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 14882:2011, 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 basis.info(inf) && exponent.info(info_flags::nonnegint);
233 case info_flags::rational_function:
234 return basis.info(inf) && exponent.info(info_flags::integer);
235 case info_flags::real:
236 return basis.info(inf) && exponent.info(info_flags::integer);
237 case info_flags::expanded:
238 return (flags & status_flags::expanded);
239 case info_flags::positive:
240 return basis.info(info_flags::positive) && exponent.info(info_flags::real);
241 case info_flags::nonnegative:
242 return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
243 (basis.info(info_flags::real) && exponent.info(info_flags::even));
244 case info_flags::has_indices: {
245 if (flags & status_flags::has_indices)
247 else if (flags & status_flags::has_no_indices)
249 else if (basis.info(info_flags::has_indices)) {
250 setflag(status_flags::has_indices);
251 clearflag(status_flags::has_no_indices);
254 clearflag(status_flags::has_indices);
255 setflag(status_flags::has_no_indices);
260 return inherited::info(inf);
263 size_t power::nops() const
268 ex power::op(size_t i) const
272 return i==0 ? basis : exponent;
275 ex power::map(map_function & f) const
277 const ex &mapped_basis = f(basis);
278 const ex &mapped_exponent = f(exponent);
280 if (!are_ex_trivially_equal(basis, mapped_basis)
281 || !are_ex_trivially_equal(exponent, mapped_exponent))
282 return dynallocate<power>(mapped_basis, mapped_exponent);
287 bool power::is_polynomial(const ex & var) const
289 if (basis.is_polynomial(var)) {
291 // basis is non-constant polynomial in var
292 return exponent.info(info_flags::nonnegint);
294 // basis is constant in var
295 return !exponent.has(var);
297 // basis is a non-polynomial function of var
301 int power::degree(const ex & s) const
303 if (is_equal(ex_to<basic>(s)))
305 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
306 if (basis.is_equal(s))
307 return ex_to<numeric>(exponent).to_int();
309 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
310 } else if (basis.has(s))
311 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
316 int power::ldegree(const ex & s) const
318 if (is_equal(ex_to<basic>(s)))
320 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
321 if (basis.is_equal(s))
322 return ex_to<numeric>(exponent).to_int();
324 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
325 } else if (basis.has(s))
326 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
331 ex power::coeff(const ex & s, int n) const
333 if (is_equal(ex_to<basic>(s)))
334 return n==1 ? _ex1 : _ex0;
335 else if (!basis.is_equal(s)) {
336 // basis not equal to s
343 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
345 int int_exp = ex_to<numeric>(exponent).to_int();
351 // non-integer exponents are treated as zero
360 /** Perform automatic term rewriting rules in this class. In the following
361 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
362 * stand for such expressions that contain a plain number.
363 * - ^(x,0) -> 1 (also handles ^(0,0))
365 * - ^(0,c) -> 0 or exception (depending on the real part of c)
367 * - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
368 * - ^(^(x,c1),c2) -> ^(x,c1*c2) if x is positive and c1 is real.
369 * - ^(^(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!)
370 * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
371 * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
372 * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
374 ex power::eval() const
376 if (flags & status_flags::evaluated)
379 const numeric *num_basis = nullptr;
380 const numeric *num_exponent = nullptr;
382 if (is_exactly_a<numeric>(basis)) {
383 num_basis = &ex_to<numeric>(basis);
385 if (is_exactly_a<numeric>(exponent)) {
386 num_exponent = &ex_to<numeric>(exponent);
389 // ^(x,0) -> 1 (0^0 also handled here)
390 if (exponent.is_zero()) {
392 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
398 if (exponent.is_equal(_ex1))
401 // ^(0,c1) -> 0 or exception (depending on real value of c1)
402 if (basis.is_zero() && num_exponent) {
403 if ((num_exponent->real()).is_zero())
404 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
405 else if ((num_exponent->real()).is_negative())
406 throw (pole_error("power::eval(): division by zero",1));
412 if (basis.is_equal(_ex1))
415 // power of a function calculated by separate rules defined for this function
416 if (is_exactly_a<function>(basis))
417 return ex_to<function>(basis).power(exponent);
419 // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
420 if (is_exactly_a<power>(basis) && basis.op(0).info(info_flags::positive) && basis.op(1).info(info_flags::real))
421 return dynallocate<power>(basis.op(0), basis.op(1) * exponent);
423 if ( num_exponent ) {
425 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
426 // except if c1,c2 are rational, but c1^c2 is not)
428 const bool basis_is_crational = num_basis->is_crational();
429 const bool exponent_is_crational = num_exponent->is_crational();
430 if (!basis_is_crational || !exponent_is_crational) {
431 // return a plain float
432 return dynallocate<numeric>(num_basis->power(*num_exponent));
435 const numeric res = num_basis->power(*num_exponent);
436 if (res.is_crational()) {
439 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
441 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
442 if (basis_is_crational && exponent_is_crational
443 && num_exponent->is_real()
444 && !num_exponent->is_integer()) {
445 const numeric n = num_exponent->numer();
446 const numeric m = num_exponent->denom();
448 numeric q = iquo(n, m, r);
449 if (r.is_negative()) {
453 if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
454 if (num_basis->is_rational() && !num_basis->is_integer()) {
455 // try it for numerator and denominator separately, in order to
456 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
457 const numeric bnum = num_basis->numer();
458 const numeric bden = num_basis->denom();
459 const numeric res_bnum = bnum.power(*num_exponent);
460 const numeric res_bden = bden.power(*num_exponent);
461 if (res_bnum.is_integer())
462 return dynallocate<mul>(dynallocate<power>(bden,-*num_exponent),res_bnum).setflag(status_flags::evaluated);
463 if (res_bden.is_integer())
464 return dynallocate<mul>(dynallocate<power>(bnum,*num_exponent),res_bden.inverse()).setflag(status_flags::evaluated);
468 // assemble resulting product, but allowing for a re-evaluation,
469 // because otherwise we'll end up with something like
470 // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
471 // instead of 7/16*7^(1/3).
472 return pow(basis, r.div(m)) * pow(basis, q);
477 // ^(^(x,c1),c2) -> ^(x,c1*c2)
478 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
479 // case c1==1 should not happen, see below!)
480 if (is_exactly_a<power>(basis)) {
481 const power & sub_power = ex_to<power>(basis);
482 const ex & sub_basis = sub_power.basis;
483 const ex & sub_exponent = sub_power.exponent;
484 if (is_exactly_a<numeric>(sub_exponent)) {
485 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
486 GINAC_ASSERT(num_sub_exponent!=numeric(1));
487 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative() ||
488 (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
489 return dynallocate<power>(sub_basis, num_sub_exponent.mul(*num_exponent));
494 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
495 if (num_exponent->is_integer() && is_exactly_a<mul>(basis)) {
496 return expand_mul(ex_to<mul>(basis), *num_exponent, false);
499 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
500 if (num_exponent->is_integer() && is_exactly_a<add>(basis)) {
501 numeric icont = basis.integer_content();
502 const numeric lead_coeff =
503 ex_to<numeric>(ex_to<add>(basis).seq.begin()->coeff).div(icont);
505 const bool canonicalizable = lead_coeff.is_integer();
506 const bool unit_normal = lead_coeff.is_pos_integer();
507 if (canonicalizable && (! unit_normal))
508 icont = icont.mul(*_num_1_p);
510 if (canonicalizable && (icont != *_num1_p)) {
511 const add& addref = ex_to<add>(basis);
512 add & addp = dynallocate<add>(addref);
513 addp.clearflag(status_flags::hash_calculated);
514 addp.overall_coeff = ex_to<numeric>(addp.overall_coeff).div_dyn(icont);
515 for (auto & i : addp.seq)
516 i.coeff = ex_to<numeric>(i.coeff).div_dyn(icont);
518 const numeric c = icont.power(*num_exponent);
519 if (likely(c != *_num1_p))
520 return dynallocate<mul>(dynallocate<power>(addp, *num_exponent), c);
522 return dynallocate<power>(addp, *num_exponent);
526 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
527 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
528 if (is_exactly_a<mul>(basis)) {
529 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
530 const mul & mulref = ex_to<mul>(basis);
531 if (!mulref.overall_coeff.is_equal(_ex1)) {
532 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
533 if (num_coeff.is_real()) {
534 if (num_coeff.is_positive()) {
535 mul & mulp = dynallocate<mul>(mulref);
536 mulp.overall_coeff = _ex1;
537 mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated);
538 return dynallocate<mul>(dynallocate<power>(mulp, exponent),
539 dynallocate<power>(num_coeff, *num_exponent));
541 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
542 if (!num_coeff.is_equal(*_num_1_p)) {
543 mul & mulp = dynallocate<mul>(mulref);
544 mulp.overall_coeff = _ex_1;
545 mulp.clearflag(status_flags::evaluated | status_flags::hash_calculated);
546 return dynallocate<mul>(dynallocate<power>(mulp, exponent),
547 dynallocate<power>(abs(num_coeff), *num_exponent));
554 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
555 if (num_exponent->is_pos_integer() &&
556 basis.return_type() != return_types::commutative &&
557 !is_a<matrix>(basis)) {
558 return ncmul(exvector(num_exponent->to_int(), basis));
565 ex power::evalf(int level) const
572 eexponent = exponent;
573 } else if (level == -max_recursion_level) {
574 throw(std::runtime_error("max recursion level reached"));
576 ebasis = basis.evalf(level-1);
577 if (!is_exactly_a<numeric>(exponent))
578 eexponent = exponent.evalf(level-1);
580 eexponent = exponent;
583 return dynallocate<power>(ebasis, eexponent);
586 ex power::evalm() const
588 const ex ebasis = basis.evalm();
589 const ex eexponent = exponent.evalm();
590 if (is_a<matrix>(ebasis)) {
591 if (is_exactly_a<numeric>(eexponent)) {
592 return dynallocate<matrix>(ex_to<matrix>(ebasis).pow(eexponent));
595 return dynallocate<power>(ebasis, eexponent);
598 bool power::has(const ex & other, unsigned options) const
600 if (!(options & has_options::algebraic))
601 return basic::has(other, options);
602 if (!is_a<power>(other))
603 return basic::has(other, options);
604 if (!exponent.info(info_flags::integer) ||
605 !other.op(1).info(info_flags::integer))
606 return basic::has(other, options);
607 if (exponent.info(info_flags::posint) &&
608 other.op(1).info(info_flags::posint) &&
609 ex_to<numeric>(exponent) > ex_to<numeric>(other.op(1)) &&
610 basis.match(other.op(0)))
612 if (exponent.info(info_flags::negint) &&
613 other.op(1).info(info_flags::negint) &&
614 ex_to<numeric>(exponent) < ex_to<numeric>(other.op(1)) &&
615 basis.match(other.op(0)))
617 return basic::has(other, options);
621 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
623 ex power::subs(const exmap & m, unsigned options) const
625 const ex &subsed_basis = basis.subs(m, options);
626 const ex &subsed_exponent = exponent.subs(m, options);
628 if (!are_ex_trivially_equal(basis, subsed_basis)
629 || !are_ex_trivially_equal(exponent, subsed_exponent))
630 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
632 if (!(options & subs_options::algebraic))
633 return subs_one_level(m, options);
635 for (auto & it : m) {
636 int nummatches = std::numeric_limits<int>::max();
638 if (tryfactsubs(*this, it.first, nummatches, repls)) {
639 ex anum = it.second.subs(repls, subs_options::no_pattern);
640 ex aden = it.first.subs(repls, subs_options::no_pattern);
641 ex result = (*this) * pow(anum/aden, nummatches);
642 return (ex_to<basic>(result)).subs_one_level(m, options);
646 return subs_one_level(m, options);
649 ex power::eval_ncmul(const exvector & v) const
651 return inherited::eval_ncmul(v);
654 ex power::conjugate() const
656 // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
657 // branch cut which runs along the negative real axis.
658 if (basis.info(info_flags::positive)) {
659 ex newexponent = exponent.conjugate();
660 if (are_ex_trivially_equal(exponent, newexponent)) {
663 return dynallocate<power>(basis, newexponent);
665 if (exponent.info(info_flags::integer)) {
666 ex newbasis = basis.conjugate();
667 if (are_ex_trivially_equal(basis, newbasis)) {
670 return dynallocate<power>(newbasis, exponent);
672 return conjugate_function(*this).hold();
675 ex power::real_part() const
677 // basis == a+I*b, exponent == c+I*d
678 const ex a = basis.real_part();
679 const ex c = exponent.real_part();
680 if (basis.is_equal(a) && exponent.is_equal(c)) {
685 const ex b = basis.imag_part();
686 if (exponent.info(info_flags::integer)) {
687 // Re((a+I*b)^c) w/ c ∈ ℤ
688 long N = ex_to<numeric>(c).to_long();
689 // Use real terms in Binomial expansion to construct
690 // Re(expand(pow(a+I*b, N))).
691 long NN = N > 0 ? N : -N;
692 ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
694 for (long n = 0; n <= NN; n += 2) {
695 ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
697 result += term; // sign: I^n w/ n == 4*m
699 result -= term; // sign: I^n w/ n == 4*m+2
705 // Re((a+I*b)^(c+I*d))
706 const ex d = exponent.imag_part();
707 return pow(abs(basis),c) * exp(-d*atan2(b,a)) * cos(c*atan2(b,a)+d*log(abs(basis)));
710 ex power::imag_part() const
712 // basis == a+I*b, exponent == c+I*d
713 const ex a = basis.real_part();
714 const ex c = exponent.real_part();
715 if (basis.is_equal(a) && exponent.is_equal(c)) {
720 const ex b = basis.imag_part();
721 if (exponent.info(info_flags::integer)) {
722 // Im((a+I*b)^c) w/ c ∈ ℤ
723 long N = ex_to<numeric>(c).to_long();
724 // Use imaginary terms in Binomial expansion to construct
725 // Im(expand(pow(a+I*b, N))).
726 long p = N > 0 ? 1 : 3; // modulus for positive sign
727 long NN = N > 0 ? N : -N;
728 ex numer = N > 0 ? _ex1 : pow(pow(a,2) + pow(b,2), NN);
730 for (long n = 1; n <= NN; n += 2) {
731 ex term = binomial(NN, n) * pow(a, NN-n) * pow(b, n) / numer;
733 result += term; // sign: I^n w/ n == 4*m+p
735 result -= term; // sign: I^n w/ n == 4*m+2+p
741 // Im((a+I*b)^(c+I*d))
742 const ex d = exponent.imag_part();
743 return pow(abs(basis),c) * exp(-d*atan2(b,a)) * sin(c*atan2(b,a)+d*log(abs(basis)));
748 /** Implementation of ex::diff() for a power.
750 ex power::derivative(const symbol & s) const
752 if (is_a<numeric>(exponent)) {
753 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
754 const epvector newseq = {expair(basis, exponent - _ex1), expair(basis.diff(s), _ex1)};
755 return dynallocate<mul>(std::move(newseq), exponent);
757 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
758 return *this * (exponent.diff(s)*log(basis) + exponent*basis.diff(s)*pow(basis, _ex_1));
762 int power::compare_same_type(const basic & other) const
764 GINAC_ASSERT(is_exactly_a<power>(other));
765 const power &o = static_cast<const power &>(other);
767 int cmpval = basis.compare(o.basis);
771 return exponent.compare(o.exponent);
774 unsigned power::return_type() const
776 return basis.return_type();
779 return_type_t power::return_type_tinfo() const
781 return basis.return_type_tinfo();
784 ex power::expand(unsigned options) const
786 if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
787 // A special case worth optimizing.
788 setflag(status_flags::expanded);
792 // (x*p)^c -> x^c * p^c, if p>0
793 // makes sense before expanding the basis
794 if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
795 const mul &m = ex_to<mul>(basis);
798 prodseq.reserve(m.seq.size() + 1);
799 powseq.reserve(m.seq.size() + 1);
802 // search for positive/negative factors
803 for (auto & cit : m.seq) {
804 ex e=m.recombine_pair_to_ex(cit);
805 if (e.info(info_flags::positive))
806 prodseq.push_back(pow(e, exponent).expand(options));
807 else if (e.info(info_flags::negative)) {
808 prodseq.push_back(pow(-e, exponent).expand(options));
811 powseq.push_back(cit);
814 // take care on the numeric coefficient
815 ex coeff=(possign? _ex1 : _ex_1);
816 if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
817 prodseq.push_back(pow(m.overall_coeff, exponent));
818 else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
819 prodseq.push_back(pow(-m.overall_coeff, exponent));
821 coeff *= m.overall_coeff;
823 // If positive/negative factors are found, then extract them.
824 // In either case we set a flag to avoid the second run on a part
825 // which does not have positive/negative terms.
826 if (prodseq.size() > 0) {
827 ex newbasis = dynallocate<mul>(std::move(powseq), coeff);
828 ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
829 return dynallocate<mul>(std::move(prodseq)) * pow(newbasis, exponent);
831 ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
834 const ex expanded_basis = basis.expand(options);
835 const ex expanded_exponent = exponent.expand(options);
837 // x^(a+b) -> x^a * x^b
838 if (is_exactly_a<add>(expanded_exponent)) {
839 const add &a = ex_to<add>(expanded_exponent);
841 distrseq.reserve(a.seq.size() + 1);
842 for (auto & cit : a.seq) {
843 distrseq.push_back(pow(expanded_basis, a.recombine_pair_to_ex(cit)));
846 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
847 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
848 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
849 long int_exponent = num_exponent.to_int();
850 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
851 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
853 distrseq.push_back(pow(expanded_basis, a.overall_coeff));
855 distrseq.push_back(pow(expanded_basis, a.overall_coeff));
857 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
858 ex r = dynallocate<mul>(distrseq);
859 return r.expand(options);
862 if (!is_exactly_a<numeric>(expanded_exponent) ||
863 !ex_to<numeric>(expanded_exponent).is_integer()) {
864 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
867 return dynallocate<power>(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0);
871 // integer numeric exponent
872 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
873 long int_exponent = num_exponent.to_long();
876 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
877 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
879 // (x*y)^n -> x^n * y^n
880 if (is_exactly_a<mul>(expanded_basis))
881 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
883 // cannot expand further
884 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
887 return dynallocate<power>(expanded_basis, expanded_exponent).setflag(options == 0 ? status_flags::expanded : 0);
891 // new virtual functions which can be overridden by derived classes
897 // non-virtual functions in this class
900 namespace { // anonymous namespace for power::expand_add() helpers
902 /** Helper class to generate all bounded combinatorial partitions of an integer
903 * n with exactly m parts (including zero parts) in non-decreasing order.
905 class partition_generator {
907 // Partitions n into m parts, not including zero parts.
908 // (Cf. OEIS sequence A008284; implementation adapted from Jörg Arndt's
912 // partition: x[1] + x[2] + ... + x[m] = n and sentinel x[0] == 0
916 mpartition2(unsigned n_, unsigned m_)
917 : x(m_+1), n(n_), m(m_)
919 for (int k=1; k<m; ++k)
923 bool next_partition()
925 int u = x[m]; // last element
934 return false; // current is last
945 int m; // number of parts 0<m<=n
946 mutable std::vector<int> partition; // current partition
948 partition_generator(unsigned n_, unsigned m_)
949 : mpgen(n_, 1), m(m_), partition(m_)
951 // returns current partition in non-decreasing order, padded with zeros
952 const std::vector<int>& current() const
954 for (int i = 0; i < m - mpgen.m; ++i)
955 partition[i] = 0; // pad with zeros
957 for (int i = m - mpgen.m; i < m; ++i)
958 partition[i] = mpgen.x[i - m + mpgen.m + 1];
964 if (!mpgen.next_partition()) {
965 if (mpgen.m == m || mpgen.m == mpgen.n)
966 return false; // current is last
967 // increment number of parts
968 mpgen = mpartition2(mpgen.n, mpgen.m + 1);
974 /** Helper class to generate all compositions of a partition of an integer n,
975 * starting with the compositions which has non-decreasing order.
977 class composition_generator {
979 // Generates all distinct permutations of a multiset.
980 // (Based on Aaron Williams' algorithm 1 from "Loopless Generation of
981 // Multiset Permutations using a Constant Number of Variables by Prefix
982 // Shifts." <http://webhome.csc.uvic.ca/~haron/CoolMulti.pdf>)
984 // element of singly linked list
988 element(int val, element* n)
989 : value(val), next(n) {}
991 { // recurses down to the end of the singly linked list
995 element *head, *i, *after_i;
996 // NB: Partition must be sorted in non-decreasing order.
997 explicit coolmulti(const std::vector<int>& partition)
998 : head(nullptr), i(nullptr), after_i(nullptr)
1000 for (unsigned n = 0; n < partition.size(); ++n) {
1001 head = new element(partition[n], head);
1008 { // deletes singly linked list
1011 void next_permutation()
1014 if (after_i->next != nullptr && i->value >= after_i->next->value)
1018 element *k = before_k->next;
1019 before_k->next = k->next;
1021 if (k->value < head->value)
1026 bool finished() const
1028 return after_i->next == nullptr && after_i->value >= head->value;
1031 bool atend; // needed for simplifying iteration over permutations
1032 bool trivial; // likewise, true if all elements are equal
1033 mutable std::vector<int> composition; // current compositions
1035 explicit composition_generator(const std::vector<int>& partition)
1036 : cmgen(partition), atend(false), trivial(true), composition(partition.size())
1038 for (unsigned i=1; i<partition.size(); ++i)
1039 trivial = trivial && (partition[0] == partition[i]);
1041 const std::vector<int>& current() const
1043 coolmulti::element* it = cmgen.head;
1045 while (it != nullptr) {
1046 composition[i] = it->value;
1054 // This ugly contortion is needed because the original coolmulti
1055 // algorithm requires code duplication of the payload procedure,
1056 // one before the loop and one inside it.
1057 if (trivial || atend)
1059 cmgen.next_permutation();
1060 atend = cmgen.finished();
1065 /** Helper function to compute the multinomial coefficient n!/(p1!*p2!*...*pk!)
1066 * where n = p1+p2+...+pk, i.e. p is a partition of n.
1069 multinomial_coefficient(const std::vector<int> & p)
1071 numeric n = 0, d = 1;
1072 for (auto & it : p) {
1074 d *= factorial(numeric(it));
1076 return factorial(numeric(n)) / d;
1079 } // anonymous namespace
1082 /** expand a^n where a is an add and n is a positive integer.
1083 * @see power::expand */
1084 ex power::expand_add(const add & a, long n, unsigned options)
1086 // The special case power(+(x,...y;x),2) can be optimized better.
1088 return expand_add_2(a, options);
1092 // Consider base as the sum of all symbolic terms and the overall numeric
1093 // coefficient and apply the binomial theorem:
1094 // S = power(+(x,...,z;c),n)
1095 // = power(+(+(x,...,z;0);c),n)
1096 // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1097 // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
1098 // The multinomial theorem is computed by an outer loop over all
1099 // partitions of the exponent and an inner loop over all compositions of
1100 // that partition. This method makes the expansion a combinatorial
1101 // problem and allows us to directly construct the expanded sum and also
1102 // to re-use the multinomial coefficients (since they depend only on the
1103 // partition, not on the composition).
1105 // multinomial power(+(x,y,z;0),3) example:
1106 // partition : compositions : multinomial coefficient
1107 // [0,0,3] : [3,0,0],[0,3,0],[0,0,3] : 3!/(3!*0!*0!) = 1
1108 // [0,1,2] : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
1109 // [1,1,1] : [1,1,1] : 3!/(1!*1!*1!) = 6
1110 // => (x + y + z)^3 =
1112 // + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
1115 // multinomial power(+(x,y,z;0),4) example:
1116 // partition : compositions : multinomial coefficient
1117 // [0,0,4] : [4,0,0],[0,4,0],[0,0,4] : 4!/(4!*0!*0!) = 1
1118 // [0,1,3] : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
1119 // [0,2,2] : [2,2,0],[2,0,2],[0,2,2] : 4!/(2!*2!*0!) = 6
1120 // [1,1,2] : [2,1,1],[1,2,1],[1,1,2] : 4!/(2!*1!*1!) = 12
1121 // (no [1,1,1,1] partition since it has too many parts)
1122 // => (x + y + z)^4 =
1124 // + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
1125 // + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
1126 // + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
1130 // for k from 0 to n:
1131 // f = c^(n-k)*binomial(n,k)
1132 // for p in all partitions of n with m parts (including zero parts):
1133 // h = f * multinomial coefficient of p
1134 // for c in all compositions of p:
1136 // for e in all elements of c:
1142 // The number of terms will be the number of combinatorial compositions,
1143 // i.e. the number of unordered arrangements of m nonnegative integers
1144 // which sum up to n. It is frequently written as C_n(m) and directly
1145 // related with binomial coefficients: binomial(n+m-1,m-1).
1146 size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_long();
1147 if (!a.overall_coeff.is_zero()) {
1148 // the result's overall_coeff is one of the terms
1151 result.reserve(result_size);
1153 // Iterate over all terms in binomial expansion of
1154 // S = power(+(x,...,z;c),n)
1155 // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1156 for (int k = 1; k <= n; ++k) {
1157 numeric binomial_coefficient; // binomial(n,k)*c^(n-k)
1158 if (a.overall_coeff.is_zero()) {
1159 // degenerate case with zero overall_coeff:
1160 // apply multinomial theorem directly to power(+(x,...z;0),n)
1161 binomial_coefficient = 1;
1166 binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
1169 // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
1170 // Iterate over all partitions of k with exactly as many parts as
1171 // there are symbolic terms in the basis (including zero parts).
1172 partition_generator partitions(k, a.seq.size());
1174 const std::vector<int>& partition = partitions.current();
1175 // All monomials of this partition have the same number of terms and the same coefficient.
1176 const unsigned msize = std::count_if(partition.begin(), partition.end(), [](int i) { return i > 0; });
1177 const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
1179 // Iterate over all compositions of the current partition.
1180 composition_generator compositions(partition);
1182 const std::vector<int>& exponent = compositions.current();
1184 monomial.reserve(msize);
1185 numeric factor = coeff;
1186 for (unsigned i = 0; i < exponent.size(); ++i) {
1187 const ex & r = a.seq[i].rest;
1188 GINAC_ASSERT(!is_exactly_a<add>(r));
1189 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1190 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1191 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1192 !is_exactly_a<add>(ex_to<power>(r).basis) ||
1193 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1194 !is_exactly_a<power>(ex_to<power>(r).basis));
1195 GINAC_ASSERT(is_exactly_a<numeric>(a.seq[i].coeff));
1196 const numeric & c = ex_to<numeric>(a.seq[i].coeff);
1197 if (exponent[i] == 0) {
1199 } else if (exponent[i] == 1) {
1201 monomial.push_back(expair(r, _ex1));
1203 factor = factor.mul(c);
1204 } else { // general case exponent[i] > 1
1205 monomial.push_back(expair(r, exponent[i]));
1207 factor = factor.mul(c.power(exponent[i]));
1210 result.push_back(expair(mul(monomial).expand(options), factor));
1211 } while (compositions.next());
1212 } while (partitions.next());
1215 GINAC_ASSERT(result.size() == result_size);
1216 if (a.overall_coeff.is_zero()) {
1217 return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
1219 return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(n)).setflag(status_flags::expanded);
1224 /** Special case of power::expand_add. Expands a^2 where a is an add.
1225 * @see power::expand_add */
1226 ex power::expand_add_2(const add & a, unsigned options)
1229 size_t result_size = (a.nops() * (a.nops()+1)) / 2;
1230 if (!a.overall_coeff.is_zero()) {
1231 // the result's overall_coeff is one of the terms
1234 result.reserve(result_size);
1236 epvector::const_iterator last = a.seq.end();
1238 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
1239 // first part: ignore overall_coeff and expand other terms
1240 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
1241 const ex & r = cit0->rest;
1242 const ex & c = cit0->coeff;
1244 GINAC_ASSERT(!is_exactly_a<add>(r));
1245 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1246 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1247 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1248 !is_exactly_a<add>(ex_to<power>(r).basis) ||
1249 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1250 !is_exactly_a<power>(ex_to<power>(r).basis));
1252 if (c.is_equal(_ex1)) {
1253 if (is_exactly_a<mul>(r)) {
1254 result.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1257 result.push_back(expair(dynallocate<power>(r, _ex2),
1261 if (is_exactly_a<mul>(r)) {
1262 result.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1263 ex_to<numeric>(c).power_dyn(*_num2_p)));
1265 result.push_back(expair(dynallocate<power>(r, _ex2),
1266 ex_to<numeric>(c).power_dyn(*_num2_p)));
1270 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1271 const ex & r1 = cit1->rest;
1272 const ex & c1 = cit1->coeff;
1273 result.push_back(expair(mul(r,r1).expand(options),
1274 _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1278 // second part: add terms coming from overall_coeff (if != 0)
1279 if (!a.overall_coeff.is_zero()) {
1280 for (auto & i : a.seq)
1281 result.push_back(a.combine_pair_with_coeff_to_pair(i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1284 GINAC_ASSERT(result.size() == result_size);
1286 if (a.overall_coeff.is_zero()) {
1287 return dynallocate<add>(std::move(result)).setflag(status_flags::expanded);
1289 return dynallocate<add>(std::move(result), ex_to<numeric>(a.overall_coeff).power(2)).setflag(status_flags::expanded);
1293 /** Expand factors of m in m^n where m is a mul and n is an integer.
1294 * @see power::expand */
1295 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand)
1297 GINAC_ASSERT(n.is_integer());
1303 // do not bother to rename indices if there are no any.
1304 if (!(options & expand_options::expand_rename_idx) &&
1305 m.info(info_flags::has_indices))
1306 options |= expand_options::expand_rename_idx;
1307 // Leave it to multiplication since dummy indices have to be renamed
1308 if ((options & expand_options::expand_rename_idx) &&
1309 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1311 exvector va = get_all_dummy_indices(m);
1312 sort(va.begin(), va.end(), ex_is_less());
1314 for (int i=1; i < n.to_int(); i++)
1315 result *= rename_dummy_indices_uniquely(va, m);
1320 distrseq.reserve(m.seq.size());
1321 bool need_reexpand = false;
1323 for (auto & cit : m.seq) {
1324 expair p = m.combine_pair_with_coeff_to_pair(cit, n);
1325 if (from_expand && is_exactly_a<add>(cit.rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1326 // this happens when e.g. (a+b)^(1/2) gets squared and
1327 // the resulting product needs to be reexpanded
1328 need_reexpand = true;
1330 distrseq.push_back(p);
1333 const mul & result = dynallocate<mul>(std::move(distrseq), ex_to<numeric>(m.overall_coeff).power_dyn(n));
1335 return ex(result).expand(options);
1337 return result.setflag(status_flags::expanded);
1341 GINAC_BIND_UNARCHIVER(power);
1343 } // namespace GiNaC