53342a9e57eef4bd751a614bda3f3d61b543524a
[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) 1999-2000 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 "archive.h"
36 #include "debugmsg.h"
37 #include "utils.h"
38
39 #ifndef NO_NAMESPACE_GINAC
40 namespace GiNaC {
41 #endif // ndef NO_NAMESPACE_GINAC
42
43 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
44
45 typedef std::vector<int> intvector;
46
47 //////////
48 // default constructor, destructor, copy constructor assignment operator and helpers
49 //////////
50
51 // public
52
53 power::power() : basic(TINFO_power)
54 {
55         debugmsg("power default constructor",LOGLEVEL_CONSTRUCT);
56 }
57
58 power::~power()
59 {
60         debugmsg("power destructor",LOGLEVEL_DESTRUCT);
61         destroy(false);
62 }
63
64 power::power(const power & other)
65 {
66         debugmsg("power copy constructor",LOGLEVEL_CONSTRUCT);
67         copy(other);
68 }
69
70 const power & power::operator=(const power & other)
71 {
72         debugmsg("power operator=",LOGLEVEL_ASSIGNMENT);
73         if (this != &other) {
74                 destroy(true);
75                 copy(other);
76         }
77         return *this;
78 }
79
80 // protected
81
82 void power::copy(const power & other)
83 {
84         inherited::copy(other);
85         basis=other.basis;
86         exponent=other.exponent;
87 }
88
89 void power::destroy(bool call_parent)
90 {
91         if (call_parent) inherited::destroy(call_parent);
92 }
93
94 //////////
95 // other constructors
96 //////////
97
98 // public
99
100 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
101 {
102         debugmsg("power constructor from ex,ex",LOGLEVEL_CONSTRUCT);
103         GINAC_ASSERT(basis.return_type()==return_types::commutative);
104 }
105
106 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
107 {
108         debugmsg("power constructor from ex,numeric",LOGLEVEL_CONSTRUCT);
109         GINAC_ASSERT(basis.return_type()==return_types::commutative);
110 }
111
112 //////////
113 // archiving
114 //////////
115
116 /** Construct object from archive_node. */
117 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
118 {
119         debugmsg("power constructor from archive_node", LOGLEVEL_CONSTRUCT);
120         n.find_ex("basis", basis, sym_lst);
121         n.find_ex("exponent", exponent, sym_lst);
122 }
123
124 /** Unarchive the object. */
125 ex power::unarchive(const archive_node &n, const lst &sym_lst)
126 {
127         return (new power(n, sym_lst))->setflag(status_flags::dynallocated);
128 }
129
130 /** Archive the object. */
131 void power::archive(archive_node &n) const
132 {
133         inherited::archive(n);
134         n.add_ex("basis", basis);
135         n.add_ex("exponent", exponent);
136 }
137
138 //////////
139 // functions overriding virtual functions from bases classes
140 //////////
141
142 // public
143
144 basic * power::duplicate() const
145 {
146         debugmsg("power duplicate",LOGLEVEL_DUPLICATE);
147         return new power(*this);
148 }
149
150 void power::print(std::ostream & os, unsigned upper_precedence) const
151 {
152         debugmsg("power print",LOGLEVEL_PRINT);
153         if (exponent.is_equal(_ex1_2())) {
154                 os << "sqrt(" << basis << ")";
155         } else {
156                 if (precedence<=upper_precedence) os << "(";
157                 basis.print(os,precedence);
158                 os << "^";
159                 exponent.print(os,precedence);
160                 if (precedence<=upper_precedence) os << ")";
161         }
162 }
163
164 void power::printraw(std::ostream & os) const
165 {
166         debugmsg("power printraw",LOGLEVEL_PRINT);
167
168         os << "power(";
169         basis.printraw(os);
170         os << ",";
171         exponent.printraw(os);
172         os << ",hash=" << hashvalue << ",flags=" << flags << ")";
173 }
174
175 void power::printtree(std::ostream & os, unsigned indent) const
176 {
177         debugmsg("power printtree",LOGLEVEL_PRINT);
178
179         os << std::string(indent,' ') << "power: "
180            << "hash=" << hashvalue
181            << " (0x" << std::hex << hashvalue << std::dec << ")"
182            << ", flags=" << flags << std::endl;
183         basis.printtree(os, indent+delta_indent);
184         exponent.printtree(os, indent+delta_indent);
185 }
186
187 static void print_sym_pow(std::ostream & os, unsigned type, const symbol &x, int exp)
188 {
189         // Optimal output of integer powers of symbols to aid compiler CSE
190         if (exp == 1) {
191                 x.printcsrc(os, type, 0);
192         } else if (exp == 2) {
193                 x.printcsrc(os, type, 0);
194                 os << "*";
195                 x.printcsrc(os, type, 0);
196         } else if (exp & 1) {
197                 x.printcsrc(os, 0);
198                 os << "*";
199                 print_sym_pow(os, type, x, exp-1);
200         } else {
201                 os << "(";
202                 print_sym_pow(os, type, x, exp >> 1);
203                 os << ")*(";
204                 print_sym_pow(os, type, x, exp >> 1);
205                 os << ")";
206         }
207 }
208
209 void power::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
210 {
211         debugmsg("power print csrc", LOGLEVEL_PRINT);
212         
213         // Integer powers of symbols are printed in a special, optimized way
214         if (exponent.info(info_flags::integer)
215          && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
216                 int exp = ex_to_numeric(exponent).to_int();
217                 if (exp > 0)
218                         os << "(";
219                 else {
220                         exp = -exp;
221                         if (type == csrc_types::ctype_cl_N)
222                                 os << "recip(";
223                         else
224                                 os << "1.0/(";
225                 }
226                 print_sym_pow(os, type, static_cast<const symbol &>(*basis.bp), exp);
227                 os << ")";
228
229         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
230         } else if (exponent.compare(_num_1()) == 0) {
231                 if (type == csrc_types::ctype_cl_N)
232                         os << "recip(";
233                 else
234                         os << "1.0/(";
235                 basis.bp->printcsrc(os, type, 0);
236                 os << ")";
237
238         // Otherwise, use the pow() or expt() (CLN) functions
239         } else {
240                 if (type == csrc_types::ctype_cl_N)
241                         os << "expt(";
242                 else
243                         os << "pow(";
244                 basis.bp->printcsrc(os, type, 0);
245                 os << ",";
246                 exponent.bp->printcsrc(os, type, 0);
247                 os << ")";
248         }
249 }
250
251 bool power::info(unsigned inf) const
252 {
253         switch (inf) {
254                 case info_flags::polynomial:
255                 case info_flags::integer_polynomial:
256                 case info_flags::cinteger_polynomial:
257                 case info_flags::rational_polynomial:
258                 case info_flags::crational_polynomial:
259                         return exponent.info(info_flags::nonnegint);
260                 case info_flags::rational_function:
261                         return exponent.info(info_flags::integer);
262                 case info_flags::algebraic:
263                         return (!exponent.info(info_flags::integer) ||
264                                         basis.info(inf));
265         }
266         return inherited::info(inf);
267 }
268
269 unsigned power::nops() const
270 {
271         return 2;
272 }
273
274 ex & power::let_op(int i)
275 {
276         GINAC_ASSERT(i>=0);
277         GINAC_ASSERT(i<2);
278
279         return i==0 ? basis : exponent;
280 }
281
282 int power::degree(const symbol & s) const
283 {
284         if (is_exactly_of_type(*exponent.bp,numeric)) {
285                 if ((*basis.bp).compare(s)==0)
286                         return ex_to_numeric(exponent).to_int();
287                 else
288                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
289         }
290         return 0;
291 }
292
293 int power::ldegree(const symbol & s) const 
294 {
295         if (is_exactly_of_type(*exponent.bp,numeric)) {
296                 if ((*basis.bp).compare(s)==0)
297                         return ex_to_numeric(exponent).to_int();
298                 else
299                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
300         }
301         return 0;
302 }
303
304 ex power::coeff(const symbol & s, int n) const
305 {
306         if ((*basis.bp).compare(s)!=0) {
307                 // basis not equal to s
308                 if (n==0) {
309                         return *this;
310                 } else {
311                         return _ex0();
312                 }
313         } else if (is_exactly_of_type(*exponent.bp,numeric)&&
314                            (static_cast<const numeric &>(*exponent.bp).compare(numeric(n))==0)) {
315                 return _ex1();
316         }
317
318         return _ex0();
319 }
320
321 ex power::eval(int level) const
322 {
323         // simplifications: ^(x,0) -> 1 (0^0 handled here)
324         //                  ^(x,1) -> x
325         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
326         //                  ^(1,x) -> 1
327         //                  ^(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)
328         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
329         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
330         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
331         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
332         
333         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
334         
335         if ((level==1) && (flags & status_flags::evaluated))
336                 return *this;
337         else if (level == -max_recursion_level)
338                 throw(std::runtime_error("max recursion level reached"));
339         
340         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
341         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
342         
343         bool basis_is_numerical = 0;
344         bool exponent_is_numerical = 0;
345         numeric * num_basis;
346         numeric * num_exponent;
347         
348         if (is_exactly_of_type(*ebasis.bp,numeric)) {
349                 basis_is_numerical = 1;
350                 num_basis = static_cast<numeric *>(ebasis.bp);
351         }
352         if (is_exactly_of_type(*eexponent.bp,numeric)) {
353                 exponent_is_numerical = 1;
354                 num_exponent = static_cast<numeric *>(eexponent.bp);
355         }
356         
357         // ^(x,0) -> 1 (0^0 also handled here)
358         if (eexponent.is_zero())
359                 if (ebasis.is_zero())
360                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
361                 else
362                         return _ex1();
363         
364         // ^(x,1) -> x
365         if (eexponent.is_equal(_ex1()))
366                 return ebasis;
367         
368         // ^(0,c1) -> 0 or exception (depending on real value of c1)
369         if (ebasis.is_zero() && exponent_is_numerical) {
370                 if ((num_exponent->real()).is_zero())
371                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
372                 else if ((num_exponent->real()).is_negative())
373                         throw (pole_error("power::eval(): division by zero",1));
374                 else
375                         return _ex0();
376         }
377         
378         // ^(1,x) -> 1
379         if (ebasis.is_equal(_ex1()))
380                 return _ex1();
381         
382         if (basis_is_numerical && exponent_is_numerical) {
383                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
384                 // except if c1,c2 are rational, but c1^c2 is not)
385                 bool basis_is_crational = num_basis->is_crational();
386                 bool exponent_is_crational = num_exponent->is_crational();
387                 numeric res = (*num_basis).power(*num_exponent);
388                 
389                 if ((!basis_is_crational || !exponent_is_crational)
390                         || res.is_crational()) {
391                         return res;
392                 }
393                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
394                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
395                 if (basis_is_crational && exponent_is_crational
396                         && num_exponent->is_real()
397                         && !num_exponent->is_integer()) {
398                         numeric n = num_exponent->numer();
399                         numeric m = num_exponent->denom();
400                         numeric r;
401                         numeric q = iquo(n, m, r);
402                         if (r.is_negative()) {
403                                 r = r.add(m);
404                                 q = q.sub(_num1());
405                         }
406                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
407                                 return this->hold();
408                         else {
409                                 epvector res;
410                                 res.push_back(expair(ebasis,r.div(m)));
411                                 return (new mul(res,ex(num_basis->power(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
412                         }
413                 }
414         }
415         
416         // ^(^(x,c1),c2) -> ^(x,c1*c2)
417         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
418         // case c1==1 should not happen, see below!)
419         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
420                 const power & sub_power = ex_to_power(ebasis);
421                 const ex & sub_basis = sub_power.basis;
422                 const ex & sub_exponent = sub_power.exponent;
423                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
424                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
425                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
426                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
427                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
428                         }
429                 }
430         }
431         
432         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
433         if (exponent_is_numerical && num_exponent->is_integer() &&
434                 is_ex_exactly_of_type(ebasis,mul)) {
435                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
436         }
437         
438         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
439         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
440         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
441                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
442                 const mul & mulref=ex_to_mul(ebasis);
443                 if (!mulref.overall_coeff.is_equal(_ex1())) {
444                         const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff);
445                         if (num_coeff.is_real()) {
446                                 if (num_coeff.is_positive()>0) {
447                                         mul * mulp=new mul(mulref);
448                                         mulp->overall_coeff=_ex1();
449                                         mulp->clearflag(status_flags::evaluated);
450                                         mulp->clearflag(status_flags::hash_calculated);
451                                         return (new mul(power(*mulp,exponent),
452                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
453                                 } else {
454                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
455                                         if (num_coeff.compare(_num_1())!=0) {
456                                                 mul * mulp=new mul(mulref);
457                                                 mulp->overall_coeff=_ex_1();
458                                                 mulp->clearflag(status_flags::evaluated);
459                                                 mulp->clearflag(status_flags::hash_calculated);
460                                                 return (new mul(power(*mulp,exponent),
461                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
462                                         }
463                                 }
464                         }
465                 }
466         }
467                 
468         if (are_ex_trivially_equal(ebasis,basis) &&
469                 are_ex_trivially_equal(eexponent,exponent)) {
470                 return this->hold();
471         }
472         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
473                                                                                                    status_flags::evaluated);
474 }
475
476 ex power::evalf(int level) const
477 {
478         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
479
480         ex ebasis;
481         ex eexponent;
482         
483         if (level==1) {
484                 ebasis = basis;
485                 eexponent = exponent;
486         } else if (level == -max_recursion_level) {
487                 throw(std::runtime_error("max recursion level reached"));
488         } else {
489                 ebasis = basis.evalf(level-1);
490                 if (!is_ex_exactly_of_type(eexponent,numeric))
491                         eexponent = exponent.evalf(level-1);
492                 else
493                         eexponent = exponent;
494         }
495
496         return power(ebasis,eexponent);
497 }
498
499 ex power::subs(const lst & ls, const lst & lr) const
500 {
501         const ex & subsed_basis=basis.subs(ls,lr);
502         const ex & subsed_exponent=exponent.subs(ls,lr);
503
504         if (are_ex_trivially_equal(basis,subsed_basis)&&
505                 are_ex_trivially_equal(exponent,subsed_exponent)) {
506                 return *this;
507         }
508         
509         return power(subsed_basis, subsed_exponent);
510 }
511
512 ex power::simplify_ncmul(const exvector & v) const
513 {
514         return inherited::simplify_ncmul(v);
515 }
516
517 // protected
518
519 /** Implementation of ex::diff() for a power.
520  *  @see ex::diff */
521 ex power::derivative(const symbol & s) const
522 {
523         if (exponent.info(info_flags::real)) {
524                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
525                 epvector newseq;
526                 newseq.reserve(2);
527                 newseq.push_back(expair(basis, exponent - _ex1()));
528                 newseq.push_back(expair(basis.diff(s), _ex1()));
529                 return mul(newseq, exponent);
530         } else {
531                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
532                 return mul(power(basis, exponent),
533                            add(mul(exponent.diff(s), log(basis)),
534                            mul(mul(exponent, basis.diff(s)), power(basis, -1))));
535         }
536 }
537
538 int power::compare_same_type(const basic & other) const
539 {
540         GINAC_ASSERT(is_exactly_of_type(other, power));
541         const power & o=static_cast<const power &>(const_cast<basic &>(other));
542
543         int cmpval;
544         cmpval=basis.compare(o.basis);
545         if (cmpval==0) {
546                 return exponent.compare(o.exponent);
547         }
548         return cmpval;
549 }
550
551 unsigned power::return_type(void) const
552 {
553         return basis.return_type();
554 }
555    
556 unsigned power::return_type_tinfo(void) const
557 {
558         return basis.return_type_tinfo();
559 }
560
561 ex power::expand(unsigned options) const
562 {
563         if (flags & status_flags::expanded)
564                 return *this;
565         
566         ex expanded_basis = basis.expand(options);
567         
568         if (!is_ex_exactly_of_type(exponent,numeric) ||
569                 !ex_to_numeric(exponent).is_integer()) {
570                 if (are_ex_trivially_equal(basis,expanded_basis)) {
571                         return this->hold();
572                 } else {
573                         return (new power(expanded_basis,exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
574                 }
575         }
576         
577         // integer numeric exponent
578         const numeric & num_exponent = ex_to_numeric(exponent);
579         int int_exponent = num_exponent.to_int();
580         
581         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
582                 return expand_add(ex_to_add(expanded_basis), int_exponent);
583         }
584         
585         if (is_ex_exactly_of_type(expanded_basis,mul)) {
586                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
587         }
588         
589         // cannot expand further
590         if (are_ex_trivially_equal(basis,expanded_basis)) {
591                 return this->hold();
592         } else {
593                 return (new power(expanded_basis,exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
594         }
595 }
596
597 //////////
598 // new virtual functions which can be overridden by derived classes
599 //////////
600
601 // none
602
603 //////////
604 // non-virtual functions in this class
605 //////////
606
607 /** expand a^n where a is an add and n is an integer.
608  *  @see power::expand */
609 ex power::expand_add(const add & a, int n) const
610 {
611         if (n==2)
612                 return expand_add_2(a);
613         
614         int m = a.nops();
615         exvector sum;
616         sum.reserve((n+1)*(m-1));
617         intvector k(m-1);
618         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
619         intvector upper_limit(m-1);
620         int l;
621         
622         for (int l=0; l<m-1; l++) {
623                 k[l] = 0;
624                 k_cum[l] = 0;
625                 upper_limit[l] = n;
626         }
627         
628         while (1) {
629                 exvector term;
630                 term.reserve(m+1);
631                 for (l=0; l<m-1; l++) {
632                         const ex & b = a.op(l);
633                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
634                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
635                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
636                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
637                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
638                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
639                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
640                         if (is_ex_exactly_of_type(b,mul)) {
641                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
642                         } else {
643                                 term.push_back(power(b,k[l]));
644                         }
645                 }
646                 
647                 const ex & b = a.op(l);
648                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
649                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
650                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
651                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
652                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
653                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
654                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
655                 if (is_ex_exactly_of_type(b,mul)) {
656                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
657                 } else {
658                         term.push_back(power(b,n-k_cum[m-2]));
659                 }
660                 
661                 numeric f = binomial(numeric(n),numeric(k[0]));
662                 for (l=1; l<m-1; l++) {
663                         f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
664                 }
665                 term.push_back(f);
666
667                 /*
668                 cout << "begin term" << endl;
669                 for (int i=0; i<m-1; i++) {
670                         cout << "k[" << i << "]=" << k[i] << endl;
671                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
672                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
673                 }
674                 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
675                         cout << *cit << endl;
676                 }
677                 cout << "end term" << endl;
678                 */
679
680                 // TODO: optimize this
681                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
682                 
683                 // increment k[]
684                 l=m-2;
685                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
686                         k[l]=0;    
687                         l--;
688                 }
689                 if (l<0) break;
690
691                 // recalc k_cum[] and upper_limit[]
692                 if (l==0) {
693                         k_cum[0]=k[0];
694                 } else {
695                         k_cum[l]=k_cum[l-1]+k[l];
696                 }
697                 for (int i=l+1; i<m-1; i++) {
698                         k_cum[i]=k_cum[i-1]+k[i];
699                 }
700
701                 for (int i=l+1; i<m-1; i++) {
702                         upper_limit[i]=n-k_cum[i-1];
703                 }   
704         }
705         return (new add(sum))->setflag(status_flags::dynallocated |
706                                                                    status_flags::expanded );
707 }
708
709
710 /** Special case of power::expand_add. Expands a^2 where a is an add.
711  *  @see power::expand_add */
712 ex power::expand_add_2(const add & a) const
713 {
714         epvector sum;
715         unsigned a_nops=a.nops();
716         sum.reserve((a_nops*(a_nops+1))/2);
717         epvector::const_iterator last=a.seq.end();
718
719         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
720         // first part: ignore overall_coeff and expand other terms
721         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
722                 const ex & r=(*cit0).rest;
723                 const ex & c=(*cit0).coeff;
724                 
725                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
726                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
727                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
728                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
729                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
730                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
731                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
732
733                 if (are_ex_trivially_equal(c,_ex1())) {
734                         if (is_ex_exactly_of_type(r,mul)) {
735                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
736                                                      _ex1()));
737                         } else {
738                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
739                                                      _ex1()));
740                         }
741                 } else {
742                         if (is_ex_exactly_of_type(r,mul)) {
743                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
744                                                      ex_to_numeric(c).power_dyn(_num2())));
745                         } else {
746                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
747                                                      ex_to_numeric(c).power_dyn(_num2())));
748                         }
749                 }
750                         
751                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
752                         const ex & r1=(*cit1).rest;
753                         const ex & c1=(*cit1).coeff;
754                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
755                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
756                 }
757         }
758
759         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
760
761         // second part: add terms coming from overall_factor (if != 0)
762         if (!a.overall_coeff.is_equal(_ex0())) {
763                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
764                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
765                 }
766                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
767         }
768                 
769         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
770         
771         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
772 }
773
774 /** Expand factors of m in m^n where m is a mul and n is and integer
775  *  @see power::expand */
776 ex power::expand_mul(const mul & m, const numeric & n) const
777 {
778         if (n.is_equal(_num0()))
779                 return _ex1();
780         
781         epvector distrseq;
782         distrseq.reserve(m.seq.size());
783         epvector::const_iterator last = m.seq.end();
784         epvector::const_iterator cit = m.seq.begin();
785         while (cit!=last) {
786                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
787                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
788                 } else {
789                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
790                         // since n is an integer
791                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
792                 }
793                 ++cit;
794         }
795         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
796 }
797
798 /*
799 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
800                                unsigned options) const
801 {
802         // obsolete
803
804         exvector distrseq;
805         epvector splitseq;
806
807         const add & addref=static_cast<const add &>(*basis.bp);
808
809         splitseq=addref.seq;
810         splitseq.pop_back();
811         ex first_operands=add(splitseq);
812         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
813         
814         int n=exponent.to_int();
815         for (int k=0; k<=n; k++) {
816                 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
817                                                  * power(last_operand,numeric(n-k)));
818         }
819         return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
820 }
821 */
822
823 /*
824 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
825                                                                 unsigned options) const
826 {
827         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
828                         expand(options | expand_options::internal_do_not_expand_power_operands);
829
830         return ex(mul(rest_power,basis),0).
831                expand(options | expand_options::internal_do_not_expand_mul_operands);
832 }
833 */
834
835 //////////
836 // static member variables
837 //////////
838
839 // protected
840
841 unsigned power::precedence = 60;
842
843 //////////
844 // global constants
845 //////////
846
847 const power some_power;
848 const std::type_info & typeid_power=typeid(some_power);
849
850 // helper function
851
852 ex sqrt(const ex & a)
853 {
854         return power(a,_ex1_2());
855 }
856
857 #ifndef NO_NAMESPACE_GINAC
858 } // namespace GiNaC
859 #endif // ndef NO_NAMESPACE_GINAC