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) ||
235 return inherited::info(inf);
238 size_t power::nops() const
243 ex power::op(size_t i) const
247 return i==0 ? basis : exponent;
250 ex power::map(map_function & f) const
252 const ex &mapped_basis = f(basis);
253 const ex &mapped_exponent = f(exponent);
255 if (!are_ex_trivially_equal(basis, mapped_basis)
256 || !are_ex_trivially_equal(exponent, mapped_exponent))
257 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
262 bool power::is_polynomial(const ex & var) const
264 if (exponent.has(var))
266 if (!exponent.info(info_flags::nonnegint))
268 return basis.is_polynomial(var);
271 int power::degree(const ex & s) const
273 if (is_equal(ex_to<basic>(s)))
275 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
276 if (basis.is_equal(s))
277 return ex_to<numeric>(exponent).to_int();
279 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
280 } else if (basis.has(s))
281 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
286 int power::ldegree(const ex & s) const
288 if (is_equal(ex_to<basic>(s)))
290 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
291 if (basis.is_equal(s))
292 return ex_to<numeric>(exponent).to_int();
294 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
295 } else if (basis.has(s))
296 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
301 ex power::coeff(const ex & s, int n) const
303 if (is_equal(ex_to<basic>(s)))
304 return n==1 ? _ex1 : _ex0;
305 else if (!basis.is_equal(s)) {
306 // basis not equal to s
313 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
315 int int_exp = ex_to<numeric>(exponent).to_int();
321 // non-integer exponents are treated as zero
330 /** Perform automatic term rewriting rules in this class. In the following
331 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
332 * stand for such expressions that contain a plain number.
333 * - ^(x,0) -> 1 (also handles ^(0,0))
335 * - ^(0,c) -> 0 or exception (depending on the real part of c)
337 * - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
338 * - ^(^(x,c1),c2) -> ^(x,c1*c2) if x is positive and c1 is real.
339 * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
340 * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
341 * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
342 * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
344 * @param level cut-off in recursive evaluation */
345 ex power::eval(int level) const
347 if ((level==1) && (flags & status_flags::evaluated))
349 else if (level == -max_recursion_level)
350 throw(std::runtime_error("max recursion level reached"));
352 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
353 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
355 bool basis_is_numerical = false;
356 bool exponent_is_numerical = false;
357 const numeric *num_basis;
358 const numeric *num_exponent;
360 if (is_exactly_a<numeric>(ebasis)) {
361 basis_is_numerical = true;
362 num_basis = &ex_to<numeric>(ebasis);
364 if (is_exactly_a<numeric>(eexponent)) {
365 exponent_is_numerical = true;
366 num_exponent = &ex_to<numeric>(eexponent);
369 // ^(x,0) -> 1 (0^0 also handled here)
370 if (eexponent.is_zero()) {
371 if (ebasis.is_zero())
372 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
378 if (eexponent.is_equal(_ex1))
381 // ^(0,c1) -> 0 or exception (depending on real value of c1)
382 if (ebasis.is_zero() && exponent_is_numerical) {
383 if ((num_exponent->real()).is_zero())
384 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
385 else if ((num_exponent->real()).is_negative())
386 throw (pole_error("power::eval(): division by zero",1));
392 if (ebasis.is_equal(_ex1))
395 // power of a function calculated by separate rules defined for this function
396 if (is_exactly_a<function>(ebasis))
397 return ex_to<function>(ebasis).power(eexponent);
399 // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
400 if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
401 return power(ebasis.op(0), ebasis.op(1) * eexponent);
403 if (exponent_is_numerical) {
405 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
406 // except if c1,c2 are rational, but c1^c2 is not)
407 if (basis_is_numerical) {
408 const bool basis_is_crational = num_basis->is_crational();
409 const bool exponent_is_crational = num_exponent->is_crational();
410 if (!basis_is_crational || !exponent_is_crational) {
411 // return a plain float
412 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
413 status_flags::evaluated |
414 status_flags::expanded);
417 const numeric res = num_basis->power(*num_exponent);
418 if (res.is_crational()) {
421 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
423 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
424 if (basis_is_crational && exponent_is_crational
425 && num_exponent->is_real()
426 && !num_exponent->is_integer()) {
427 const numeric n = num_exponent->numer();
428 const numeric m = num_exponent->denom();
430 numeric q = iquo(n, m, r);
431 if (r.is_negative()) {
435 if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
436 if (num_basis->is_rational() && !num_basis->is_integer()) {
437 // try it for numerator and denominator separately, in order to
438 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
439 const numeric bnum = num_basis->numer();
440 const numeric bden = num_basis->denom();
441 const numeric res_bnum = bnum.power(*num_exponent);
442 const numeric res_bden = bden.power(*num_exponent);
443 if (res_bnum.is_integer())
444 return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
445 if (res_bden.is_integer())
446 return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
450 // assemble resulting product, but allowing for a re-evaluation,
451 // because otherwise we'll end up with something like
452 // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
453 // instead of 7/16*7^(1/3).
454 ex prod = power(*num_basis,r.div(m));
455 return prod*power(*num_basis,q);
460 // ^(^(x,c1),c2) -> ^(x,c1*c2)
461 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
462 // case c1==1 should not happen, see below!)
463 if (is_exactly_a<power>(ebasis)) {
464 const power & sub_power = ex_to<power>(ebasis);
465 const ex & sub_basis = sub_power.basis;
466 const ex & sub_exponent = sub_power.exponent;
467 if (is_exactly_a<numeric>(sub_exponent)) {
468 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
469 GINAC_ASSERT(num_sub_exponent!=numeric(1));
470 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()) {
471 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
476 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
477 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
478 return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
481 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
482 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
483 const numeric icont = ebasis.integer_content();
484 const numeric& lead_coeff =
485 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div_dyn(icont);
487 const bool canonicalizable = lead_coeff.is_integer();
488 const bool unit_normal = lead_coeff.is_pos_integer();
490 if (icont != *_num1_p) {
491 return (new mul(power(ebasis/icont, *num_exponent), power(icont, *num_exponent))
492 )->setflag(status_flags::dynallocated);
495 if (canonicalizable && (! unit_normal)) {
496 if (num_exponent->is_even()) {
497 return power(-ebasis, *num_exponent);
499 return (new mul(power(-ebasis, *num_exponent), *_num_1_p)
500 )->setflag(status_flags::dynallocated);
505 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
506 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
507 if (is_exactly_a<mul>(ebasis)) {
508 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
509 const mul & mulref = ex_to<mul>(ebasis);
510 if (!mulref.overall_coeff.is_equal(_ex1)) {
511 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
512 if (num_coeff.is_real()) {
513 if (num_coeff.is_positive()) {
514 mul *mulp = new mul(mulref);
515 mulp->overall_coeff = _ex1;
516 mulp->clearflag(status_flags::evaluated);
517 mulp->clearflag(status_flags::hash_calculated);
518 return (new mul(power(*mulp,exponent),
519 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
521 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
522 if (!num_coeff.is_equal(*_num_1_p)) {
523 mul *mulp = new mul(mulref);
524 mulp->overall_coeff = _ex_1;
525 mulp->clearflag(status_flags::evaluated);
526 mulp->clearflag(status_flags::hash_calculated);
527 return (new mul(power(*mulp,exponent),
528 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
535 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
536 if (num_exponent->is_pos_integer() &&
537 ebasis.return_type() != return_types::commutative &&
538 !is_a<matrix>(ebasis)) {
539 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
543 if (are_ex_trivially_equal(ebasis,basis) &&
544 are_ex_trivially_equal(eexponent,exponent)) {
547 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
548 status_flags::evaluated);
551 ex power::evalf(int level) const
558 eexponent = exponent;
559 } else if (level == -max_recursion_level) {
560 throw(std::runtime_error("max recursion level reached"));
562 ebasis = basis.evalf(level-1);
563 if (!is_exactly_a<numeric>(exponent))
564 eexponent = exponent.evalf(level-1);
566 eexponent = exponent;
569 return power(ebasis,eexponent);
572 ex power::evalm() const
574 const ex ebasis = basis.evalm();
575 const ex eexponent = exponent.evalm();
576 if (is_a<matrix>(ebasis)) {
577 if (is_exactly_a<numeric>(eexponent)) {
578 return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
581 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
584 bool power::has(const ex & other, unsigned options) const
586 if (!(options & has_options::algebraic))
587 return basic::has(other, options);
588 if (!is_a<power>(other))
589 return basic::has(other, options);
590 if (!exponent.info(info_flags::integer)
591 || !other.op(1).info(info_flags::integer))
592 return basic::has(other, options);
593 if (exponent.info(info_flags::posint)
594 && other.op(1).info(info_flags::posint)
595 && ex_to<numeric>(exponent).to_int()
596 > ex_to<numeric>(other.op(1)).to_int()
597 && basis.match(other.op(0)))
599 if (exponent.info(info_flags::negint)
600 && other.op(1).info(info_flags::negint)
601 && ex_to<numeric>(exponent).to_int()
602 < ex_to<numeric>(other.op(1)).to_int()
603 && basis.match(other.op(0)))
605 return basic::has(other, options);
609 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
611 ex power::subs(const exmap & m, unsigned options) const
613 const ex &subsed_basis = basis.subs(m, options);
614 const ex &subsed_exponent = exponent.subs(m, options);
616 if (!are_ex_trivially_equal(basis, subsed_basis)
617 || !are_ex_trivially_equal(exponent, subsed_exponent))
618 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
620 if (!(options & subs_options::algebraic))
621 return subs_one_level(m, options);
623 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
624 int nummatches = std::numeric_limits<int>::max();
626 if (tryfactsubs(*this, it->first, nummatches, repls))
627 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);
630 return subs_one_level(m, options);
633 ex power::eval_ncmul(const exvector & v) const
635 return inherited::eval_ncmul(v);
638 ex power::conjugate() const
640 ex newbasis = basis.conjugate();
641 ex newexponent = exponent.conjugate();
642 if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
645 return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
648 ex power::real_part() const
650 if (exponent.info(info_flags::integer)) {
651 ex basis_real = basis.real_part();
652 if (basis_real == basis)
654 realsymbol a("a"),b("b");
656 if (exponent.info(info_flags::posint))
657 result = power(a+I*b,exponent);
659 result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
660 result = result.expand();
661 result = result.real_part();
662 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
666 ex a = basis.real_part();
667 ex b = basis.imag_part();
668 ex c = exponent.real_part();
669 ex d = exponent.imag_part();
670 return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
673 ex power::imag_part() const
675 if (exponent.info(info_flags::integer)) {
676 ex basis_real = basis.real_part();
677 if (basis_real == basis)
679 realsymbol a("a"),b("b");
681 if (exponent.info(info_flags::posint))
682 result = power(a+I*b,exponent);
684 result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
685 result = result.expand();
686 result = result.imag_part();
687 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
691 ex a=basis.real_part();
692 ex b=basis.imag_part();
693 ex c=exponent.real_part();
694 ex d=exponent.imag_part();
696 power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
703 /** Implementation of ex::diff() for a power.
705 ex power::derivative(const symbol & s) const
707 if (is_a<numeric>(exponent)) {
708 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
711 newseq.push_back(expair(basis, exponent - _ex1));
712 newseq.push_back(expair(basis.diff(s), _ex1));
713 return mul(newseq, exponent);
715 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
717 add(mul(exponent.diff(s), log(basis)),
718 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
722 int power::compare_same_type(const basic & other) const
724 GINAC_ASSERT(is_exactly_a<power>(other));
725 const power &o = static_cast<const power &>(other);
727 int cmpval = basis.compare(o.basis);
731 return exponent.compare(o.exponent);
734 unsigned power::return_type() const
736 return basis.return_type();
739 tinfo_t power::return_type_tinfo() const
741 return basis.return_type_tinfo();
744 ex power::expand(unsigned options) const
746 if (options == 0 && (flags & status_flags::expanded))
749 const ex expanded_basis = basis.expand(options);
750 const ex expanded_exponent = exponent.expand(options);
752 // x^(a+b) -> x^a * x^b
753 if (is_exactly_a<add>(expanded_exponent)) {
754 const add &a = ex_to<add>(expanded_exponent);
756 distrseq.reserve(a.seq.size() + 1);
757 epvector::const_iterator last = a.seq.end();
758 epvector::const_iterator cit = a.seq.begin();
760 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
764 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
765 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
766 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
767 int int_exponent = num_exponent.to_int();
768 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
769 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
771 distrseq.push_back(power(expanded_basis, a.overall_coeff));
773 distrseq.push_back(power(expanded_basis, a.overall_coeff));
775 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
776 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
777 return r.expand(options);
780 if (!is_exactly_a<numeric>(expanded_exponent) ||
781 !ex_to<numeric>(expanded_exponent).is_integer()) {
782 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
785 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
789 // integer numeric exponent
790 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
791 int int_exponent = num_exponent.to_int();
794 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
795 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
797 // (x*y)^n -> x^n * y^n
798 if (is_exactly_a<mul>(expanded_basis))
799 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
801 // cannot expand further
802 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
805 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
809 // new virtual functions which can be overridden by derived classes
815 // non-virtual functions in this class
818 /** expand a^n where a is an add and n is a positive integer.
819 * @see power::expand */
820 ex power::expand_add(const add & a, int n, unsigned options) const
823 return expand_add_2(a, options);
825 const size_t m = a.nops();
827 // The number of terms will be the number of combinatorial compositions,
828 // i.e. the number of unordered arrangements of m nonnegative integers
829 // which sum up to n. It is frequently written as C_n(m) and directly
830 // related with binomial coefficients:
831 result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
833 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
834 intvector upper_limit(m-1);
837 for (size_t l=0; l<m-1; ++l) {
846 for (l=0; l<m-1; ++l) {
847 const ex & b = a.op(l);
848 GINAC_ASSERT(!is_exactly_a<add>(b));
849 GINAC_ASSERT(!is_exactly_a<power>(b) ||
850 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
851 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
852 !is_exactly_a<add>(ex_to<power>(b).basis) ||
853 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
854 !is_exactly_a<power>(ex_to<power>(b).basis));
855 if (is_exactly_a<mul>(b))
856 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
858 term.push_back(power(b,k[l]));
861 const ex & b = a.op(l);
862 GINAC_ASSERT(!is_exactly_a<add>(b));
863 GINAC_ASSERT(!is_exactly_a<power>(b) ||
864 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
865 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
866 !is_exactly_a<add>(ex_to<power>(b).basis) ||
867 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
868 !is_exactly_a<power>(ex_to<power>(b).basis));
869 if (is_exactly_a<mul>(b))
870 term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
872 term.push_back(power(b,n-k_cum[m-2]));
874 numeric f = binomial(numeric(n),numeric(k[0]));
875 for (l=1; l<m-1; ++l)
876 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
880 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
884 while ((l>=0) && ((++k[l])>upper_limit[l])) {
890 // recalc k_cum[] and upper_limit[]
891 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
893 for (size_t i=l+1; i<m-1; ++i)
894 k_cum[i] = k_cum[i-1]+k[i];
896 for (size_t i=l+1; i<m-1; ++i)
897 upper_limit[i] = n-k_cum[i-1];
900 return (new add(result))->setflag(status_flags::dynallocated |
901 status_flags::expanded);
905 /** Special case of power::expand_add. Expands a^2 where a is an add.
906 * @see power::expand_add */
907 ex power::expand_add_2(const add & a, unsigned options) const
910 size_t a_nops = a.nops();
911 sum.reserve((a_nops*(a_nops+1))/2);
912 epvector::const_iterator last = a.seq.end();
914 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
915 // first part: ignore overall_coeff and expand other terms
916 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
917 const ex & r = cit0->rest;
918 const ex & c = cit0->coeff;
920 GINAC_ASSERT(!is_exactly_a<add>(r));
921 GINAC_ASSERT(!is_exactly_a<power>(r) ||
922 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
923 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
924 !is_exactly_a<add>(ex_to<power>(r).basis) ||
925 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
926 !is_exactly_a<power>(ex_to<power>(r).basis));
928 if (c.is_equal(_ex1)) {
929 if (is_exactly_a<mul>(r)) {
930 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
933 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
937 if (is_exactly_a<mul>(r)) {
938 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
939 ex_to<numeric>(c).power_dyn(*_num2_p)));
941 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
942 ex_to<numeric>(c).power_dyn(*_num2_p)));
946 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
947 const ex & r1 = cit1->rest;
948 const ex & c1 = cit1->coeff;
949 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
950 _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
954 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
956 // second part: add terms coming from overall_factor (if != 0)
957 if (!a.overall_coeff.is_zero()) {
958 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
960 sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
963 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
966 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
968 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
971 /** Expand factors of m in m^n where m is a mul and n is an integer.
972 * @see power::expand */
973 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
975 GINAC_ASSERT(n.is_integer());
981 // Leave it to multiplication since dummy indices have to be renamed
982 if (get_all_dummy_indices(m).size() > 0 && n.is_positive()) {
984 exvector va = get_all_dummy_indices(m);
985 sort(va.begin(), va.end(), ex_is_less());
987 for (int i=1; i < n.to_int(); i++)
988 result *= rename_dummy_indices_uniquely(va, m);
993 distrseq.reserve(m.seq.size());
994 bool need_reexpand = false;
996 epvector::const_iterator last = m.seq.end();
997 epvector::const_iterator cit = m.seq.begin();
999 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1000 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1001 // this happens when e.g. (a+b)^(1/2) gets squared and
1002 // the resulting product needs to be reexpanded
1003 need_reexpand = true;
1005 distrseq.push_back(p);
1009 const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1011 return ex(result).expand(options);
1013 return result.setflag(status_flags::expanded);
1017 } // namespace GiNaC