use new-style print methods
[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) 1999-2003 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., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
21  */
22
23 #include <vector>
24 #include <iostream>
25 #include <stdexcept>
26 #include <limits>
27
28 #include "power.h"
29 #include "expairseq.h"
30 #include "add.h"
31 #include "mul.h"
32 #include "ncmul.h"
33 #include "numeric.h"
34 #include "constant.h"
35 #include "operators.h"
36 #include "inifcns.h" // for log() in power::derivative()
37 #include "matrix.h"
38 #include "indexed.h"
39 #include "symbol.h"
40 #include "lst.h"
41 #include "archive.h"
42 #include "utils.h"
43
44 namespace GiNaC {
45
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))
52
53 typedef std::vector<int> intvector;
54
55 //////////
56 // default constructor
57 //////////
58
59 power::power() : inherited(TINFO_power) { }
60
61 //////////
62 // other constructors
63 //////////
64
65 // all inlined
66
67 //////////
68 // archiving
69 //////////
70
71 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
72 {
73         n.find_ex("basis", basis, sym_lst);
74         n.find_ex("exponent", exponent, sym_lst);
75 }
76
77 void power::archive(archive_node &n) const
78 {
79         inherited::archive(n);
80         n.add_ex("basis", basis);
81         n.add_ex("exponent", exponent);
82 }
83
84 DEFAULT_UNARCHIVE(power)
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(const print_csrc & c, unsigned level) const
164 {
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();
169                 if (exp > 0)
170                         c.s << '(';
171                 else {
172                         exp = -exp;
173                         if (is_a<print_csrc_cl_N>(c))
174                                 c.s << "recip(";
175                         else
176                                 c.s << "1.0/(";
177                 }
178                 print_sym_pow(c, ex_to<symbol>(basis), exp);
179                 c.s << ')';
180
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))
184                         c.s << "recip(";
185                 else
186                         c.s << "1.0/(";
187                 basis.print(c);
188                 c.s << ')';
189
190         // Otherwise, use the pow() or expt() (CLN) functions
191         } else {
192                 if (is_a<print_csrc_cl_N>(c))
193                         c.s << "expt(";
194                 else
195                         c.s << "pow(";
196                 basis.print(c);
197                 c.s << ',';
198                 exponent.print(c);
199                 c.s << ')';
200         }
201 }
202
203 void power::do_print_python(const print_python & c, unsigned level) const
204 {
205         print_power(c, "**", "", "", level);
206 }
207
208 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
209 {
210         c.s << class_name() << '(';
211         basis.print(c);
212         c.s << ',';
213         exponent.print(c);
214         c.s << ')';
215 }
216
217 bool power::info(unsigned inf) const
218 {
219         switch (inf) {
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);
226                 case info_flags::rational_function:
227                         return exponent.info(info_flags::integer);
228                 case info_flags::algebraic:
229                         return (!exponent.info(info_flags::integer) ||
230                                         basis.info(inf));
231         }
232         return inherited::info(inf);
233 }
234
235 size_t power::nops() const
236 {
237         return 2;
238 }
239
240 ex power::op(size_t i) const
241 {
242         GINAC_ASSERT(i<2);
243
244         return i==0 ? basis : exponent;
245 }
246
247 ex power::map(map_function & f) const
248 {
249         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
250 }
251
252 int power::degree(const ex & s) const
253 {
254         if (is_equal(ex_to<basic>(s)))
255                 return 1;
256         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
257                 if (basis.is_equal(s))
258                         return ex_to<numeric>(exponent).to_int();
259                 else
260                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
261         } else if (basis.has(s))
262                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
263         else
264                 return 0;
265 }
266
267 int power::ldegree(const ex & s) const 
268 {
269         if (is_equal(ex_to<basic>(s)))
270                 return 1;
271         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
272                 if (basis.is_equal(s))
273                         return ex_to<numeric>(exponent).to_int();
274                 else
275                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
276         } else if (basis.has(s))
277                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
278         else
279                 return 0;
280 }
281
282 ex power::coeff(const ex & s, int n) const
283 {
284         if (is_equal(ex_to<basic>(s)))
285                 return n==1 ? _ex1 : _ex0;
286         else if (!basis.is_equal(s)) {
287                 // basis not equal to s
288                 if (n == 0)
289                         return *this;
290                 else
291                         return _ex0;
292         } else {
293                 // basis equal to s
294                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
295                         // integer exponent
296                         int int_exp = ex_to<numeric>(exponent).to_int();
297                         if (n == int_exp)
298                                 return _ex1;
299                         else
300                                 return _ex0;
301                 } else {
302                         // non-integer exponents are treated as zero
303                         if (n == 0)
304                                 return *this;
305                         else
306                                 return _ex0;
307                 }
308         }
309 }
310
311 /** Perform automatic term rewriting rules in this class.  In the following
312  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
313  *  stand for such expressions that contain a plain number.
314  *  - ^(x,0) -> 1  (also handles ^(0,0))
315  *  - ^(x,1) -> x
316  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
317  *  - ^(1,x) -> 1
318  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
319  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
320  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
321  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
322  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
323  *
324  *  @param level cut-off in recursive evaluation */
325 ex power::eval(int level) const
326 {
327         if ((level==1) && (flags & status_flags::evaluated))
328                 return *this;
329         else if (level == -max_recursion_level)
330                 throw(std::runtime_error("max recursion level reached"));
331         
332         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
333         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
334         
335         bool basis_is_numerical = false;
336         bool exponent_is_numerical = false;
337         const numeric *num_basis;
338         const numeric *num_exponent;
339         
340         if (is_exactly_a<numeric>(ebasis)) {
341                 basis_is_numerical = true;
342                 num_basis = &ex_to<numeric>(ebasis);
343         }
344         if (is_exactly_a<numeric>(eexponent)) {
345                 exponent_is_numerical = true;
346                 num_exponent = &ex_to<numeric>(eexponent);
347         }
348         
349         // ^(x,0) -> 1  (0^0 also handled here)
350         if (eexponent.is_zero()) {
351                 if (ebasis.is_zero())
352                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
353                 else
354                         return _ex1;
355         }
356         
357         // ^(x,1) -> x
358         if (eexponent.is_equal(_ex1))
359                 return ebasis;
360
361         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
362         if (ebasis.is_zero() && exponent_is_numerical) {
363                 if ((num_exponent->real()).is_zero())
364                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
365                 else if ((num_exponent->real()).is_negative())
366                         throw (pole_error("power::eval(): division by zero",1));
367                 else
368                         return _ex0;
369         }
370
371         // ^(1,x) -> 1
372         if (ebasis.is_equal(_ex1))
373                 return _ex1;
374
375         if (exponent_is_numerical) {
376
377                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
378                 // except if c1,c2 are rational, but c1^c2 is not)
379                 if (basis_is_numerical) {
380                         const bool basis_is_crational = num_basis->is_crational();
381                         const bool exponent_is_crational = num_exponent->is_crational();
382                         if (!basis_is_crational || !exponent_is_crational) {
383                                 // return a plain float
384                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
385                                                                                                status_flags::evaluated |
386                                                                                                status_flags::expanded);
387                         }
388
389                         const numeric res = num_basis->power(*num_exponent);
390                         if (res.is_crational()) {
391                                 return res;
392                         }
393                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
394
395                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
396                         if (basis_is_crational && exponent_is_crational
397                             && num_exponent->is_real()
398                             && !num_exponent->is_integer()) {
399                                 const numeric n = num_exponent->numer();
400                                 const numeric m = num_exponent->denom();
401                                 numeric r;
402                                 numeric q = iquo(n, m, r);
403                                 if (r.is_negative()) {
404                                         r += m;
405                                         --q;
406                                 }
407                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
408                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
409                                                 // try it for numerator and denominator separately, in order to
410                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
411                                                 const numeric bnum = num_basis->numer();
412                                                 const numeric bden = num_basis->denom();
413                                                 const numeric res_bnum = bnum.power(*num_exponent);
414                                                 const numeric res_bden = bden.power(*num_exponent);
415                                                 if (res_bnum.is_integer())
416                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
417                                                 if (res_bden.is_integer())
418                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
419                                         }
420                                         return this->hold();
421                                 } else {
422                                         // assemble resulting product, but allowing for a re-evaluation,
423                                         // because otherwise we'll end up with something like
424                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
425                                         // instead of 7/16*7^(1/3).
426                                         ex prod = power(*num_basis,r.div(m));
427                                         return prod*power(*num_basis,q);
428                                 }
429                         }
430                 }
431         
432                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
433                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
434                 // case c1==1 should not happen, see below!)
435                 if (is_exactly_a<power>(ebasis)) {
436                         const power & sub_power = ex_to<power>(ebasis);
437                         const ex & sub_basis = sub_power.basis;
438                         const ex & sub_exponent = sub_power.exponent;
439                         if (is_exactly_a<numeric>(sub_exponent)) {
440                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
441                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
442                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1).is_negative())
443                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
444                         }
445                 }
446         
447                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
448                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
449                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
450                 }
451         
452                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
453                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
454                 if (is_exactly_a<mul>(ebasis)) {
455                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
456                         const mul & mulref = ex_to<mul>(ebasis);
457                         if (!mulref.overall_coeff.is_equal(_ex1)) {
458                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
459                                 if (num_coeff.is_real()) {
460                                         if (num_coeff.is_positive()) {
461                                                 mul *mulp = new mul(mulref);
462                                                 mulp->overall_coeff = _ex1;
463                                                 mulp->clearflag(status_flags::evaluated);
464                                                 mulp->clearflag(status_flags::hash_calculated);
465                                                 return (new mul(power(*mulp,exponent),
466                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
467                                         } else {
468                                                 GINAC_ASSERT(num_coeff.compare(_num0)<0);
469                                                 if (!num_coeff.is_equal(_num_1)) {
470                                                         mul *mulp = new mul(mulref);
471                                                         mulp->overall_coeff = _ex_1;
472                                                         mulp->clearflag(status_flags::evaluated);
473                                                         mulp->clearflag(status_flags::hash_calculated);
474                                                         return (new mul(power(*mulp,exponent),
475                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
476                                                 }
477                                         }
478                                 }
479                         }
480                 }
481
482                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
483                 if (num_exponent->is_pos_integer() &&
484                     ebasis.return_type() != return_types::commutative &&
485                     !is_a<matrix>(ebasis)) {
486                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
487                 }
488         }
489         
490         if (are_ex_trivially_equal(ebasis,basis) &&
491             are_ex_trivially_equal(eexponent,exponent)) {
492                 return this->hold();
493         }
494         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
495                                                        status_flags::evaluated);
496 }
497
498 ex power::evalf(int level) const
499 {
500         ex ebasis;
501         ex eexponent;
502         
503         if (level==1) {
504                 ebasis = basis;
505                 eexponent = exponent;
506         } else if (level == -max_recursion_level) {
507                 throw(std::runtime_error("max recursion level reached"));
508         } else {
509                 ebasis = basis.evalf(level-1);
510                 if (!is_exactly_a<numeric>(exponent))
511                         eexponent = exponent.evalf(level-1);
512                 else
513                         eexponent = exponent;
514         }
515
516         return power(ebasis,eexponent);
517 }
518
519 ex power::evalm() const
520 {
521         const ex ebasis = basis.evalm();
522         const ex eexponent = exponent.evalm();
523         if (is_a<matrix>(ebasis)) {
524                 if (is_exactly_a<numeric>(eexponent)) {
525                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
526                 }
527         }
528         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
529 }
530
531 // from mul.cpp
532 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
533
534 ex power::subs(const exmap & m, unsigned options) const
535 {       
536         const ex &subsed_basis = basis.subs(m, options);
537         const ex &subsed_exponent = exponent.subs(m, options);
538
539         if (!are_ex_trivially_equal(basis, subsed_basis)
540          || !are_ex_trivially_equal(exponent, subsed_exponent)) 
541                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
542
543         if (!(options & subs_options::algebraic))
544                 return subs_one_level(m, options);
545
546         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
547                 int nummatches = std::numeric_limits<int>::max();
548                 lst repls;
549                 if (tryfactsubs(*this, it->first, nummatches, repls))
550                         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);
551         }
552
553         return subs_one_level(m, options);
554 }
555
556 ex power::eval_ncmul(const exvector & v) const
557 {
558         return inherited::eval_ncmul(v);
559 }
560
561 // protected
562
563 /** Implementation of ex::diff() for a power.
564  *  @see ex::diff */
565 ex power::derivative(const symbol & s) const
566 {
567         if (exponent.info(info_flags::real)) {
568                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
569                 epvector newseq;
570                 newseq.reserve(2);
571                 newseq.push_back(expair(basis, exponent - _ex1));
572                 newseq.push_back(expair(basis.diff(s), _ex1));
573                 return mul(newseq, exponent);
574         } else {
575                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
576                 return mul(*this,
577                            add(mul(exponent.diff(s), log(basis)),
578                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
579         }
580 }
581
582 int power::compare_same_type(const basic & other) const
583 {
584         GINAC_ASSERT(is_exactly_a<power>(other));
585         const power &o = static_cast<const power &>(other);
586
587         int cmpval = basis.compare(o.basis);
588         if (cmpval)
589                 return cmpval;
590         else
591                 return exponent.compare(o.exponent);
592 }
593
594 unsigned power::return_type() const
595 {
596         return basis.return_type();
597 }
598    
599 unsigned power::return_type_tinfo() const
600 {
601         return basis.return_type_tinfo();
602 }
603
604 ex power::expand(unsigned options) const
605 {
606         if (options == 0 && (flags & status_flags::expanded))
607                 return *this;
608         
609         const ex expanded_basis = basis.expand(options);
610         const ex expanded_exponent = exponent.expand(options);
611         
612         // x^(a+b) -> x^a * x^b
613         if (is_exactly_a<add>(expanded_exponent)) {
614                 const add &a = ex_to<add>(expanded_exponent);
615                 exvector distrseq;
616                 distrseq.reserve(a.seq.size() + 1);
617                 epvector::const_iterator last = a.seq.end();
618                 epvector::const_iterator cit = a.seq.begin();
619                 while (cit!=last) {
620                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
621                         ++cit;
622                 }
623                 
624                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
625                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
626                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
627                         int int_exponent = num_exponent.to_int();
628                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
629                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
630                         else
631                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
632                 } else
633                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
634                 
635                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
636                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
637                 return r.expand();
638         }
639         
640         if (!is_exactly_a<numeric>(expanded_exponent) ||
641                 !ex_to<numeric>(expanded_exponent).is_integer()) {
642                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
643                         return this->hold();
644                 } else {
645                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
646                 }
647         }
648         
649         // integer numeric exponent
650         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
651         int int_exponent = num_exponent.to_int();
652         
653         // (x+y)^n, n>0
654         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
655                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
656         
657         // (x*y)^n -> x^n * y^n
658         if (is_exactly_a<mul>(expanded_basis))
659                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
660         
661         // cannot expand further
662         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
663                 return this->hold();
664         else
665                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
666 }
667
668 //////////
669 // new virtual functions which can be overridden by derived classes
670 //////////
671
672 // none
673
674 //////////
675 // non-virtual functions in this class
676 //////////
677
678 /** expand a^n where a is an add and n is a positive integer.
679  *  @see power::expand */
680 ex power::expand_add(const add & a, int n) const
681 {
682         if (n==2)
683                 return expand_add_2(a);
684
685         const size_t m = a.nops();
686         exvector result;
687         // The number of terms will be the number of combinatorial compositions,
688         // i.e. the number of unordered arrangement of m nonnegative integers
689         // which sum up to n.  It is frequently written as C_n(m) and directly
690         // related with binomial coefficients:
691         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
692         intvector k(m-1);
693         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
694         intvector upper_limit(m-1);
695         int l;
696
697         for (size_t l=0; l<m-1; ++l) {
698                 k[l] = 0;
699                 k_cum[l] = 0;
700                 upper_limit[l] = n;
701         }
702
703         while (true) {
704                 exvector term;
705                 term.reserve(m+1);
706                 for (l=0; l<m-1; ++l) {
707                         const ex & b = a.op(l);
708                         GINAC_ASSERT(!is_exactly_a<add>(b));
709                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
710                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
711                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
712                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
713                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
714                                      !is_exactly_a<power>(ex_to<power>(b).basis));
715                         if (is_exactly_a<mul>(b))
716                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
717                         else
718                                 term.push_back(power(b,k[l]));
719                 }
720
721                 const ex & b = a.op(l);
722                 GINAC_ASSERT(!is_exactly_a<add>(b));
723                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
724                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
725                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
726                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
727                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
728                              !is_exactly_a<power>(ex_to<power>(b).basis));
729                 if (is_exactly_a<mul>(b))
730                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
731                 else
732                         term.push_back(power(b,n-k_cum[m-2]));
733
734                 numeric f = binomial(numeric(n),numeric(k[0]));
735                 for (l=1; l<m-1; ++l)
736                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
737
738                 term.push_back(f);
739
740                 result.push_back((new mul(term))->setflag(status_flags::dynallocated));
741
742                 // increment k[]
743                 l = m-2;
744                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
745                         k[l] = 0;
746                         --l;
747                 }
748                 if (l<0) break;
749
750                 // recalc k_cum[] and upper_limit[]
751                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
752
753                 for (size_t i=l+1; i<m-1; ++i)
754                         k_cum[i] = k_cum[i-1]+k[i];
755
756                 for (size_t i=l+1; i<m-1; ++i)
757                         upper_limit[i] = n-k_cum[i-1];
758         }
759
760         return (new add(result))->setflag(status_flags::dynallocated |
761                                           status_flags::expanded);
762 }
763
764
765 /** Special case of power::expand_add. Expands a^2 where a is an add.
766  *  @see power::expand_add */
767 ex power::expand_add_2(const add & a) const
768 {
769         epvector sum;
770         size_t a_nops = a.nops();
771         sum.reserve((a_nops*(a_nops+1))/2);
772         epvector::const_iterator last = a.seq.end();
773
774         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
775         // first part: ignore overall_coeff and expand other terms
776         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
777                 const ex & r = cit0->rest;
778                 const ex & c = cit0->coeff;
779                 
780                 GINAC_ASSERT(!is_exactly_a<add>(r));
781                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
782                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
783                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
784                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
785                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
786                              !is_exactly_a<power>(ex_to<power>(r).basis));
787                 
788                 if (c.is_equal(_ex1)) {
789                         if (is_exactly_a<mul>(r)) {
790                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
791                                                      _ex1));
792                         } else {
793                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
794                                                      _ex1));
795                         }
796                 } else {
797                         if (is_exactly_a<mul>(r)) {
798                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r),_num2),
799                                                      ex_to<numeric>(c).power_dyn(_num2)));
800                         } else {
801                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
802                                                      ex_to<numeric>(c).power_dyn(_num2)));
803                         }
804                 }
805
806                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
807                         const ex & r1 = cit1->rest;
808                         const ex & c1 = cit1->coeff;
809                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
810                                                                       _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
811                 }
812         }
813         
814         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
815         
816         // second part: add terms coming from overall_factor (if != 0)
817         if (!a.overall_coeff.is_zero()) {
818                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
819                 while (i != end) {
820                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
821                         ++i;
822                 }
823                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
824         }
825         
826         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
827         
828         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
829 }
830
831 /** Expand factors of m in m^n where m is a mul and n is and integer.
832  *  @see power::expand */
833 ex power::expand_mul(const mul & m, const numeric & n) const
834 {
835         GINAC_ASSERT(n.is_integer());
836
837         if (n.is_zero())
838                 return _ex1;
839
840         epvector distrseq;
841         distrseq.reserve(m.seq.size());
842         epvector::const_iterator last = m.seq.end();
843         epvector::const_iterator cit = m.seq.begin();
844         while (cit!=last) {
845                 if (is_exactly_a<numeric>(cit->rest)) {
846                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
847                 } else {
848                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
849                         // since n is an integer
850                         distrseq.push_back(expair(cit->rest, ex_to<numeric>(cit->coeff).mul(n)));
851                 }
852                 ++cit;
853         }
854         return (new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
855 }
856
857 } // namespace GiNaC