the destructor, copy constructor, and assignment operator (which were the
[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 #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 // protected
59
60 void power::copy(const power & other)
61 {
62         inherited::copy(other);
63         basis=other.basis;
64         exponent=other.exponent;
65 }
66
67 void power::destroy(bool call_parent)
68 {
69         if (call_parent) inherited::destroy(call_parent);
70 }
71
72 //////////
73 // other constructors
74 //////////
75
76 // public
77
78 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
79 {
80         debugmsg("power constructor from ex,ex",LOGLEVEL_CONSTRUCT);
81         GINAC_ASSERT(basis.return_type()==return_types::commutative);
82 }
83
84 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
85 {
86         debugmsg("power constructor from ex,numeric",LOGLEVEL_CONSTRUCT);
87         GINAC_ASSERT(basis.return_type()==return_types::commutative);
88 }
89
90 //////////
91 // archiving
92 //////////
93
94 /** Construct object from archive_node. */
95 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
96 {
97         debugmsg("power constructor from archive_node", LOGLEVEL_CONSTRUCT);
98         n.find_ex("basis", basis, sym_lst);
99         n.find_ex("exponent", exponent, sym_lst);
100 }
101
102 /** Unarchive the object. */
103 ex power::unarchive(const archive_node &n, const lst &sym_lst)
104 {
105         return (new power(n, sym_lst))->setflag(status_flags::dynallocated);
106 }
107
108 /** Archive the object. */
109 void power::archive(archive_node &n) const
110 {
111         inherited::archive(n);
112         n.add_ex("basis", basis);
113         n.add_ex("exponent", exponent);
114 }
115
116 //////////
117 // functions overriding virtual functions from bases classes
118 //////////
119
120 // public
121
122 basic * power::duplicate() const
123 {
124         debugmsg("power duplicate",LOGLEVEL_DUPLICATE);
125         return new power(*this);
126 }
127
128 void power::print(std::ostream & os, unsigned upper_precedence) const
129 {
130         debugmsg("power print",LOGLEVEL_PRINT);
131         if (exponent.is_equal(_ex1_2())) {
132                 os << "sqrt(" << basis << ")";
133         } else {
134                 if (precedence<=upper_precedence) os << "(";
135                 basis.print(os,precedence);
136                 os << "^";
137                 exponent.print(os,precedence);
138                 if (precedence<=upper_precedence) os << ")";
139         }
140 }
141
142 void power::printraw(std::ostream & os) const
143 {
144         debugmsg("power printraw",LOGLEVEL_PRINT);
145
146         os << "power(";
147         basis.printraw(os);
148         os << ",";
149         exponent.printraw(os);
150         os << ",hash=" << hashvalue << ",flags=" << flags << ")";
151 }
152
153 void power::printtree(std::ostream & os, unsigned indent) const
154 {
155         debugmsg("power printtree",LOGLEVEL_PRINT);
156
157         os << std::string(indent,' ') << "power: "
158            << "hash=" << hashvalue
159            << " (0x" << std::hex << hashvalue << std::dec << ")"
160            << ", flags=" << flags << std::endl;
161         basis.printtree(os, indent+delta_indent);
162         exponent.printtree(os, indent+delta_indent);
163 }
164
165 static void print_sym_pow(std::ostream & os, unsigned type, const symbol &x, int exp)
166 {
167         // Optimal output of integer powers of symbols to aid compiler CSE
168         if (exp == 1) {
169                 x.printcsrc(os, type, 0);
170         } else if (exp == 2) {
171                 x.printcsrc(os, type, 0);
172                 os << "*";
173                 x.printcsrc(os, type, 0);
174         } else if (exp & 1) {
175                 x.printcsrc(os, 0);
176                 os << "*";
177                 print_sym_pow(os, type, x, exp-1);
178         } else {
179                 os << "(";
180                 print_sym_pow(os, type, x, exp >> 1);
181                 os << ")*(";
182                 print_sym_pow(os, type, x, exp >> 1);
183                 os << ")";
184         }
185 }
186
187 void power::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
188 {
189         debugmsg("power print csrc", LOGLEVEL_PRINT);
190         
191         // Integer powers of symbols are printed in a special, optimized way
192         if (exponent.info(info_flags::integer)
193          && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
194                 int exp = ex_to_numeric(exponent).to_int();
195                 if (exp > 0)
196                         os << "(";
197                 else {
198                         exp = -exp;
199                         if (type == csrc_types::ctype_cl_N)
200                                 os << "recip(";
201                         else
202                                 os << "1.0/(";
203                 }
204                 print_sym_pow(os, type, static_cast<const symbol &>(*basis.bp), exp);
205                 os << ")";
206
207         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
208         } else if (exponent.compare(_num_1()) == 0) {
209                 if (type == csrc_types::ctype_cl_N)
210                         os << "recip(";
211                 else
212                         os << "1.0/(";
213                 basis.bp->printcsrc(os, type, 0);
214                 os << ")";
215
216         // Otherwise, use the pow() or expt() (CLN) functions
217         } else {
218                 if (type == csrc_types::ctype_cl_N)
219                         os << "expt(";
220                 else
221                         os << "pow(";
222                 basis.bp->printcsrc(os, type, 0);
223                 os << ",";
224                 exponent.bp->printcsrc(os, type, 0);
225                 os << ")";
226         }
227 }
228
229 bool power::info(unsigned inf) const
230 {
231         switch (inf) {
232                 case info_flags::polynomial:
233                 case info_flags::integer_polynomial:
234                 case info_flags::cinteger_polynomial:
235                 case info_flags::rational_polynomial:
236                 case info_flags::crational_polynomial:
237                         return exponent.info(info_flags::nonnegint);
238                 case info_flags::rational_function:
239                         return exponent.info(info_flags::integer);
240                 case info_flags::algebraic:
241                         return (!exponent.info(info_flags::integer) ||
242                                         basis.info(inf));
243         }
244         return inherited::info(inf);
245 }
246
247 unsigned power::nops() const
248 {
249         return 2;
250 }
251
252 ex & power::let_op(int i)
253 {
254         GINAC_ASSERT(i>=0);
255         GINAC_ASSERT(i<2);
256
257         return i==0 ? basis : exponent;
258 }
259
260 int power::degree(const symbol & s) const
261 {
262         if (is_exactly_of_type(*exponent.bp,numeric)) {
263                 if ((*basis.bp).compare(s)==0) {
264                         if (ex_to_numeric(exponent).is_integer())
265                                 return ex_to_numeric(exponent).to_int();
266                         else
267                                 return 0;
268                 } else
269                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
270         }
271         return 0;
272 }
273
274 int power::ldegree(const symbol & s) const 
275 {
276         if (is_exactly_of_type(*exponent.bp,numeric)) {
277                 if ((*basis.bp).compare(s)==0) {
278                         if (ex_to_numeric(exponent).is_integer())
279                                 return ex_to_numeric(exponent).to_int();
280                         else
281                                 return 0;
282                 } else
283                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
284         }
285         return 0;
286 }
287
288 ex power::coeff(const symbol & s, int n) const
289 {
290         if ((*basis.bp).compare(s)!=0) {
291                 // basis not equal to s
292                 if (n == 0)
293                         return *this;
294                 else
295                         return _ex0();
296         } else {
297                 // basis equal to s
298                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
299                         // integer exponent
300                         int int_exp = ex_to_numeric(exponent).to_int();
301                         if (n == int_exp)
302                                 return _ex1();
303                         else
304                                 return _ex0();
305                 } else {
306                         // non-integer exponents are treated as zero
307                         if (n == 0)
308                                 return *this;
309                         else
310                                 return _ex0();
311                 }
312         }
313 }
314
315 ex power::eval(int level) const
316 {
317         // simplifications: ^(x,0) -> 1 (0^0 handled here)
318         //                  ^(x,1) -> x
319         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
320         //                  ^(1,x) -> 1
321         //                  ^(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)
322         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
323         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
324         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
325         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
326         
327         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
328         
329         if ((level==1) && (flags & status_flags::evaluated))
330                 return *this;
331         else if (level == -max_recursion_level)
332                 throw(std::runtime_error("max recursion level reached"));
333         
334         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
335         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
336         
337         bool basis_is_numerical = 0;
338         bool exponent_is_numerical = 0;
339         numeric * num_basis;
340         numeric * num_exponent;
341         
342         if (is_exactly_of_type(*ebasis.bp,numeric)) {
343                 basis_is_numerical = 1;
344                 num_basis = static_cast<numeric *>(ebasis.bp);
345         }
346         if (is_exactly_of_type(*eexponent.bp,numeric)) {
347                 exponent_is_numerical = 1;
348                 num_exponent = static_cast<numeric *>(eexponent.bp);
349         }
350         
351         // ^(x,0) -> 1 (0^0 also handled here)
352         if (eexponent.is_zero())
353                 if (ebasis.is_zero())
354                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
355                 else
356                         return _ex1();
357         
358         // ^(x,1) -> x
359         if (eexponent.is_equal(_ex1()))
360                 return ebasis;
361         
362         // ^(0,c1) -> 0 or exception (depending on real value of c1)
363         if (ebasis.is_zero() && exponent_is_numerical) {
364                 if ((num_exponent->real()).is_zero())
365                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
366                 else if ((num_exponent->real()).is_negative())
367                         throw (pole_error("power::eval(): division by zero",1));
368                 else
369                         return _ex0();
370         }
371         
372         // ^(1,x) -> 1
373         if (ebasis.is_equal(_ex1()))
374                 return _ex1();
375         
376         if (basis_is_numerical && exponent_is_numerical) {
377                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
378                 // except if c1,c2 are rational, but c1^c2 is not)
379                 bool basis_is_crational = num_basis->is_crational();
380                 bool exponent_is_crational = num_exponent->is_crational();
381                 numeric res = (*num_basis).power(*num_exponent);
382                 
383                 if ((!basis_is_crational || !exponent_is_crational)
384                         || res.is_crational()) {
385                         return res;
386                 }
387                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
388                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
389                 if (basis_is_crational && exponent_is_crational
390                         && num_exponent->is_real()
391                         && !num_exponent->is_integer()) {
392                         numeric n = num_exponent->numer();
393                         numeric m = num_exponent->denom();
394                         numeric r;
395                         numeric q = iquo(n, m, r);
396                         if (r.is_negative()) {
397                                 r = r.add(m);
398                                 q = q.sub(_num1());
399                         }
400                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
401                                 return this->hold();
402                         else {
403                                 epvector res;
404                                 res.push_back(expair(ebasis,r.div(m)));
405                                 return (new mul(res,ex(num_basis->power(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
406                         }
407                 }
408         }
409         
410         // ^(^(x,c1),c2) -> ^(x,c1*c2)
411         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
412         // case c1==1 should not happen, see below!)
413         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
414                 const power & sub_power = ex_to_power(ebasis);
415                 const ex & sub_basis = sub_power.basis;
416                 const ex & sub_exponent = sub_power.exponent;
417                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
418                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
419                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
420                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
421                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
422                         }
423                 }
424         }
425         
426         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
427         if (exponent_is_numerical && num_exponent->is_integer() &&
428                 is_ex_exactly_of_type(ebasis,mul)) {
429                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
430         }
431         
432         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
433         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
434         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
435                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
436                 const mul & mulref=ex_to_mul(ebasis);
437                 if (!mulref.overall_coeff.is_equal(_ex1())) {
438                         const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff);
439                         if (num_coeff.is_real()) {
440                                 if (num_coeff.is_positive()>0) {
441                                         mul * mulp=new mul(mulref);
442                                         mulp->overall_coeff=_ex1();
443                                         mulp->clearflag(status_flags::evaluated);
444                                         mulp->clearflag(status_flags::hash_calculated);
445                                         return (new mul(power(*mulp,exponent),
446                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
447                                 } else {
448                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
449                                         if (num_coeff.compare(_num_1())!=0) {
450                                                 mul * mulp=new mul(mulref);
451                                                 mulp->overall_coeff=_ex_1();
452                                                 mulp->clearflag(status_flags::evaluated);
453                                                 mulp->clearflag(status_flags::hash_calculated);
454                                                 return (new mul(power(*mulp,exponent),
455                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
456                                         }
457                                 }
458                         }
459                 }
460         }
461                 
462         if (are_ex_trivially_equal(ebasis,basis) &&
463                 are_ex_trivially_equal(eexponent,exponent)) {
464                 return this->hold();
465         }
466         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
467                                                                                                    status_flags::evaluated);
468 }
469
470 ex power::evalf(int level) const
471 {
472         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
473
474         ex ebasis;
475         ex eexponent;
476         
477         if (level==1) {
478                 ebasis = basis;
479                 eexponent = exponent;
480         } else if (level == -max_recursion_level) {
481                 throw(std::runtime_error("max recursion level reached"));
482         } else {
483                 ebasis = basis.evalf(level-1);
484                 if (!is_ex_exactly_of_type(eexponent,numeric))
485                         eexponent = exponent.evalf(level-1);
486                 else
487                         eexponent = exponent;
488         }
489
490         return power(ebasis,eexponent);
491 }
492
493 ex power::subs(const lst & ls, const lst & lr) const
494 {
495         const ex & subsed_basis=basis.subs(ls,lr);
496         const ex & subsed_exponent=exponent.subs(ls,lr);
497
498         if (are_ex_trivially_equal(basis,subsed_basis)&&
499                 are_ex_trivially_equal(exponent,subsed_exponent)) {
500                 return *this;
501         }
502         
503         return power(subsed_basis, subsed_exponent);
504 }
505
506 ex power::simplify_ncmul(const exvector & v) const
507 {
508         return inherited::simplify_ncmul(v);
509 }
510
511 // protected
512
513 /** Implementation of ex::diff() for a power.
514  *  @see ex::diff */
515 ex power::derivative(const symbol & s) const
516 {
517         if (exponent.info(info_flags::real)) {
518                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
519                 epvector newseq;
520                 newseq.reserve(2);
521                 newseq.push_back(expair(basis, exponent - _ex1()));
522                 newseq.push_back(expair(basis.diff(s), _ex1()));
523                 return mul(newseq, exponent);
524         } else {
525                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
526                 return mul(*this,
527                            add(mul(exponent.diff(s), log(basis)),
528                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
529         }
530 }
531
532 int power::compare_same_type(const basic & other) const
533 {
534         GINAC_ASSERT(is_exactly_of_type(other, power));
535         const power & o=static_cast<const power &>(const_cast<basic &>(other));
536
537         int cmpval;
538         cmpval=basis.compare(o.basis);
539         if (cmpval==0) {
540                 return exponent.compare(o.exponent);
541         }
542         return cmpval;
543 }
544
545 unsigned power::return_type(void) const
546 {
547         return basis.return_type();
548 }
549    
550 unsigned power::return_type_tinfo(void) const
551 {
552         return basis.return_type_tinfo();
553 }
554
555 ex power::expand(unsigned options) const
556 {
557         if (flags & status_flags::expanded)
558                 return *this;
559         
560         ex expanded_basis = basis.expand(options);
561         ex expanded_exponent = exponent.expand(options);
562
563         // x^(a+b) -> x^a * x^b
564         if (is_ex_exactly_of_type(expanded_exponent, add)) {
565                 const add &a = ex_to_add(expanded_exponent);
566                 exvector distrseq;
567                 distrseq.reserve(a.seq.size() + 1);
568                 epvector::const_iterator last = a.seq.end();
569                 epvector::const_iterator cit = a.seq.begin();
570                 while (cit!=last) {
571                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
572                         cit++;
573                 }
574
575                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
576                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
577                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
578                         int int_exponent = num_exponent.to_int();
579                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
580                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
581                         else
582                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
583                 } else
584                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
585
586                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
587                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
588                 return r.expand();
589         }
590
591         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
592                 !ex_to_numeric(expanded_exponent).is_integer()) {
593                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
594                         return this->hold();
595                 } else {
596                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
597                 }
598         }
599         
600         // integer numeric exponent
601         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
602         int int_exponent = num_exponent.to_int();
603         
604         // (x+y)^n, n>0
605         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
606                 return expand_add(ex_to_add(expanded_basis), int_exponent);
607         }
608         
609         // (x*y)^n -> x^n * y^n
610         if (is_ex_exactly_of_type(expanded_basis,mul)) {
611                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
612         }
613         
614         // cannot expand further
615         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
616                 return this->hold();
617         } else {
618                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
619         }
620 }
621
622 //////////
623 // new virtual functions which can be overridden by derived classes
624 //////////
625
626 // none
627
628 //////////
629 // non-virtual functions in this class
630 //////////
631
632 /** expand a^n where a is an add and n is an integer.
633  *  @see power::expand */
634 ex power::expand_add(const add & a, int n) const
635 {
636         if (n==2)
637                 return expand_add_2(a);
638         
639         int m = a.nops();
640         exvector sum;
641         sum.reserve((n+1)*(m-1));
642         intvector k(m-1);
643         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
644         intvector upper_limit(m-1);
645         int l;
646         
647         for (int l=0; l<m-1; l++) {
648                 k[l] = 0;
649                 k_cum[l] = 0;
650                 upper_limit[l] = n;
651         }
652         
653         while (1) {
654                 exvector term;
655                 term.reserve(m+1);
656                 for (l=0; l<m-1; l++) {
657                         const ex & b = a.op(l);
658                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
659                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
660                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
661                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
662                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
663                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
664                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
665                         if (is_ex_exactly_of_type(b,mul)) {
666                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
667                         } else {
668                                 term.push_back(power(b,k[l]));
669                         }
670                 }
671                 
672                 const ex & b = a.op(l);
673                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
674                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
675                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
676                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
677                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
678                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
679                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
680                 if (is_ex_exactly_of_type(b,mul)) {
681                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
682                 } else {
683                         term.push_back(power(b,n-k_cum[m-2]));
684                 }
685                 
686                 numeric f = binomial(numeric(n),numeric(k[0]));
687                 for (l=1; l<m-1; l++) {
688                         f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
689                 }
690                 term.push_back(f);
691
692                 /*
693                 cout << "begin term" << endl;
694                 for (int i=0; i<m-1; i++) {
695                         cout << "k[" << i << "]=" << k[i] << endl;
696                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
697                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
698                 }
699                 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
700                         cout << *cit << endl;
701                 }
702                 cout << "end term" << endl;
703                 */
704
705                 // TODO: optimize this
706                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
707                 
708                 // increment k[]
709                 l=m-2;
710                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
711                         k[l]=0;    
712                         l--;
713                 }
714                 if (l<0) break;
715
716                 // recalc k_cum[] and upper_limit[]
717                 if (l==0) {
718                         k_cum[0]=k[0];
719                 } else {
720                         k_cum[l]=k_cum[l-1]+k[l];
721                 }
722                 for (int i=l+1; i<m-1; i++) {
723                         k_cum[i]=k_cum[i-1]+k[i];
724                 }
725
726                 for (int i=l+1; i<m-1; i++) {
727                         upper_limit[i]=n-k_cum[i-1];
728                 }   
729         }
730         return (new add(sum))->setflag(status_flags::dynallocated |
731                                                                    status_flags::expanded );
732 }
733
734
735 /** Special case of power::expand_add. Expands a^2 where a is an add.
736  *  @see power::expand_add */
737 ex power::expand_add_2(const add & a) const
738 {
739         epvector sum;
740         unsigned a_nops=a.nops();
741         sum.reserve((a_nops*(a_nops+1))/2);
742         epvector::const_iterator last=a.seq.end();
743
744         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
745         // first part: ignore overall_coeff and expand other terms
746         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
747                 const ex & r=(*cit0).rest;
748                 const ex & c=(*cit0).coeff;
749                 
750                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
751                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
752                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
753                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
754                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
755                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
756                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
757
758                 if (are_ex_trivially_equal(c,_ex1())) {
759                         if (is_ex_exactly_of_type(r,mul)) {
760                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
761                                                      _ex1()));
762                         } else {
763                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
764                                                      _ex1()));
765                         }
766                 } else {
767                         if (is_ex_exactly_of_type(r,mul)) {
768                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
769                                                      ex_to_numeric(c).power_dyn(_num2())));
770                         } else {
771                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
772                                                      ex_to_numeric(c).power_dyn(_num2())));
773                         }
774                 }
775                         
776                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
777                         const ex & r1=(*cit1).rest;
778                         const ex & c1=(*cit1).coeff;
779                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
780                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
781                 }
782         }
783
784         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
785
786         // second part: add terms coming from overall_factor (if != 0)
787         if (!a.overall_coeff.is_equal(_ex0())) {
788                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
789                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
790                 }
791                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
792         }
793                 
794         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
795         
796         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
797 }
798
799 /** Expand factors of m in m^n where m is a mul and n is and integer
800  *  @see power::expand */
801 ex power::expand_mul(const mul & m, const numeric & n) const
802 {
803         if (n.is_equal(_num0()))
804                 return _ex1();
805         
806         epvector distrseq;
807         distrseq.reserve(m.seq.size());
808         epvector::const_iterator last = m.seq.end();
809         epvector::const_iterator cit = m.seq.begin();
810         while (cit!=last) {
811                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
812                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
813                 } else {
814                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
815                         // since n is an integer
816                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
817                 }
818                 ++cit;
819         }
820         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
821 }
822
823 /*
824 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
825                                unsigned options) const
826 {
827         // obsolete
828
829         exvector distrseq;
830         epvector splitseq;
831
832         const add & addref=static_cast<const add &>(*basis.bp);
833
834         splitseq=addref.seq;
835         splitseq.pop_back();
836         ex first_operands=add(splitseq);
837         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
838         
839         int n=exponent.to_int();
840         for (int k=0; k<=n; k++) {
841                 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
842                                                  * power(last_operand,numeric(n-k)));
843         }
844         return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
845 }
846 */
847
848 /*
849 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
850                                                                 unsigned options) const
851 {
852         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
853                         expand(options | expand_options::internal_do_not_expand_power_operands);
854
855         return ex(mul(rest_power,basis),0).
856                expand(options | expand_options::internal_do_not_expand_mul_operands);
857 }
858 */
859
860 //////////
861 // static member variables
862 //////////
863
864 // protected
865
866 unsigned power::precedence = 60;
867
868 // helper function
869
870 ex sqrt(const ex & a)
871 {
872         return power(a,_ex1_2());
873 }
874
875 #ifndef NO_NAMESPACE_GINAC
876 } // namespace GiNaC
877 #endif // ndef NO_NAMESPACE_GINAC