eefd438f9bb334415694e313bf0018e0bae48df7
[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 "inifcns.h"
34 #include "matrix.h"
35 #include "symbol.h"
36 #include "print.h"
37 #include "archive.h"
38 #include "debugmsg.h"
39 #include "utils.h"
40
41 namespace GiNaC {
42
43 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
44
45 typedef std::vector<int> intvector;
46
47 //////////
48 // default ctor, dtor, copy ctor assignment operator and helpers
49 //////////
50
51 power::power() : basic(TINFO_power)
52 {
53         debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
54 }
55
56 void power::copy(const power & other)
57 {
58         inherited::copy(other);
59         basis = other.basis;
60         exponent = other.exponent;
61 }
62
63 DEFAULT_DESTROY(power)
64
65 //////////
66 // other ctors
67 //////////
68
69 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
70 {
71         debugmsg("power ctor from ex,ex",LOGLEVEL_CONSTRUCT);
72 }
73
74 /** Ctor from an ex and a bare numeric.  This is somewhat more efficient than
75  *  the normal ctor from two ex whenever it can be used. */
76 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
77 {
78         debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
79 }
80
81 //////////
82 // archiving
83 //////////
84
85 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
86 {
87         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
88         n.find_ex("basis", basis, sym_lst);
89         n.find_ex("exponent", exponent, sym_lst);
90 }
91
92 void power::archive(archive_node &n) const
93 {
94         inherited::archive(n);
95         n.add_ex("basis", basis);
96         n.add_ex("exponent", exponent);
97 }
98
99 DEFAULT_UNARCHIVE(power)
100
101 //////////
102 // functions overriding virtual functions from bases classes
103 //////////
104
105 // public
106
107 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
108 {
109         // Optimal output of integer powers of symbols to aid compiler CSE.
110         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
111         // to learn why such a hack is really necessary.
112         if (exp == 1) {
113                 x.print(c);
114         } else if (exp == 2) {
115                 x.print(c);
116                 c.s << "*";
117                 x.print(c);
118         } else if (exp & 1) {
119                 x.print(c);
120                 c.s << "*";
121                 print_sym_pow(c, x, exp-1);
122         } else {
123                 c.s << "(";
124                 print_sym_pow(c, x, exp >> 1);
125                 c.s << ")*(";
126                 print_sym_pow(c, x, exp >> 1);
127                 c.s << ")";
128         }
129 }
130
131 void power::print(const print_context & c, unsigned level) const
132 {
133         debugmsg("power print", LOGLEVEL_PRINT);
134
135         if (is_of_type(c, print_tree)) {
136
137                 inherited::print(c, level);
138
139         } else if (is_of_type(c, print_csrc)) {
140
141                 // Integer powers of symbols are printed in a special, optimized way
142                 if (exponent.info(info_flags::integer)
143                  && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
144                         int exp = ex_to_numeric(exponent).to_int();
145                         if (exp > 0)
146                                 c.s << '(';
147                         else {
148                                 exp = -exp;
149                                 if (is_of_type(c, print_csrc_cl_N))
150                                         c.s << "recip(";
151                                 else
152                                         c.s << "1.0/(";
153                         }
154                         print_sym_pow(c, ex_to_symbol(basis), exp);
155                         c.s << ')';
156
157                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
158                 } else if (exponent.compare(_num_1()) == 0) {
159                         if (is_of_type(c, print_csrc_cl_N))
160                                 c.s << "recip(";
161                         else
162                                 c.s << "1.0/(";
163                         basis.print(c);
164                         c.s << ')';
165
166                 // Otherwise, use the pow() or expt() (CLN) functions
167                 } else {
168                         if (is_of_type(c, print_csrc_cl_N))
169                                 c.s << "expt(";
170                         else
171                                 c.s << "pow(";
172                         basis.print(c);
173                         c.s << ',';
174                         exponent.print(c);
175                         c.s << ')';
176                 }
177
178         } else {
179
180                 if (exponent.is_equal(_ex1_2())) {
181                         if (is_of_type(c, print_latex))
182                                 c.s << "\\sqrt{";
183                         else
184                                 c.s << "sqrt(";
185                         basis.print(c);
186                         if (is_of_type(c, print_latex))
187                                 c.s << '}';
188                         else
189                                 c.s << ')';
190                 } else {
191                         if (precedence() <= level) {
192                                 if (is_of_type(c, print_latex))
193                                         c.s << "{(";
194                                 else
195                                         c.s << "(";
196                         }
197                         basis.print(c, precedence());
198                         c.s << '^';
199                         if (is_of_type(c, print_latex))
200                                 c.s << '{';
201                         exponent.print(c, precedence());
202                         if (is_of_type(c, print_latex))
203                                 c.s << '}';
204                         if (precedence() <= level) {
205                                 if (is_of_type(c, print_latex))
206                                         c.s << ")}";
207                                 else
208                                         c.s << ')';
209                         }
210                 }
211         }
212 }
213
214 bool power::info(unsigned inf) const
215 {
216         switch (inf) {
217                 case info_flags::polynomial:
218                 case info_flags::integer_polynomial:
219                 case info_flags::cinteger_polynomial:
220                 case info_flags::rational_polynomial:
221                 case info_flags::crational_polynomial:
222                         return exponent.info(info_flags::nonnegint);
223                 case info_flags::rational_function:
224                         return exponent.info(info_flags::integer);
225                 case info_flags::algebraic:
226                         return (!exponent.info(info_flags::integer) ||
227                                         basis.info(inf));
228         }
229         return inherited::info(inf);
230 }
231
232 unsigned power::nops() const
233 {
234         return 2;
235 }
236
237 ex & power::let_op(int i)
238 {
239         GINAC_ASSERT(i>=0);
240         GINAC_ASSERT(i<2);
241
242         return i==0 ? basis : exponent;
243 }
244
245 ex power::map(map_func f) const
246 {
247         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
248 }
249
250 int power::degree(const ex & s) const
251 {
252         if (is_exactly_of_type(*exponent.bp,numeric)) {
253                 if (basis.is_equal(s)) {
254                         if (ex_to_numeric(exponent).is_integer())
255                                 return ex_to_numeric(exponent).to_int();
256                         else
257                                 return 0;
258                 } else
259                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
260         }
261         return 0;
262 }
263
264 int power::ldegree(const ex & s) const 
265 {
266         if (is_exactly_of_type(*exponent.bp,numeric)) {
267                 if (basis.is_equal(s)) {
268                         if (ex_to_numeric(exponent).is_integer())
269                                 return ex_to_numeric(exponent).to_int();
270                         else
271                                 return 0;
272                 } else
273                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
274         }
275         return 0;
276 }
277
278 ex power::coeff(const ex & s, int n) const
279 {
280         if (!basis.is_equal(s)) {
281                 // basis not equal to s
282                 if (n == 0)
283                         return *this;
284                 else
285                         return _ex0();
286         } else {
287                 // basis equal to s
288                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
289                         // integer exponent
290                         int int_exp = ex_to_numeric(exponent).to_int();
291                         if (n == int_exp)
292                                 return _ex1();
293                         else
294                                 return _ex0();
295                 } else {
296                         // non-integer exponents are treated as zero
297                         if (n == 0)
298                                 return *this;
299                         else
300                                 return _ex0();
301                 }
302         }
303 }
304
305 ex power::eval(int level) const
306 {
307         // simplifications: ^(x,0) -> 1 (0^0 handled here)
308         //                  ^(x,1) -> x
309         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
310         //                  ^(1,x) -> 1
311         //                  ^(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)
312         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
313         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
314         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
315         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
316         
317         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
318         
319         if ((level==1) && (flags & status_flags::evaluated))
320                 return *this;
321         else if (level == -max_recursion_level)
322                 throw(std::runtime_error("max recursion level reached"));
323         
324         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
325         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
326         
327         bool basis_is_numerical = false;
328         bool exponent_is_numerical = false;
329         numeric * num_basis;
330         numeric * num_exponent;
331         
332         if (is_exactly_of_type(*ebasis.bp,numeric)) {
333                 basis_is_numerical = true;
334                 num_basis = static_cast<numeric *>(ebasis.bp);
335         }
336         if (is_exactly_of_type(*eexponent.bp,numeric)) {
337                 exponent_is_numerical = true;
338                 num_exponent = static_cast<numeric *>(eexponent.bp);
339         }
340         
341         // ^(x,0) -> 1 (0^0 also handled here)
342         if (eexponent.is_zero()) {
343                 if (ebasis.is_zero())
344                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
345                 else
346                         return _ex1();
347         }
348         
349         // ^(x,1) -> x
350         if (eexponent.is_equal(_ex1()))
351                 return ebasis;
352         
353         // ^(0,c1) -> 0 or exception (depending on real value of c1)
354         if (ebasis.is_zero() && exponent_is_numerical) {
355                 if ((num_exponent->real()).is_zero())
356                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
357                 else if ((num_exponent->real()).is_negative())
358                         throw (pole_error("power::eval(): division by zero",1));
359                 else
360                         return _ex0();
361         }
362         
363         // ^(1,x) -> 1
364         if (ebasis.is_equal(_ex1()))
365                 return _ex1();
366         
367         if (exponent_is_numerical) {
368
369                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
370                 // except if c1,c2 are rational, but c1^c2 is not)
371                 if (basis_is_numerical) {
372                         bool basis_is_crational = num_basis->is_crational();
373                         bool exponent_is_crational = num_exponent->is_crational();
374                         numeric res = num_basis->power(*num_exponent);
375                 
376                         if ((!basis_is_crational || !exponent_is_crational)
377                                 || res.is_crational()) {
378                                 return res;
379                         }
380                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
381
382                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
383                         if (basis_is_crational && exponent_is_crational
384                                 && num_exponent->is_real()
385                                 && !num_exponent->is_integer()) {
386                                 numeric n = num_exponent->numer();
387                                 numeric m = num_exponent->denom();
388                                 numeric r;
389                                 numeric q = iquo(n, m, r);
390                                 if (r.is_negative()) {
391                                         r = r.add(m);
392                                         q = q.sub(_num1());
393                                 }
394                                 if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
395                                         return this->hold();
396                                 else {
397                                         epvector res;
398                                         res.push_back(expair(ebasis,r.div(m)));
399                                         return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
400                                 }
401                         }
402                 }
403         
404                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
405                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
406                 // case c1==1 should not happen, see below!)
407                 if (is_ex_exactly_of_type(ebasis,power)) {
408                         const power & sub_power = ex_to_power(ebasis);
409                         const ex & sub_basis = sub_power.basis;
410                         const ex & sub_exponent = sub_power.exponent;
411                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
412                                 const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
413                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
414                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1()).is_negative())
415                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
416                         }
417                 }
418         
419                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
420                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
421                         return expand_mul(ex_to_mul(ebasis), *num_exponent);
422                 }
423         
424                 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
425                 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
426                 if (is_ex_exactly_of_type(ebasis,mul)) {
427                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
428                         const mul & mulref = ex_to_mul(ebasis);
429                         if (!mulref.overall_coeff.is_equal(_ex1())) {
430                                 const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
431                                 if (num_coeff.is_real()) {
432                                         if (num_coeff.is_positive()) {
433                                                 mul * mulp = new mul(mulref);
434                                                 mulp->overall_coeff = _ex1();
435                                                 mulp->clearflag(status_flags::evaluated);
436                                                 mulp->clearflag(status_flags::hash_calculated);
437                                                 return (new mul(power(*mulp,exponent),
438                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
439                                         } else {
440                                                 GINAC_ASSERT(num_coeff.compare(_num0())<0);
441                                                 if (num_coeff.compare(_num_1())!=0) {
442                                                         mul * mulp = new mul(mulref);
443                                                         mulp->overall_coeff = _ex_1();
444                                                         mulp->clearflag(status_flags::evaluated);
445                                                         mulp->clearflag(status_flags::hash_calculated);
446                                                         return (new mul(power(*mulp,exponent),
447                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
448                                                 }
449                                         }
450                                 }
451                         }
452                 }
453
454                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
455                 if (num_exponent->is_pos_integer() &&
456                     ebasis.return_type() != return_types::commutative &&
457                     !is_ex_of_type(ebasis,matrix)) {
458                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
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::evalm(void) const
494 {
495         ex ebasis = basis.evalm();
496         ex eexponent = exponent.evalm();
497         if (is_ex_of_type(ebasis,matrix)) {
498                 if (is_ex_of_type(eexponent,numeric)) {
499                         return (new matrix(ex_to_matrix(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
500                 }
501         }
502         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
503 }
504
505 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
506 {
507         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
508         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
509
510         if (are_ex_trivially_equal(basis, subsed_basis)
511          && are_ex_trivially_equal(exponent, subsed_exponent))
512                 return basic::subs(ls, lr, no_pattern);
513         else
514                 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
515 }
516
517 ex power::simplify_ncmul(const exvector & v) const
518 {
519         return inherited::simplify_ncmul(v);
520 }
521
522 // protected
523
524 /** Implementation of ex::diff() for a power.
525  *  @see ex::diff */
526 ex power::derivative(const symbol & s) const
527 {
528         if (exponent.info(info_flags::real)) {
529                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
530                 epvector newseq;
531                 newseq.reserve(2);
532                 newseq.push_back(expair(basis, exponent - _ex1()));
533                 newseq.push_back(expair(basis.diff(s), _ex1()));
534                 return mul(newseq, exponent);
535         } else {
536                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
537                 return mul(*this,
538                            add(mul(exponent.diff(s), log(basis)),
539                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
540         }
541 }
542
543 int power::compare_same_type(const basic & other) const
544 {
545         GINAC_ASSERT(is_exactly_of_type(other, power));
546         const power & o=static_cast<const power &>(const_cast<basic &>(other));
547
548         int cmpval;
549         cmpval=basis.compare(o.basis);
550         if (cmpval==0) {
551                 return exponent.compare(o.exponent);
552         }
553         return cmpval;
554 }
555
556 unsigned power::return_type(void) const
557 {
558         return basis.return_type();
559 }
560    
561 unsigned power::return_type_tinfo(void) const
562 {
563         return basis.return_type_tinfo();
564 }
565
566 ex power::expand(unsigned options) const
567 {
568         if (flags & status_flags::expanded)
569                 return *this;
570         
571         ex expanded_basis = basis.expand(options);
572         ex expanded_exponent = exponent.expand(options);
573         
574         // x^(a+b) -> x^a * x^b
575         if (is_ex_exactly_of_type(expanded_exponent, add)) {
576                 const add &a = ex_to_add(expanded_exponent);
577                 exvector distrseq;
578                 distrseq.reserve(a.seq.size() + 1);
579                 epvector::const_iterator last = a.seq.end();
580                 epvector::const_iterator cit = a.seq.begin();
581                 while (cit!=last) {
582                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
583                         cit++;
584                 }
585                 
586                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
587                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
588                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
589                         int int_exponent = num_exponent.to_int();
590                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
591                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
592                         else
593                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
594                 } else
595                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
596                 
597                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
598                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
599                 return r.expand();
600         }
601         
602         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
603                 !ex_to_numeric(expanded_exponent).is_integer()) {
604                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
605                         return this->hold();
606                 } else {
607                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
608                 }
609         }
610         
611         // integer numeric exponent
612         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
613         int int_exponent = num_exponent.to_int();
614         
615         // (x+y)^n, n>0
616         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
617                 return expand_add(ex_to_add(expanded_basis), int_exponent);
618         
619         // (x*y)^n -> x^n * y^n
620         if (is_ex_exactly_of_type(expanded_basis,mul))
621                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
622         
623         // cannot expand further
624         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
625                 return this->hold();
626         else
627                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
628 }
629
630 //////////
631 // new virtual functions which can be overridden by derived classes
632 //////////
633
634 // none
635
636 //////////
637 // non-virtual functions in this class
638 //////////
639
640 /** expand a^n where a is an add and n is an integer.
641  *  @see power::expand */
642 ex power::expand_add(const add & a, int n) const
643 {
644         if (n==2)
645                 return expand_add_2(a);
646         
647         int m = a.nops();
648         exvector sum;
649         sum.reserve((n+1)*(m-1));
650         intvector k(m-1);
651         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
652         intvector upper_limit(m-1);
653         int l;
654         
655         for (int l=0; l<m-1; l++) {
656                 k[l] = 0;
657                 k_cum[l] = 0;
658                 upper_limit[l] = n;
659         }
660         
661         while (1) {
662                 exvector term;
663                 term.reserve(m+1);
664                 for (l=0; l<m-1; l++) {
665                         const ex & b = a.op(l);
666                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
667                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
668                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
669                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
670                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
671                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
672                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
673                         if (is_ex_exactly_of_type(b,mul))
674                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
675                         else
676                                 term.push_back(power(b,k[l]));
677                 }
678                 
679                 const ex & b = a.op(l);
680                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
681                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
682                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
683                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
684                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
685                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
686                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
687                 if (is_ex_exactly_of_type(b,mul))
688                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
689                 else
690                         term.push_back(power(b,n-k_cum[m-2]));
691                 
692                 numeric f = binomial(numeric(n),numeric(k[0]));
693                 for (l=1; l<m-1; l++)
694                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
695                 
696                 term.push_back(f);
697                 
698                 /*
699                 cout << "begin term" << endl;
700                 for (int i=0; i<m-1; i++) {
701                         cout << "k[" << i << "]=" << k[i] << endl;
702                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
703                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
704                 }
705                 for_each(term.begin(), term.end(), ostream_iterator<ex>(cout, "\n"));
706                 cout << "end term" << endl;
707                 */
708                 
709                 // TODO: optimize this
710                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
711                 
712                 // increment k[]
713                 l = m-2;
714                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
715                         k[l] = 0;    
716                         l--;
717                 }
718                 if (l<0) break;
719                 
720                 // recalc k_cum[] and upper_limit[]
721                 if (l==0)
722                         k_cum[0] = k[0];
723                 else
724                         k_cum[l] = k_cum[l-1]+k[l];
725                 
726                 for (int i=l+1; i<m-1; i++)
727                         k_cum[i] = k_cum[i-1]+k[i];
728                 
729                 for (int i=l+1; i<m-1; i++)
730                         upper_limit[i] = n-k_cum[i-1];
731         }
732         return (new add(sum))->setflag(status_flags::dynallocated |
733                                                                    status_flags::expanded );
734 }
735
736
737 /** Special case of power::expand_add. Expands a^2 where a is an add.
738  *  @see power::expand_add */
739 ex power::expand_add_2(const add & a) const
740 {
741         epvector sum;
742         unsigned a_nops = a.nops();
743         sum.reserve((a_nops*(a_nops+1))/2);
744         epvector::const_iterator last = a.seq.end();
745         
746         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
747         // first part: ignore overall_coeff and expand other terms
748         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
749                 const ex & r = (*cit0).rest;
750                 const ex & c = (*cit0).coeff;
751                 
752                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
753                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
754                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
755                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
756                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
757                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
758                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
759                 
760                 if (are_ex_trivially_equal(c,_ex1())) {
761                         if (is_ex_exactly_of_type(r,mul)) {
762                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
763                                                      _ex1()));
764                         } else {
765                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
766                                                      _ex1()));
767                         }
768                 } else {
769                         if (is_ex_exactly_of_type(r,mul)) {
770                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
771                                                      ex_to_numeric(c).power_dyn(_num2())));
772                         } else {
773                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
774                                                      ex_to_numeric(c).power_dyn(_num2())));
775                         }
776                 }
777                         
778                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
779                         const ex & r1 = (*cit1).rest;
780                         const ex & c1 = (*cit1).coeff;
781                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
782                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
783                 }
784         }
785         
786         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
787         
788         // second part: add terms coming from overall_factor (if != 0)
789         if (!a.overall_coeff.is_zero()) {
790                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
791                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
792                 }
793                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
794         }
795         
796         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
797         
798         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
799 }
800
801 /** Expand factors of m in m^n where m is a mul and n is and integer
802  *  @see power::expand */
803 ex power::expand_mul(const mul & m, const numeric & n) const
804 {
805         if (n.is_zero())
806                 return _ex1();
807         
808         epvector distrseq;
809         distrseq.reserve(m.seq.size());
810         epvector::const_iterator last = m.seq.end();
811         epvector::const_iterator cit = m.seq.begin();
812         while (cit!=last) {
813                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
814                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
815                 } else {
816                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
817                         // since n is an integer
818                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
819                 }
820                 ++cit;
821         }
822         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
823 }
824
825 /*
826 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
827                                                                 unsigned options) const
828 {
829         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
830                         expand(options | expand_options::internal_do_not_expand_power_operands);
831
832         return ex(mul(rest_power,basis),0).
833                expand(options | expand_options::internal_do_not_expand_mul_operands);
834 }
835 */
836
837 // helper function
838
839 ex sqrt(const ex & a)
840 {
841         return power(a,_ex1_2());
842 }
843
844 } // namespace GiNaC