3 * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
6 * GiNaC Copyright (C) 1999-2006 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()
46 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
47 print_func<print_dflt>(&power::do_print_dflt).
48 print_func<print_latex>(&power::do_print_latex).
49 print_func<print_csrc>(&power::do_print_csrc).
50 print_func<print_python>(&power::do_print_python).
51 print_func<print_python_repr>(&power::do_print_python_repr))
53 typedef std::vector<int> intvector;
56 // default constructor
59 power::power() : inherited(&power::tinfo_static) { }
71 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
73 n.find_ex("basis", basis, sym_lst);
74 n.find_ex("exponent", exponent, sym_lst);
77 void power::archive(archive_node &n) const
79 inherited::archive(n);
80 n.add_ex("basis", basis);
81 n.add_ex("exponent", exponent);
84 DEFAULT_UNARCHIVE(power)
87 // functions overriding virtual functions from base classes
92 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
94 // Ordinary output of powers using '^' or '**'
95 if (precedence() <= level)
96 c.s << openbrace << '(';
97 basis.print(c, precedence());
100 exponent.print(c, precedence());
102 if (precedence() <= level)
103 c.s << ')' << closebrace;
106 void power::do_print_dflt(const print_dflt & c, unsigned level) const
108 if (exponent.is_equal(_ex1_2)) {
110 // Square roots are printed in a special way
116 print_power(c, "^", "", "", level);
119 void power::do_print_latex(const print_latex & c, unsigned level) const
121 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
123 // Powers with negative numeric exponents are printed as fractions
125 power(basis, -exponent).eval().print(c);
128 } else if (exponent.is_equal(_ex1_2)) {
130 // Square roots are printed in a special way
136 print_power(c, "^", "{", "}", level);
139 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
141 // Optimal output of integer powers of symbols to aid compiler CSE.
142 // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
143 // to learn why such a parenthesation is really necessary.
146 } else if (exp == 2) {
150 } else if (exp & 1) {
153 print_sym_pow(c, x, exp-1);
156 print_sym_pow(c, x, exp >> 1);
158 print_sym_pow(c, x, exp >> 1);
163 void power::do_print_csrc(const print_csrc & c, unsigned level) const
165 // Integer powers of symbols are printed in a special, optimized way
166 if (exponent.info(info_flags::integer)
167 && (is_a<symbol>(basis) || is_a<constant>(basis))) {
168 int exp = ex_to<numeric>(exponent).to_int();
173 if (is_a<print_csrc_cl_N>(c))
178 print_sym_pow(c, ex_to<symbol>(basis), exp);
181 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
182 } else if (exponent.is_equal(_ex_1)) {
183 if (is_a<print_csrc_cl_N>(c))
190 // Otherwise, use the pow() or expt() (CLN) functions
192 if (is_a<print_csrc_cl_N>(c))
203 void power::do_print_python(const print_python & c, unsigned level) const
205 print_power(c, "**", "", "", level);
208 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
210 c.s << class_name() << '(';
217 bool power::info(unsigned inf) const
220 case info_flags::polynomial:
221 case info_flags::integer_polynomial:
222 case info_flags::cinteger_polynomial:
223 case info_flags::rational_polynomial:
224 case info_flags::crational_polynomial:
225 return exponent.info(info_flags::nonnegint) &&
227 case info_flags::rational_function:
228 return exponent.info(info_flags::integer) &&
230 case info_flags::algebraic:
231 return !exponent.info(info_flags::integer) ||
234 return inherited::info(inf);
237 size_t power::nops() const
242 ex power::op(size_t i) const
246 return i==0 ? basis : exponent;
249 ex power::map(map_function & f) const
251 const ex &mapped_basis = f(basis);
252 const ex &mapped_exponent = f(exponent);
254 if (!are_ex_trivially_equal(basis, mapped_basis)
255 || !are_ex_trivially_equal(exponent, mapped_exponent))
256 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
261 bool power::is_polynomial(const ex & var) const
263 if (exponent.has(var))
265 if (!exponent.info(info_flags::nonnegint))
267 return basis.is_polynomial(var);
270 int power::degree(const ex & s) const
272 if (is_equal(ex_to<basic>(s)))
274 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
275 if (basis.is_equal(s))
276 return ex_to<numeric>(exponent).to_int();
278 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
279 } else if (basis.has(s))
280 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
285 int power::ldegree(const ex & s) const
287 if (is_equal(ex_to<basic>(s)))
289 else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
290 if (basis.is_equal(s))
291 return ex_to<numeric>(exponent).to_int();
293 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
294 } else if (basis.has(s))
295 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
300 ex power::coeff(const ex & s, int n) const
302 if (is_equal(ex_to<basic>(s)))
303 return n==1 ? _ex1 : _ex0;
304 else if (!basis.is_equal(s)) {
305 // basis not equal to s
312 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
314 int int_exp = ex_to<numeric>(exponent).to_int();
320 // non-integer exponents are treated as zero
329 /** Perform automatic term rewriting rules in this class. In the following
330 * x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
331 * stand for such expressions that contain a plain number.
332 * - ^(x,0) -> 1 (also handles ^(0,0))
334 * - ^(0,c) -> 0 or exception (depending on the real part of c)
336 * - ^(c1,c2) -> *(c1^n,c1^(c2-n)) (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
337 * - ^(^(x,c1),c2) -> ^(x,c1*c2) (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
338 * - ^(*(x,y,z),c) -> *(x^c,y^c,z^c) (if c integer)
339 * - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1>0)
340 * - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1<0)
342 * @param level cut-off in recursive evaluation */
343 ex power::eval(int level) const
345 if ((level==1) && (flags & status_flags::evaluated))
347 else if (level == -max_recursion_level)
348 throw(std::runtime_error("max recursion level reached"));
350 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
351 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
353 bool basis_is_numerical = false;
354 bool exponent_is_numerical = false;
355 const numeric *num_basis;
356 const numeric *num_exponent;
358 if (is_exactly_a<numeric>(ebasis)) {
359 basis_is_numerical = true;
360 num_basis = &ex_to<numeric>(ebasis);
362 if (is_exactly_a<numeric>(eexponent)) {
363 exponent_is_numerical = true;
364 num_exponent = &ex_to<numeric>(eexponent);
367 // ^(x,0) -> 1 (0^0 also handled here)
368 if (eexponent.is_zero()) {
369 if (ebasis.is_zero())
370 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
376 if (eexponent.is_equal(_ex1))
379 // ^(0,c1) -> 0 or exception (depending on real value of c1)
380 if (ebasis.is_zero() && exponent_is_numerical) {
381 if ((num_exponent->real()).is_zero())
382 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
383 else if ((num_exponent->real()).is_negative())
384 throw (pole_error("power::eval(): division by zero",1));
390 if (ebasis.is_equal(_ex1))
393 // power of a function calculated by separate rules defined for this function
394 if (is_exactly_a<function>(ebasis))
395 return ex_to<function>(ebasis).power(eexponent);
397 if (exponent_is_numerical) {
399 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
400 // except if c1,c2 are rational, but c1^c2 is not)
401 if (basis_is_numerical) {
402 const bool basis_is_crational = num_basis->is_crational();
403 const bool exponent_is_crational = num_exponent->is_crational();
404 if (!basis_is_crational || !exponent_is_crational) {
405 // return a plain float
406 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
407 status_flags::evaluated |
408 status_flags::expanded);
411 const numeric res = num_basis->power(*num_exponent);
412 if (res.is_crational()) {
415 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
417 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
418 if (basis_is_crational && exponent_is_crational
419 && num_exponent->is_real()
420 && !num_exponent->is_integer()) {
421 const numeric n = num_exponent->numer();
422 const numeric m = num_exponent->denom();
424 numeric q = iquo(n, m, r);
425 if (r.is_negative()) {
429 if (q.is_zero()) { // the exponent was in the allowed range 0<(n/m)<1
430 if (num_basis->is_rational() && !num_basis->is_integer()) {
431 // try it for numerator and denominator separately, in order to
432 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
433 const numeric bnum = num_basis->numer();
434 const numeric bden = num_basis->denom();
435 const numeric res_bnum = bnum.power(*num_exponent);
436 const numeric res_bden = bden.power(*num_exponent);
437 if (res_bnum.is_integer())
438 return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
439 if (res_bden.is_integer())
440 return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
444 // assemble resulting product, but allowing for a re-evaluation,
445 // because otherwise we'll end up with something like
446 // (7/8)^(4/3) -> 7/8*(1/2*7^(1/3))
447 // instead of 7/16*7^(1/3).
448 ex prod = power(*num_basis,r.div(m));
449 return prod*power(*num_basis,q);
454 // ^(^(x,c1),c2) -> ^(x,c1*c2)
455 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
456 // case c1==1 should not happen, see below!)
457 if (is_exactly_a<power>(ebasis)) {
458 const power & sub_power = ex_to<power>(ebasis);
459 const ex & sub_basis = sub_power.basis;
460 const ex & sub_exponent = sub_power.exponent;
461 if (is_exactly_a<numeric>(sub_exponent)) {
462 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
463 GINAC_ASSERT(num_sub_exponent!=numeric(1));
464 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative())
465 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
469 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
470 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
471 return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
474 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2) (c1, c2 numeric(), c1>0)
475 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2) (c1, c2 numeric(), c1<0)
476 if (is_exactly_a<mul>(ebasis)) {
477 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
478 const mul & mulref = ex_to<mul>(ebasis);
479 if (!mulref.overall_coeff.is_equal(_ex1)) {
480 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
481 if (num_coeff.is_real()) {
482 if (num_coeff.is_positive()) {
483 mul *mulp = new mul(mulref);
484 mulp->overall_coeff = _ex1;
485 mulp->clearflag(status_flags::evaluated);
486 mulp->clearflag(status_flags::hash_calculated);
487 return (new mul(power(*mulp,exponent),
488 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
490 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
491 if (!num_coeff.is_equal(*_num_1_p)) {
492 mul *mulp = new mul(mulref);
493 mulp->overall_coeff = _ex_1;
494 mulp->clearflag(status_flags::evaluated);
495 mulp->clearflag(status_flags::hash_calculated);
496 return (new mul(power(*mulp,exponent),
497 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
504 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
505 if (num_exponent->is_pos_integer() &&
506 ebasis.return_type() != return_types::commutative &&
507 !is_a<matrix>(ebasis)) {
508 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
512 if (are_ex_trivially_equal(ebasis,basis) &&
513 are_ex_trivially_equal(eexponent,exponent)) {
516 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
517 status_flags::evaluated);
520 ex power::evalf(int level) const
527 eexponent = exponent;
528 } else if (level == -max_recursion_level) {
529 throw(std::runtime_error("max recursion level reached"));
531 ebasis = basis.evalf(level-1);
532 if (!is_exactly_a<numeric>(exponent))
533 eexponent = exponent.evalf(level-1);
535 eexponent = exponent;
538 return power(ebasis,eexponent);
541 ex power::evalm() const
543 const ex ebasis = basis.evalm();
544 const ex eexponent = exponent.evalm();
545 if (is_a<matrix>(ebasis)) {
546 if (is_exactly_a<numeric>(eexponent)) {
547 return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
550 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
553 bool power::has(const ex & other, unsigned options) const
555 if (!(options & has_options::algebraic))
556 return basic::has(other, options);
557 if (!is_a<power>(other))
558 return basic::has(other, options);
559 if (!exponent.info(info_flags::integer)
560 || !other.op(1).info(info_flags::integer))
561 return basic::has(other, options);
562 if (exponent.info(info_flags::posint)
563 && other.op(1).info(info_flags::posint)
564 && ex_to<numeric>(exponent).to_int()
565 > ex_to<numeric>(other.op(1)).to_int()
566 && basis.match(other.op(0)))
568 if (exponent.info(info_flags::negint)
569 && other.op(1).info(info_flags::negint)
570 && ex_to<numeric>(exponent).to_int()
571 < ex_to<numeric>(other.op(1)).to_int()
572 && basis.match(other.op(0)))
574 return basic::has(other, options);
578 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
580 ex power::subs(const exmap & m, unsigned options) const
582 const ex &subsed_basis = basis.subs(m, options);
583 const ex &subsed_exponent = exponent.subs(m, options);
585 if (!are_ex_trivially_equal(basis, subsed_basis)
586 || !are_ex_trivially_equal(exponent, subsed_exponent))
587 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
589 if (!(options & subs_options::algebraic))
590 return subs_one_level(m, options);
592 for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
593 int nummatches = std::numeric_limits<int>::max();
595 if (tryfactsubs(*this, it->first, nummatches, repls))
596 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);
599 return subs_one_level(m, options);
602 ex power::eval_ncmul(const exvector & v) const
604 return inherited::eval_ncmul(v);
607 ex power::conjugate() const
609 ex newbasis = basis.conjugate();
610 ex newexponent = exponent.conjugate();
611 if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
614 return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
619 /** Implementation of ex::diff() for a power.
621 ex power::derivative(const symbol & s) const
623 if (is_a<numeric>(exponent)) {
624 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
627 newseq.push_back(expair(basis, exponent - _ex1));
628 newseq.push_back(expair(basis.diff(s), _ex1));
629 return mul(newseq, exponent);
631 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
633 add(mul(exponent.diff(s), log(basis)),
634 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
638 int power::compare_same_type(const basic & other) const
640 GINAC_ASSERT(is_exactly_a<power>(other));
641 const power &o = static_cast<const power &>(other);
643 int cmpval = basis.compare(o.basis);
647 return exponent.compare(o.exponent);
650 unsigned power::return_type() const
652 return basis.return_type();
655 tinfo_t power::return_type_tinfo() const
657 return basis.return_type_tinfo();
660 ex power::expand(unsigned options) const
662 if (options == 0 && (flags & status_flags::expanded))
665 const ex expanded_basis = basis.expand(options);
666 const ex expanded_exponent = exponent.expand(options);
668 // x^(a+b) -> x^a * x^b
669 if (is_exactly_a<add>(expanded_exponent)) {
670 const add &a = ex_to<add>(expanded_exponent);
672 distrseq.reserve(a.seq.size() + 1);
673 epvector::const_iterator last = a.seq.end();
674 epvector::const_iterator cit = a.seq.begin();
676 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
680 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
681 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
682 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
683 int int_exponent = num_exponent.to_int();
684 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
685 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
687 distrseq.push_back(power(expanded_basis, a.overall_coeff));
689 distrseq.push_back(power(expanded_basis, a.overall_coeff));
691 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
692 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
693 return r.expand(options);
696 if (!is_exactly_a<numeric>(expanded_exponent) ||
697 !ex_to<numeric>(expanded_exponent).is_integer()) {
698 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
701 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
705 // integer numeric exponent
706 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
707 int int_exponent = num_exponent.to_int();
710 if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
711 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
713 // (x*y)^n -> x^n * y^n
714 if (is_exactly_a<mul>(expanded_basis))
715 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
717 // cannot expand further
718 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
721 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
725 // new virtual functions which can be overridden by derived classes
731 // non-virtual functions in this class
734 /** expand a^n where a is an add and n is a positive integer.
735 * @see power::expand */
736 ex power::expand_add(const add & a, int n, unsigned options) const
739 return expand_add_2(a, options);
741 const size_t m = a.nops();
743 // The number of terms will be the number of combinatorial compositions,
744 // i.e. the number of unordered arrangements of m nonnegative integers
745 // which sum up to n. It is frequently written as C_n(m) and directly
746 // related with binomial coefficients:
747 result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
749 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
750 intvector upper_limit(m-1);
753 for (size_t l=0; l<m-1; ++l) {
762 for (l=0; l<m-1; ++l) {
763 const ex & b = a.op(l);
764 GINAC_ASSERT(!is_exactly_a<add>(b));
765 GINAC_ASSERT(!is_exactly_a<power>(b) ||
766 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
767 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
768 !is_exactly_a<add>(ex_to<power>(b).basis) ||
769 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
770 !is_exactly_a<power>(ex_to<power>(b).basis));
771 if (is_exactly_a<mul>(b))
772 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
774 term.push_back(power(b,k[l]));
777 const ex & b = a.op(l);
778 GINAC_ASSERT(!is_exactly_a<add>(b));
779 GINAC_ASSERT(!is_exactly_a<power>(b) ||
780 !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
781 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
782 !is_exactly_a<add>(ex_to<power>(b).basis) ||
783 !is_exactly_a<mul>(ex_to<power>(b).basis) ||
784 !is_exactly_a<power>(ex_to<power>(b).basis));
785 if (is_exactly_a<mul>(b))
786 term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
788 term.push_back(power(b,n-k_cum[m-2]));
790 numeric f = binomial(numeric(n),numeric(k[0]));
791 for (l=1; l<m-1; ++l)
792 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
796 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
800 while ((l>=0) && ((++k[l])>upper_limit[l])) {
806 // recalc k_cum[] and upper_limit[]
807 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
809 for (size_t i=l+1; i<m-1; ++i)
810 k_cum[i] = k_cum[i-1]+k[i];
812 for (size_t i=l+1; i<m-1; ++i)
813 upper_limit[i] = n-k_cum[i-1];
816 return (new add(result))->setflag(status_flags::dynallocated |
817 status_flags::expanded);
821 /** Special case of power::expand_add. Expands a^2 where a is an add.
822 * @see power::expand_add */
823 ex power::expand_add_2(const add & a, unsigned options) const
826 size_t a_nops = a.nops();
827 sum.reserve((a_nops*(a_nops+1))/2);
828 epvector::const_iterator last = a.seq.end();
830 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
831 // first part: ignore overall_coeff and expand other terms
832 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
833 const ex & r = cit0->rest;
834 const ex & c = cit0->coeff;
836 GINAC_ASSERT(!is_exactly_a<add>(r));
837 GINAC_ASSERT(!is_exactly_a<power>(r) ||
838 !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
839 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
840 !is_exactly_a<add>(ex_to<power>(r).basis) ||
841 !is_exactly_a<mul>(ex_to<power>(r).basis) ||
842 !is_exactly_a<power>(ex_to<power>(r).basis));
844 if (c.is_equal(_ex1)) {
845 if (is_exactly_a<mul>(r)) {
846 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
849 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
853 if (is_exactly_a<mul>(r)) {
854 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
855 ex_to<numeric>(c).power_dyn(*_num2_p)));
857 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
858 ex_to<numeric>(c).power_dyn(*_num2_p)));
862 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
863 const ex & r1 = cit1->rest;
864 const ex & c1 = cit1->coeff;
865 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
866 _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
870 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
872 // second part: add terms coming from overall_factor (if != 0)
873 if (!a.overall_coeff.is_zero()) {
874 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
876 sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
879 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
882 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
884 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
887 /** Expand factors of m in m^n where m is a mul and n is and integer.
888 * @see power::expand */
889 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
891 GINAC_ASSERT(n.is_integer());
897 // Leave it to multiplication since dummy indices have to be renamed
898 if (get_all_dummy_indices(m).size() > 0 && n.is_positive()) {
900 exvector va = get_all_dummy_indices(m);
901 sort(va.begin(), va.end(), ex_is_less());
903 for (int i=1; i < n.to_int(); i++)
904 result *= rename_dummy_indices_uniquely(va, m);
909 distrseq.reserve(m.seq.size());
910 bool need_reexpand = false;
912 epvector::const_iterator last = m.seq.end();
913 epvector::const_iterator cit = m.seq.begin();
915 if (is_exactly_a<numeric>(cit->rest)) {
916 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
918 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
919 // since n is an integer
920 numeric new_coeff = ex_to<numeric>(cit->coeff).mul(n);
921 if (from_expand && is_exactly_a<add>(cit->rest) && new_coeff.is_pos_integer()) {
922 // this happens when e.g. (a+b)^(1/2) gets squared and
923 // the resulting product needs to be reexpanded
924 need_reexpand = true;
926 distrseq.push_back(expair(cit->rest, new_coeff));
931 const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
933 return ex(result).expand(options);
935 return result.setflag(status_flags::expanded);