ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2015 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 //////////
  57 // default constructor
  58 //////////
  59
  60 power::power() { }
  61
  62 //////////
  63 // other constructors
  64 //////////
  65
  66 // all inlined
  67
  68 //////////
  69 // archiving
  70 //////////
  71
  72 void power::read_archive(const archive_node &n, lst &sym_lst)
  73 {
  74         inherited::read_archive(n, sym_lst);
  75         n.find_ex("basis", basis, sym_lst);
  76         n.find_ex("exponent", exponent, sym_lst);
  77 }
  78
  79 void power::archive(archive_node &n) const
  80 {
  81         inherited::archive(n);
  82         n.add_ex("basis", basis);
  83         n.add_ex("exponent", exponent);
  84 }
  85
  86 //////////
  87 // functions overriding virtual functions from base classes
  88 //////////
  89
  90 // public
  91
  92 void power::print_power(const print_context & c, const char *powersymbol, const char *openbrace, const char *closebrace, unsigned level) const
  93 {
  94         // Ordinary output of powers using '^' or '**'
  95         if (precedence() <= level)
  96                 c.s << openbrace << '(';
  97         basis.print(c, precedence());
  98         c.s << powersymbol;
  99         c.s << openbrace;
 100         exponent.print(c, precedence());
 101         c.s << closebrace;
 102         if (precedence() <= level)
 103                 c.s << ')' << closebrace;
 104 }
 105
 106 void power::do_print_dflt(const print_dflt & c, unsigned level) const
 107 {
 108         if (exponent.is_equal(_ex1_2)) {
 109
 110                 // Square roots are printed in a special way
 111                 c.s << "sqrt(";
 112                 basis.print(c);
 113                 c.s << ')';
 114
 115         } else
 116                 print_power(c, "^", "", "", level);
 117 }
 118
 119 void power::do_print_latex(const print_latex & c, unsigned level) const
 120 {
 121         if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
 122
 123                 // Powers with negative numeric exponents are printed as fractions
 124                 c.s << "\\frac{1}{";
 125                 power(basis, -exponent).eval().print(c);
 126                 c.s << '}';
 127
 128         } else if (exponent.is_equal(_ex1_2)) {
 129
 130                 // Square roots are printed in a special way
 131                 c.s << "\\sqrt{";
 132                 basis.print(c);
 133                 c.s << '}';
 134
 135         } else
 136                 print_power(c, "^", "{", "}", level);
 137 }
 138
 139 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 140 {
 141         // Optimal output of integer powers of symbols to aid compiler CSE.
 142         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 143         // to learn why such a parenthesation is really necessary.
 144         if (exp == 1) {
 145                 x.print(c);
 146         } else if (exp == 2) {
 147                 x.print(c);
 148                 c.s << "*";
 149                 x.print(c);
 150         } else if (exp & 1) {
 151                 x.print(c);
 152                 c.s << "*";
 153                 print_sym_pow(c, x, exp-1);
 154         } else {
 155                 c.s << "(";
 156                 print_sym_pow(c, x, exp >> 1);
 157                 c.s << ")*(";
 158                 print_sym_pow(c, x, exp >> 1);
 159                 c.s << ")";
 160         }
 161 }
 162
 163 void power::do_print_csrc_cl_N(const print_csrc_cl_N& c, unsigned level) const
 164 {
 165         if (exponent.is_equal(_ex_1)) {
 166                 c.s << "recip(";
 167                 basis.print(c);
 168                 c.s << ')';
 169                 return;
 170         }
 171         c.s << "expt(";
 172         basis.print(c);
 173         c.s << ", ";
 174         exponent.print(c);
 175         c.s << ')';
 176 }
 177
 178 void power::do_print_csrc(const print_csrc & c, unsigned level) const
 179 {
 180         // Integer powers of symbols are printed in a special, optimized way
 181         if (exponent.info(info_flags::integer) &&
 182             (is_a<symbol>(basis) || is_a<constant>(basis))) {
 183                 int exp = ex_to<numeric>(exponent).to_int();
 184                 if (exp > 0)
 185                         c.s << '(';
 186                 else {
 187                         exp = -exp;
 188                         c.s << "1.0/(";
 189                 }
 190                 print_sym_pow(c, ex_to<symbol>(basis), exp);
 191                 c.s << ')';
 192
 193         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 194         } else if (exponent.is_equal(_ex_1)) {
 195                 c.s << "1.0/(";
 196                 basis.print(c);
 197                 c.s << ')';
 198
 199         // Otherwise, use the pow() function
 200         } else {
 201                 c.s << "pow(";
 202                 basis.print(c);
 203                 c.s << ',';
 204                 exponent.print(c);
 205                 c.s << ')';
 206         }
 207 }
 208
 209 void power::do_print_python(const print_python & c, unsigned level) const
 210 {
 211         print_power(c, "**", "", "", level);
 212 }
 213
 214 void power::do_print_python_repr(const print_python_repr & c, unsigned level) const
 215 {
 216         c.s << class_name() << '(';
 217         basis.print(c);
 218         c.s << ',';
 219         exponent.print(c);
 220         c.s << ')';
 221 }
 222
 223 bool power::info(unsigned inf) const
 224 {
 225         switch (inf) {
 226                 case info_flags::polynomial:
 227                 case info_flags::integer_polynomial:
 228                 case info_flags::cinteger_polynomial:
 229                 case info_flags::rational_polynomial:
 230                 case info_flags::crational_polynomial:
 231                         return exponent.info(info_flags::nonnegint) &&
 232                                basis.info(inf);
 233                 case info_flags::rational_function:
 234                         return exponent.info(info_flags::integer) &&
 235                                basis.info(inf);
 236                 case info_flags::algebraic:
 237                         return !exponent.info(info_flags::integer) ||
 238                                basis.info(inf);
 239                 case info_flags::expanded:
 240                         return (flags & status_flags::expanded);
 241                 case info_flags::positive:
 242                         return basis.info(info_flags::positive) && exponent.info(info_flags::real);
 243                 case info_flags::nonnegative:
 244                         return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
 245                                (basis.info(info_flags::real) && exponent.info(info_flags::integer) && exponent.info(info_flags::even));
 246                 case info_flags::has_indices: {
 247                         if (flags & status_flags::has_indices)
 248                                 return true;
 249                         else if (flags & status_flags::has_no_indices)
 250                                 return false;
 251                         else if (basis.info(info_flags::has_indices)) {
 252                                 setflag(status_flags::has_indices);
 253                                 clearflag(status_flags::has_no_indices);
 254                                 return true;
 255                         } else {
 256                                 clearflag(status_flags::has_indices);
 257                                 setflag(status_flags::has_no_indices);
 258                                 return false;
 259                         }
 260                 }
 261         }
 262         return inherited::info(inf);
 263 }
 264
 265 size_t power::nops() const
 266 {
 267         return 2;
 268 }
 269
 270 ex power::op(size_t i) const
 271 {
 272         GINAC_ASSERT(i<2);
 273
 274         return i==0 ? basis : exponent;
 275 }
 276
 277 ex power::map(map_function & f) const
 278 {
 279         const ex &mapped_basis = f(basis);
 280         const ex &mapped_exponent = f(exponent);
 281
 282         if (!are_ex_trivially_equal(basis, mapped_basis)
 283          || !are_ex_trivially_equal(exponent, mapped_exponent))
 284                 return (new power(mapped_basis, mapped_exponent))->setflag(status_flags::dynallocated);
 285         else
 286                 return *this;
 287 }
 288
 289 bool power::is_polynomial(const ex & var) const
 290 {
 291         if (basis.is_polynomial(var)) {
 292                 if (basis.has(var))
 293                         // basis is non-constant polynomial in var
 294                         return exponent.info(info_flags::nonnegint);
 295                 else
 296                         // basis is constant in var
 297                         return !exponent.has(var);
 298         }
 299         // basis is a non-polynomial function of var
 300         return false;
 301 }
 302
 303 int power::degree(const ex & s) const
 304 {
 305         if (is_equal(ex_to<basic>(s)))
 306                 return 1;
 307         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 308                 if (basis.is_equal(s))
 309                         return ex_to<numeric>(exponent).to_int();
 310                 else
 311                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 312         } else if (basis.has(s))
 313                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 314         else
 315                 return 0;
 316 }
 317
 318 int power::ldegree(const ex & s) const
 319 {
 320         if (is_equal(ex_to<basic>(s)))
 321                 return 1;
 322         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 323                 if (basis.is_equal(s))
 324                         return ex_to<numeric>(exponent).to_int();
 325                 else
 326                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 327         } else if (basis.has(s))
 328                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 329         else
 330                 return 0;
 331 }
 332
 333 ex power::coeff(const ex & s, int n) const
 334 {
 335         if (is_equal(ex_to<basic>(s)))
 336                 return n==1 ? _ex1 : _ex0;
 337         else if (!basis.is_equal(s)) {
 338                 // basis not equal to s
 339                 if (n == 0)
 340                         return *this;
 341                 else
 342                         return _ex0;
 343         } else {
 344                 // basis equal to s
 345                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 346                         // integer exponent
 347                         int int_exp = ex_to<numeric>(exponent).to_int();
 348                         if (n == int_exp)
 349                                 return _ex1;
 350                         else
 351                                 return _ex0;
 352                 } else {
 353                         // non-integer exponents are treated as zero
 354                         if (n == 0)
 355                                 return *this;
 356                         else
 357                                 return _ex0;
 358                 }
 359         }
 360 }
 361
 362 /** Perform automatic term rewriting rules in this class.  In the following
 363  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 364  *  stand for such expressions that contain a plain number.
 365  *  - ^(x,0) -> 1  (also handles ^(0,0))
 366  *  - ^(x,1) -> x
 367  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 368  *  - ^(1,x) -> 1
 369  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 370  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  if x is positive and c1 is real.
 371  *  - ^(^(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!)
 372  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 373  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 374  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 375  *
 376  *  @param level cut-off in recursive evaluation */
 377 ex power::eval(int level) const
 378 {
 379         if ((level==1) && (flags & status_flags::evaluated))
 380                 return *this;
 381         else if (level == -max_recursion_level)
 382                 throw(std::runtime_error("max recursion level reached"));
 383
 384         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 385         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 386
 387         const numeric *num_basis = NULL;
 388         const numeric *num_exponent = NULL;
 389
 390         if (is_exactly_a<numeric>(ebasis)) {
 391                 num_basis = &ex_to<numeric>(ebasis);
 392         }
 393         if (is_exactly_a<numeric>(eexponent)) {
 394                 num_exponent = &ex_to<numeric>(eexponent);
 395         }
 396
 397         // ^(x,0) -> 1  (0^0 also handled here)
 398         if (eexponent.is_zero()) {
 399                 if (ebasis.is_zero())
 400                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 401                 else
 402                         return _ex1;
 403         }
 404
 405         // ^(x,1) -> x
 406         if (eexponent.is_equal(_ex1))
 407                 return ebasis;
 408
 409         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 410         if ( ebasis.is_zero() && num_exponent ) {
 411                 if ((num_exponent->real()).is_zero())
 412                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 413                 else if ((num_exponent->real()).is_negative())
 414                         throw (pole_error("power::eval(): division by zero",1));
 415                 else
 416                         return _ex0;
 417         }
 418
 419         // ^(1,x) -> 1
 420         if (ebasis.is_equal(_ex1))
 421                 return _ex1;
 422
 423         // power of a function calculated by separate rules defined for this function
 424         if (is_exactly_a<function>(ebasis))
 425                 return ex_to<function>(ebasis).power(eexponent);
 426
 427         // Turn (x^c)^d into x^(c*d) in the case that x is positive and c is real.
 428         if (is_exactly_a<power>(ebasis) && ebasis.op(0).info(info_flags::positive) && ebasis.op(1).info(info_flags::real))
 429                 return power(ebasis.op(0), ebasis.op(1) * eexponent);
 430
 431         if ( num_exponent ) {
 432
 433                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 434                 // except if c1,c2 are rational, but c1^c2 is not)
 435                 if ( num_basis ) {
 436                         const bool basis_is_crational = num_basis->is_crational();
 437                         const bool exponent_is_crational = num_exponent->is_crational();
 438                         if (!basis_is_crational || !exponent_is_crational) {
 439                                 // return a plain float
 440                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 441                                                                                                status_flags::evaluated |
 442                                                                                                status_flags::expanded);
 443                         }
 444
 445                         const numeric res = num_basis->power(*num_exponent);
 446                         if (res.is_crational()) {
 447                                 return res;
 448                         }
 449                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 450
 451                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 452                         if (basis_is_crational && exponent_is_crational
 453                             && num_exponent->is_real()
 454                             && !num_exponent->is_integer()) {
 455                                 const numeric n = num_exponent->numer();
 456                                 const numeric m = num_exponent->denom();
 457                                 numeric r;
 458                                 numeric q = iquo(n, m, r);
 459                                 if (r.is_negative()) {
 460                                         r += m;
 461                                         --q;
 462                                 }
 463                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 464                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 465                                                 // try it for numerator and denominator separately, in order to
 466                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 467                                                 const numeric bnum = num_basis->numer();
 468                                                 const numeric bden = num_basis->denom();
 469                                                 const numeric res_bnum = bnum.power(*num_exponent);
 470                                                 const numeric res_bden = bden.power(*num_exponent);
 471                                                 if (res_bnum.is_integer())
 472                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 473                                                 if (res_bden.is_integer())
 474                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 475                                         }
 476                                         return this->hold();
 477                                 } else {
 478                                         // assemble resulting product, but allowing for a re-evaluation,
 479                                         // because otherwise we'll end up with something like
 480                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 481                                         // instead of 7/16*7^(1/3).
 482                                         ex prod = power(*num_basis,r.div(m));
 483                                         return prod*power(*num_basis,q);
 484                                 }
 485                         }
 486                 }
 487
 488                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 489                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1 or (c1=-1 and c2>0),
 490                 // case c1==1 should not happen, see below!)
 491                 if (is_exactly_a<power>(ebasis)) {
 492                         const power & sub_power = ex_to<power>(ebasis);
 493                         const ex & sub_basis = sub_power.basis;
 494                         const ex & sub_exponent = sub_power.exponent;
 495                         if (is_exactly_a<numeric>(sub_exponent)) {
 496                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 497                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 498                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - (*_num1_p)).is_negative() ||
 499                                     (num_sub_exponent == *_num_1_p && num_exponent->is_positive())) {
 500                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 501                                 }
 502                         }
 503                 }
 504
 505                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 506                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 507                         return expand_mul(ex_to<mul>(ebasis), *num_exponent, 0);
 508                 }
 509
 510                 // (2*x + 6*y)^(-4) -> 1/16*(x + 3*y)^(-4)
 511                 if (num_exponent->is_integer() && is_exactly_a<add>(ebasis)) {
 512                         numeric icont = ebasis.integer_content();
 513                         const numeric lead_coeff =
 514                                 ex_to<numeric>(ex_to<add>(ebasis).seq.begin()->coeff).div(icont);
 515
 516                         const bool canonicalizable = lead_coeff.is_integer();
 517                         const bool unit_normal = lead_coeff.is_pos_integer();
 518                         if (canonicalizable && (! unit_normal))
 519                                 icont = icont.mul(*_num_1_p);
 520
 521                         if (canonicalizable && (icont != *_num1_p)) {
 522                                 const add& addref = ex_to<add>(ebasis);
 523                                 add* addp = new add(addref);
 524                                 addp->setflag(status_flags::dynallocated);
 525                                 addp->clearflag(status_flags::hash_calculated);
 526                                 addp->overall_coeff = ex_to<numeric>(addp->overall_coeff).div_dyn(icont);
 527                                 for (epvector::iterator i = addp->seq.begin(); i != addp->seq.end(); ++i)
 528                                         i->coeff = ex_to<numeric>(i->coeff).div_dyn(icont);
 529
 530                                 const numeric c = icont.power(*num_exponent);
 531                                 if (likely(c != *_num1_p))
 532                                         return (new mul(power(*addp, *num_exponent), c))->setflag(status_flags::dynallocated);
 533                                 else
 534                                         return power(*addp, *num_exponent);
 535                         }
 536                 }
 537
 538                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 539                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 540                 if (is_exactly_a<mul>(ebasis)) {
 541                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 542                         const mul & mulref = ex_to<mul>(ebasis);
 543                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 544                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 545                                 if (num_coeff.is_real()) {
 546                                         if (num_coeff.is_positive()) {
 547                                                 mul *mulp = new mul(mulref);
 548                                                 mulp->overall_coeff = _ex1;
 549                                                 mulp->setflag(status_flags::dynallocated);
 550                                                 mulp->clearflag(status_flags::evaluated);
 551                                                 mulp->clearflag(status_flags::hash_calculated);
 552                                                 return (new mul(power(*mulp,exponent),
 553                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 554                                         } else {
 555                                                 GINAC_ASSERT(num_coeff.compare(*_num0_p)<0);
 556                                                 if (!num_coeff.is_equal(*_num_1_p)) {
 557                                                         mul *mulp = new mul(mulref);
 558                                                         mulp->overall_coeff = _ex_1;
 559                                                         mulp->setflag(status_flags::dynallocated);
 560                                                         mulp->clearflag(status_flags::evaluated);
 561                                                         mulp->clearflag(status_flags::hash_calculated);
 562                                                         return (new mul(power(*mulp,exponent),
 563                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 564                                                 }
 565                                         }
 566                                 }
 567                         }
 568                 }
 569
 570                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 571                 if (num_exponent->is_pos_integer() &&
 572                     ebasis.return_type() != return_types::commutative &&
 573                     !is_a<matrix>(ebasis)) {
 574                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 575                 }
 576         }
 577
 578         if (are_ex_trivially_equal(ebasis,basis) &&
 579             are_ex_trivially_equal(eexponent,exponent)) {
 580                 return this->hold();
 581         }
 582         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 583                                                        status_flags::evaluated);
 584 }
 585
 586 ex power::evalf(int level) const
 587 {
 588         ex ebasis;
 589         ex eexponent;
 590
 591         if (level==1) {
 592                 ebasis = basis;
 593                 eexponent = exponent;
 594         } else if (level == -max_recursion_level) {
 595                 throw(std::runtime_error("max recursion level reached"));
 596         } else {
 597                 ebasis = basis.evalf(level-1);
 598                 if (!is_exactly_a<numeric>(exponent))
 599                         eexponent = exponent.evalf(level-1);
 600                 else
 601                         eexponent = exponent;
 602         }
 603
 604         return power(ebasis,eexponent);
 605 }
 606
 607 ex power::evalm() const
 608 {
 609         const ex ebasis = basis.evalm();
 610         const ex eexponent = exponent.evalm();
 611         if (is_a<matrix>(ebasis)) {
 612                 if (is_exactly_a<numeric>(eexponent)) {
 613                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 614                 }
 615         }
 616         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 617 }
 618
 619 bool power::has(const ex & other, unsigned options) const
 620 {
 621         if (!(options & has_options::algebraic))
 622                 return basic::has(other, options);
 623         if (!is_a<power>(other))
 624                 return basic::has(other, options);
 625         if (!exponent.info(info_flags::integer) ||
 626             !other.op(1).info(info_flags::integer))
 627                 return basic::has(other, options);
 628         if (exponent.info(info_flags::posint) &&
 629             other.op(1).info(info_flags::posint) &&
 630             ex_to<numeric>(exponent) > ex_to<numeric>(other.op(1)) &&
 631             basis.match(other.op(0)))
 632                 return true;
 633         if (exponent.info(info_flags::negint) &&
 634             other.op(1).info(info_flags::negint) &&
 635             ex_to<numeric>(exponent) < ex_to<numeric>(other.op(1)) &&
 636             basis.match(other.op(0)))
 637                 return true;
 638         return basic::has(other, options);
 639 }
 640
 641 // from mul.cpp
 642 extern bool tryfactsubs(const ex &, const ex &, int &, exmap&);
 643
 644 ex power::subs(const exmap & m, unsigned options) const
 645 {
 646         const ex &subsed_basis = basis.subs(m, options);
 647         const ex &subsed_exponent = exponent.subs(m, options);
 648
 649         if (!are_ex_trivially_equal(basis, subsed_basis)
 650          || !are_ex_trivially_equal(exponent, subsed_exponent))
 651                 return power(subsed_basis, subsed_exponent).subs_one_level(m, options);
 652
 653         if (!(options & subs_options::algebraic))
 654                 return subs_one_level(m, options);
 655
 656         for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) {
 657                 int nummatches = std::numeric_limits<int>::max();
 658                 exmap repls;
 659                 if (tryfactsubs(*this, it->first, nummatches, repls)) {
 660                         ex anum = it->second.subs(repls, subs_options::no_pattern);
 661                         ex aden = it->first.subs(repls, subs_options::no_pattern);
 662                         ex result = (*this)*power(anum/aden, nummatches);
 663                         return (ex_to<basic>(result)).subs_one_level(m, options);
 664                 }
 665         }
 666
 667         return subs_one_level(m, options);
 668 }
 669
 670 ex power::eval_ncmul(const exvector & v) const
 671 {
 672         return inherited::eval_ncmul(v);
 673 }
 674
 675 ex power::conjugate() const
 676 {
 677         // conjugate(pow(x,y))==pow(conjugate(x),conjugate(y)) unless on the
 678         // branch cut which runs along the negative real axis.
 679         if (basis.info(info_flags::positive)) {
 680                 ex newexponent = exponent.conjugate();
 681                 if (are_ex_trivially_equal(exponent, newexponent)) {
 682                         return *this;
 683                 }
 684                 return (new power(basis, newexponent))->setflag(status_flags::dynallocated);
 685         }
 686         if (exponent.info(info_flags::integer)) {
 687                 ex newbasis = basis.conjugate();
 688                 if (are_ex_trivially_equal(basis, newbasis)) {
 689                         return *this;
 690                 }
 691                 return (new power(newbasis, exponent))->setflag(status_flags::dynallocated);
 692         }
 693         return conjugate_function(*this).hold();
 694 }
 695
 696 ex power::real_part() const
 697 {
 698         // basis == a+I*b, exponent == c+I*d
 699         const ex a = basis.real_part();
 700         const ex c = exponent.real_part();
 701         if (basis.is_equal(a) && exponent.is_equal(c)) {
 702                 // Re(a^c)
 703                 return *this;
 704         }
 705
 706         const ex b = basis.imag_part();
 707         if (exponent.info(info_flags::integer)) {
 708                 // Re((a+I*b)^c)  w/  c ∈ ℤ
 709                 long N = ex_to<numeric>(c).to_long();
 710                 // Use real terms in Binomial expansion to construct
 711                 // Re(expand(power(a+I*b, N))).
 712                 long NN = N > 0 ? N : -N;
 713                 ex numer = N > 0 ? _ex1 : power(power(a,2) + power(b,2), NN);
 714                 ex result = 0;
 715                 for (long n = 0; n <= NN; n += 2) {
 716                         ex term = binomial(NN, n) * power(a, NN-n) * power(b, n) / numer;
 717                         if (n % 4 == 0) {
 718                                 result += term;  // sign: I^n w/ n == 4*m
 719                         } else {
 720                                 result -= term;  // sign: I^n w/ n == 4*m+2
 721                         }
 722                 }
 723                 return result;
 724         }
 725
 726         // Re((a+I*b)^(c+I*d))
 727         const ex d = exponent.imag_part();
 728         return power(abs(basis),c)*exp(-d*atan2(b,a))*cos(c*atan2(b,a)+d*log(abs(basis)));
 729 }
 730
 731 ex power::imag_part() const
 732 {
 733         const ex a = basis.real_part();
 734         const ex c = exponent.real_part();
 735         if (basis.is_equal(a) && exponent.is_equal(c)) {
 736                 // Im(a^c)
 737                 return 0;
 738         }
 739
 740         const ex b = basis.imag_part();
 741         if (exponent.info(info_flags::integer)) {
 742                 // Im((a+I*b)^c)  w/  c ∈ ℤ
 743                 long N = ex_to<numeric>(c).to_long();
 744                 // Use imaginary terms in Binomial expansion to construct
 745                 // Im(expand(power(a+I*b, N))).
 746                 long p = N > 0 ? 1 : 3;  // modulus for positive sign
 747                 long NN = N > 0 ? N : -N;
 748                 ex numer = N > 0 ? _ex1 : power(power(a,2) + power(b,2), NN);
 749                 ex result = 0;
 750                 for (long n = 1; n <= NN; n += 2) {
 751                         ex term = binomial(NN, n) * power(a, NN-n) * power(b, n) / numer;
 752                         if (n % 4 == p) {
 753                                 result += term;  // sign: I^n w/ n == 4*m+p
 754                         } else {
 755                                 result -= term;  // sign: I^n w/ n == 4*m+2+p
 756                         }
 757                 }
 758                 return result;
 759         }
 760
 761         // Im((a+I*b)^(c+I*d))
 762         const ex d = exponent.imag_part();
 763         return power(abs(basis),c)*exp(-d*atan2(b,a))*sin(c*atan2(b,a)+d*log(abs(basis)));
 764 }
 765
 766 // protected
 767
 768 /** Implementation of ex::diff() for a power.
 769  *  @see ex::diff */
 770 ex power::derivative(const symbol & s) const
 771 {
 772         if (is_a<numeric>(exponent)) {
 773                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 774                 epvector newseq;
 775                 newseq.reserve(2);
 776                 newseq.push_back(expair(basis, exponent - _ex1));
 777                 newseq.push_back(expair(basis.diff(s), _ex1));
 778                 return mul(newseq, exponent);
 779         } else {
 780                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 781                 return mul(*this,
 782                            add(mul(exponent.diff(s), log(basis)),
 783                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 784         }
 785 }
 786
 787 int power::compare_same_type(const basic & other) const
 788 {
 789         GINAC_ASSERT(is_exactly_a<power>(other));
 790         const power &o = static_cast<const power &>(other);
 791
 792         int cmpval = basis.compare(o.basis);
 793         if (cmpval)
 794                 return cmpval;
 795         else
 796                 return exponent.compare(o.exponent);
 797 }
 798
 799 unsigned power::return_type() const
 800 {
 801         return basis.return_type();
 802 }
 803
 804 return_type_t power::return_type_tinfo() const
 805 {
 806         return basis.return_type_tinfo();
 807 }
 808
 809 ex power::expand(unsigned options) const
 810 {
 811         if (is_a<symbol>(basis) && exponent.info(info_flags::integer)) {
 812                 // A special case worth optimizing.
 813                 setflag(status_flags::expanded);
 814                 return *this;
 815         }
 816
 817         // (x*p)^c -> x^c * p^c, if p>0
 818         // makes sense before expanding the basis
 819         if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
 820                 const mul &m = ex_to<mul>(basis);
 821                 exvector prodseq;
 822                 epvector powseq;
 823                 prodseq.reserve(m.seq.size() + 1);
 824                 powseq.reserve(m.seq.size() + 1);
 825                 epvector::const_iterator last = m.seq.end();
 826                 epvector::const_iterator cit = m.seq.begin();
 827                 bool possign = true;
 828
 829                 // search for positive/negative factors
 830                 while (cit!=last) {
 831                         ex e=m.recombine_pair_to_ex(*cit);
 832                         if (e.info(info_flags::positive))
 833                                 prodseq.push_back(pow(e, exponent).expand(options));
 834                         else if (e.info(info_flags::negative)) {
 835                                 prodseq.push_back(pow(-e, exponent).expand(options));
 836                                 possign = !possign;
 837                         } else
 838                                 powseq.push_back(*cit);
 839                         ++cit;
 840                 }
 841
 842                 // take care on the numeric coefficient
 843                 ex coeff=(possign? _ex1 : _ex_1);
 844                 if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
 845                         prodseq.push_back(power(m.overall_coeff, exponent));
 846                 else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
 847                         prodseq.push_back(power(-m.overall_coeff, exponent));
 848                 else
 849                         coeff *= m.overall_coeff;
 850
 851                 // If positive/negative factors are found, then extract them.
 852                 // In either case we set a flag to avoid the second run on a part
 853                 // which does not have positive/negative terms.
 854                 if (prodseq.size() > 0) {
 855                         ex newbasis = coeff*mul(powseq);
 856                         ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
 857                         return ((new mul(prodseq))->setflag(status_flags::dynallocated)*(new power(newbasis, exponent))->setflag(status_flags::dynallocated).expand(options)).expand(options);
 858                 } else
 859                         ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
 860         }
 861
 862         const ex expanded_basis = basis.expand(options);
 863         const ex expanded_exponent = exponent.expand(options);
 864
 865         // x^(a+b) -> x^a * x^b
 866         if (is_exactly_a<add>(expanded_exponent)) {
 867                 const add &a = ex_to<add>(expanded_exponent);
 868                 exvector distrseq;
 869                 distrseq.reserve(a.seq.size() + 1);
 870                 epvector::const_iterator last = a.seq.end();
 871                 epvector::const_iterator cit = a.seq.begin();
 872                 while (cit!=last) {
 873                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 874                         ++cit;
 875                 }
 876
 877                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 878                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 879                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 880                         int int_exponent = num_exponent.to_int();
 881                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 882                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent, options));
 883                         else
 884                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 885                 } else
 886                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 887
 888                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 889                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 890                 return r.expand(options);
 891         }
 892
 893         if (!is_exactly_a<numeric>(expanded_exponent) ||
 894                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 895                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 896                         return this->hold();
 897                 } else {
 898                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 899                 }
 900         }
 901
 902         // integer numeric exponent
 903         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 904         int int_exponent = num_exponent.to_int();
 905
 906         // (x+y)^n, n>0
 907         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 908                 return expand_add(ex_to<add>(expanded_basis), int_exponent, options);
 909
 910         // (x*y)^n -> x^n * y^n
 911         if (is_exactly_a<mul>(expanded_basis))
 912                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent, options, true);
 913
 914         // cannot expand further
 915         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 916                 return this->hold();
 917         else
 918                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 919 }
 920
 921 //////////
 922 // new virtual functions which can be overridden by derived classes
 923 //////////
 924
 925 // none
 926
 927 //////////
 928 // non-virtual functions in this class
 929 //////////
 930
 931 namespace {  // anonymous namespace for power::expand_add() helpers
 932
 933 /** Helper class to generate all bounded combinatorial partitions of an integer
 934  *  n with exactly m parts (including zero parts) in non-decreasing order.
 935  */
 936 class partition_generator {
 937 private:
 938         // Partitions n into m parts, not including zero parts.
 939         // (Cf. OEIS sequence A008284; implementation adapted from Jörg Arndt's
 940         // FXT library)
 941         struct mpartition2
 942         {
 943                 // partition: x[1] + x[2] + ... + x[m] = n and sentinel x[0] == 0
 944                 std::vector<int> x;
 945                 int n;   // n>0
 946                 int m;   // 0<m<=n
 947                 mpartition2(unsigned n_, unsigned m_)
 948                   : x(m_+1), n(n_), m(m_)
 949                 {
 950                         for (int k=1; k<m; ++k)
 951                                 x[k] = 1;
 952                         x[m] = n - m + 1;
 953                 }
 954                 bool next_partition()
 955                 {
 956                         int u = x[m];  // last element
 957                         int k = m;
 958                         int s = u;
 959                         while (--k) {
 960                                 s += x[k];
 961                                 if (x[k] + 2 <= u)
 962                                         break;
 963                         }
 964                         if (k==0)
 965                                 return false;  // current is last
 966                         int f = x[k] + 1;
 967                         while (k < m) {
 968                                 x[k] = f;
 969                                 s -= f;
 970                                 ++k;
 971                         }
 972                         x[m] = s;
 973                         return true;
 974                 }
 975         } mpgen;
 976         int m;  // number of parts 0<m<=n
 977         mutable std::vector<int> partition;  // current partition
 978 public:
 979         partition_generator(unsigned n_, unsigned m_)
 980           : mpgen(n_, 1), m(m_), partition(m_)
 981         { }
 982         // returns current partition in non-decreasing order, padded with zeros
 983         const std::vector<int>& current() const
 984         {
 985                 for (int i = 0; i < m - mpgen.m; ++i)
 986                         partition[i] = 0;  // pad with zeros
 987
 988                 for (int i = m - mpgen.m; i < m; ++i)
 989                         partition[i] = mpgen.x[i - m + mpgen.m + 1];
 990
 991                 return partition;
 992         }
 993         bool next()
 994         {
 995                 if (!mpgen.next_partition()) {
 996                         if (mpgen.m == m || mpgen.m == mpgen.n)
 997                                 return false;  // current is last
 998                         // increment number of parts
 999                         mpgen = mpartition2(mpgen.n, mpgen.m + 1);
1000                 }
1001                 return true;
1002         }
1003 };
1004
1005 /** Helper class to generate all compositions of a partition of an integer n,
1006  *  starting with the compositions which has non-decreasing order.
1007  */
1008 class composition_generator {
1009 private:
1010         // Generates all distinct permutations of a multiset.
1011         // (Based on Aaron Williams' algorithm 1 from "Loopless Generation of
1012         // Multiset Permutations using a Constant Number of Variables by Prefix
1013         // Shifts." <http://webhome.csc.uvic.ca/~haron/CoolMulti.pdf>)
1014         struct coolmulti {
1015                 // element of singly linked list
1016                 struct element {
1017                         int value;
1018                         element* next;
1019                         element(int val, element* n)
1020                           : value(val), next(n) {}
1021                         ~element()
1022                         {   // recurses down to the end of the singly linked list
1023                                 delete next;
1024                         }
1025                 };
1026                 element *head, *i, *after_i;
1027                 // NB: Partition must be sorted in non-decreasing order.
1028                 explicit coolmulti(const std::vector<int>& partition)
1029                 {
1030                         head = NULL;
1031                         for (unsigned n = 0; n < partition.size(); ++n) {
1032                                 head = new element(partition[n], head);
1033                                 if (n <= 1)
1034                                         i = head;
1035                         }
1036                         after_i = i->next;
1037                 }
1038                 ~coolmulti()
1039                 {   // deletes singly linked list
1040                         delete head;
1041                 }
1042                 void next_permutation()
1043                 {
1044                         element *before_k;
1045                         if (after_i->next != NULL && i->value >= after_i->next->value)
1046                                 before_k = after_i;
1047                         else
1048                                 before_k = i;
1049                         element *k = before_k->next;
1050                         before_k->next = k->next;
1051                         k->next = head;
1052                         if (k->value < head->value)
1053                                 i = k;
1054                         after_i = i->next;
1055                         head = k;
1056                 }
1057                 bool finished() const
1058                 {
1059                         return after_i->next == NULL && after_i->value >= head->value;
1060                 }
1061         } cmgen;
1062         bool atend;  // needed for simplifying iteration over permutations
1063         bool trivial;  // likewise, true if all elements are equal
1064         mutable std::vector<int> composition;  // current compositions
1065 public:
1066         explicit composition_generator(const std::vector<int>& partition)
1067           : cmgen(partition), atend(false), trivial(true), composition(partition.size())
1068         {
1069                 for (unsigned i=1; i<partition.size(); ++i)
1070                         trivial = trivial && (partition[0] == partition[i]);
1071         }
1072         const std::vector<int>& current() const
1073         {
1074                 coolmulti::element* it = cmgen.head;
1075                 size_t i = 0;
1076                 while (it != NULL) {
1077                         composition[i] = it->value;
1078                         it = it->next;
1079                         ++i;
1080                 }
1081                 return composition;
1082         }
1083         bool next()
1084         {
1085                 // This ugly contortion is needed because the original coolmulti
1086                 // algorithm requires code duplication of the payload procedure,
1087                 // one before the loop and one inside it.
1088                 if (trivial || atend)
1089                         return false;
1090                 cmgen.next_permutation();
1091                 atend = cmgen.finished();
1092                 return true;
1093         }
1094 };
1095
1096 /** Helper function to compute the multinomial coefficient n!/(p1!*p2!*...*pk!)
1097  *  where n = p1+p2+...+pk, i.e. p is a partition of n.
1098  */
1099 const numeric
1100 multinomial_coefficient(const std::vector<int> & p)
1101 {
1102         numeric n = 0, d = 1;
1103         std::vector<int>::const_iterator it = p.begin(), itend = p.end();
1104         while (it != itend) {
1105                 n += numeric(*it);
1106                 d *= factorial(numeric(*it));
1107                 ++it;
1108         }
1109         return factorial(numeric(n)) / d;
1110 }
1111
1112 }  // anonymous namespace
1113
1114 /** expand a^n where a is an add and n is a positive integer.
1115  *  @see power::expand */
1116 ex power::expand_add(const add & a, int n, unsigned options) const
1117 {
1118         // The special case power(+(x,...y;x),2) can be optimized better.
1119         if (n==2)
1120                 return expand_add_2(a, options);
1121
1122         // method:
1123         //
1124         // Consider base as the sum of all symbolic terms and the overall numeric
1125         // coefficient and apply the binomial theorem:
1126         // S = power(+(x,...,z;c),n)
1127         //   = power(+(+(x,...,z;0);c),n)
1128         //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1129         // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
1130         // The multinomial theorem is computed by an outer loop over all
1131         // partitions of the exponent and an inner loop over all compositions of
1132         // that partition. This method makes the expansion a combinatorial
1133         // problem and allows us to directly construct the expanded sum and also
1134         // to re-use the multinomial coefficients (since they depend only on the
1135         // partition, not on the composition).
1136         //
1137         // multinomial power(+(x,y,z;0),3) example:
1138         // partition : compositions                : multinomial coefficient
1139         // [0,0,3]   : [3,0,0],[0,3,0],[0,0,3]     : 3!/(3!*0!*0!) = 1
1140         // [0,1,2]   : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
1141         // [1,1,1]   : [1,1,1]                     : 3!/(1!*1!*1!) = 6
1142         //  =>  (x + y + z)^3 =
1143         //        x^3 + y^3 + z^3
1144         //      + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
1145         //      + 6*x*y*z
1146         //
1147         // multinomial power(+(x,y,z;0),4) example:
1148         // partition : compositions                : multinomial coefficient
1149         // [0,0,4]   : [4,0,0],[0,4,0],[0,0,4]     : 4!/(4!*0!*0!) = 1
1150         // [0,1,3]   : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
1151         // [0,2,2]   : [2,2,0],[2,0,2],[0,2,2]     : 4!/(2!*2!*0!) = 6
1152         // [1,1,2]   : [2,1,1],[1,2,1],[1,1,2]     : 4!/(2!*1!*1!) = 12
1153         // (no [1,1,1,1] partition since it has too many parts)
1154         //  =>  (x + y + z)^4 =
1155         //        x^4 + y^4 + z^4
1156         //      + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
1157         //      + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
1158         //      + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
1159         //
1160         // Summary:
1161         // r = 0
1162         // for k from 0 to n:
1163         //     f = c^(n-k)*binomial(n,k)
1164         //     for p in all partitions of n with m parts (including zero parts):
1165         //         h = f * multinomial coefficient of p
1166         //         for c in all compositions of p:
1167         //             t = 1
1168         //             for e in all elements of c:
1169         //                 t = t * a[e]^e
1170         //             r = r + h*t
1171         // return r
1172
1173         epvector result;
1174         // The number of terms will be the number of combinatorial compositions,
1175         // i.e. the number of unordered arrangements of m nonnegative integers
1176         // which sum up to n.  It is frequently written as C_n(m) and directly
1177         // related with binomial coefficients: binomial(n+m-1,m-1).
1178         size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_int();
1179         if (!a.overall_coeff.is_zero()) {
1180                 // the result's overall_coeff is one of the terms
1181                 --result_size;
1182         }
1183         result.reserve(result_size);
1184
1185         // Iterate over all terms in binomial expansion of
1186         // S = power(+(x,...,z;c),n)
1187         //   = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
1188         for (int k = 1; k <= n; ++k) {
1189                 numeric binomial_coefficient;  // binomial(n,k)*c^(n-k)
1190                 if (a.overall_coeff.is_zero()) {
1191                         // degenerate case with zero overall_coeff:
1192                         // apply multinomial theorem directly to power(+(x,...z;0),n)
1193                         binomial_coefficient = 1;
1194                         if (k < n) {
1195                                 continue;
1196                         }
1197                 } else {
1198                         binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
1199                 }
1200
1201                 // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
1202                 // Iterate over all partitions of k with exactly as many parts as
1203                 // there are symbolic terms in the basis (including zero parts).
1204                 partition_generator partitions(k, a.seq.size());
1205                 do {
1206                         const std::vector<int>& partition = partitions.current();
1207                         const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
1208
1209                         // Iterate over all compositions of the current partition.
1210                         composition_generator compositions(partition);
1211                         do {
1212                                 const std::vector<int>& exponent = compositions.current();
1213                                 exvector term;
1214                                 term.reserve(n);
1215                                 numeric factor = coeff;
1216                                 for (unsigned i = 0; i < exponent.size(); ++i) {
1217                                         const ex & r = a.seq[i].rest;
1218                                         GINAC_ASSERT(!is_exactly_a<add>(r));
1219                                         GINAC_ASSERT(!is_exactly_a<power>(r) ||
1220                                                      !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1221                                                      !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1222                                                      !is_exactly_a<add>(ex_to<power>(r).basis) ||
1223                                                      !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1224                                                      !is_exactly_a<power>(ex_to<power>(r).basis));
1225                                         GINAC_ASSERT(is_exactly_a<numeric>(a.seq[i].coeff));
1226                                         const numeric & c = ex_to<numeric>(a.seq[i].coeff);
1227                                         if (exponent[i] == 0) {
1228                                                 // optimize away
1229                                         } else if (exponent[i] == 1) {
1230                                                 // optimized
1231                                                 term.push_back(r);
1232                                                 if (c != *_num1_p)
1233                                                         factor = factor.mul(c);
1234                                         } else { // general case exponent[i] > 1
1235                                                 term.push_back((new power(r, exponent[i]))->setflag(status_flags::dynallocated));
1236                                                 if (c != *_num1_p)
1237                                                         factor = factor.mul(c.power(exponent[i]));
1238                                         }
1239                                 }
1240                                 result.push_back(a.combine_ex_with_coeff_to_pair(mul(term).expand(options), factor));
1241                         } while (compositions.next());
1242                 } while (partitions.next());
1243         }
1244
1245         GINAC_ASSERT(result.size() == result_size);
1246
1247         if (a.overall_coeff.is_zero()) {
1248                 return (new add(result))->setflag(status_flags::dynallocated |
1249                                                   status_flags::expanded);
1250         } else {
1251                 return (new add(result, ex_to<numeric>(a.overall_coeff).power(n)))->setflag(status_flags::dynallocated |
1252                                                                                             status_flags::expanded);
1253         }
1254 }
1255
1256
1257 /** Special case of power::expand_add. Expands a^2 where a is an add.
1258  *  @see power::expand_add */
1259 ex power::expand_add_2(const add & a, unsigned options) const
1260 {
1261         epvector sum;
1262         size_t a_nops = a.nops();
1263         sum.reserve((a_nops*(a_nops+1))/2);
1264         epvector::const_iterator last = a.seq.end();
1265
1266         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
1267         // first part: ignore overall_coeff and expand other terms
1268         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
1269                 const ex & r = cit0->rest;
1270                 const ex & c = cit0->coeff;
1271
1272                 GINAC_ASSERT(!is_exactly_a<add>(r));
1273                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
1274                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
1275                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
1276                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
1277                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
1278                              !is_exactly_a<power>(ex_to<power>(r).basis));
1279
1280                 if (c.is_equal(_ex1)) {
1281                         if (is_exactly_a<mul>(r)) {
1282                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1283                                                                               _ex1));
1284                         } else {
1285                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1286                                                                               _ex1));
1287                         }
1288                 } else {
1289                         if (is_exactly_a<mul>(r)) {
1290                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
1291                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
1292                         } else {
1293                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
1294                                                      ex_to<numeric>(c).power_dyn(*_num2_p)));
1295                         }
1296                 }
1297
1298                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
1299                         const ex & r1 = cit1->rest;
1300                         const ex & c1 = cit1->coeff;
1301                         sum.push_back(a.combine_ex_with_coeff_to_pair(mul(r,r1).expand(options),
1302                                                                       _num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
1303                 }
1304         }
1305
1306         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
1307
1308         // second part: add terms coming from overall_coeff (if != 0)
1309         if (!a.overall_coeff.is_zero()) {
1310                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
1311                 while (i != end) {
1312                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(*_num2_p)));
1313                         ++i;
1314                 }
1315                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(*_num2_p),_ex1));
1316         }
1317
1318         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
1319
1320         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
1321 }
1322
1323 /** Expand factors of m in m^n where m is a mul and n is an integer.
1324  *  @see power::expand */
1325 ex power::expand_mul(const mul & m, const numeric & n, unsigned options, bool from_expand) const
1326 {
1327         GINAC_ASSERT(n.is_integer());
1328
1329         if (n.is_zero()) {
1330                 return _ex1;
1331         }
1332
1333         // do not bother to rename indices if there are no any.
1334         if (!(options & expand_options::expand_rename_idx) &&
1335             m.info(info_flags::has_indices))
1336                 options |= expand_options::expand_rename_idx;
1337         // Leave it to multiplication since dummy indices have to be renamed
1338         if ((options & expand_options::expand_rename_idx) &&
1339             (get_all_dummy_indices(m).size() > 0) && n.is_positive()) {
1340                 ex result = m;
1341                 exvector va = get_all_dummy_indices(m);
1342                 sort(va.begin(), va.end(), ex_is_less());
1343
1344                 for (int i=1; i < n.to_int(); i++)
1345                         result *= rename_dummy_indices_uniquely(va, m);
1346                 return result;
1347         }
1348
1349         epvector distrseq;
1350         distrseq.reserve(m.seq.size());
1351         bool need_reexpand = false;
1352
1353         epvector::const_iterator last = m.seq.end();
1354         epvector::const_iterator cit = m.seq.begin();
1355         while (cit!=last) {
1356                 expair p = m.combine_pair_with_coeff_to_pair(*cit, n);
1357                 if (from_expand && is_exactly_a<add>(cit->rest) && ex_to<numeric>(p.coeff).is_pos_integer()) {
1358                         // this happens when e.g. (a+b)^(1/2) gets squared and
1359                         // the resulting product needs to be reexpanded
1360                         need_reexpand = true;
1361                 }
1362                 distrseq.push_back(p);
1363                 ++cit;
1364         }
1365
1366         const mul & result = static_cast<const mul &>((new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated));
1367         if (need_reexpand)
1368                 return ex(result).expand(options);
1369         if (from_expand)
1370                 return result.setflag(status_flags::expanded);
1371         return result;
1372 }
1373
1374 GINAC_BIND_UNARCHIVER(power);
1375
1376 } // namespace GiNaC