3 * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
6 * GiNaC Copyright (C) 1999-2007 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
29 #include "expairseq.h"
35 #include "operators.h"
36 #include "inifcns.h" // for log() in power::derivative()
43 #include "relational.h"
47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
48 print_func<print_dflt>(&power::do_print_dflt).
49 print_func<print_latex>(&power::do_print_latex).
50 print_func<print_csrc>(&power::do_print_csrc).
51 print_func<print_python>(&power::do_print_python).
52 print_func<print_python_repr>(&power::do_print_python_repr))
54 typedef std::vector<int> intvector;
57 // default constructor
60 power::power() : inherited(&power::tinfo_static) { }
72 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
74 n.find_ex("basis", basis, sym_lst);
75 n.find_ex("exponent", exponent, sym_lst);
78 void power::archive(archive_node &n) const
80 inherited::archive(n);
81 n.add_ex("basis", basis);
82 n.add_ex("exponent", exponent);
85 DEFAULT_UNARCHIVE(power)
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(const print_csrc & c, unsigned level) const
166 // Integer powers of symbols are printed in a special, optimized way
167 if (exponent.info(info_flags::integer)
168 && (is_a<symbol>(basis) || is_a<constant>(basis))) {
169 int exp = ex_to<numeric>(exponent).to_int();
174 if (is_a<print_csrc_cl_N>(c))
179 print_sym_pow(c, ex_to<symbol>(basis), exp);
182 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
183 } else if (exponent.is_equal(_ex_1)) {
184 if (is_a<print_csrc_cl_N>(c))
191 // Otherwise, use the pow() or expt() (CLN) functions
193 if (is_a<print_csrc_cl_N>(c))
204 void power::do_print_python(const print_python & c, unsigned level) const
206 print_power(c, "**", "", "", level);
209 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
211 c.s << class_name() << '(';
218 bool power::info(unsigned inf) const
221 case info_flags::polynomial:
222 case info_flags::integer_polynomial:
223 case info_flags::cinteger_polynomial:
224 case info_flags::rational_polynomial:
225 case info_flags::crational_polynomial:
226 return exponent.info(info_flags::nonnegint) &&
228 case info_flags::rational_function:
229 return exponent.info(info_flags::integer) &&
231 case info_flags::algebraic:
232 return !exponent.info(info_flags::integer) ||
234 case info_flags::expanded:
235 return (flags & status_flags::expanded);
237 return inherited::info(inf);
240 size_t power::nops() const
245 ex power::op(size_t i) const
249 return i==0 ? basis : exponent;
252 ex power::map(map_function & f) const
254 const ex &mapped_basis = f(basis);
255 const ex &mapped_exponent = f(exponent);
257 if (!are_ex_trivially_equal(basis, mapped_basis)
258 || !are_ex_trivially_equal(exponent, mapped_exponent))
259 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
264 bool power::is_polynomial(const ex & var) const
266 if (exponent.has(var))
268 if (!exponent.info(info_flags::nonnegint))
270 return basis.is_polynomial(var);
273 int power::degree(const ex & s) const
275 if (is_equal(ex_to<basic>(s)))
277 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
278 if (basis.is_equal(s))
279 return ex_to<numeric>(exponent).to_int();
281 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
282 } else if (basis.has(s))
283 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
288 int power::ldegree(const ex & s) const
290 if (is_equal(ex_to<basic>(s)))
292 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
293 if (basis.is_equal(s))
294 return ex_to<numeric>(exponent).to_int();
296 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
297 } else if (basis.has(s))
298 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
303 ex power::coeff(const ex & s, int n) const
305 if (is_equal(ex_to<basic>(s)))
306 return n==1 ? _ex1 : _ex0;
307 else if (!basis.is_equal(s)) {
308 // basis not equal to s
315 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
317 int int_exp = ex_to<numeric>(exponent).to_int();
323 // non-integer exponents are treated as zero
332 /** Perform automatic term rewriting rules in this class. In the following
333 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
334 * stand for such expressions that contain a plain number.
335 * - ^(x,0) -> 1 (also handles ^(0,0))
337 * - ^(0,c) -> 0 or exception (depending on the real part of c)
339 * - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
340 * - ^(^(x,c1),c2) -> ^(x,c1*c2) if x is positive and c1 is real.
341 * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
342 * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
343 * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
344 * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
346 * @param level cut-off in recursive evaluation */
347 ex power::eval(int level) const
349 if ((level==1) && (flags & status_flags::evaluated))
351 else if (level == -max_recursion_level)
352 throw(std::runtime_error("max recursion level reached"));
354 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
355 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
357 bool basis_is_numerical = false;
358 bool exponent_is_numerical = false;
359 const numeric *num_basis;
360 const numeric *num_exponent;
362 if (is_exactly_a<numeric>(ebasis)) {
363 basis_is_numerical = true;
364 num_basis = &ex_to<numeric>(ebasis);
366 if (is_exactly_a<numeric>(eexponent)) {
367 exponent_is_numerical = true;
368 num_exponent = &ex_to<numeric>(eexponent);
371 // ^(x,0) -> 1 (0^0 also handled here)
372 if (eexponent.is_zero()) {
373 if (ebasis.is_zero())
374 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
380 if (eexponent.is_equal(_ex1))
383 // ^(0,c1) -> 0 or exception (depending on real value of c1)
384 if (ebasis.is_zero() && exponent_is_numerical) {
385 if ((num_exponent->real()).is_zero())
386 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
387 else if ((num_exponent->real()).is_negative())
388 throw (pole_error("power::eval(): division by zero",1));
394 if (ebasis.is_equal(_ex1))
397 // power of a function calculated by separate rules defined for this function
398 if (is_exactly_a<function>(ebasis))
399 return ex_to<function>(ebasis).power(eexponent);
401 // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
402 if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
403 return power(ebasis.op(0), ebasis.op(1) * eexponent);
405 if (exponent_is_numerical) {
407 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
408 // except if c1,c2 are rational, but c1^c2 is not)
409 if (basis_is_numerical) {
410 const bool basis_is_crational = num_basis->is_crational();
411 const bool exponent_is_crational = num_exponent->is_crational();
412 if (!basis_is_crational || !exponent_is_crational) {
413 // return a plain float
414 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
415 status_flags::evaluated |
416 status_flags::expanded);
419 const numeric res = num_basis->power(*num_exponent);
420 if (res.is_crational()) {
423 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
425 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
426 if (basis_is_crational && exponent_is_crational
427 && num_exponent->is_real()
428 && !num_exponent->is_integer()) {
429 const numeric n = num_exponent->numer();
430 const numeric m = num_exponent->denom();
432 numeric q = iquo(n, m, r);
433 if (r.is_negative()) {
437 if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
438 if (num_basis->is_rational() && !num_basis->is_integer()) {
439 // try it for numerator and denominator separately, in order to
440 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
441 const numeric bnum = num_basis->numer();
442 const numeric bden = num_basis->denom();
443 const numeric res_bnum = bnum.power(*num_exponent);
444 const numeric res_bden = bden.power(*num_exponent);
445 if (res_bnum.is_integer())
446 return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
447 if (res_bden.is_integer())
448 return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
452 // assemble resulting product, but allowing for a re-evaluation,
453 // because otherwise we'll end up with something like
454 // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
455 // instead of 7/16*7^(1/3).
456 ex prod = power(*num_basis,r.div(m));
457 return prod*power(*num_basis,q);
462 // ^(^(x,c1),c2) -> ^(x,c1*c2)
463 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
464 // case c1==1 should not happen, see below!)
465 if (is_exactly_a<power>(ebasis)) {
466 const power & sub_power = ex_to<power>(ebasis);
467 const ex & sub_basis = sub_power.basis;
468 const ex & sub_exponent = sub_power.exponent;
469 if (is_exactly_a<numeric>(sub_exponent)) {
470 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
471 GINAC_ASSERT(num_sub_exponent!=numeric(1));
472 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()) {
473 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
478 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
479 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
480 return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
483 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
484 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
485 const numeric icont = ebasis.integer_content();
486 const numeric& lead_coeff =
487 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div_dyn(icont);
489 const bool canonicalizable = lead_coeff.is_integer();
490 const bool unit_normal = lead_coeff.is_pos_integer();
492 if (icont != *_num1_p) {
493 return (new mul(power(ebasis/icont, *num_exponent), power(icont, *num_exponent))
494 )->setflag(status_flags::dynallocated);
497 if (canonicalizable && (! unit_normal)) {
498 if (num_exponent->is_even()) {
499 return power(-ebasis, *num_exponent);
501 return (new mul(power(-ebasis, *num_exponent), *_num_1_p)
502 )->setflag(status_flags::dynallocated);
507 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
508 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
509 if (is_exactly_a<mul>(ebasis)) {
510 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
511 const mul & mulref = ex_to<mul>(ebasis);
512 if (!mulref.overall_coeff.is_equal(_ex1)) {
513 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
514 if (num_coeff.is_real()) {
515 if (num_coeff.is_positive()) {
516 mul *mulp = new mul(mulref);
517 mulp->overall_coeff = _ex1;
518 mulp->clearflag(status_flags::evaluated);
519 mulp->clearflag(status_flags::hash_calculated);
520 return (new mul(power(*mulp,exponent),
521 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
523 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
524 if (!num_coeff.is_equal(*_num_1_p)) {
525 mul *mulp = new mul(mulref);
526 mulp->overall_coeff = _ex_1;
527 mulp->clearflag(status_flags::evaluated);
528 mulp->clearflag(status_flags::hash_calculated);
529 return (new mul(power(*mulp,exponent),
530 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
537 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
538 if (num_exponent->is_pos_integer() &&
539 ebasis.return_type() != return_types::commutative &&
540 !is_a<matrix>(ebasis)) {
541 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
545 if (are_ex_trivially_equal(ebasis,basis) &&
546 are_ex_trivially_equal(eexponent,exponent)) {
549 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
550 status_flags::evaluated);
553 ex power::evalf(int level) const
560 eexponent = exponent;
561 } else if (level == -max_recursion_level) {
562 throw(std::runtime_error("max recursion level reached"));
564 ebasis = basis.evalf(level-1);
565 if (!is_exactly_a<numeric>(exponent))
566 eexponent = exponent.evalf(level-1);
568 eexponent = exponent;
571 return power(ebasis,eexponent);
574 ex power::evalm() const
576 const ex ebasis = basis.evalm();
577 const ex eexponent = exponent.evalm();
578 if (is_a<matrix>(ebasis)) {
579 if (is_exactly_a<numeric>(eexponent)) {
580 return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
583 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
586 bool power::has(const ex & other, unsigned options) const
588 if (!(options & has_options::algebraic))
589 return basic::has(other, options);
590 if (!is_a<power>(other))
591 return basic::has(other, options);
592 if (!exponent.info(info_flags::integer)
593 || !other.op(1).info(info_flags::integer))
594 return basic::has(other, options);
595 if (exponent.info(info_flags::posint)
596 && other.op(1).info(info_flags::posint)
597 && ex_to<numeric>(exponent).to_int()
598 > ex_to<numeric>(other.op(1)).to_int()
599 && basis.match(other.op(0)))
601 if (exponent.info(info_flags::negint)
602 && other.op(1).info(info_flags::negint)
603 && ex_to<numeric>(exponent).to_int()
604 < ex_to<numeric>(other.op(1)).to_int()
605 && basis.match(other.op(0)))
607 return basic::has(other, options);
611 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
613 ex power::subs(const exmap & m, unsigned options) const
615 const ex &subsed_basis = basis.subs(m, options);
616 const ex &subsed_exponent = exponent.subs(m, options);
618 if (!are_ex_trivially_equal(basis, subsed_basis)
619 || !are_ex_trivially_equal(exponent, subsed_exponent))
620 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
622 if (!(options & subs_options::algebraic))
623 return subs_one_level(m, options);
625 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
626 int nummatches = std::numeric_limits<int>::max();
628 if (tryfactsubs(*this, it->first, nummatches, repls))
629 return (ex_to<basic>((*this) * power(it->second.subs(ex(repls), subs_options::no_pattern) / it->first.subs(ex(repls), subs_options::no_pattern), nummatches))).subs_one_level(m, options);
632 return subs_one_level(m, options);
635 ex power::eval_ncmul(const exvector & v) const
637 return inherited::eval_ncmul(v);
640 ex power::conjugate() const
642 ex newbasis = basis.conjugate();
643 ex newexponent = exponent.conjugate();
644 if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
647 return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
650 ex power::real_part() const
652 if (exponent.info(info_flags::integer)) {
653 ex basis_real = basis.real_part();
654 if (basis_real == basis)
656 realsymbol a("a"),b("b");
658 if (exponent.info(info_flags::posint))
659 result = power(a+I*b,exponent);
661 result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
662 result = result.expand();
663 result = result.real_part();
664 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
668 ex a = basis.real_part();
669 ex b = basis.imag_part();
670 ex c = exponent.real_part();
671 ex d = exponent.imag_part();
672 return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
675 ex power::imag_part() const
677 if (exponent.info(info_flags::integer)) {
678 ex basis_real = basis.real_part();
679 if (basis_real == basis)
681 realsymbol a("a"),b("b");
683 if (exponent.info(info_flags::posint))
684 result = power(a+I*b,exponent);
686 result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
687 result = result.expand();
688 result = result.imag_part();
689 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
693 ex a=basis.real_part();
694 ex b=basis.imag_part();
695 ex c=exponent.real_part();
696 ex d=exponent.imag_part();
698 power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
705 /** Implementation of ex::diff() for a power.
707 ex power::derivative(const symbol & s) const
709 if (is_a<numeric>(exponent)) {
710 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
713 newseq.push_back(expair(basis, exponent - _ex1));
714 newseq.push_back(expair(basis.diff(s), _ex1));
715 return mul(newseq, exponent);
717 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
719 add(mul(exponent.diff(s), log(basis)),
720 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
724 int power::compare_same_type(const basic & other) const
726 GINAC_ASSERT(is_exactly_a<power>(other));
727 const power &o = static_cast<const power &>(other);
729 int cmpval = basis.compare(o.basis);
733 return exponent.compare(o.exponent);
736 unsigned power::return_type() const
738 return basis.return_type();
741 tinfo_t power::return_type_tinfo() const
743 return basis.return_type_tinfo();
746 ex power::expand(unsigned options) const
748 if (options == 0 && (flags & status_flags::expanded))
751 const ex expanded_basis = basis.expand(options);
752 const ex expanded_exponent = exponent.expand(options);
754 // x^(a+b) -> x^a * x^b
755 if (is_exactly_a<add>(expanded_exponent)) {
756 const add &a = ex_to<add>(expanded_exponent);
758 distrseq.reserve(a.seq.size() + 1);
759 epvector::const_iterator last = a.seq.end();
760 epvector::const_iterator cit = a.seq.begin();
762 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
766 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
767 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
768 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
769 int int_exponent = num_exponent.to_int();
770 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
771 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
773 distrseq.push_back(power(expanded_basis, a.overall_coeff));
775 distrseq.push_back(power(expanded_basis, a.overall_coeff));
777 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
778 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
779 return r.expand(options);
782 if (!is_exactly_a<numeric>(expanded_exponent) ||
783 !ex_to<numeric>(expanded_exponent).is_integer()) {
784 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
787 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
791 // integer numeric exponent
792 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
793 int int_exponent = num_exponent.to_int();
796 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
797 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
799 // (x*y)^n -> x^n * y^n
800 if (is_exactly_a<mul>(expanded_basis))
801 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
803 // cannot expand further
804 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
807 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
811 // new virtual functions which can be overridden by derived classes
817 // non-virtual functions in this class
820 /** expand a^n where a is an add and n is a positive integer.
821 * @see power::expand */
822 ex power::expand_add(const add & a, int n, unsigned options) const
825 return expand_add_2(a, options);
827 const size_t m = a.nops();
829 // The number of terms will be the number of combinatorial compositions,
830 // i.e. the number of unordered arrangements of m nonnegative integers
831 // which sum up to n. It is frequently written as C_n(m) and directly
832 // related with binomial coefficients:
833 result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
835 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
836 intvector upper_limit(m-1);
839 for (size_t l=0; l<m-1; ++l) {
848 for (l=0; l<m-1; ++l) {
849 const ex & b = a.op(l);
850 GINAC_ASSERT(!is_exactly_a<add>(b));
851 GINAC_ASSERT(!is_exactly_a<power>(b) ||
852 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
853 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
854 !is_exactly_a<add>(ex_to<power>(b).basis) ||
855 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
856 !is_exactly_a<power>(ex_to<power>(b).basis));
857 if (is_exactly_a<mul>(b))
858 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
860 term.push_back(power(b,k[l]));
863 const ex & b = a.op(l);
864 GINAC_ASSERT(!is_exactly_a<add>(b));
865 GINAC_ASSERT(!is_exactly_a<power>(b) ||
866 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
867 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
868 !is_exactly_a<add>(ex_to<power>(b).basis) ||
869 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
870 !is_exactly_a<power>(ex_to<power>(b).basis));
871 if (is_exactly_a<mul>(b))
872 term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
874 term.push_back(power(b,n-k_cum[m-2]));
876 numeric f = binomial(numeric(n),numeric(k[0]));
877 for (l=1; l<m-1; ++l)
878 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
882 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
886 while ((l>=0) && ((++k[l])>upper_limit[l])) {
892 // recalc k_cum[] and upper_limit[]
893 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
895 for (size_t i=l+1; i<m-1; ++i)
896 k_cum[i] = k_cum[i-1]+k[i];
898 for (size_t i=l+1; i<m-1; ++i)
899 upper_limit[i] = n-k_cum[i-1];
902 return (new add(result))->setflag(status_flags::dynallocated |
903 status_flags::expanded);
907 /** Special case of power::expand_add. Expands a^2 where a is an add.
908 * @see power::expand_add */
909 ex power::expand_add_2(const add & a, unsigned options) const
912 size_t a_nops = a.nops();
913 sum.reserve((a_nops*(a_nops+1))/2);
914 epvector::const_iterator last = a.seq.end();
916 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
917 // first part: ignore overall_coeff and expand other terms
918 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
919 const ex & r = cit0->rest;
920 const ex & c = cit0->coeff;
922 GINAC_ASSERT(!is_exactly_a<add>(r));
923 GINAC_ASSERT(!is_exactly_a<power>(r) ||
924 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
925 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
926 !is_exactly_a<add>(ex_to<power>(r).basis) ||
927 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
928 !is_exactly_a<power>(ex_to<power>(r).basis));
930 if (c.is_equal(_ex1)) {
931 if (is_exactly_a<mul>(r)) {
932 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
935 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
939 if (is_exactly_a<mul>(r)) {
940 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
941 ex_to<numeric>(c).power_dyn(*_num2_p)));
943 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
944 ex_to<numeric>(c).power_dyn(*_num2_p)));
948 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
949 const ex & r1 = cit1->rest;
950 const ex & c1 = cit1->coeff;
951 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
952 _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
956 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
958 // second part: add terms coming from overall_factor (if != 0)
959 if (!a.overall_coeff.is_zero()) {
960 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
962 sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
965 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
968 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
970 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
973 /** Expand factors of m in m^n where m is a mul and n is an integer.
974 * @see power::expand */
975 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
977 GINAC_ASSERT(n.is_integer());
983 // Leave it to multiplication since dummy indices have to be renamed
984 if (get_all_dummy_indices(m).size() > 0 && n.is_positive()) {
986 exvector va = get_all_dummy_indices(m);
987 sort(va.begin(), va.end(), ex_is_less());
989 for (int i=1; i < n.to_int(); i++)
990 result *= rename_dummy_indices_uniquely(va, m);
995 distrseq.reserve(m.seq.size());
996 bool need_reexpand = false;
998 epvector::const_iterator last = m.seq.end();
999 epvector::const_iterator cit = m.seq.begin();
1001 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1002 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1003 // this happens when e.g. (a+b)^(1/2) gets squared and
1004 // the resulting product needs to be reexpanded
1005 need_reexpand = true;
1007 distrseq.push_back(p);
1011 const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1013 return ex(result).expand(options);
1015 return result.setflag(status_flags::expanded);
1019 } // namespace GiNaC