d4c12c17d8da76dfc2b599148fc9ecc585d5f286
[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 "print.h"
36 #include "archive.h"
37 #include "debugmsg.h"
38 #include "utils.h"
39
40 namespace GiNaC {
41
42 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
43
44 typedef std::vector<int> intvector;
45
46 //////////
47 // default ctor, dtor, copy ctor assignment operator and helpers
48 //////////
49
50 power::power() : basic(TINFO_power)
51 {
52         debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
53 }
54
55 void power::copy(const power & other)
56 {
57         inherited::copy(other);
58         basis = other.basis;
59         exponent = other.exponent;
60 }
61
62 DEFAULT_DESTROY(power)
63
64 //////////
65 // other ctors
66 //////////
67
68 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
69 {
70         debugmsg("power ctor from ex,ex",LOGLEVEL_CONSTRUCT);
71         GINAC_ASSERT(basis.return_type()==return_types::commutative);
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         GINAC_ASSERT(basis.return_type()==return_types::commutative);
80 }
81
82 //////////
83 // archiving
84 //////////
85
86 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
87 {
88         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
89         n.find_ex("basis", basis, sym_lst);
90         n.find_ex("exponent", exponent, sym_lst);
91 }
92
93 void power::archive(archive_node &n) const
94 {
95         inherited::archive(n);
96         n.add_ex("basis", basis);
97         n.add_ex("exponent", exponent);
98 }
99
100 DEFAULT_UNARCHIVE(power)
101
102 //////////
103 // functions overriding virtual functions from bases classes
104 //////////
105
106 // public
107
108 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
109 {
110         // Optimal output of integer powers of symbols to aid compiler CSE.
111         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
112         // to learn why such a hack is really necessary.
113         if (exp == 1) {
114                 x.print(c);
115         } else if (exp == 2) {
116                 x.print(c);
117                 c.s << "*";
118                 x.print(c);
119         } else if (exp & 1) {
120                 x.print(c);
121                 c.s << "*";
122                 print_sym_pow(c, x, exp-1);
123         } else {
124                 c.s << "(";
125                 print_sym_pow(c, x, exp >> 1);
126                 c.s << ")*(";
127                 print_sym_pow(c, x, exp >> 1);
128                 c.s << ")";
129         }
130 }
131
132 void power::print(const print_context & c, unsigned level) const
133 {
134         debugmsg("power print", LOGLEVEL_PRINT);
135
136         if (is_of_type(c, print_tree)) {
137
138                 inherited::print(c, level);
139
140         } else if (is_of_type(c, print_csrc)) {
141
142                 // Integer powers of symbols are printed in a special, optimized way
143                 if (exponent.info(info_flags::integer)
144                  && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
145                         int exp = ex_to_numeric(exponent).to_int();
146                         if (exp > 0)
147                                 c.s << "(";
148                         else {
149                                 exp = -exp;
150                                 if (is_of_type(c, print_csrc_cl_N))
151                                         c.s << "recip(";
152                                 else
153                                         c.s << "1.0/(";
154                         }
155                         print_sym_pow(c, ex_to_symbol(basis), exp);
156                         c.s << ")";
157
158                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
159                 } else if (exponent.compare(_num_1()) == 0) {
160                         if (is_of_type(c, print_csrc_cl_N))
161                                 c.s << "recip(";
162                         else
163                                 c.s << "1.0/(";
164                         basis.print(c);
165                         c.s << ")";
166
167                 // Otherwise, use the pow() or expt() (CLN) functions
168                 } else {
169                         if (is_of_type(c, print_csrc_cl_N))
170                                 c.s << "expt(";
171                         else
172                                 c.s << "pow(";
173                         basis.print(c);
174                         c.s << ",";
175                         exponent.print(c);
176                         c.s << ")";
177                 }
178
179         } else {
180
181                 if (exponent.is_equal(_ex1_2())) {
182                         if (is_of_type(c, print_latex))
183                                 c.s << "\\sqrt{";
184                         else
185                                 c.s << "sqrt(";
186                         basis.print(c);
187                         if (is_of_type(c, print_latex))
188                                 c.s << "}";
189                         else
190                                 c.s << ")";
191                 } else {
192                         if (precedence() <= level) {
193                                 if (is_of_type(c, print_latex))
194                                         c.s << "{(";
195                                 else
196                                         c.s << "(";
197                         }
198                         basis.print(c, precedence());
199                         c.s << "^";
200                         exponent.print(c, precedence());
201                         if (precedence() <= level) {
202                                 if (is_of_type(c, print_latex))
203                                         c.s << ")}";
204                                 else
205                                         c.s << ")";
206                         }
207                 }
208         }
209 }
210
211 bool power::info(unsigned inf) const
212 {
213         switch (inf) {
214                 case info_flags::polynomial:
215                 case info_flags::integer_polynomial:
216                 case info_flags::cinteger_polynomial:
217                 case info_flags::rational_polynomial:
218                 case info_flags::crational_polynomial:
219                         return exponent.info(info_flags::nonnegint);
220                 case info_flags::rational_function:
221                         return exponent.info(info_flags::integer);
222                 case info_flags::algebraic:
223                         return (!exponent.info(info_flags::integer) ||
224                                         basis.info(inf));
225         }
226         return inherited::info(inf);
227 }
228
229 unsigned power::nops() const
230 {
231         return 2;
232 }
233
234 ex & power::let_op(int i)
235 {
236         GINAC_ASSERT(i>=0);
237         GINAC_ASSERT(i<2);
238
239         return i==0 ? basis : exponent;
240 }
241
242 int power::degree(const ex & s) const
243 {
244         if (is_exactly_of_type(*exponent.bp,numeric)) {
245                 if (basis.is_equal(s)) {
246                         if (ex_to_numeric(exponent).is_integer())
247                                 return ex_to_numeric(exponent).to_int();
248                         else
249                                 return 0;
250                 } else
251                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
252         }
253         return 0;
254 }
255
256 int power::ldegree(const ex & s) const 
257 {
258         if (is_exactly_of_type(*exponent.bp,numeric)) {
259                 if (basis.is_equal(s)) {
260                         if (ex_to_numeric(exponent).is_integer())
261                                 return ex_to_numeric(exponent).to_int();
262                         else
263                                 return 0;
264                 } else
265                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
266         }
267         return 0;
268 }
269
270 ex power::coeff(const ex & s, int n) const
271 {
272         if (!basis.is_equal(s)) {
273                 // basis not equal to s
274                 if (n == 0)
275                         return *this;
276                 else
277                         return _ex0();
278         } else {
279                 // basis equal to s
280                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
281                         // integer exponent
282                         int int_exp = ex_to_numeric(exponent).to_int();
283                         if (n == int_exp)
284                                 return _ex1();
285                         else
286                                 return _ex0();
287                 } else {
288                         // non-integer exponents are treated as zero
289                         if (n == 0)
290                                 return *this;
291                         else
292                                 return _ex0();
293                 }
294         }
295 }
296
297 ex power::eval(int level) const
298 {
299         // simplifications: ^(x,0) -> 1 (0^0 handled here)
300         //                  ^(x,1) -> x
301         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
302         //                  ^(1,x) -> 1
303         //                  ^(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)
304         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
305         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
306         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
307         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
308         
309         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
310         
311         if ((level==1) && (flags & status_flags::evaluated))
312                 return *this;
313         else if (level == -max_recursion_level)
314                 throw(std::runtime_error("max recursion level reached"));
315         
316         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
317         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
318         
319         bool basis_is_numerical = 0;
320         bool exponent_is_numerical = 0;
321         numeric * num_basis;
322         numeric * num_exponent;
323         
324         if (is_exactly_of_type(*ebasis.bp,numeric)) {
325                 basis_is_numerical = 1;
326                 num_basis = static_cast<numeric *>(ebasis.bp);
327         }
328         if (is_exactly_of_type(*eexponent.bp,numeric)) {
329                 exponent_is_numerical = 1;
330                 num_exponent = static_cast<numeric *>(eexponent.bp);
331         }
332         
333         // ^(x,0) -> 1 (0^0 also handled here)
334         if (eexponent.is_zero()) {
335                 if (ebasis.is_zero())
336                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
337                 else
338                         return _ex1();
339         }
340         
341         // ^(x,1) -> x
342         if (eexponent.is_equal(_ex1()))
343                 return ebasis;
344         
345         // ^(0,c1) -> 0 or exception (depending on real value of c1)
346         if (ebasis.is_zero() && exponent_is_numerical) {
347                 if ((num_exponent->real()).is_zero())
348                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
349                 else if ((num_exponent->real()).is_negative())
350                         throw (pole_error("power::eval(): division by zero",1));
351                 else
352                         return _ex0();
353         }
354         
355         // ^(1,x) -> 1
356         if (ebasis.is_equal(_ex1()))
357                 return _ex1();
358         
359         if (basis_is_numerical && exponent_is_numerical) {
360                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
361                 // except if c1,c2 are rational, but c1^c2 is not)
362                 bool basis_is_crational = num_basis->is_crational();
363                 bool exponent_is_crational = num_exponent->is_crational();
364                 numeric res = num_basis->power(*num_exponent);
365                 
366                 if ((!basis_is_crational || !exponent_is_crational)
367                         || res.is_crational()) {
368                         return res;
369                 }
370                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
371                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
372                 if (basis_is_crational && exponent_is_crational
373                         && num_exponent->is_real()
374                         && !num_exponent->is_integer()) {
375                         numeric n = num_exponent->numer();
376                         numeric m = num_exponent->denom();
377                         numeric r;
378                         numeric q = iquo(n, m, r);
379                         if (r.is_negative()) {
380                                 r = r.add(m);
381                                 q = q.sub(_num1());
382                         }
383                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
384                                 return this->hold();
385                         else {
386                                 epvector res;
387                                 res.push_back(expair(ebasis,r.div(m)));
388                                 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
389                         }
390                 }
391         }
392         
393         // ^(^(x,c1),c2) -> ^(x,c1*c2)
394         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
395         // case c1==1 should not happen, see below!)
396         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
397                 const power & sub_power = ex_to_power(ebasis);
398                 const ex & sub_basis = sub_power.basis;
399                 const ex & sub_exponent = sub_power.exponent;
400                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
401                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
402                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
403                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1)
404                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
405                 }
406         }
407         
408         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
409         if (exponent_is_numerical && num_exponent->is_integer() &&
410                 is_ex_exactly_of_type(ebasis,mul)) {
411                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
412         }
413         
414         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
415         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
416         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
417                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
418                 const mul & mulref = ex_to_mul(ebasis);
419                 if (!mulref.overall_coeff.is_equal(_ex1())) {
420                         const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
421                         if (num_coeff.is_real()) {
422                                 if (num_coeff.is_positive()) {
423                                         mul * mulp = new mul(mulref);
424                                         mulp->overall_coeff = _ex1();
425                                         mulp->clearflag(status_flags::evaluated);
426                                         mulp->clearflag(status_flags::hash_calculated);
427                                         return (new mul(power(*mulp,exponent),
428                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
429                                 } else {
430                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
431                                         if (num_coeff.compare(_num_1())!=0) {
432                                                 mul * mulp = new mul(mulref);
433                                                 mulp->overall_coeff = _ex_1();
434                                                 mulp->clearflag(status_flags::evaluated);
435                                                 mulp->clearflag(status_flags::hash_calculated);
436                                                 return (new mul(power(*mulp,exponent),
437                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
438                                         }
439                                 }
440                         }
441                 }
442         }
443         
444         if (are_ex_trivially_equal(ebasis,basis) &&
445                 are_ex_trivially_equal(eexponent,exponent)) {
446                 return this->hold();
447         }
448         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
449                                                                                                    status_flags::evaluated);
450 }
451
452 ex power::evalf(int level) const
453 {
454         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
455
456         ex ebasis;
457         ex eexponent;
458         
459         if (level==1) {
460                 ebasis = basis;
461                 eexponent = exponent;
462         } else if (level == -max_recursion_level) {
463                 throw(std::runtime_error("max recursion level reached"));
464         } else {
465                 ebasis = basis.evalf(level-1);
466                 if (!is_ex_exactly_of_type(eexponent,numeric))
467                         eexponent = exponent.evalf(level-1);
468                 else
469                         eexponent = exponent;
470         }
471
472         return power(ebasis,eexponent);
473 }
474
475 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
476 {
477         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
478         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
479
480         if (are_ex_trivially_equal(basis, subsed_basis)
481          && are_ex_trivially_equal(exponent, subsed_exponent))
482                 return basic::subs(ls, lr, no_pattern);
483         else
484                 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
485 }
486
487 ex power::simplify_ncmul(const exvector & v) const
488 {
489         return inherited::simplify_ncmul(v);
490 }
491
492 // protected
493
494 /** Implementation of ex::diff() for a power.
495  *  @see ex::diff */
496 ex power::derivative(const symbol & s) const
497 {
498         if (exponent.info(info_flags::real)) {
499                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
500                 epvector newseq;
501                 newseq.reserve(2);
502                 newseq.push_back(expair(basis, exponent - _ex1()));
503                 newseq.push_back(expair(basis.diff(s), _ex1()));
504                 return mul(newseq, exponent);
505         } else {
506                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
507                 return mul(*this,
508                            add(mul(exponent.diff(s), log(basis)),
509                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
510         }
511 }
512
513 int power::compare_same_type(const basic & other) const
514 {
515         GINAC_ASSERT(is_exactly_of_type(other, power));
516         const power & o=static_cast<const power &>(const_cast<basic &>(other));
517
518         int cmpval;
519         cmpval=basis.compare(o.basis);
520         if (cmpval==0) {
521                 return exponent.compare(o.exponent);
522         }
523         return cmpval;
524 }
525
526 unsigned power::return_type(void) const
527 {
528         return basis.return_type();
529 }
530    
531 unsigned power::return_type_tinfo(void) const
532 {
533         return basis.return_type_tinfo();
534 }
535
536 ex power::expand(unsigned options) const
537 {
538         if (flags & status_flags::expanded)
539                 return *this;
540         
541         ex expanded_basis = basis.expand(options);
542         ex expanded_exponent = exponent.expand(options);
543         
544         // x^(a+b) -> x^a * x^b
545         if (is_ex_exactly_of_type(expanded_exponent, add)) {
546                 const add &a = ex_to_add(expanded_exponent);
547                 exvector distrseq;
548                 distrseq.reserve(a.seq.size() + 1);
549                 epvector::const_iterator last = a.seq.end();
550                 epvector::const_iterator cit = a.seq.begin();
551                 while (cit!=last) {
552                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
553                         cit++;
554                 }
555                 
556                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
557                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
558                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
559                         int int_exponent = num_exponent.to_int();
560                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
561                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
562                         else
563                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
564                 } else
565                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
566                 
567                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
568                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
569                 return r.expand();
570         }
571         
572         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
573                 !ex_to_numeric(expanded_exponent).is_integer()) {
574                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
575                         return this->hold();
576                 } else {
577                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
578                 }
579         }
580         
581         // integer numeric exponent
582         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
583         int int_exponent = num_exponent.to_int();
584         
585         // (x+y)^n, n>0
586         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
587                 return expand_add(ex_to_add(expanded_basis), int_exponent);
588         
589         // (x*y)^n -> x^n * y^n
590         if (is_ex_exactly_of_type(expanded_basis,mul))
591                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
592         
593         // cannot expand further
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 | status_flags::expanded);
598 }
599
600 //////////
601 // new virtual functions which can be overridden by derived classes
602 //////////
603
604 // none
605
606 //////////
607 // non-virtual functions in this class
608 //////////
609
610 /** expand a^n where a is an add and n is an integer.
611  *  @see power::expand */
612 ex power::expand_add(const add & a, int n) const
613 {
614         if (n==2)
615                 return expand_add_2(a);
616         
617         int m = a.nops();
618         exvector sum;
619         sum.reserve((n+1)*(m-1));
620         intvector k(m-1);
621         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
622         intvector upper_limit(m-1);
623         int l;
624         
625         for (int l=0; l<m-1; l++) {
626                 k[l] = 0;
627                 k_cum[l] = 0;
628                 upper_limit[l] = n;
629         }
630         
631         while (1) {
632                 exvector term;
633                 term.reserve(m+1);
634                 for (l=0; l<m-1; l++) {
635                         const ex & b = a.op(l);
636                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
637                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
638                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
639                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
640                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
641                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
642                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
643                         if (is_ex_exactly_of_type(b,mul))
644                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
645                         else
646                                 term.push_back(power(b,k[l]));
647                 }
648                 
649                 const ex & b = a.op(l);
650                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
651                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
652                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
653                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
654                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
655                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
656                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
657                 if (is_ex_exactly_of_type(b,mul))
658                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
659                 else
660                         term.push_back(power(b,n-k_cum[m-2]));
661                 
662                 numeric f = binomial(numeric(n),numeric(k[0]));
663                 for (l=1; l<m-1; l++)
664                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
665                 
666                 term.push_back(f);
667                 
668                 /*
669                 cout << "begin term" << endl;
670                 for (int i=0; i<m-1; i++) {
671                         cout << "k[" << i << "]=" << k[i] << endl;
672                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
673                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
674                 }
675                 for_each(term.begin(), term.end(), ostream_iterator<ex>(cout, "\n"));
676                 cout << "end term" << endl;
677                 */
678                 
679                 // TODO: optimize this
680                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
681                 
682                 // increment k[]
683                 l = m-2;
684                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
685                         k[l] = 0;    
686                         l--;
687                 }
688                 if (l<0) break;
689                 
690                 // recalc k_cum[] and upper_limit[]
691                 if (l==0)
692                         k_cum[0] = k[0];
693                 else
694                         k_cum[l] = k_cum[l-1]+k[l];
695                 
696                 for (int i=l+1; i<m-1; i++)
697                         k_cum[i] = k_cum[i-1]+k[i];
698                 
699                 for (int i=l+1; i<m-1; i++)
700                         upper_limit[i] = n-k_cum[i-1];
701         }
702         return (new add(sum))->setflag(status_flags::dynallocated |
703                                                                    status_flags::expanded );
704 }
705
706
707 /** Special case of power::expand_add. Expands a^2 where a is an add.
708  *  @see power::expand_add */
709 ex power::expand_add_2(const add & a) const
710 {
711         epvector sum;
712         unsigned a_nops = a.nops();
713         sum.reserve((a_nops*(a_nops+1))/2);
714         epvector::const_iterator last = a.seq.end();
715         
716         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
717         // first part: ignore overall_coeff and expand other terms
718         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
719                 const ex & r = (*cit0).rest;
720                 const ex & c = (*cit0).coeff;
721                 
722                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
723                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
724                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
725                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
726                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
727                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
728                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
729                 
730                 if (are_ex_trivially_equal(c,_ex1())) {
731                         if (is_ex_exactly_of_type(r,mul)) {
732                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
733                                                      _ex1()));
734                         } else {
735                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
736                                                      _ex1()));
737                         }
738                 } else {
739                         if (is_ex_exactly_of_type(r,mul)) {
740                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
741                                                      ex_to_numeric(c).power_dyn(_num2())));
742                         } else {
743                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
744                                                      ex_to_numeric(c).power_dyn(_num2())));
745                         }
746                 }
747                         
748                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
749                         const ex & r1 = (*cit1).rest;
750                         const ex & c1 = (*cit1).coeff;
751                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
752                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
753                 }
754         }
755         
756         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
757         
758         // second part: add terms coming from overall_factor (if != 0)
759         if (!a.overall_coeff.is_zero()) {
760                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
761                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
762                 }
763                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
764         }
765         
766         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
767         
768         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
769 }
770
771 /** Expand factors of m in m^n where m is a mul and n is and integer
772  *  @see power::expand */
773 ex power::expand_mul(const mul & m, const numeric & n) const
774 {
775         if (n.is_zero())
776                 return _ex1();
777         
778         epvector distrseq;
779         distrseq.reserve(m.seq.size());
780         epvector::const_iterator last = m.seq.end();
781         epvector::const_iterator cit = m.seq.begin();
782         while (cit!=last) {
783                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
784                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
785                 } else {
786                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
787                         // since n is an integer
788                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
789                 }
790                 ++cit;
791         }
792         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
793 }
794
795 /*
796 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
797                                unsigned options) const
798 {
799         // obsolete
800
801         exvector distrseq;
802         epvector splitseq;
803
804         const add & addref=static_cast<const add &>(*basis.bp);
805
806         splitseq=addref.seq;
807         splitseq.pop_back();
808         ex first_operands=add(splitseq);
809         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
810         
811         int n=exponent.to_int();
812         for (int k=0; k<=n; k++) {
813                 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
814                                                  * power(last_operand,numeric(n-k)));
815         }
816         return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
817 }
818 */
819
820 /*
821 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
822                                                                 unsigned options) const
823 {
824         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
825                         expand(options | expand_options::internal_do_not_expand_power_operands);
826
827         return ex(mul(rest_power,basis),0).
828                expand(options | expand_options::internal_do_not_expand_mul_operands);
829 }
830 */
831
832 // helper function
833
834 ex sqrt(const ex & a)
835 {
836         return power(a,_ex1_2());
837 }
838
839 } // namespace GiNaC