2d4360dea0cce94416d4f136872a2c5d7da9561e
[ginac.git] / ginac / power.cpp
1 /** @file power.cpp
2  *
3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
7  *
8  *  This program is free software; you can redistribute it and/or modify
9  *  it under the terms of the GNU General Public License as published by
10  *  the Free Software Foundation; either version 2 of the License, or
11  *  (at your option) any later version.
12  *
13  *  This program is distributed in the hope that it will be useful,
14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
16  *  GNU General Public License for more details.
17  *
18  *  You should have received a copy of the GNU General Public License
19  *  along with this program; if not, write to the Free Software
20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
21  */
22
23 #include <vector>
24 #include <iostream>
25 #include <stdexcept>
26
27 #include "power.h"
28 #include "expairseq.h"
29 #include "add.h"
30 #include "mul.h"
31 #include "ncmul.h"
32 #include "numeric.h"
33 #include "constant.h"
34 #include "inifcns.h" // for log() in power::derivative()
35 #include "matrix.h"
36 #include "symbol.h"
37 #include "print.h"
38 #include "archive.h"
39 #include "debugmsg.h"
40 #include "utils.h"
41
42 namespace GiNaC {
43
44 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
45
46 typedef std::vector<int> intvector;
47
48 //////////
49 // default ctor, dtor, copy ctor assignment operator and helpers
50 //////////
51
52 power::power() : inherited(TINFO_power)
53 {
54         debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
55 }
56
57 void power::copy(const power & other)
58 {
59         inherited::copy(other);
60         basis = other.basis;
61         exponent = other.exponent;
62 }
63
64 DEFAULT_DESTROY(power)
65
66 //////////
67 // other ctors
68 //////////
69
70 // all inlined
71
72 //////////
73 // archiving
74 //////////
75
76 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
77 {
78         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
79         n.find_ex("basis", basis, sym_lst);
80         n.find_ex("exponent", exponent, sym_lst);
81 }
82
83 void power::archive(archive_node &n) const
84 {
85         inherited::archive(n);
86         n.add_ex("basis", basis);
87         n.add_ex("exponent", exponent);
88 }
89
90 DEFAULT_UNARCHIVE(power)
91
92 //////////
93 // functions overriding virtual functions from base classes
94 //////////
95
96 // public
97
98 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
99 {
100         // Optimal output of integer powers of symbols to aid compiler CSE.
101         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
102         // to learn why such a hack is really necessary.
103         if (exp == 1) {
104                 x.print(c);
105         } else if (exp == 2) {
106                 x.print(c);
107                 c.s << "*";
108                 x.print(c);
109         } else if (exp & 1) {
110                 x.print(c);
111                 c.s << "*";
112                 print_sym_pow(c, x, exp-1);
113         } else {
114                 c.s << "(";
115                 print_sym_pow(c, x, exp >> 1);
116                 c.s << ")*(";
117                 print_sym_pow(c, x, exp >> 1);
118                 c.s << ")";
119         }
120 }
121
122 void power::print(const print_context & c, unsigned level) const
123 {
124         debugmsg("power print", LOGLEVEL_PRINT);
125
126         if (is_a<print_tree>(c)) {
127
128                 inherited::print(c, level);
129
130         } else if (is_a<print_csrc>(c)) {
131
132                 // Integer powers of symbols are printed in a special, optimized way
133                 if (exponent.info(info_flags::integer)
134                  && (is_exactly_a<symbol>(basis) || is_exactly_a<constant>(basis))) {
135                         int exp = ex_to<numeric>(exponent).to_int();
136                         if (exp > 0)
137                                 c.s << '(';
138                         else {
139                                 exp = -exp;
140                                 if (is_a<print_csrc_cl_N>(c))
141                                         c.s << "recip(";
142                                 else
143                                         c.s << "1.0/(";
144                         }
145                         print_sym_pow(c, ex_to<symbol>(basis), exp);
146                         c.s << ')';
147
148                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
149                 } else if (exponent.compare(_num_1()) == 0) {
150                         if (is_a<print_csrc_cl_N>(c))
151                                 c.s << "recip(";
152                         else
153                                 c.s << "1.0/(";
154                         basis.print(c);
155                         c.s << ')';
156
157                 // Otherwise, use the pow() or expt() (CLN) functions
158                 } else {
159                         if (is_a<print_csrc_cl_N>(c))
160                                 c.s << "expt(";
161                         else
162                                 c.s << "pow(";
163                         basis.print(c);
164                         c.s << ',';
165                         exponent.print(c);
166                         c.s << ')';
167                 }
168
169         } else {
170
171                 if (exponent.is_equal(_ex1_2())) {
172                         if (is_a<print_latex>(c))
173                                 c.s << "\\sqrt{";
174                         else
175                                 c.s << "sqrt(";
176                         basis.print(c);
177                         if (is_a<print_latex>(c))
178                                 c.s << '}';
179                         else
180                                 c.s << ')';
181                 } else {
182                         if (precedence() <= level) {
183                                 if (is_a<print_latex>(c))
184                                         c.s << "{(";
185                                 else
186                                         c.s << "(";
187                         }
188                         basis.print(c, precedence());
189                         c.s << '^';
190                         if (is_a<print_latex>(c))
191                                 c.s << '{';
192                         exponent.print(c, precedence());
193                         if (is_a<print_latex>(c))
194                                 c.s << '}';
195                         if (precedence() <= level) {
196                                 if (is_a<print_latex>(c))
197                                         c.s << ")}";
198                                 else
199                                         c.s << ')';
200                         }
201                 }
202         }
203 }
204
205 bool power::info(unsigned inf) const
206 {
207         switch (inf) {
208                 case info_flags::polynomial:
209                 case info_flags::integer_polynomial:
210                 case info_flags::cinteger_polynomial:
211                 case info_flags::rational_polynomial:
212                 case info_flags::crational_polynomial:
213                         return exponent.info(info_flags::nonnegint);
214                 case info_flags::rational_function:
215                         return exponent.info(info_flags::integer);
216                 case info_flags::algebraic:
217                         return (!exponent.info(info_flags::integer) ||
218                                         basis.info(inf));
219         }
220         return inherited::info(inf);
221 }
222
223 unsigned power::nops() const
224 {
225         return 2;
226 }
227
228 ex & power::let_op(int i)
229 {
230         GINAC_ASSERT(i>=0);
231         GINAC_ASSERT(i<2);
232
233         return i==0 ? basis : exponent;
234 }
235
236 ex power::map(map_function & f) const
237 {
238         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
239 }
240
241 int power::degree(const ex & s) const
242 {
243         if (is_exactly_of_type(*exponent.bp,numeric)) {
244                 if (basis.is_equal(s)) {
245                         if (ex_to<numeric>(exponent).is_integer())
246                                 return ex_to<numeric>(exponent).to_int();
247                         else
248                                 return 0;
249                 } else
250                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
251         }
252         return 0;
253 }
254
255 int power::ldegree(const ex & s) const 
256 {
257         if (is_exactly_of_type(*exponent.bp,numeric)) {
258                 if (basis.is_equal(s)) {
259                         if (ex_to<numeric>(exponent).is_integer())
260                                 return ex_to<numeric>(exponent).to_int();
261                         else
262                                 return 0;
263                 } else
264                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
265         }
266         return 0;
267 }
268
269 ex power::coeff(const ex & s, int n) const
270 {
271         if (!basis.is_equal(s)) {
272                 // basis not equal to s
273                 if (n == 0)
274                         return *this;
275                 else
276                         return _ex0();
277         } else {
278                 // basis equal to s
279                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to<numeric>(exponent).is_integer()) {
280                         // integer exponent
281                         int int_exp = ex_to<numeric>(exponent).to_int();
282                         if (n == int_exp)
283                                 return _ex1();
284                         else
285                                 return _ex0();
286                 } else {
287                         // non-integer exponents are treated as zero
288                         if (n == 0)
289                                 return *this;
290                         else
291                                 return _ex0();
292                 }
293         }
294 }
295
296 ex power::eval(int level) const
297 {
298         // simplifications: ^(x,0) -> 1 (0^0 handled here)
299         //                  ^(x,1) -> x
300         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
301         //                  ^(1,x) -> 1
302         //                  ^(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)
303         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
304         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
305         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
306         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
307         
308         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
309         
310         if ((level==1) && (flags & status_flags::evaluated))
311                 return *this;
312         else if (level == -max_recursion_level)
313                 throw(std::runtime_error("max recursion level reached"));
314         
315         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
316         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
317         
318         bool basis_is_numerical = false;
319         bool exponent_is_numerical = false;
320         numeric * num_basis;
321         numeric * num_exponent;
322         
323         if (is_exactly_of_type(*ebasis.bp,numeric)) {
324                 basis_is_numerical = true;
325                 num_basis = static_cast<numeric *>(ebasis.bp);
326         }
327         if (is_exactly_of_type(*eexponent.bp,numeric)) {
328                 exponent_is_numerical = true;
329                 num_exponent = static_cast<numeric *>(eexponent.bp);
330         }
331         
332         // ^(x,0) -> 1 (0^0 also handled here)
333         if (eexponent.is_zero()) {
334                 if (ebasis.is_zero())
335                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
336                 else
337                         return _ex1();
338         }
339         
340         // ^(x,1) -> x
341         if (eexponent.is_equal(_ex1()))
342                 return ebasis;
343         
344         // ^(0,c1) -> 0 or exception (depending on real value of c1)
345         if (ebasis.is_zero() && exponent_is_numerical) {
346                 if ((num_exponent->real()).is_zero())
347                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
348                 else if ((num_exponent->real()).is_negative())
349                         throw (pole_error("power::eval(): division by zero",1));
350                 else
351                         return _ex0();
352         }
353         
354         // ^(1,x) -> 1
355         if (ebasis.is_equal(_ex1()))
356                 return _ex1();
357         
358         if (exponent_is_numerical) {
359
360                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
361                 // except if c1,c2 are rational, but c1^c2 is not)
362                 if (basis_is_numerical) {
363                         bool basis_is_crational = num_basis->is_crational();
364                         bool exponent_is_crational = num_exponent->is_crational();
365                         numeric res = num_basis->power(*num_exponent);
366                 
367                         if ((!basis_is_crational || !exponent_is_crational)
368                                 || res.is_crational()) {
369                                 return res;
370                         }
371                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
372
373                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
374                         if (basis_is_crational && exponent_is_crational
375                                 && num_exponent->is_real()
376                                 && !num_exponent->is_integer()) {
377                                 numeric n = num_exponent->numer();
378                                 numeric m = num_exponent->denom();
379                                 numeric r;
380                                 numeric q = iquo(n, m, r);
381                                 if (r.is_negative()) {
382                                         r = r.add(m);
383                                         q = q.sub(_num1());
384                                 }
385                                 if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
386                                         return this->hold();
387                                 else {
388                                         epvector res;
389                                         res.push_back(expair(ebasis,r.div(m)));
390                                         return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
391                                 }
392                         }
393                 }
394         
395                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
396                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
397                 // case c1==1 should not happen, see below!)
398                 if (is_ex_exactly_of_type(ebasis,power)) {
399                         const power & sub_power = ex_to<power>(ebasis);
400                         const ex & sub_basis = sub_power.basis;
401                         const ex & sub_exponent = sub_power.exponent;
402                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
403                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
404                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
405                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1()).is_negative())
406                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
407                         }
408                 }
409         
410                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
411                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
412                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
413                 }
414         
415                 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
416                 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
417                 if (is_ex_exactly_of_type(ebasis,mul)) {
418                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
419                         const mul & mulref = ex_to<mul>(ebasis);
420                         if (!mulref.overall_coeff.is_equal(_ex1())) {
421                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
422                                 if (num_coeff.is_real()) {
423                                         if (num_coeff.is_positive()) {
424                                                 mul * mulp = new mul(mulref);
425                                                 mulp->overall_coeff = _ex1();
426                                                 mulp->clearflag(status_flags::evaluated);
427                                                 mulp->clearflag(status_flags::hash_calculated);
428                                                 return (new mul(power(*mulp,exponent),
429                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
430                                         } else {
431                                                 GINAC_ASSERT(num_coeff.compare(_num0())<0);
432                                                 if (num_coeff.compare(_num_1())!=0) {
433                                                         mul * mulp = new mul(mulref);
434                                                         mulp->overall_coeff = _ex_1();
435                                                         mulp->clearflag(status_flags::evaluated);
436                                                         mulp->clearflag(status_flags::hash_calculated);
437                                                         return (new mul(power(*mulp,exponent),
438                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
439                                                 }
440                                         }
441                                 }
442                         }
443                 }
444
445                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
446                 if (num_exponent->is_pos_integer() &&
447                     ebasis.return_type() != return_types::commutative &&
448                     !is_ex_of_type(ebasis,matrix)) {
449                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
450                 }
451         }
452         
453         if (are_ex_trivially_equal(ebasis,basis) &&
454                 are_ex_trivially_equal(eexponent,exponent)) {
455                 return this->hold();
456         }
457         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
458                                                                                                    status_flags::evaluated);
459 }
460
461 ex power::evalf(int level) const
462 {
463         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
464
465         ex ebasis;
466         ex eexponent;
467         
468         if (level==1) {
469                 ebasis = basis;
470                 eexponent = exponent;
471         } else if (level == -max_recursion_level) {
472                 throw(std::runtime_error("max recursion level reached"));
473         } else {
474                 ebasis = basis.evalf(level-1);
475                 if (!is_ex_exactly_of_type(eexponent,numeric))
476                         eexponent = exponent.evalf(level-1);
477                 else
478                         eexponent = exponent;
479         }
480
481         return power(ebasis,eexponent);
482 }
483
484 ex power::evalm(void) const
485 {
486         const ex ebasis = basis.evalm();
487         const ex eexponent = exponent.evalm();
488         if (is_ex_of_type(ebasis,matrix)) {
489                 if (is_ex_of_type(eexponent,numeric)) {
490                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
491                 }
492         }
493         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
494 }
495
496 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
497 {
498         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
499         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
500
501         if (are_ex_trivially_equal(basis, subsed_basis)
502          && are_ex_trivially_equal(exponent, subsed_exponent))
503                 return basic::subs(ls, lr, no_pattern);
504         else
505                 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
506 }
507
508 ex power::simplify_ncmul(const exvector & v) const
509 {
510         return inherited::simplify_ncmul(v);
511 }
512
513 // protected
514
515 /** Implementation of ex::diff() for a power.
516  *  @see ex::diff */
517 ex power::derivative(const symbol & s) const
518 {
519         if (exponent.info(info_flags::real)) {
520                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
521                 epvector newseq;
522                 newseq.reserve(2);
523                 newseq.push_back(expair(basis, exponent - _ex1()));
524                 newseq.push_back(expair(basis.diff(s), _ex1()));
525                 return mul(newseq, exponent);
526         } else {
527                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
528                 return mul(*this,
529                            add(mul(exponent.diff(s), log(basis)),
530                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
531         }
532 }
533
534 int power::compare_same_type(const basic & other) const
535 {
536         GINAC_ASSERT(is_exactly_of_type(other, power));
537         const power &o = static_cast<const power &>(other);
538
539         int cmpval = basis.compare(o.basis);
540         if (cmpval)
541                 return cmpval;
542         else
543                 return exponent.compare(o.exponent);
544 }
545
546 unsigned power::return_type(void) const
547 {
548         return basis.return_type();
549 }
550    
551 unsigned power::return_type_tinfo(void) const
552 {
553         return basis.return_type_tinfo();
554 }
555
556 ex power::expand(unsigned options) const
557 {
558         if (options == 0 && (flags & status_flags::expanded))
559                 return *this;
560         
561         ex expanded_basis = basis.expand(options);
562         ex expanded_exponent = exponent.expand(options);
563         
564         // x^(a+b) -> x^a * x^b
565         if (is_ex_exactly_of_type(expanded_exponent, add)) {
566                 const add &a = ex_to<add>(expanded_exponent);
567                 exvector distrseq;
568                 distrseq.reserve(a.seq.size() + 1);
569                 epvector::const_iterator last = a.seq.end();
570                 epvector::const_iterator cit = a.seq.begin();
571                 while (cit!=last) {
572                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
573                         ++cit;
574                 }
575                 
576                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
577                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
578                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
579                         int int_exponent = num_exponent.to_int();
580                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
581                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
582                         else
583                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
584                 } else
585                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
586                 
587                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
588                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
589                 return r.expand();
590         }
591         
592         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
593                 !ex_to<numeric>(expanded_exponent).is_integer()) {
594                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
595                         return this->hold();
596                 } else {
597                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
598                 }
599         }
600         
601         // integer numeric exponent
602         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
603         int int_exponent = num_exponent.to_int();
604         
605         // (x+y)^n, n>0
606         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
607                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
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         // cannot expand further
614         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
615                 return this->hold();
616         else
617                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
618 }
619
620 //////////
621 // new virtual functions which can be overridden by derived classes
622 //////////
623
624 // none
625
626 //////////
627 // non-virtual functions in this class
628 //////////
629
630 /** expand a^n where a is an add and n is an integer.
631  *  @see power::expand */
632 ex power::expand_add(const add & a, int n) const
633 {
634         if (n==2)
635                 return expand_add_2(a);
636         
637         int m = a.nops();
638         exvector sum;
639         sum.reserve((n+1)*(m-1));
640         intvector k(m-1);
641         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
642         intvector upper_limit(m-1);
643         int l;
644         
645         for (int l=0; l<m-1; l++) {
646                 k[l] = 0;
647                 k_cum[l] = 0;
648                 upper_limit[l] = n;
649         }
650         
651         while (true) {
652                 exvector term;
653                 term.reserve(m+1);
654                 for (l=0; l<m-1; l++) {
655                         const ex & b = a.op(l);
656                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
657                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
658                                      !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
659                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
660                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
661                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
662                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
663                         if (is_ex_exactly_of_type(b,mul))
664                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
665                         else
666                                 term.push_back(power(b,k[l]));
667                 }
668                 
669                 const ex & b = a.op(l);
670                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
671                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
672                              !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
673                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
674                              !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
675                              !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
676                              !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
677                 if (is_ex_exactly_of_type(b,mul))
678                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
679                 else
680                         term.push_back(power(b,n-k_cum[m-2]));
681                 
682                 numeric f = binomial(numeric(n),numeric(k[0]));
683                 for (l=1; l<m-1; l++)
684                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
685                 
686                 term.push_back(f);
687                 
688                 // TODO: Can we optimize this?  Alex seemed to think so...
689                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
690                 
691                 // increment k[]
692                 l = m-2;
693                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
694                         k[l] = 0;    
695                         --l;
696                 }
697                 if (l<0) break;
698                 
699                 // recalc k_cum[] and upper_limit[]
700                 if (l==0)
701                         k_cum[0] = k[0];
702                 else
703                         k_cum[l] = k_cum[l-1]+k[l];
704                 
705                 for (int i=l+1; i<m-1; i++)
706                         k_cum[i] = k_cum[i-1]+k[i];
707                 
708                 for (int i=l+1; i<m-1; i++)
709                         upper_limit[i] = n-k_cum[i-1];
710         }
711         return (new add(sum))->setflag(status_flags::dynallocated |
712                                        status_flags::expanded );
713 }
714
715
716 /** Special case of power::expand_add. Expands a^2 where a is an add.
717  *  @see power::expand_add */
718 ex power::expand_add_2(const add & a) const
719 {
720         epvector sum;
721         unsigned a_nops = a.nops();
722         sum.reserve((a_nops*(a_nops+1))/2);
723         epvector::const_iterator last = a.seq.end();
724         
725         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
726         // first part: ignore overall_coeff and expand other terms
727         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
728                 const ex & r = cit0->rest;
729                 const ex & c = cit0->coeff;
730                 
731                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
732                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
733                              !is_ex_exactly_of_type(ex_to<power>(r).exponent,numeric) ||
734                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
735                              !is_ex_exactly_of_type(ex_to<power>(r).basis,add) ||
736                              !is_ex_exactly_of_type(ex_to<power>(r).basis,mul) ||
737                              !is_ex_exactly_of_type(ex_to<power>(r).basis,power));
738                 
739                 if (are_ex_trivially_equal(c,_ex1())) {
740                         if (is_ex_exactly_of_type(r,mul)) {
741                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
742                                                      _ex1()));
743                         } else {
744                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
745                                                      _ex1()));
746                         }
747                 } else {
748                         if (is_ex_exactly_of_type(r,mul)) {
749                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
750                                                      ex_to<numeric>(c).power_dyn(_num2())));
751                         } else {
752                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
753                                                      ex_to<numeric>(c).power_dyn(_num2())));
754                         }
755                 }
756                         
757                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
758                         const ex & r1 = cit1->rest;
759                         const ex & c1 = cit1->coeff;
760                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
761                                                                       _num2().mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
762                 }
763         }
764         
765         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
766         
767         // second part: add terms coming from overall_factor (if != 0)
768         if (!a.overall_coeff.is_zero()) {
769                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
770                 while (i != end) {
771                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2())));
772                         ++i;
773                 }
774                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2()),_ex1()));
775         }
776         
777         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
778         
779         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
780 }
781
782 /** Expand factors of m in m^n where m is a mul and n is and integer
783  *  @see power::expand */
784 ex power::expand_mul(const mul & m, const numeric & n) const
785 {
786         if (n.is_zero())
787                 return _ex1();
788         
789         epvector distrseq;
790         distrseq.reserve(m.seq.size());
791         epvector::const_iterator last = m.seq.end();
792         epvector::const_iterator cit = m.seq.begin();
793         while (cit!=last) {
794                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
795                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
796                 } else {
797                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
798                         // since n is an integer
799                         distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
800                 }
801                 ++cit;
802         }
803         return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
804 }
805
806 // helper function
807
808 ex sqrt(const ex & a)
809 {
810         return power(a,_ex1_2());
811 }
812
813 } // namespace GiNaC