ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2010 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::positive:
 244                         return basis.info(info_flags::positive) && exponent.info(info_flags::real);
 245                 case info_flags::has_indices: {
 246                         if (flags & status_flags::has_indices)
 247                                 return true;
 248                         else if (flags & status_flags::has_no_indices)
 249                                 return false;
 250                         else if (basis.info(info_flags::has_indices)) {
 251                                 setflag(status_flags::has_indices);
 252                                 clearflag(status_flags::has_no_indices);
 253                                 return true;
 254                         } else {
 255                                 clearflag(status_flags::has_indices);
 256                                 setflag(status_flags::has_no_indices);
 257                                 return false;
 258                         }
 259                 }
 260         }
 261         return inherited::info(inf);
 262 }
 263
 264 size_t power::nops() const
 265 {
 266         return 2;
 267 }
 268
 269 ex power::op(size_t i) const
 270 {
 271         GINAC_ASSERT(i<2);
 272
 273         return i==0 ? basis : exponent;
 274 }
 275
 276 ex power::map(map_function & f) const
 277 {
 278         const ex &mapped_basis = f(basis);
 279         const ex &mapped_exponent = f(exponent);
 280
 281         if (!are_ex_trivially_equal(basis, mapped_basis)
 282          || !are_ex_trivially_equal(exponent, mapped_exponent))
 283                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
 284         else
 285                 return *this;
 286 }
 287
 288 bool power::is_polynomial(const ex & var) const
 289 {
 290         if (exponent.has(var))
 291                 return false;
 292         if (!exponent.info(info_flags::nonnegint))
 293                 return false;
 294         return basis.is_polynomial(var);
 295 }
 296
 297 int power::degree(const ex & s) const
 298 {
 299         if (is_equal(ex_to<basic>(s)))
 300                 return 1;
 301         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 302                 if (basis.is_equal(s))
 303                         return ex_to<numeric>(exponent).to_int();
 304                 else
 305                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 306         } else if (basis.has(s))
 307                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 308         else
 309                 return 0;
 310 }
 311
 312 int power::ldegree(const ex & s) const
 313 {
 314         if (is_equal(ex_to<basic>(s)))
 315                 return 1;
 316         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 317                 if (basis.is_equal(s))
 318                         return ex_to<numeric>(exponent).to_int();
 319                 else
 320                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 321         } else if (basis.has(s))
 322                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 323         else
 324                 return 0;
 325 }
 326
 327 ex power::coeff(const ex & s, int n) const
 328 {
 329         if (is_equal(ex_to<basic>(s)))
 330                 return n==1 ? _ex1 : _ex0;
 331         else if (!basis.is_equal(s)) {
 332                 // basis not equal to s
 333                 if (n == 0)
 334                         return *this;
 335                 else
 336                         return _ex0;
 337         } else {
 338                 // basis equal to s
 339                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 340                         // integer exponent
 341                         int int_exp = ex_to<numeric>(exponent).to_int();
 342                         if (n == int_exp)
 343                                 return _ex1;
 344                         else
 345                                 return _ex0;
 346                 } else {
 347                         // non-integer exponents are treated as zero
 348                         if (n == 0)
 349                                 return *this;
 350                         else
 351                                 return _ex0;
 352                 }
 353         }
 354 }
 355
 356 /** Perform automatic term rewriting rules in this class.  In the following
 357  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 358  *  stand for such expressions that contain a plain number.
 359  *  - ^(x,0) -> 1  (also handles ^(0,0))
 360  *  - ^(x,1) -> x
 361  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 362  *  - ^(1,x) -> 1
 363  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 364  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
 365  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 366  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 367  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 368  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 369  *
 370  *  @param level cut-off in recursive evaluation */
 371 ex power::eval(int level) const
 372 {
 373         if ((level==1) && (flags & status_flags::evaluated))
 374                 return *this;
 375         else if (level == -max_recursion_level)
 376                 throw(std::runtime_error("max recursion level reached"));
 377
 378         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 379         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 380
 381         const numeric *num_basis = NULL;
 382         const numeric *num_exponent = NULL;
 383
 384         if (is_exactly_a<numeric>(ebasis)) {
 385                 num_basis = &ex_to<numeric>(ebasis);
 386         }
 387         if (is_exactly_a<numeric>(eexponent)) {
 388                 num_exponent = &ex_to<numeric>(eexponent);
 389         }
 390
 391         // ^(x,0) -> 1  (0^0 also handled here)
 392         if (eexponent.is_zero()) {
 393                 if (ebasis.is_zero())
 394                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 395                 else
 396                         return _ex1;
 397         }
 398
 399         // ^(x,1) -> x
 400         if (eexponent.is_equal(_ex1))
 401                 return ebasis;
 402
 403         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 404         if ( ebasis.is_zero() && num_exponent ) {
 405                 if ((num_exponent->real()).is_zero())
 406                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 407                 else if ((num_exponent->real()).is_negative())
 408                         throw (pole_error("power::eval(): division by zero",1));
 409                 else
 410                         return _ex0;
 411         }
 412
 413         // ^(1,x) -> 1
 414         if (ebasis.is_equal(_ex1))
 415                 return _ex1;
 416
 417         // power of a function calculated by separate rules defined for this function
 418         if (is_exactly_a<function>(ebasis))
 419                 return ex_to<function>(ebasis).power(eexponent);
 420
 421         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
 422         if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
 423                 return power(ebasis.op(0), ebasis.op(1) * eexponent);
 424
 425         if ( num_exponent ) {
 426
 427                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 428                 // except if c1,c2 are rational, but c1^c2 is not)
 429                 if ( num_basis ) {
 430                         const bool basis_is_crational = num_basis->is_crational();
 431                         const bool exponent_is_crational = num_exponent->is_crational();
 432                         if (!basis_is_crational || !exponent_is_crational) {
 433                                 // return a plain float
 434                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 435                                                                                                status_flags::evaluated |
 436                                                                                                status_flags::expanded);
 437                         }
 438
 439                         const numeric res = num_basis->power(*num_exponent);
 440                         if (res.is_crational()) {
 441                                 return res;
 442                         }
 443                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 444
 445                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 446                         if (basis_is_crational && exponent_is_crational
 447                             && num_exponent->is_real()
 448                             && !num_exponent->is_integer()) {
 449                                 const numeric n = num_exponent->numer();
 450                                 const numeric m = num_exponent->denom();
 451                                 numeric r;
 452                                 numeric q = iquo(n, m, r);
 453                                 if (r.is_negative()) {
 454                                         r += m;
 455                                         --q;
 456                                 }
 457                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 458                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 459                                                 // try it for numerator and denominator separately, in order to
 460                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 461                                                 const numeric bnum = num_basis->numer();
 462                                                 const numeric bden = num_basis->denom();
 463                                                 const numeric res_bnum = bnum.power(*num_exponent);
 464                                                 const numeric res_bden = bden.power(*num_exponent);
 465                                                 if (res_bnum.is_integer())
 466                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 467                                                 if (res_bden.is_integer())
 468                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 469                                         }
 470                                         return this->hold();
 471                                 } else {
 472                                         // assemble resulting product, but allowing for a re-evaluation,
 473                                         // because otherwise we'll end up with something like
 474                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 475                                         // instead of 7/16*7^(1/3).
 476                                         ex prod = power(*num_basis,r.div(m));
 477                                         return prod*power(*num_basis,q);
 478                                 }
 479                         }
 480                 }
 481
 482                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 483                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 484                 // case c1==1 should not happen, see below!)
 485                 if (is_exactly_a<power>(ebasis)) {
 486                         const power & sub_power = ex_to<power>(ebasis);
 487                         const ex & sub_basis = sub_power.basis;
 488                         const ex & sub_exponent = sub_power.exponent;
 489                         if (is_exactly_a<numeric>(sub_exponent)) {
 490                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 491                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 492                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative()) {
 493                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 494                                 }
 495                         }
 496                 }
 497
 498                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 499                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 500                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 501                 }
 502
 503                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
 504                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
 505                         numeric icont = ebasis.integer_content();
 506                         const numeric lead_coeff =
 507                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
 508
 509                         const bool canonicalizable = lead_coeff.is_integer();
 510                         const bool unit_normal = lead_coeff.is_pos_integer();
 511                         if (canonicalizable && (! unit_normal))
 512                                 icont = icont.mul(*_num_1_p);
 513
 514                         if (canonicalizable && (icont != *_num1_p)) {
 515                                 const add& addref = ex_to<add>(ebasis);
 516                                 add* addp = new add(addref);
 517                                 addp->setflag(status_flags::dynallocated);
 518                                 addp->clearflag(status_flags::hash_calculated);
 519                                 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
 520                                 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
 521                                         i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
 522
 523                                 const numeric c = icont.power(*num_exponent);
 524                                 if (likely(c != *_num1_p))
 525                                         return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
 526                                 else
 527                                         return power(*addp, *num_exponent);
 528                         }
 529                 }
 530
 531                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 532                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 533                 if (is_exactly_a<mul>(ebasis)) {
 534                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 535                         const mul & mulref = ex_to<mul>(ebasis);
 536                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 537                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 538                                 if (num_coeff.is_real()) {
 539                                         if (num_coeff.is_positive()) {
 540                                                 mul *mulp = new mul(mulref);
 541                                                 mulp->overall_coeff = _ex1;
 542                                                 mulp->clearflag(status_flags::evaluated);
 543                                                 mulp->clearflag(status_flags::hash_calculated);
 544                                                 return (new mul(power(*mulp,exponent),
 545                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 546                                         } else {
 547                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 548                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 549                                                         mul *mulp = new mul(mulref);
 550                                                         mulp->overall_coeff = _ex_1;
 551                                                         mulp->clearflag(status_flags::evaluated);
 552                                                         mulp->clearflag(status_flags::hash_calculated);
 553                                                         return (new mul(power(*mulp,exponent),
 554                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 555                                                 }
 556                                         }
 557                                 }
 558                         }
 559                 }
 560
 561                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 562                 if (num_exponent->is_pos_integer() &&
 563                     ebasis.return_type() != return_types::commutative &&
 564                     !is_a<matrix>(ebasis)) {
 565                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 566                 }
 567         }
 568
 569         if (are_ex_trivially_equal(ebasis,basis) &&
 570             are_ex_trivially_equal(eexponent,exponent)) {
 571                 return this->hold();
 572         }
 573         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 574                                                        status_flags::evaluated);
 575 }
 576
 577 ex power::evalf(int level) const
 578 {
 579         ex ebasis;
 580         ex eexponent;
 581
 582         if (level==1) {
 583                 ebasis = basis;
 584                 eexponent = exponent;
 585         } else if (level == -max_recursion_level) {
 586                 throw(std::runtime_error("max recursion level reached"));
 587         } else {
 588                 ebasis = basis.evalf(level-1);
 589                 if (!is_exactly_a<numeric>(exponent))
 590                         eexponent = exponent.evalf(level-1);
 591                 else
 592                         eexponent = exponent;
 593         }
 594
 595         return power(ebasis,eexponent);
 596 }
 597
 598 ex power::evalm() const
 599 {
 600         const ex ebasis = basis.evalm();
 601         const ex eexponent = exponent.evalm();
 602         if (is_a<matrix>(ebasis)) {
 603                 if (is_exactly_a<numeric>(eexponent)) {
 604                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 605                 }
 606         }
 607         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 608 }
 609
 610 bool power::has(const ex & other, unsigned options) const
 611 {
 612         if (!(options & has_options::algebraic))
 613                 return basic::has(other, options);
 614         if (!is_a<power>(other))
 615                 return basic::has(other, options);
 616         if (!exponent.info(info_flags::integer)
 617                         || !other.op(1).info(info_flags::integer))
 618                 return basic::has(other, options);
 619         if (exponent.info(info_flags::posint)
 620                         && other.op(1).info(info_flags::posint)
 621                         && ex_to<numeric>(exponent).to_int()
 622                                         > ex_to<numeric>(other.op(1)).to_int()
 623                         && basis.match(other.op(0)))
 624                 return true;
 625         if (exponent.info(info_flags::negint)
 626                         && other.op(1).info(info_flags::negint)
 627                         && ex_to<numeric>(exponent).to_int()
 628                                         < ex_to<numeric>(other.op(1)).to_int()
 629                         && basis.match(other.op(0)))
 630                 return true;
 631         return basic::has(other, options);
 632 }
 633
 634 // from mul.cpp
 635 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
 636
 637 ex power::subs(const exmap & m, unsigned options) const
 638 {
 639         const ex &subsed_basis = basis.subs(m, options);
 640         const ex &subsed_exponent = exponent.subs(m, options);
 641
 642         if (!are_ex_trivially_equal(basis, subsed_basis)
 643          || !are_ex_trivially_equal(exponent, subsed_exponent))
 644                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 645
 646         if (!(options & subs_options::algebraic))
 647                 return subs_one_level(m, options);
 648
 649         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 650                 int nummatches = std::numeric_limits<int>::max();
 651                 exmap repls;
 652                 if (tryfactsubs(*this, it->first, nummatches, repls)) {
 653                         ex anum = it->second.subs(repls, subs_options::no_pattern);
 654                         ex aden = it->first.subs(repls, subs_options::no_pattern);
 655                         ex result = (*this)*power(anum/aden, nummatches);
 656                         return (ex_to<basic>(result)).subs_one_level(m, options);
 657                 }
 658         }
 659
 660         return subs_one_level(m, options);
 661 }
 662
 663 ex power::eval_ncmul(const exvector & v) const
 664 {
 665         return inherited::eval_ncmul(v);
 666 }
 667
 668 ex power::conjugate() const
 669 {
 670         // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
 671         // branch cut which runs along the negative real axis.
 672         if (basis.info(info_flags::positive)) {
 673                 ex newexponent = exponent.conjugate();
 674                 if (are_ex_trivially_equal(exponent, newexponent)) {
 675                         return *this;
 676                 }
 677                 return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
 678         }
 679         if (exponent.info(info_flags::integer)) {
 680                 ex newbasis = basis.conjugate();
 681                 if (are_ex_trivially_equal(basis, newbasis)) {
 682                         return *this;
 683                 }
 684                 return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
 685         }
 686         return conjugate_function(*this).hold();
 687 }
 688
 689 ex power::real_part() const
 690 {
 691         if (exponent.info(info_flags::integer)) {
 692                 ex basis_real = basis.real_part();
 693                 if (basis_real == basis)
 694                         return *this;
 695                 realsymbol a("a"),b("b");
 696                 ex result;
 697                 if (exponent.info(info_flags::posint))
 698                         result = power(a+I*b,exponent);
 699                 else
 700                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 701                 result = result.expand();
 702                 result = result.real_part();
 703                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 704                 return result;
 705         }
 706
 707         ex a = basis.real_part();
 708         ex b = basis.imag_part();
 709         ex c = exponent.real_part();
 710         ex d = exponent.imag_part();
 711         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 712 }
 713
 714 ex power::imag_part() const
 715 {
 716         if (exponent.info(info_flags::integer)) {
 717                 ex basis_real = basis.real_part();
 718                 if (basis_real == basis)
 719                         return 0;
 720                 realsymbol a("a"),b("b");
 721                 ex result;
 722                 if (exponent.info(info_flags::posint))
 723                         result = power(a+I*b,exponent);
 724                 else
 725                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 726                 result = result.expand();
 727                 result = result.imag_part();
 728                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 729                 return result;
 730         }
 731
 732         ex a=basis.real_part();
 733         ex b=basis.imag_part();
 734         ex c=exponent.real_part();
 735         ex d=exponent.imag_part();
 736         return
 737                 power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 738 }
 739
 740 // protected
 741
 742 // protected
 743
 744 /** Implementation of ex::diff() for a power.
 745  *  @see ex::diff */
 746 ex power::derivative(const symbol & s) const
 747 {
 748         if (is_a<numeric>(exponent)) {
 749                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 750                 epvector newseq;
 751                 newseq.reserve(2);
 752                 newseq.push_back(expair(basis, exponent - _ex1));
 753                 newseq.push_back(expair(basis.diff(s), _ex1));
 754                 return mul(newseq, exponent);
 755         } else {
 756                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 757                 return mul(*this,
 758                            add(mul(exponent.diff(s), log(basis)),
 759                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 760         }
 761 }
 762
 763 int power::compare_same_type(const basic & other) const
 764 {
 765         GINAC_ASSERT(is_exactly_a<power>(other));
 766         const power &o = static_cast<const power &>(other);
 767
 768         int cmpval = basis.compare(o.basis);
 769         if (cmpval)
 770                 return cmpval;
 771         else
 772                 return exponent.compare(o.exponent);
 773 }
 774
 775 unsigned power::return_type() const
 776 {
 777         return basis.return_type();
 778 }
 779
 780 return_type_t power::return_type_tinfo() const
 781 {
 782         return basis.return_type_tinfo();
 783 }
 784
 785 ex power::expand(unsigned options) const
 786 {
 787         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
 788                 // A special case worth optimizing.
 789                 setflag(status_flags::expanded);
 790                 return *this;
 791         }
 792
 793         const ex expanded_basis = basis.expand(options);
 794         const ex expanded_exponent = exponent.expand(options);
 795
 796         // x^(a+b) -> x^a * x^b
 797         if (is_exactly_a<add>(expanded_exponent)) {
 798                 const add &a = ex_to<add>(expanded_exponent);
 799                 exvector distrseq;
 800                 distrseq.reserve(a.seq.size() + 1);
 801                 epvector::const_iterator last = a.seq.end();
 802                 epvector::const_iterator cit = a.seq.begin();
 803                 while (cit!=last) {
 804                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 805                         ++cit;
 806                 }
 807
 808                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 809                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 810                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 811                         int int_exponent = num_exponent.to_int();
 812                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 813                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 814                         else
 815                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 816                 } else
 817                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 818
 819                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 820                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 821                 return r.expand(options);
 822         }
 823
 824         if (!is_exactly_a<numeric>(expanded_exponent) ||
 825                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 826                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 827                         return this->hold();
 828                 } else {
 829                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 830                 }
 831         }
 832
 833         // integer numeric exponent
 834         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 835         int int_exponent = num_exponent.to_int();
 836
 837         // (x+y)^n, n>0
 838         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 839                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 840
 841         // (x*y)^n -> x^n * y^n
 842         if (is_exactly_a<mul>(expanded_basis))
 843                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 844
 845         // cannot expand further
 846         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 847                 return this->hold();
 848         else
 849                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 850 }
 851
 852 //////////
 853 // new virtual functions which can be overridden by derived classes
 854 //////////
 855
 856 // none
 857
 858 //////////
 859 // non-virtual functions in this class
 860 //////////
 861
 862 /** expand a^n where a is an add and n is a positive integer.
 863  *  @see power::expand */
 864 ex power::expand_add(const add & a, int n, unsigned options) const
 865 {
 866         if (n==2)
 867                 return expand_add_2(a, options);
 868
 869         const size_t m = a.nops();
 870         exvector result;
 871         // The number of terms will be the number of combinatorial compositions,
 872         // i.e. the number of unordered arrangements of m nonnegative integers
 873         // which sum up to n.  It is frequently written as C_n(m) and directly
 874         // related with binomial coefficients:
 875         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 876         intvector k(m-1);
 877         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 878         intvector upper_limit(m-1);
 879
 880         for (size_t l=0; l<m-1; ++l) {
 881                 k[l] = 0;
 882                 k_cum[l] = 0;
 883                 upper_limit[l] = n;
 884         }
 885
 886         while (true) {
 887                 exvector term;
 888                 term.reserve(m+1);
 889                 for (std::size_t l = 0; l < m - 1; ++l) {
 890                         const ex & b = a.op(l);
 891                         GINAC_ASSERT(!is_exactly_a<add>(b));
 892                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 893                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 894                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 895                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 896                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 897                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 898                         if (is_exactly_a<mul>(b))
 899                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 900                         else
 901                                 term.push_back(power(b,k[l]));
 902                 }
 903
 904                 const ex & b = a.op(m - 1);
 905                 GINAC_ASSERT(!is_exactly_a<add>(b));
 906                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 907                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 908                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 909                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 910                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 911                              !is_exactly_a<power>(ex_to<power>(b).basis));
 912                 if (is_exactly_a<mul>(b))
 913                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 914                 else
 915                         term.push_back(power(b,n-k_cum[m-2]));
 916
 917                 numeric f = binomial(numeric(n),numeric(k[0]));
 918                 for (std::size_t l = 1; l < m - 1; ++l)
 919                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 920
 921                 term.push_back(f);
 922
 923                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 924
 925                 // increment k[]
 926                 bool done = false;
 927                 std::size_t l = m - 2;
 928                 while ((++k[l]) > upper_limit[l]) {
 929                         k[l] = 0;
 930                         if (l != 0)
 931                                 --l;
 932                         else {
 933                                 done = true;
 934                                 break;
 935                         }
 936                 }
 937                 if (done)
 938                         break;
 939
 940                 // recalc k_cum[] and upper_limit[]
 941                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 942
 943                 for (size_t i=l+1; i<m-1; ++i)
 944                         k_cum[i] = k_cum[i-1]+k[i];
 945
 946                 for (size_t i=l+1; i<m-1; ++i)
 947                         upper_limit[i] = n-k_cum[i-1];
 948         }
 949
 950         return (new add(result))->setflag(status_flags::dynallocated |
 951                                           status_flags::expanded);
 952 }
 953
 954
 955 /** Special case of power::expand_add. Expands a^2 where a is an add.
 956  *  @see power::expand_add */
 957 ex power::expand_add_2(const add & a, unsigned options) const
 958 {
 959         epvector sum;
 960         size_t a_nops = a.nops();
 961         sum.reserve((a_nops*(a_nops+1))/2);
 962         epvector::const_iterator last = a.seq.end();
 963
 964         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 965         // first part: ignore overall_coeff and expand other terms
 966         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 967                 const ex & r = cit0->rest;
 968                 const ex & c = cit0->coeff;
 969
 970                 GINAC_ASSERT(!is_exactly_a<add>(r));
 971                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 972                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 973                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 974                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 975                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 976                              !is_exactly_a<power>(ex_to<power>(r).basis));
 977
 978                 if (c.is_equal(_ex1)) {
 979                         if (is_exactly_a<mul>(r)) {
 980                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 981                                                      _ex1));
 982                         } else {
 983                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 984                                                      _ex1));
 985                         }
 986                 } else {
 987                         if (is_exactly_a<mul>(r)) {
 988                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 989                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 990                         } else {
 991                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 992                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 993                         }
 994                 }
 995
 996                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 997                         const ex & r1 = cit1->rest;
 998                         const ex & c1 = cit1->coeff;
 999                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
1000                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1001                 }
1002         }
1003
1004         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
1005
1006         // second part: add terms coming from overall_factor (if != 0)
1007         if (!a.overall_coeff.is_zero()) {
1008                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1009                 while (i != end) {
1010                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1011                         ++i;
1012                 }
1013                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1014         }
1015
1016         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1017
1018         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1019 }
1020
1021 /** Expand factors of m in m^n where m is a mul and n is an integer.
1022  *  @see power::expand */
1023 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1024 {
1025         GINAC_ASSERT(n.is_integer());
1026
1027         if (n.is_zero()) {
1028                 return _ex1;
1029         }
1030
1031         // do not bother to rename indices if there are no any.
1032         if ((!(options & expand_options::expand_rename_idx))
1033                         && m.info(info_flags::has_indices))
1034                 options |= expand_options::expand_rename_idx;
1035         // Leave it to multiplication since dummy indices have to be renamed
1036         if ((options & expand_options::expand_rename_idx) &&
1037                 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1038                 ex result = m;
1039                 exvector va = get_all_dummy_indices(m);
1040                 sort(va.begin(), va.end(), ex_is_less());
1041
1042                 for (int i=1; i < n.to_int(); i++)
1043                         result *= rename_dummy_indices_uniquely(va, m);
1044                 return result;
1045         }
1046
1047         epvector distrseq;
1048         distrseq.reserve(m.seq.size());
1049         bool need_reexpand = false;
1050
1051         epvector::const_iterator last = m.seq.end();
1052         epvector::const_iterator cit = m.seq.begin();
1053         while (cit!=last) {
1054                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1055                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1056                         // this happens when e.g. (a+b)^(1/2) gets squared and
1057                         // the resulting product needs to be reexpanded
1058                         need_reexpand = true;
1059                 }
1060                 distrseq.push_back(p);
1061                 ++cit;
1062         }
1063
1064         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1065         if (need_reexpand)
1066                 return ex(result).expand(options);
1067         if (from_expand)
1068                 return result.setflag(status_flags::expanded);
1069         return result;
1070 }
1071
1072 GINAC_BIND_UNARCHIVER(power);
1073
1074 } // namespace GiNaC