* Added output-support for Python bindings and LaTeX printing for
[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) 1999-2001 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.compare(_num_1) == 0) {
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_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
248                 if (basis.is_equal(s))
249                         return ex_to<numeric>(exponent).to_int();
250                 else
251                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
252         } else if (basis.has(s))
253                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
254         else
255                 return 0;
256 }
257
258 int power::ldegree(const ex & s) const 
259 {
260         if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
261                 if (basis.is_equal(s))
262                         return ex_to<numeric>(exponent).to_int();
263                 else
264                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
265         } else if (basis.has(s))
266                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
267         else
268                 return 0;
269 }
270
271 ex power::coeff(const ex & s, int n) const
272 {
273         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_ex_exactly_of_type(exponent, numeric) && 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_ex_exactly_of_type(ebasis, numeric)) {
328                 basis_is_numerical = true;
329                 num_basis = &ex_to<numeric>(ebasis);
330         }
331         if (is_ex_exactly_of_type(eexponent, numeric)) {
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_ex_exactly_of_type(ebasis,power)) {
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_ex_exactly_of_type(sub_exponent,numeric)) {
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_ex_exactly_of_type(ebasis,mul)) {
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_ex_exactly_of_type(ebasis,mul)) {
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_ex_of_type(ebasis,matrix)) {
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(void) const
507 {
508         const ex ebasis = basis.evalm();
509         const ex eexponent = exponent.evalm();
510         if (is_ex_of_type(ebasis,matrix)) {
511                 if (is_ex_of_type(eexponent,numeric)) {
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 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
519 {
520         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
521         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
522
523         if (are_ex_trivially_equal(basis, subsed_basis)
524          && are_ex_trivially_equal(exponent, subsed_exponent))
525                 return basic::subs(ls, lr, no_pattern);
526         else
527                 return power(subsed_basis, subsed_exponent).basic::subs(ls, lr, no_pattern);
528 }
529
530 ex power::simplify_ncmul(const exvector & v) const
531 {
532         return inherited::simplify_ncmul(v);
533 }
534
535 // protected
536
537 /** Implementation of ex::diff() for a power.
538  *  @see ex::diff */
539 ex power::derivative(const symbol & s) const
540 {
541         if (exponent.info(info_flags::real)) {
542                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
543                 epvector newseq;
544                 newseq.reserve(2);
545                 newseq.push_back(expair(basis, exponent - _ex1));
546                 newseq.push_back(expair(basis.diff(s), _ex1));
547                 return mul(newseq, exponent);
548         } else {
549                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
550                 return mul(*this,
551                            add(mul(exponent.diff(s), log(basis)),
552                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
553         }
554 }
555
556 int power::compare_same_type(const basic & other) const
557 {
558         GINAC_ASSERT(is_exactly_a<power>(other));
559         const power &o = static_cast<const power &>(other);
560
561         int cmpval = basis.compare(o.basis);
562         if (cmpval)
563                 return cmpval;
564         else
565                 return exponent.compare(o.exponent);
566 }
567
568 unsigned power::return_type(void) const
569 {
570         return basis.return_type();
571 }
572    
573 unsigned power::return_type_tinfo(void) const
574 {
575         return basis.return_type_tinfo();
576 }
577
578 ex power::expand(unsigned options) const
579 {
580         if (options == 0 && (flags & status_flags::expanded))
581                 return *this;
582         
583         const ex expanded_basis = basis.expand(options);
584         const ex expanded_exponent = exponent.expand(options);
585         
586         // x^(a+b) -> x^a * x^b
587         if (is_ex_exactly_of_type(expanded_exponent, add)) {
588                 const add &a = ex_to<add>(expanded_exponent);
589                 exvector distrseq;
590                 distrseq.reserve(a.seq.size() + 1);
591                 epvector::const_iterator last = a.seq.end();
592                 epvector::const_iterator cit = a.seq.begin();
593                 while (cit!=last) {
594                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
595                         ++cit;
596                 }
597                 
598                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
599                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
600                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
601                         int int_exponent = num_exponent.to_int();
602                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
603                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
604                         else
605                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
606                 } else
607                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
608                 
609                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
610                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
611                 return r.expand();
612         }
613         
614         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
615                 !ex_to<numeric>(expanded_exponent).is_integer()) {
616                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
617                         return this->hold();
618                 } else {
619                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
620                 }
621         }
622         
623         // integer numeric exponent
624         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
625         int int_exponent = num_exponent.to_int();
626         
627         // (x+y)^n, n>0
628         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
629                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
630         
631         // (x*y)^n -> x^n * y^n
632         if (is_ex_exactly_of_type(expanded_basis,mul))
633                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
634         
635         // cannot expand further
636         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
637                 return this->hold();
638         else
639                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
640 }
641
642 //////////
643 // new virtual functions which can be overridden by derived classes
644 //////////
645
646 // none
647
648 //////////
649 // non-virtual functions in this class
650 //////////
651
652 /** expand a^n where a is an add and n is an integer.
653  *  @see power::expand */
654 ex power::expand_add(const add & a, int n) const
655 {
656         if (n==2)
657                 return expand_add_2(a);
658         
659         int m = a.nops();
660         exvector sum;
661         sum.reserve((n+1)*(m-1));
662         intvector k(m-1);
663         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
664         intvector upper_limit(m-1);
665         int l;
666         
667         for (int l=0; l<m-1; l++) {
668                 k[l] = 0;
669                 k_cum[l] = 0;
670                 upper_limit[l] = n;
671         }
672         
673         while (true) {
674                 exvector term;
675                 term.reserve(m+1);
676                 for (l=0; l<m-1; l++) {
677                         const ex & b = a.op(l);
678                         GINAC_ASSERT(!is_exactly_a<add>(b));
679                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
680                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
681                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
682                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
683                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
684                                      !is_exactly_a<power>(ex_to<power>(b).basis));
685                         if (is_ex_exactly_of_type(b,mul))
686                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
687                         else
688                                 term.push_back(power(b,k[l]));
689                 }
690                 
691                 const ex & b = a.op(l);
692                 GINAC_ASSERT(!is_exactly_a<add>(b));
693                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
694                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
695                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
696                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
697                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
698                              !is_exactly_a<power>(ex_to<power>(b).basis));
699                 if (is_ex_exactly_of_type(b,mul))
700                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
701                 else
702                         term.push_back(power(b,n-k_cum[m-2]));
703                 
704                 numeric f = binomial(numeric(n),numeric(k[0]));
705                 for (l=1; l<m-1; l++)
706                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
707                 
708                 term.push_back(f);
709                 
710                 // TODO: Can we optimize this?  Alex seemed to think so...
711                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
712                 
713                 // increment k[]
714                 l = m-2;
715                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
716                         k[l] = 0;    
717                         --l;
718                 }
719                 if (l<0) break;
720                 
721                 // recalc k_cum[] and upper_limit[]
722                 if (l==0)
723                         k_cum[0] = k[0];
724                 else
725                         k_cum[l] = k_cum[l-1]+k[l];
726                 
727                 for (int i=l+1; i<m-1; i++)
728                         k_cum[i] = k_cum[i-1]+k[i];
729                 
730                 for (int i=l+1; i<m-1; i++)
731                         upper_limit[i] = n-k_cum[i-1];
732         }
733         return (new add(sum))->setflag(status_flags::dynallocated |
734                                        status_flags::expanded );
735 }
736
737
738 /** Special case of power::expand_add. Expands a^2 where a is an add.
739  *  @see power::expand_add */
740 ex power::expand_add_2(const add & a) const
741 {
742         epvector sum;
743         unsigned a_nops = a.nops();
744         sum.reserve((a_nops*(a_nops+1))/2);
745         epvector::const_iterator last = a.seq.end();
746         
747         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
748         // first part: ignore overall_coeff and expand other terms
749         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
750                 const ex & r = cit0->rest;
751                 const ex & c = cit0->coeff;
752                 
753                 GINAC_ASSERT(!is_exactly_a<add>(r));
754                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
755                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
756                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
757                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
758                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
759                              !is_exactly_a<power>(ex_to<power>(r).basis));
760                 
761                 if (are_ex_trivially_equal(c,_ex1)) {
762                         if (is_ex_exactly_of_type(r,mul)) {
763                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
764                                                      _ex1));
765                         } else {
766                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
767                                                      _ex1));
768                         }
769                 } else {
770                         if (is_ex_exactly_of_type(r,mul)) {
771                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
772                                                      ex_to<numeric>(c).power_dyn(_num2)));
773                         } else {
774                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
775                                                      ex_to<numeric>(c).power_dyn(_num2)));
776                         }
777                 }
778                         
779                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
780                         const ex & r1 = cit1->rest;
781                         const ex & c1 = cit1->coeff;
782                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
783                                                                       _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
784                 }
785         }
786         
787         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
788         
789         // second part: add terms coming from overall_factor (if != 0)
790         if (!a.overall_coeff.is_zero()) {
791                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
792                 while (i != end) {
793                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
794                         ++i;
795                 }
796                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
797         }
798         
799         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
800         
801         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
802 }
803
804 /** Expand factors of m in m^n where m is a mul and n is and integer
805  *  @see power::expand */
806 ex power::expand_mul(const mul & m, const numeric & n) const
807 {
808         if (n.is_zero())
809                 return _ex1;
810         
811         epvector distrseq;
812         distrseq.reserve(m.seq.size());
813         epvector::const_iterator last = m.seq.end();
814         epvector::const_iterator cit = m.seq.begin();
815         while (cit!=last) {
816                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
817                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
818                 } else {
819                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
820                         // since n is an integer
821                         distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
822                 }
823                 ++cit;
824         }
825         return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
826 }
827
828 } // namespace GiNaC