* Some internal reorganization WRT flyweight handling and initialization,
[ginac.git] / ginac / power.cpp
1 /** @file power.cpp
2  *
3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
7  *
8  *  This program is free software; you can redistribute it and/or modify
9  *  it under the terms of the GNU General Public License as published by
10  *  the Free Software Foundation; either version 2 of the License, or
11  *  (at your option) any later version.
12  *
13  *  This program is distributed in the hope that it will be useful,
14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
16  *  GNU General Public License for more details.
17  *
18  *  You should have received a copy of the GNU General Public License
19  *  along with this program; if not, write to the Free Software
20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
21  */
22
23 #include <vector>
24 #include <iostream>
25 #include <stdexcept>
26
27 #include "power.h"
28 #include "expairseq.h"
29 #include "add.h"
30 #include "mul.h"
31 #include "ncmul.h"
32 #include "numeric.h"
33 #include "constant.h"
34 #include "inifcns.h" // for log() in power::derivative()
35 #include "matrix.h"
36 #include "symbol.h"
37 #include "print.h"
38 #include "archive.h"
39 #include "debugmsg.h"
40 #include "utils.h"
41
42 namespace GiNaC {
43
44 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
45
46 typedef std::vector<int> intvector;
47
48 //////////
49 // default ctor, dtor, copy ctor assignment operator and helpers
50 //////////
51
52 power::power() : inherited(TINFO_power)
53 {
54         debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
55 }
56
57 void power::copy(const power & other)
58 {
59         inherited::copy(other);
60         basis = other.basis;
61         exponent = other.exponent;
62 }
63
64 DEFAULT_DESTROY(power)
65
66 //////////
67 // other ctors
68 //////////
69
70 // all inlined
71
72 //////////
73 // archiving
74 //////////
75
76 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
77 {
78         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
79         n.find_ex("basis", basis, sym_lst);
80         n.find_ex("exponent", exponent, sym_lst);
81 }
82
83 void power::archive(archive_node &n) const
84 {
85         inherited::archive(n);
86         n.add_ex("basis", basis);
87         n.add_ex("exponent", exponent);
88 }
89
90 DEFAULT_UNARCHIVE(power)
91
92 //////////
93 // functions overriding virtual functions from base classes
94 //////////
95
96 // public
97
98 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
99 {
100         // Optimal output of integer powers of symbols to aid compiler CSE.
101         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
102         // to learn why such a parenthisation is really necessary.
103         if (exp == 1) {
104                 x.print(c);
105         } else if (exp == 2) {
106                 x.print(c);
107                 c.s << "*";
108                 x.print(c);
109         } else if (exp & 1) {
110                 x.print(c);
111                 c.s << "*";
112                 print_sym_pow(c, x, exp-1);
113         } else {
114                 c.s << "(";
115                 print_sym_pow(c, x, exp >> 1);
116                 c.s << ")*(";
117                 print_sym_pow(c, x, exp >> 1);
118                 c.s << ")";
119         }
120 }
121
122 void power::print(const print_context & c, unsigned level) const
123 {
124         debugmsg("power print", LOGLEVEL_PRINT);
125
126         if (is_a<print_tree>(c)) {
127
128                 inherited::print(c, level);
129
130         } else if (is_a<print_csrc>(c)) {
131
132                 // Integer powers of symbols are printed in a special, optimized way
133                 if (exponent.info(info_flags::integer)
134                  && (is_exactly_a<symbol>(basis) || is_exactly_a<constant>(basis))) {
135                         int exp = ex_to<numeric>(exponent).to_int();
136                         if (exp > 0)
137                                 c.s << '(';
138                         else {
139                                 exp = -exp;
140                                 if (is_a<print_csrc_cl_N>(c))
141                                         c.s << "recip(";
142                                 else
143                                         c.s << "1.0/(";
144                         }
145                         print_sym_pow(c, ex_to<symbol>(basis), exp);
146                         c.s << ')';
147
148                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
149                 } else if (exponent.compare(_num_1) == 0) {
150                         if (is_a<print_csrc_cl_N>(c))
151                                 c.s << "recip(";
152                         else
153                                 c.s << "1.0/(";
154                         basis.print(c);
155                         c.s << ')';
156
157                 // Otherwise, use the pow() or expt() (CLN) functions
158                 } else {
159                         if (is_a<print_csrc_cl_N>(c))
160                                 c.s << "expt(";
161                         else
162                                 c.s << "pow(";
163                         basis.print(c);
164                         c.s << ',';
165                         exponent.print(c);
166                         c.s << ')';
167                 }
168
169         } else {
170
171                 if (exponent.is_equal(_ex1_2)) {
172                         if (is_a<print_latex>(c))
173                                 c.s << "\\sqrt{";
174                         else
175                                 c.s << "sqrt(";
176                         basis.print(c);
177                         if (is_a<print_latex>(c))
178                                 c.s << '}';
179                         else
180                                 c.s << ')';
181                 } else {
182                         if (precedence() <= level) {
183                                 if (is_a<print_latex>(c))
184                                         c.s << "{(";
185                                 else
186                                         c.s << "(";
187                         }
188                         basis.print(c, precedence());
189                         c.s << '^';
190                         if (is_a<print_latex>(c))
191                                 c.s << '{';
192                         exponent.print(c, precedence());
193                         if (is_a<print_latex>(c))
194                                 c.s << '}';
195                         if (precedence() <= level) {
196                                 if (is_a<print_latex>(c))
197                                         c.s << ")}";
198                                 else
199                                         c.s << ')';
200                         }
201                 }
202         }
203 }
204
205 bool power::info(unsigned inf) const
206 {
207         switch (inf) {
208                 case info_flags::polynomial:
209                 case info_flags::integer_polynomial:
210                 case info_flags::cinteger_polynomial:
211                 case info_flags::rational_polynomial:
212                 case info_flags::crational_polynomial:
213                         return exponent.info(info_flags::nonnegint);
214                 case info_flags::rational_function:
215                         return exponent.info(info_flags::integer);
216                 case info_flags::algebraic:
217                         return (!exponent.info(info_flags::integer) ||
218                                         basis.info(inf));
219         }
220         return inherited::info(inf);
221 }
222
223 unsigned power::nops() const
224 {
225         return 2;
226 }
227
228 ex & power::let_op(int i)
229 {
230         GINAC_ASSERT(i>=0);
231         GINAC_ASSERT(i<2);
232
233         return i==0 ? basis : exponent;
234 }
235
236 ex power::map(map_function & f) const
237 {
238         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
239 }
240
241 int power::degree(const ex & s) const
242 {
243         if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
244                 if (basis.is_equal(s))
245                         return ex_to<numeric>(exponent).to_int();
246                 else
247                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
248         }
249         return 0;
250 }
251
252 int power::ldegree(const ex & s) const 
253 {
254         if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
255                 if (basis.is_equal(s))
256                         return ex_to<numeric>(exponent).to_int();
257                 else
258                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
259         }
260         return 0;
261 }
262
263 ex power::coeff(const ex & s, int n) const
264 {
265         if (!basis.is_equal(s)) {
266                 // basis not equal to s
267                 if (n == 0)
268                         return *this;
269                 else
270                         return _ex0;
271         } else {
272                 // basis equal to s
273                 if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
274                         // integer exponent
275                         int int_exp = ex_to<numeric>(exponent).to_int();
276                         if (n == int_exp)
277                                 return _ex1;
278                         else
279                                 return _ex0;
280                 } else {
281                         // non-integer exponents are treated as zero
282                         if (n == 0)
283                                 return *this;
284                         else
285                                 return _ex0;
286                 }
287         }
288 }
289
290 /** Perform automatic term rewriting rules in this class.  In the following
291  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
292  *  stand for such expressions that contain a plain number.
293  *  - ^(x,0) -> 1  (also handles ^(0,0))
294  *  - ^(x,1) -> x
295  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
296  *  - ^(1,x) -> 1
297  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
298  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
299  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
300  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
301  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
302  *
303  *  @param level cut-off in recursive evaluation */
304 ex power::eval(int level) const
305 {
306         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
307         
308         if ((level==1) && (flags & status_flags::evaluated))
309                 return *this;
310         else if (level == -max_recursion_level)
311                 throw(std::runtime_error("max recursion level reached"));
312         
313         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
314         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
315         
316         bool basis_is_numerical = false;
317         bool exponent_is_numerical = false;
318         const numeric *num_basis;
319         const numeric *num_exponent;
320         
321         if (is_ex_exactly_of_type(ebasis, numeric)) {
322                 basis_is_numerical = true;
323                 num_basis = &ex_to<numeric>(ebasis);
324         }
325         if (is_ex_exactly_of_type(eexponent, numeric)) {
326                 exponent_is_numerical = true;
327                 num_exponent = &ex_to<numeric>(eexponent);
328         }
329         
330         // ^(x,0) -> 1  (0^0 also handled here)
331         if (eexponent.is_zero()) {
332                 if (ebasis.is_zero())
333                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
334                 else
335                         return _ex1;
336         }
337         
338         // ^(x,1) -> x
339         if (eexponent.is_equal(_ex1))
340                 return ebasis;
341
342         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
343         if (ebasis.is_zero() && exponent_is_numerical) {
344                 if ((num_exponent->real()).is_zero())
345                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
346                 else if ((num_exponent->real()).is_negative())
347                         throw (pole_error("power::eval(): division by zero",1));
348                 else
349                         return _ex0;
350         }
351
352         // ^(1,x) -> 1
353         if (ebasis.is_equal(_ex1))
354                 return _ex1;
355
356         if (exponent_is_numerical) {
357
358                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
359                 // except if c1,c2 are rational, but c1^c2 is not)
360                 if (basis_is_numerical) {
361                         const bool basis_is_crational = num_basis->is_crational();
362                         const bool exponent_is_crational = num_exponent->is_crational();
363                         if (!basis_is_crational || !exponent_is_crational) {
364                                 // return a plain float
365                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
366                                                                                                status_flags::evaluated |
367                                                                                                status_flags::expanded);
368                         }
369
370                         const numeric res = num_basis->power(*num_exponent);
371                         if (res.is_crational()) {
372                                 return res;
373                         }
374                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
375
376                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
377                         if (basis_is_crational && exponent_is_crational
378                             && num_exponent->is_real()
379                             && !num_exponent->is_integer()) {
380                                 const numeric n = num_exponent->numer();
381                                 const numeric m = num_exponent->denom();
382                                 numeric r;
383                                 numeric q = iquo(n, m, r);
384                                 if (r.is_negative()) {
385                                         r += m;
386                                         --q;
387                                 }
388                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
389                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
390                                                 // try it for numerator and denominator separately, in order to
391                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
392                                                 const numeric bnum = num_basis->numer();
393                                                 const numeric bden = num_basis->denom();
394                                                 const numeric res_bnum = bnum.power(*num_exponent);
395                                                 const numeric res_bden = bden.power(*num_exponent);
396                                                 if (res_bnum.is_integer())
397                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
398                                                 if (res_bden.is_integer())
399                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
400                                         }
401                                         return this->hold();
402                                 } else {
403                                         // assemble resulting product, but allowing for a re-evaluation,
404                                         // because otherwise we'll end up with something like
405                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
406                                         // instead of 7/16*7^(1/3).
407                                         ex prod = power(*num_basis,r.div(m));
408                                         return prod*power(*num_basis,q);
409                                 }
410                         }
411                 }
412         
413                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
414                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
415                 // case c1==1 should not happen, see below!)
416                 if (is_ex_exactly_of_type(ebasis,power)) {
417                         const power & sub_power = ex_to<power>(ebasis);
418                         const ex & sub_basis = sub_power.basis;
419                         const ex & sub_exponent = sub_power.exponent;
420                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
421                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
422                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
423                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1).is_negative())
424                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
425                         }
426                 }
427         
428                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
429                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
430                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
431                 }
432         
433                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
434                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
435                 if (is_ex_exactly_of_type(ebasis,mul)) {
436                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
437                         const mul & mulref = ex_to<mul>(ebasis);
438                         if (!mulref.overall_coeff.is_equal(_ex1)) {
439                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
440                                 if (num_coeff.is_real()) {
441                                         if (num_coeff.is_positive()) {
442                                                 mul *mulp = new mul(mulref);
443                                                 mulp->overall_coeff = _ex1;
444                                                 mulp->clearflag(status_flags::evaluated);
445                                                 mulp->clearflag(status_flags::hash_calculated);
446                                                 return (new mul(power(*mulp,exponent),
447                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
448                                         } else {
449                                                 GINAC_ASSERT(num_coeff.compare(_num0)<0);
450                                                 if (!num_coeff.is_equal(_num_1)) {
451                                                         mul *mulp = new mul(mulref);
452                                                         mulp->overall_coeff = _ex_1;
453                                                         mulp->clearflag(status_flags::evaluated);
454                                                         mulp->clearflag(status_flags::hash_calculated);
455                                                         return (new mul(power(*mulp,exponent),
456                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
457                                                 }
458                                         }
459                                 }
460                         }
461                 }
462
463                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
464                 if (num_exponent->is_pos_integer() &&
465                     ebasis.return_type() != return_types::commutative &&
466                     !is_ex_of_type(ebasis,matrix)) {
467                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
468                 }
469         }
470         
471         if (are_ex_trivially_equal(ebasis,basis) &&
472             are_ex_trivially_equal(eexponent,exponent)) {
473                 return this->hold();
474         }
475         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
476                                                        status_flags::evaluated);
477 }
478
479 ex power::evalf(int level) const
480 {
481         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
482         
483         ex ebasis;
484         ex eexponent;
485         
486         if (level==1) {
487                 ebasis = basis;
488                 eexponent = exponent;
489         } else if (level == -max_recursion_level) {
490                 throw(std::runtime_error("max recursion level reached"));
491         } else {
492                 ebasis = basis.evalf(level-1);
493                 if (!is_ex_exactly_of_type(exponent,numeric))
494                         eexponent = exponent.evalf(level-1);
495                 else
496                         eexponent = exponent;
497         }
498
499         return power(ebasis,eexponent);
500 }
501
502 ex power::evalm(void) const
503 {
504         const ex ebasis = basis.evalm();
505         const ex eexponent = exponent.evalm();
506         if (is_ex_of_type(ebasis,matrix)) {
507                 if (is_ex_of_type(eexponent,numeric)) {
508                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
509                 }
510         }
511         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
512 }
513
514 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
515 {
516         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
517         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
518
519         if (are_ex_trivially_equal(basis, subsed_basis)
520          && are_ex_trivially_equal(exponent, subsed_exponent))
521                 return basic::subs(ls, lr, no_pattern);
522         else
523                 return power(subsed_basis, subsed_exponent).basic::subs(ls, lr, no_pattern);
524 }
525
526 ex power::simplify_ncmul(const exvector & v) const
527 {
528         return inherited::simplify_ncmul(v);
529 }
530
531 // protected
532
533 /** Implementation of ex::diff() for a power.
534  *  @see ex::diff */
535 ex power::derivative(const symbol & s) const
536 {
537         if (exponent.info(info_flags::real)) {
538                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
539                 epvector newseq;
540                 newseq.reserve(2);
541                 newseq.push_back(expair(basis, exponent - _ex1));
542                 newseq.push_back(expair(basis.diff(s), _ex1));
543                 return mul(newseq, exponent);
544         } else {
545                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
546                 return mul(*this,
547                            add(mul(exponent.diff(s), log(basis)),
548                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
549         }
550 }
551
552 int power::compare_same_type(const basic & other) const
553 {
554         GINAC_ASSERT(is_exactly_a<power>(other));
555         const power &o = static_cast<const power &>(other);
556
557         int cmpval = basis.compare(o.basis);
558         if (cmpval)
559                 return cmpval;
560         else
561                 return exponent.compare(o.exponent);
562 }
563
564 unsigned power::return_type(void) const
565 {
566         return basis.return_type();
567 }
568    
569 unsigned power::return_type_tinfo(void) const
570 {
571         return basis.return_type_tinfo();
572 }
573
574 ex power::expand(unsigned options) const
575 {
576         if (options == 0 && (flags & status_flags::expanded))
577                 return *this;
578         
579         const ex expanded_basis = basis.expand(options);
580         const ex expanded_exponent = exponent.expand(options);
581         
582         // x^(a+b) -> x^a * x^b
583         if (is_ex_exactly_of_type(expanded_exponent, add)) {
584                 const add &a = ex_to<add>(expanded_exponent);
585                 exvector distrseq;
586                 distrseq.reserve(a.seq.size() + 1);
587                 epvector::const_iterator last = a.seq.end();
588                 epvector::const_iterator cit = a.seq.begin();
589                 while (cit!=last) {
590                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
591                         ++cit;
592                 }
593                 
594                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
595                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
596                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
597                         int int_exponent = num_exponent.to_int();
598                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
599                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
600                         else
601                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
602                 } else
603                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
604                 
605                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
606                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
607                 return r.expand();
608         }
609         
610         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
611                 !ex_to<numeric>(expanded_exponent).is_integer()) {
612                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
613                         return this->hold();
614                 } else {
615                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
616                 }
617         }
618         
619         // integer numeric exponent
620         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
621         int int_exponent = num_exponent.to_int();
622         
623         // (x+y)^n, n>0
624         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
625                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
626         
627         // (x*y)^n -> x^n * y^n
628         if (is_ex_exactly_of_type(expanded_basis,mul))
629                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
630         
631         // cannot expand further
632         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
633                 return this->hold();
634         else
635                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
636 }
637
638 //////////
639 // new virtual functions which can be overridden by derived classes
640 //////////
641
642 // none
643
644 //////////
645 // non-virtual functions in this class
646 //////////
647
648 /** expand a^n where a is an add and n is an integer.
649  *  @see power::expand */
650 ex power::expand_add(const add & a, int n) const
651 {
652         if (n==2)
653                 return expand_add_2(a);
654         
655         int m = a.nops();
656         exvector sum;
657         sum.reserve((n+1)*(m-1));
658         intvector k(m-1);
659         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
660         intvector upper_limit(m-1);
661         int l;
662         
663         for (int l=0; l<m-1; l++) {
664                 k[l] = 0;
665                 k_cum[l] = 0;
666                 upper_limit[l] = n;
667         }
668         
669         while (true) {
670                 exvector term;
671                 term.reserve(m+1);
672                 for (l=0; l<m-1; l++) {
673                         const ex & b = a.op(l);
674                         GINAC_ASSERT(!is_exactly_a<add>(b));
675                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
676                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
677                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
678                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
679                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
680                                      !is_exactly_a<power>(ex_to<power>(b).basis));
681                         if (is_ex_exactly_of_type(b,mul))
682                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
683                         else
684                                 term.push_back(power(b,k[l]));
685                 }
686                 
687                 const ex & b = a.op(l);
688                 GINAC_ASSERT(!is_exactly_a<add>(b));
689                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
690                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
691                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
692                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
693                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
694                              !is_exactly_a<power>(ex_to<power>(b).basis));
695                 if (is_ex_exactly_of_type(b,mul))
696                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
697                 else
698                         term.push_back(power(b,n-k_cum[m-2]));
699                 
700                 numeric f = binomial(numeric(n),numeric(k[0]));
701                 for (l=1; l<m-1; l++)
702                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
703                 
704                 term.push_back(f);
705                 
706                 // TODO: Can we optimize this?  Alex seemed to think so...
707                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
708                 
709                 // increment k[]
710                 l = m-2;
711                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
712                         k[l] = 0;    
713                         --l;
714                 }
715                 if (l<0) break;
716                 
717                 // recalc k_cum[] and upper_limit[]
718                 if (l==0)
719                         k_cum[0] = k[0];
720                 else
721                         k_cum[l] = k_cum[l-1]+k[l];
722                 
723                 for (int i=l+1; i<m-1; i++)
724                         k_cum[i] = k_cum[i-1]+k[i];
725                 
726                 for (int i=l+1; i<m-1; i++)
727                         upper_limit[i] = n-k_cum[i-1];
728         }
729         return (new add(sum))->setflag(status_flags::dynallocated |
730                                        status_flags::expanded );
731 }
732
733
734 /** Special case of power::expand_add. Expands a^2 where a is an add.
735  *  @see power::expand_add */
736 ex power::expand_add_2(const add & a) const
737 {
738         epvector sum;
739         unsigned a_nops = a.nops();
740         sum.reserve((a_nops*(a_nops+1))/2);
741         epvector::const_iterator last = a.seq.end();
742         
743         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
744         // first part: ignore overall_coeff and expand other terms
745         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
746                 const ex & r = cit0->rest;
747                 const ex & c = cit0->coeff;
748                 
749                 GINAC_ASSERT(!is_exactly_a<add>(r));
750                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
751                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
752                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
753                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
754                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
755                              !is_exactly_a<power>(ex_to<power>(r).basis));
756                 
757                 if (are_ex_trivially_equal(c,_ex1)) {
758                         if (is_ex_exactly_of_type(r,mul)) {
759                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
760                                                      _ex1));
761                         } else {
762                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
763                                                      _ex1));
764                         }
765                 } else {
766                         if (is_ex_exactly_of_type(r,mul)) {
767                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
768                                                      ex_to<numeric>(c).power_dyn(_num2)));
769                         } else {
770                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
771                                                      ex_to<numeric>(c).power_dyn(_num2)));
772                         }
773                 }
774                         
775                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
776                         const ex & r1 = cit1->rest;
777                         const ex & c1 = cit1->coeff;
778                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
779                                                                       _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
780                 }
781         }
782         
783         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
784         
785         // second part: add terms coming from overall_factor (if != 0)
786         if (!a.overall_coeff.is_zero()) {
787                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
788                 while (i != end) {
789                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
790                         ++i;
791                 }
792                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
793         }
794         
795         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
796         
797         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
798 }
799
800 /** Expand factors of m in m^n where m is a mul and n is and integer
801  *  @see power::expand */
802 ex power::expand_mul(const mul & m, const numeric & n) const
803 {
804         if (n.is_zero())
805                 return _ex1;
806         
807         epvector distrseq;
808         distrseq.reserve(m.seq.size());
809         epvector::const_iterator last = m.seq.end();
810         epvector::const_iterator cit = m.seq.begin();
811         while (cit!=last) {
812                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
813                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
814                 } else {
815                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
816                         // since n is an integer
817                         distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
818                 }
819                 ++cit;
820         }
821         return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
822 }
823
824 } // namespace GiNaC