999d9b147284f894f1e18298f6bd9a5465c2e9e8
[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) 1999-2009 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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
21  */
22
23 #include "power.h"
24 #include "expairseq.h"
25 #include "add.h"
26 #include "mul.h"
27 #include "ncmul.h"
28 #include "numeric.h"
29 #include "constant.h"
30 #include "operators.h"
31 #include "inifcns.h" // for log() in power::derivative()
32 #include "matrix.h"
33 #include "indexed.h"
34 #include "symbol.h"
35 #include "lst.h"
36 #include "archive.h"
37 #include "utils.h"
38 #include "relational.h"
39 #include "compiler.h"
40
41 #include <iostream>
42 #include <limits>
43 #include <stdexcept>
44 #include <vector>
45
46 namespace GiNaC {
47
48 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(power, basic,
49   print_func<print_dflt>(&power::do_print_dflt).
50   print_func<print_latex>(&power::do_print_latex).
51   print_func<print_csrc>(&power::do_print_csrc).
52   print_func<print_python>(&power::do_print_python).
53   print_func<print_python_repr>(&power::do_print_python_repr).
54   print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))
55
56 typedef std::vector<int> intvector;
57
58 //////////
59 // default constructor
60 //////////
61
62 power::power() { }
63
64 //////////
65 // other constructors
66 //////////
67
68 // all inlined
69
70 //////////
71 // archiving
72 //////////
73
74 void power::read_archive(const archive_node &n, lst &sym_lst)
75 {
76         inherited::read_archive(n, sym_lst);
77         n.find_ex("basis", basis, sym_lst);
78         n.find_ex("exponent", exponent, sym_lst);
79 }
80
81 void power::archive(archive_node &n) const
82 {
83         inherited::archive(n);
84         n.add_ex("basis", basis);
85         n.add_ex("exponent", exponent);
86 }
87
88 //////////
89 // functions overriding virtual functions from base classes
90 //////////
91
92 // public
93
94 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
95 {
96         // Ordinary output of powers using '^' or '**'
97         if (precedence() <= level)
98                 c.s << openbrace << '(';
99         basis.print(c, precedence());
100         c.s << powersymbol;
101         c.s << openbrace;
102         exponent.print(c, precedence());
103         c.s << closebrace;
104         if (precedence() <= level)
105                 c.s << ')' << closebrace;
106 }
107
108 void power::do_print_dflt(const print_dflt & c, unsigned level) const
109 {
110         if (exponent.is_equal(_ex1_2)) {
111
112                 // Square roots are printed in a special way
113                 c.s << "sqrt(";
114                 basis.print(c);
115                 c.s << ')';
116
117         } else
118                 print_power(c, "^", "", "", level);
119 }
120
121 void power::do_print_latex(const print_latex & c, unsigned level) const
122 {
123         if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
124
125                 // Powers with negative numeric exponents are printed as fractions
126                 c.s << "\\frac{1}{";
127                 power(basis, -exponent).eval().print(c);
128                 c.s << '}';
129
130         } else if (exponent.is_equal(_ex1_2)) {
131
132                 // Square roots are printed in a special way
133                 c.s << "\\sqrt{";
134                 basis.print(c);
135                 c.s << '}';
136
137         } else
138                 print_power(c, "^", "{", "}", level);
139 }
140
141 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
142 {
143         // Optimal output of integer powers of symbols to aid compiler CSE.
144         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
145         // to learn why such a parenthesation is really necessary.
146         if (exp == 1) {
147                 x.print(c);
148         } else if (exp == 2) {
149                 x.print(c);
150                 c.s << "*";
151                 x.print(c);
152         } else if (exp & 1) {
153                 x.print(c);
154                 c.s << "*";
155                 print_sym_pow(c, x, exp-1);
156         } else {
157                 c.s << "(";
158                 print_sym_pow(c, x, exp >> 1);
159                 c.s << ")*(";
160                 print_sym_pow(c, x, exp >> 1);
161                 c.s << ")";
162         }
163 }
164
165 void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const
166 {
167         if (exponent.is_equal(_ex_1)) {
168                 c.s << "recip(";
169                 basis.print(c);
170                 c.s << ')';
171                 return;
172         }
173         c.s << "expt(";
174         basis.print(c);
175         c.s << ", ";
176         exponent.print(c);
177         c.s << ')';
178 }
179
180 void power::do_print_csrc(const print_csrc & c, unsigned level) const
181 {
182         // Integer powers of symbols are printed in a special, optimized way
183         if (exponent.info(info_flags::integer)
184          && (is_a<symbol>(basis) || is_a<constant>(basis))) {
185                 int exp = ex_to<numeric>(exponent).to_int();
186                 if (exp > 0)
187                         c.s << '(';
188                 else {
189                         exp = -exp;
190                         c.s << "1.0/(";
191                 }
192                 print_sym_pow(c, ex_to<symbol>(basis), exp);
193                 c.s << ')';
194
195         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
196         } else if (exponent.is_equal(_ex_1)) {
197                 c.s << "1.0/(";
198                 basis.print(c);
199                 c.s << ')';
200
201         // Otherwise, use the pow() function
202         } else {
203                 c.s << "pow(";
204                 basis.print(c);
205                 c.s << ',';
206                 exponent.print(c);
207                 c.s << ')';
208         }
209 }
210
211 void power::do_print_python(const print_python & c, unsigned level) const
212 {
213         print_power(c, "**", "", "", level);
214 }
215
216 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
217 {
218         c.s << class_name() << '(';
219         basis.print(c);
220         c.s << ',';
221         exponent.print(c);
222         c.s << ')';
223 }
224
225 bool power::info(unsigned inf) const
226 {
227         switch (inf) {
228                 case info_flags::polynomial:
229                 case info_flags::integer_polynomial:
230                 case info_flags::cinteger_polynomial:
231                 case info_flags::rational_polynomial:
232                 case info_flags::crational_polynomial:
233                         return exponent.info(info_flags::nonnegint) &&
234                                basis.info(inf);
235                 case info_flags::rational_function:
236                         return exponent.info(info_flags::integer) &&
237                                basis.info(inf);
238                 case info_flags::algebraic:
239                         return !exponent.info(info_flags::integer) ||
240                                basis.info(inf);
241                 case info_flags::expanded:
242                         return (flags & status_flags::expanded);
243                 case info_flags::has_indices: {
244                         if (flags & status_flags::has_indices)
245                                 return true;
246                         else if (flags & status_flags::has_no_indices)
247                                 return false;
248                         else if (basis.info(info_flags::has_indices)) {
249                                 setflag(status_flags::has_indices);
250                                 clearflag(status_flags::has_no_indices);
251                                 return true;
252                         } else {
253                                 clearflag(status_flags::has_indices);
254                                 setflag(status_flags::has_no_indices);
255                                 return false;
256                         }
257                 }
258         }
259         return inherited::info(inf);
260 }
261
262 size_t power::nops() const
263 {
264         return 2;
265 }
266
267 ex power::op(size_t i) const
268 {
269         GINAC_ASSERT(i<2);
270
271         return i==0 ? basis : exponent;
272 }
273
274 ex power::map(map_function & f) const
275 {
276         const ex &mapped_basis = f(basis);
277         const ex &mapped_exponent = f(exponent);
278
279         if (!are_ex_trivially_equal(basis, mapped_basis)
280          || !are_ex_trivially_equal(exponent, mapped_exponent))
281                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
282         else
283                 return *this;
284 }
285
286 bool power::is_polynomial(const ex & var) const
287 {
288         if (exponent.has(var))
289                 return false;
290         if (!exponent.info(info_flags::nonnegint))
291                 return false;
292         return basis.is_polynomial(var);
293 }
294
295 int power::degree(const ex & s) const
296 {
297         if (is_equal(ex_to<basic>(s)))
298                 return 1;
299         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
300                 if (basis.is_equal(s))
301                         return ex_to<numeric>(exponent).to_int();
302                 else
303                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
304         } else if (basis.has(s))
305                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
306         else
307                 return 0;
308 }
309
310 int power::ldegree(const ex & s) const 
311 {
312         if (is_equal(ex_to<basic>(s)))
313                 return 1;
314         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
315                 if (basis.is_equal(s))
316                         return ex_to<numeric>(exponent).to_int();
317                 else
318                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
319         } else if (basis.has(s))
320                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
321         else
322                 return 0;
323 }
324
325 ex power::coeff(const ex & s, int n) const
326 {
327         if (is_equal(ex_to<basic>(s)))
328                 return n==1 ? _ex1 : _ex0;
329         else if (!basis.is_equal(s)) {
330                 // basis not equal to s
331                 if (n == 0)
332                         return *this;
333                 else
334                         return _ex0;
335         } else {
336                 // basis equal to s
337                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
338                         // integer exponent
339                         int int_exp = ex_to<numeric>(exponent).to_int();
340                         if (n == int_exp)
341                                 return _ex1;
342                         else
343                                 return _ex0;
344                 } else {
345                         // non-integer exponents are treated as zero
346                         if (n == 0)
347                                 return *this;
348                         else
349                                 return _ex0;
350                 }
351         }
352 }
353
354 /** Perform automatic term rewriting rules in this class.  In the following
355  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
356  *  stand for such expressions that contain a plain number.
357  *  - ^(x,0) -> 1  (also handles ^(0,0))
358  *  - ^(x,1) -> x
359  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
360  *  - ^(1,x) -> 1
361  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
362  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
363  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
364  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
365  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
366  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
367  *
368  *  @param level cut-off in recursive evaluation */
369 ex power::eval(int level) const
370 {
371         if ((level==1) && (flags & status_flags::evaluated))
372                 return *this;
373         else if (level == -max_recursion_level)
374                 throw(std::runtime_error("max recursion level reached"));
375         
376         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
377         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
378         
379         const numeric *num_basis = NULL;
380         const numeric *num_exponent = NULL;
381         
382         if (is_exactly_a<numeric>(ebasis)) {
383                 num_basis = &ex_to<numeric>(ebasis);
384         }
385         if (is_exactly_a<numeric>(eexponent)) {
386                 num_exponent = &ex_to<numeric>(eexponent);
387         }
388         
389         // ^(x,0) -> 1  (0^0 also handled here)
390         if (eexponent.is_zero()) {
391                 if (ebasis.is_zero())
392                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
393                 else
394                         return _ex1;
395         }
396         
397         // ^(x,1) -> x
398         if (eexponent.is_equal(_ex1))
399                 return ebasis;
400
401         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
402         if ( ebasis.is_zero() && num_exponent ) {
403                 if ((num_exponent->real()).is_zero())
404                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
405                 else if ((num_exponent->real()).is_negative())
406                         throw (pole_error("power::eval(): division by zero",1));
407                 else
408                         return _ex0;
409         }
410
411         // ^(1,x) -> 1
412         if (ebasis.is_equal(_ex1))
413                 return _ex1;
414
415         // power of a function calculated by separate rules defined for this function
416         if (is_exactly_a<function>(ebasis))
417                 return ex_to<function>(ebasis).power(eexponent);
418
419         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
420         if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
421                 return power(ebasis.op(0), ebasis.op(1) * eexponent);
422
423         if ( num_exponent ) {
424
425                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
426                 // except if c1,c2 are rational, but c1^c2 is not)
427                 if ( num_basis ) {
428                         const bool basis_is_crational = num_basis->is_crational();
429                         const bool exponent_is_crational = num_exponent->is_crational();
430                         if (!basis_is_crational || !exponent_is_crational) {
431                                 // return a plain float
432                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
433                                                                                                status_flags::evaluated |
434                                                                                                status_flags::expanded);
435                         }
436
437                         const numeric res = num_basis->power(*num_exponent);
438                         if (res.is_crational()) {
439                                 return res;
440                         }
441                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
442
443                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
444                         if (basis_is_crational && exponent_is_crational
445                             && num_exponent->is_real()
446                             && !num_exponent->is_integer()) {
447                                 const numeric n = num_exponent->numer();
448                                 const numeric m = num_exponent->denom();
449                                 numeric r;
450                                 numeric q = iquo(n, m, r);
451                                 if (r.is_negative()) {
452                                         r += m;
453                                         --q;
454                                 }
455                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
456                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
457                                                 // try it for numerator and denominator separately, in order to
458                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
459                                                 const numeric bnum = num_basis->numer();
460                                                 const numeric bden = num_basis->denom();
461                                                 const numeric res_bnum = bnum.power(*num_exponent);
462                                                 const numeric res_bden = bden.power(*num_exponent);
463                                                 if (res_bnum.is_integer())
464                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
465                                                 if (res_bden.is_integer())
466                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
467                                         }
468                                         return this->hold();
469                                 } else {
470                                         // assemble resulting product, but allowing for a re-evaluation,
471                                         // because otherwise we'll end up with something like
472                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
473                                         // instead of 7/16*7^(1/3).
474                                         ex prod = power(*num_basis,r.div(m));
475                                         return prod*power(*num_basis,q);
476                                 }
477                         }
478                 }
479         
480                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
481                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
482                 // case c1==1 should not happen, see below!)
483                 if (is_exactly_a<power>(ebasis)) {
484                         const power & sub_power = ex_to<power>(ebasis);
485                         const ex & sub_basis = sub_power.basis;
486                         const ex & sub_exponent = sub_power.exponent;
487                         if (is_exactly_a<numeric>(sub_exponent)) {
488                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
489                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
490                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()) {
491                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
492                                 }
493                         }
494                 }
495         
496                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
497                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
498                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
499                 }
500
501                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
502                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
503                         numeric icont = ebasis.integer_content();
504                         const numeric lead_coeff = 
505                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
506
507                         const bool canonicalizable = lead_coeff.is_integer();
508                         const bool unit_normal = lead_coeff.is_pos_integer();
509                         if (canonicalizable && (! unit_normal))
510                                 icont = icont.mul(*_num_1_p);
511                         
512                         if (canonicalizable && (icont != *_num1_p)) {
513                                 const add& addref = ex_to<add>(ebasis);
514                                 add* addp = new add(addref);
515                                 addp->setflag(status_flags::dynallocated);
516                                 addp->clearflag(status_flags::hash_calculated);
517                                 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
518                                 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
519                                         i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
520
521                                 const numeric c = icont.power(*num_exponent);
522                                 if (likely(c != *_num1_p))
523                                         return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
524                                 else
525                                         return power(*addp, *num_exponent);
526                         }
527                 }
528
529                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
530                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
531                 if (is_exactly_a<mul>(ebasis)) {
532                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
533                         const mul & mulref = ex_to<mul>(ebasis);
534                         if (!mulref.overall_coeff.is_equal(_ex1)) {
535                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
536                                 if (num_coeff.is_real()) {
537                                         if (num_coeff.is_positive()) {
538                                                 mul *mulp = new mul(mulref);
539                                                 mulp->overall_coeff = _ex1;
540                                                 mulp->clearflag(status_flags::evaluated);
541                                                 mulp->clearflag(status_flags::hash_calculated);
542                                                 return (new mul(power(*mulp,exponent),
543                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
544                                         } else {
545                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
546                                                 if (!num_coeff.is_equal(*_num_1_p)) {
547                                                         mul *mulp = new mul(mulref);
548                                                         mulp->overall_coeff = _ex_1;
549                                                         mulp->clearflag(status_flags::evaluated);
550                                                         mulp->clearflag(status_flags::hash_calculated);
551                                                         return (new mul(power(*mulp,exponent),
552                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
553                                                 }
554                                         }
555                                 }
556                         }
557                 }
558
559                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
560                 if (num_exponent->is_pos_integer() &&
561                     ebasis.return_type() != return_types::commutative &&
562                     !is_a<matrix>(ebasis)) {
563                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
564                 }
565         }
566         
567         if (are_ex_trivially_equal(ebasis,basis) &&
568             are_ex_trivially_equal(eexponent,exponent)) {
569                 return this->hold();
570         }
571         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
572                                                        status_flags::evaluated);
573 }
574
575 ex power::evalf(int level) const
576 {
577         ex ebasis;
578         ex eexponent;
579         
580         if (level==1) {
581                 ebasis = basis;
582                 eexponent = exponent;
583         } else if (level == -max_recursion_level) {
584                 throw(std::runtime_error("max recursion level reached"));
585         } else {
586                 ebasis = basis.evalf(level-1);
587                 if (!is_exactly_a<numeric>(exponent))
588                         eexponent = exponent.evalf(level-1);
589                 else
590                         eexponent = exponent;
591         }
592
593         return power(ebasis,eexponent);
594 }
595
596 ex power::evalm() const
597 {
598         const ex ebasis = basis.evalm();
599         const ex eexponent = exponent.evalm();
600         if (is_a<matrix>(ebasis)) {
601                 if (is_exactly_a<numeric>(eexponent)) {
602                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
603                 }
604         }
605         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
606 }
607
608 bool power::has(const ex & other, unsigned options) const
609 {
610         if (!(options & has_options::algebraic))
611                 return basic::has(other, options);
612         if (!is_a<power>(other))
613                 return basic::has(other, options);
614         if (!exponent.info(info_flags::integer)
615                         || !other.op(1).info(info_flags::integer))
616                 return basic::has(other, options);
617         if (exponent.info(info_flags::posint)
618                         && other.op(1).info(info_flags::posint)
619                         && ex_to<numeric>(exponent).to_int()
620                                         > ex_to<numeric>(other.op(1)).to_int()
621                         && basis.match(other.op(0)))
622                 return true;
623         if (exponent.info(info_flags::negint)
624                         && other.op(1).info(info_flags::negint)
625                         && ex_to<numeric>(exponent).to_int()
626                                         < ex_to<numeric>(other.op(1)).to_int()
627                         && basis.match(other.op(0)))
628                 return true;
629         return basic::has(other, options);
630 }
631
632 // from mul.cpp
633 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
634
635 ex power::subs(const exmap & m, unsigned options) const
636 {       
637         const ex &subsed_basis = basis.subs(m, options);
638         const ex &subsed_exponent = exponent.subs(m, options);
639
640         if (!are_ex_trivially_equal(basis, subsed_basis)
641          || !are_ex_trivially_equal(exponent, subsed_exponent)) 
642                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
643
644         if (!(options & subs_options::algebraic))
645                 return subs_one_level(m, options);
646
647         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
648                 int nummatches = std::numeric_limits<int>::max();
649                 exmap repls;
650                 if (tryfactsubs(*this, it->first, nummatches, repls)) {
651                         ex anum = it->second.subs(repls, subs_options::no_pattern);
652                         ex aden = it->first.subs(repls, subs_options::no_pattern);
653                         ex result = (*this)*power(anum/aden, nummatches);
654                         return (ex_to<basic>(result)).subs_one_level(m, options);
655                 }
656         }
657
658         return subs_one_level(m, options);
659 }
660
661 ex power::eval_ncmul(const exvector & v) const
662 {
663         return inherited::eval_ncmul(v);
664 }
665
666 ex power::conjugate() const
667 {
668         ex newbasis = basis.conjugate();
669         ex newexponent = exponent.conjugate();
670         if (are_ex_trivially_equal(basis, newbasis) && are_ex_trivially_equal(exponent, newexponent)) {
671                 return *this;
672         }
673         return (new power(newbasis, newexponent))->setflag(status_flags::dynallocated);
674 }
675
676 ex power::real_part() const
677 {
678         if (exponent.info(info_flags::integer)) {
679                 ex basis_real = basis.real_part();
680                 if (basis_real == basis)
681                         return *this;
682                 realsymbol a("a"),b("b");
683                 ex result;
684                 if (exponent.info(info_flags::posint))
685                         result = power(a+I*b,exponent);
686                 else
687                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
688                 result = result.expand();
689                 result = result.real_part();
690                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
691                 return result;
692         }
693         
694         ex a = basis.real_part();
695         ex b = basis.imag_part();
696         ex c = exponent.real_part();
697         ex d = exponent.imag_part();
698         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
699 }
700
701 ex power::imag_part() const
702 {
703         if (exponent.info(info_flags::integer)) {
704                 ex basis_real = basis.real_part();
705                 if (basis_real == basis)
706                         return 0;
707                 realsymbol a("a"),b("b");
708                 ex result;
709                 if (exponent.info(info_flags::posint))
710                         result = power(a+I*b,exponent);
711                 else
712                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
713                 result = result.expand();
714                 result = result.imag_part();
715                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
716                 return result;
717         }
718         
719         ex a=basis.real_part();
720         ex b=basis.imag_part();
721         ex c=exponent.real_part();
722         ex d=exponent.imag_part();
723         return
724                 power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
725 }
726
727 // protected
728
729 // protected
730
731 /** Implementation of ex::diff() for a power.
732  *  @see ex::diff */
733 ex power::derivative(const symbol & s) const
734 {
735         if (is_a<numeric>(exponent)) {
736                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
737                 epvector newseq;
738                 newseq.reserve(2);
739                 newseq.push_back(expair(basis, exponent - _ex1));
740                 newseq.push_back(expair(basis.diff(s), _ex1));
741                 return mul(newseq, exponent);
742         } else {
743                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
744                 return mul(*this,
745                            add(mul(exponent.diff(s), log(basis)),
746                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
747         }
748 }
749
750 int power::compare_same_type(const basic & other) const
751 {
752         GINAC_ASSERT(is_exactly_a<power>(other));
753         const power &o = static_cast<const power &>(other);
754
755         int cmpval = basis.compare(o.basis);
756         if (cmpval)
757                 return cmpval;
758         else
759                 return exponent.compare(o.exponent);
760 }
761
762 unsigned power::return_type() const
763 {
764         return basis.return_type();
765 }
766
767 return_type_t power::return_type_tinfo() const
768 {
769         return basis.return_type_tinfo();
770 }
771
772 ex power::expand(unsigned options) const
773 {
774         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
775                 // A special case worth optimizing.
776                 setflag(status_flags::expanded);
777                 return *this;
778         }
779
780         const ex expanded_basis = basis.expand(options);
781         const ex expanded_exponent = exponent.expand(options);
782         
783         // x^(a+b) -> x^a * x^b
784         if (is_exactly_a<add>(expanded_exponent)) {
785                 const add &a = ex_to<add>(expanded_exponent);
786                 exvector distrseq;
787                 distrseq.reserve(a.seq.size() + 1);
788                 epvector::const_iterator last = a.seq.end();
789                 epvector::const_iterator cit = a.seq.begin();
790                 while (cit!=last) {
791                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
792                         ++cit;
793                 }
794                 
795                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
796                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
797                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
798                         int int_exponent = num_exponent.to_int();
799                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
800                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
801                         else
802                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
803                 } else
804                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
805                 
806                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
807                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
808                 return r.expand(options);
809         }
810         
811         if (!is_exactly_a<numeric>(expanded_exponent) ||
812                 !ex_to<numeric>(expanded_exponent).is_integer()) {
813                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
814                         return this->hold();
815                 } else {
816                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
817                 }
818         }
819         
820         // integer numeric exponent
821         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
822         int int_exponent = num_exponent.to_int();
823         
824         // (x+y)^n, n>0
825         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
826                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
827         
828         // (x*y)^n -> x^n * y^n
829         if (is_exactly_a<mul>(expanded_basis))
830                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
831         
832         // cannot expand further
833         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
834                 return this->hold();
835         else
836                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
837 }
838
839 //////////
840 // new virtual functions which can be overridden by derived classes
841 //////////
842
843 // none
844
845 //////////
846 // non-virtual functions in this class
847 //////////
848
849 /** expand a^n where a is an add and n is a positive integer.
850  *  @see power::expand */
851 ex power::expand_add(const add & a, int n, unsigned options) const
852 {
853         if (n==2)
854                 return expand_add_2(a, options);
855
856         const size_t m = a.nops();
857         exvector result;
858         // The number of terms will be the number of combinatorial compositions,
859         // i.e. the number of unordered arrangements of m nonnegative integers
860         // which sum up to n.  It is frequently written as C_n(m) and directly
861         // related with binomial coefficients:
862         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
863         intvector k(m-1);
864         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
865         intvector upper_limit(m-1);
866
867         for (size_t l=0; l<m-1; ++l) {
868                 k[l] = 0;
869                 k_cum[l] = 0;
870                 upper_limit[l] = n;
871         }
872
873         while (true) {
874                 exvector term;
875                 term.reserve(m+1);
876                 for (std::size_t l = 0; l < m - 1; ++l) {
877                         const ex & b = a.op(l);
878                         GINAC_ASSERT(!is_exactly_a<add>(b));
879                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
880                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
881                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
882                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
883                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
884                                      !is_exactly_a<power>(ex_to<power>(b).basis));
885                         if (is_exactly_a<mul>(b))
886                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
887                         else
888                                 term.push_back(power(b,k[l]));
889                 }
890
891                 const ex & b = a.op(m - 1);
892                 GINAC_ASSERT(!is_exactly_a<add>(b));
893                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
894                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
895                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
896                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
897                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
898                              !is_exactly_a<power>(ex_to<power>(b).basis));
899                 if (is_exactly_a<mul>(b))
900                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
901                 else
902                         term.push_back(power(b,n-k_cum[m-2]));
903
904                 numeric f = binomial(numeric(n),numeric(k[0]));
905                 for (std::size_t l = 1; l < m - 1; ++l)
906                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
907
908                 term.push_back(f);
909
910                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
911
912                 // increment k[]
913                 bool done = false;
914                 std::size_t l = m - 2;
915                 while ((++k[l]) > upper_limit[l]) {
916                         k[l] = 0;
917                         if (l != 0)
918                                 --l;
919                         else {
920                                 done = true;
921                                 break;
922                         }
923                 }
924                 if (done)
925                         break;
926
927                 // recalc k_cum[] and upper_limit[]
928                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
929
930                 for (size_t i=l+1; i<m-1; ++i)
931                         k_cum[i] = k_cum[i-1]+k[i];
932
933                 for (size_t i=l+1; i<m-1; ++i)
934                         upper_limit[i] = n-k_cum[i-1];
935         }
936
937         return (new add(result))->setflag(status_flags::dynallocated |
938                                           status_flags::expanded);
939 }
940
941
942 /** Special case of power::expand_add. Expands a^2 where a is an add.
943  *  @see power::expand_add */
944 ex power::expand_add_2(const add & a, unsigned options) const
945 {
946         epvector sum;
947         size_t a_nops = a.nops();
948         sum.reserve((a_nops*(a_nops+1))/2);
949         epvector::const_iterator last = a.seq.end();
950
951         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
952         // first part: ignore overall_coeff and expand other terms
953         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
954                 const ex & r = cit0->rest;
955                 const ex & c = cit0->coeff;
956                 
957                 GINAC_ASSERT(!is_exactly_a<add>(r));
958                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
959                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
960                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
961                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
962                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
963                              !is_exactly_a<power>(ex_to<power>(r).basis));
964                 
965                 if (c.is_equal(_ex1)) {
966                         if (is_exactly_a<mul>(r)) {
967                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
968                                                      _ex1));
969                         } else {
970                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
971                                                      _ex1));
972                         }
973                 } else {
974                         if (is_exactly_a<mul>(r)) {
975                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
976                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
977                         } else {
978                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
979                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
980                         }
981                 }
982
983                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
984                         const ex & r1 = cit1->rest;
985                         const ex & c1 = cit1->coeff;
986                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
987                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
988                 }
989         }
990         
991         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
992         
993         // second part: add terms coming from overall_factor (if != 0)
994         if (!a.overall_coeff.is_zero()) {
995                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
996                 while (i != end) {
997                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
998                         ++i;
999                 }
1000                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1001         }
1002         
1003         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1004         
1005         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1006 }
1007
1008 /** Expand factors of m in m^n where m is a mul and n is an integer.
1009  *  @see power::expand */
1010 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1011 {
1012         GINAC_ASSERT(n.is_integer());
1013
1014         if (n.is_zero()) {
1015                 return _ex1;
1016         }
1017
1018         // do not bother to rename indices if there are no any.
1019         if ((!(options & expand_options::expand_rename_idx)) 
1020                         && m.info(info_flags::has_indices))
1021                 options |= expand_options::expand_rename_idx;
1022         // Leave it to multiplication since dummy indices have to be renamed
1023         if ((options & expand_options::expand_rename_idx) &&
1024                 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1025                 ex result = m;
1026                 exvector va = get_all_dummy_indices(m);
1027                 sort(va.begin(), va.end(), ex_is_less());
1028
1029                 for (int i=1; i < n.to_int(); i++)
1030                         result *= rename_dummy_indices_uniquely(va, m);
1031                 return result;
1032         }
1033
1034         epvector distrseq;
1035         distrseq.reserve(m.seq.size());
1036         bool need_reexpand = false;
1037
1038         epvector::const_iterator last = m.seq.end();
1039         epvector::const_iterator cit = m.seq.begin();
1040         while (cit!=last) {
1041                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1042                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1043                         // this happens when e.g. (a+b)^(1/2) gets squared and
1044                         // the resulting product needs to be reexpanded
1045                         need_reexpand = true;
1046                 }
1047                 distrseq.push_back(p);
1048                 ++cit;
1049         }
1050
1051         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1052         if (need_reexpand)
1053                 return ex(result).expand(options);
1054         if (from_expand)
1055                 return result.setflag(status_flags::expanded);
1056         return result;
1057 }
1058
1059 GINAC_BIND_UNARCHIVER(power);
1060
1061 } // namespace GiNaC