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))
57 // default constructor
72 void power::read_archive(const archive_node &n, lst &sym_lst)
74 inherited::read_archive(n, sym_lst);
75 n.find_ex("basis", basis, sym_lst);
76 n.find_ex("exponent", exponent, sym_lst);
79 void power::archive(archive_node &n) const
81 inherited::archive(n);
82 n.add_ex("basis", basis);
83 n.add_ex("exponent", exponent);
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_cl_N(const print_csrc_cl_N& c, unsigned level) const
165 if (exponent.is_equal(_ex_1)) {
178 void power::do_print_csrc(const print_csrc & c, unsigned level) const
180 // Integer powers of symbols are printed in a special, optimized way
181 if (exponent.info(info_flags::integer)
182 && (is_a<symbol>(basis) || is_a<constant>(basis))) {
183 int exp = ex_to<numeric>(exponent).to_int();
190 print_sym_pow(c, ex_to<symbol>(basis), exp);
193 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
194 } else if (exponent.is_equal(_ex_1)) {
199 // Otherwise, use the pow() function
209 void power::do_print_python(const print_python & c, unsigned level) const
211 print_power(c, "**", "", "", level);
214 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
216 c.s << class_name() << '(';
223 bool power::info(unsigned inf) const
226 case info_flags::polynomial:
227 case info_flags::integer_polynomial:
228 case info_flags::cinteger_polynomial:
229 case info_flags::rational_polynomial:
230 case info_flags::crational_polynomial:
231 return exponent.info(info_flags::nonnegint) &&
233 case info_flags::rational_function:
234 return exponent.info(info_flags::integer) &&
236 case info_flags::algebraic:
237 return !exponent.info(info_flags::integer) ||
239 case info_flags::expanded:
240 return (flags & status_flags::expanded);
241 case info_flags::positive:
242 return basis.info(info_flags::positive) && exponent.info(info_flags::real);
243 case info_flags::nonnegative:
244 return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
245 (basis.info(info_flags::real) && exponent.info(info_flags::integer) && exponent.info(info_flags::even));
246 case info_flags::has_indices: {
247 if (flags & status_flags::has_indices)
249 else if (flags & status_flags::has_no_indices)
251 else if (basis.info(info_flags::has_indices)) {
252 setflag(status_flags::has_indices);
253 clearflag(status_flags::has_no_indices);
256 clearflag(status_flags::has_indices);
257 setflag(status_flags::has_no_indices);
262 return inherited::info(inf);
265 size_t power::nops() const
270 ex power::op(size_t i) const
274 return i==0 ? basis : exponent;
277 ex power::map(map_function & f) const
279 const ex &mapped_basis = f(basis);
280 const ex &mapped_exponent = f(exponent);
282 if (!are_ex_trivially_equal(basis, mapped_basis)
283 || !are_ex_trivially_equal(exponent, mapped_exponent))
284 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
289 bool power::is_polynomial(const ex & var) const
291 if (basis.is_polynomial(var)) {
293 // basis is non-constant polynomial in var
294 return exponent.info(info_flags::nonnegint);
296 // basis is constant in var
297 return !exponent.has(var);
299 // basis is a non-polynomial function of var
303 int power::degree(const ex & s) const
305 if (is_equal(ex_to<basic>(s)))
307 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
308 if (basis.is_equal(s))
309 return ex_to<numeric>(exponent).to_int();
311 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
312 } else if (basis.has(s))
313 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
318 int power::ldegree(const ex & s) const
320 if (is_equal(ex_to<basic>(s)))
322 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
323 if (basis.is_equal(s))
324 return ex_to<numeric>(exponent).to_int();
326 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
327 } else if (basis.has(s))
328 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
333 ex power::coeff(const ex & s, int n) const
335 if (is_equal(ex_to<basic>(s)))
336 return n==1 ? _ex1 : _ex0;
337 else if (!basis.is_equal(s)) {
338 // basis not equal to s
345 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
347 int int_exp = ex_to<numeric>(exponent).to_int();
353 // non-integer exponents are treated as zero
362 /** Perform automatic term rewriting rules in this class. In the following
363 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
364 * stand for such expressions that contain a plain number.
365 * - ^(x,0) -> 1 (also handles ^(0,0))
367 * - ^(0,c) -> 0 or exception (depending on the real part of c)
369 * - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
370 * - ^(^(x,c1),c2) -> ^(x,c1*c2) if x is positive and c1 is real.
371 * - ^(^(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!)
372 * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
373 * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
374 * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
376 * @param level cut-off in recursive evaluation */
377 ex power::eval(int level) const
379 if ((level==1) && (flags & status_flags::evaluated))
381 else if (level == -max_recursion_level)
382 throw(std::runtime_error("max recursion level reached"));
384 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
385 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
387 const numeric *num_basis = NULL;
388 const numeric *num_exponent = NULL;
390 if (is_exactly_a<numeric>(ebasis)) {
391 num_basis = &ex_to<numeric>(ebasis);
393 if (is_exactly_a<numeric>(eexponent)) {
394 num_exponent = &ex_to<numeric>(eexponent);
397 // ^(x,0) -> 1 (0^0 also handled here)
398 if (eexponent.is_zero()) {
399 if (ebasis.is_zero())
400 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
406 if (eexponent.is_equal(_ex1))
409 // ^(0,c1) -> 0 or exception (depending on real value of c1)
410 if ( ebasis.is_zero() && num_exponent ) {
411 if ((num_exponent->real()).is_zero())
412 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
413 else if ((num_exponent->real()).is_negative())
414 throw (pole_error("power::eval(): division by zero",1));
420 if (ebasis.is_equal(_ex1))
423 // power of a function calculated by separate rules defined for this function
424 if (is_exactly_a<function>(ebasis))
425 return ex_to<function>(ebasis).power(eexponent);
427 // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
428 if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
429 return power(ebasis.op(0), ebasis.op(1) * eexponent);
431 if ( num_exponent ) {
433 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
434 // except if c1,c2 are rational, but c1^c2 is not)
436 const bool basis_is_crational = num_basis->is_crational();
437 const bool exponent_is_crational = num_exponent->is_crational();
438 if (!basis_is_crational || !exponent_is_crational) {
439 // return a plain float
440 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
441 status_flags::evaluated |
442 status_flags::expanded);
445 const numeric res = num_basis->power(*num_exponent);
446 if (res.is_crational()) {
449 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
451 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
452 if (basis_is_crational && exponent_is_crational
453 && num_exponent->is_real()
454 && !num_exponent->is_integer()) {
455 const numeric n = num_exponent->numer();
456 const numeric m = num_exponent->denom();
458 numeric q = iquo(n, m, r);
459 if (r.is_negative()) {
463 if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
464 if (num_basis->is_rational() && !num_basis->is_integer()) {
465 // try it for numerator and denominator separately, in order to
466 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
467 const numeric bnum = num_basis->numer();
468 const numeric bden = num_basis->denom();
469 const numeric res_bnum = bnum.power(*num_exponent);
470 const numeric res_bden = bden.power(*num_exponent);
471 if (res_bnum.is_integer())
472 return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
473 if (res_bden.is_integer())
474 return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
478 // assemble resulting product, but allowing for a re-evaluation,
479 // because otherwise we'll end up with something like
480 // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
481 // instead of 7/16*7^(1/3).
482 ex prod = power(*num_basis,r.div(m));
483 return prod*power(*num_basis,q);
488 // ^(^(x,c1),c2) -> ^(x,c1*c2)
489 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
490 // case c1==1 should not happen, see below!)
491 if (is_exactly_a<power>(ebasis)) {
492 const power & sub_power = ex_to<power>(ebasis);
493 const ex & sub_basis = sub_power.basis;
494 const ex & sub_exponent = sub_power.exponent;
495 if (is_exactly_a<numeric>(sub_exponent)) {
496 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
497 GINAC_ASSERT(num_sub_exponent!=numeric(1));
498 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()
499 || (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
500 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
505 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
506 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
507 return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
510 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
511 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
512 numeric icont = ebasis.integer_content();
513 const numeric lead_coeff =
514 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
516 const bool canonicalizable = lead_coeff.is_integer();
517 const bool unit_normal = lead_coeff.is_pos_integer();
518 if (canonicalizable && (! unit_normal))
519 icont = icont.mul(*_num_1_p);
521 if (canonicalizable && (icont != *_num1_p)) {
522 const add& addref = ex_to<add>(ebasis);
523 add* addp = new add(addref);
524 addp->setflag(status_flags::dynallocated);
525 addp->clearflag(status_flags::hash_calculated);
526 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
527 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
528 i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
530 const numeric c = icont.power(*num_exponent);
531 if (likely(c != *_num1_p))
532 return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
534 return power(*addp, *num_exponent);
538 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
539 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
540 if (is_exactly_a<mul>(ebasis)) {
541 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
542 const mul & mulref = ex_to<mul>(ebasis);
543 if (!mulref.overall_coeff.is_equal(_ex1)) {
544 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
545 if (num_coeff.is_real()) {
546 if (num_coeff.is_positive()) {
547 mul *mulp = new mul(mulref);
548 mulp->overall_coeff = _ex1;
549 mulp->setflag(status_flags::dynallocated);
550 mulp->clearflag(status_flags::evaluated);
551 mulp->clearflag(status_flags::hash_calculated);
552 return (new mul(power(*mulp,exponent),
553 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
555 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
556 if (!num_coeff.is_equal(*_num_1_p)) {
557 mul *mulp = new mul(mulref);
558 mulp->overall_coeff = _ex_1;
559 mulp->setflag(status_flags::dynallocated);
560 mulp->clearflag(status_flags::evaluated);
561 mulp->clearflag(status_flags::hash_calculated);
562 return (new mul(power(*mulp,exponent),
563 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
570 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
571 if (num_exponent->is_pos_integer() &&
572 ebasis.return_type() != return_types::commutative &&
573 !is_a<matrix>(ebasis)) {
574 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
578 if (are_ex_trivially_equal(ebasis,basis) &&
579 are_ex_trivially_equal(eexponent,exponent)) {
582 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
583 status_flags::evaluated);
586 ex power::evalf(int level) const
593 eexponent = exponent;
594 } else if (level == -max_recursion_level) {
595 throw(std::runtime_error("max recursion level reached"));
597 ebasis = basis.evalf(level-1);
598 if (!is_exactly_a<numeric>(exponent))
599 eexponent = exponent.evalf(level-1);
601 eexponent = exponent;
604 return power(ebasis,eexponent);
607 ex power::evalm() const
609 const ex ebasis = basis.evalm();
610 const ex eexponent = exponent.evalm();
611 if (is_a<matrix>(ebasis)) {
612 if (is_exactly_a<numeric>(eexponent)) {
613 return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
616 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
619 bool power::has(const ex & other, unsigned options) const
621 if (!(options & has_options::algebraic))
622 return basic::has(other, options);
623 if (!is_a<power>(other))
624 return basic::has(other, options);
625 if (!exponent.info(info_flags::integer)
626 || !other.op(1).info(info_flags::integer))
627 return basic::has(other, options);
628 if (exponent.info(info_flags::posint)
629 && other.op(1).info(info_flags::posint)
630 && ex_to<numeric>(exponent).to_int()
631 > ex_to<numeric>(other.op(1)).to_int()
632 && basis.match(other.op(0)))
634 if (exponent.info(info_flags::negint)
635 && other.op(1).info(info_flags::negint)
636 && ex_to<numeric>(exponent).to_int()
637 < ex_to<numeric>(other.op(1)).to_int()
638 && basis.match(other.op(0)))
640 return basic::has(other, options);
644 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
646 ex power::subs(const exmap & m, unsigned options) const
648 const ex &subsed_basis = basis.subs(m, options);
649 const ex &subsed_exponent = exponent.subs(m, options);
651 if (!are_ex_trivially_equal(basis, subsed_basis)
652 || !are_ex_trivially_equal(exponent, subsed_exponent))
653 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
655 if (!(options & subs_options::algebraic))
656 return subs_one_level(m, options);
658 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
659 int nummatches = std::numeric_limits<int>::max();
661 if (tryfactsubs(*this, it->first, nummatches, repls)) {
662 ex anum = it->second.subs(repls, subs_options::no_pattern);
663 ex aden = it->first.subs(repls, subs_options::no_pattern);
664 ex result = (*this)*power(anum/aden, nummatches);
665 return (ex_to<basic>(result)).subs_one_level(m, options);
669 return subs_one_level(m, options);
672 ex power::eval_ncmul(const exvector & v) const
674 return inherited::eval_ncmul(v);
677 ex power::conjugate() const
679 // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
680 // branch cut which runs along the negative real axis.
681 if (basis.info(info_flags::positive)) {
682 ex newexponent = exponent.conjugate();
683 if (are_ex_trivially_equal(exponent, newexponent)) {
686 return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
688 if (exponent.info(info_flags::integer)) {
689 ex newbasis = basis.conjugate();
690 if (are_ex_trivially_equal(basis, newbasis)) {
693 return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
695 return conjugate_function(*this).hold();
698 ex power::real_part() const
700 if (exponent.info(info_flags::integer)) {
701 ex basis_real = basis.real_part();
702 if (basis_real == basis)
704 realsymbol a("a"),b("b");
706 if (exponent.info(info_flags::posint))
707 result = power(a+I*b,exponent);
709 result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
710 result = result.expand();
711 result = result.real_part();
712 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
716 ex a = basis.real_part();
717 ex b = basis.imag_part();
718 ex c = exponent.real_part();
719 ex d = exponent.imag_part();
720 return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
723 ex power::imag_part() const
725 if (exponent.info(info_flags::integer)) {
726 ex basis_real = basis.real_part();
727 if (basis_real == basis)
729 realsymbol a("a"),b("b");
731 if (exponent.info(info_flags::posint))
732 result = power(a+I*b,exponent);
734 result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
735 result = result.expand();
736 result = result.imag_part();
737 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
741 ex a=basis.real_part();
742 ex b=basis.imag_part();
743 ex c=exponent.real_part();
744 ex d=exponent.imag_part();
745 return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
750 /** Implementation of ex::diff() for a power.
752 ex power::derivative(const symbol & s) const
754 if (is_a<numeric>(exponent)) {
755 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
758 newseq.push_back(expair(basis, exponent - _ex1));
759 newseq.push_back(expair(basis.diff(s), _ex1));
760 return mul(newseq, exponent);
762 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
764 add(mul(exponent.diff(s), log(basis)),
765 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
769 int power::compare_same_type(const basic & other) const
771 GINAC_ASSERT(is_exactly_a<power>(other));
772 const power &o = static_cast<const power &>(other);
774 int cmpval = basis.compare(o.basis);
778 return exponent.compare(o.exponent);
781 unsigned power::return_type() const
783 return basis.return_type();
786 return_type_t power::return_type_tinfo() const
788 return basis.return_type_tinfo();
791 ex power::expand(unsigned options) const
793 if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
794 // A special case worth optimizing.
795 setflag(status_flags::expanded);
799 // (x*p)^c -> x^c * p^c, if p>0
800 // makes sense before expanding the basis
801 if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
802 const mul &m = ex_to<mul>(basis);
805 prodseq.reserve(m.seq.size() + 1);
806 powseq.reserve(m.seq.size() + 1);
807 epvector::const_iterator last = m.seq.end();
808 epvector::const_iterator cit = m.seq.begin();
811 // search for positive/negative factors
813 ex e=m.recombine_pair_to_ex(*cit);
814 if (e.info(info_flags::positive))
815 prodseq.push_back(pow(e, exponent).expand(options));
816 else if (e.info(info_flags::negative)) {
817 prodseq.push_back(pow(-e, exponent).expand(options));
820 powseq.push_back(*cit);
824 // take care on the numeric coefficient
825 ex coeff=(possign? _ex1 : _ex_1);
826 if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
827 prodseq.push_back(power(m.overall_coeff, exponent));
828 else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
829 prodseq.push_back(power(-m.overall_coeff, exponent));
831 coeff *= m.overall_coeff;
833 // If positive/negative factors are found, then extract them.
834 // In either case we set a flag to avoid the second run on a part
835 // which does not have positive/negative terms.
836 if (prodseq.size() > 0) {
837 ex newbasis = coeff*mul(powseq);
838 ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
839 return ((new mul(prodseq))->setflag(status_flags::dynallocated)*(new power(newbasis, exponent))->setflag(status_flags::dynallocated).expand(options)).expand(options);
841 ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
844 const ex expanded_basis = basis.expand(options);
845 const ex expanded_exponent = exponent.expand(options);
847 // x^(a+b) -> x^a * x^b
848 if (is_exactly_a<add>(expanded_exponent)) {
849 const add &a = ex_to<add>(expanded_exponent);
851 distrseq.reserve(a.seq.size() + 1);
852 epvector::const_iterator last = a.seq.end();
853 epvector::const_iterator cit = a.seq.begin();
855 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
859 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
860 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
861 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
862 int int_exponent = num_exponent.to_int();
863 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
864 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
866 distrseq.push_back(power(expanded_basis, a.overall_coeff));
868 distrseq.push_back(power(expanded_basis, a.overall_coeff));
870 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
871 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
872 return r.expand(options);
875 if (!is_exactly_a<numeric>(expanded_exponent) ||
876 !ex_to<numeric>(expanded_exponent).is_integer()) {
877 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
880 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
884 // integer numeric exponent
885 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
886 int int_exponent = num_exponent.to_int();
889 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
890 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
892 // (x*y)^n -> x^n * y^n
893 if (is_exactly_a<mul>(expanded_basis))
894 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
896 // cannot expand further
897 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
900 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
904 // new virtual functions which can be overridden by derived classes
910 // non-virtual functions in this class
913 namespace { // anonymous namespace for power::expand_add() helpers
915 /** Helper class to generate all bounded combinatorial partitions of an integer
916 * n with exactly m parts (including zero parts) in non-decreasing order.
918 class partition_generator {
920 // Partitions n into m parts, not including zero parts.
921 // (Cf. OEIS sequence A008284; implementation adapted from Jörg Arndt's
925 // partition: x[1] + x[2] + ... + x[m] = n and sentinel x[0] == 0
929 mpartition2(unsigned n_, unsigned m_)
930 : x(m_+1), n(n_), m(m_)
932 for (int k=1; k<m; ++k)
936 bool next_partition()
938 int u = x[m]; // last element
947 return false; // current is last
958 int m; // number of parts 0<m<=n
959 mutable std::vector<int> partition; // current partition
961 partition_generator(unsigned n_, unsigned m_)
962 : mpgen(n_, 1), m(m_), partition(m_)
964 // returns current partition in non-decreasing order, padded with zeros
965 const std::vector<int>& current() const
967 for (int i = 0; i < m - mpgen.m; ++i)
968 partition[i] = 0; // pad with zeros
970 for (int i = m - mpgen.m; i < m; ++i)
971 partition[i] = mpgen.x[i - m + mpgen.m + 1];
977 if (!mpgen.next_partition()) {
978 if (mpgen.m == m || mpgen.m == mpgen.n)
979 return false; // current is last
980 // increment number of parts
981 mpgen = mpartition2(mpgen.n, mpgen.m + 1);
987 /** Helper class to generate all compositions of a partition of an integer n,
988 * starting with the compositions which has non-decreasing order.
990 class composition_generator {
992 // Generates all distinct permutations of a multiset.
993 // (Based on Aaron Williams' algorithm 1 from "Loopless Generation of
994 // Multiset Permutations using a Constant Number of Variables by Prefix
995 // Shifts." <http://webhome.csc.uvic.ca/~haron/CoolMulti.pdf>)
997 // element of singly linked list
1001 element(int val, element* n)
1002 : value(val), next(n) {}
1004 { // recurses down to the end of the singly linked list
1008 element *head, *i, *after_i;
1009 // NB: Partition must be sorted in non-decreasing order.
1010 explicit coolmulti(const std::vector<int>& partition)
1013 for (unsigned n = 0; n < partition.size(); ++n) {
1014 head = new element(partition[n], head);
1021 { // deletes singly linked list
1024 void next_permutation()
1027 if (after_i->next != NULL && i->value >= after_i->next->value)
1031 element *k = before_k->next;
1032 before_k->next = k->next;
1034 if (k->value < head->value)
1039 bool finished() const
1041 return after_i->next == NULL && after_i->value >= head->value;
1044 bool atend; // needed for simplifying iteration over permutations
1045 bool trivial; // likewise, true if all elements are equal
1046 mutable std::vector<int> composition; // current compositions
1048 explicit composition_generator(const std::vector<int>& partition)
1049 : cmgen(partition), atend(false), trivial(true), composition(partition.size())
1051 for (unsigned i=1; i<partition.size(); ++i)
1052 trivial = trivial && (partition[0] == partition[i]);
1054 const std::vector<int>& current() const
1056 coolmulti::element* it = cmgen.head;
1058 while (it != NULL) {
1059 composition[i] = it->value;
1067 // This ugly contortion is needed because the original coolmulti
1068 // algorithm requires code duplication of the payload procedure,
1069 // one before the loop and one inside it.
1070 if (trivial || atend)
1072 cmgen.next_permutation();
1073 atend = cmgen.finished();
1078 /** Helper function to compute the multinomial coefficient n!/(p1!*p2!*...*pk!)
1079 * where n = p1+p2+...+pk, i.e. p is a partition of n.
1082 multinomial_coefficient(const std::vector<int> p)
1084 numeric n = 0, d = 1;
1085 std::vector<int>::const_iterator it = p.begin(), itend = p.end();
1086 while (it != itend) {
1088 d *= factorial(numeric(*it));
1091 return factorial(numeric(n)) / d;
1094 } // anonymous namespace
1096 /** expand a^n where a is an add and n is a positive integer.
1097 * @see power::expand */
1098 ex power::expand_add(const add & a, int n, unsigned options) const
1100 // The special case power(+(x,...y;x),2) can be optimized better.
1102 return expand_add_2(a, options);
1106 // Consider base as the sum of all symbolic terms and the overall numeric
1107 // coefficient and apply the binomial theorem:
1108 // S = power(+(x,...,z;c),n)
1109 // = power(+(+(x,...,z;0);c),n)
1110 // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1111 // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
1112 // The multinomial theorem is computed by an outer loop over all
1113 // partitions of the exponent and an inner loop over all compositions of
1114 // that partition. This method makes the expansion a combinatorial
1115 // problem and allows us to directly construct the expanded sum and also
1116 // to re-use the multinomial coefficients (since they depend only on the
1117 // partition, not on the composition).
1119 // multinomial power(+(x,y,z;0),3) example:
1120 // partition : compositions : multinomial coefficient
1121 // [0,0,3] : [3,0,0],[0,3,0],[0,0,3] : 3!/(3!*0!*0!) = 1
1122 // [0,1,2] : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
1123 // [1,1,1] : [1,1,1] : 3!/(1!*1!*1!) = 6
1124 // => (x + y + z)^3 =
1126 // + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
1129 // multinomial power(+(x,y,z;0),4) example:
1130 // partition : compositions : multinomial coefficient
1131 // [0,0,4] : [4,0,0],[0,4,0],[0,0,4] : 4!/(4!*0!*0!) = 1
1132 // [0,1,3] : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
1133 // [0,2,2] : [2,2,0],[2,0,2],[0,2,2] : 4!/(2!*2!*0!) = 6
1134 // [1,1,2] : [2,1,1],[1,2,1],[1,1,2] : 4!/(2!*1!*1!) = 12
1135 // (no [1,1,1,1] partition since it has too many parts)
1136 // => (x + y + z)^4 =
1138 // + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
1139 // + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
1140 // + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
1144 // for k from 0 to n:
1145 // f = c^(n-k)*binomial(n,k)
1146 // for p in all partitions of n with m parts (including zero parts):
1147 // h = f * multinomial coefficient of p
1148 // for c in all compositions of p:
1150 // for e in all elements of c:
1156 // The number of terms will be the number of combinatorial compositions,
1157 // i.e. the number of unordered arrangements of m nonnegative integers
1158 // which sum up to n. It is frequently written as C_n(m) and directly
1159 // related with binomial coefficients: binomial(n+m-1,m-1).
1160 size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_int();
1161 if (!a.overall_coeff.is_zero()) {
1162 // the result's overall_coeff is one of the terms
1165 result.reserve(result_size);
1167 // Iterate over all terms in binomial expansion of
1168 // S = power(+(x,...,z;c),n)
1169 // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1170 for (int k = 1; k <= n; ++k) {
1171 numeric binomial_coefficient; // binomial(n,k)*c^(n-k)
1172 if (a.overall_coeff.is_zero()) {
1173 // degenerate case with zero overall_coeff:
1174 // apply multinomial theorem directly to power(+(x,...z;0),n)
1175 binomial_coefficient = 1;
1180 binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
1183 // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
1184 // Iterate over all partitions of k with exactly as many parts as
1185 // there are symbolic terms in the basis (including zero parts).
1186 partition_generator partitions(k, a.seq.size());
1188 const std::vector<int>& partition = partitions.current();
1189 const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
1191 // Iterate over all compositions of the current partition.
1192 composition_generator compositions(partition);
1194 const std::vector<int>& exponent = compositions.current();
1197 numeric factor = coeff;
1198 for (unsigned i = 0; i < exponent.size(); ++i) {
1199 const ex & r = a.seq[i].rest;
1200 const ex & c = a.seq[i].coeff;
1201 GINAC_ASSERT(!is_exactly_a<add>(r));
1202 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1203 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1204 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1205 !is_exactly_a<add>(ex_to<power>(r).basis) ||
1206 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1207 !is_exactly_a<power>(ex_to<power>(r).basis));
1208 if (exponent[i] == 0) {
1210 } else if (exponent[i] == 1) {
1213 factor = factor.mul(ex_to<numeric>(c));
1214 } else { // general case exponent[i] > 1
1215 term.push_back((new power(r, exponent[i]))->setflag(status_flags::dynallocated));
1216 factor = factor.mul(ex_to<numeric>(c).power(exponent[i]));
1219 result.push_back(a.combine_ex_with_coeff_to_pair(mul(term).expand(options), factor));
1220 } while (compositions.next());
1221 } while (partitions.next());
1224 GINAC_ASSERT(result.size() == result_size);
1226 if (a.overall_coeff.is_zero()) {
1227 return (new add(result))->setflag(status_flags::dynallocated |
1228 status_flags::expanded);
1230 return (new add(result, ex_to<numeric>(a.overall_coeff).power(n)))->setflag(status_flags::dynallocated |
1231 status_flags::expanded);
1236 /** Special case of power::expand_add. Expands a^2 where a is an add.
1237 * @see power::expand_add */
1238 ex power::expand_add_2(const add & a, unsigned options) const
1241 size_t a_nops = a.nops();
1242 sum.reserve((a_nops*(a_nops+1))/2);
1243 epvector::const_iterator last = a.seq.end();
1245 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
1246 // first part: ignore overall_coeff and expand other terms
1247 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
1248 const ex & r = cit0->rest;
1249 const ex & c = cit0->coeff;
1251 GINAC_ASSERT(!is_exactly_a<add>(r));
1252 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1253 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1254 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1255 !is_exactly_a<add>(ex_to<power>(r).basis) ||
1256 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1257 !is_exactly_a<power>(ex_to<power>(r).basis));
1259 if (c.is_equal(_ex1)) {
1260 if (is_exactly_a<mul>(r)) {
1261 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1264 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1268 if (is_exactly_a<mul>(r)) {
1269 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1270 ex_to<numeric>(c).power_dyn(*_num2_p)));
1272 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1273 ex_to<numeric>(c).power_dyn(*_num2_p)));
1277 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1278 const ex & r1 = cit1->rest;
1279 const ex & c1 = cit1->coeff;
1280 sum.push_back(a.combine_ex_with_coeff_to_pair(mul(r,r1).expand(options),
1281 _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1285 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
1287 // second part: add terms coming from overall_coeff (if != 0)
1288 if (!a.overall_coeff.is_zero()) {
1289 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1291 sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1294 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1297 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1299 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1302 /** Expand factors of m in m^n where m is a mul and n is an integer.
1303 * @see power::expand */
1304 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1306 GINAC_ASSERT(n.is_integer());
1312 // do not bother to rename indices if there are no any.
1313 if ((!(options & expand_options::expand_rename_idx))
1314 && m.info(info_flags::has_indices))
1315 options |= expand_options::expand_rename_idx;
1316 // Leave it to multiplication since dummy indices have to be renamed
1317 if ((options & expand_options::expand_rename_idx) &&
1318 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1320 exvector va = get_all_dummy_indices(m);
1321 sort(va.begin(), va.end(), ex_is_less());
1323 for (int i=1; i < n.to_int(); i++)
1324 result *= rename_dummy_indices_uniquely(va, m);
1329 distrseq.reserve(m.seq.size());
1330 bool need_reexpand = false;
1332 epvector::const_iterator last = m.seq.end();
1333 epvector::const_iterator cit = m.seq.begin();
1335 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1336 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1337 // this happens when e.g. (a+b)^(1/2) gets squared and
1338 // the resulting product needs to be reexpanded
1339 need_reexpand = true;
1341 distrseq.push_back(p);
1345 const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1347 return ex(result).expand(options);
1349 return result.setflag(status_flags::expanded);
1353 GINAC_BIND_UNARCHIVER(power);
1355 } // namespace GiNaC