- non-integer powers are treated as 0 by (l)degree() and coeff()
[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                         if (ex_to_numeric(exponent).is_integer())
287                                 return ex_to_numeric(exponent).to_int();
288                         else
289                                 return 0;
290                 } else
291                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
292         }
293         return 0;
294 }
295
296 int power::ldegree(const symbol & s) const 
297 {
298         if (is_exactly_of_type(*exponent.bp,numeric)) {
299                 if ((*basis.bp).compare(s)==0) {
300                         if (ex_to_numeric(exponent).is_integer())
301                                 return ex_to_numeric(exponent).to_int();
302                         else
303                                 return 0;
304                 } else
305                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
306         }
307         return 0;
308 }
309
310 ex power::coeff(const symbol & s, int n) const
311 {
312         if ((*basis.bp).compare(s)!=0) {
313                 // basis not equal to s
314                 if (n == 0)
315                         return *this;
316                 else
317                         return _ex0();
318         } else {
319                 // basis equal to s
320                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
321                         // integer exponent
322                         int int_exp = ex_to_numeric(exponent).to_int();
323                         if (n == int_exp)
324                                 return _ex1();
325                         else
326                                 return _ex0();
327                 } else {
328                         // non-integer exponents are treated as zero
329                         if (n == 0)
330                                 return *this;
331                         else
332                                 return _ex0();
333                 }
334         }
335 }
336
337 ex power::eval(int level) const
338 {
339         // simplifications: ^(x,0) -> 1 (0^0 handled here)
340         //                  ^(x,1) -> x
341         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
342         //                  ^(1,x) -> 1
343         //                  ^(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)
344         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
345         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
346         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
347         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
348         
349         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
350         
351         if ((level==1) && (flags & status_flags::evaluated))
352                 return *this;
353         else if (level == -max_recursion_level)
354                 throw(std::runtime_error("max recursion level reached"));
355         
356         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
357         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
358         
359         bool basis_is_numerical = 0;
360         bool exponent_is_numerical = 0;
361         numeric * num_basis;
362         numeric * num_exponent;
363         
364         if (is_exactly_of_type(*ebasis.bp,numeric)) {
365                 basis_is_numerical = 1;
366                 num_basis = static_cast<numeric *>(ebasis.bp);
367         }
368         if (is_exactly_of_type(*eexponent.bp,numeric)) {
369                 exponent_is_numerical = 1;
370                 num_exponent = static_cast<numeric *>(eexponent.bp);
371         }
372         
373         // ^(x,0) -> 1 (0^0 also handled here)
374         if (eexponent.is_zero())
375                 if (ebasis.is_zero())
376                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
377                 else
378                         return _ex1();
379         
380         // ^(x,1) -> x
381         if (eexponent.is_equal(_ex1()))
382                 return ebasis;
383         
384         // ^(0,c1) -> 0 or exception (depending on real value of c1)
385         if (ebasis.is_zero() && exponent_is_numerical) {
386                 if ((num_exponent->real()).is_zero())
387                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
388                 else if ((num_exponent->real()).is_negative())
389                         throw (pole_error("power::eval(): division by zero",1));
390                 else
391                         return _ex0();
392         }
393         
394         // ^(1,x) -> 1
395         if (ebasis.is_equal(_ex1()))
396                 return _ex1();
397         
398         if (basis_is_numerical && exponent_is_numerical) {
399                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
400                 // except if c1,c2 are rational, but c1^c2 is not)
401                 bool basis_is_crational = num_basis->is_crational();
402                 bool exponent_is_crational = num_exponent->is_crational();
403                 numeric res = (*num_basis).power(*num_exponent);
404                 
405                 if ((!basis_is_crational || !exponent_is_crational)
406                         || res.is_crational()) {
407                         return res;
408                 }
409                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
410                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
411                 if (basis_is_crational && exponent_is_crational
412                         && num_exponent->is_real()
413                         && !num_exponent->is_integer()) {
414                         numeric n = num_exponent->numer();
415                         numeric m = num_exponent->denom();
416                         numeric r;
417                         numeric q = iquo(n, m, r);
418                         if (r.is_negative()) {
419                                 r = r.add(m);
420                                 q = q.sub(_num1());
421                         }
422                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
423                                 return this->hold();
424                         else {
425                                 epvector res;
426                                 res.push_back(expair(ebasis,r.div(m)));
427                                 return (new mul(res,ex(num_basis->power(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
428                         }
429                 }
430         }
431         
432         // ^(^(x,c1),c2) -> ^(x,c1*c2)
433         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
434         // case c1==1 should not happen, see below!)
435         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
436                 const power & sub_power = ex_to_power(ebasis);
437                 const ex & sub_basis = sub_power.basis;
438                 const ex & sub_exponent = sub_power.exponent;
439                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
440                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
441                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
442                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
443                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
444                         }
445                 }
446         }
447         
448         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
449         if (exponent_is_numerical && num_exponent->is_integer() &&
450                 is_ex_exactly_of_type(ebasis,mul)) {
451                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
452         }
453         
454         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
455         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
456         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
457                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
458                 const mul & mulref=ex_to_mul(ebasis);
459                 if (!mulref.overall_coeff.is_equal(_ex1())) {
460                         const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff);
461                         if (num_coeff.is_real()) {
462                                 if (num_coeff.is_positive()>0) {
463                                         mul * mulp=new mul(mulref);
464                                         mulp->overall_coeff=_ex1();
465                                         mulp->clearflag(status_flags::evaluated);
466                                         mulp->clearflag(status_flags::hash_calculated);
467                                         return (new mul(power(*mulp,exponent),
468                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
469                                 } else {
470                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
471                                         if (num_coeff.compare(_num_1())!=0) {
472                                                 mul * mulp=new mul(mulref);
473                                                 mulp->overall_coeff=_ex_1();
474                                                 mulp->clearflag(status_flags::evaluated);
475                                                 mulp->clearflag(status_flags::hash_calculated);
476                                                 return (new mul(power(*mulp,exponent),
477                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
478                                         }
479                                 }
480                         }
481                 }
482         }
483                 
484         if (are_ex_trivially_equal(ebasis,basis) &&
485                 are_ex_trivially_equal(eexponent,exponent)) {
486                 return this->hold();
487         }
488         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
489                                                                                                    status_flags::evaluated);
490 }
491
492 ex power::evalf(int level) const
493 {
494         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
495
496         ex ebasis;
497         ex eexponent;
498         
499         if (level==1) {
500                 ebasis = basis;
501                 eexponent = exponent;
502         } else if (level == -max_recursion_level) {
503                 throw(std::runtime_error("max recursion level reached"));
504         } else {
505                 ebasis = basis.evalf(level-1);
506                 if (!is_ex_exactly_of_type(eexponent,numeric))
507                         eexponent = exponent.evalf(level-1);
508                 else
509                         eexponent = exponent;
510         }
511
512         return power(ebasis,eexponent);
513 }
514
515 ex power::subs(const lst & ls, const lst & lr) const
516 {
517         const ex & subsed_basis=basis.subs(ls,lr);
518         const ex & subsed_exponent=exponent.subs(ls,lr);
519
520         if (are_ex_trivially_equal(basis,subsed_basis)&&
521                 are_ex_trivially_equal(exponent,subsed_exponent)) {
522                 return *this;
523         }
524         
525         return power(subsed_basis, subsed_exponent);
526 }
527
528 ex power::simplify_ncmul(const exvector & v) const
529 {
530         return inherited::simplify_ncmul(v);
531 }
532
533 // protected
534
535 /** Implementation of ex::diff() for a power.
536  *  @see ex::diff */
537 ex power::derivative(const symbol & s) const
538 {
539         if (exponent.info(info_flags::real)) {
540                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
541                 epvector newseq;
542                 newseq.reserve(2);
543                 newseq.push_back(expair(basis, exponent - _ex1()));
544                 newseq.push_back(expair(basis.diff(s), _ex1()));
545                 return mul(newseq, exponent);
546         } else {
547                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
548                 return mul(power(basis, exponent),
549                            add(mul(exponent.diff(s), log(basis)),
550                            mul(mul(exponent, basis.diff(s)), power(basis, -1))));
551         }
552 }
553
554 int power::compare_same_type(const basic & other) const
555 {
556         GINAC_ASSERT(is_exactly_of_type(other, power));
557         const power & o=static_cast<const power &>(const_cast<basic &>(other));
558
559         int cmpval;
560         cmpval=basis.compare(o.basis);
561         if (cmpval==0) {
562                 return exponent.compare(o.exponent);
563         }
564         return cmpval;
565 }
566
567 unsigned power::return_type(void) const
568 {
569         return basis.return_type();
570 }
571    
572 unsigned power::return_type_tinfo(void) const
573 {
574         return basis.return_type_tinfo();
575 }
576
577 ex power::expand(unsigned options) const
578 {
579         if (flags & status_flags::expanded)
580                 return *this;
581         
582         ex expanded_basis = basis.expand(options);
583         
584         if (!is_ex_exactly_of_type(exponent,numeric) ||
585                 !ex_to_numeric(exponent).is_integer()) {
586                 if (are_ex_trivially_equal(basis,expanded_basis)) {
587                         return this->hold();
588                 } else {
589                         return (new power(expanded_basis,exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
590                 }
591         }
592         
593         // integer numeric exponent
594         const numeric & num_exponent = ex_to_numeric(exponent);
595         int int_exponent = num_exponent.to_int();
596         
597         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
598                 return expand_add(ex_to_add(expanded_basis), int_exponent);
599         }
600         
601         if (is_ex_exactly_of_type(expanded_basis,mul)) {
602                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
603         }
604         
605         // cannot expand further
606         if (are_ex_trivially_equal(basis,expanded_basis)) {
607                 return this->hold();
608         } else {
609                 return (new power(expanded_basis,exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
610         }
611 }
612
613 //////////
614 // new virtual functions which can be overridden by derived classes
615 //////////
616
617 // none
618
619 //////////
620 // non-virtual functions in this class
621 //////////
622
623 /** expand a^n where a is an add and n is an integer.
624  *  @see power::expand */
625 ex power::expand_add(const add & a, int n) const
626 {
627         if (n==2)
628                 return expand_add_2(a);
629         
630         int m = a.nops();
631         exvector sum;
632         sum.reserve((n+1)*(m-1));
633         intvector k(m-1);
634         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
635         intvector upper_limit(m-1);
636         int l;
637         
638         for (int l=0; l<m-1; l++) {
639                 k[l] = 0;
640                 k_cum[l] = 0;
641                 upper_limit[l] = n;
642         }
643         
644         while (1) {
645                 exvector term;
646                 term.reserve(m+1);
647                 for (l=0; l<m-1; l++) {
648                         const ex & b = a.op(l);
649                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
650                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
651                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
652                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
653                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
654                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
655                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
656                         if (is_ex_exactly_of_type(b,mul)) {
657                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
658                         } else {
659                                 term.push_back(power(b,k[l]));
660                         }
661                 }
662                 
663                 const ex & b = a.op(l);
664                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
665                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
666                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
667                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
668                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
669                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
670                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
671                 if (is_ex_exactly_of_type(b,mul)) {
672                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
673                 } else {
674                         term.push_back(power(b,n-k_cum[m-2]));
675                 }
676                 
677                 numeric f = binomial(numeric(n),numeric(k[0]));
678                 for (l=1; l<m-1; l++) {
679                         f=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 (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
691                         cout << *cit << endl;
692                 }
693                 cout << "end term" << endl;
694                 */
695
696                 // TODO: optimize this
697                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
698                 
699                 // increment k[]
700                 l=m-2;
701                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
702                         k[l]=0;    
703                         l--;
704                 }
705                 if (l<0) break;
706
707                 // recalc k_cum[] and upper_limit[]
708                 if (l==0) {
709                         k_cum[0]=k[0];
710                 } else {
711                         k_cum[l]=k_cum[l-1]+k[l];
712                 }
713                 for (int i=l+1; i<m-1; i++) {
714                         k_cum[i]=k_cum[i-1]+k[i];
715                 }
716
717                 for (int i=l+1; i<m-1; i++) {
718                         upper_limit[i]=n-k_cum[i-1];
719                 }   
720         }
721         return (new add(sum))->setflag(status_flags::dynallocated |
722                                                                    status_flags::expanded );
723 }
724
725
726 /** Special case of power::expand_add. Expands a^2 where a is an add.
727  *  @see power::expand_add */
728 ex power::expand_add_2(const add & a) const
729 {
730         epvector sum;
731         unsigned a_nops=a.nops();
732         sum.reserve((a_nops*(a_nops+1))/2);
733         epvector::const_iterator last=a.seq.end();
734
735         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
736         // first part: ignore overall_coeff and expand other terms
737         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
738                 const ex & r=(*cit0).rest;
739                 const ex & c=(*cit0).coeff;
740                 
741                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
742                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
743                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
744                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
745                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
746                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
747                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
748
749                 if (are_ex_trivially_equal(c,_ex1())) {
750                         if (is_ex_exactly_of_type(r,mul)) {
751                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
752                                                      _ex1()));
753                         } else {
754                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
755                                                      _ex1()));
756                         }
757                 } else {
758                         if (is_ex_exactly_of_type(r,mul)) {
759                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
760                                                      ex_to_numeric(c).power_dyn(_num2())));
761                         } else {
762                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
763                                                      ex_to_numeric(c).power_dyn(_num2())));
764                         }
765                 }
766                         
767                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
768                         const ex & r1=(*cit1).rest;
769                         const ex & c1=(*cit1).coeff;
770                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
771                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
772                 }
773         }
774
775         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
776
777         // second part: add terms coming from overall_factor (if != 0)
778         if (!a.overall_coeff.is_equal(_ex0())) {
779                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
780                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
781                 }
782                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
783         }
784                 
785         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
786         
787         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
788 }
789
790 /** Expand factors of m in m^n where m is a mul and n is and integer
791  *  @see power::expand */
792 ex power::expand_mul(const mul & m, const numeric & n) const
793 {
794         if (n.is_equal(_num0()))
795                 return _ex1();
796         
797         epvector distrseq;
798         distrseq.reserve(m.seq.size());
799         epvector::const_iterator last = m.seq.end();
800         epvector::const_iterator cit = m.seq.begin();
801         while (cit!=last) {
802                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
803                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
804                 } else {
805                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
806                         // since n is an integer
807                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
808                 }
809                 ++cit;
810         }
811         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
812 }
813
814 /*
815 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
816                                unsigned options) const
817 {
818         // obsolete
819
820         exvector distrseq;
821         epvector splitseq;
822
823         const add & addref=static_cast<const add &>(*basis.bp);
824
825         splitseq=addref.seq;
826         splitseq.pop_back();
827         ex first_operands=add(splitseq);
828         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
829         
830         int n=exponent.to_int();
831         for (int k=0; k<=n; k++) {
832                 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
833                                                  * power(last_operand,numeric(n-k)));
834         }
835         return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
836 }
837 */
838
839 /*
840 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
841                                                                 unsigned options) const
842 {
843         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
844                         expand(options | expand_options::internal_do_not_expand_power_operands);
845
846         return ex(mul(rest_power,basis),0).
847                expand(options | expand_options::internal_do_not_expand_mul_operands);
848 }
849 */
850
851 //////////
852 // static member variables
853 //////////
854
855 // protected
856
857 unsigned power::precedence = 60;
858
859 //////////
860 // global constants
861 //////////
862
863 const power some_power;
864 const std::type_info & typeid_power=typeid(some_power);
865
866 // helper function
867
868 ex sqrt(const ex & a)
869 {
870         return power(a,_ex1_2());
871 }
872
873 #ifndef NO_NAMESPACE_GINAC
874 } // namespace GiNaC
875 #endif // ndef NO_NAMESPACE_GINAC