0730e3c8bf7fc385c5d4d1c1a97b6c565997cfab
[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
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 "indexed.h"
37 #include "symbol.h"
38 #include "print.h"
39 #include "archive.h"
40 #include "utils.h"
41
42 namespace GiNaC {
43
44 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
45
46 typedef std::vector<int> intvector;
47
48 //////////
49 // default ctor, dtor, copy ctor, assignment operator and helpers
50 //////////
51
52 power::power() : inherited(TINFO_power) { }
53
54 void power::copy(const power & other)
55 {
56         inherited::copy(other);
57         basis = other.basis;
58         exponent = other.exponent;
59 }
60
61 DEFAULT_DESTROY(power)
62
63 //////////
64 // other ctors
65 //////////
66
67 // all inlined
68
69 //////////
70 // archiving
71 //////////
72
73 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
74 {
75         n.find_ex("basis", basis, sym_lst);
76         n.find_ex("exponent", exponent, sym_lst);
77 }
78
79 void power::archive(archive_node &n) const
80 {
81         inherited::archive(n);
82         n.add_ex("basis", basis);
83         n.add_ex("exponent", exponent);
84 }
85
86 DEFAULT_UNARCHIVE(power)
87
88 //////////
89 // functions overriding virtual functions from base classes
90 //////////
91
92 // public
93
94 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
95 {
96         // Optimal output of integer powers of symbols to aid compiler CSE.
97         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
98         // to learn why such a parenthisation is really necessary.
99         if (exp == 1) {
100                 x.print(c);
101         } else if (exp == 2) {
102                 x.print(c);
103                 c.s << "*";
104                 x.print(c);
105         } else if (exp & 1) {
106                 x.print(c);
107                 c.s << "*";
108                 print_sym_pow(c, x, exp-1);
109         } else {
110                 c.s << "(";
111                 print_sym_pow(c, x, exp >> 1);
112                 c.s << ")*(";
113                 print_sym_pow(c, x, exp >> 1);
114                 c.s << ")";
115         }
116 }
117
118 void power::print(const print_context & c, unsigned level) const
119 {
120         if (is_a<print_tree>(c)) {
121
122                 inherited::print(c, level);
123
124         } else if (is_a<print_csrc>(c)) {
125
126                 // Integer powers of symbols are printed in a special, optimized way
127                 if (exponent.info(info_flags::integer)
128                  && (is_a<symbol>(basis) || is_a<constant>(basis))) {
129                         int exp = ex_to<numeric>(exponent).to_int();
130                         if (exp > 0)
131                                 c.s << '(';
132                         else {
133                                 exp = -exp;
134                                 if (is_a<print_csrc_cl_N>(c))
135                                         c.s << "recip(";
136                                 else
137                                         c.s << "1.0/(";
138                         }
139                         print_sym_pow(c, ex_to<symbol>(basis), exp);
140                         c.s << ')';
141
142                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
143                 } else if (exponent.is_equal(_ex_1)) {
144                         if (is_a<print_csrc_cl_N>(c))
145                                 c.s << "recip(";
146                         else
147                                 c.s << "1.0/(";
148                         basis.print(c);
149                         c.s << ')';
150
151                 // Otherwise, use the pow() or expt() (CLN) functions
152                 } else {
153                         if (is_a<print_csrc_cl_N>(c))
154                                 c.s << "expt(";
155                         else
156                                 c.s << "pow(";
157                         basis.print(c);
158                         c.s << ',';
159                         exponent.print(c);
160                         c.s << ')';
161                 }
162
163         } else if (is_a<print_python_repr>(c)) {
164
165                 c.s << class_name() << '(';
166                 basis.print(c);
167                 c.s << ',';
168                 exponent.print(c);
169                 c.s << ')';
170
171         } else {
172
173                 bool is_tex = is_a<print_latex>(c);
174
175                 if (is_tex && is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
176
177                         // Powers with negative numeric exponents are printed as fractions in TeX
178                         c.s << "\\frac{1}{";
179                         power(basis, -exponent).eval().print(c);
180                         c.s << "}";
181
182                 } else if (exponent.is_equal(_ex1_2)) {
183
184                         // Square roots are printed in a special way
185                         c.s << (is_tex ? "\\sqrt{" : "sqrt(");
186                         basis.print(c);
187                         c.s << (is_tex ? '}' : ')');
188
189                 } else {
190
191                         // Ordinary output of powers using '^' or '**'
192                         if (precedence() <= level)
193                                 c.s << (is_tex ? "{(" : "(");
194                         basis.print(c, precedence());
195                         if (is_a<print_python>(c))
196                                 c.s << "**";
197                         else
198                                 c.s << '^';
199                         if (is_tex)
200                                 c.s << '{';
201                         exponent.print(c, precedence());
202                         if (is_tex)
203                                 c.s << '}';
204                         if (precedence() <= level)
205                                 c.s << (is_tex ? ")}" : ")");
206                 }
207         }
208 }
209
210 bool power::info(unsigned inf) const
211 {
212         switch (inf) {
213                 case info_flags::polynomial:
214                 case info_flags::integer_polynomial:
215                 case info_flags::cinteger_polynomial:
216                 case info_flags::rational_polynomial:
217                 case info_flags::crational_polynomial:
218                         return exponent.info(info_flags::nonnegint);
219                 case info_flags::rational_function:
220                         return exponent.info(info_flags::integer);
221                 case info_flags::algebraic:
222                         return (!exponent.info(info_flags::integer) ||
223                                         basis.info(inf));
224         }
225         return inherited::info(inf);
226 }
227
228 unsigned power::nops() const
229 {
230         return 2;
231 }
232
233 ex & power::let_op(int i)
234 {
235         GINAC_ASSERT(i>=0);
236         GINAC_ASSERT(i<2);
237
238         return i==0 ? basis : exponent;
239 }
240
241 ex power::map(map_function & f) const
242 {
243         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
244 }
245
246 int power::degree(const ex & s) const
247 {
248         if (is_equal(ex_to<basic>(s)))
249                 return 1;
250         else if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
251                 if (basis.is_equal(s))
252                         return ex_to<numeric>(exponent).to_int();
253                 else
254                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
255         } else if (basis.has(s))
256                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
257         else
258                 return 0;
259 }
260
261 int power::ldegree(const ex & s) const 
262 {
263         if (is_equal(ex_to<basic>(s)))
264                 return 1;
265         else if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
266                 if (basis.is_equal(s))
267                         return ex_to<numeric>(exponent).to_int();
268                 else
269                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
270         } else if (basis.has(s))
271                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
272         else
273                 return 0;
274 }
275
276 ex power::coeff(const ex & s, int n) const
277 {
278         if (is_equal(ex_to<basic>(s)))
279                 return n==1 ? _ex1 : _ex0;
280         else if (!basis.is_equal(s)) {
281                 // basis not equal to s
282                 if (n == 0)
283                         return *this;
284                 else
285                         return _ex0;
286         } else {
287                 // basis equal to s
288                 if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
289                         // integer exponent
290                         int int_exp = ex_to<numeric>(exponent).to_int();
291                         if (n == int_exp)
292                                 return _ex1;
293                         else
294                                 return _ex0;
295                 } else {
296                         // non-integer exponents are treated as zero
297                         if (n == 0)
298                                 return *this;
299                         else
300                                 return _ex0;
301                 }
302         }
303 }
304
305 /** Perform automatic term rewriting rules in this class.  In the following
306  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
307  *  stand for such expressions that contain a plain number.
308  *  - ^(x,0) -> 1  (also handles ^(0,0))
309  *  - ^(x,1) -> x
310  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
311  *  - ^(1,x) -> 1
312  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
313  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
314  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
315  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
316  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
317  *
318  *  @param level cut-off in recursive evaluation */
319 ex power::eval(int level) const
320 {
321         if ((level==1) && (flags & status_flags::evaluated))
322                 return *this;
323         else if (level == -max_recursion_level)
324                 throw(std::runtime_error("max recursion level reached"));
325         
326         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
327         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
328         
329         bool basis_is_numerical = false;
330         bool exponent_is_numerical = false;
331         const numeric *num_basis;
332         const numeric *num_exponent;
333         
334         if (is_ex_exactly_of_type(ebasis, numeric)) {
335                 basis_is_numerical = true;
336                 num_basis = &ex_to<numeric>(ebasis);
337         }
338         if (is_ex_exactly_of_type(eexponent, numeric)) {
339                 exponent_is_numerical = true;
340                 num_exponent = &ex_to<numeric>(eexponent);
341         }
342         
343         // ^(x,0) -> 1  (0^0 also handled here)
344         if (eexponent.is_zero()) {
345                 if (ebasis.is_zero())
346                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
347                 else
348                         return _ex1;
349         }
350         
351         // ^(x,1) -> x
352         if (eexponent.is_equal(_ex1))
353                 return ebasis;
354
355         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
356         if (ebasis.is_zero() && exponent_is_numerical) {
357                 if ((num_exponent->real()).is_zero())
358                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
359                 else if ((num_exponent->real()).is_negative())
360                         throw (pole_error("power::eval(): division by zero",1));
361                 else
362                         return _ex0;
363         }
364
365         // ^(1,x) -> 1
366         if (ebasis.is_equal(_ex1))
367                 return _ex1;
368
369         if (exponent_is_numerical) {
370
371                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
372                 // except if c1,c2 are rational, but c1^c2 is not)
373                 if (basis_is_numerical) {
374                         const bool basis_is_crational = num_basis->is_crational();
375                         const bool exponent_is_crational = num_exponent->is_crational();
376                         if (!basis_is_crational || !exponent_is_crational) {
377                                 // return a plain float
378                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
379                                                                                                status_flags::evaluated |
380                                                                                                status_flags::expanded);
381                         }
382
383                         const numeric res = num_basis->power(*num_exponent);
384                         if (res.is_crational()) {
385                                 return res;
386                         }
387                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
388
389                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
390                         if (basis_is_crational && exponent_is_crational
391                             && num_exponent->is_real()
392                             && !num_exponent->is_integer()) {
393                                 const numeric n = num_exponent->numer();
394                                 const numeric m = num_exponent->denom();
395                                 numeric r;
396                                 numeric q = iquo(n, m, r);
397                                 if (r.is_negative()) {
398                                         r += m;
399                                         --q;
400                                 }
401                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
402                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
403                                                 // try it for numerator and denominator separately, in order to
404                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
405                                                 const numeric bnum = num_basis->numer();
406                                                 const numeric bden = num_basis->denom();
407                                                 const numeric res_bnum = bnum.power(*num_exponent);
408                                                 const numeric res_bden = bden.power(*num_exponent);
409                                                 if (res_bnum.is_integer())
410                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
411                                                 if (res_bden.is_integer())
412                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
413                                         }
414                                         return this->hold();
415                                 } else {
416                                         // assemble resulting product, but allowing for a re-evaluation,
417                                         // because otherwise we'll end up with something like
418                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
419                                         // instead of 7/16*7^(1/3).
420                                         ex prod = power(*num_basis,r.div(m));
421                                         return prod*power(*num_basis,q);
422                                 }
423                         }
424                 }
425         
426                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
427                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
428                 // case c1==1 should not happen, see below!)
429                 if (is_ex_exactly_of_type(ebasis,power)) {
430                         const power & sub_power = ex_to<power>(ebasis);
431                         const ex & sub_basis = sub_power.basis;
432                         const ex & sub_exponent = sub_power.exponent;
433                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
434                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
435                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
436                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1).is_negative())
437                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
438                         }
439                 }
440         
441                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
442                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
443                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
444                 }
445         
446                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
447                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
448                 if (is_ex_exactly_of_type(ebasis,mul)) {
449                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
450                         const mul & mulref = ex_to<mul>(ebasis);
451                         if (!mulref.overall_coeff.is_equal(_ex1)) {
452                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
453                                 if (num_coeff.is_real()) {
454                                         if (num_coeff.is_positive()) {
455                                                 mul *mulp = new mul(mulref);
456                                                 mulp->overall_coeff = _ex1;
457                                                 mulp->clearflag(status_flags::evaluated);
458                                                 mulp->clearflag(status_flags::hash_calculated);
459                                                 return (new mul(power(*mulp,exponent),
460                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
461                                         } else {
462                                                 GINAC_ASSERT(num_coeff.compare(_num0)<0);
463                                                 if (!num_coeff.is_equal(_num_1)) {
464                                                         mul *mulp = new mul(mulref);
465                                                         mulp->overall_coeff = _ex_1;
466                                                         mulp->clearflag(status_flags::evaluated);
467                                                         mulp->clearflag(status_flags::hash_calculated);
468                                                         return (new mul(power(*mulp,exponent),
469                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
470                                                 }
471                                         }
472                                 }
473                         }
474                 }
475
476                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
477                 if (num_exponent->is_pos_integer() &&
478                     ebasis.return_type() != return_types::commutative &&
479                     !is_ex_of_type(ebasis,matrix)) {
480                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
481                 }
482         }
483         
484         if (are_ex_trivially_equal(ebasis,basis) &&
485             are_ex_trivially_equal(eexponent,exponent)) {
486                 return this->hold();
487         }
488         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
489                                                        status_flags::evaluated);
490 }
491
492 ex power::evalf(int level) const
493 {
494         ex ebasis;
495         ex eexponent;
496         
497         if (level==1) {
498                 ebasis = basis;
499                 eexponent = exponent;
500         } else if (level == -max_recursion_level) {
501                 throw(std::runtime_error("max recursion level reached"));
502         } else {
503                 ebasis = basis.evalf(level-1);
504                 if (!is_exactly_a<numeric>(exponent))
505                         eexponent = exponent.evalf(level-1);
506                 else
507                         eexponent = exponent;
508         }
509
510         return power(ebasis,eexponent);
511 }
512
513 ex power::evalm(void) const
514 {
515         const ex ebasis = basis.evalm();
516         const ex eexponent = exponent.evalm();
517         if (is_ex_of_type(ebasis,matrix)) {
518                 if (is_ex_of_type(eexponent,numeric)) {
519                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
520                 }
521         }
522         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
523 }
524
525 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
526 {
527         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
528         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
529
530         if (are_ex_trivially_equal(basis, subsed_basis)
531          && are_ex_trivially_equal(exponent, subsed_exponent))
532                 return basic::subs(ls, lr, no_pattern);
533         else
534                 return power(subsed_basis, subsed_exponent).basic::subs(ls, lr, no_pattern);
535 }
536
537 ex power::simplify_ncmul(const exvector & v) const
538 {
539         return inherited::simplify_ncmul(v);
540 }
541
542 // protected
543
544 /** Implementation of ex::diff() for a power.
545  *  @see ex::diff */
546 ex power::derivative(const symbol & s) const
547 {
548         if (exponent.info(info_flags::real)) {
549                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
550                 epvector newseq;
551                 newseq.reserve(2);
552                 newseq.push_back(expair(basis, exponent - _ex1));
553                 newseq.push_back(expair(basis.diff(s), _ex1));
554                 return mul(newseq, exponent);
555         } else {
556                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
557                 return mul(*this,
558                            add(mul(exponent.diff(s), log(basis)),
559                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
560         }
561 }
562
563 int power::compare_same_type(const basic & other) const
564 {
565         GINAC_ASSERT(is_exactly_a<power>(other));
566         const power &o = static_cast<const power &>(other);
567
568         int cmpval = basis.compare(o.basis);
569         if (cmpval)
570                 return cmpval;
571         else
572                 return exponent.compare(o.exponent);
573 }
574
575 unsigned power::return_type(void) const
576 {
577         return basis.return_type();
578 }
579    
580 unsigned power::return_type_tinfo(void) const
581 {
582         return basis.return_type_tinfo();
583 }
584
585 ex power::expand(unsigned options) const
586 {
587         if (options == 0 && (flags & status_flags::expanded))
588                 return *this;
589         
590         const ex expanded_basis = basis.expand(options);
591         const ex expanded_exponent = exponent.expand(options);
592         
593         // x^(a+b) -> x^a * x^b
594         if (is_ex_exactly_of_type(expanded_exponent, add)) {
595                 const add &a = ex_to<add>(expanded_exponent);
596                 exvector distrseq;
597                 distrseq.reserve(a.seq.size() + 1);
598                 epvector::const_iterator last = a.seq.end();
599                 epvector::const_iterator cit = a.seq.begin();
600                 while (cit!=last) {
601                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
602                         ++cit;
603                 }
604                 
605                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
606                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
607                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
608                         int int_exponent = num_exponent.to_int();
609                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
610                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
611                         else
612                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
613                 } else
614                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
615                 
616                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
617                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
618                 return r.expand();
619         }
620         
621         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
622                 !ex_to<numeric>(expanded_exponent).is_integer()) {
623                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
624                         return this->hold();
625                 } else {
626                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
627                 }
628         }
629         
630         // integer numeric exponent
631         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
632         int int_exponent = num_exponent.to_int();
633         
634         // (x+y)^n, n>0
635         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
636                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
637         
638         // (x*y)^n -> x^n * y^n
639         if (is_ex_exactly_of_type(expanded_basis,mul))
640                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
641         
642         // cannot expand further
643         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
644                 return this->hold();
645         else
646                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
647 }
648
649 //////////
650 // new virtual functions which can be overridden by derived classes
651 //////////
652
653 // none
654
655 //////////
656 // non-virtual functions in this class
657 //////////
658
659 /** expand a^n where a is an add and n is a positive integer.
660  *  @see power::expand */
661 ex power::expand_add(const add & a, int n) const
662 {
663         if (n==2)
664                 return expand_add_2(a);
665
666         const int m = a.nops();
667         exvector result;
668         // The number of terms will be the number of combinatorial compositions,
669         // i.e. the number of unordered arrangement of m nonnegative integers
670         // which sum up to n.  It is frequently written as C_n(m) and directly
671         // related with binomial coefficients:
672         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
673         intvector k(m-1);
674         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
675         intvector upper_limit(m-1);
676         int l;
677
678         for (int l=0; l<m-1; ++l) {
679                 k[l] = 0;
680                 k_cum[l] = 0;
681                 upper_limit[l] = n;
682         }
683
684         while (true) {
685                 exvector term;
686                 term.reserve(m+1);
687                 for (l=0; l<m-1; ++l) {
688                         const ex & b = a.op(l);
689                         GINAC_ASSERT(!is_exactly_a<add>(b));
690                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
691                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
692                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
693                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
694                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
695                                      !is_exactly_a<power>(ex_to<power>(b).basis));
696                         if (is_ex_exactly_of_type(b,mul))
697                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
698                         else
699                                 term.push_back(power(b,k[l]));
700                 }
701
702                 const ex & b = a.op(l);
703                 GINAC_ASSERT(!is_exactly_a<add>(b));
704                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
705                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
706                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
707                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
708                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
709                              !is_exactly_a<power>(ex_to<power>(b).basis));
710                 if (is_ex_exactly_of_type(b,mul))
711                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
712                 else
713                         term.push_back(power(b,n-k_cum[m-2]));
714
715                 numeric f = binomial(numeric(n),numeric(k[0]));
716                 for (l=1; l<m-1; ++l)
717                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
718
719                 term.push_back(f);
720
721                 result.push_back((new mul(term))->setflag(status_flags::dynallocated));
722
723                 // increment k[]
724                 l = m-2;
725                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
726                         k[l] = 0;
727                         --l;
728                 }
729                 if (l<0) break;
730
731                 // recalc k_cum[] and upper_limit[]
732                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
733
734                 for (int i=l+1; i<m-1; ++i)
735                         k_cum[i] = k_cum[i-1]+k[i];
736
737                 for (int i=l+1; i<m-1; ++i)
738                         upper_limit[i] = n-k_cum[i-1];
739         }
740
741         return (new add(result))->setflag(status_flags::dynallocated |
742                                           status_flags::expanded);
743 }
744
745
746 /** Special case of power::expand_add. Expands a^2 where a is an add.
747  *  @see power::expand_add */
748 ex power::expand_add_2(const add & a) const
749 {
750         epvector sum;
751         unsigned a_nops = a.nops();
752         sum.reserve((a_nops*(a_nops+1))/2);
753         epvector::const_iterator last = a.seq.end();
754
755         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
756         // first part: ignore overall_coeff and expand other terms
757         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
758                 const ex & r = cit0->rest;
759                 const ex & c = cit0->coeff;
760                 
761                 GINAC_ASSERT(!is_exactly_a<add>(r));
762                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
763                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
764                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
765                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
766                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
767                              !is_exactly_a<power>(ex_to<power>(r).basis));
768                 
769                 if (are_ex_trivially_equal(c,_ex1)) {
770                         if (is_ex_exactly_of_type(r,mul)) {
771                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
772                                                      _ex1));
773                         } else {
774                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
775                                                      _ex1));
776                         }
777                 } else {
778                         if (is_ex_exactly_of_type(r,mul)) {
779                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r),_num2),
780                                                      ex_to<numeric>(c).power_dyn(_num2)));
781                         } else {
782                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
783                                                      ex_to<numeric>(c).power_dyn(_num2)));
784                         }
785                 }
786                         
787                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
788                         const ex & r1 = cit1->rest;
789                         const ex & c1 = cit1->coeff;
790                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
791                                                                       _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
792                 }
793         }
794         
795         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
796         
797         // second part: add terms coming from overall_factor (if != 0)
798         if (!a.overall_coeff.is_zero()) {
799                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
800                 while (i != end) {
801                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
802                         ++i;
803                 }
804                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
805         }
806         
807         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
808         
809         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
810 }
811
812 /** Expand factors of m in m^n where m is a mul and n is and integer.
813  *  @see power::expand */
814 ex power::expand_mul(const mul & m, const numeric & n) const
815 {
816         GINAC_ASSERT(n.is_integer());
817
818         if (n.is_zero())
819                 return _ex1;
820
821         epvector distrseq;
822         distrseq.reserve(m.seq.size());
823         epvector::const_iterator last = m.seq.end();
824         epvector::const_iterator cit = m.seq.begin();
825         while (cit!=last) {
826                 if (is_ex_exactly_of_type(cit->rest,numeric)) {
827                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
828                 } else {
829                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
830                         // since n is an integer
831                         distrseq.push_back(expair(cit->rest, ex_to<numeric>(cit->coeff).mul(n)));
832                 }
833                 ++cit;
834         }
835         return (new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
836 }
837
838 } // namespace GiNaC