Replace use of NULL by C++11 nullptr.
[ginac.git] / ginac / power.cpp
1 /** @file power.cpp
2  *
3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2015 Johannes Gutenberg University Mainz, Germany
7  *
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.
12  *
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.
17  *
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
21  */
22
23 #include "power.h"
24 #include "expairseq.h"
25 #include "add.h"
26 #include "mul.h"
27 #include "ncmul.h"
28 #include "numeric.h"
29 #include "constant.h"
30 #include "operators.h"
31 #include "inifcns.h" // for log() in power::derivative()
32 #include "matrix.h"
33 #include "indexed.h"
34 #include "symbol.h"
35 #include "lst.h"
36 #include "archive.h"
37 #include "utils.h"
38 #include "relational.h"
39 #include "compiler.h"
40
41 #include <iostream>
42 #include <limits>
43 #include <stdexcept>
44 #include <vector>
45
46 namespace GiNaC {
47
48 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
49   print_func<print_dflt>(&power::do_print_dflt).
50   print_func<print_latex>(&power::do_print_latex).
51   print_func<print_csrc>(&power::do_print_csrc).
52   print_func<print_python>(&power::do_print_python).
53   print_func<print_python_repr>(&power::do_print_python_repr).
54   print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))
55
56 //////////
57 // default constructor
58 //////////
59
60 power::power() { }
61
62 //////////
63 // other constructors
64 //////////
65
66 // all inlined
67
68 //////////
69 // archiving
70 //////////
71
72 void power::read_archive(const archive_node &n, lst &sym_lst)
73 {
74         inherited::read_archive(n, sym_lst);
75         n.find_ex("basis", basis, sym_lst);
76         n.find_ex("exponent", exponent, sym_lst);
77 }
78
79 void power::archive(archive_node &n) const
80 {
81         inherited::archive(n);
82         n.add_ex("basis", basis);
83         n.add_ex("exponent", exponent);
84 }
85
86 //////////
87 // functions overriding virtual functions from base classes
88 //////////
89
90 // public
91
92 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
93 {
94         // Ordinary output of powers using '^' or '**'
95         if (precedence() <= level)
96                 c.s << openbrace << '(';
97         basis.print(c, precedence());
98         c.s << powersymbol;
99         c.s << openbrace;
100         exponent.print(c, precedence());
101         c.s << closebrace;
102         if (precedence() <= level)
103                 c.s << ')' << closebrace;
104 }
105
106 void power::do_print_dflt(const print_dflt & c, unsigned level) const
107 {
108         if (exponent.is_equal(_ex1_2)) {
109
110                 // Square roots are printed in a special way
111                 c.s << "sqrt(";
112                 basis.print(c);
113                 c.s << ')';
114
115         } else
116                 print_power(c, "^", "", "", level);
117 }
118
119 void power::do_print_latex(const print_latex & c, unsigned level) const
120 {
121         if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
122
123                 // Powers with negative numeric exponents are printed as fractions
124                 c.s << "\\frac{1}{";
125                 power(basis, -exponent).eval().print(c);
126                 c.s << '}';
127
128         } else if (exponent.is_equal(_ex1_2)) {
129
130                 // Square roots are printed in a special way
131                 c.s << "\\sqrt{";
132                 basis.print(c);
133                 c.s << '}';
134
135         } else
136                 print_power(c, "^", "{", "}", level);
137 }
138
139 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
140 {
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.
144         if (exp == 1) {
145                 x.print(c);
146         } else if (exp == 2) {
147                 x.print(c);
148                 c.s << "*";
149                 x.print(c);
150         } else if (exp & 1) {
151                 x.print(c);
152                 c.s << "*";
153                 print_sym_pow(c, x, exp-1);
154         } else {
155                 c.s << "(";
156                 print_sym_pow(c, x, exp >> 1);
157                 c.s << ")*(";
158                 print_sym_pow(c, x, exp >> 1);
159                 c.s << ")";
160         }
161 }
162
163 void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const
164 {
165         if (exponent.is_equal(_ex_1)) {
166                 c.s << "recip(";
167                 basis.print(c);
168                 c.s << ')';
169                 return;
170         }
171         c.s << "expt(";
172         basis.print(c);
173         c.s << ", ";
174         exponent.print(c);
175         c.s << ')';
176 }
177
178 void power::do_print_csrc(const print_csrc & c, unsigned level) const
179 {
180         // Integer powers of symbols are printed in a special, optimized way
181         if (exponent.info(info_flags::integer) &&
182             (is_a<symbol>(basis) || is_a<constant>(basis))) {
183                 int exp = ex_to<numeric>(exponent).to_int();
184                 if (exp > 0)
185                         c.s << '(';
186                 else {
187                         exp = -exp;
188                         c.s << "1.0/(";
189                 }
190                 print_sym_pow(c, ex_to<symbol>(basis), exp);
191                 c.s << ')';
192
193         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
194         } else if (exponent.is_equal(_ex_1)) {
195                 c.s << "1.0/(";
196                 basis.print(c);
197                 c.s << ')';
198
199         // Otherwise, use the pow() function
200         } else {
201                 c.s << "pow(";
202                 basis.print(c);
203                 c.s << ',';
204                 exponent.print(c);
205                 c.s << ')';
206         }
207 }
208
209 void power::do_print_python(const print_python & c, unsigned level) const
210 {
211         print_power(c, "**", "", "", level);
212 }
213
214 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
215 {
216         c.s << class_name() << '(';
217         basis.print(c);
218         c.s << ',';
219         exponent.print(c);
220         c.s << ')';
221 }
222
223 bool power::info(unsigned inf) const
224 {
225         switch (inf) {
226                 case info_flags::polynomial:
227                 case info_flags::integer_polynomial:
228                 case info_flags::cinteger_polynomial:
229                 case info_flags::rational_polynomial:
230                 case info_flags::crational_polynomial:
231                         return exponent.info(info_flags::nonnegint) &&
232                                basis.info(inf);
233                 case info_flags::rational_function:
234                         return exponent.info(info_flags::integer) &&
235                                basis.info(inf);
236                 case info_flags::algebraic:
237                         return !exponent.info(info_flags::integer) ||
238                                basis.info(inf);
239                 case info_flags::expanded:
240                         return (flags & status_flags::expanded);
241                 case info_flags::positive:
242                         return basis.info(info_flags::positive) && exponent.info(info_flags::real);
243                 case info_flags::nonnegative:
244                         return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
245                                (basis.info(info_flags::real) && exponent.info(info_flags::integer) && exponent.info(info_flags::even));
246                 case info_flags::has_indices: {
247                         if (flags & status_flags::has_indices)
248                                 return true;
249                         else if (flags & status_flags::has_no_indices)
250                                 return false;
251                         else if (basis.info(info_flags::has_indices)) {
252                                 setflag(status_flags::has_indices);
253                                 clearflag(status_flags::has_no_indices);
254                                 return true;
255                         } else {
256                                 clearflag(status_flags::has_indices);
257                                 setflag(status_flags::has_no_indices);
258                                 return false;
259                         }
260                 }
261         }
262         return inherited::info(inf);
263 }
264
265 size_t power::nops() const
266 {
267         return 2;
268 }
269
270 ex power::op(size_t i) const
271 {
272         GINAC_ASSERT(i<2);
273
274         return i==0 ? basis : exponent;
275 }
276
277 ex power::map(map_function & f) const
278 {
279         const ex &mapped_basis = f(basis);
280         const ex &mapped_exponent = f(exponent);
281
282         if (!are_ex_trivially_equal(basis, mapped_basis)
283          || !are_ex_trivially_equal(exponent, mapped_exponent))
284                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
285         else
286                 return *this;
287 }
288
289 bool power::is_polynomial(const ex & var) const
290 {
291         if (basis.is_polynomial(var)) {
292                 if (basis.has(var))
293                         // basis is non-constant polynomial in var
294                         return exponent.info(info_flags::nonnegint);
295                 else
296                         // basis is constant in var
297                         return !exponent.has(var);
298         }
299         // basis is a non-polynomial function of var
300         return false;
301 }
302
303 int power::degree(const ex & s) const
304 {
305         if (is_equal(ex_to<basic>(s)))
306                 return 1;
307         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
308                 if (basis.is_equal(s))
309                         return ex_to<numeric>(exponent).to_int();
310                 else
311                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
312         } else if (basis.has(s))
313                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
314         else
315                 return 0;
316 }
317
318 int power::ldegree(const ex & s) const 
319 {
320         if (is_equal(ex_to<basic>(s)))
321                 return 1;
322         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
323                 if (basis.is_equal(s))
324                         return ex_to<numeric>(exponent).to_int();
325                 else
326                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
327         } else if (basis.has(s))
328                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
329         else
330                 return 0;
331 }
332
333 ex power::coeff(const ex & s, int n) const
334 {
335         if (is_equal(ex_to<basic>(s)))
336                 return n==1 ? _ex1 : _ex0;
337         else if (!basis.is_equal(s)) {
338                 // basis not equal to s
339                 if (n == 0)
340                         return *this;
341                 else
342                         return _ex0;
343         } else {
344                 // basis equal to s
345                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
346                         // integer exponent
347                         int int_exp = ex_to<numeric>(exponent).to_int();
348                         if (n == int_exp)
349                                 return _ex1;
350                         else
351                                 return _ex0;
352                 } else {
353                         // non-integer exponents are treated as zero
354                         if (n == 0)
355                                 return *this;
356                         else
357                                 return _ex0;
358                 }
359         }
360 }
361
362 /** Perform automatic term rewriting rules in this class.  In the following
363  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
364  *  stand for such expressions that contain a plain number.
365  *  - ^(x,0) -> 1  (also handles ^(0,0))
366  *  - ^(x,1) -> x
367  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
368  *  - ^(1,x) -> 1
369  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
370  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
371  *  - ^(^(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!)
372  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
373  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
374  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
375  *
376  *  @param level cut-off in recursive evaluation */
377 ex power::eval(int level) const
378 {
379         if ((level==1) && (flags & status_flags::evaluated))
380                 return *this;
381         else if (level == -max_recursion_level)
382                 throw(std::runtime_error("max recursion level reached"));
383         
384         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
385         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
386         
387         const numeric *num_basis = nullptr;
388         const numeric *num_exponent = nullptr;
389         
390         if (is_exactly_a<numeric>(ebasis)) {
391                 num_basis = &ex_to<numeric>(ebasis);
392         }
393         if (is_exactly_a<numeric>(eexponent)) {
394                 num_exponent = &ex_to<numeric>(eexponent);
395         }
396         
397         // ^(x,0) -> 1  (0^0 also handled here)
398         if (eexponent.is_zero()) {
399                 if (ebasis.is_zero())
400                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
401                 else
402                         return _ex1;
403         }
404         
405         // ^(x,1) -> x
406         if (eexponent.is_equal(_ex1))
407                 return ebasis;
408
409         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
410         if ( ebasis.is_zero() && num_exponent ) {
411                 if ((num_exponent->real()).is_zero())
412                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
413                 else if ((num_exponent->real()).is_negative())
414                         throw (pole_error("power::eval(): division by zero",1));
415                 else
416                         return _ex0;
417         }
418
419         // ^(1,x) -> 1
420         if (ebasis.is_equal(_ex1))
421                 return _ex1;
422
423         // power of a function calculated by separate rules defined for this function
424         if (is_exactly_a<function>(ebasis))
425                 return ex_to<function>(ebasis).power(eexponent);
426
427         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
428         if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
429                 return power(ebasis.op(0), ebasis.op(1) * eexponent);
430
431         if ( num_exponent ) {
432
433                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
434                 // except if c1,c2 are rational, but c1^c2 is not)
435                 if ( num_basis ) {
436                         const bool basis_is_crational = num_basis->is_crational();
437                         const bool exponent_is_crational = num_exponent->is_crational();
438                         if (!basis_is_crational || !exponent_is_crational) {
439                                 // return a plain float
440                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
441                                                                                                status_flags::evaluated |
442                                                                                                status_flags::expanded);
443                         }
444
445                         const numeric res = num_basis->power(*num_exponent);
446                         if (res.is_crational()) {
447                                 return res;
448                         }
449                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
450
451                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
452                         if (basis_is_crational && exponent_is_crational
453                             && num_exponent->is_real()
454                             && !num_exponent->is_integer()) {
455                                 const numeric n = num_exponent->numer();
456                                 const numeric m = num_exponent->denom();
457                                 numeric r;
458                                 numeric q = iquo(n, m, r);
459                                 if (r.is_negative()) {
460                                         r += m;
461                                         --q;
462                                 }
463                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
464                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
465                                                 // try it for numerator and denominator separately, in order to
466                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
467                                                 const numeric bnum = num_basis->numer();
468                                                 const numeric bden = num_basis->denom();
469                                                 const numeric res_bnum = bnum.power(*num_exponent);
470                                                 const numeric res_bden = bden.power(*num_exponent);
471                                                 if (res_bnum.is_integer())
472                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
473                                                 if (res_bden.is_integer())
474                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
475                                         }
476                                         return this->hold();
477                                 } else {
478                                         // assemble resulting product, but allowing for a re-evaluation,
479                                         // because otherwise we'll end up with something like
480                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
481                                         // instead of 7/16*7^(1/3).
482                                         ex prod = power(*num_basis,r.div(m));
483                                         return prod*power(*num_basis,q);
484                                 }
485                         }
486                 }
487         
488                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
489                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
490                 // case c1==1 should not happen, see below!)
491                 if (is_exactly_a<power>(ebasis)) {
492                         const power & sub_power = ex_to<power>(ebasis);
493                         const ex & sub_basis = sub_power.basis;
494                         const ex & sub_exponent = sub_power.exponent;
495                         if (is_exactly_a<numeric>(sub_exponent)) {
496                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
497                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
498                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative() ||
499                                     (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
500                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
501                                 }
502                         }
503                 }
504         
505                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
506                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
507                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
508                 }
509
510                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
511                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
512                         numeric icont = ebasis.integer_content();
513                         const numeric lead_coeff = 
514                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
515
516                         const bool canonicalizable = lead_coeff.is_integer();
517                         const bool unit_normal = lead_coeff.is_pos_integer();
518                         if (canonicalizable && (! unit_normal))
519                                 icont = icont.mul(*_num_1_p);
520                         
521                         if (canonicalizable && (icont != *_num1_p)) {
522                                 const add& addref = ex_to<add>(ebasis);
523                                 add* addp = new add(addref);
524                                 addp->setflag(status_flags::dynallocated);
525                                 addp->clearflag(status_flags::hash_calculated);
526                                 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
527                                 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
528                                         i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
529
530                                 const numeric c = icont.power(*num_exponent);
531                                 if (likely(c != *_num1_p))
532                                         return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
533                                 else
534                                         return power(*addp, *num_exponent);
535                         }
536                 }
537
538                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
539                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
540                 if (is_exactly_a<mul>(ebasis)) {
541                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
542                         const mul & mulref = ex_to<mul>(ebasis);
543                         if (!mulref.overall_coeff.is_equal(_ex1)) {
544                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
545                                 if (num_coeff.is_real()) {
546                                         if (num_coeff.is_positive()) {
547                                                 mul *mulp = new mul(mulref);
548                                                 mulp->overall_coeff = _ex1;
549                                                 mulp->setflag(status_flags::dynallocated);
550                                                 mulp->clearflag(status_flags::evaluated);
551                                                 mulp->clearflag(status_flags::hash_calculated);
552                                                 return (new mul(power(*mulp,exponent),
553                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
554                                         } else {
555                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
556                                                 if (!num_coeff.is_equal(*_num_1_p)) {
557                                                         mul *mulp = new mul(mulref);
558                                                         mulp->overall_coeff = _ex_1;
559                                                         mulp->setflag(status_flags::dynallocated);
560                                                         mulp->clearflag(status_flags::evaluated);
561                                                         mulp->clearflag(status_flags::hash_calculated);
562                                                         return (new mul(power(*mulp,exponent),
563                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
564                                                 }
565                                         }
566                                 }
567                         }
568                 }
569
570                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
571                 if (num_exponent->is_pos_integer() &&
572                     ebasis.return_type() != return_types::commutative &&
573                     !is_a<matrix>(ebasis)) {
574                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
575                 }
576         }
577         
578         if (are_ex_trivially_equal(ebasis,basis) &&
579             are_ex_trivially_equal(eexponent,exponent)) {
580                 return this->hold();
581         }
582         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
583                                                        status_flags::evaluated);
584 }
585
586 ex power::evalf(int level) const
587 {
588         ex ebasis;
589         ex eexponent;
590         
591         if (level==1) {
592                 ebasis = basis;
593                 eexponent = exponent;
594         } else if (level == -max_recursion_level) {
595                 throw(std::runtime_error("max recursion level reached"));
596         } else {
597                 ebasis = basis.evalf(level-1);
598                 if (!is_exactly_a<numeric>(exponent))
599                         eexponent = exponent.evalf(level-1);
600                 else
601                         eexponent = exponent;
602         }
603
604         return power(ebasis,eexponent);
605 }
606
607 ex power::evalm() const
608 {
609         const ex ebasis = basis.evalm();
610         const ex eexponent = exponent.evalm();
611         if (is_a<matrix>(ebasis)) {
612                 if (is_exactly_a<numeric>(eexponent)) {
613                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
614                 }
615         }
616         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
617 }
618
619 bool power::has(const ex & other, unsigned options) const
620 {
621         if (!(options & has_options::algebraic))
622                 return basic::has(other, options);
623         if (!is_a<power>(other))
624                 return basic::has(other, options);
625         if (!exponent.info(info_flags::integer) ||
626             !other.op(1).info(info_flags::integer))
627                 return basic::has(other, options);
628         if (exponent.info(info_flags::posint) &&
629             other.op(1).info(info_flags::posint) &&
630             ex_to<numeric>(exponent) > ex_to<numeric>(other.op(1)) &&
631             basis.match(other.op(0)))
632                 return true;
633         if (exponent.info(info_flags::negint) &&
634             other.op(1).info(info_flags::negint) &&
635             ex_to<numeric>(exponent) < ex_to<numeric>(other.op(1)) &&
636             basis.match(other.op(0)))
637                 return true;
638         return basic::has(other, options);
639 }
640
641 // from mul.cpp
642 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
643
644 ex power::subs(const exmap & m, unsigned options) const
645 {       
646         const ex &subsed_basis = basis.subs(m, options);
647         const ex &subsed_exponent = exponent.subs(m, options);
648
649         if (!are_ex_trivially_equal(basis, subsed_basis)
650          || !are_ex_trivially_equal(exponent, subsed_exponent)) 
651                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
652
653         if (!(options & subs_options::algebraic))
654                 return subs_one_level(m, options);
655
656         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
657                 int nummatches = std::numeric_limits<int>::max();
658                 exmap repls;
659                 if (tryfactsubs(*this, it->first, nummatches, repls)) {
660                         ex anum = it->second.subs(repls, subs_options::no_pattern);
661                         ex aden = it->first.subs(repls, subs_options::no_pattern);
662                         ex result = (*this)*power(anum/aden, nummatches);
663                         return (ex_to<basic>(result)).subs_one_level(m, options);
664                 }
665         }
666
667         return subs_one_level(m, options);
668 }
669
670 ex power::eval_ncmul(const exvector & v) const
671 {
672         return inherited::eval_ncmul(v);
673 }
674
675 ex power::conjugate() const
676 {
677         // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
678         // branch cut which runs along the negative real axis.
679         if (basis.info(info_flags::positive)) {
680                 ex newexponent = exponent.conjugate();
681                 if (are_ex_trivially_equal(exponent, newexponent)) {
682                         return *this;
683                 }
684                 return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
685         }
686         if (exponent.info(info_flags::integer)) {
687                 ex newbasis = basis.conjugate();
688                 if (are_ex_trivially_equal(basis, newbasis)) {
689                         return *this;
690                 }
691                 return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
692         }
693         return conjugate_function(*this).hold();
694 }
695
696 ex power::real_part() const
697 {
698         if (exponent.info(info_flags::integer)) {
699                 ex basis_real = basis.real_part();
700                 if (basis_real == basis)
701                         return *this;
702                 realsymbol a("a"),b("b");
703                 ex result;
704                 if (exponent.info(info_flags::posint))
705                         result = power(a+I*b,exponent);
706                 else
707                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
708                 result = result.expand();
709                 result = result.real_part();
710                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
711                 return result;
712         }
713         
714         ex a = basis.real_part();
715         ex b = basis.imag_part();
716         ex c = exponent.real_part();
717         ex d = exponent.imag_part();
718         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
719 }
720
721 ex power::imag_part() const
722 {
723         if (exponent.info(info_flags::integer)) {
724                 ex basis_real = basis.real_part();
725                 if (basis_real == basis)
726                         return 0;
727                 realsymbol a("a"),b("b");
728                 ex result;
729                 if (exponent.info(info_flags::posint))
730                         result = power(a+I*b,exponent);
731                 else
732                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
733                 result = result.expand();
734                 result = result.imag_part();
735                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
736                 return result;
737         }
738         
739         ex a=basis.real_part();
740         ex b=basis.imag_part();
741         ex c=exponent.real_part();
742         ex d=exponent.imag_part();
743         return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
744 }
745
746 // protected
747
748 /** Implementation of ex::diff() for a power.
749  *  @see ex::diff */
750 ex power::derivative(const symbol & s) const
751 {
752         if (is_a<numeric>(exponent)) {
753                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
754                 epvector newseq;
755                 newseq.reserve(2);
756                 newseq.push_back(expair(basis, exponent - _ex1));
757                 newseq.push_back(expair(basis.diff(s), _ex1));
758                 return mul(newseq, exponent);
759         } else {
760                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
761                 return mul(*this,
762                            add(mul(exponent.diff(s), log(basis)),
763                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
764         }
765 }
766
767 int power::compare_same_type(const basic & other) const
768 {
769         GINAC_ASSERT(is_exactly_a<power>(other));
770         const power &o = static_cast<const power &>(other);
771
772         int cmpval = basis.compare(o.basis);
773         if (cmpval)
774                 return cmpval;
775         else
776                 return exponent.compare(o.exponent);
777 }
778
779 unsigned power::return_type() const
780 {
781         return basis.return_type();
782 }
783
784 return_type_t power::return_type_tinfo() const
785 {
786         return basis.return_type_tinfo();
787 }
788
789 ex power::expand(unsigned options) const
790 {
791         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
792                 // A special case worth optimizing.
793                 setflag(status_flags::expanded);
794                 return *this;
795         }
796
797         // (x*p)^c -> x^c * p^c, if p>0
798         // makes sense before expanding the basis
799         if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
800                 const mul &m = ex_to<mul>(basis);
801                 exvector prodseq;
802                 epvector powseq;
803                 prodseq.reserve(m.seq.size() + 1);
804                 powseq.reserve(m.seq.size() + 1);
805                 epvector::const_iterator last = m.seq.end();
806                 epvector::const_iterator cit = m.seq.begin();
807                 bool possign = true;
808
809                 // search for positive/negative factors
810                 while (cit!=last) {
811                         ex e=m.recombine_pair_to_ex(*cit);
812                         if (e.info(info_flags::positive))
813                                 prodseq.push_back(pow(e, exponent).expand(options));
814                         else if (e.info(info_flags::negative)) {
815                                 prodseq.push_back(pow(-e, exponent).expand(options));
816                                 possign = !possign;
817                         } else
818                                 powseq.push_back(*cit);
819                         ++cit;
820                 }
821
822                 // take care on the numeric coefficient
823                 ex coeff=(possign? _ex1 : _ex_1);
824                 if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
825                         prodseq.push_back(power(m.overall_coeff, exponent));
826                 else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
827                         prodseq.push_back(power(-m.overall_coeff, exponent));
828                 else
829                         coeff *= m.overall_coeff;
830
831                 // If positive/negative factors are found, then extract them.
832                 // In either case we set a flag to avoid the second run on a part
833                 // which does not have positive/negative terms.
834                 if (prodseq.size() > 0) {
835                         ex newbasis = coeff*mul(powseq);
836                         ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
837                         return ((new mul(prodseq))->setflag(status_flags::dynallocated)*(new power(newbasis, exponent))->setflag(status_flags::dynallocated).expand(options)).expand(options);
838                 } else
839                         ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
840         }
841
842         const ex expanded_basis = basis.expand(options);
843         const ex expanded_exponent = exponent.expand(options);
844         
845         // x^(a+b) -> x^a * x^b
846         if (is_exactly_a<add>(expanded_exponent)) {
847                 const add &a = ex_to<add>(expanded_exponent);
848                 exvector distrseq;
849                 distrseq.reserve(a.seq.size() + 1);
850                 epvector::const_iterator last = a.seq.end();
851                 epvector::const_iterator cit = a.seq.begin();
852                 while (cit!=last) {
853                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
854                         ++cit;
855                 }
856                 
857                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
858                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
859                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
860                         int int_exponent = num_exponent.to_int();
861                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
862                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
863                         else
864                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
865                 } else
866                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
867                 
868                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
869                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
870                 return r.expand(options);
871         }
872         
873         if (!is_exactly_a<numeric>(expanded_exponent) ||
874                 !ex_to<numeric>(expanded_exponent).is_integer()) {
875                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
876                         return this->hold();
877                 } else {
878                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
879                 }
880         }
881         
882         // integer numeric exponent
883         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
884         int int_exponent = num_exponent.to_int();
885         
886         // (x+y)^n, n>0
887         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
888                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
889         
890         // (x*y)^n -> x^n * y^n
891         if (is_exactly_a<mul>(expanded_basis))
892                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
893         
894         // cannot expand further
895         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
896                 return this->hold();
897         else
898                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
899 }
900
901 //////////
902 // new virtual functions which can be overridden by derived classes
903 //////////
904
905 // none
906
907 //////////
908 // non-virtual functions in this class
909 //////////
910
911 namespace {  // anonymous namespace for power::expand_add() helpers
912
913 /** Helper class to generate all bounded combinatorial partitions of an integer
914  *  n with exactly m parts (including zero parts) in non-decreasing order.
915  */
916 class partition_generator {
917 private:
918         // Partitions n into m parts, not including zero parts.
919         // (Cf. OEIS sequence A008284; implementation adapted from Jörg Arndt's
920         // FXT library)
921         struct mpartition2
922         {
923                 // partition: x[1] + x[2] + ... + x[m] = n and sentinel x[0] == 0
924                 std::vector<int> x;
925                 int n;   // n>0
926                 int m;   // 0<m<=n
927                 mpartition2(unsigned n_, unsigned m_)
928                   : x(m_+1), n(n_), m(m_)
929                 {
930                         for (int k=1; k<m; ++k)
931                                 x[k] = 1;
932                         x[m] = n - m + 1;
933                 }
934                 bool next_partition()
935                 {
936                         int u = x[m];  // last element
937                         int k = m;
938                         int s = u;
939                         while (--k) {
940                                 s += x[k];
941                                 if (x[k] + 2 <= u)
942                                         break;
943                         }
944                         if (k==0)
945                                 return false;  // current is last
946                         int f = x[k] + 1;
947                         while (k < m) {
948                                 x[k] = f;
949                                 s -= f;
950                                 ++k;
951                         }
952                         x[m] = s;
953                         return true;
954                 }
955         } mpgen;
956         int m;  // number of parts 0<m<=n
957         mutable std::vector<int> partition;  // current partition
958 public:
959         partition_generator(unsigned n_, unsigned m_)
960           : mpgen(n_, 1), m(m_), partition(m_)
961         { }
962         // returns current partition in non-decreasing order, padded with zeros
963         const std::vector<int>& current() const
964         {
965                 for (int i = 0; i < m - mpgen.m; ++i)
966                         partition[i] = 0;  // pad with zeros
967
968                 for (int i = m - mpgen.m; i < m; ++i)
969                         partition[i] = mpgen.x[i - m + mpgen.m + 1];
970
971                 return partition;
972         }
973         bool next()
974         {
975                 if (!mpgen.next_partition()) {
976                         if (mpgen.m == m || mpgen.m == mpgen.n)
977                                 return false;  // current is last
978                         // increment number of parts
979                         mpgen = mpartition2(mpgen.n, mpgen.m + 1);
980                 }
981                 return true;
982         }
983 };
984
985 /** Helper class to generate all compositions of a partition of an integer n,
986  *  starting with the compositions which has non-decreasing order.
987  */
988 class composition_generator {
989 private:
990         // Generates all distinct permutations of a multiset.
991         // (Based on Aaron Williams' algorithm 1 from "Loopless Generation of
992         // Multiset Permutations using a Constant Number of Variables by Prefix
993         // Shifts." <http://webhome.csc.uvic.ca/~haron/CoolMulti.pdf>)
994         struct coolmulti {
995                 // element of singly linked list
996                 struct element {
997                         int value;
998                         element* next;
999                         element(int val, element* n)
1000                           : value(val), next(n) {}
1001                         ~element()
1002                         {   // recurses down to the end of the singly linked list
1003                                 delete next;
1004                         }
1005                 };
1006                 element *head, *i, *after_i;
1007                 // NB: Partition must be sorted in non-decreasing order.
1008                 explicit coolmulti(const std::vector<int>& partition)
1009                 {
1010                         head = nullptr;
1011                         for (unsigned n = 0; n < partition.size(); ++n) {
1012                                 head = new element(partition[n], head);
1013                                 if (n <= 1)
1014                                         i = head;
1015                         }
1016                         after_i = i->next;
1017                 }
1018                 ~coolmulti()
1019                 {   // deletes singly linked list
1020                         delete head;
1021                 }
1022                 void next_permutation()
1023                 {
1024                         element *before_k;
1025                         if (after_i->next != nullptr && i->value >= after_i->next->value)
1026                                 before_k = after_i;
1027                         else
1028                                 before_k = i;
1029                         element *k = before_k->next;
1030                         before_k->next = k->next;
1031                         k->next = head;
1032                         if (k->value < head->value)
1033                                 i = k;
1034                         after_i = i->next;
1035                         head = k;
1036                 }
1037                 bool finished() const
1038                 {
1039                         return after_i->next == nullptr && after_i->value >= head->value;
1040                 }
1041         } cmgen;
1042         bool atend;  // needed for simplifying iteration over permutations
1043         bool trivial;  // likewise, true if all elements are equal
1044         mutable std::vector<int> composition;  // current compositions
1045 public:
1046         explicit composition_generator(const std::vector<int>& partition)
1047           : cmgen(partition), atend(false), trivial(true), composition(partition.size())
1048         {
1049                 for (unsigned i=1; i<partition.size(); ++i)
1050                         trivial = trivial && (partition[0] == partition[i]);
1051         }
1052         const std::vector<int>& current() const
1053         {
1054                 coolmulti::element* it = cmgen.head;
1055                 size_t i = 0;
1056                 while (it != nullptr) {
1057                         composition[i] = it->value;
1058                         it = it->next;
1059                         ++i;
1060                 }
1061                 return composition;
1062         }
1063         bool next()
1064         {
1065                 // This ugly contortion is needed because the original coolmulti
1066                 // algorithm requires code duplication of the payload procedure,
1067                 // one before the loop and one inside it.
1068                 if (trivial || atend)
1069                         return false;
1070                 cmgen.next_permutation();
1071                 atend = cmgen.finished();
1072                 return true;
1073         }
1074 };
1075
1076 /** Helper function to compute the multinomial coefficient n!/(p1!*p2!*...*pk!)
1077  *  where n = p1+p2+...+pk, i.e. p is a partition of n.
1078  */
1079 const numeric
1080 multinomial_coefficient(const std::vector<int> p)
1081 {
1082         numeric n = 0, d = 1;
1083         std::vector<int>::const_iterator it = p.begin(), itend = p.end();
1084         while (it != itend) {
1085                 n += numeric(*it);
1086                 d *= factorial(numeric(*it));
1087                 ++it;
1088         }
1089         return factorial(numeric(n)) / d;
1090 }
1091
1092 }  // anonymous namespace
1093
1094 /** expand a^n where a is an add and n is a positive integer.
1095  *  @see power::expand */
1096 ex power::expand_add(const add & a, int n, unsigned options) const
1097 {
1098         // The special case power(+(x,...y;x),2) can be optimized better.
1099         if (n==2)
1100                 return expand_add_2(a, options);
1101
1102         // method:
1103         //
1104         // Consider base as the sum of all symbolic terms and the overall numeric
1105         // coefficient and apply the binomial theorem:
1106         // S = power(+(x,...,z;c),n)
1107         //   = power(+(+(x,...,z;0);c),n)
1108         //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1109         // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
1110         // The multinomial theorem is computed by an outer loop over all
1111         // partitions of the exponent and an inner loop over all compositions of
1112         // that partition. This method makes the expansion a combinatorial
1113         // problem and allows us to directly construct the expanded sum and also
1114         // to re-use the multinomial coefficients (since they depend only on the
1115         // partition, not on the composition).
1116         // 
1117         // multinomial power(+(x,y,z;0),3) example:
1118         // partition : compositions                : multinomial coefficient
1119         // [0,0,3]   : [3,0,0],[0,3,0],[0,0,3]     : 3!/(3!*0!*0!) = 1
1120         // [0,1,2]   : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
1121         // [1,1,1]   : [1,1,1]                     : 3!/(1!*1!*1!) = 6
1122         //  =>  (x + y + z)^3 =
1123         //        x^3 + y^3 + z^3
1124         //      + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
1125         //      + 6*x*y*z
1126         //
1127         // multinomial power(+(x,y,z;0),4) example:
1128         // partition : compositions                : multinomial coefficient
1129         // [0,0,4]   : [4,0,0],[0,4,0],[0,0,4]     : 4!/(4!*0!*0!) = 1
1130         // [0,1,3]   : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
1131         // [0,2,2]   : [2,2,0],[2,0,2],[0,2,2]     : 4!/(2!*2!*0!) = 6
1132         // [1,1,2]   : [2,1,1],[1,2,1],[1,1,2]     : 4!/(2!*1!*1!) = 12
1133         // (no [1,1,1,1] partition since it has too many parts)
1134         //  =>  (x + y + z)^4 =
1135         //        x^4 + y^4 + z^4
1136         //      + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
1137         //      + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
1138         //      + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
1139         //
1140         // Summary:
1141         // r = 0
1142         // for k from 0 to n:
1143         //     f = c^(n-k)*binomial(n,k)
1144         //     for p in all partitions of n with m parts (including zero parts):
1145         //         h = f * multinomial coefficient of p
1146         //         for c in all compositions of p:
1147         //             t = 1
1148         //             for e in all elements of c:
1149         //                 t = t * a[e]^e
1150         //             r = r + h*t
1151         // return r
1152
1153         epvector result;
1154         // The number of terms will be the number of combinatorial compositions,
1155         // i.e. the number of unordered arrangements of m nonnegative integers
1156         // which sum up to n.  It is frequently written as C_n(m) and directly
1157         // related with binomial coefficients: binomial(n+m-1,m-1).
1158         size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_int();
1159         if (!a.overall_coeff.is_zero()) {
1160                 // the result's overall_coeff is one of the terms
1161                 --result_size;
1162         }
1163         result.reserve(result_size);
1164
1165         // Iterate over all terms in binomial expansion of
1166         // S = power(+(x,...,z;c),n)
1167         //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1168         for (int k = 1; k <= n; ++k) {
1169                 numeric binomial_coefficient;  // binomial(n,k)*c^(n-k)
1170                 if (a.overall_coeff.is_zero()) {
1171                         // degenerate case with zero overall_coeff:
1172                         // apply multinomial theorem directly to power(+(x,...z;0),n)
1173                         binomial_coefficient = 1;
1174                         if (k < n) {
1175                                 continue;
1176                         }
1177                 } else {
1178                         binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
1179                 }
1180
1181                 // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
1182                 // Iterate over all partitions of k with exactly as many parts as
1183                 // there are symbolic terms in the basis (including zero parts).
1184                 partition_generator partitions(k, a.seq.size());
1185                 do {
1186                         const std::vector<int>& partition = partitions.current();
1187                         const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
1188
1189                         // Iterate over all compositions of the current partition.
1190                         composition_generator compositions(partition);
1191                         do {
1192                                 const std::vector<int>& exponent = compositions.current();
1193                                 exvector term;
1194                                 term.reserve(n);
1195                                 numeric factor = coeff;
1196                                 for (unsigned i = 0; i < exponent.size(); ++i) {
1197                                         const ex & r = a.seq[i].rest;
1198                                         const ex & c = a.seq[i].coeff;
1199                                         GINAC_ASSERT(!is_exactly_a<add>(r));
1200                                         GINAC_ASSERT(!is_exactly_a<power>(r) ||
1201                                                      !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1202                                                      !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1203                                                      !is_exactly_a<add>(ex_to<power>(r).basis) ||
1204                                                      !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1205                                                      !is_exactly_a<power>(ex_to<power>(r).basis));
1206                                         if (exponent[i] == 0) {
1207                                                 // optimize away
1208                                         } else if (exponent[i] == 1) {
1209                                                 // optimized
1210                                                 term.push_back(r);
1211                                                 factor = factor.mul(ex_to<numeric>(c));
1212                                         } else { // general case exponent[i] > 1
1213                                                 term.push_back((new power(r, exponent[i]))->setflag(status_flags::dynallocated));
1214                                                 factor = factor.mul(ex_to<numeric>(c).power(exponent[i]));
1215                                         }
1216                                 }
1217                                 result.push_back(a.combine_ex_with_coeff_to_pair(mul(term).expand(options), factor));
1218                         } while (compositions.next());
1219                 } while (partitions.next());
1220         }
1221
1222         GINAC_ASSERT(result.size() == result_size);
1223
1224         if (a.overall_coeff.is_zero()) {
1225                 return (new add(result))->setflag(status_flags::dynallocated |
1226                                                   status_flags::expanded);
1227         } else {
1228                 return (new add(result, ex_to<numeric>(a.overall_coeff).power(n)))->setflag(status_flags::dynallocated |
1229                                                                                             status_flags::expanded);
1230         }
1231 }
1232
1233
1234 /** Special case of power::expand_add. Expands a^2 where a is an add.
1235  *  @see power::expand_add */
1236 ex power::expand_add_2(const add & a, unsigned options) const
1237 {
1238         epvector sum;
1239         size_t a_nops = a.nops();
1240         sum.reserve((a_nops*(a_nops+1))/2);
1241         epvector::const_iterator last = a.seq.end();
1242
1243         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
1244         // first part: ignore overall_coeff and expand other terms
1245         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
1246                 const ex & r = cit0->rest;
1247                 const ex & c = cit0->coeff;
1248                 
1249                 GINAC_ASSERT(!is_exactly_a<add>(r));
1250                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1251                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1252                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1253                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
1254                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1255                              !is_exactly_a<power>(ex_to<power>(r).basis));
1256                 
1257                 if (c.is_equal(_ex1)) {
1258                         if (is_exactly_a<mul>(r)) {
1259                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1260                                                                               _ex1));
1261                         } else {
1262                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1263                                                                               _ex1));
1264                         }
1265                 } else {
1266                         if (is_exactly_a<mul>(r)) {
1267                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1268                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
1269                         } else {
1270                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1271                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
1272                         }
1273                 }
1274
1275                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1276                         const ex & r1 = cit1->rest;
1277                         const ex & c1 = cit1->coeff;
1278                         sum.push_back(a.combine_ex_with_coeff_to_pair(mul(r,r1).expand(options),
1279                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1280                 }
1281         }
1282         
1283         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
1284         
1285         // second part: add terms coming from overall_coeff (if != 0)
1286         if (!a.overall_coeff.is_zero()) {
1287                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1288                 while (i != end) {
1289                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1290                         ++i;
1291                 }
1292                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1293         }
1294         
1295         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1296         
1297         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1298 }
1299
1300 /** Expand factors of m in m^n where m is a mul and n is an integer.
1301  *  @see power::expand */
1302 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1303 {
1304         GINAC_ASSERT(n.is_integer());
1305
1306         if (n.is_zero()) {
1307                 return _ex1;
1308         }
1309
1310         // do not bother to rename indices if there are no any.
1311         if (!(options & expand_options::expand_rename_idx) &&
1312             m.info(info_flags::has_indices))
1313                 options |= expand_options::expand_rename_idx;
1314         // Leave it to multiplication since dummy indices have to be renamed
1315         if ((options & expand_options::expand_rename_idx) &&
1316             (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1317                 ex result = m;
1318                 exvector va = get_all_dummy_indices(m);
1319                 sort(va.begin(), va.end(), ex_is_less());
1320
1321                 for (int i=1; i < n.to_int(); i++)
1322                         result *= rename_dummy_indices_uniquely(va, m);
1323                 return result;
1324         }
1325
1326         epvector distrseq;
1327         distrseq.reserve(m.seq.size());
1328         bool need_reexpand = false;
1329
1330         epvector::const_iterator last = m.seq.end();
1331         epvector::const_iterator cit = m.seq.begin();
1332         while (cit!=last) {
1333                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1334                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1335                         // this happens when e.g. (a+b)^(1/2) gets squared and
1336                         // the resulting product needs to be reexpanded
1337                         need_reexpand = true;
1338                 }
1339                 distrseq.push_back(p);
1340                 ++cit;
1341         }
1342
1343         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1344         if (need_reexpand)
1345                 return ex(result).expand(options);
1346         if (from_expand)
1347                 return result.setflag(status_flags::expanded);
1348         return result;
1349 }
1350
1351 GINAC_BIND_UNARCHIVER(power);
1352
1353 } // namespace GiNaC