- renamed ncpower() to ncpow()
[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                         if (is_of_type(c, print_latex))
201                                 c.s << '{';
202                         exponent.print(c, precedence());
203                         if (is_of_type(c, print_latex))
204                                 c.s << '}';
205                         if (precedence() <= level) {
206                                 if (is_of_type(c, print_latex))
207                                         c.s << ")}";
208                                 else
209                                         c.s << ')';
210                         }
211                 }
212         }
213 }
214
215 bool power::info(unsigned inf) const
216 {
217         switch (inf) {
218                 case info_flags::polynomial:
219                 case info_flags::integer_polynomial:
220                 case info_flags::cinteger_polynomial:
221                 case info_flags::rational_polynomial:
222                 case info_flags::crational_polynomial:
223                         return exponent.info(info_flags::nonnegint);
224                 case info_flags::rational_function:
225                         return exponent.info(info_flags::integer);
226                 case info_flags::algebraic:
227                         return (!exponent.info(info_flags::integer) ||
228                                         basis.info(inf));
229         }
230         return inherited::info(inf);
231 }
232
233 unsigned power::nops() const
234 {
235         return 2;
236 }
237
238 ex & power::let_op(int i)
239 {
240         GINAC_ASSERT(i>=0);
241         GINAC_ASSERT(i<2);
242
243         return i==0 ? basis : exponent;
244 }
245
246 int power::degree(const ex & s) const
247 {
248         if (is_exactly_of_type(*exponent.bp,numeric)) {
249                 if (basis.is_equal(s)) {
250                         if (ex_to_numeric(exponent).is_integer())
251                                 return ex_to_numeric(exponent).to_int();
252                         else
253                                 return 0;
254                 } else
255                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
256         }
257         return 0;
258 }
259
260 int power::ldegree(const ex & s) const 
261 {
262         if (is_exactly_of_type(*exponent.bp,numeric)) {
263                 if (basis.is_equal(s)) {
264                         if (ex_to_numeric(exponent).is_integer())
265                                 return ex_to_numeric(exponent).to_int();
266                         else
267                                 return 0;
268                 } else
269                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
270         }
271         return 0;
272 }
273
274 ex power::coeff(const ex & s, int n) const
275 {
276         if (!basis.is_equal(s)) {
277                 // basis not equal to s
278                 if (n == 0)
279                         return *this;
280                 else
281                         return _ex0();
282         } else {
283                 // basis equal to s
284                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
285                         // integer exponent
286                         int int_exp = ex_to_numeric(exponent).to_int();
287                         if (n == int_exp)
288                                 return _ex1();
289                         else
290                                 return _ex0();
291                 } else {
292                         // non-integer exponents are treated as zero
293                         if (n == 0)
294                                 return *this;
295                         else
296                                 return _ex0();
297                 }
298         }
299 }
300
301 ex power::eval(int level) const
302 {
303         // simplifications: ^(x,0) -> 1 (0^0 handled here)
304         //                  ^(x,1) -> x
305         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
306         //                  ^(1,x) -> 1
307         //                  ^(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)
308         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
309         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
310         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
311         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
312         
313         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
314         
315         if ((level==1) && (flags & status_flags::evaluated))
316                 return *this;
317         else if (level == -max_recursion_level)
318                 throw(std::runtime_error("max recursion level reached"));
319         
320         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
321         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
322         
323         bool basis_is_numerical = 0;
324         bool exponent_is_numerical = 0;
325         numeric * num_basis;
326         numeric * num_exponent;
327         
328         if (is_exactly_of_type(*ebasis.bp,numeric)) {
329                 basis_is_numerical = 1;
330                 num_basis = static_cast<numeric *>(ebasis.bp);
331         }
332         if (is_exactly_of_type(*eexponent.bp,numeric)) {
333                 exponent_is_numerical = 1;
334                 num_exponent = static_cast<numeric *>(eexponent.bp);
335         }
336         
337         // ^(x,0) -> 1 (0^0 also handled here)
338         if (eexponent.is_zero()) {
339                 if (ebasis.is_zero())
340                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
341                 else
342                         return _ex1();
343         }
344         
345         // ^(x,1) -> x
346         if (eexponent.is_equal(_ex1()))
347                 return ebasis;
348         
349         // ^(0,c1) -> 0 or exception (depending on real value of c1)
350         if (ebasis.is_zero() && exponent_is_numerical) {
351                 if ((num_exponent->real()).is_zero())
352                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
353                 else if ((num_exponent->real()).is_negative())
354                         throw (pole_error("power::eval(): division by zero",1));
355                 else
356                         return _ex0();
357         }
358         
359         // ^(1,x) -> 1
360         if (ebasis.is_equal(_ex1()))
361                 return _ex1();
362         
363         if (basis_is_numerical && exponent_is_numerical) {
364                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
365                 // except if c1,c2 are rational, but c1^c2 is not)
366                 bool basis_is_crational = num_basis->is_crational();
367                 bool exponent_is_crational = num_exponent->is_crational();
368                 numeric res = num_basis->power(*num_exponent);
369                 
370                 if ((!basis_is_crational || !exponent_is_crational)
371                         || res.is_crational()) {
372                         return res;
373                 }
374                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
375                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
376                 if (basis_is_crational && exponent_is_crational
377                         && num_exponent->is_real()
378                         && !num_exponent->is_integer()) {
379                         numeric n = num_exponent->numer();
380                         numeric m = num_exponent->denom();
381                         numeric r;
382                         numeric q = iquo(n, m, r);
383                         if (r.is_negative()) {
384                                 r = r.add(m);
385                                 q = q.sub(_num1());
386                         }
387                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
388                                 return this->hold();
389                         else {
390                                 epvector res;
391                                 res.push_back(expair(ebasis,r.div(m)));
392                                 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
393                         }
394                 }
395         }
396         
397         // ^(^(x,c1),c2) -> ^(x,c1*c2)
398         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
399         // case c1==1 should not happen, see below!)
400         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
401                 const power & sub_power = ex_to_power(ebasis);
402                 const ex & sub_basis = sub_power.basis;
403                 const ex & sub_exponent = sub_power.exponent;
404                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
405                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
406                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
407                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1)
408                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
409                 }
410         }
411         
412         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
413         if (exponent_is_numerical && num_exponent->is_integer() &&
414                 is_ex_exactly_of_type(ebasis,mul)) {
415                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
416         }
417         
418         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
419         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
420         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
421                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
422                 const mul & mulref = ex_to_mul(ebasis);
423                 if (!mulref.overall_coeff.is_equal(_ex1())) {
424                         const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
425                         if (num_coeff.is_real()) {
426                                 if (num_coeff.is_positive()) {
427                                         mul * mulp = new mul(mulref);
428                                         mulp->overall_coeff = _ex1();
429                                         mulp->clearflag(status_flags::evaluated);
430                                         mulp->clearflag(status_flags::hash_calculated);
431                                         return (new mul(power(*mulp,exponent),
432                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
433                                 } else {
434                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
435                                         if (num_coeff.compare(_num_1())!=0) {
436                                                 mul * mulp = new mul(mulref);
437                                                 mulp->overall_coeff = _ex_1();
438                                                 mulp->clearflag(status_flags::evaluated);
439                                                 mulp->clearflag(status_flags::hash_calculated);
440                                                 return (new mul(power(*mulp,exponent),
441                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
442                                         }
443                                 }
444                         }
445                 }
446         }
447         
448         if (are_ex_trivially_equal(ebasis,basis) &&
449                 are_ex_trivially_equal(eexponent,exponent)) {
450                 return this->hold();
451         }
452         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
453                                                                                                    status_flags::evaluated);
454 }
455
456 ex power::evalf(int level) const
457 {
458         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
459
460         ex ebasis;
461         ex eexponent;
462         
463         if (level==1) {
464                 ebasis = basis;
465                 eexponent = exponent;
466         } else if (level == -max_recursion_level) {
467                 throw(std::runtime_error("max recursion level reached"));
468         } else {
469                 ebasis = basis.evalf(level-1);
470                 if (!is_ex_exactly_of_type(eexponent,numeric))
471                         eexponent = exponent.evalf(level-1);
472                 else
473                         eexponent = exponent;
474         }
475
476         return power(ebasis,eexponent);
477 }
478
479 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
480 {
481         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
482         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
483
484         if (are_ex_trivially_equal(basis, subsed_basis)
485          && are_ex_trivially_equal(exponent, subsed_exponent))
486                 return basic::subs(ls, lr, no_pattern);
487         else
488                 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
489 }
490
491 ex power::simplify_ncmul(const exvector & v) const
492 {
493         return inherited::simplify_ncmul(v);
494 }
495
496 // protected
497
498 /** Implementation of ex::diff() for a power.
499  *  @see ex::diff */
500 ex power::derivative(const symbol & s) const
501 {
502         if (exponent.info(info_flags::real)) {
503                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
504                 epvector newseq;
505                 newseq.reserve(2);
506                 newseq.push_back(expair(basis, exponent - _ex1()));
507                 newseq.push_back(expair(basis.diff(s), _ex1()));
508                 return mul(newseq, exponent);
509         } else {
510                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
511                 return mul(*this,
512                            add(mul(exponent.diff(s), log(basis)),
513                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
514         }
515 }
516
517 int power::compare_same_type(const basic & other) const
518 {
519         GINAC_ASSERT(is_exactly_of_type(other, power));
520         const power & o=static_cast<const power &>(const_cast<basic &>(other));
521
522         int cmpval;
523         cmpval=basis.compare(o.basis);
524         if (cmpval==0) {
525                 return exponent.compare(o.exponent);
526         }
527         return cmpval;
528 }
529
530 unsigned power::return_type(void) const
531 {
532         return basis.return_type();
533 }
534    
535 unsigned power::return_type_tinfo(void) const
536 {
537         return basis.return_type_tinfo();
538 }
539
540 ex power::expand(unsigned options) const
541 {
542         if (flags & status_flags::expanded)
543                 return *this;
544         
545         ex expanded_basis = basis.expand(options);
546         ex expanded_exponent = exponent.expand(options);
547         
548         // x^(a+b) -> x^a * x^b
549         if (is_ex_exactly_of_type(expanded_exponent, add)) {
550                 const add &a = ex_to_add(expanded_exponent);
551                 exvector distrseq;
552                 distrseq.reserve(a.seq.size() + 1);
553                 epvector::const_iterator last = a.seq.end();
554                 epvector::const_iterator cit = a.seq.begin();
555                 while (cit!=last) {
556                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
557                         cit++;
558                 }
559                 
560                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
561                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
562                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
563                         int int_exponent = num_exponent.to_int();
564                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
565                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
566                         else
567                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
568                 } else
569                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
570                 
571                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
572                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
573                 return r.expand();
574         }
575         
576         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
577                 !ex_to_numeric(expanded_exponent).is_integer()) {
578                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
579                         return this->hold();
580                 } else {
581                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
582                 }
583         }
584         
585         // integer numeric exponent
586         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
587         int int_exponent = num_exponent.to_int();
588         
589         // (x+y)^n, n>0
590         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
591                 return expand_add(ex_to_add(expanded_basis), int_exponent);
592         
593         // (x*y)^n -> x^n * y^n
594         if (is_ex_exactly_of_type(expanded_basis,mul))
595                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
596         
597         // cannot expand further
598         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
599                 return this->hold();
600         else
601                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
602 }
603
604 //////////
605 // new virtual functions which can be overridden by derived classes
606 //////////
607
608 // none
609
610 //////////
611 // non-virtual functions in this class
612 //////////
613
614 /** expand a^n where a is an add and n is an integer.
615  *  @see power::expand */
616 ex power::expand_add(const add & a, int n) const
617 {
618         if (n==2)
619                 return expand_add_2(a);
620         
621         int m = a.nops();
622         exvector sum;
623         sum.reserve((n+1)*(m-1));
624         intvector k(m-1);
625         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
626         intvector upper_limit(m-1);
627         int l;
628         
629         for (int l=0; l<m-1; l++) {
630                 k[l] = 0;
631                 k_cum[l] = 0;
632                 upper_limit[l] = n;
633         }
634         
635         while (1) {
636                 exvector term;
637                 term.reserve(m+1);
638                 for (l=0; l<m-1; l++) {
639                         const ex & b = a.op(l);
640                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
641                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
642                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
643                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
644                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
645                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
646                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
647                         if (is_ex_exactly_of_type(b,mul))
648                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
649                         else
650                                 term.push_back(power(b,k[l]));
651                 }
652                 
653                 const ex & b = a.op(l);
654                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
655                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
656                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
657                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
658                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
659                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
660                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
661                 if (is_ex_exactly_of_type(b,mul))
662                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
663                 else
664                         term.push_back(power(b,n-k_cum[m-2]));
665                 
666                 numeric f = binomial(numeric(n),numeric(k[0]));
667                 for (l=1; l<m-1; l++)
668                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
669                 
670                 term.push_back(f);
671                 
672                 /*
673                 cout << "begin term" << endl;
674                 for (int i=0; i<m-1; i++) {
675                         cout << "k[" << i << "]=" << k[i] << endl;
676                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
677                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
678                 }
679                 for_each(term.begin(), term.end(), ostream_iterator<ex>(cout, "\n"));
680                 cout << "end term" << endl;
681                 */
682                 
683                 // TODO: optimize this
684                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
685                 
686                 // increment k[]
687                 l = m-2;
688                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
689                         k[l] = 0;    
690                         l--;
691                 }
692                 if (l<0) break;
693                 
694                 // recalc k_cum[] and upper_limit[]
695                 if (l==0)
696                         k_cum[0] = k[0];
697                 else
698                         k_cum[l] = k_cum[l-1]+k[l];
699                 
700                 for (int i=l+1; i<m-1; i++)
701                         k_cum[i] = k_cum[i-1]+k[i];
702                 
703                 for (int i=l+1; i<m-1; i++)
704                         upper_limit[i] = n-k_cum[i-1];
705         }
706         return (new add(sum))->setflag(status_flags::dynallocated |
707                                                                    status_flags::expanded );
708 }
709
710
711 /** Special case of power::expand_add. Expands a^2 where a is an add.
712  *  @see power::expand_add */
713 ex power::expand_add_2(const add & a) const
714 {
715         epvector sum;
716         unsigned a_nops = a.nops();
717         sum.reserve((a_nops*(a_nops+1))/2);
718         epvector::const_iterator last = a.seq.end();
719         
720         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
721         // first part: ignore overall_coeff and expand other terms
722         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
723                 const ex & r = (*cit0).rest;
724                 const ex & c = (*cit0).coeff;
725                 
726                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
727                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
728                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
729                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
730                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
731                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
732                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
733                 
734                 if (are_ex_trivially_equal(c,_ex1())) {
735                         if (is_ex_exactly_of_type(r,mul)) {
736                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
737                                                      _ex1()));
738                         } else {
739                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
740                                                      _ex1()));
741                         }
742                 } else {
743                         if (is_ex_exactly_of_type(r,mul)) {
744                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
745                                                      ex_to_numeric(c).power_dyn(_num2())));
746                         } else {
747                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
748                                                      ex_to_numeric(c).power_dyn(_num2())));
749                         }
750                 }
751                         
752                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
753                         const ex & r1 = (*cit1).rest;
754                         const ex & c1 = (*cit1).coeff;
755                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
756                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
757                 }
758         }
759         
760         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
761         
762         // second part: add terms coming from overall_factor (if != 0)
763         if (!a.overall_coeff.is_zero()) {
764                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
765                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
766                 }
767                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
768         }
769         
770         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
771         
772         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
773 }
774
775 /** Expand factors of m in m^n where m is a mul and n is and integer
776  *  @see power::expand */
777 ex power::expand_mul(const mul & m, const numeric & n) const
778 {
779         if (n.is_zero())
780                 return _ex1();
781         
782         epvector distrseq;
783         distrseq.reserve(m.seq.size());
784         epvector::const_iterator last = m.seq.end();
785         epvector::const_iterator cit = m.seq.begin();
786         while (cit!=last) {
787                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
788                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
789                 } else {
790                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
791                         // since n is an integer
792                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
793                 }
794                 ++cit;
795         }
796         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
797 }
798
799 /*
800 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
801                                                                 unsigned options) const
802 {
803         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
804                         expand(options | expand_options::internal_do_not_expand_power_operands);
805
806         return ex(mul(rest_power,basis),0).
807                expand(options | expand_options::internal_do_not_expand_mul_operands);
808 }
809 */
810
811 // helper function
812
813 ex sqrt(const ex & a)
814 {
815         return power(a,_ex1_2());
816 }
817
818 } // namespace GiNaC