* Supplement some (now deprecated) macros by inlined template functions:
[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() : basic(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 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
71 {
72         debugmsg("power ctor from ex,ex",LOGLEVEL_CONSTRUCT);
73 }
74
75 /** Ctor from an ex and a bare numeric.  This is somewhat more efficient than
76  *  the normal ctor from two ex whenever it can be used. */
77 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
78 {
79         debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
80 }
81
82 //////////
83 // archiving
84 //////////
85
86 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
87 {
88         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
89         n.find_ex("basis", basis, sym_lst);
90         n.find_ex("exponent", exponent, sym_lst);
91 }
92
93 void power::archive(archive_node &n) const
94 {
95         inherited::archive(n);
96         n.add_ex("basis", basis);
97         n.add_ex("exponent", exponent);
98 }
99
100 DEFAULT_UNARCHIVE(power)
101
102 //////////
103 // functions overriding virtual functions from bases classes
104 //////////
105
106 // public
107
108 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
109 {
110         // Optimal output of integer powers of symbols to aid compiler CSE.
111         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
112         // to learn why such a hack is really necessary.
113         if (exp == 1) {
114                 x.print(c);
115         } else if (exp == 2) {
116                 x.print(c);
117                 c.s << "*";
118                 x.print(c);
119         } else if (exp & 1) {
120                 x.print(c);
121                 c.s << "*";
122                 print_sym_pow(c, x, exp-1);
123         } else {
124                 c.s << "(";
125                 print_sym_pow(c, x, exp >> 1);
126                 c.s << ")*(";
127                 print_sym_pow(c, x, exp >> 1);
128                 c.s << ")";
129         }
130 }
131
132 void power::print(const print_context & c, unsigned level) const
133 {
134         debugmsg("power print", LOGLEVEL_PRINT);
135
136         if (is_a<print_tree>(c)) {
137
138                 inherited::print(c, level);
139
140         } else if (is_a<print_csrc>(c)) {
141
142                 // Integer powers of symbols are printed in a special, optimized way
143                 if (exponent.info(info_flags::integer)
144                  && (is_exactly_a<symbol>(basis) || is_exactly_a<constant>(basis))) {
145                         int exp = ex_to<numeric>(exponent).to_int();
146                         if (exp > 0)
147                                 c.s << '(';
148                         else {
149                                 exp = -exp;
150                                 if (is_a<print_csrc_cl_N>(c))
151                                         c.s << "recip(";
152                                 else
153                                         c.s << "1.0/(";
154                         }
155                         print_sym_pow(c, ex_to<symbol>(basis), exp);
156                         c.s << ')';
157
158                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
159                 } else if (exponent.compare(_num_1()) == 0) {
160                         if (is_a<print_csrc_cl_N>(c))
161                                 c.s << "recip(";
162                         else
163                                 c.s << "1.0/(";
164                         basis.print(c);
165                         c.s << ')';
166
167                 // Otherwise, use the pow() or expt() (CLN) functions
168                 } else {
169                         if (is_a<print_csrc_cl_N>(c))
170                                 c.s << "expt(";
171                         else
172                                 c.s << "pow(";
173                         basis.print(c);
174                         c.s << ',';
175                         exponent.print(c);
176                         c.s << ')';
177                 }
178
179         } else {
180
181                 if (exponent.is_equal(_ex1_2())) {
182                         if (is_a<print_latex>(c))
183                                 c.s << "\\sqrt{";
184                         else
185                                 c.s << "sqrt(";
186                         basis.print(c);
187                         if (is_a<print_latex>(c))
188                                 c.s << '}';
189                         else
190                                 c.s << ')';
191                 } else {
192                         if (precedence() <= level) {
193                                 if (is_a<print_latex>(c))
194                                         c.s << "{(";
195                                 else
196                                         c.s << "(";
197                         }
198                         basis.print(c, precedence());
199                         c.s << '^';
200                         if (is_a<print_latex>(c))
201                                 c.s << '{';
202                         exponent.print(c, precedence());
203                         if (is_a<print_latex>(c))
204                                 c.s << '}';
205                         if (precedence() <= level) {
206                                 if (is_a<print_latex>(c))
207                                         c.s << ")}";
208                                 else
209                                         c.s << ')';
210                         }
211                 }
212         }
213 }
214
215 bool power::info(unsigned inf) const
216 {
217         switch (inf) {
218                 case info_flags::polynomial:
219                 case info_flags::integer_polynomial:
220                 case info_flags::cinteger_polynomial:
221                 case info_flags::rational_polynomial:
222                 case info_flags::crational_polynomial:
223                         return exponent.info(info_flags::nonnegint);
224                 case info_flags::rational_function:
225                         return exponent.info(info_flags::integer);
226                 case info_flags::algebraic:
227                         return (!exponent.info(info_flags::integer) ||
228                                         basis.info(inf));
229         }
230         return inherited::info(inf);
231 }
232
233 unsigned power::nops() const
234 {
235         return 2;
236 }
237
238 ex & power::let_op(int i)
239 {
240         GINAC_ASSERT(i>=0);
241         GINAC_ASSERT(i<2);
242
243         return i==0 ? basis : exponent;
244 }
245
246 ex power::map(map_function & f) const
247 {
248         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
249 }
250
251 int power::degree(const ex & s) const
252 {
253         if (is_exactly_of_type(*exponent.bp,numeric)) {
254                 if (basis.is_equal(s)) {
255                         if (ex_to<numeric>(exponent).is_integer())
256                                 return ex_to<numeric>(exponent).to_int();
257                         else
258                                 return 0;
259                 } else
260                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
261         }
262         return 0;
263 }
264
265 int power::ldegree(const ex & s) const 
266 {
267         if (is_exactly_of_type(*exponent.bp,numeric)) {
268                 if (basis.is_equal(s)) {
269                         if (ex_to<numeric>(exponent).is_integer())
270                                 return ex_to<numeric>(exponent).to_int();
271                         else
272                                 return 0;
273                 } else
274                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
275         }
276         return 0;
277 }
278
279 ex power::coeff(const ex & s, int n) const
280 {
281         if (!basis.is_equal(s)) {
282                 // basis not equal to s
283                 if (n == 0)
284                         return *this;
285                 else
286                         return _ex0();
287         } else {
288                 // basis equal to s
289                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to<numeric>(exponent).is_integer()) {
290                         // integer exponent
291                         int int_exp = ex_to<numeric>(exponent).to_int();
292                         if (n == int_exp)
293                                 return _ex1();
294                         else
295                                 return _ex0();
296                 } else {
297                         // non-integer exponents are treated as zero
298                         if (n == 0)
299                                 return *this;
300                         else
301                                 return _ex0();
302                 }
303         }
304 }
305
306 ex power::eval(int level) const
307 {
308         // simplifications: ^(x,0) -> 1 (0^0 handled here)
309         //                  ^(x,1) -> x
310         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
311         //                  ^(1,x) -> 1
312         //                  ^(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)
313         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
314         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
315         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
316         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
317         
318         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
319         
320         if ((level==1) && (flags & status_flags::evaluated))
321                 return *this;
322         else if (level == -max_recursion_level)
323                 throw(std::runtime_error("max recursion level reached"));
324         
325         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
326         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
327         
328         bool basis_is_numerical = false;
329         bool exponent_is_numerical = false;
330         numeric * num_basis;
331         numeric * num_exponent;
332         
333         if (is_exactly_of_type(*ebasis.bp,numeric)) {
334                 basis_is_numerical = true;
335                 num_basis = static_cast<numeric *>(ebasis.bp);
336         }
337         if (is_exactly_of_type(*eexponent.bp,numeric)) {
338                 exponent_is_numerical = true;
339                 num_exponent = static_cast<numeric *>(eexponent.bp);
340         }
341         
342         // ^(x,0) -> 1 (0^0 also handled here)
343         if (eexponent.is_zero()) {
344                 if (ebasis.is_zero())
345                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
346                 else
347                         return _ex1();
348         }
349         
350         // ^(x,1) -> x
351         if (eexponent.is_equal(_ex1()))
352                 return ebasis;
353         
354         // ^(0,c1) -> 0 or exception (depending on real value of c1)
355         if (ebasis.is_zero() && exponent_is_numerical) {
356                 if ((num_exponent->real()).is_zero())
357                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
358                 else if ((num_exponent->real()).is_negative())
359                         throw (pole_error("power::eval(): division by zero",1));
360                 else
361                         return _ex0();
362         }
363         
364         // ^(1,x) -> 1
365         if (ebasis.is_equal(_ex1()))
366                 return _ex1();
367         
368         if (exponent_is_numerical) {
369
370                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
371                 // except if c1,c2 are rational, but c1^c2 is not)
372                 if (basis_is_numerical) {
373                         bool basis_is_crational = num_basis->is_crational();
374                         bool exponent_is_crational = num_exponent->is_crational();
375                         numeric res = num_basis->power(*num_exponent);
376                 
377                         if ((!basis_is_crational || !exponent_is_crational)
378                                 || res.is_crational()) {
379                                 return res;
380                         }
381                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
382
383                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
384                         if (basis_is_crational && exponent_is_crational
385                                 && num_exponent->is_real()
386                                 && !num_exponent->is_integer()) {
387                                 numeric n = num_exponent->numer();
388                                 numeric m = num_exponent->denom();
389                                 numeric r;
390                                 numeric q = iquo(n, m, r);
391                                 if (r.is_negative()) {
392                                         r = r.add(m);
393                                         q = q.sub(_num1());
394                                 }
395                                 if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
396                                         return this->hold();
397                                 else {
398                                         epvector res;
399                                         res.push_back(expair(ebasis,r.div(m)));
400                                         return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
401                                 }
402                         }
403                 }
404         
405                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
406                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
407                 // case c1==1 should not happen, see below!)
408                 if (is_ex_exactly_of_type(ebasis,power)) {
409                         const power & sub_power = ex_to<power>(ebasis);
410                         const ex & sub_basis = sub_power.basis;
411                         const ex & sub_exponent = sub_power.exponent;
412                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
413                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
414                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
415                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1()).is_negative())
416                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
417                         }
418                 }
419         
420                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
421                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
422                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
423                 }
424         
425                 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
426                 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
427                 if (is_ex_exactly_of_type(ebasis,mul)) {
428                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
429                         const mul & mulref = ex_to<mul>(ebasis);
430                         if (!mulref.overall_coeff.is_equal(_ex1())) {
431                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
432                                 if (num_coeff.is_real()) {
433                                         if (num_coeff.is_positive()) {
434                                                 mul * mulp = new mul(mulref);
435                                                 mulp->overall_coeff = _ex1();
436                                                 mulp->clearflag(status_flags::evaluated);
437                                                 mulp->clearflag(status_flags::hash_calculated);
438                                                 return (new mul(power(*mulp,exponent),
439                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
440                                         } else {
441                                                 GINAC_ASSERT(num_coeff.compare(_num0())<0);
442                                                 if (num_coeff.compare(_num_1())!=0) {
443                                                         mul * mulp = new mul(mulref);
444                                                         mulp->overall_coeff = _ex_1();
445                                                         mulp->clearflag(status_flags::evaluated);
446                                                         mulp->clearflag(status_flags::hash_calculated);
447                                                         return (new mul(power(*mulp,exponent),
448                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
449                                                 }
450                                         }
451                                 }
452                         }
453                 }
454
455                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
456                 if (num_exponent->is_pos_integer() &&
457                     ebasis.return_type() != return_types::commutative &&
458                     !is_ex_of_type(ebasis,matrix)) {
459                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
460                 }
461         }
462         
463         if (are_ex_trivially_equal(ebasis,basis) &&
464                 are_ex_trivially_equal(eexponent,exponent)) {
465                 return this->hold();
466         }
467         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
468                                                                                                    status_flags::evaluated);
469 }
470
471 ex power::evalf(int level) const
472 {
473         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
474
475         ex ebasis;
476         ex eexponent;
477         
478         if (level==1) {
479                 ebasis = basis;
480                 eexponent = exponent;
481         } else if (level == -max_recursion_level) {
482                 throw(std::runtime_error("max recursion level reached"));
483         } else {
484                 ebasis = basis.evalf(level-1);
485                 if (!is_ex_exactly_of_type(eexponent,numeric))
486                         eexponent = exponent.evalf(level-1);
487                 else
488                         eexponent = exponent;
489         }
490
491         return power(ebasis,eexponent);
492 }
493
494 ex power::evalm(void) const
495 {
496         ex ebasis = basis.evalm();
497         ex eexponent = exponent.evalm();
498         if (is_ex_of_type(ebasis,matrix)) {
499                 if (is_ex_of_type(eexponent,numeric)) {
500                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
501                 }
502         }
503         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
504 }
505
506 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
507 {
508         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
509         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
510
511         if (are_ex_trivially_equal(basis, subsed_basis)
512          && are_ex_trivially_equal(exponent, subsed_exponent))
513                 return basic::subs(ls, lr, no_pattern);
514         else
515                 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
516 }
517
518 ex power::simplify_ncmul(const exvector & v) const
519 {
520         return inherited::simplify_ncmul(v);
521 }
522
523 // protected
524
525 /** Implementation of ex::diff() for a power.
526  *  @see ex::diff */
527 ex power::derivative(const symbol & s) const
528 {
529         if (exponent.info(info_flags::real)) {
530                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
531                 epvector newseq;
532                 newseq.reserve(2);
533                 newseq.push_back(expair(basis, exponent - _ex1()));
534                 newseq.push_back(expair(basis.diff(s), _ex1()));
535                 return mul(newseq, exponent);
536         } else {
537                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
538                 return mul(*this,
539                            add(mul(exponent.diff(s), log(basis)),
540                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
541         }
542 }
543
544 int power::compare_same_type(const basic & other) const
545 {
546         GINAC_ASSERT(is_exactly_of_type(other, power));
547         const power & o=static_cast<const power &>(const_cast<basic &>(other));
548
549         int cmpval;
550         cmpval=basis.compare(o.basis);
551         if (cmpval==0) {
552                 return exponent.compare(o.exponent);
553         }
554         return cmpval;
555 }
556
557 unsigned power::return_type(void) const
558 {
559         return basis.return_type();
560 }
561    
562 unsigned power::return_type_tinfo(void) const
563 {
564         return basis.return_type_tinfo();
565 }
566
567 ex power::expand(unsigned options) const
568 {
569         if (flags & status_flags::expanded)
570                 return *this;
571         
572         ex expanded_basis = basis.expand(options);
573         ex expanded_exponent = exponent.expand(options);
574         
575         // x^(a+b) -> x^a * x^b
576         if (is_ex_exactly_of_type(expanded_exponent, add)) {
577                 const add &a = ex_to<add>(expanded_exponent);
578                 exvector distrseq;
579                 distrseq.reserve(a.seq.size() + 1);
580                 epvector::const_iterator last = a.seq.end();
581                 epvector::const_iterator cit = a.seq.begin();
582                 while (cit!=last) {
583                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
584                         cit++;
585                 }
586                 
587                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
588                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
589                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
590                         int int_exponent = num_exponent.to_int();
591                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
592                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
593                         else
594                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
595                 } else
596                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
597                 
598                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
599                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
600                 return r.expand();
601         }
602         
603         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
604                 !ex_to<numeric>(expanded_exponent).is_integer()) {
605                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
606                         return this->hold();
607                 } else {
608                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
609                 }
610         }
611         
612         // integer numeric exponent
613         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
614         int int_exponent = num_exponent.to_int();
615         
616         // (x+y)^n, n>0
617         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
618                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
619         
620         // (x*y)^n -> x^n * y^n
621         if (is_ex_exactly_of_type(expanded_basis,mul))
622                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
623         
624         // cannot expand further
625         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
626                 return this->hold();
627         else
628                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
629 }
630
631 //////////
632 // new virtual functions which can be overridden by derived classes
633 //////////
634
635 // none
636
637 //////////
638 // non-virtual functions in this class
639 //////////
640
641 /** expand a^n where a is an add and n is an integer.
642  *  @see power::expand */
643 ex power::expand_add(const add & a, int n) const
644 {
645         if (n==2)
646                 return expand_add_2(a);
647         
648         int m = a.nops();
649         exvector sum;
650         sum.reserve((n+1)*(m-1));
651         intvector k(m-1);
652         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
653         intvector upper_limit(m-1);
654         int l;
655         
656         for (int l=0; l<m-1; l++) {
657                 k[l] = 0;
658                 k_cum[l] = 0;
659                 upper_limit[l] = n;
660         }
661         
662         while (1) {
663                 exvector term;
664                 term.reserve(m+1);
665                 for (l=0; l<m-1; l++) {
666                         const ex & b = a.op(l);
667                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
668                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
669                                      !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
670                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
671                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
672                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
673                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
674                         if (is_ex_exactly_of_type(b,mul))
675                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
676                         else
677                                 term.push_back(power(b,k[l]));
678                 }
679                 
680                 const ex & b = a.op(l);
681                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
682                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
683                              !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
684                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
685                              !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
686                              !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
687                              !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
688                 if (is_ex_exactly_of_type(b,mul))
689                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
690                 else
691                         term.push_back(power(b,n-k_cum[m-2]));
692                 
693                 numeric f = binomial(numeric(n),numeric(k[0]));
694                 for (l=1; l<m-1; l++)
695                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
696                 
697                 term.push_back(f);
698                 
699                 /*
700                 cout << "begin term" << endl;
701                 for (int i=0; i<m-1; i++) {
702                         cout << "k[" << i << "]=" << k[i] << endl;
703                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
704                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
705                 }
706                 for_each(term.begin(), term.end(), ostream_iterator<ex>(cout, "\n"));
707                 cout << "end term" << endl;
708                 */
709                 
710                 // TODO: optimize this
711                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
712                 
713                 // increment k[]
714                 l = m-2;
715                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
716                         k[l] = 0;    
717                         l--;
718                 }
719                 if (l<0) break;
720                 
721                 // recalc k_cum[] and upper_limit[]
722                 if (l==0)
723                         k_cum[0] = k[0];
724                 else
725                         k_cum[l] = k_cum[l-1]+k[l];
726                 
727                 for (int i=l+1; i<m-1; i++)
728                         k_cum[i] = k_cum[i-1]+k[i];
729                 
730                 for (int i=l+1; i<m-1; i++)
731                         upper_limit[i] = n-k_cum[i-1];
732         }
733         return (new add(sum))->setflag(status_flags::dynallocated |
734                                                                    status_flags::expanded );
735 }
736
737
738 /** Special case of power::expand_add. Expands a^2 where a is an add.
739  *  @see power::expand_add */
740 ex power::expand_add_2(const add & a) const
741 {
742         epvector sum;
743         unsigned a_nops = a.nops();
744         sum.reserve((a_nops*(a_nops+1))/2);
745         epvector::const_iterator last = a.seq.end();
746         
747         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
748         // first part: ignore overall_coeff and expand other terms
749         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
750                 const ex & r = (*cit0).rest;
751                 const ex & c = (*cit0).coeff;
752                 
753                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
754                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
755                              !is_ex_exactly_of_type(ex_to<power>(r).exponent,numeric) ||
756                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
757                              !is_ex_exactly_of_type(ex_to<power>(r).basis,add) ||
758                              !is_ex_exactly_of_type(ex_to<power>(r).basis,mul) ||
759                              !is_ex_exactly_of_type(ex_to<power>(r).basis,power));
760                 
761                 if (are_ex_trivially_equal(c,_ex1())) {
762                         if (is_ex_exactly_of_type(r,mul)) {
763                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
764                                                      _ex1()));
765                         } else {
766                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
767                                                      _ex1()));
768                         }
769                 } else {
770                         if (is_ex_exactly_of_type(r,mul)) {
771                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
772                                                      ex_to<numeric>(c).power_dyn(_num2())));
773                         } else {
774                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
775                                                      ex_to<numeric>(c).power_dyn(_num2())));
776                         }
777                 }
778                         
779                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
780                         const ex & r1 = (*cit1).rest;
781                         const ex & c1 = (*cit1).coeff;
782                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
783                                                                       _num2().mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
784                 }
785         }
786         
787         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
788         
789         // second part: add terms coming from overall_factor (if != 0)
790         if (!a.overall_coeff.is_zero()) {
791                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
792                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to<numeric>(a.overall_coeff).mul_dyn(_num2())));
793                 }
794                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2()),_ex1()));
795         }
796         
797         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
798         
799         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
800 }
801
802 /** Expand factors of m in m^n where m is a mul and n is and integer
803  *  @see power::expand */
804 ex power::expand_mul(const mul & m, const numeric & n) const
805 {
806         if (n.is_zero())
807                 return _ex1();
808         
809         epvector distrseq;
810         distrseq.reserve(m.seq.size());
811         epvector::const_iterator last = m.seq.end();
812         epvector::const_iterator cit = m.seq.begin();
813         while (cit!=last) {
814                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
815                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
816                 } else {
817                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
818                         // since n is an integer
819                         distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
820                 }
821                 ++cit;
822         }
823         return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
824 }
825
826 /*
827 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
828                                                                 unsigned options) const
829 {
830         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
831                         expand(options | expand_options::internal_do_not_expand_power_operands);
832
833         return ex(mul(rest_power,basis),0).
834                expand(options | expand_options::internal_do_not_expand_mul_operands);
835 }
836 */
837
838 // helper function
839
840 ex sqrt(const ex & a)
841 {
842         return power(a,_ex1_2());
843 }
844
845 } // namespace GiNaC