3 * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
6 * GiNaC Copyright (C) 1999-2016 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"
49 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
50 print_func<print_dflt>(&power::do_print_dflt).
51 print_func<print_latex>(&power::do_print_latex).
52 print_func<print_csrc>(&power::do_print_csrc).
53 print_func<print_python>(&power::do_print_python).
54 print_func<print_python_repr>(&power::do_print_python_repr).
55 print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))
58 // default constructor
73 void power::read_archive(const archive_node &n, lst &sym_lst)
75 inherited::read_archive(n, sym_lst);
76 n.find_ex("basis", basis, sym_lst);
77 n.find_ex("exponent", exponent, sym_lst);
80 void power::archive(archive_node &n) const
82 inherited::archive(n);
83 n.add_ex("basis", basis);
84 n.add_ex("exponent", exponent);
88 // functions overriding virtual functions from base classes
93 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
95 // Ordinary output of powers using '^' or '**'
96 if (precedence() <= level)
97 c.s << openbrace << '(';
98 basis.print(c, precedence());
101 exponent.print(c, precedence());
103 if (precedence() <= level)
104 c.s << ')' << closebrace;
107 void power::do_print_dflt(const print_dflt & c, unsigned level) const
109 if (exponent.is_equal(_ex1_2)) {
111 // Square roots are printed in a special way
117 print_power(c, "^", "", "", level);
120 void power::do_print_latex(const print_latex & c, unsigned level) const
122 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
124 // Powers with negative numeric exponents are printed as fractions
126 power(basis, -exponent).eval().print(c);
129 } else if (exponent.is_equal(_ex1_2)) {
131 // Square roots are printed in a special way
137 print_power(c, "^", "{", "}", level);
140 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
142 // Optimal output of integer powers of symbols to aid compiler CSE.
143 // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
144 // to learn why such a parenthesation is really necessary.
147 } else if (exp == 2) {
151 } else if (exp & 1) {
154 print_sym_pow(c, x, exp-1);
157 print_sym_pow(c, x, exp >> 1);
159 print_sym_pow(c, x, exp >> 1);
164 void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const
166 if (exponent.is_equal(_ex_1)) {
179 void power::do_print_csrc(const print_csrc & c, unsigned level) const
181 // Integer powers of symbols are printed in a special, optimized way
182 if (exponent.info(info_flags::integer) &&
183 (is_a<symbol>(basis) || is_a<constant>(basis))) {
184 int exp = ex_to<numeric>(exponent).to_int();
191 print_sym_pow(c, ex_to<symbol>(basis), exp);
194 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
195 } else if (exponent.is_equal(_ex_1)) {
200 // Otherwise, use the pow() function
210 void power::do_print_python(const print_python & c, unsigned level) const
212 print_power(c, "**", "", "", level);
215 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
217 c.s << class_name() << '(';
224 bool power::info(unsigned inf) const
227 case info_flags::polynomial:
228 case info_flags::integer_polynomial:
229 case info_flags::cinteger_polynomial:
230 case info_flags::rational_polynomial:
231 case info_flags::crational_polynomial:
232 return exponent.info(info_flags::nonnegint) &&
234 case info_flags::rational_function:
235 return exponent.info(info_flags::integer) &&
237 case info_flags::algebraic:
238 return !exponent.info(info_flags::integer) ||
240 case info_flags::expanded:
241 return (flags & status_flags::expanded);
242 case info_flags::positive:
243 return basis.info(info_flags::positive) && exponent.info(info_flags::real);
244 case info_flags::nonnegative:
245 return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
246 (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) > ex_to<numeric>(other.op(1)) &&
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) < ex_to<numeric>(other.op(1)) &&
637 basis.match(other.op(0)))
639 return basic::has(other, options);
643 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
645 ex power::subs(const exmap & m, unsigned options) const
647 const ex &subsed_basis = basis.subs(m, options);
648 const ex &subsed_exponent = exponent.subs(m, options);
650 if (!are_ex_trivially_equal(basis, subsed_basis)
651 || !are_ex_trivially_equal(exponent, subsed_exponent))
652 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
654 if (!(options & subs_options::algebraic))
655 return subs_one_level(m, options);
657 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
658 int nummatches = std::numeric_limits<int>::max();
660 if (tryfactsubs(*this, it->first, nummatches, repls)) {
661 ex anum = it->second.subs(repls, subs_options::no_pattern);
662 ex aden = it->first.subs(repls, subs_options::no_pattern);
663 ex result = (*this)*power(anum/aden, nummatches);
664 return (ex_to<basic>(result)).subs_one_level(m, options);
668 return subs_one_level(m, options);
671 ex power::eval_ncmul(const exvector & v) const
673 return inherited::eval_ncmul(v);
676 ex power::conjugate() const
678 // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
679 // branch cut which runs along the negative real axis.
680 if (basis.info(info_flags::positive)) {
681 ex newexponent = exponent.conjugate();
682 if (are_ex_trivially_equal(exponent, newexponent)) {
685 return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
687 if (exponent.info(info_flags::integer)) {
688 ex newbasis = basis.conjugate();
689 if (are_ex_trivially_equal(basis, newbasis)) {
692 return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
694 return conjugate_function(*this).hold();
697 ex power::real_part() const
699 // basis == a+I*b, exponent == c+I*d
700 const ex a = basis.real_part();
701 const ex c = exponent.real_part();
702 if (basis.is_equal(a) && exponent.is_equal(c)) {
707 const ex b = basis.imag_part();
708 if (exponent.info(info_flags::integer)) {
709 // Re((a+I*b)^c) w/ c ∈ ℤ
710 long N = ex_to<numeric>(c).to_long();
711 // Use real terms in Binomial expansion to construct
712 // Re(expand(power(a+I*b, N))).
713 long NN = N > 0 ? N : -N;
714 ex numer = N > 0 ? _ex1 : power(power(a,2) + power(b,2), NN);
716 for (long n = 0; n <= NN; n += 2) {
717 ex term = binomial(NN, n) * power(a, NN-n) * power(b, n) / numer;
719 result += term; // sign: I^n w/ n == 4*m
721 result -= term; // sign: I^n w/ n == 4*m+2
727 // Re((a+I*b)^(c+I*d))
728 const ex d = exponent.imag_part();
729 return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
732 ex power::imag_part() const
734 const ex a = basis.real_part();
735 const ex c = exponent.real_part();
736 if (basis.is_equal(a) && exponent.is_equal(c)) {
741 const ex b = basis.imag_part();
742 if (exponent.info(info_flags::integer)) {
743 // Im((a+I*b)^c) w/ c ∈ ℤ
744 long N = ex_to<numeric>(c).to_long();
745 // Use imaginary terms in Binomial expansion to construct
746 // Im(expand(power(a+I*b, N))).
747 long p = N > 0 ? 1 : 3; // modulus for positive sign
748 long NN = N > 0 ? N : -N;
749 ex numer = N > 0 ? _ex1 : power(power(a,2) + power(b,2), NN);
751 for (long n = 1; n <= NN; n += 2) {
752 ex term = binomial(NN, n) * power(a, NN-n) * power(b, n) / numer;
754 result += term; // sign: I^n w/ n == 4*m+p
756 result -= term; // sign: I^n w/ n == 4*m+2+p
762 // Im((a+I*b)^(c+I*d))
763 const ex d = exponent.imag_part();
764 return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
769 /** Implementation of ex::diff() for a power.
771 ex power::derivative(const symbol & s) const
773 if (is_a<numeric>(exponent)) {
774 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
777 newseq.push_back(expair(basis, exponent - _ex1));
778 newseq.push_back(expair(basis.diff(s), _ex1));
779 return mul(newseq, exponent);
781 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
783 add(mul(exponent.diff(s), log(basis)),
784 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
788 int power::compare_same_type(const basic & other) const
790 GINAC_ASSERT(is_exactly_a<power>(other));
791 const power &o = static_cast<const power &>(other);
793 int cmpval = basis.compare(o.basis);
797 return exponent.compare(o.exponent);
800 unsigned power::return_type() const
802 return basis.return_type();
805 return_type_t power::return_type_tinfo() const
807 return basis.return_type_tinfo();
810 ex power::expand(unsigned options) const
812 if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
813 // A special case worth optimizing.
814 setflag(status_flags::expanded);
818 // (x*p)^c -> x^c * p^c, if p>0
819 // makes sense before expanding the basis
820 if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
821 const mul &m = ex_to<mul>(basis);
824 prodseq.reserve(m.seq.size() + 1);
825 powseq.reserve(m.seq.size() + 1);
826 epvector::const_iterator last = m.seq.end();
827 epvector::const_iterator cit = m.seq.begin();
830 // search for positive/negative factors
832 ex e=m.recombine_pair_to_ex(*cit);
833 if (e.info(info_flags::positive))
834 prodseq.push_back(pow(e, exponent).expand(options));
835 else if (e.info(info_flags::negative)) {
836 prodseq.push_back(pow(-e, exponent).expand(options));
839 powseq.push_back(*cit);
843 // take care on the numeric coefficient
844 ex coeff=(possign? _ex1 : _ex_1);
845 if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
846 prodseq.push_back(power(m.overall_coeff, exponent));
847 else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
848 prodseq.push_back(power(-m.overall_coeff, exponent));
850 coeff *= m.overall_coeff;
852 // If positive/negative factors are found, then extract them.
853 // In either case we set a flag to avoid the second run on a part
854 // which does not have positive/negative terms.
855 if (prodseq.size() > 0) {
856 ex newbasis = coeff*mul(powseq);
857 ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
858 return ((new mul(prodseq))->setflag(status_flags::dynallocated)*(new power(newbasis, exponent))->setflag(status_flags::dynallocated).expand(options)).expand(options);
860 ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
863 const ex expanded_basis = basis.expand(options);
864 const ex expanded_exponent = exponent.expand(options);
866 // x^(a+b) -> x^a * x^b
867 if (is_exactly_a<add>(expanded_exponent)) {
868 const add &a = ex_to<add>(expanded_exponent);
870 distrseq.reserve(a.seq.size() + 1);
871 epvector::const_iterator last = a.seq.end();
872 epvector::const_iterator cit = a.seq.begin();
874 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
878 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
879 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
880 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
881 int int_exponent = num_exponent.to_int();
882 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
883 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
885 distrseq.push_back(power(expanded_basis, a.overall_coeff));
887 distrseq.push_back(power(expanded_basis, a.overall_coeff));
889 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
890 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
891 return r.expand(options);
894 if (!is_exactly_a<numeric>(expanded_exponent) ||
895 !ex_to<numeric>(expanded_exponent).is_integer()) {
896 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
899 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
903 // integer numeric exponent
904 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
905 int int_exponent = num_exponent.to_int();
908 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
909 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
911 // (x*y)^n -> x^n * y^n
912 if (is_exactly_a<mul>(expanded_basis))
913 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
915 // cannot expand further
916 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
919 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
923 // new virtual functions which can be overridden by derived classes
929 // non-virtual functions in this class
932 namespace { // anonymous namespace for power::expand_add() helpers
934 /** Helper class to generate all bounded combinatorial partitions of an integer
935 * n with exactly m parts (including zero parts) in non-decreasing order.
937 class partition_generator {
939 // Partitions n into m parts, not including zero parts.
940 // (Cf. OEIS sequence A008284; implementation adapted from Jörg Arndt's
944 // partition: x[1] + x[2] + ... + x[m] = n and sentinel x[0] == 0
948 mpartition2(unsigned n_, unsigned m_)
949 : x(m_+1), n(n_), m(m_)
951 for (int k=1; k<m; ++k)
955 bool next_partition()
957 int u = x[m]; // last element
966 return false; // current is last
977 int m; // number of parts 0<m<=n
978 mutable std::vector<int> partition; // current partition
980 partition_generator(unsigned n_, unsigned m_)
981 : mpgen(n_, 1), m(m_), partition(m_)
983 // returns current partition in non-decreasing order, padded with zeros
984 const std::vector<int>& current() const
986 for (int i = 0; i < m - mpgen.m; ++i)
987 partition[i] = 0; // pad with zeros
989 for (int i = m - mpgen.m; i < m; ++i)
990 partition[i] = mpgen.x[i - m + mpgen.m + 1];
996 if (!mpgen.next_partition()) {
997 if (mpgen.m == m || mpgen.m == mpgen.n)
998 return false; // current is last
999 // increment number of parts
1000 mpgen = mpartition2(mpgen.n, mpgen.m + 1);
1006 /** Helper class to generate all compositions of a partition of an integer n,
1007 * starting with the compositions which has non-decreasing order.
1009 class composition_generator {
1011 // Generates all distinct permutations of a multiset.
1012 // (Based on Aaron Williams' algorithm 1 from "Loopless Generation of
1013 // Multiset Permutations using a Constant Number of Variables by Prefix
1014 // Shifts." <http://webhome.csc.uvic.ca/~haron/CoolMulti.pdf>)
1016 // element of singly linked list
1020 element(int val, element* n)
1021 : value(val), next(n) {}
1023 { // recurses down to the end of the singly linked list
1027 element *head, *i, *after_i;
1028 // NB: Partition must be sorted in non-decreasing order.
1029 explicit coolmulti(const std::vector<int>& partition)
1032 for (unsigned n = 0; n < partition.size(); ++n) {
1033 head = new element(partition[n], head);
1040 { // deletes singly linked list
1043 void next_permutation()
1046 if (after_i->next != NULL && i->value >= after_i->next->value)
1050 element *k = before_k->next;
1051 before_k->next = k->next;
1053 if (k->value < head->value)
1058 bool finished() const
1060 return after_i->next == NULL && after_i->value >= head->value;
1063 bool atend; // needed for simplifying iteration over permutations
1064 bool trivial; // likewise, true if all elements are equal
1065 mutable std::vector<int> composition; // current compositions
1067 explicit composition_generator(const std::vector<int>& partition)
1068 : cmgen(partition), atend(false), trivial(true), composition(partition.size())
1070 for (unsigned i=1; i<partition.size(); ++i)
1071 trivial = trivial && (partition[0] == partition[i]);
1073 const std::vector<int>& current() const
1075 coolmulti::element* it = cmgen.head;
1077 while (it != NULL) {
1078 composition[i] = it->value;
1086 // This ugly contortion is needed because the original coolmulti
1087 // algorithm requires code duplication of the payload procedure,
1088 // one before the loop and one inside it.
1089 if (trivial || atend)
1091 cmgen.next_permutation();
1092 atend = cmgen.finished();
1097 /** Helper function to compute the multinomial coefficient n!/(p1!*p2!*...*pk!)
1098 * where n = p1+p2+...+pk, i.e. p is a partition of n.
1101 multinomial_coefficient(const std::vector<int> & p)
1103 numeric n = 0, d = 1;
1104 std::vector<int>::const_iterator it = p.begin(), itend = p.end();
1105 while (it != itend) {
1107 d *= factorial(numeric(*it));
1110 return factorial(numeric(n)) / d;
1113 } // anonymous namespace
1115 /** expand a^n where a is an add and n is a positive integer.
1116 * @see power::expand */
1117 ex power::expand_add(const add & a, int n, unsigned options) const
1119 // The special case power(+(x,...y;x),2) can be optimized better.
1121 return expand_add_2(a, options);
1125 // Consider base as the sum of all symbolic terms and the overall numeric
1126 // coefficient and apply the binomial theorem:
1127 // S = power(+(x,...,z;c),n)
1128 // = power(+(+(x,...,z;0);c),n)
1129 // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1130 // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
1131 // The multinomial theorem is computed by an outer loop over all
1132 // partitions of the exponent and an inner loop over all compositions of
1133 // that partition. This method makes the expansion a combinatorial
1134 // problem and allows us to directly construct the expanded sum and also
1135 // to re-use the multinomial coefficients (since they depend only on the
1136 // partition, not on the composition).
1138 // multinomial power(+(x,y,z;0),3) example:
1139 // partition : compositions : multinomial coefficient
1140 // [0,0,3] : [3,0,0],[0,3,0],[0,0,3] : 3!/(3!*0!*0!) = 1
1141 // [0,1,2] : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
1142 // [1,1,1] : [1,1,1] : 3!/(1!*1!*1!) = 6
1143 // => (x + y + z)^3 =
1145 // + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
1148 // multinomial power(+(x,y,z;0),4) example:
1149 // partition : compositions : multinomial coefficient
1150 // [0,0,4] : [4,0,0],[0,4,0],[0,0,4] : 4!/(4!*0!*0!) = 1
1151 // [0,1,3] : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
1152 // [0,2,2] : [2,2,0],[2,0,2],[0,2,2] : 4!/(2!*2!*0!) = 6
1153 // [1,1,2] : [2,1,1],[1,2,1],[1,1,2] : 4!/(2!*1!*1!) = 12
1154 // (no [1,1,1,1] partition since it has too many parts)
1155 // => (x + y + z)^4 =
1157 // + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
1158 // + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
1159 // + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
1163 // for k from 0 to n:
1164 // f = c^(n-k)*binomial(n,k)
1165 // for p in all partitions of n with m parts (including zero parts):
1166 // h = f * multinomial coefficient of p
1167 // for c in all compositions of p:
1169 // for e in all elements of c:
1175 // The number of terms will be the number of combinatorial compositions,
1176 // i.e. the number of unordered arrangements of m nonnegative integers
1177 // which sum up to n. It is frequently written as C_n(m) and directly
1178 // related with binomial coefficients: binomial(n+m-1,m-1).
1179 size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_int();
1180 if (!a.overall_coeff.is_zero()) {
1181 // the result's overall_coeff is one of the terms
1184 result.reserve(result_size);
1186 // Iterate over all terms in binomial expansion of
1187 // S = power(+(x,...,z;c),n)
1188 // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1189 for (int k = 1; k <= n; ++k) {
1190 numeric binomial_coefficient; // binomial(n,k)*c^(n-k)
1191 if (a.overall_coeff.is_zero()) {
1192 // degenerate case with zero overall_coeff:
1193 // apply multinomial theorem directly to power(+(x,...z;0),n)
1194 binomial_coefficient = 1;
1199 binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
1202 // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
1203 // Iterate over all partitions of k with exactly as many parts as
1204 // there are symbolic terms in the basis (including zero parts).
1205 partition_generator partitions(k, a.seq.size());
1207 const std::vector<int>& partition = partitions.current();
1208 // All monomials of this partition have the same number of terms and the same coefficient.
1209 const unsigned msize = count_if(partition.begin(), partition.end(), bind2nd(std::greater<int>(), 0));
1210 const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
1212 // Iterate over all compositions of the current partition.
1213 composition_generator compositions(partition);
1215 const std::vector<int>& exponent = compositions.current();
1217 monomial.reserve(msize);
1218 numeric factor = coeff;
1219 for (unsigned i = 0; i < exponent.size(); ++i) {
1220 const ex & r = a.seq[i].rest;
1221 GINAC_ASSERT(!is_exactly_a<add>(r));
1222 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1223 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1224 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1225 !is_exactly_a<add>(ex_to<power>(r).basis) ||
1226 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1227 !is_exactly_a<power>(ex_to<power>(r).basis));
1228 GINAC_ASSERT(is_exactly_a<numeric>(a.seq[i].coeff));
1229 const numeric & c = ex_to<numeric>(a.seq[i].coeff);
1230 if (exponent[i] == 0) {
1232 } else if (exponent[i] == 1) {
1234 monomial.push_back(r);
1236 factor = factor.mul(c);
1237 } else { // general case exponent[i] > 1
1238 monomial.push_back((new power(r, exponent[i]))->setflag(status_flags::dynallocated));
1240 factor = factor.mul(c.power(exponent[i]));
1243 result.push_back(a.combine_ex_with_coeff_to_pair(mul(monomial).expand(options), factor));
1244 } while (compositions.next());
1245 } while (partitions.next());
1248 GINAC_ASSERT(result.size() == result_size);
1250 if (a.overall_coeff.is_zero()) {
1251 return (new add(result))->setflag(status_flags::dynallocated |
1252 status_flags::expanded);
1254 return (new add(result, ex_to<numeric>(a.overall_coeff).power(n)))->setflag(status_flags::dynallocated |
1255 status_flags::expanded);
1260 /** Special case of power::expand_add. Expands a^2 where a is an add.
1261 * @see power::expand_add */
1262 ex power::expand_add_2(const add & a, unsigned options) const
1265 size_t a_nops = a.nops();
1266 sum.reserve((a_nops*(a_nops+1))/2);
1267 epvector::const_iterator last = a.seq.end();
1269 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
1270 // first part: ignore overall_coeff and expand other terms
1271 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
1272 const ex & r = cit0->rest;
1273 const ex & c = cit0->coeff;
1275 GINAC_ASSERT(!is_exactly_a<add>(r));
1276 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1277 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1278 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1279 !is_exactly_a<add>(ex_to<power>(r).basis) ||
1280 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1281 !is_exactly_a<power>(ex_to<power>(r).basis));
1283 if (c.is_equal(_ex1)) {
1284 if (is_exactly_a<mul>(r)) {
1285 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1288 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1292 if (is_exactly_a<mul>(r)) {
1293 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1294 ex_to<numeric>(c).power_dyn(*_num2_p)));
1296 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1297 ex_to<numeric>(c).power_dyn(*_num2_p)));
1301 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1302 const ex & r1 = cit1->rest;
1303 const ex & c1 = cit1->coeff;
1304 sum.push_back(a.combine_ex_with_coeff_to_pair(mul(r,r1).expand(options),
1305 _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1309 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
1311 // second part: add terms coming from overall_coeff (if != 0)
1312 if (!a.overall_coeff.is_zero()) {
1313 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1315 sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1318 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1321 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1323 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1326 /** Expand factors of m in m^n where m is a mul and n is an integer.
1327 * @see power::expand */
1328 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1330 GINAC_ASSERT(n.is_integer());
1336 // do not bother to rename indices if there are no any.
1337 if (!(options & expand_options::expand_rename_idx) &&
1338 m.info(info_flags::has_indices))
1339 options |= expand_options::expand_rename_idx;
1340 // Leave it to multiplication since dummy indices have to be renamed
1341 if ((options & expand_options::expand_rename_idx) &&
1342 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1344 exvector va = get_all_dummy_indices(m);
1345 sort(va.begin(), va.end(), ex_is_less());
1347 for (int i=1; i < n.to_int(); i++)
1348 result *= rename_dummy_indices_uniquely(va, m);
1353 distrseq.reserve(m.seq.size());
1354 bool need_reexpand = false;
1356 epvector::const_iterator last = m.seq.end();
1357 epvector::const_iterator cit = m.seq.begin();
1359 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1360 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1361 // this happens when e.g. (a+b)^(1/2) gets squared and
1362 // the resulting product needs to be reexpanded
1363 need_reexpand = true;
1365 distrseq.push_back(p);
1369 const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1371 return ex(result).expand(options);
1373 return result.setflag(status_flags::expanded);
1377 GINAC_BIND_UNARCHIVER(power);
1379 } // namespace GiNaC