ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2011 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 or (c1=-1 and c2>0), 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 or (c1=-1 and c2>0),
 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                                                 || (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
 494                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 495                                 }
 496                         }
 497                 }
 498
 499                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 500                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 501                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 502                 }
 503
 504                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
 505                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
 506                         numeric icont = ebasis.integer_content();
 507                         const numeric lead_coeff =
 508                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
 509
 510                         const bool canonicalizable = lead_coeff.is_integer();
 511                         const bool unit_normal = lead_coeff.is_pos_integer();
 512                         if (canonicalizable && (! unit_normal))
 513                                 icont = icont.mul(*_num_1_p);
 514
 515                         if (canonicalizable && (icont != *_num1_p)) {
 516                                 const add& addref = ex_to<add>(ebasis);
 517                                 add* addp = new add(addref);
 518                                 addp->setflag(status_flags::dynallocated);
 519                                 addp->clearflag(status_flags::hash_calculated);
 520                                 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
 521                                 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
 522                                         i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
 523
 524                                 const numeric c = icont.power(*num_exponent);
 525                                 if (likely(c != *_num1_p))
 526                                         return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
 527                                 else
 528                                         return power(*addp, *num_exponent);
 529                         }
 530                 }
 531
 532                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 533                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 534                 if (is_exactly_a<mul>(ebasis)) {
 535                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 536                         const mul & mulref = ex_to<mul>(ebasis);
 537                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 538                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 539                                 if (num_coeff.is_real()) {
 540                                         if (num_coeff.is_positive()) {
 541                                                 mul *mulp = new mul(mulref);
 542                                                 mulp->overall_coeff = _ex1;
 543                                                 mulp->setflag(status_flags::dynallocated);
 544                                                 mulp->clearflag(status_flags::evaluated);
 545                                                 mulp->clearflag(status_flags::hash_calculated);
 546                                                 return (new mul(power(*mulp,exponent),
 547                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 548                                         } else {
 549                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 550                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 551                                                         mul *mulp = new mul(mulref);
 552                                                         mulp->overall_coeff = _ex_1;
 553                                                         mulp->setflag(status_flags::dynallocated);
 554                                                         mulp->clearflag(status_flags::evaluated);
 555                                                         mulp->clearflag(status_flags::hash_calculated);
 556                                                         return (new mul(power(*mulp,exponent),
 557                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 558                                                 }
 559                                         }
 560                                 }
 561                         }
 562                 }
 563
 564                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 565                 if (num_exponent->is_pos_integer() &&
 566                     ebasis.return_type() != return_types::commutative &&
 567                     !is_a<matrix>(ebasis)) {
 568                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 569                 }
 570         }
 571
 572         if (are_ex_trivially_equal(ebasis,basis) &&
 573             are_ex_trivially_equal(eexponent,exponent)) {
 574                 return this->hold();
 575         }
 576         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 577                                                        status_flags::evaluated);
 578 }
 579
 580 ex power::evalf(int level) const
 581 {
 582         ex ebasis;
 583         ex eexponent;
 584
 585         if (level==1) {
 586                 ebasis = basis;
 587                 eexponent = exponent;
 588         } else if (level == -max_recursion_level) {
 589                 throw(std::runtime_error("max recursion level reached"));
 590         } else {
 591                 ebasis = basis.evalf(level-1);
 592                 if (!is_exactly_a<numeric>(exponent))
 593                         eexponent = exponent.evalf(level-1);
 594                 else
 595                         eexponent = exponent;
 596         }
 597
 598         return power(ebasis,eexponent);
 599 }
 600
 601 ex power::evalm() const
 602 {
 603         const ex ebasis = basis.evalm();
 604         const ex eexponent = exponent.evalm();
 605         if (is_a<matrix>(ebasis)) {
 606                 if (is_exactly_a<numeric>(eexponent)) {
 607                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 608                 }
 609         }
 610         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 611 }
 612
 613 bool power::has(const ex & other, unsigned options) const
 614 {
 615         if (!(options & has_options::algebraic))
 616                 return basic::has(other, options);
 617         if (!is_a<power>(other))
 618                 return basic::has(other, options);
 619         if (!exponent.info(info_flags::integer)
 620                         || !other.op(1).info(info_flags::integer))
 621                 return basic::has(other, options);
 622         if (exponent.info(info_flags::posint)
 623                         && other.op(1).info(info_flags::posint)
 624                         && ex_to<numeric>(exponent).to_int()
 625                                         > ex_to<numeric>(other.op(1)).to_int()
 626                         && basis.match(other.op(0)))
 627                 return true;
 628         if (exponent.info(info_flags::negint)
 629                         && other.op(1).info(info_flags::negint)
 630                         && ex_to<numeric>(exponent).to_int()
 631                                         < ex_to<numeric>(other.op(1)).to_int()
 632                         && basis.match(other.op(0)))
 633                 return true;
 634         return basic::has(other, options);
 635 }
 636
 637 // from mul.cpp
 638 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
 639
 640 ex power::subs(const exmap & m, unsigned options) const
 641 {
 642         const ex &subsed_basis = basis.subs(m, options);
 643         const ex &subsed_exponent = exponent.subs(m, options);
 644
 645         if (!are_ex_trivially_equal(basis, subsed_basis)
 646          || !are_ex_trivially_equal(exponent, subsed_exponent))
 647                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 648
 649         if (!(options & subs_options::algebraic))
 650                 return subs_one_level(m, options);
 651
 652         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 653                 int nummatches = std::numeric_limits<int>::max();
 654                 exmap repls;
 655                 if (tryfactsubs(*this, it->first, nummatches, repls)) {
 656                         ex anum = it->second.subs(repls, subs_options::no_pattern);
 657                         ex aden = it->first.subs(repls, subs_options::no_pattern);
 658                         ex result = (*this)*power(anum/aden, nummatches);
 659                         return (ex_to<basic>(result)).subs_one_level(m, options);
 660                 }
 661         }
 662
 663         return subs_one_level(m, options);
 664 }
 665
 666 ex power::eval_ncmul(const exvector & v) const
 667 {
 668         return inherited::eval_ncmul(v);
 669 }
 670
 671 ex power::conjugate() const
 672 {
 673         // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
 674         // branch cut which runs along the negative real axis.
 675         if (basis.info(info_flags::positive)) {
 676                 ex newexponent = exponent.conjugate();
 677                 if (are_ex_trivially_equal(exponent, newexponent)) {
 678                         return *this;
 679                 }
 680                 return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
 681         }
 682         if (exponent.info(info_flags::integer)) {
 683                 ex newbasis = basis.conjugate();
 684                 if (are_ex_trivially_equal(basis, newbasis)) {
 685                         return *this;
 686                 }
 687                 return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
 688         }
 689         return conjugate_function(*this).hold();
 690 }
 691
 692 ex power::real_part() const
 693 {
 694         if (exponent.info(info_flags::integer)) {
 695                 ex basis_real = basis.real_part();
 696                 if (basis_real == basis)
 697                         return *this;
 698                 realsymbol a("a"),b("b");
 699                 ex result;
 700                 if (exponent.info(info_flags::posint))
 701                         result = power(a+I*b,exponent);
 702                 else
 703                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 704                 result = result.expand();
 705                 result = result.real_part();
 706                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 707                 return result;
 708         }
 709
 710         ex a = basis.real_part();
 711         ex b = basis.imag_part();
 712         ex c = exponent.real_part();
 713         ex d = exponent.imag_part();
 714         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 715 }
 716
 717 ex power::imag_part() const
 718 {
 719         if (exponent.info(info_flags::integer)) {
 720                 ex basis_real = basis.real_part();
 721                 if (basis_real == basis)
 722                         return 0;
 723                 realsymbol a("a"),b("b");
 724                 ex result;
 725                 if (exponent.info(info_flags::posint))
 726                         result = power(a+I*b,exponent);
 727                 else
 728                         result = power(a/(a*a+b*b)-I*b/(a*a+b*b),-exponent);
 729                 result = result.expand();
 730                 result = result.imag_part();
 731                 result = result.subs(lst( a==basis_real, b==basis.imag_part() ));
 732                 return result;
 733         }
 734
 735         ex a=basis.real_part();
 736         ex b=basis.imag_part();
 737         ex c=exponent.real_part();
 738         ex d=exponent.imag_part();
 739         return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 740 }
 741
 742 // protected
 743
 744 // protected
 745
 746 /** Implementation of ex::diff() for a power.
 747  *  @see ex::diff */
 748 ex power::derivative(const symbol & s) const
 749 {
 750         if (is_a<numeric>(exponent)) {
 751                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 752                 epvector newseq;
 753                 newseq.reserve(2);
 754                 newseq.push_back(expair(basis, exponent - _ex1));
 755                 newseq.push_back(expair(basis.diff(s), _ex1));
 756                 return mul(newseq, exponent);
 757         } else {
 758                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 759                 return mul(*this,
 760                            add(mul(exponent.diff(s), log(basis)),
 761                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 762         }
 763 }
 764
 765 int power::compare_same_type(const basic & other) const
 766 {
 767         GINAC_ASSERT(is_exactly_a<power>(other));
 768         const power &o = static_cast<const power &>(other);
 769
 770         int cmpval = basis.compare(o.basis);
 771         if (cmpval)
 772                 return cmpval;
 773         else
 774                 return exponent.compare(o.exponent);
 775 }
 776
 777 unsigned power::return_type() const
 778 {
 779         return basis.return_type();
 780 }
 781
 782 return_type_t power::return_type_tinfo() const
 783 {
 784         return basis.return_type_tinfo();
 785 }
 786
 787 ex power::expand(unsigned options) const
 788 {
 789         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
 790                 // A special case worth optimizing.
 791                 setflag(status_flags::expanded);
 792                 return *this;
 793         }
 794
 795         const ex expanded_basis = basis.expand(options);
 796         const ex expanded_exponent = exponent.expand(options);
 797
 798         // x^(a+b) -> x^a * x^b
 799         if (is_exactly_a<add>(expanded_exponent)) {
 800                 const add &a = ex_to<add>(expanded_exponent);
 801                 exvector distrseq;
 802                 distrseq.reserve(a.seq.size() + 1);
 803                 epvector::const_iterator last = a.seq.end();
 804                 epvector::const_iterator cit = a.seq.begin();
 805                 while (cit!=last) {
 806                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 807                         ++cit;
 808                 }
 809
 810                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 811                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 812                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 813                         int int_exponent = num_exponent.to_int();
 814                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 815                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 816                         else
 817                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 818                 } else
 819                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 820
 821                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 822                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 823                 return r.expand(options);
 824         }
 825
 826         if (!is_exactly_a<numeric>(expanded_exponent) ||
 827                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 828                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 829                         return this->hold();
 830                 } else {
 831                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 832                 }
 833         }
 834
 835         // integer numeric exponent
 836         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 837         int int_exponent = num_exponent.to_int();
 838
 839         // (x+y)^n, n>0
 840         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 841                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 842
 843         // (x*y)^n -> x^n * y^n
 844         if (is_exactly_a<mul>(expanded_basis))
 845                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 846
 847         // cannot expand further
 848         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 849                 return this->hold();
 850         else
 851                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 852 }
 853
 854 //////////
 855 // new virtual functions which can be overridden by derived classes
 856 //////////
 857
 858 // none
 859
 860 //////////
 861 // non-virtual functions in this class
 862 //////////
 863
 864 /** expand a^n where a is an add and n is a positive integer.
 865  *  @see power::expand */
 866 ex power::expand_add(const add & a, int n, unsigned options) const
 867 {
 868         if (n==2)
 869                 return expand_add_2(a, options);
 870
 871         const size_t m = a.nops();
 872         exvector result;
 873         // The number of terms will be the number of combinatorial compositions,
 874         // i.e. the number of unordered arrangements of m nonnegative integers
 875         // which sum up to n.  It is frequently written as C_n(m) and directly
 876         // related with binomial coefficients:
 877         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 878         intvector k(m-1);
 879         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 880         intvector upper_limit(m-1);
 881
 882         for (size_t l=0; l<m-1; ++l) {
 883                 k[l] = 0;
 884                 k_cum[l] = 0;
 885                 upper_limit[l] = n;
 886         }
 887
 888         while (true) {
 889                 exvector term;
 890                 term.reserve(m+1);
 891                 for (std::size_t l = 0; l < m - 1; ++l) {
 892                         const ex & b = a.op(l);
 893                         GINAC_ASSERT(!is_exactly_a<add>(b));
 894                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 895                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 896                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 897                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 898                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 899                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 900                         if (is_exactly_a<mul>(b))
 901                                 term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
 902                         else
 903                                 term.push_back(power(b,k[l]));
 904                 }
 905
 906                 const ex & b = a.op(m - 1);
 907                 GINAC_ASSERT(!is_exactly_a<add>(b));
 908                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 909                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 910                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 911                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 912                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 913                              !is_exactly_a<power>(ex_to<power>(b).basis));
 914                 if (is_exactly_a<mul>(b))
 915                         term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
 916                 else
 917                         term.push_back(power(b,n-k_cum[m-2]));
 918
 919                 numeric f = binomial(numeric(n),numeric(k[0]));
 920                 for (std::size_t l = 1; l < m - 1; ++l)
 921                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 922
 923                 term.push_back(f);
 924
 925                 result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
 926
 927                 // increment k[]
 928                 bool done = false;
 929                 std::size_t l = m - 2;
 930                 while ((++k[l]) > upper_limit[l]) {
 931                         k[l] = 0;
 932                         if (l != 0)
 933                                 --l;
 934                         else {
 935                                 done = true;
 936                                 break;
 937                         }
 938                 }
 939                 if (done)
 940                         break;
 941
 942                 // recalc k_cum[] and upper_limit[]
 943                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 944
 945                 for (size_t i=l+1; i<m-1; ++i)
 946                         k_cum[i] = k_cum[i-1]+k[i];
 947
 948                 for (size_t i=l+1; i<m-1; ++i)
 949                         upper_limit[i] = n-k_cum[i-1];
 950         }
 951
 952         return (new add(result))->setflag(status_flags::dynallocated |
 953                                           status_flags::expanded);
 954 }
 955
 956
 957 /** Special case of power::expand_add. Expands a^2 where a is an add.
 958  *  @see power::expand_add */
 959 ex power::expand_add_2(const add & a, unsigned options) const
 960 {
 961         epvector sum;
 962         size_t a_nops = a.nops();
 963         sum.reserve((a_nops*(a_nops+1))/2);
 964         epvector::const_iterator last = a.seq.end();
 965
 966         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 967         // first part: ignore overall_coeff and expand other terms
 968         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 969                 const ex & r = cit0->rest;
 970                 const ex & c = cit0->coeff;
 971
 972                 GINAC_ASSERT(!is_exactly_a<add>(r));
 973                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 974                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 975                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 976                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 977                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 978                              !is_exactly_a<power>(ex_to<power>(r).basis));
 979
 980                 if (c.is_equal(_ex1)) {
 981                         if (is_exactly_a<mul>(r)) {
 982                                 sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 983                                                      _ex1));
 984                         } else {
 985                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 986                                                      _ex1));
 987                         }
 988                 } else {
 989                         if (is_exactly_a<mul>(r)) {
 990                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
 991                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 992                         } else {
 993                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 994                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
 995                         }
 996                 }
 997
 998                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 999                         const ex & r1 = cit1->rest;
1000                         const ex & c1 = cit1->coeff;
1001                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
1002                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1003                 }
1004         }
1005
1006         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
1007
1008         // second part: add terms coming from overall_factor (if != 0)
1009         if (!a.overall_coeff.is_zero()) {
1010                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1011                 while (i != end) {
1012                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1013                         ++i;
1014                 }
1015                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1016         }
1017
1018         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1019
1020         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1021 }
1022
1023 /** Expand factors of m in m^n where m is a mul and n is an integer.
1024  *  @see power::expand */
1025 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1026 {
1027         GINAC_ASSERT(n.is_integer());
1028
1029         if (n.is_zero()) {
1030                 return _ex1;
1031         }
1032
1033         // do not bother to rename indices if there are no any.
1034         if ((!(options & expand_options::expand_rename_idx))
1035                         && m.info(info_flags::has_indices))
1036                 options |= expand_options::expand_rename_idx;
1037         // Leave it to multiplication since dummy indices have to be renamed
1038         if ((options & expand_options::expand_rename_idx) &&
1039                 (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1040                 ex result = m;
1041                 exvector va = get_all_dummy_indices(m);
1042                 sort(va.begin(), va.end(), ex_is_less());
1043
1044                 for (int i=1; i < n.to_int(); i++)
1045                         result *= rename_dummy_indices_uniquely(va, m);
1046                 return result;
1047         }
1048
1049         epvector distrseq;
1050         distrseq.reserve(m.seq.size());
1051         bool need_reexpand = false;
1052
1053         epvector::const_iterator last = m.seq.end();
1054         epvector::const_iterator cit = m.seq.begin();
1055         while (cit!=last) {
1056                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1057                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1058                         // this happens when e.g. (a+b)^(1/2) gets squared and
1059                         // the resulting product needs to be reexpanded
1060                         need_reexpand = true;
1061                 }
1062                 distrseq.push_back(p);
1063                 ++cit;
1064         }
1065
1066         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1067         if (need_reexpand)
1068                 return ex(result).expand(options);
1069         if (from_expand)
1070                 return result.setflag(status_flags::expanded);
1071         return result;
1072 }
1073
1074 GINAC_BIND_UNARCHIVER(power);
1075
1076 } // namespace GiNaC