(is_exactly_a<power>(other));
776 const power &o = static_cast<const power &>(other);
778 int cmpval = basis.compare(o.basis);
782 return exponent.compare(o.exponent);
785 unsigned power::return_type() const
787 return basis.return_type();
790 return_type_t power::return_type_tinfo() const
792 return basis.return_type_tinfo();
795 ex power::expand(unsigned options) const
797 if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
798 // A special case worth optimizing.
799 setflag(status_flags::expanded);
803 // (x*p)^c -> x^c * p^c, if p>0
804 // makes sense before expanding the basis
805 if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
806 const mul &m = ex_to<mul>(basis);
809 prodseq.reserve(m.seq.size() + 1);
810 powseq.reserve(m.seq.size() + 1);
811 epvector::const_iterator last = m.seq.end();
812 epvector::const_iterator cit = m.seq.begin();
815 // search for positive/negative factors
817 ex e=m.recombine_pair_to_ex(*cit);
818 if (e.info(info_flags::positive))
819 prodseq.push_back(pow(e, exponent).expand(options));
820 else if (e.info(info_flags::negative)) {
821 prodseq.push_back(pow(-e, exponent).expand(options));
824 powseq.push_back(*cit);
828 // take care on the numeric coefficient
829 ex coeff=(possign? _ex1 : _ex_1);
830 if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
831 prodseq.push_back(power(m.overall_coeff, exponent));
832 else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
833 prodseq.push_back(power(-m.overall_coeff, exponent));
835 coeff *= m.overall_coeff;
837 // If positive/negative factors are found, then extract them.
838 // In either case we set a flag to avoid the second run on a part
839 // which does not have positive/negative terms.
840 if (prodseq.size() > 0) {
841 ex newbasis = coeff*mul(powseq);
842 ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
843 return ((new mul(prodseq))->setflag(status_flags::dynallocated)*(new power(newbasis, exponent))->setflag(status_flags::dynallocated).expand(options)).expand(options);
845 ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
848 const ex expanded_basis = basis.expand(options);
849 const ex expanded_exponent = exponent.expand(options);
851 // x^(a+b) -> x^a * x^b
852 if (is_exactly_a<add>(expanded_exponent)) {
853 const add &a = ex_to<add>(expanded_exponent);
855 distrseq.reserve(a.seq.size() + 1);
856 epvector::const_iterator last = a.seq.end();
857 epvector::const_iterator cit = a.seq.begin();
859 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
863 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
864 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
865 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
866 int int_exponent = num_exponent.to_int();
867 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
868 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
870 distrseq.push_back(power(expanded_basis, a.overall_coeff));
872 distrseq.push_back(power(expanded_basis, a.overall_coeff));
874 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
875 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
876 return r.expand(options);
879 if (!is_exactly_a<numeric>(expanded_exponent) ||
880 !ex_to<numeric>(expanded_exponent).is_integer()) {
881 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
884 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
888 // integer numeric exponent
889 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
890 int int_exponent = num_exponent.to_int();
893 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
894 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
896 // (x*y)^n -> x^n * y^n
897 if (is_exactly_a<mul>(expanded_basis))
898 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
900 // cannot expand further
901 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
904 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
908 // new virtual functions which can be overridden by derived classes
914 // non-virtual functions in this class
917 /** expand a^n where a is an add and n is a positive integer.
918 * @see power::expand */
919 ex power::expand_add(const add & a, int n, unsigned options) const
922 return expand_add_2(a, options);
924 const size_t m = a.nops();
926 // The number of terms will be the number of combinatorial compositions,
927 // i.e. the number of unordered arrangements of m nonnegative integers
928 // which sum up to n. It is frequently written as C_n(m) and directly
929 // related with binomial coefficients:
930 result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
932 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
933 intvector upper_limit(m-1);
935 for (size_t l=0; l<m-1; ++l) {
944 for (std::size_t l = 0; l < m - 1; ++l) {
945 const ex & b = a.op(l);
946 GINAC_ASSERT(!is_exactly_a<add>(b));
947 GINAC_ASSERT(!is_exactly_a<power>(b) ||
948 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
949 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
950 !is_exactly_a<add>(ex_to<power>(b).basis) ||
951 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
952 !is_exactly_a<power>(ex_to<power>(b).basis));
953 if (is_exactly_a<mul>(b))
954 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
956 term.push_back(power(b,k[l]));
959 const ex & b = a.op(m - 1);
960 GINAC_ASSERT(!is_exactly_a<add>(b));
961 GINAC_ASSERT(!is_exactly_a<power>(b) ||
962 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
963 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
964 !is_exactly_a<add>(ex_to<power>(b).basis) ||
965 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
966 !is_exactly_a<power>(ex_to<power>(b).basis));
967 if (is_exactly_a<mul>(b))
968 term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
970 term.push_back(power(b,n-k_cum[m-2]));
972 numeric f = binomial(numeric(n),numeric(k[0]));
973 for (std::size_t l = 1; l < m - 1; ++l)
974 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
978 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
982 std::size_t l = m - 2;
983 while ((++k[l]) > upper_limit[l]) {
995 // recalc k_cum[] and upper_limit[]
996 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
998 for (size_t i=l+1; i<m-1; ++i)
999 k_cum[i] = k_cum[i-1]+k[i];
1001 for (size_t i=l+1; i<m-1; ++i)
1002 upper_limit[i] = n-k_cum[i-1];
1005 return (new add(result))->setflag(status_flags::dynallocated |
1006 status_flags::expanded);
1010 /** Special case of power::expand_add. Expands a^2 where a is an add.
1011 * @see power::expand_add */
1012 ex power::expand_add_2(const add & a, unsigned options) const
1015 size_t a_nops = a.nops();
1016 sum.reserve((a_nops*(a_nops+1))/2);
1017 epvector::const_iterator last = a.seq.end();
1019 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
1020 // first part: ignore overall_coeff and expand other terms
1021 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
1022 const ex & r = cit0->rest;
1023 const ex & c = cit0->coeff;
1025 GINAC_ASSERT(!is_exactly_a<add>(r));
1026 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1027 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1028 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1029 !is_exactly_a<add>(ex_to<power>(r).basis) ||
1030 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1031 !is_exactly_a<power>(ex_to<power>(r).basis));
1033 if (c.is_equal(_ex1)) {
1034 if (is_exactly_a<mul>(r)) {
1035 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1038 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1042 if (is_exactly_a<mul>(r)) {
1043 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1044 ex_to<numeric>(c).power_dyn(*_num2_p)));
1046 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1047 ex_to<numeric>(c).power_dyn(*_num2_p)));
1051 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1052 const ex & r1 = cit1->rest;
1053 const ex & c1 = cit1->coeff;
1054 sum.push_back(a.combine_ex_with_coeff_to_pair(mul(r,r1).expand(options),
1055 _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1059 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
1061 // second part: add terms coming from overall_coeff (if != 0)
1062 if (!a.overall_coeff.is_zero()) {
1063 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1065 sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1068 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1071 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1073 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1076 /** Expand factors of m in m^n where m is a mul and n is an integer.
1077 * @see power::expand */
1078 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1080 GINAC_ASSERT(n.is_integer());
1086 // do not bother to rename indices if there are no any.
1087 if ((!(options & expand_options::expand_rename_idx))
1088 && m.info(info_flags::has_indices))
1089 options |= expand_options::expand_rename_idx;
1090 // Leave it to multiplication since dummy indices have to be renamed
1091 if ((options & expand_options::expand_rename_idx) &&
1092 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1094 exvector va = get_all_dummy_indices(m);
1095 sort(va.begin(), va.end(), ex_is_less());
1097 for (int i=1; i < n.to_int(); i++)
1098 result *= rename_dummy_indices_uniquely(va, m);
1103 distrseq.reserve(m.seq.size());
1104 bool need_reexpand = false;
1106 epvector::const_iterator last = m.seq.end();
1107 epvector::const_iterator cit = m.seq.begin();
1109 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1110 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1111 // this happens when e.g. (a+b)^(1/2) gets squared and
1112 // the resulting product needs to be reexpanded
1113 need_reexpand = true;
1115 distrseq.push_back(p);
1119 const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1121 return ex(result).expand(options);
1123 return result.setflag(status_flags::expanded);
1127 GINAC_BIND_UNARCHIVER(power);
1129 } // namespace GiNaC