f292ff8b0daecc4f4d50d710d2358128eb8f2551
[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 "debugmsg.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         debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
55 }
56
57 void power::copy(const power & other)
58 {
59         inherited::copy(other);
60         basis = other.basis;
61         exponent = other.exponent;
62 }
63
64 DEFAULT_DESTROY(power)
65
66 //////////
67 // other ctors
68 //////////
69
70 // all inlined
71
72 //////////
73 // archiving
74 //////////
75
76 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
77 {
78         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
79         n.find_ex("basis", basis, sym_lst);
80         n.find_ex("exponent", exponent, sym_lst);
81 }
82
83 void power::archive(archive_node &n) const
84 {
85         inherited::archive(n);
86         n.add_ex("basis", basis);
87         n.add_ex("exponent", exponent);
88 }
89
90 DEFAULT_UNARCHIVE(power)
91
92 //////////
93 // functions overriding virtual functions from base classes
94 //////////
95
96 // public
97
98 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
99 {
100         // Optimal output of integer powers of symbols to aid compiler CSE.
101         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
102         // to learn why such a parenthisation is really necessary.
103         if (exp == 1) {
104                 x.print(c);
105         } else if (exp == 2) {
106                 x.print(c);
107                 c.s << "*";
108                 x.print(c);
109         } else if (exp & 1) {
110                 x.print(c);
111                 c.s << "*";
112                 print_sym_pow(c, x, exp-1);
113         } else {
114                 c.s << "(";
115                 print_sym_pow(c, x, exp >> 1);
116                 c.s << ")*(";
117                 print_sym_pow(c, x, exp >> 1);
118                 c.s << ")";
119         }
120 }
121
122 void power::print(const print_context & c, unsigned level) const
123 {
124         debugmsg("power print", LOGLEVEL_PRINT);
125
126         if (is_a<print_tree>(c)) {
127
128                 inherited::print(c, level);
129
130         } else if (is_a<print_csrc>(c)) {
131
132                 // Integer powers of symbols are printed in a special, optimized way
133                 if (exponent.info(info_flags::integer)
134                  && (is_exactly_a<symbol>(basis) || is_exactly_a<constant>(basis))) {
135                         int exp = ex_to<numeric>(exponent).to_int();
136                         if (exp > 0)
137                                 c.s << '(';
138                         else {
139                                 exp = -exp;
140                                 if (is_a<print_csrc_cl_N>(c))
141                                         c.s << "recip(";
142                                 else
143                                         c.s << "1.0/(";
144                         }
145                         print_sym_pow(c, ex_to<symbol>(basis), exp);
146                         c.s << ')';
147
148                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
149                 } else if (exponent.compare(_num_1()) == 0) {
150                         if (is_a<print_csrc_cl_N>(c))
151                                 c.s << "recip(";
152                         else
153                                 c.s << "1.0/(";
154                         basis.print(c);
155                         c.s << ')';
156
157                 // Otherwise, use the pow() or expt() (CLN) functions
158                 } else {
159                         if (is_a<print_csrc_cl_N>(c))
160                                 c.s << "expt(";
161                         else
162                                 c.s << "pow(";
163                         basis.print(c);
164                         c.s << ',';
165                         exponent.print(c);
166                         c.s << ')';
167                 }
168
169         } else {
170
171                 if (exponent.is_equal(_ex1_2())) {
172                         if (is_a<print_latex>(c))
173                                 c.s << "\\sqrt{";
174                         else
175                                 c.s << "sqrt(";
176                         basis.print(c);
177                         if (is_a<print_latex>(c))
178                                 c.s << '}';
179                         else
180                                 c.s << ')';
181                 } else {
182                         if (precedence() <= level) {
183                                 if (is_a<print_latex>(c))
184                                         c.s << "{(";
185                                 else
186                                         c.s << "(";
187                         }
188                         basis.print(c, precedence());
189                         c.s << '^';
190                         if (is_a<print_latex>(c))
191                                 c.s << '{';
192                         exponent.print(c, precedence());
193                         if (is_a<print_latex>(c))
194                                 c.s << '}';
195                         if (precedence() <= level) {
196                                 if (is_a<print_latex>(c))
197                                         c.s << ")}";
198                                 else
199                                         c.s << ')';
200                         }
201                 }
202         }
203 }
204
205 bool power::info(unsigned inf) const
206 {
207         switch (inf) {
208                 case info_flags::polynomial:
209                 case info_flags::integer_polynomial:
210                 case info_flags::cinteger_polynomial:
211                 case info_flags::rational_polynomial:
212                 case info_flags::crational_polynomial:
213                         return exponent.info(info_flags::nonnegint);
214                 case info_flags::rational_function:
215                         return exponent.info(info_flags::integer);
216                 case info_flags::algebraic:
217                         return (!exponent.info(info_flags::integer) ||
218                                         basis.info(inf));
219         }
220         return inherited::info(inf);
221 }
222
223 unsigned power::nops() const
224 {
225         return 2;
226 }
227
228 ex & power::let_op(int i)
229 {
230         GINAC_ASSERT(i>=0);
231         GINAC_ASSERT(i<2);
232
233         return i==0 ? basis : exponent;
234 }
235
236 ex power::map(map_function & f) const
237 {
238         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
239 }
240
241 int power::degree(const ex & s) const
242 {
243         if (is_exactly_of_type(*exponent.bp,numeric)) {
244                 if (basis.is_equal(s)) {
245                         if (ex_to<numeric>(exponent).is_integer())
246                                 return ex_to<numeric>(exponent).to_int();
247                         else
248                                 return 0;
249                 } else
250                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
251         }
252         return 0;
253 }
254
255 int power::ldegree(const ex & s) const 
256 {
257         if (is_exactly_of_type(*exponent.bp,numeric)) {
258                 if (basis.is_equal(s)) {
259                         if (ex_to<numeric>(exponent).is_integer())
260                                 return ex_to<numeric>(exponent).to_int();
261                         else
262                                 return 0;
263                 } else
264                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
265         }
266         return 0;
267 }
268
269 ex power::coeff(const ex & s, int n) const
270 {
271         if (!basis.is_equal(s)) {
272                 // basis not equal to s
273                 if (n == 0)
274                         return *this;
275                 else
276                         return _ex0();
277         } else {
278                 // basis equal to s
279                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to<numeric>(exponent).is_integer()) {
280                         // integer exponent
281                         int int_exp = ex_to<numeric>(exponent).to_int();
282                         if (n == int_exp)
283                                 return _ex1();
284                         else
285                                 return _ex0();
286                 } else {
287                         // non-integer exponents are treated as zero
288                         if (n == 0)
289                                 return *this;
290                         else
291                                 return _ex0();
292                 }
293         }
294 }
295
296 /** Perform automatic term rewriting rules in this class.  In the following
297  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
298  *  stand for such expressions that contain a plain number.
299  *  - ^(x,0) -> 1  (also handles ^(0,0))
300  *  - ^(x,1) -> x
301  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
302  *  - ^(1,x) -> 1
303  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
304  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
305  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
306  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
307  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
308  *
309  *  @param level cut-off in recursive evaluation */
310 ex power::eval(int level) const
311 {
312         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
313         
314         if ((level==1) && (flags & status_flags::evaluated))
315                 return *this;
316         else if (level == -max_recursion_level)
317                 throw(std::runtime_error("max recursion level reached"));
318         
319         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
320         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
321         
322         bool basis_is_numerical = false;
323         bool exponent_is_numerical = false;
324         const numeric *num_basis;
325         const numeric *num_exponent;
326         
327         if (is_exactly_of_type(*ebasis.bp,numeric)) {
328                 basis_is_numerical = true;
329                 num_basis = static_cast<const numeric *>(ebasis.bp);
330         }
331         if (is_exactly_of_type(*eexponent.bp,numeric)) {
332                 exponent_is_numerical = true;
333                 num_exponent = static_cast<const numeric *>(eexponent.bp);
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.compare(_num_1())!=0) {
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         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
488         
489         ex ebasis;
490         ex eexponent;
491         
492         if (level==1) {
493                 ebasis = basis;
494                 eexponent = exponent;
495         } else if (level == -max_recursion_level) {
496                 throw(std::runtime_error("max recursion level reached"));
497         } else {
498                 ebasis = basis.evalf(level-1);
499                 if (!is_ex_exactly_of_type(eexponent,numeric))
500                         eexponent = exponent.evalf(level-1);
501                 else
502                         eexponent = exponent;
503         }
504
505         return power(ebasis,eexponent);
506 }
507
508 ex power::evalm(void) const
509 {
510         const ex ebasis = basis.evalm();
511         const ex eexponent = exponent.evalm();
512         if (is_ex_of_type(ebasis,matrix)) {
513                 if (is_ex_of_type(eexponent,numeric)) {
514                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
515                 }
516         }
517         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
518 }
519
520 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
521 {
522         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
523         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
524
525         if (are_ex_trivially_equal(basis, subsed_basis)
526          && are_ex_trivially_equal(exponent, subsed_exponent))
527                 return basic::subs(ls, lr, no_pattern);
528         else
529                 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
530 }
531
532 ex power::simplify_ncmul(const exvector & v) const
533 {
534         return inherited::simplify_ncmul(v);
535 }
536
537 // protected
538
539 /** Implementation of ex::diff() for a power.
540  *  @see ex::diff */
541 ex power::derivative(const symbol & s) const
542 {
543         if (exponent.info(info_flags::real)) {
544                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
545                 epvector newseq;
546                 newseq.reserve(2);
547                 newseq.push_back(expair(basis, exponent - _ex1()));
548                 newseq.push_back(expair(basis.diff(s), _ex1()));
549                 return mul(newseq, exponent);
550         } else {
551                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
552                 return mul(*this,
553                            add(mul(exponent.diff(s), log(basis)),
554                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
555         }
556 }
557
558 int power::compare_same_type(const basic & other) const
559 {
560         GINAC_ASSERT(is_exactly_of_type(other, power));
561         const power &o = static_cast<const power &>(other);
562
563         int cmpval = basis.compare(o.basis);
564         if (cmpval)
565                 return cmpval;
566         else
567                 return exponent.compare(o.exponent);
568 }
569
570 unsigned power::return_type(void) const
571 {
572         return basis.return_type();
573 }
574    
575 unsigned power::return_type_tinfo(void) const
576 {
577         return basis.return_type_tinfo();
578 }
579
580 ex power::expand(unsigned options) const
581 {
582         if (options == 0 && (flags & status_flags::expanded))
583                 return *this;
584         
585         const ex expanded_basis = basis.expand(options);
586         const ex expanded_exponent = exponent.expand(options);
587         
588         // x^(a+b) -> x^a * x^b
589         if (is_ex_exactly_of_type(expanded_exponent, add)) {
590                 const add &a = ex_to<add>(expanded_exponent);
591                 exvector distrseq;
592                 distrseq.reserve(a.seq.size() + 1);
593                 epvector::const_iterator last = a.seq.end();
594                 epvector::const_iterator cit = a.seq.begin();
595                 while (cit!=last) {
596                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
597                         ++cit;
598                 }
599                 
600                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
601                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
602                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
603                         int int_exponent = num_exponent.to_int();
604                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
605                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
606                         else
607                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
608                 } else
609                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
610                 
611                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
612                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
613                 return r.expand();
614         }
615         
616         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
617                 !ex_to<numeric>(expanded_exponent).is_integer()) {
618                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
619                         return this->hold();
620                 } else {
621                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
622                 }
623         }
624         
625         // integer numeric exponent
626         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
627         int int_exponent = num_exponent.to_int();
628         
629         // (x+y)^n, n>0
630         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
631                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
632         
633         // (x*y)^n -> x^n * y^n
634         if (is_ex_exactly_of_type(expanded_basis,mul))
635                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
636         
637         // cannot expand further
638         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
639                 return this->hold();
640         else
641                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
642 }
643
644 //////////
645 // new virtual functions which can be overridden by derived classes
646 //////////
647
648 // none
649
650 //////////
651 // non-virtual functions in this class
652 //////////
653
654 /** expand a^n where a is an add and n is an integer.
655  *  @see power::expand */
656 ex power::expand_add(const add & a, int n) const
657 {
658         if (n==2)
659                 return expand_add_2(a);
660         
661         int m = a.nops();
662         exvector sum;
663         sum.reserve((n+1)*(m-1));
664         intvector k(m-1);
665         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
666         intvector upper_limit(m-1);
667         int l;
668         
669         for (int l=0; l<m-1; l++) {
670                 k[l] = 0;
671                 k_cum[l] = 0;
672                 upper_limit[l] = n;
673         }
674         
675         while (true) {
676                 exvector term;
677                 term.reserve(m+1);
678                 for (l=0; l<m-1; l++) {
679                         const ex & b = a.op(l);
680                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
681                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
682                                      !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
683                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
684                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
685                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
686                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
687                         if (is_ex_exactly_of_type(b,mul))
688                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
689                         else
690                                 term.push_back(power(b,k[l]));
691                 }
692                 
693                 const ex & b = a.op(l);
694                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
695                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
696                              !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
697                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
698                              !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
699                              !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
700                              !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
701                 if (is_ex_exactly_of_type(b,mul))
702                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
703                 else
704                         term.push_back(power(b,n-k_cum[m-2]));
705                 
706                 numeric f = binomial(numeric(n),numeric(k[0]));
707                 for (l=1; l<m-1; l++)
708                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
709                 
710                 term.push_back(f);
711                 
712                 // TODO: Can we optimize this?  Alex seemed to think so...
713                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
714                 
715                 // increment k[]
716                 l = m-2;
717                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
718                         k[l] = 0;    
719                         --l;
720                 }
721                 if (l<0) break;
722                 
723                 // recalc k_cum[] and upper_limit[]
724                 if (l==0)
725                         k_cum[0] = k[0];
726                 else
727                         k_cum[l] = k_cum[l-1]+k[l];
728                 
729                 for (int i=l+1; i<m-1; i++)
730                         k_cum[i] = k_cum[i-1]+k[i];
731                 
732                 for (int i=l+1; i<m-1; i++)
733                         upper_limit[i] = n-k_cum[i-1];
734         }
735         return (new add(sum))->setflag(status_flags::dynallocated |
736                                        status_flags::expanded );
737 }
738
739
740 /** Special case of power::expand_add. Expands a^2 where a is an add.
741  *  @see power::expand_add */
742 ex power::expand_add_2(const add & a) const
743 {
744         epvector sum;
745         unsigned a_nops = a.nops();
746         sum.reserve((a_nops*(a_nops+1))/2);
747         epvector::const_iterator last = a.seq.end();
748         
749         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
750         // first part: ignore overall_coeff and expand other terms
751         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
752                 const ex & r = cit0->rest;
753                 const ex & c = cit0->coeff;
754                 
755                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
756                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
757                              !is_ex_exactly_of_type(ex_to<power>(r).exponent,numeric) ||
758                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
759                              !is_ex_exactly_of_type(ex_to<power>(r).basis,add) ||
760                              !is_ex_exactly_of_type(ex_to<power>(r).basis,mul) ||
761                              !is_ex_exactly_of_type(ex_to<power>(r).basis,power));
762                 
763                 if (are_ex_trivially_equal(c,_ex1())) {
764                         if (is_ex_exactly_of_type(r,mul)) {
765                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
766                                                      _ex1()));
767                         } else {
768                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
769                                                      _ex1()));
770                         }
771                 } else {
772                         if (is_ex_exactly_of_type(r,mul)) {
773                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
774                                                      ex_to<numeric>(c).power_dyn(_num2())));
775                         } else {
776                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
777                                                      ex_to<numeric>(c).power_dyn(_num2())));
778                         }
779                 }
780                         
781                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
782                         const ex & r1 = cit1->rest;
783                         const ex & c1 = cit1->coeff;
784                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
785                                                                       _num2().mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
786                 }
787         }
788         
789         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
790         
791         // second part: add terms coming from overall_factor (if != 0)
792         if (!a.overall_coeff.is_zero()) {
793                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
794                 while (i != end) {
795                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2())));
796                         ++i;
797                 }
798                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2()),_ex1()));
799         }
800         
801         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
802         
803         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
804 }
805
806 /** Expand factors of m in m^n where m is a mul and n is and integer
807  *  @see power::expand */
808 ex power::expand_mul(const mul & m, const numeric & n) const
809 {
810         if (n.is_zero())
811                 return _ex1();
812         
813         epvector distrseq;
814         distrseq.reserve(m.seq.size());
815         epvector::const_iterator last = m.seq.end();
816         epvector::const_iterator cit = m.seq.begin();
817         while (cit!=last) {
818                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
819                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
820                 } else {
821                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
822                         // since n is an integer
823                         distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
824                 }
825                 ++cit;
826         }
827         return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
828 }
829
830 // helper function
831
832 ex sqrt(const ex & a)
833 {
834         return power(a,_ex1_2());
835 }
836
837 } // namespace GiNaC