109eacfeae4e4b7c025ec8bfab2c4930a490d672
[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 "numeric.h"
32 #include "inifcns.h"
33 #include "relational.h"
34 #include "symbol.h"
35 #include "print.h"
36 #include "archive.h"
37 #include "debugmsg.h"
38 #include "utils.h"
39
40 namespace GiNaC {
41
42 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
43
44 typedef std::vector<int> intvector;
45
46 //////////
47 // default ctor, dtor, copy ctor assignment operator and helpers
48 //////////
49
50 power::power() : basic(TINFO_power)
51 {
52         debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
53 }
54
55 void power::copy(const power & other)
56 {
57         inherited::copy(other);
58         basis = other.basis;
59         exponent = other.exponent;
60 }
61
62 DEFAULT_DESTROY(power)
63
64 //////////
65 // other ctors
66 //////////
67
68 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
69 {
70         debugmsg("power ctor from ex,ex",LOGLEVEL_CONSTRUCT);
71         GINAC_ASSERT(basis.return_type()==return_types::commutative);
72 }
73
74 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
75 {
76         debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
77         GINAC_ASSERT(basis.return_type()==return_types::commutative);
78 }
79
80 //////////
81 // archiving
82 //////////
83
84 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
85 {
86         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
87         n.find_ex("basis", basis, sym_lst);
88         n.find_ex("exponent", exponent, sym_lst);
89 }
90
91 void power::archive(archive_node &n) const
92 {
93         inherited::archive(n);
94         n.add_ex("basis", basis);
95         n.add_ex("exponent", exponent);
96 }
97
98 DEFAULT_UNARCHIVE(power)
99
100 //////////
101 // functions overriding virtual functions from bases classes
102 //////////
103
104 // public
105
106 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
107 {
108         // Optimal output of integer powers of symbols to aid compiler CSE.
109         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
110         // to learn why such a hack is really necessary.
111         if (exp == 1) {
112                 x.print(c);
113         } else if (exp == 2) {
114                 x.print(c);
115                 c.s << "*";
116                 x.print(c);
117         } else if (exp & 1) {
118                 x.print(c);
119                 c.s << "*";
120                 print_sym_pow(c, x, exp-1);
121         } else {
122                 c.s << "(";
123                 print_sym_pow(c, x, exp >> 1);
124                 c.s << ")*(";
125                 print_sym_pow(c, x, exp >> 1);
126                 c.s << ")";
127         }
128 }
129
130 void power::print(const print_context & c, unsigned level) const
131 {
132         debugmsg("power print", LOGLEVEL_PRINT);
133
134         if (is_of_type(c, print_tree)) {
135
136                 inherited::print(c, level);
137
138         } else if (is_of_type(c, print_csrc)) {
139
140                 // Integer powers of symbols are printed in a special, optimized way
141                 if (exponent.info(info_flags::integer)
142                  && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
143                         int exp = ex_to_numeric(exponent).to_int();
144                         if (exp > 0)
145                                 c.s << "(";
146                         else {
147                                 exp = -exp;
148                                 if (is_of_type(c, print_csrc_cl_N))
149                                         c.s << "recip(";
150                                 else
151                                         c.s << "1.0/(";
152                         }
153                         print_sym_pow(c, ex_to_symbol(basis), exp);
154                         c.s << ")";
155
156                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
157                 } else if (exponent.compare(_num_1()) == 0) {
158                         if (is_of_type(c, print_csrc_cl_N))
159                                 c.s << "recip(";
160                         else
161                                 c.s << "1.0/(";
162                         basis.print(c);
163                         c.s << ")";
164
165                 // Otherwise, use the pow() or expt() (CLN) functions
166                 } else {
167                         if (is_of_type(c, print_csrc_cl_N))
168                                 c.s << "expt(";
169                         else
170                                 c.s << "pow(";
171                         basis.print(c);
172                         c.s << ",";
173                         exponent.print(c);
174                         c.s << ")";
175                 }
176
177         } else {
178
179                 if (exponent.is_equal(_ex1_2())) {
180                         c.s << "sqrt(";
181                         basis.print(c);
182                         c.s << ")";
183                 } else {
184                         if (precedence <= level)
185                                 c.s << "(";
186                         basis.print(c, precedence);
187                         c.s << "^";
188                         exponent.print(c, precedence);
189                         if (precedence <= level)
190                                 c.s << ")";
191                 }
192         }
193 }
194
195 bool power::info(unsigned inf) const
196 {
197         switch (inf) {
198                 case info_flags::polynomial:
199                 case info_flags::integer_polynomial:
200                 case info_flags::cinteger_polynomial:
201                 case info_flags::rational_polynomial:
202                 case info_flags::crational_polynomial:
203                         return exponent.info(info_flags::nonnegint);
204                 case info_flags::rational_function:
205                         return exponent.info(info_flags::integer);
206                 case info_flags::algebraic:
207                         return (!exponent.info(info_flags::integer) ||
208                                         basis.info(inf));
209         }
210         return inherited::info(inf);
211 }
212
213 unsigned power::nops() const
214 {
215         return 2;
216 }
217
218 ex & power::let_op(int i)
219 {
220         GINAC_ASSERT(i>=0);
221         GINAC_ASSERT(i<2);
222
223         return i==0 ? basis : exponent;
224 }
225
226 int power::degree(const ex & s) const
227 {
228         if (is_exactly_of_type(*exponent.bp,numeric)) {
229                 if (basis.is_equal(s)) {
230                         if (ex_to_numeric(exponent).is_integer())
231                                 return ex_to_numeric(exponent).to_int();
232                         else
233                                 return 0;
234                 } else
235                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
236         }
237         return 0;
238 }
239
240 int power::ldegree(const ex & s) const 
241 {
242         if (is_exactly_of_type(*exponent.bp,numeric)) {
243                 if (basis.is_equal(s)) {
244                         if (ex_to_numeric(exponent).is_integer())
245                                 return ex_to_numeric(exponent).to_int();
246                         else
247                                 return 0;
248                 } else
249                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
250         }
251         return 0;
252 }
253
254 ex power::coeff(const ex & s, int n) const
255 {
256         if (!basis.is_equal(s)) {
257                 // basis not equal to s
258                 if (n == 0)
259                         return *this;
260                 else
261                         return _ex0();
262         } else {
263                 // basis equal to s
264                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
265                         // integer exponent
266                         int int_exp = ex_to_numeric(exponent).to_int();
267                         if (n == int_exp)
268                                 return _ex1();
269                         else
270                                 return _ex0();
271                 } else {
272                         // non-integer exponents are treated as zero
273                         if (n == 0)
274                                 return *this;
275                         else
276                                 return _ex0();
277                 }
278         }
279 }
280
281 ex power::eval(int level) const
282 {
283         // simplifications: ^(x,0) -> 1 (0^0 handled here)
284         //                  ^(x,1) -> x
285         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
286         //                  ^(1,x) -> 1
287         //                  ^(c1,c2) -> *(c1^n,c1^(c2-n)) (c1, c2 numeric(), 0<(c2-n)<1 except if c1,c2 are rational, but c1^c2 is not)
288         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
289         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
290         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
291         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
292         
293         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
294         
295         if ((level==1) && (flags & status_flags::evaluated))
296                 return *this;
297         else if (level == -max_recursion_level)
298                 throw(std::runtime_error("max recursion level reached"));
299         
300         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
301         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
302         
303         bool basis_is_numerical = 0;
304         bool exponent_is_numerical = 0;
305         numeric * num_basis;
306         numeric * num_exponent;
307         
308         if (is_exactly_of_type(*ebasis.bp,numeric)) {
309                 basis_is_numerical = 1;
310                 num_basis = static_cast<numeric *>(ebasis.bp);
311         }
312         if (is_exactly_of_type(*eexponent.bp,numeric)) {
313                 exponent_is_numerical = 1;
314                 num_exponent = static_cast<numeric *>(eexponent.bp);
315         }
316         
317         // ^(x,0) -> 1 (0^0 also handled here)
318         if (eexponent.is_zero()) {
319                 if (ebasis.is_zero())
320                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
321                 else
322                         return _ex1();
323         }
324         
325         // ^(x,1) -> x
326         if (eexponent.is_equal(_ex1()))
327                 return ebasis;
328         
329         // ^(0,c1) -> 0 or exception (depending on real value of c1)
330         if (ebasis.is_zero() && exponent_is_numerical) {
331                 if ((num_exponent->real()).is_zero())
332                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
333                 else if ((num_exponent->real()).is_negative())
334                         throw (pole_error("power::eval(): division by zero",1));
335                 else
336                         return _ex0();
337         }
338         
339         // ^(1,x) -> 1
340         if (ebasis.is_equal(_ex1()))
341                 return _ex1();
342         
343         if (basis_is_numerical && exponent_is_numerical) {
344                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
345                 // except if c1,c2 are rational, but c1^c2 is not)
346                 bool basis_is_crational = num_basis->is_crational();
347                 bool exponent_is_crational = num_exponent->is_crational();
348                 numeric res = num_basis->power(*num_exponent);
349                 
350                 if ((!basis_is_crational || !exponent_is_crational)
351                         || res.is_crational()) {
352                         return res;
353                 }
354                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
355                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
356                 if (basis_is_crational && exponent_is_crational
357                         && num_exponent->is_real()
358                         && !num_exponent->is_integer()) {
359                         numeric n = num_exponent->numer();
360                         numeric m = num_exponent->denom();
361                         numeric r;
362                         numeric q = iquo(n, m, r);
363                         if (r.is_negative()) {
364                                 r = r.add(m);
365                                 q = q.sub(_num1());
366                         }
367                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
368                                 return this->hold();
369                         else {
370                                 epvector res;
371                                 res.push_back(expair(ebasis,r.div(m)));
372                                 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
373                         }
374                 }
375         }
376         
377         // ^(^(x,c1),c2) -> ^(x,c1*c2)
378         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
379         // case c1==1 should not happen, see below!)
380         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
381                 const power & sub_power = ex_to_power(ebasis);
382                 const ex & sub_basis = sub_power.basis;
383                 const ex & sub_exponent = sub_power.exponent;
384                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
385                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
386                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
387                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1)
388                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
389                 }
390         }
391         
392         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
393         if (exponent_is_numerical && num_exponent->is_integer() &&
394                 is_ex_exactly_of_type(ebasis,mul)) {
395                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
396         }
397         
398         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
399         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
400         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
401                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
402                 const mul & mulref = ex_to_mul(ebasis);
403                 if (!mulref.overall_coeff.is_equal(_ex1())) {
404                         const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
405                         if (num_coeff.is_real()) {
406                                 if (num_coeff.is_positive()) {
407                                         mul * mulp = new mul(mulref);
408                                         mulp->overall_coeff = _ex1();
409                                         mulp->clearflag(status_flags::evaluated);
410                                         mulp->clearflag(status_flags::hash_calculated);
411                                         return (new mul(power(*mulp,exponent),
412                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
413                                 } else {
414                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
415                                         if (num_coeff.compare(_num_1())!=0) {
416                                                 mul * mulp = new mul(mulref);
417                                                 mulp->overall_coeff = _ex_1();
418                                                 mulp->clearflag(status_flags::evaluated);
419                                                 mulp->clearflag(status_flags::hash_calculated);
420                                                 return (new mul(power(*mulp,exponent),
421                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
422                                         }
423                                 }
424                         }
425                 }
426         }
427         
428         if (are_ex_trivially_equal(ebasis,basis) &&
429                 are_ex_trivially_equal(eexponent,exponent)) {
430                 return this->hold();
431         }
432         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
433                                                                                                    status_flags::evaluated);
434 }
435
436 ex power::evalf(int level) const
437 {
438         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
439
440         ex ebasis;
441         ex eexponent;
442         
443         if (level==1) {
444                 ebasis = basis;
445                 eexponent = exponent;
446         } else if (level == -max_recursion_level) {
447                 throw(std::runtime_error("max recursion level reached"));
448         } else {
449                 ebasis = basis.evalf(level-1);
450                 if (!is_ex_exactly_of_type(eexponent,numeric))
451                         eexponent = exponent.evalf(level-1);
452                 else
453                         eexponent = exponent;
454         }
455
456         return power(ebasis,eexponent);
457 }
458
459 ex power::subs(const lst & ls, const lst & lr) const
460 {
461         const ex & subsed_basis=basis.subs(ls,lr);
462         const ex & subsed_exponent=exponent.subs(ls,lr);
463
464         if (are_ex_trivially_equal(basis,subsed_basis)&&
465                 are_ex_trivially_equal(exponent,subsed_exponent)) {
466                 return inherited::subs(ls, lr);
467         }
468         
469         return power(subsed_basis, subsed_exponent);
470 }
471
472 ex power::simplify_ncmul(const exvector & v) const
473 {
474         return inherited::simplify_ncmul(v);
475 }
476
477 // protected
478
479 /** Implementation of ex::diff() for a power.
480  *  @see ex::diff */
481 ex power::derivative(const symbol & s) const
482 {
483         if (exponent.info(info_flags::real)) {
484                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
485                 epvector newseq;
486                 newseq.reserve(2);
487                 newseq.push_back(expair(basis, exponent - _ex1()));
488                 newseq.push_back(expair(basis.diff(s), _ex1()));
489                 return mul(newseq, exponent);
490         } else {
491                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
492                 return mul(*this,
493                            add(mul(exponent.diff(s), log(basis)),
494                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
495         }
496 }
497
498 int power::compare_same_type(const basic & other) const
499 {
500         GINAC_ASSERT(is_exactly_of_type(other, power));
501         const power & o=static_cast<const power &>(const_cast<basic &>(other));
502
503         int cmpval;
504         cmpval=basis.compare(o.basis);
505         if (cmpval==0) {
506                 return exponent.compare(o.exponent);
507         }
508         return cmpval;
509 }
510
511 unsigned power::return_type(void) const
512 {
513         return basis.return_type();
514 }
515    
516 unsigned power::return_type_tinfo(void) const
517 {
518         return basis.return_type_tinfo();
519 }
520
521 ex power::expand(unsigned options) const
522 {
523         if (flags & status_flags::expanded)
524                 return *this;
525         
526         ex expanded_basis = basis.expand(options);
527         ex expanded_exponent = exponent.expand(options);
528         
529         // x^(a+b) -> x^a * x^b
530         if (is_ex_exactly_of_type(expanded_exponent, add)) {
531                 const add &a = ex_to_add(expanded_exponent);
532                 exvector distrseq;
533                 distrseq.reserve(a.seq.size() + 1);
534                 epvector::const_iterator last = a.seq.end();
535                 epvector::const_iterator cit = a.seq.begin();
536                 while (cit!=last) {
537                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
538                         cit++;
539                 }
540                 
541                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
542                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
543                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
544                         int int_exponent = num_exponent.to_int();
545                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
546                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
547                         else
548                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
549                 } else
550                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
551                 
552                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
553                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
554                 return r.expand();
555         }
556         
557         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
558                 !ex_to_numeric(expanded_exponent).is_integer()) {
559                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
560                         return this->hold();
561                 } else {
562                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
563                 }
564         }
565         
566         // integer numeric exponent
567         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
568         int int_exponent = num_exponent.to_int();
569         
570         // (x+y)^n, n>0
571         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
572                 return expand_add(ex_to_add(expanded_basis), int_exponent);
573         
574         // (x*y)^n -> x^n * y^n
575         if (is_ex_exactly_of_type(expanded_basis,mul))
576                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
577         
578         // cannot expand further
579         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
580                 return this->hold();
581         else
582                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
583 }
584
585 //////////
586 // new virtual functions which can be overridden by derived classes
587 //////////
588
589 // none
590
591 //////////
592 // non-virtual functions in this class
593 //////////
594
595 /** expand a^n where a is an add and n is an integer.
596  *  @see power::expand */
597 ex power::expand_add(const add & a, int n) const
598 {
599         if (n==2)
600                 return expand_add_2(a);
601         
602         int m = a.nops();
603         exvector sum;
604         sum.reserve((n+1)*(m-1));
605         intvector k(m-1);
606         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
607         intvector upper_limit(m-1);
608         int l;
609         
610         for (int l=0; l<m-1; l++) {
611                 k[l] = 0;
612                 k_cum[l] = 0;
613                 upper_limit[l] = n;
614         }
615         
616         while (1) {
617                 exvector term;
618                 term.reserve(m+1);
619                 for (l=0; l<m-1; l++) {
620                         const ex & b = a.op(l);
621                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
622                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
623                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
624                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
625                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
626                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
627                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
628                         if (is_ex_exactly_of_type(b,mul))
629                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
630                         else
631                                 term.push_back(power(b,k[l]));
632                 }
633                 
634                 const ex & b = a.op(l);
635                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
636                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
637                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
638                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
639                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
640                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
641                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
642                 if (is_ex_exactly_of_type(b,mul))
643                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
644                 else
645                         term.push_back(power(b,n-k_cum[m-2]));
646                 
647                 numeric f = binomial(numeric(n),numeric(k[0]));
648                 for (l=1; l<m-1; l++)
649                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
650                 
651                 term.push_back(f);
652                 
653                 /*
654                 cout << "begin term" << endl;
655                 for (int i=0; i<m-1; i++) {
656                         cout << "k[" << i << "]=" << k[i] << endl;
657                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
658                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
659                 }
660                 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
661                         cout << *cit << endl;
662                 }
663                 cout << "end term" << endl;
664                 */
665                 
666                 // TODO: optimize this
667                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
668                 
669                 // increment k[]
670                 l = m-2;
671                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
672                         k[l] = 0;    
673                         l--;
674                 }
675                 if (l<0) break;
676                 
677                 // recalc k_cum[] and upper_limit[]
678                 if (l==0)
679                         k_cum[0] = k[0];
680                 else
681                         k_cum[l] = k_cum[l-1]+k[l];
682                 
683                 for (int i=l+1; i<m-1; i++)
684                         k_cum[i] = k_cum[i-1]+k[i];
685                 
686                 for (int i=l+1; i<m-1; i++)
687                         upper_limit[i] = n-k_cum[i-1];
688         }
689         return (new add(sum))->setflag(status_flags::dynallocated |
690                                                                    status_flags::expanded );
691 }
692
693
694 /** Special case of power::expand_add. Expands a^2 where a is an add.
695  *  @see power::expand_add */
696 ex power::expand_add_2(const add & a) const
697 {
698         epvector sum;
699         unsigned a_nops = a.nops();
700         sum.reserve((a_nops*(a_nops+1))/2);
701         epvector::const_iterator last = a.seq.end();
702         
703         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
704         // first part: ignore overall_coeff and expand other terms
705         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
706                 const ex & r = (*cit0).rest;
707                 const ex & c = (*cit0).coeff;
708                 
709                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
710                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
711                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
712                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
713                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
714                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
715                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
716                 
717                 if (are_ex_trivially_equal(c,_ex1())) {
718                         if (is_ex_exactly_of_type(r,mul)) {
719                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
720                                                      _ex1()));
721                         } else {
722                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
723                                                      _ex1()));
724                         }
725                 } else {
726                         if (is_ex_exactly_of_type(r,mul)) {
727                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
728                                                      ex_to_numeric(c).power_dyn(_num2())));
729                         } else {
730                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
731                                                      ex_to_numeric(c).power_dyn(_num2())));
732                         }
733                 }
734                         
735                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
736                         const ex & r1 = (*cit1).rest;
737                         const ex & c1 = (*cit1).coeff;
738                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
739                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
740                 }
741         }
742         
743         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
744         
745         // second part: add terms coming from overall_factor (if != 0)
746         if (!a.overall_coeff.is_zero()) {
747                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
748                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
749                 }
750                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
751         }
752         
753         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
754         
755         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
756 }
757
758 /** Expand factors of m in m^n where m is a mul and n is and integer
759  *  @see power::expand */
760 ex power::expand_mul(const mul & m, const numeric & n) const
761 {
762         if (n.is_zero())
763                 return _ex1();
764         
765         epvector distrseq;
766         distrseq.reserve(m.seq.size());
767         epvector::const_iterator last = m.seq.end();
768         epvector::const_iterator cit = m.seq.begin();
769         while (cit!=last) {
770                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
771                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
772                 } else {
773                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
774                         // since n is an integer
775                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
776                 }
777                 ++cit;
778         }
779         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
780 }
781
782 /*
783 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
784                                unsigned options) const
785 {
786         // obsolete
787
788         exvector distrseq;
789         epvector splitseq;
790
791         const add & addref=static_cast<const add &>(*basis.bp);
792
793         splitseq=addref.seq;
794         splitseq.pop_back();
795         ex first_operands=add(splitseq);
796         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
797         
798         int n=exponent.to_int();
799         for (int k=0; k<=n; k++) {
800                 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
801                                                  * power(last_operand,numeric(n-k)));
802         }
803         return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
804 }
805 */
806
807 /*
808 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
809                                                                 unsigned options) const
810 {
811         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
812                         expand(options | expand_options::internal_do_not_expand_power_operands);
813
814         return ex(mul(rest_power,basis),0).
815                expand(options | expand_options::internal_do_not_expand_mul_operands);
816 }
817 */
818
819 //////////
820 // static member variables
821 //////////
822
823 // protected
824
825 unsigned power::precedence = 60;
826
827 // helper function
828
829 ex sqrt(const ex & a)
830 {
831         return power(a,_ex1_2());
832 }
833
834 } // namespace GiNaC