- in the output, list delimiters are now { } and matrix delimiters are [ ]
[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 "inifcns.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 }
72
73 /** Ctor from an ex and a bare numeric.  This is somewhat more efficient than
74  *  the normal ctor from two ex whenever it can be used. */
75 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
76 {
77         debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
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                         if (is_of_type(c, print_latex))
181                                 c.s << "\\sqrt{";
182                         else
183                                 c.s << "sqrt(";
184                         basis.print(c);
185                         if (is_of_type(c, print_latex))
186                                 c.s << '}';
187                         else
188                                 c.s << ')';
189                 } else {
190                         if (precedence() <= level) {
191                                 if (is_of_type(c, print_latex))
192                                         c.s << "{(";
193                                 else
194                                         c.s << "(";
195                         }
196                         basis.print(c, precedence());
197                         c.s << '^';
198                         if (is_of_type(c, print_latex))
199                                 c.s << '{';
200                         exponent.print(c, precedence());
201                         if (is_of_type(c, print_latex))
202                                 c.s << '}';
203                         if (precedence() <= level) {
204                                 if (is_of_type(c, print_latex))
205                                         c.s << ")}";
206                                 else
207                                         c.s << ')';
208                         }
209                 }
210         }
211 }
212
213 bool power::info(unsigned inf) const
214 {
215         switch (inf) {
216                 case info_flags::polynomial:
217                 case info_flags::integer_polynomial:
218                 case info_flags::cinteger_polynomial:
219                 case info_flags::rational_polynomial:
220                 case info_flags::crational_polynomial:
221                         return exponent.info(info_flags::nonnegint);
222                 case info_flags::rational_function:
223                         return exponent.info(info_flags::integer);
224                 case info_flags::algebraic:
225                         return (!exponent.info(info_flags::integer) ||
226                                         basis.info(inf));
227         }
228         return inherited::info(inf);
229 }
230
231 unsigned power::nops() const
232 {
233         return 2;
234 }
235
236 ex & power::let_op(int i)
237 {
238         GINAC_ASSERT(i>=0);
239         GINAC_ASSERT(i<2);
240
241         return i==0 ? basis : exponent;
242 }
243
244 ex power::map(map_func f) const
245 {
246         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
247 }
248
249 int power::degree(const ex & s) const
250 {
251         if (is_exactly_of_type(*exponent.bp,numeric)) {
252                 if (basis.is_equal(s)) {
253                         if (ex_to_numeric(exponent).is_integer())
254                                 return ex_to_numeric(exponent).to_int();
255                         else
256                                 return 0;
257                 } else
258                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
259         }
260         return 0;
261 }
262
263 int power::ldegree(const ex & s) const 
264 {
265         if (is_exactly_of_type(*exponent.bp,numeric)) {
266                 if (basis.is_equal(s)) {
267                         if (ex_to_numeric(exponent).is_integer())
268                                 return ex_to_numeric(exponent).to_int();
269                         else
270                                 return 0;
271                 } else
272                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
273         }
274         return 0;
275 }
276
277 ex power::coeff(const ex & s, int n) const
278 {
279         if (!basis.is_equal(s)) {
280                 // basis not equal to s
281                 if (n == 0)
282                         return *this;
283                 else
284                         return _ex0();
285         } else {
286                 // basis equal to s
287                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
288                         // integer exponent
289                         int int_exp = ex_to_numeric(exponent).to_int();
290                         if (n == int_exp)
291                                 return _ex1();
292                         else
293                                 return _ex0();
294                 } else {
295                         // non-integer exponents are treated as zero
296                         if (n == 0)
297                                 return *this;
298                         else
299                                 return _ex0();
300                 }
301         }
302 }
303
304 ex power::eval(int level) const
305 {
306         // simplifications: ^(x,0) -> 1 (0^0 handled here)
307         //                  ^(x,1) -> x
308         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
309         //                  ^(1,x) -> 1
310         //                  ^(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)
311         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
312         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
313         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
314         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
315         
316         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
317         
318         if ((level==1) && (flags & status_flags::evaluated))
319                 return *this;
320         else if (level == -max_recursion_level)
321                 throw(std::runtime_error("max recursion level reached"));
322         
323         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
324         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
325         
326         bool basis_is_numerical = false;
327         bool exponent_is_numerical = false;
328         numeric * num_basis;
329         numeric * num_exponent;
330         
331         if (is_exactly_of_type(*ebasis.bp,numeric)) {
332                 basis_is_numerical = true;
333                 num_basis = static_cast<numeric *>(ebasis.bp);
334         }
335         if (is_exactly_of_type(*eexponent.bp,numeric)) {
336                 exponent_is_numerical = true;
337                 num_exponent = static_cast<numeric *>(eexponent.bp);
338         }
339         
340         // ^(x,0) -> 1 (0^0 also handled here)
341         if (eexponent.is_zero()) {
342                 if (ebasis.is_zero())
343                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
344                 else
345                         return _ex1();
346         }
347         
348         // ^(x,1) -> x
349         if (eexponent.is_equal(_ex1()))
350                 return ebasis;
351         
352         // ^(0,c1) -> 0 or exception (depending on real value of c1)
353         if (ebasis.is_zero() && exponent_is_numerical) {
354                 if ((num_exponent->real()).is_zero())
355                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
356                 else if ((num_exponent->real()).is_negative())
357                         throw (pole_error("power::eval(): division by zero",1));
358                 else
359                         return _ex0();
360         }
361         
362         // ^(1,x) -> 1
363         if (ebasis.is_equal(_ex1()))
364                 return _ex1();
365         
366         if (exponent_is_numerical) {
367
368                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
369                 // except if c1,c2 are rational, but c1^c2 is not)
370                 if (basis_is_numerical) {
371                         bool basis_is_crational = num_basis->is_crational();
372                         bool exponent_is_crational = num_exponent->is_crational();
373                         numeric res = num_basis->power(*num_exponent);
374                 
375                         if ((!basis_is_crational || !exponent_is_crational)
376                                 || res.is_crational()) {
377                                 return res;
378                         }
379                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
380
381                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
382                         if (basis_is_crational && exponent_is_crational
383                                 && num_exponent->is_real()
384                                 && !num_exponent->is_integer()) {
385                                 numeric n = num_exponent->numer();
386                                 numeric m = num_exponent->denom();
387                                 numeric r;
388                                 numeric q = iquo(n, m, r);
389                                 if (r.is_negative()) {
390                                         r = r.add(m);
391                                         q = q.sub(_num1());
392                                 }
393                                 if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
394                                         return this->hold();
395                                 else {
396                                         epvector res;
397                                         res.push_back(expair(ebasis,r.div(m)));
398                                         return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
399                                 }
400                         }
401                 }
402         
403                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
404                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
405                 // case c1==1 should not happen, see below!)
406                 if (is_ex_exactly_of_type(ebasis,power)) {
407                         const power & sub_power = ex_to_power(ebasis);
408                         const ex & sub_basis = sub_power.basis;
409                         const ex & sub_exponent = sub_power.exponent;
410                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
411                                 const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
412                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
413                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1()).is_negative())
414                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
415                         }
416                 }
417         
418                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
419                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
420                         return expand_mul(ex_to_mul(ebasis), *num_exponent);
421                 }
422         
423                 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
424                 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
425                 if (is_ex_exactly_of_type(ebasis,mul)) {
426                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
427                         const mul & mulref = ex_to_mul(ebasis);
428                         if (!mulref.overall_coeff.is_equal(_ex1())) {
429                                 const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
430                                 if (num_coeff.is_real()) {
431                                         if (num_coeff.is_positive()) {
432                                                 mul * mulp = new mul(mulref);
433                                                 mulp->overall_coeff = _ex1();
434                                                 mulp->clearflag(status_flags::evaluated);
435                                                 mulp->clearflag(status_flags::hash_calculated);
436                                                 return (new mul(power(*mulp,exponent),
437                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
438                                         } else {
439                                                 GINAC_ASSERT(num_coeff.compare(_num0())<0);
440                                                 if (num_coeff.compare(_num_1())!=0) {
441                                                         mul * mulp = new mul(mulref);
442                                                         mulp->overall_coeff = _ex_1();
443                                                         mulp->clearflag(status_flags::evaluated);
444                                                         mulp->clearflag(status_flags::hash_calculated);
445                                                         return (new mul(power(*mulp,exponent),
446                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
447                                                 }
448                                         }
449                                 }
450                         }
451                 }
452
453                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer)
454                 if (num_exponent->is_pos_integer() && ebasis.return_type() != return_types::commutative) {
455                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
456                 }
457         }
458         
459         if (are_ex_trivially_equal(ebasis,basis) &&
460                 are_ex_trivially_equal(eexponent,exponent)) {
461                 return this->hold();
462         }
463         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
464                                                                                                    status_flags::evaluated);
465 }
466
467 ex power::evalf(int level) const
468 {
469         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
470
471         ex ebasis;
472         ex eexponent;
473         
474         if (level==1) {
475                 ebasis = basis;
476                 eexponent = exponent;
477         } else if (level == -max_recursion_level) {
478                 throw(std::runtime_error("max recursion level reached"));
479         } else {
480                 ebasis = basis.evalf(level-1);
481                 if (!is_ex_exactly_of_type(eexponent,numeric))
482                         eexponent = exponent.evalf(level-1);
483                 else
484                         eexponent = exponent;
485         }
486
487         return power(ebasis,eexponent);
488 }
489
490 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
491 {
492         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
493         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
494
495         if (are_ex_trivially_equal(basis, subsed_basis)
496          && are_ex_trivially_equal(exponent, subsed_exponent))
497                 return basic::subs(ls, lr, no_pattern);
498         else
499                 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
500 }
501
502 ex power::simplify_ncmul(const exvector & v) const
503 {
504         return inherited::simplify_ncmul(v);
505 }
506
507 // protected
508
509 /** Implementation of ex::diff() for a power.
510  *  @see ex::diff */
511 ex power::derivative(const symbol & s) const
512 {
513         if (exponent.info(info_flags::real)) {
514                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
515                 epvector newseq;
516                 newseq.reserve(2);
517                 newseq.push_back(expair(basis, exponent - _ex1()));
518                 newseq.push_back(expair(basis.diff(s), _ex1()));
519                 return mul(newseq, exponent);
520         } else {
521                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
522                 return mul(*this,
523                            add(mul(exponent.diff(s), log(basis)),
524                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
525         }
526 }
527
528 int power::compare_same_type(const basic & other) const
529 {
530         GINAC_ASSERT(is_exactly_of_type(other, power));
531         const power & o=static_cast<const power &>(const_cast<basic &>(other));
532
533         int cmpval;
534         cmpval=basis.compare(o.basis);
535         if (cmpval==0) {
536                 return exponent.compare(o.exponent);
537         }
538         return cmpval;
539 }
540
541 unsigned power::return_type(void) const
542 {
543         return basis.return_type();
544 }
545    
546 unsigned power::return_type_tinfo(void) const
547 {
548         return basis.return_type_tinfo();
549 }
550
551 ex power::expand(unsigned options) const
552 {
553         if (flags & status_flags::expanded)
554                 return *this;
555         
556         ex expanded_basis = basis.expand(options);
557         ex expanded_exponent = exponent.expand(options);
558         
559         // x^(a+b) -> x^a * x^b
560         if (is_ex_exactly_of_type(expanded_exponent, add)) {
561                 const add &a = ex_to_add(expanded_exponent);
562                 exvector distrseq;
563                 distrseq.reserve(a.seq.size() + 1);
564                 epvector::const_iterator last = a.seq.end();
565                 epvector::const_iterator cit = a.seq.begin();
566                 while (cit!=last) {
567                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
568                         cit++;
569                 }
570                 
571                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
572                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
573                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
574                         int int_exponent = num_exponent.to_int();
575                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
576                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
577                         else
578                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
579                 } else
580                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
581                 
582                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
583                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
584                 return r.expand();
585         }
586         
587         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
588                 !ex_to_numeric(expanded_exponent).is_integer()) {
589                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
590                         return this->hold();
591                 } else {
592                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
593                 }
594         }
595         
596         // integer numeric exponent
597         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
598         int int_exponent = num_exponent.to_int();
599         
600         // (x+y)^n, n>0
601         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
602                 return expand_add(ex_to_add(expanded_basis), int_exponent);
603         
604         // (x*y)^n -> x^n * y^n
605         if (is_ex_exactly_of_type(expanded_basis,mul))
606                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
607         
608         // cannot expand further
609         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
610                 return this->hold();
611         else
612                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
613 }
614
615 //////////
616 // new virtual functions which can be overridden by derived classes
617 //////////
618
619 // none
620
621 //////////
622 // non-virtual functions in this class
623 //////////
624
625 /** expand a^n where a is an add and n is an integer.
626  *  @see power::expand */
627 ex power::expand_add(const add & a, int n) const
628 {
629         if (n==2)
630                 return expand_add_2(a);
631         
632         int m = a.nops();
633         exvector sum;
634         sum.reserve((n+1)*(m-1));
635         intvector k(m-1);
636         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
637         intvector upper_limit(m-1);
638         int l;
639         
640         for (int l=0; l<m-1; l++) {
641                 k[l] = 0;
642                 k_cum[l] = 0;
643                 upper_limit[l] = n;
644         }
645         
646         while (1) {
647                 exvector term;
648                 term.reserve(m+1);
649                 for (l=0; l<m-1; l++) {
650                         const ex & b = a.op(l);
651                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
652                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
653                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
654                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
655                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
656                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
657                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
658                         if (is_ex_exactly_of_type(b,mul))
659                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
660                         else
661                                 term.push_back(power(b,k[l]));
662                 }
663                 
664                 const ex & b = a.op(l);
665                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
666                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
667                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
668                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
669                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
670                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
671                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
672                 if (is_ex_exactly_of_type(b,mul))
673                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
674                 else
675                         term.push_back(power(b,n-k_cum[m-2]));
676                 
677                 numeric f = binomial(numeric(n),numeric(k[0]));
678                 for (l=1; l<m-1; l++)
679                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
680                 
681                 term.push_back(f);
682                 
683                 /*
684                 cout << "begin term" << endl;
685                 for (int i=0; i<m-1; i++) {
686                         cout << "k[" << i << "]=" << k[i] << endl;
687                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
688                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
689                 }
690                 for_each(term.begin(), term.end(), ostream_iterator<ex>(cout, "\n"));
691                 cout << "end term" << endl;
692                 */
693                 
694                 // TODO: optimize this
695                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
696                 
697                 // increment k[]
698                 l = m-2;
699                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
700                         k[l] = 0;    
701                         l--;
702                 }
703                 if (l<0) break;
704                 
705                 // recalc k_cum[] and upper_limit[]
706                 if (l==0)
707                         k_cum[0] = k[0];
708                 else
709                         k_cum[l] = k_cum[l-1]+k[l];
710                 
711                 for (int i=l+1; i<m-1; i++)
712                         k_cum[i] = k_cum[i-1]+k[i];
713                 
714                 for (int i=l+1; i<m-1; i++)
715                         upper_limit[i] = n-k_cum[i-1];
716         }
717         return (new add(sum))->setflag(status_flags::dynallocated |
718                                                                    status_flags::expanded );
719 }
720
721
722 /** Special case of power::expand_add. Expands a^2 where a is an add.
723  *  @see power::expand_add */
724 ex power::expand_add_2(const add & a) const
725 {
726         epvector sum;
727         unsigned a_nops = a.nops();
728         sum.reserve((a_nops*(a_nops+1))/2);
729         epvector::const_iterator last = a.seq.end();
730         
731         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
732         // first part: ignore overall_coeff and expand other terms
733         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
734                 const ex & r = (*cit0).rest;
735                 const ex & c = (*cit0).coeff;
736                 
737                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
738                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
739                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
740                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
741                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
742                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
743                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
744                 
745                 if (are_ex_trivially_equal(c,_ex1())) {
746                         if (is_ex_exactly_of_type(r,mul)) {
747                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
748                                                      _ex1()));
749                         } else {
750                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
751                                                      _ex1()));
752                         }
753                 } else {
754                         if (is_ex_exactly_of_type(r,mul)) {
755                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
756                                                      ex_to_numeric(c).power_dyn(_num2())));
757                         } else {
758                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
759                                                      ex_to_numeric(c).power_dyn(_num2())));
760                         }
761                 }
762                         
763                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
764                         const ex & r1 = (*cit1).rest;
765                         const ex & c1 = (*cit1).coeff;
766                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
767                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
768                 }
769         }
770         
771         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
772         
773         // second part: add terms coming from overall_factor (if != 0)
774         if (!a.overall_coeff.is_zero()) {
775                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
776                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
777                 }
778                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
779         }
780         
781         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
782         
783         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
784 }
785
786 /** Expand factors of m in m^n where m is a mul and n is and integer
787  *  @see power::expand */
788 ex power::expand_mul(const mul & m, const numeric & n) const
789 {
790         if (n.is_zero())
791                 return _ex1();
792         
793         epvector distrseq;
794         distrseq.reserve(m.seq.size());
795         epvector::const_iterator last = m.seq.end();
796         epvector::const_iterator cit = m.seq.begin();
797         while (cit!=last) {
798                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
799                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
800                 } else {
801                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
802                         // since n is an integer
803                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
804                 }
805                 ++cit;
806         }
807         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
808 }
809
810 /*
811 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
812                                                                 unsigned options) const
813 {
814         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
815                         expand(options | expand_options::internal_do_not_expand_power_operands);
816
817         return ex(mul(rest_power,basis),0).
818                expand(options | expand_options::internal_do_not_expand_mul_operands);
819 }
820 */
821
822 // helper function
823
824 ex sqrt(const ex & a)
825 {
826         return power(a,_ex1_2());
827 }
828
829 } // namespace GiNaC