- (l)degree(s), coeff(s, n) and collect(s) were extended to accept expressions
[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) 1999-2002 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
27 #include "power.h"
28 #include "expairseq.h"
29 #include "add.h"
30 #include "mul.h"
31 #include "ncmul.h"
32 #include "numeric.h"
33 #include "constant.h"
34 #include "inifcns.h" // for log() in power::derivative()
35 #include "matrix.h"
36 #include "symbol.h"
37 #include "print.h"
38 #include "archive.h"
39 #include "utils.h"
40
41 namespace GiNaC {
42
43 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
44
45 typedef std::vector<int> intvector;
46
47 //////////
48 // default ctor, dtor, copy ctor, assignment operator and helpers
49 //////////
50
51 power::power() : inherited(TINFO_power) { }
52
53 void power::copy(const power & other)
54 {
55         inherited::copy(other);
56         basis = other.basis;
57         exponent = other.exponent;
58 }
59
60 DEFAULT_DESTROY(power)
61
62 //////////
63 // other ctors
64 //////////
65
66 // all inlined
67
68 //////////
69 // archiving
70 //////////
71
72 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
73 {
74         n.find_ex("basis", basis, sym_lst);
75         n.find_ex("exponent", exponent, sym_lst);
76 }
77
78 void power::archive(archive_node &n) const
79 {
80         inherited::archive(n);
81         n.add_ex("basis", basis);
82         n.add_ex("exponent", exponent);
83 }
84
85 DEFAULT_UNARCHIVE(power)
86
87 //////////
88 // functions overriding virtual functions from base classes
89 //////////
90
91 // public
92
93 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
94 {
95         // Optimal output of integer powers of symbols to aid compiler CSE.
96         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
97         // to learn why such a parenthisation is really necessary.
98         if (exp == 1) {
99                 x.print(c);
100         } else if (exp == 2) {
101                 x.print(c);
102                 c.s << "*";
103                 x.print(c);
104         } else if (exp & 1) {
105                 x.print(c);
106                 c.s << "*";
107                 print_sym_pow(c, x, exp-1);
108         } else {
109                 c.s << "(";
110                 print_sym_pow(c, x, exp >> 1);
111                 c.s << ")*(";
112                 print_sym_pow(c, x, exp >> 1);
113                 c.s << ")";
114         }
115 }
116
117 void power::print(const print_context & c, unsigned level) const
118 {
119         if (is_a<print_tree>(c)) {
120
121                 inherited::print(c, level);
122
123         } else if (is_a<print_csrc>(c)) {
124
125                 // Integer powers of symbols are printed in a special, optimized way
126                 if (exponent.info(info_flags::integer)
127                  && (is_exactly_a<symbol>(basis) || is_exactly_a<constant>(basis))) {
128                         int exp = ex_to<numeric>(exponent).to_int();
129                         if (exp > 0)
130                                 c.s << '(';
131                         else {
132                                 exp = -exp;
133                                 if (is_a<print_csrc_cl_N>(c))
134                                         c.s << "recip(";
135                                 else
136                                         c.s << "1.0/(";
137                         }
138                         print_sym_pow(c, ex_to<symbol>(basis), exp);
139                         c.s << ')';
140
141                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
142                 } else if (exponent.is_equal(_ex_1)) {
143                         if (is_a<print_csrc_cl_N>(c))
144                                 c.s << "recip(";
145                         else
146                                 c.s << "1.0/(";
147                         basis.print(c);
148                         c.s << ')';
149
150                 // Otherwise, use the pow() or expt() (CLN) functions
151                 } else {
152                         if (is_a<print_csrc_cl_N>(c))
153                                 c.s << "expt(";
154                         else
155                                 c.s << "pow(";
156                         basis.print(c);
157                         c.s << ',';
158                         exponent.print(c);
159                         c.s << ')';
160                 }
161
162         } else if (is_a<print_python_repr>(c)) {
163
164                 c.s << class_name() << '(';
165                 basis.print(c);
166                 c.s << ',';
167                 exponent.print(c);
168                 c.s << ')';
169
170         } else {
171
172                 if (exponent.is_equal(_ex1_2)) {
173                         if (is_a<print_latex>(c))
174                                 c.s << "\\sqrt{";
175                         else
176                                 c.s << "sqrt(";
177                         basis.print(c);
178                         if (is_a<print_latex>(c))
179                                 c.s << '}';
180                         else
181                                 c.s << ')';
182                 } else {
183                         if (precedence() <= level) {
184                                 if (is_a<print_latex>(c))
185                                         c.s << "{(";
186                                 else
187                                         c.s << "(";
188                         }
189                         basis.print(c, precedence());
190                         if (is_a<print_python>(c))
191                                 c.s << "**";
192                         else
193                                 c.s << '^';
194                         if (is_a<print_latex>(c))
195                                 c.s << '{';
196                         exponent.print(c, precedence());
197                         if (is_a<print_latex>(c))
198                                 c.s << '}';
199                         if (precedence() <= level) {
200                                 if (is_a<print_latex>(c))
201                                         c.s << ")}";
202                                 else
203                                         c.s << ')';
204                         }
205                 }
206         }
207 }
208
209 bool power::info(unsigned inf) const
210 {
211         switch (inf) {
212                 case info_flags::polynomial:
213                 case info_flags::integer_polynomial:
214                 case info_flags::cinteger_polynomial:
215                 case info_flags::rational_polynomial:
216                 case info_flags::crational_polynomial:
217                         return exponent.info(info_flags::nonnegint);
218                 case info_flags::rational_function:
219                         return exponent.info(info_flags::integer);
220                 case info_flags::algebraic:
221                         return (!exponent.info(info_flags::integer) ||
222                                         basis.info(inf));
223         }
224         return inherited::info(inf);
225 }
226
227 unsigned power::nops() const
228 {
229         return 2;
230 }
231
232 ex & power::let_op(int i)
233 {
234         GINAC_ASSERT(i>=0);
235         GINAC_ASSERT(i<2);
236
237         return i==0 ? basis : exponent;
238 }
239
240 ex power::map(map_function & f) const
241 {
242         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
243 }
244
245 int power::degree(const ex & s) const
246 {
247         if (is_equal(ex_to<basic>(s)))
248                 return 1;
249         else if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
250                 if (basis.is_equal(s))
251                         return ex_to<numeric>(exponent).to_int();
252                 else
253                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
254         } else if (basis.has(s))
255                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
256         else
257                 return 0;
258 }
259
260 int power::ldegree(const ex & s) const 
261 {
262         if (is_equal(ex_to<basic>(s)))
263                 return 1;
264         else if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
265                 if (basis.is_equal(s))
266                         return ex_to<numeric>(exponent).to_int();
267                 else
268                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
269         } else if (basis.has(s))
270                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
271         else
272                 return 0;
273 }
274
275 ex power::coeff(const ex & s, int n) const
276 {
277         if (is_equal(ex_to<basic>(s)))
278                 return n==1 ? _ex1 : _ex0;
279         else if (!basis.is_equal(s)) {
280                 // basis not equal to s
281                 if (n == 0)
282                         return *this;
283                 else
284                         return _ex0;
285         } else {
286                 // basis equal to s
287                 if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
288                         // integer exponent
289                         int int_exp = ex_to<numeric>(exponent).to_int();
290                         if (n == int_exp)
291                                 return _ex1;
292                         else
293                                 return _ex0;
294                 } else {
295                         // non-integer exponents are treated as zero
296                         if (n == 0)
297                                 return *this;
298                         else
299                                 return _ex0;
300                 }
301         }
302 }
303
304 /** Perform automatic term rewriting rules in this class.  In the following
305  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
306  *  stand for such expressions that contain a plain number.
307  *  - ^(x,0) -> 1  (also handles ^(0,0))
308  *  - ^(x,1) -> x
309  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
310  *  - ^(1,x) -> 1
311  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
312  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
313  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
314  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
315  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
316  *
317  *  @param level cut-off in recursive evaluation */
318 ex power::eval(int level) const
319 {
320         if ((level==1) && (flags & status_flags::evaluated))
321                 return *this;
322         else if (level == -max_recursion_level)
323                 throw(std::runtime_error("max recursion level reached"));
324         
325         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
326         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
327         
328         bool basis_is_numerical = false;
329         bool exponent_is_numerical = false;
330         const numeric *num_basis;
331         const numeric *num_exponent;
332         
333         if (is_ex_exactly_of_type(ebasis, numeric)) {
334                 basis_is_numerical = true;
335                 num_basis = &ex_to<numeric>(ebasis);
336         }
337         if (is_ex_exactly_of_type(eexponent, numeric)) {
338                 exponent_is_numerical = true;
339                 num_exponent = &ex_to<numeric>(eexponent);
340         }
341         
342         // ^(x,0) -> 1  (0^0 also handled here)
343         if (eexponent.is_zero()) {
344                 if (ebasis.is_zero())
345                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
346                 else
347                         return _ex1;
348         }
349         
350         // ^(x,1) -> x
351         if (eexponent.is_equal(_ex1))
352                 return ebasis;
353
354         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
355         if (ebasis.is_zero() && exponent_is_numerical) {
356                 if ((num_exponent->real()).is_zero())
357                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
358                 else if ((num_exponent->real()).is_negative())
359                         throw (pole_error("power::eval(): division by zero",1));
360                 else
361                         return _ex0;
362         }
363
364         // ^(1,x) -> 1
365         if (ebasis.is_equal(_ex1))
366                 return _ex1;
367
368         if (exponent_is_numerical) {
369
370                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
371                 // except if c1,c2 are rational, but c1^c2 is not)
372                 if (basis_is_numerical) {
373                         const bool basis_is_crational = num_basis->is_crational();
374                         const bool exponent_is_crational = num_exponent->is_crational();
375                         if (!basis_is_crational || !exponent_is_crational) {
376                                 // return a plain float
377                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
378                                                                                                status_flags::evaluated |
379                                                                                                status_flags::expanded);
380                         }
381
382                         const numeric res = num_basis->power(*num_exponent);
383                         if (res.is_crational()) {
384                                 return res;
385                         }
386                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
387
388                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
389                         if (basis_is_crational && exponent_is_crational
390                             && num_exponent->is_real()
391                             && !num_exponent->is_integer()) {
392                                 const numeric n = num_exponent->numer();
393                                 const numeric m = num_exponent->denom();
394                                 numeric r;
395                                 numeric q = iquo(n, m, r);
396                                 if (r.is_negative()) {
397                                         r += m;
398                                         --q;
399                                 }
400                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
401                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
402                                                 // try it for numerator and denominator separately, in order to
403                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
404                                                 const numeric bnum = num_basis->numer();
405                                                 const numeric bden = num_basis->denom();
406                                                 const numeric res_bnum = bnum.power(*num_exponent);
407                                                 const numeric res_bden = bden.power(*num_exponent);
408                                                 if (res_bnum.is_integer())
409                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
410                                                 if (res_bden.is_integer())
411                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
412                                         }
413                                         return this->hold();
414                                 } else {
415                                         // assemble resulting product, but allowing for a re-evaluation,
416                                         // because otherwise we'll end up with something like
417                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
418                                         // instead of 7/16*7^(1/3).
419                                         ex prod = power(*num_basis,r.div(m));
420                                         return prod*power(*num_basis,q);
421                                 }
422                         }
423                 }
424         
425                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
426                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
427                 // case c1==1 should not happen, see below!)
428                 if (is_ex_exactly_of_type(ebasis,power)) {
429                         const power & sub_power = ex_to<power>(ebasis);
430                         const ex & sub_basis = sub_power.basis;
431                         const ex & sub_exponent = sub_power.exponent;
432                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
433                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
434                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
435                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1).is_negative())
436                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
437                         }
438                 }
439         
440                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
441                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
442                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
443                 }
444         
445                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
446                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
447                 if (is_ex_exactly_of_type(ebasis,mul)) {
448                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
449                         const mul & mulref = ex_to<mul>(ebasis);
450                         if (!mulref.overall_coeff.is_equal(_ex1)) {
451                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
452                                 if (num_coeff.is_real()) {
453                                         if (num_coeff.is_positive()) {
454                                                 mul *mulp = new mul(mulref);
455                                                 mulp->overall_coeff = _ex1;
456                                                 mulp->clearflag(status_flags::evaluated);
457                                                 mulp->clearflag(status_flags::hash_calculated);
458                                                 return (new mul(power(*mulp,exponent),
459                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
460                                         } else {
461                                                 GINAC_ASSERT(num_coeff.compare(_num0)<0);
462                                                 if (!num_coeff.is_equal(_num_1)) {
463                                                         mul *mulp = new mul(mulref);
464                                                         mulp->overall_coeff = _ex_1;
465                                                         mulp->clearflag(status_flags::evaluated);
466                                                         mulp->clearflag(status_flags::hash_calculated);
467                                                         return (new mul(power(*mulp,exponent),
468                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
469                                                 }
470                                         }
471                                 }
472                         }
473                 }
474
475                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
476                 if (num_exponent->is_pos_integer() &&
477                     ebasis.return_type() != return_types::commutative &&
478                     !is_ex_of_type(ebasis,matrix)) {
479                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
480                 }
481         }
482         
483         if (are_ex_trivially_equal(ebasis,basis) &&
484             are_ex_trivially_equal(eexponent,exponent)) {
485                 return this->hold();
486         }
487         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
488                                                        status_flags::evaluated);
489 }
490
491 ex power::evalf(int level) const
492 {
493         ex ebasis;
494         ex eexponent;
495         
496         if (level==1) {
497                 ebasis = basis;
498                 eexponent = exponent;
499         } else if (level == -max_recursion_level) {
500                 throw(std::runtime_error("max recursion level reached"));
501         } else {
502                 ebasis = basis.evalf(level-1);
503                 if (!is_exactly_a<numeric>(exponent))
504                         eexponent = exponent.evalf(level-1);
505                 else
506                         eexponent = exponent;
507         }
508
509         return power(ebasis,eexponent);
510 }
511
512 ex power::evalm(void) const
513 {
514         const ex ebasis = basis.evalm();
515         const ex eexponent = exponent.evalm();
516         if (is_ex_of_type(ebasis,matrix)) {
517                 if (is_ex_of_type(eexponent,numeric)) {
518                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
519                 }
520         }
521         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
522 }
523
524 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
525 {
526         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
527         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
528
529         if (are_ex_trivially_equal(basis, subsed_basis)
530          && are_ex_trivially_equal(exponent, subsed_exponent))
531                 return basic::subs(ls, lr, no_pattern);
532         else
533                 return power(subsed_basis, subsed_exponent).basic::subs(ls, lr, no_pattern);
534 }
535
536 ex power::simplify_ncmul(const exvector & v) const
537 {
538         return inherited::simplify_ncmul(v);
539 }
540
541 // protected
542
543 /** Implementation of ex::diff() for a power.
544  *  @see ex::diff */
545 ex power::derivative(const symbol & s) const
546 {
547         if (exponent.info(info_flags::real)) {
548                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
549                 epvector newseq;
550                 newseq.reserve(2);
551                 newseq.push_back(expair(basis, exponent - _ex1));
552                 newseq.push_back(expair(basis.diff(s), _ex1));
553                 return mul(newseq, exponent);
554         } else {
555                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
556                 return mul(*this,
557                            add(mul(exponent.diff(s), log(basis)),
558                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
559         }
560 }
561
562 int power::compare_same_type(const basic & other) const
563 {
564         GINAC_ASSERT(is_exactly_a<power>(other));
565         const power &o = static_cast<const power &>(other);
566
567         int cmpval = basis.compare(o.basis);
568         if (cmpval)
569                 return cmpval;
570         else
571                 return exponent.compare(o.exponent);
572 }
573
574 unsigned power::return_type(void) const
575 {
576         return basis.return_type();
577 }
578    
579 unsigned power::return_type_tinfo(void) const
580 {
581         return basis.return_type_tinfo();
582 }
583
584 ex power::expand(unsigned options) const
585 {
586         if (options == 0 && (flags & status_flags::expanded))
587                 return *this;
588         
589         const ex expanded_basis = basis.expand(options);
590         const ex expanded_exponent = exponent.expand(options);
591         
592         // x^(a+b) -> x^a * x^b
593         if (is_ex_exactly_of_type(expanded_exponent, add)) {
594                 const add &a = ex_to<add>(expanded_exponent);
595                 exvector distrseq;
596                 distrseq.reserve(a.seq.size() + 1);
597                 epvector::const_iterator last = a.seq.end();
598                 epvector::const_iterator cit = a.seq.begin();
599                 while (cit!=last) {
600                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
601                         ++cit;
602                 }
603                 
604                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
605                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
606                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
607                         int int_exponent = num_exponent.to_int();
608                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
609                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
610                         else
611                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
612                 } else
613                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
614                 
615                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
616                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
617                 return r.expand();
618         }
619         
620         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
621                 !ex_to<numeric>(expanded_exponent).is_integer()) {
622                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
623                         return this->hold();
624                 } else {
625                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
626                 }
627         }
628         
629         // integer numeric exponent
630         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
631         int int_exponent = num_exponent.to_int();
632         
633         // (x+y)^n, n>0
634         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
635                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
636         
637         // (x*y)^n -> x^n * y^n
638         if (is_ex_exactly_of_type(expanded_basis,mul))
639                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
640         
641         // cannot expand further
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 // new virtual functions which can be overridden by derived classes
650 //////////
651
652 // none
653
654 //////////
655 // non-virtual functions in this class
656 //////////
657
658 /** expand a^n where a is an add and n is an integer.
659  *  @see power::expand */
660 ex power::expand_add(const add & a, int n) const
661 {
662         if (n==2)
663                 return expand_add_2(a);
664         
665         int m = a.nops();
666         exvector sum;
667         sum.reserve((n+1)*(m-1));
668         intvector k(m-1);
669         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
670         intvector upper_limit(m-1);
671         int l;
672         
673         for (int l=0; l<m-1; l++) {
674                 k[l] = 0;
675                 k_cum[l] = 0;
676                 upper_limit[l] = n;
677         }
678         
679         while (true) {
680                 exvector term;
681                 term.reserve(m+1);
682                 for (l=0; l<m-1; l++) {
683                         const ex & b = a.op(l);
684                         GINAC_ASSERT(!is_exactly_a<add>(b));
685                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
686                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
687                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
688                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
689                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
690                                      !is_exactly_a<power>(ex_to<power>(b).basis));
691                         if (is_ex_exactly_of_type(b,mul))
692                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
693                         else
694                                 term.push_back(power(b,k[l]));
695                 }
696                 
697                 const ex & b = a.op(l);
698                 GINAC_ASSERT(!is_exactly_a<add>(b));
699                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
700                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
701                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
702                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
703                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
704                              !is_exactly_a<power>(ex_to<power>(b).basis));
705                 if (is_ex_exactly_of_type(b,mul))
706                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
707                 else
708                         term.push_back(power(b,n-k_cum[m-2]));
709                 
710                 numeric f = binomial(numeric(n),numeric(k[0]));
711                 for (l=1; l<m-1; l++)
712                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
713                 
714                 term.push_back(f);
715                 
716                 // TODO: Can we optimize this?  Alex seemed to think so...
717                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
718                 
719                 // increment k[]
720                 l = m-2;
721                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
722                         k[l] = 0;    
723                         --l;
724                 }
725                 if (l<0) break;
726                 
727                 // recalc k_cum[] and upper_limit[]
728                 if (l==0)
729                         k_cum[0] = k[0];
730                 else
731                         k_cum[l] = k_cum[l-1]+k[l];
732                 
733                 for (int i=l+1; i<m-1; i++)
734                         k_cum[i] = k_cum[i-1]+k[i];
735                 
736                 for (int i=l+1; i<m-1; i++)
737                         upper_limit[i] = n-k_cum[i-1];
738         }
739         return (new add(sum))->setflag(status_flags::dynallocated |
740                                        status_flags::expanded );
741 }
742
743
744 /** Special case of power::expand_add. Expands a^2 where a is an add.
745  *  @see power::expand_add */
746 ex power::expand_add_2(const add & a) const
747 {
748         epvector sum;
749         unsigned a_nops = a.nops();
750         sum.reserve((a_nops*(a_nops+1))/2);
751         epvector::const_iterator last = a.seq.end();
752         
753         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
754         // first part: ignore overall_coeff and expand other terms
755         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
756                 const ex & r = cit0->rest;
757                 const ex & c = cit0->coeff;
758                 
759                 GINAC_ASSERT(!is_exactly_a<add>(r));
760                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
761                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
762                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
763                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
764                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
765                              !is_exactly_a<power>(ex_to<power>(r).basis));
766                 
767                 if (are_ex_trivially_equal(c,_ex1)) {
768                         if (is_ex_exactly_of_type(r,mul)) {
769                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
770                                                      _ex1));
771                         } else {
772                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
773                                                      _ex1));
774                         }
775                 } else {
776                         if (is_ex_exactly_of_type(r,mul)) {
777                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
778                                                      ex_to<numeric>(c).power_dyn(_num2)));
779                         } else {
780                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
781                                                      ex_to<numeric>(c).power_dyn(_num2)));
782                         }
783                 }
784                         
785                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
786                         const ex & r1 = cit1->rest;
787                         const ex & c1 = cit1->coeff;
788                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
789                                                                       _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
790                 }
791         }
792         
793         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
794         
795         // second part: add terms coming from overall_factor (if != 0)
796         if (!a.overall_coeff.is_zero()) {
797                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
798                 while (i != end) {
799                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
800                         ++i;
801                 }
802                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
803         }
804         
805         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
806         
807         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
808 }
809
810 /** Expand factors of m in m^n where m is a mul and n is and integer
811  *  @see power::expand */
812 ex power::expand_mul(const mul & m, const numeric & n) const
813 {
814         if (n.is_zero())
815                 return _ex1;
816         
817         epvector distrseq;
818         distrseq.reserve(m.seq.size());
819         epvector::const_iterator last = m.seq.end();
820         epvector::const_iterator cit = m.seq.begin();
821         while (cit!=last) {
822                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
823                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
824                 } else {
825                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
826                         // since n is an integer
827                         distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
828                 }
829                 ++cit;
830         }
831         return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
832 }
833
834 } // namespace GiNaC