0d24a04f950f312c54934ae5b8d2a0e994e7e117
[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 "print.h"
42 #include "archive.h"
43 #include "utils.h"
44
45 namespace GiNaC {
46
47 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
48
49 typedef std::vector<int> intvector;
50
51 //////////
52 // default constructor
53 //////////
54
55 power::power() : inherited(TINFO_power) { }
56
57 //////////
58 // other constructors
59 //////////
60
61 // all inlined
62
63 //////////
64 // archiving
65 //////////
66
67 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
68 {
69         n.find_ex("basis", basis, sym_lst);
70         n.find_ex("exponent", exponent, sym_lst);
71 }
72
73 void power::archive(archive_node &n) const
74 {
75         inherited::archive(n);
76         n.add_ex("basis", basis);
77         n.add_ex("exponent", exponent);
78 }
79
80 DEFAULT_UNARCHIVE(power)
81
82 //////////
83 // functions overriding virtual functions from base classes
84 //////////
85
86 // public
87
88 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
89 {
90         // Optimal output of integer powers of symbols to aid compiler CSE.
91         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
92         // to learn why such a parenthisation is really necessary.
93         if (exp == 1) {
94                 x.print(c);
95         } else if (exp == 2) {
96                 x.print(c);
97                 c.s << "*";
98                 x.print(c);
99         } else if (exp & 1) {
100                 x.print(c);
101                 c.s << "*";
102                 print_sym_pow(c, x, exp-1);
103         } else {
104                 c.s << "(";
105                 print_sym_pow(c, x, exp >> 1);
106                 c.s << ")*(";
107                 print_sym_pow(c, x, exp >> 1);
108                 c.s << ")";
109         }
110 }
111
112 void power::print(const print_context & c, unsigned level) const
113 {
114         if (is_a<print_tree>(c)) {
115
116                 inherited::print(c, level);
117
118         } else if (is_a<print_csrc>(c)) {
119
120                 // Integer powers of symbols are printed in a special, optimized way
121                 if (exponent.info(info_flags::integer)
122                  && (is_a<symbol>(basis) || is_a<constant>(basis))) {
123                         int exp = ex_to<numeric>(exponent).to_int();
124                         if (exp > 0)
125                                 c.s << '(';
126                         else {
127                                 exp = -exp;
128                                 if (is_a<print_csrc_cl_N>(c))
129                                         c.s << "recip(";
130                                 else
131                                         c.s << "1.0/(";
132                         }
133                         print_sym_pow(c, ex_to<symbol>(basis), exp);
134                         c.s << ')';
135
136                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
137                 } else if (exponent.is_equal(_ex_1)) {
138                         if (is_a<print_csrc_cl_N>(c))
139                                 c.s << "recip(";
140                         else
141                                 c.s << "1.0/(";
142                         basis.print(c);
143                         c.s << ')';
144
145                 // Otherwise, use the pow() or expt() (CLN) functions
146                 } else {
147                         if (is_a<print_csrc_cl_N>(c))
148                                 c.s << "expt(";
149                         else
150                                 c.s << "pow(";
151                         basis.print(c);
152                         c.s << ',';
153                         exponent.print(c);
154                         c.s << ')';
155                 }
156
157         } else if (is_a<print_python_repr>(c)) {
158
159                 c.s << class_name() << '(';
160                 basis.print(c);
161                 c.s << ',';
162                 exponent.print(c);
163                 c.s << ')';
164
165         } else {
166
167                 bool is_tex = is_a<print_latex>(c);
168
169                 if (is_tex && is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
170
171                         // Powers with negative numeric exponents are printed as fractions in TeX
172                         c.s << "\\frac{1}{";
173                         power(basis, -exponent).eval().print(c);
174                         c.s << "}";
175
176                 } else if (exponent.is_equal(_ex1_2)) {
177
178                         // Square roots are printed in a special way
179                         c.s << (is_tex ? "\\sqrt{" : "sqrt(");
180                         basis.print(c);
181                         c.s << (is_tex ? '}' : ')');
182
183                 } else {
184
185                         // Ordinary output of powers using '^' or '**'
186                         if (precedence() <= level)
187                                 c.s << (is_tex ? "{(" : "(");
188                         basis.print(c, precedence());
189                         if (is_a<print_python>(c))
190                                 c.s << "**";
191                         else
192                                 c.s << '^';
193                         if (is_tex)
194                                 c.s << '{';
195                         exponent.print(c, precedence());
196                         if (is_tex)
197                                 c.s << '}';
198                         if (precedence() <= level)
199                                 c.s << (is_tex ? ")}" : ")");
200                 }
201         }
202 }
203
204 bool power::info(unsigned inf) const
205 {
206         switch (inf) {
207                 case info_flags::polynomial:
208                 case info_flags::integer_polynomial:
209                 case info_flags::cinteger_polynomial:
210                 case info_flags::rational_polynomial:
211                 case info_flags::crational_polynomial:
212                         return exponent.info(info_flags::nonnegint);
213                 case info_flags::rational_function:
214                         return exponent.info(info_flags::integer);
215                 case info_flags::algebraic:
216                         return (!exponent.info(info_flags::integer) ||
217                                         basis.info(inf));
218         }
219         return inherited::info(inf);
220 }
221
222 size_t power::nops() const
223 {
224         return 2;
225 }
226
227 ex power::op(size_t i) const
228 {
229         GINAC_ASSERT(i<2);
230
231         return i==0 ? basis : exponent;
232 }
233
234 ex power::map(map_function & f) const
235 {
236         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
237 }
238
239 int power::degree(const ex & s) const
240 {
241         if (is_equal(ex_to<basic>(s)))
242                 return 1;
243         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
244                 if (basis.is_equal(s))
245                         return ex_to<numeric>(exponent).to_int();
246                 else
247                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
248         } else if (basis.has(s))
249                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
250         else
251                 return 0;
252 }
253
254 int power::ldegree(const ex & s) const 
255 {
256         if (is_equal(ex_to<basic>(s)))
257                 return 1;
258         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
259                 if (basis.is_equal(s))
260                         return ex_to<numeric>(exponent).to_int();
261                 else
262                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
263         } else if (basis.has(s))
264                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
265         else
266                 return 0;
267 }
268
269 ex power::coeff(const ex & s, int n) const
270 {
271         if (is_equal(ex_to<basic>(s)))
272                 return n==1 ? _ex1 : _ex0;
273         else if (!basis.is_equal(s)) {
274                 // basis not equal to s
275                 if (n == 0)
276                         return *this;
277                 else
278                         return _ex0;
279         } else {
280                 // basis equal to s
281                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
282                         // integer exponent
283                         int int_exp = ex_to<numeric>(exponent).to_int();
284                         if (n == int_exp)
285                                 return _ex1;
286                         else
287                                 return _ex0;
288                 } else {
289                         // non-integer exponents are treated as zero
290                         if (n == 0)
291                                 return *this;
292                         else
293                                 return _ex0;
294                 }
295         }
296 }
297
298 /** Perform automatic term rewriting rules in this class.  In the following
299  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
300  *  stand for such expressions that contain a plain number.
301  *  - ^(x,0) -> 1  (also handles ^(0,0))
302  *  - ^(x,1) -> x
303  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
304  *  - ^(1,x) -> 1
305  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
306  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
307  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
308  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
309  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
310  *
311  *  @param level cut-off in recursive evaluation */
312 ex power::eval(int level) const
313 {
314         if ((level==1) && (flags & status_flags::evaluated))
315                 return *this;
316         else if (level == -max_recursion_level)
317                 throw(std::runtime_error("max recursion level reached"));
318         
319         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
320         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
321         
322         bool basis_is_numerical = false;
323         bool exponent_is_numerical = false;
324         const numeric *num_basis;
325         const numeric *num_exponent;
326         
327         if (is_exactly_a<numeric>(ebasis)) {
328                 basis_is_numerical = true;
329                 num_basis = &ex_to<numeric>(ebasis);
330         }
331         if (is_exactly_a<numeric>(eexponent)) {
332                 exponent_is_numerical = true;
333                 num_exponent = &ex_to<numeric>(eexponent);
334         }
335         
336         // ^(x,0) -> 1  (0^0 also handled here)
337         if (eexponent.is_zero()) {
338                 if (ebasis.is_zero())
339                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
340                 else
341                         return _ex1;
342         }
343         
344         // ^(x,1) -> x
345         if (eexponent.is_equal(_ex1))
346                 return ebasis;
347
348         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
349         if (ebasis.is_zero() && exponent_is_numerical) {
350                 if ((num_exponent->real()).is_zero())
351                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
352                 else if ((num_exponent->real()).is_negative())
353                         throw (pole_error("power::eval(): division by zero",1));
354                 else
355                         return _ex0;
356         }
357
358         // ^(1,x) -> 1
359         if (ebasis.is_equal(_ex1))
360                 return _ex1;
361
362         if (exponent_is_numerical) {
363
364                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
365                 // except if c1,c2 are rational, but c1^c2 is not)
366                 if (basis_is_numerical) {
367                         const bool basis_is_crational = num_basis->is_crational();
368                         const bool exponent_is_crational = num_exponent->is_crational();
369                         if (!basis_is_crational || !exponent_is_crational) {
370                                 // return a plain float
371                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
372                                                                                                status_flags::evaluated |
373                                                                                                status_flags::expanded);
374                         }
375
376                         const numeric res = num_basis->power(*num_exponent);
377                         if (res.is_crational()) {
378                                 return res;
379                         }
380                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
381
382                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
383                         if (basis_is_crational && exponent_is_crational
384                             && num_exponent->is_real()
385                             && !num_exponent->is_integer()) {
386                                 const numeric n = num_exponent->numer();
387                                 const numeric m = num_exponent->denom();
388                                 numeric r;
389                                 numeric q = iquo(n, m, r);
390                                 if (r.is_negative()) {
391                                         r += m;
392                                         --q;
393                                 }
394                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
395                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
396                                                 // try it for numerator and denominator separately, in order to
397                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
398                                                 const numeric bnum = num_basis->numer();
399                                                 const numeric bden = num_basis->denom();
400                                                 const numeric res_bnum = bnum.power(*num_exponent);
401                                                 const numeric res_bden = bden.power(*num_exponent);
402                                                 if (res_bnum.is_integer())
403                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
404                                                 if (res_bden.is_integer())
405                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
406                                         }
407                                         return this->hold();
408                                 } else {
409                                         // assemble resulting product, but allowing for a re-evaluation,
410                                         // because otherwise we'll end up with something like
411                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
412                                         // instead of 7/16*7^(1/3).
413                                         ex prod = power(*num_basis,r.div(m));
414                                         return prod*power(*num_basis,q);
415                                 }
416                         }
417                 }
418         
419                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
420                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
421                 // case c1==1 should not happen, see below!)
422                 if (is_exactly_a<power>(ebasis)) {
423                         const power & sub_power = ex_to<power>(ebasis);
424                         const ex & sub_basis = sub_power.basis;
425                         const ex & sub_exponent = sub_power.exponent;
426                         if (is_exactly_a<numeric>(sub_exponent)) {
427                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
428                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
429                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1).is_negative())
430                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
431                         }
432                 }
433         
434                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
435                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
436                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
437                 }
438         
439                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
440                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
441                 if (is_exactly_a<mul>(ebasis)) {
442                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
443                         const mul & mulref = ex_to<mul>(ebasis);
444                         if (!mulref.overall_coeff.is_equal(_ex1)) {
445                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
446                                 if (num_coeff.is_real()) {
447                                         if (num_coeff.is_positive()) {
448                                                 mul *mulp = new mul(mulref);
449                                                 mulp->overall_coeff = _ex1;
450                                                 mulp->clearflag(status_flags::evaluated);
451                                                 mulp->clearflag(status_flags::hash_calculated);
452                                                 return (new mul(power(*mulp,exponent),
453                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
454                                         } else {
455                                                 GINAC_ASSERT(num_coeff.compare(_num0)<0);
456                                                 if (!num_coeff.is_equal(_num_1)) {
457                                                         mul *mulp = new mul(mulref);
458                                                         mulp->overall_coeff = _ex_1;
459                                                         mulp->clearflag(status_flags::evaluated);
460                                                         mulp->clearflag(status_flags::hash_calculated);
461                                                         return (new mul(power(*mulp,exponent),
462                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
463                                                 }
464                                         }
465                                 }
466                         }
467                 }
468
469                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
470                 if (num_exponent->is_pos_integer() &&
471                     ebasis.return_type() != return_types::commutative &&
472                     !is_a<matrix>(ebasis)) {
473                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
474                 }
475         }
476         
477         if (are_ex_trivially_equal(ebasis,basis) &&
478             are_ex_trivially_equal(eexponent,exponent)) {
479                 return this->hold();
480         }
481         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
482                                                        status_flags::evaluated);
483 }
484
485 ex power::evalf(int level) const
486 {
487         ex ebasis;
488         ex eexponent;
489         
490         if (level==1) {
491                 ebasis = basis;
492                 eexponent = exponent;
493         } else if (level == -max_recursion_level) {
494                 throw(std::runtime_error("max recursion level reached"));
495         } else {
496                 ebasis = basis.evalf(level-1);
497                 if (!is_exactly_a<numeric>(exponent))
498                         eexponent = exponent.evalf(level-1);
499                 else
500                         eexponent = exponent;
501         }
502
503         return power(ebasis,eexponent);
504 }
505
506 ex power::evalm() const
507 {
508         const ex ebasis = basis.evalm();
509         const ex eexponent = exponent.evalm();
510         if (is_a<matrix>(ebasis)) {
511                 if (is_exactly_a<numeric>(eexponent)) {
512                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
513                 }
514         }
515         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
516 }
517
518 // from mul.cpp
519 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
520
521 ex power::subs(const lst & ls, const lst & lr, unsigned options) const
522 {       
523         const ex &subsed_basis = basis.subs(ls, lr, options);
524         const ex &subsed_exponent = exponent.subs(ls, lr, options);
525
526         if (!are_ex_trivially_equal(basis, subsed_basis)
527          || !are_ex_trivially_equal(exponent, subsed_exponent)) 
528                 return power(subsed_basis, subsed_exponent).subs_one_level(ls, lr, options);
529
530         if (!(options & subs_options::subs_algebraic))
531                 return subs_one_level(ls, lr, options);
532
533         lst::const_iterator its, itr;
534         for (its = ls.begin(), itr = lr.begin(); its != ls.end(); ++its, ++itr) {
535                 int nummatches = std::numeric_limits<int>::max();
536                 lst repls;
537                 if (tryfactsubs(*this, *its, nummatches, repls))
538                         return (ex_to<basic>((*this) * power(itr->subs(ex(repls), subs_options::subs_no_pattern) / its->subs(ex(repls), subs_options::subs_no_pattern), nummatches))).subs_one_level(ls, lr, options);
539         }
540
541         return subs_one_level(ls, lr, options);
542 }
543
544 ex power::eval_ncmul(const exvector & v) const
545 {
546         return inherited::eval_ncmul(v);
547 }
548
549 // protected
550
551 /** Implementation of ex::diff() for a power.
552  *  @see ex::diff */
553 ex power::derivative(const symbol & s) const
554 {
555         if (exponent.info(info_flags::real)) {
556                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
557                 epvector newseq;
558                 newseq.reserve(2);
559                 newseq.push_back(expair(basis, exponent - _ex1));
560                 newseq.push_back(expair(basis.diff(s), _ex1));
561                 return mul(newseq, exponent);
562         } else {
563                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
564                 return mul(*this,
565                            add(mul(exponent.diff(s), log(basis)),
566                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
567         }
568 }
569
570 int power::compare_same_type(const basic & other) const
571 {
572         GINAC_ASSERT(is_exactly_a<power>(other));
573         const power &o = static_cast<const power &>(other);
574
575         int cmpval = basis.compare(o.basis);
576         if (cmpval)
577                 return cmpval;
578         else
579                 return exponent.compare(o.exponent);
580 }
581
582 unsigned power::return_type() const
583 {
584         return basis.return_type();
585 }
586    
587 unsigned power::return_type_tinfo() const
588 {
589         return basis.return_type_tinfo();
590 }
591
592 ex power::expand(unsigned options) const
593 {
594         if (options == 0 && (flags & status_flags::expanded))
595                 return *this;
596         
597         const ex expanded_basis = basis.expand(options);
598         const ex expanded_exponent = exponent.expand(options);
599         
600         // x^(a+b) -> x^a * x^b
601         if (is_exactly_a<add>(expanded_exponent)) {
602                 const add &a = ex_to<add>(expanded_exponent);
603                 exvector distrseq;
604                 distrseq.reserve(a.seq.size() + 1);
605                 epvector::const_iterator last = a.seq.end();
606                 epvector::const_iterator cit = a.seq.begin();
607                 while (cit!=last) {
608                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
609                         ++cit;
610                 }
611                 
612                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
613                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
614                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
615                         int int_exponent = num_exponent.to_int();
616                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
617                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
618                         else
619                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
620                 } else
621                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
622                 
623                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
624                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
625                 return r.expand();
626         }
627         
628         if (!is_exactly_a<numeric>(expanded_exponent) ||
629                 !ex_to<numeric>(expanded_exponent).is_integer()) {
630                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
631                         return this->hold();
632                 } else {
633                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
634                 }
635         }
636         
637         // integer numeric exponent
638         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
639         int int_exponent = num_exponent.to_int();
640         
641         // (x+y)^n, n>0
642         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
643                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
644         
645         // (x*y)^n -> x^n * y^n
646         if (is_exactly_a<mul>(expanded_basis))
647                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
648         
649         // cannot expand further
650         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
651                 return this->hold();
652         else
653                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
654 }
655
656 //////////
657 // new virtual functions which can be overridden by derived classes
658 //////////
659
660 // none
661
662 //////////
663 // non-virtual functions in this class
664 //////////
665
666 /** expand a^n where a is an add and n is a positive integer.
667  *  @see power::expand */
668 ex power::expand_add(const add & a, int n) const
669 {
670         if (n==2)
671                 return expand_add_2(a);
672
673         const size_t m = a.nops();
674         exvector result;
675         // The number of terms will be the number of combinatorial compositions,
676         // i.e. the number of unordered arrangement of m nonnegative integers
677         // which sum up to n.  It is frequently written as C_n(m) and directly
678         // related with binomial coefficients:
679         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
680         intvector k(m-1);
681         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
682         intvector upper_limit(m-1);
683         int l;
684
685         for (size_t l=0; l<m-1; ++l) {
686                 k[l] = 0;
687                 k_cum[l] = 0;
688                 upper_limit[l] = n;
689         }
690
691         while (true) {
692                 exvector term;
693                 term.reserve(m+1);
694                 for (l=0; l<m-1; ++l) {
695                         const ex & b = a.op(l);
696                         GINAC_ASSERT(!is_exactly_a<add>(b));
697                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
698                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
699                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
700                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
701                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
702                                      !is_exactly_a<power>(ex_to<power>(b).basis));
703                         if (is_exactly_a<mul>(b))
704                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
705                         else
706                                 term.push_back(power(b,k[l]));
707                 }
708
709                 const ex & b = a.op(l);
710                 GINAC_ASSERT(!is_exactly_a<add>(b));
711                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
712                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
713                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
714                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
715                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
716                              !is_exactly_a<power>(ex_to<power>(b).basis));
717                 if (is_exactly_a<mul>(b))
718                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
719                 else
720                         term.push_back(power(b,n-k_cum[m-2]));
721
722                 numeric f = binomial(numeric(n),numeric(k[0]));
723                 for (l=1; l<m-1; ++l)
724                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
725
726                 term.push_back(f);
727
728                 result.push_back((new mul(term))->setflag(status_flags::dynallocated));
729
730                 // increment k[]
731                 l = m-2;
732                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
733                         k[l] = 0;
734                         --l;
735                 }
736                 if (l<0) break;
737
738                 // recalc k_cum[] and upper_limit[]
739                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
740
741                 for (size_t i=l+1; i<m-1; ++i)
742                         k_cum[i] = k_cum[i-1]+k[i];
743
744                 for (size_t i=l+1; i<m-1; ++i)
745                         upper_limit[i] = n-k_cum[i-1];
746         }
747
748         return (new add(result))->setflag(status_flags::dynallocated |
749                                           status_flags::expanded);
750 }
751
752
753 /** Special case of power::expand_add. Expands a^2 where a is an add.
754  *  @see power::expand_add */
755 ex power::expand_add_2(const add & a) const
756 {
757         epvector sum;
758         size_t a_nops = a.nops();
759         sum.reserve((a_nops*(a_nops+1))/2);
760         epvector::const_iterator last = a.seq.end();
761
762         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
763         // first part: ignore overall_coeff and expand other terms
764         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
765                 const ex & r = cit0->rest;
766                 const ex & c = cit0->coeff;
767                 
768                 GINAC_ASSERT(!is_exactly_a<add>(r));
769                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
770                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
771                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
772                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
773                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
774                              !is_exactly_a<power>(ex_to<power>(r).basis));
775                 
776                 if (c.is_equal(_ex1)) {
777                         if (is_exactly_a<mul>(r)) {
778                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
779                                                      _ex1));
780                         } else {
781                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
782                                                      _ex1));
783                         }
784                 } else {
785                         if (is_exactly_a<mul>(r)) {
786                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r),_num2),
787                                                      ex_to<numeric>(c).power_dyn(_num2)));
788                         } else {
789                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
790                                                      ex_to<numeric>(c).power_dyn(_num2)));
791                         }
792                 }
793
794                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
795                         const ex & r1 = cit1->rest;
796                         const ex & c1 = cit1->coeff;
797                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
798                                                                       _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
799                 }
800         }
801         
802         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
803         
804         // second part: add terms coming from overall_factor (if != 0)
805         if (!a.overall_coeff.is_zero()) {
806                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
807                 while (i != end) {
808                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
809                         ++i;
810                 }
811                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
812         }
813         
814         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
815         
816         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
817 }
818
819 /** Expand factors of m in m^n where m is a mul and n is and integer.
820  *  @see power::expand */
821 ex power::expand_mul(const mul & m, const numeric & n) const
822 {
823         GINAC_ASSERT(n.is_integer());
824
825         if (n.is_zero())
826                 return _ex1;
827
828         epvector distrseq;
829         distrseq.reserve(m.seq.size());
830         epvector::const_iterator last = m.seq.end();
831         epvector::const_iterator cit = m.seq.begin();
832         while (cit!=last) {
833                 if (is_exactly_a<numeric>(cit->rest)) {
834                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
835                 } else {
836                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
837                         // since n is an integer
838                         distrseq.push_back(expair(cit->rest, ex_to<numeric>(cit->coeff).mul(n)));
839                 }
840                 ++cit;
841         }
842         return (new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
843 }
844
845 } // namespace GiNaC