* ginac/registrar.h: dtor is inlined now.
[ginac.git] / ginac / power.cpp
1 /** @file power.cpp
2  *
3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
7  *
8  *  This program is free software; you can redistribute it and/or modify
9  *  it under the terms of the GNU General Public License as published by
10  *  the Free Software Foundation; either version 2 of the License, or
11  *  (at your option) any later version.
12  *
13  *  This program is distributed in the hope that it will be useful,
14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
16  *  GNU General Public License for more details.
17  *
18  *  You should have received a copy of the GNU General Public License
19  *  along with this program; if not, write to the Free Software
20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
21  */
22
23 #include <vector>
24 #include <iostream>
25 #include <stdexcept>
26
27 #include "power.h"
28 #include "expairseq.h"
29 #include "add.h"
30 #include "mul.h"
31 #include "numeric.h"
32 #include "inifcns.h"
33 #include "relational.h"
34 #include "symbol.h"
35 #include "archive.h"
36 #include "debugmsg.h"
37 #include "utils.h"
38
39 namespace GiNaC {
40
41 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
42
43 typedef std::vector<int> intvector;
44
45 //////////
46 // default ctor, dtor, copy ctor assignment operator and helpers
47 //////////
48
49 // public
50
51 power::power() : basic(TINFO_power)
52 {
53         debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
54 }
55
56 // protected
57
58 void power::copy(const power & other)
59 {
60         inherited::copy(other);
61         basis = other.basis;
62         exponent = other.exponent;
63 }
64
65 void power::destroy(bool call_parent)
66 {
67         if (call_parent) inherited::destroy(call_parent);
68 }
69
70 //////////
71 // other ctors
72 //////////
73
74 // public
75
76 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
77 {
78         debugmsg("power ctor from ex,ex",LOGLEVEL_CONSTRUCT);
79         GINAC_ASSERT(basis.return_type()==return_types::commutative);
80 }
81
82 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
83 {
84         debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
85         GINAC_ASSERT(basis.return_type()==return_types::commutative);
86 }
87
88 //////////
89 // archiving
90 //////////
91
92 /** Construct object from archive_node. */
93 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
94 {
95         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
96         n.find_ex("basis", basis, sym_lst);
97         n.find_ex("exponent", exponent, sym_lst);
98 }
99
100 /** Unarchive the object. */
101 ex power::unarchive(const archive_node &n, const lst &sym_lst)
102 {
103         return (new power(n, sym_lst))->setflag(status_flags::dynallocated);
104 }
105
106 /** Archive the object. */
107 void power::archive(archive_node &n) const
108 {
109         inherited::archive(n);
110         n.add_ex("basis", basis);
111         n.add_ex("exponent", exponent);
112 }
113
114 //////////
115 // functions overriding virtual functions from bases classes
116 //////////
117
118 // public
119
120 void power::print(std::ostream & os, unsigned upper_precedence) const
121 {
122         debugmsg("power print",LOGLEVEL_PRINT);
123         if (exponent.is_equal(_ex1_2())) {
124                 os << "sqrt(" << basis << ")";
125         } else {
126                 if (precedence<=upper_precedence) os << "(";
127                 basis.print(os,precedence);
128                 os << "^";
129                 exponent.print(os,precedence);
130                 if (precedence<=upper_precedence) os << ")";
131         }
132 }
133
134 void power::printraw(std::ostream & os) const
135 {
136         debugmsg("power printraw",LOGLEVEL_PRINT);
137
138         os << "power(";
139         basis.printraw(os);
140         os << ",";
141         exponent.printraw(os);
142         os << ",hash=" << hashvalue << ",flags=" << flags << ")";
143 }
144
145 void power::printtree(std::ostream & os, unsigned indent) const
146 {
147         debugmsg("power printtree",LOGLEVEL_PRINT);
148
149         os << std::string(indent,' ') << "power: "
150            << "hash=" << hashvalue
151            << " (0x" << std::hex << hashvalue << std::dec << ")"
152            << ", flags=" << flags << std::endl;
153         basis.printtree(os, indent+delta_indent);
154         exponent.printtree(os, indent+delta_indent);
155 }
156
157 static void print_sym_pow(std::ostream & os, unsigned type, const symbol &x, int exp)
158 {
159         // Optimal output of integer powers of symbols to aid compiler CSE.
160         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
161         // to learn why such a hack is really necessary.
162         if (exp == 1) {
163                 x.printcsrc(os, type, 0);
164         } else if (exp == 2) {
165                 x.printcsrc(os, type, 0);
166                 os << "*";
167                 x.printcsrc(os, type, 0);
168         } else if (exp & 1) {
169                 x.printcsrc(os, 0);
170                 os << "*";
171                 print_sym_pow(os, type, x, exp-1);
172         } else {
173                 os << "(";
174                 print_sym_pow(os, type, x, exp >> 1);
175                 os << ")*(";
176                 print_sym_pow(os, type, x, exp >> 1);
177                 os << ")";
178         }
179 }
180
181 void power::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
182 {
183         debugmsg("power print csrc", LOGLEVEL_PRINT);
184         
185         // Integer powers of symbols are printed in a special, optimized way
186         if (exponent.info(info_flags::integer)
187          && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
188                 int exp = ex_to_numeric(exponent).to_int();
189                 if (exp > 0)
190                         os << "(";
191                 else {
192                         exp = -exp;
193                         if (type == csrc_types::ctype_cl_N)
194                                 os << "recip(";
195                         else
196                                 os << "1.0/(";
197                 }
198                 print_sym_pow(os, type, static_cast<const symbol &>(*basis.bp), exp);
199                 os << ")";
200
201         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
202         } else if (exponent.compare(_num_1()) == 0) {
203                 if (type == csrc_types::ctype_cl_N)
204                         os << "recip(";
205                 else
206                         os << "1.0/(";
207                 basis.bp->printcsrc(os, type, 0);
208                 os << ")";
209
210         // Otherwise, use the pow() or expt() (CLN) functions
211         } else {
212                 if (type == csrc_types::ctype_cl_N)
213                         os << "expt(";
214                 else
215                         os << "pow(";
216                 basis.bp->printcsrc(os, type, 0);
217                 os << ",";
218                 exponent.bp->printcsrc(os, type, 0);
219                 os << ")";
220         }
221 }
222
223 bool power::info(unsigned inf) const
224 {
225         switch (inf) {
226                 case info_flags::polynomial:
227                 case info_flags::integer_polynomial:
228                 case info_flags::cinteger_polynomial:
229                 case info_flags::rational_polynomial:
230                 case info_flags::crational_polynomial:
231                         return exponent.info(info_flags::nonnegint);
232                 case info_flags::rational_function:
233                         return exponent.info(info_flags::integer);
234                 case info_flags::algebraic:
235                         return (!exponent.info(info_flags::integer) ||
236                                         basis.info(inf));
237         }
238         return inherited::info(inf);
239 }
240
241 unsigned power::nops() const
242 {
243         return 2;
244 }
245
246 ex & power::let_op(int i)
247 {
248         GINAC_ASSERT(i>=0);
249         GINAC_ASSERT(i<2);
250
251         return i==0 ? basis : exponent;
252 }
253
254 int power::degree(const symbol & s) const
255 {
256         if (is_exactly_of_type(*exponent.bp,numeric)) {
257                 if ((*basis.bp).compare(s)==0) {
258                         if (ex_to_numeric(exponent).is_integer())
259                                 return ex_to_numeric(exponent).to_int();
260                         else
261                                 return 0;
262                 } else
263                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
264         }
265         return 0;
266 }
267
268 int power::ldegree(const symbol & s) const 
269 {
270         if (is_exactly_of_type(*exponent.bp,numeric)) {
271                 if ((*basis.bp).compare(s)==0) {
272                         if (ex_to_numeric(exponent).is_integer())
273                                 return ex_to_numeric(exponent).to_int();
274                         else
275                                 return 0;
276                 } else
277                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
278         }
279         return 0;
280 }
281
282 ex power::coeff(const symbol & s, int n) const
283 {
284         if ((*basis.bp).compare(s)!=0) {
285                 // basis not equal to s
286                 if (n == 0)
287                         return *this;
288                 else
289                         return _ex0();
290         } else {
291                 // basis equal to s
292                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
293                         // integer exponent
294                         int int_exp = ex_to_numeric(exponent).to_int();
295                         if (n == int_exp)
296                                 return _ex1();
297                         else
298                                 return _ex0();
299                 } else {
300                         // non-integer exponents are treated as zero
301                         if (n == 0)
302                                 return *this;
303                         else
304                                 return _ex0();
305                 }
306         }
307 }
308
309 ex power::eval(int level) const
310 {
311         // simplifications: ^(x,0) -> 1 (0^0 handled here)
312         //                  ^(x,1) -> x
313         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
314         //                  ^(1,x) -> 1
315         //                  ^(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)
316         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
317         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
318         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
319         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
320         
321         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
322         
323         if ((level==1) && (flags & status_flags::evaluated))
324                 return *this;
325         else if (level == -max_recursion_level)
326                 throw(std::runtime_error("max recursion level reached"));
327         
328         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
329         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
330         
331         bool basis_is_numerical = 0;
332         bool exponent_is_numerical = 0;
333         numeric * num_basis;
334         numeric * num_exponent;
335         
336         if (is_exactly_of_type(*ebasis.bp,numeric)) {
337                 basis_is_numerical = 1;
338                 num_basis = static_cast<numeric *>(ebasis.bp);
339         }
340         if (is_exactly_of_type(*eexponent.bp,numeric)) {
341                 exponent_is_numerical = 1;
342                 num_exponent = static_cast<numeric *>(eexponent.bp);
343         }
344         
345         // ^(x,0) -> 1 (0^0 also handled here)
346         if (eexponent.is_zero()) {
347                 if (ebasis.is_zero())
348                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
349                 else
350                         return _ex1();
351         }
352         
353         // ^(x,1) -> x
354         if (eexponent.is_equal(_ex1()))
355                 return ebasis;
356         
357         // ^(0,c1) -> 0 or exception (depending on real value of c1)
358         if (ebasis.is_zero() && exponent_is_numerical) {
359                 if ((num_exponent->real()).is_zero())
360                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
361                 else if ((num_exponent->real()).is_negative())
362                         throw (pole_error("power::eval(): division by zero",1));
363                 else
364                         return _ex0();
365         }
366         
367         // ^(1,x) -> 1
368         if (ebasis.is_equal(_ex1()))
369                 return _ex1();
370         
371         if (basis_is_numerical && exponent_is_numerical) {
372                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
373                 // except if c1,c2 are rational, but c1^c2 is not)
374                 bool basis_is_crational = num_basis->is_crational();
375                 bool exponent_is_crational = num_exponent->is_crational();
376                 numeric res = num_basis->power(*num_exponent);
377                 
378                 if ((!basis_is_crational || !exponent_is_crational)
379                         || res.is_crational()) {
380                         return res;
381                 }
382                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
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 (exponent_is_numerical && 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)<1)
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 (exponent_is_numerical && num_exponent->is_integer() &&
422                 is_ex_exactly_of_type(ebasis,mul)) {
423                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
424         }
425         
426         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
427         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
428         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
429                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
430                 const mul & mulref = ex_to_mul(ebasis);
431                 if (!mulref.overall_coeff.is_equal(_ex1())) {
432                         const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
433                         if (num_coeff.is_real()) {
434                                 if (num_coeff.is_positive()) {
435                                         mul * mulp = new mul(mulref);
436                                         mulp->overall_coeff = _ex1();
437                                         mulp->clearflag(status_flags::evaluated);
438                                         mulp->clearflag(status_flags::hash_calculated);
439                                         return (new mul(power(*mulp,exponent),
440                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
441                                 } else {
442                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
443                                         if (num_coeff.compare(_num_1())!=0) {
444                                                 mul * mulp = new mul(mulref);
445                                                 mulp->overall_coeff = _ex_1();
446                                                 mulp->clearflag(status_flags::evaluated);
447                                                 mulp->clearflag(status_flags::hash_calculated);
448                                                 return (new mul(power(*mulp,exponent),
449                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
450                                         }
451                                 }
452                         }
453                 }
454         }
455         
456         if (are_ex_trivially_equal(ebasis,basis) &&
457                 are_ex_trivially_equal(eexponent,exponent)) {
458                 return this->hold();
459         }
460         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
461                                                                                                    status_flags::evaluated);
462 }
463
464 ex power::evalf(int level) const
465 {
466         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
467
468         ex ebasis;
469         ex eexponent;
470         
471         if (level==1) {
472                 ebasis = basis;
473                 eexponent = exponent;
474         } else if (level == -max_recursion_level) {
475                 throw(std::runtime_error("max recursion level reached"));
476         } else {
477                 ebasis = basis.evalf(level-1);
478                 if (!is_ex_exactly_of_type(eexponent,numeric))
479                         eexponent = exponent.evalf(level-1);
480                 else
481                         eexponent = exponent;
482         }
483
484         return power(ebasis,eexponent);
485 }
486
487 ex power::subs(const lst & ls, const lst & lr) const
488 {
489         const ex & subsed_basis=basis.subs(ls,lr);
490         const ex & subsed_exponent=exponent.subs(ls,lr);
491
492         if (are_ex_trivially_equal(basis,subsed_basis)&&
493                 are_ex_trivially_equal(exponent,subsed_exponent)) {
494                 return *this;
495         }
496         
497         return power(subsed_basis, subsed_exponent);
498 }
499
500 ex power::simplify_ncmul(const exvector & v) const
501 {
502         return inherited::simplify_ncmul(v);
503 }
504
505 // protected
506
507 /** Implementation of ex::diff() for a power.
508  *  @see ex::diff */
509 ex power::derivative(const symbol & s) const
510 {
511         if (exponent.info(info_flags::real)) {
512                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
513                 epvector newseq;
514                 newseq.reserve(2);
515                 newseq.push_back(expair(basis, exponent - _ex1()));
516                 newseq.push_back(expair(basis.diff(s), _ex1()));
517                 return mul(newseq, exponent);
518         } else {
519                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
520                 return mul(*this,
521                            add(mul(exponent.diff(s), log(basis)),
522                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
523         }
524 }
525
526 int power::compare_same_type(const basic & other) const
527 {
528         GINAC_ASSERT(is_exactly_of_type(other, power));
529         const power & o=static_cast<const power &>(const_cast<basic &>(other));
530
531         int cmpval;
532         cmpval=basis.compare(o.basis);
533         if (cmpval==0) {
534                 return exponent.compare(o.exponent);
535         }
536         return cmpval;
537 }
538
539 unsigned power::return_type(void) const
540 {
541         return basis.return_type();
542 }
543    
544 unsigned power::return_type_tinfo(void) const
545 {
546         return basis.return_type_tinfo();
547 }
548
549 ex power::expand(unsigned options) const
550 {
551         if (flags & status_flags::expanded)
552                 return *this;
553         
554         ex expanded_basis = basis.expand(options);
555         ex expanded_exponent = exponent.expand(options);
556         
557         // x^(a+b) -> x^a * x^b
558         if (is_ex_exactly_of_type(expanded_exponent, add)) {
559                 const add &a = ex_to_add(expanded_exponent);
560                 exvector distrseq;
561                 distrseq.reserve(a.seq.size() + 1);
562                 epvector::const_iterator last = a.seq.end();
563                 epvector::const_iterator cit = a.seq.begin();
564                 while (cit!=last) {
565                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
566                         cit++;
567                 }
568                 
569                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
570                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
571                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
572                         int int_exponent = num_exponent.to_int();
573                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
574                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
575                         else
576                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
577                 } else
578                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
579                 
580                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
581                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
582                 return r.expand();
583         }
584         
585         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
586                 !ex_to_numeric(expanded_exponent).is_integer()) {
587                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
588                         return this->hold();
589                 } else {
590                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
591                 }
592         }
593         
594         // integer numeric exponent
595         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
596         int int_exponent = num_exponent.to_int();
597         
598         // (x+y)^n, n>0
599         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
600                 return expand_add(ex_to_add(expanded_basis), int_exponent);
601         
602         // (x*y)^n -> x^n * y^n
603         if (is_ex_exactly_of_type(expanded_basis,mul))
604                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
605         
606         // cannot expand further
607         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
608                 return this->hold();
609         else
610                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
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                 const ex & b = a.op(l);
663                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
664                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
665                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
666                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
667                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
668                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
669                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
670                 if (is_ex_exactly_of_type(b,mul))
671                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
672                 else
673                         term.push_back(power(b,n-k_cum[m-2]));
674                 
675                 numeric f = binomial(numeric(n),numeric(k[0]));
676                 for (l=1; l<m-1; l++)
677                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
678                 
679                 term.push_back(f);
680                 
681                 /*
682                 cout << "begin term" << endl;
683                 for (int i=0; i<m-1; i++) {
684                         cout << "k[" << i << "]=" << k[i] << endl;
685                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
686                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
687                 }
688                 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
689                         cout << *cit << endl;
690                 }
691                 cout << "end term" << endl;
692                 */
693                 
694                 // TODO: optimize this
695                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
696                 
697                 // increment k[]
698                 l = m-2;
699                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
700                         k[l] = 0;    
701                         l--;
702                 }
703                 if (l<0) break;
704                 
705                 // recalc k_cum[] and upper_limit[]
706                 if (l==0)
707                         k_cum[0] = k[0];
708                 else
709                         k_cum[l] = k_cum[l-1]+k[l];
710                 
711                 for (int i=l+1; i<m-1; i++)
712                         k_cum[i] = k_cum[i-1]+k[i];
713                 
714                 for (int i=l+1; i<m-1; i++)
715                         upper_limit[i] = n-k_cum[i-1];
716         }
717         return (new add(sum))->setflag(status_flags::dynallocated |
718                                                                    status_flags::expanded );
719 }
720
721
722 /** Special case of power::expand_add. Expands a^2 where a is an add.
723  *  @see power::expand_add */
724 ex power::expand_add_2(const add & a) const
725 {
726         epvector sum;
727         unsigned a_nops = a.nops();
728         sum.reserve((a_nops*(a_nops+1))/2);
729         epvector::const_iterator last = a.seq.end();
730         
731         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
732         // first part: ignore overall_coeff and expand other terms
733         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
734                 const ex & r = (*cit0).rest;
735                 const ex & c = (*cit0).coeff;
736                 
737                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
738                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
739                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
740                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
741                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
742                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
743                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
744                 
745                 if (are_ex_trivially_equal(c,_ex1())) {
746                         if (is_ex_exactly_of_type(r,mul)) {
747                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
748                                                      _ex1()));
749                         } else {
750                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
751                                                      _ex1()));
752                         }
753                 } else {
754                         if (is_ex_exactly_of_type(r,mul)) {
755                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
756                                                      ex_to_numeric(c).power_dyn(_num2())));
757                         } else {
758                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
759                                                      ex_to_numeric(c).power_dyn(_num2())));
760                         }
761                 }
762                         
763                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
764                         const ex & r1 = (*cit1).rest;
765                         const ex & c1 = (*cit1).coeff;
766                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
767                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
768                 }
769         }
770         
771         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
772         
773         // second part: add terms coming from overall_factor (if != 0)
774         if (!a.overall_coeff.is_zero()) {
775                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
776                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
777                 }
778                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
779         }
780         
781         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
782         
783         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
784 }
785
786 /** Expand factors of m in m^n where m is a mul and n is and integer
787  *  @see power::expand */
788 ex power::expand_mul(const mul & m, const numeric & n) const
789 {
790         if (n.is_zero())
791                 return _ex1();
792         
793         epvector distrseq;
794         distrseq.reserve(m.seq.size());
795         epvector::const_iterator last = m.seq.end();
796         epvector::const_iterator cit = m.seq.begin();
797         while (cit!=last) {
798                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
799                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
800                 } else {
801                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
802                         // since n is an integer
803                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
804                 }
805                 ++cit;
806         }
807         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
808 }
809
810 /*
811 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
812                                unsigned options) const
813 {
814         // obsolete
815
816         exvector distrseq;
817         epvector splitseq;
818
819         const add & addref=static_cast<const add &>(*basis.bp);
820
821         splitseq=addref.seq;
822         splitseq.pop_back();
823         ex first_operands=add(splitseq);
824         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
825         
826         int n=exponent.to_int();
827         for (int k=0; k<=n; k++) {
828                 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
829                                                  * power(last_operand,numeric(n-k)));
830         }
831         return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
832 }
833 */
834
835 /*
836 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
837                                                                 unsigned options) const
838 {
839         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
840                         expand(options | expand_options::internal_do_not_expand_power_operands);
841
842         return ex(mul(rest_power,basis),0).
843                expand(options | expand_options::internal_do_not_expand_mul_operands);
844 }
845 */
846
847 //////////
848 // static member variables
849 //////////
850
851 // protected
852
853 unsigned power::precedence = 60;
854
855 // helper function
856
857 ex sqrt(const ex & a)
858 {
859         return power(a,_ex1_2());
860 }
861
862 } // namespace GiNaC