ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <vector>
  24 #include <iostream>
  25 #include <stdexcept>
  26 #include <limits>
  27
  28 #include "power.h"
  29 #include "expairseq.h"
  30 #include "add.h"
  31 #include "mul.h"
  32 #include "ncmul.h"
  33 #include "numeric.h"
  34 #include "constant.h"
  35 #include "operators.h"
  36 #include "inifcns.h" // for log() in power::derivative()
  37 #include "matrix.h"
  38 #include "indexed.h"
  39 #include "symbol.h"
  40 #include "lst.h"
  41 #include "print.h"
  42 #include "archive.h"
  43 #include "utils.h"
  44
  45 namespace GiNaC {
  46
  47 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
  48
  49 typedef std::vector<int> intvector;
  50
  51 //////////
  52 // default ctor, dtor, copy ctor, assignment operator and helpers
  53 //////////
  54
  55 power::power() : inherited(TINFO_power) { }
  56
  57 void power::copy(const power & other)
  58 {
  59         inherited::copy(other);
  60         basis = other.basis;
  61         exponent = other.exponent;
  62 }
  63
  64 DEFAULT_DESTROY(power)
  65
  66 //////////
  67 // other ctors
  68 //////////
  69
  70 // all inlined
  71
  72 //////////
  73 // archiving
  74 //////////
  75
  76 power::power(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  77 {
  78         n.find_ex("basis", basis, sym_lst);
  79         n.find_ex("exponent", exponent, sym_lst);
  80 }
  81
  82 void power::archive(archive_node &n) const
  83 {
  84         inherited::archive(n);
  85         n.add_ex("basis", basis);
  86         n.add_ex("exponent", exponent);
  87 }
  88
  89 DEFAULT_UNARCHIVE(power)
  90
  91 //////////
  92 // functions overriding virtual functions from base classes
  93 //////////
  94
  95 // public
  96
  97 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
  98 {
  99         // Optimal output of integer powers of symbols to aid compiler CSE.
 100         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 101         // to learn why such a parenthisation is really necessary.
 102         if (exp == 1) {
 103                 x.print(c);
 104         } else if (exp == 2) {
 105                 x.print(c);
 106                 c.s << "*";
 107                 x.print(c);
 108         } else if (exp & 1) {
 109                 x.print(c);
 110                 c.s << "*";
 111                 print_sym_pow(c, x, exp-1);
 112         } else {
 113                 c.s << "(";
 114                 print_sym_pow(c, x, exp >> 1);
 115                 c.s << ")*(";
 116                 print_sym_pow(c, x, exp >> 1);
 117                 c.s << ")";
 118         }
 119 }
 120
 121 void power::print(const print_context & c, unsigned level) const
 122 {
 123         if (is_a<print_tree>(c)) {
 124
 125                 inherited::print(c, level);
 126
 127         } else if (is_a<print_csrc>(c)) {
 128
 129                 // Integer powers of symbols are printed in a special, optimized way
 130                 if (exponent.info(info_flags::integer)
 131                  && (is_a<symbol>(basis) || is_a<constant>(basis))) {
 132                         int exp = ex_to<numeric>(exponent).to_int();
 133                         if (exp > 0)
 134                                 c.s << '(';
 135                         else {
 136                                 exp = -exp;
 137                                 if (is_a<print_csrc_cl_N>(c))
 138                                         c.s << "recip(";
 139                                 else
 140                                         c.s << "1.0/(";
 141                         }
 142                         print_sym_pow(c, ex_to<symbol>(basis), exp);
 143                         c.s << ')';
 144
 145                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 146                 } else if (exponent.is_equal(_ex_1)) {
 147                         if (is_a<print_csrc_cl_N>(c))
 148                                 c.s << "recip(";
 149                         else
 150                                 c.s << "1.0/(";
 151                         basis.print(c);
 152                         c.s << ')';
 153
 154                 // Otherwise, use the pow() or expt() (CLN) functions
 155                 } else {
 156                         if (is_a<print_csrc_cl_N>(c))
 157                                 c.s << "expt(";
 158                         else
 159                                 c.s << "pow(";
 160                         basis.print(c);
 161                         c.s << ',';
 162                         exponent.print(c);
 163                         c.s << ')';
 164                 }
 165
 166         } else if (is_a<print_python_repr>(c)) {
 167
 168                 c.s << class_name() << '(';
 169                 basis.print(c);
 170                 c.s << ',';
 171                 exponent.print(c);
 172                 c.s << ')';
 173
 174         } else {
 175
 176                 bool is_tex = is_a<print_latex>(c);
 177
 178                 if (is_tex && is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_negative()) {
 179
 180                         // Powers with negative numeric exponents are printed as fractions in TeX
 181                         c.s << "\\frac{1}{";
 182                         power(basis, -exponent).eval().print(c);
 183                         c.s << "}";
 184
 185                 } else if (exponent.is_equal(_ex1_2)) {
 186
 187                         // Square roots are printed in a special way
 188                         c.s << (is_tex ? "\\sqrt{" : "sqrt(");
 189                         basis.print(c);
 190                         c.s << (is_tex ? '}' : ')');
 191
 192                 } else {
 193
 194                         // Ordinary output of powers using '^' or '**'
 195                         if (precedence() <= level)
 196                                 c.s << (is_tex ? "{(" : "(");
 197                         basis.print(c, precedence());
 198                         if (is_a<print_python>(c))
 199                                 c.s << "**";
 200                         else
 201                                 c.s << '^';
 202                         if (is_tex)
 203                                 c.s << '{';
 204                         exponent.print(c, precedence());
 205                         if (is_tex)
 206                                 c.s << '}';
 207                         if (precedence() <= level)
 208                                 c.s << (is_tex ? ")}" : ")");
 209                 }
 210         }
 211 }
 212
 213 bool power::info(unsigned inf) const
 214 {
 215         switch (inf) {
 216                 case info_flags::polynomial:
 217                 case info_flags::integer_polynomial:
 218                 case info_flags::cinteger_polynomial:
 219                 case info_flags::rational_polynomial:
 220                 case info_flags::crational_polynomial:
 221                         return exponent.info(info_flags::nonnegint);
 222                 case info_flags::rational_function:
 223                         return exponent.info(info_flags::integer);
 224                 case info_flags::algebraic:
 225                         return (!exponent.info(info_flags::integer) ||
 226                                         basis.info(inf));
 227         }
 228         return inherited::info(inf);
 229 }
 230
 231 size_t power::nops() const
 232 {
 233         return 2;
 234 }
 235
 236 ex power::op(size_t i) const
 237 {
 238         GINAC_ASSERT(i<2);
 239
 240         return i==0 ? basis : exponent;
 241 }
 242
 243 ex power::map(map_function & f) const
 244 {
 245         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
 246 }
 247
 248 int power::degree(const ex & s) const
 249 {
 250         if (is_equal(ex_to<basic>(s)))
 251                 return 1;
 252         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 253                 if (basis.is_equal(s))
 254                         return ex_to<numeric>(exponent).to_int();
 255                 else
 256                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 257         } else if (basis.has(s))
 258                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 259         else
 260                 return 0;
 261 }
 262
 263 int power::ldegree(const ex & s) const
 264 {
 265         if (is_equal(ex_to<basic>(s)))
 266                 return 1;
 267         else if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 268                 if (basis.is_equal(s))
 269                         return ex_to<numeric>(exponent).to_int();
 270                 else
 271                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 272         } else if (basis.has(s))
 273                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 274         else
 275                 return 0;
 276 }
 277
 278 ex power::coeff(const ex & s, int n) const
 279 {
 280         if (is_equal(ex_to<basic>(s)))
 281                 return n==1 ? _ex1 : _ex0;
 282         else if (!basis.is_equal(s)) {
 283                 // basis not equal to s
 284                 if (n == 0)
 285                         return *this;
 286                 else
 287                         return _ex0;
 288         } else {
 289                 // basis equal to s
 290                 if (is_exactly_a<numeric>(exponent) && ex_to<numeric>(exponent).is_integer()) {
 291                         // integer exponent
 292                         int int_exp = ex_to<numeric>(exponent).to_int();
 293                         if (n == int_exp)
 294                                 return _ex1;
 295                         else
 296                                 return _ex0;
 297                 } else {
 298                         // non-integer exponents are treated as zero
 299                         if (n == 0)
 300                                 return *this;
 301                         else
 302                                 return _ex0;
 303                 }
 304         }
 305 }
 306
 307 /** Perform automatic term rewriting rules in this class.  In the following
 308  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 309  *  stand for such expressions that contain a plain number.
 310  *  - ^(x,0) -> 1  (also handles ^(0,0))
 311  *  - ^(x,1) -> x
 312  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 313  *  - ^(1,x) -> 1
 314  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 315  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 316  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 317  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 318  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 319  *
 320  *  @param level cut-off in recursive evaluation */
 321 ex power::eval(int level) const
 322 {
 323         if ((level==1) && (flags & status_flags::evaluated))
 324                 return *this;
 325         else if (level == -max_recursion_level)
 326                 throw(std::runtime_error("max recursion level reached"));
 327
 328         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 329         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 330
 331         bool basis_is_numerical = false;
 332         bool exponent_is_numerical = false;
 333         const numeric *num_basis;
 334         const numeric *num_exponent;
 335
 336         if (is_exactly_a<numeric>(ebasis)) {
 337                 basis_is_numerical = true;
 338                 num_basis = &ex_to<numeric>(ebasis);
 339         }
 340         if (is_exactly_a<numeric>(eexponent)) {
 341                 exponent_is_numerical = true;
 342                 num_exponent = &ex_to<numeric>(eexponent);
 343         }
 344
 345         // ^(x,0) -> 1  (0^0 also handled here)
 346         if (eexponent.is_zero()) {
 347                 if (ebasis.is_zero())
 348                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 349                 else
 350                         return _ex1;
 351         }
 352
 353         // ^(x,1) -> x
 354         if (eexponent.is_equal(_ex1))
 355                 return ebasis;
 356
 357         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 358         if (ebasis.is_zero() && exponent_is_numerical) {
 359                 if ((num_exponent->real()).is_zero())
 360                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 361                 else if ((num_exponent->real()).is_negative())
 362                         throw (pole_error("power::eval(): division by zero",1));
 363                 else
 364                         return _ex0;
 365         }
 366
 367         // ^(1,x) -> 1
 368         if (ebasis.is_equal(_ex1))
 369                 return _ex1;
 370
 371         if (exponent_is_numerical) {
 372
 373                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 374                 // except if c1,c2 are rational, but c1^c2 is not)
 375                 if (basis_is_numerical) {
 376                         const bool basis_is_crational = num_basis->is_crational();
 377                         const bool exponent_is_crational = num_exponent->is_crational();
 378                         if (!basis_is_crational || !exponent_is_crational) {
 379                                 // return a plain float
 380                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 381                                                                                                status_flags::evaluated |
 382                                                                                                status_flags::expanded);
 383                         }
 384
 385                         const numeric res = num_basis->power(*num_exponent);
 386                         if (res.is_crational()) {
 387                                 return res;
 388                         }
 389                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 390
 391                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 392                         if (basis_is_crational && exponent_is_crational
 393                             && num_exponent->is_real()
 394                             && !num_exponent->is_integer()) {
 395                                 const numeric n = num_exponent->numer();
 396                                 const numeric m = num_exponent->denom();
 397                                 numeric r;
 398                                 numeric q = iquo(n, m, r);
 399                                 if (r.is_negative()) {
 400                                         r += m;
 401                                         --q;
 402                                 }
 403                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 404                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 405                                                 // try it for numerator and denominator separately, in order to
 406                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 407                                                 const numeric bnum = num_basis->numer();
 408                                                 const numeric bden = num_basis->denom();
 409                                                 const numeric res_bnum = bnum.power(*num_exponent);
 410                                                 const numeric res_bden = bden.power(*num_exponent);
 411                                                 if (res_bnum.is_integer())
 412                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 413                                                 if (res_bden.is_integer())
 414                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 415                                         }
 416                                         return this->hold();
 417                                 } else {
 418                                         // assemble resulting product, but allowing for a re-evaluation,
 419                                         // because otherwise we'll end up with something like
 420                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 421                                         // instead of 7/16*7^(1/3).
 422                                         ex prod = power(*num_basis,r.div(m));
 423                                         return prod*power(*num_basis,q);
 424                                 }
 425                         }
 426                 }
 427
 428                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 429                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 430                 // case c1==1 should not happen, see below!)
 431                 if (is_exactly_a<power>(ebasis)) {
 432                         const power & sub_power = ex_to<power>(ebasis);
 433                         const ex & sub_basis = sub_power.basis;
 434                         const ex & sub_exponent = sub_power.exponent;
 435                         if (is_exactly_a<numeric>(sub_exponent)) {
 436                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 437                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 438                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1).is_negative())
 439                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 440                         }
 441                 }
 442
 443                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 444                 if (num_exponent->is_integer() && is_exactly_a<mul>(ebasis)) {
 445                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
 446                 }
 447
 448                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 449                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 450                 if (is_exactly_a<mul>(ebasis)) {
 451                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 452                         const mul & mulref = ex_to<mul>(ebasis);
 453                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 454                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 455                                 if (num_coeff.is_real()) {
 456                                         if (num_coeff.is_positive()) {
 457                                                 mul *mulp = new mul(mulref);
 458                                                 mulp->overall_coeff = _ex1;
 459                                                 mulp->clearflag(status_flags::evaluated);
 460                                                 mulp->clearflag(status_flags::hash_calculated);
 461                                                 return (new mul(power(*mulp,exponent),
 462                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 463                                         } else {
 464                                                 GINAC_ASSERT(num_coeff.compare(_num0)<0);
 465                                                 if (!num_coeff.is_equal(_num_1)) {
 466                                                         mul *mulp = new mul(mulref);
 467                                                         mulp->overall_coeff = _ex_1;
 468                                                         mulp->clearflag(status_flags::evaluated);
 469                                                         mulp->clearflag(status_flags::hash_calculated);
 470                                                         return (new mul(power(*mulp,exponent),
 471                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 472                                                 }
 473                                         }
 474                                 }
 475                         }
 476                 }
 477
 478                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 479                 if (num_exponent->is_pos_integer() &&
 480                     ebasis.return_type() != return_types::commutative &&
 481                     !is_a<matrix>(ebasis)) {
 482                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 483                 }
 484         }
 485
 486         if (are_ex_trivially_equal(ebasis,basis) &&
 487             are_ex_trivially_equal(eexponent,exponent)) {
 488                 return this->hold();
 489         }
 490         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 491                                                        status_flags::evaluated);
 492 }
 493
 494 ex power::evalf(int level) const
 495 {
 496         ex ebasis;
 497         ex eexponent;
 498
 499         if (level==1) {
 500                 ebasis = basis;
 501                 eexponent = exponent;
 502         } else if (level == -max_recursion_level) {
 503                 throw(std::runtime_error("max recursion level reached"));
 504         } else {
 505                 ebasis = basis.evalf(level-1);
 506                 if (!is_exactly_a<numeric>(exponent))
 507                         eexponent = exponent.evalf(level-1);
 508                 else
 509                         eexponent = exponent;
 510         }
 511
 512         return power(ebasis,eexponent);
 513 }
 514
 515 ex power::evalm(void) const
 516 {
 517         const ex ebasis = basis.evalm();
 518         const ex eexponent = exponent.evalm();
 519         if (is_a<matrix>(ebasis)) {
 520                 if (is_exactly_a<numeric>(eexponent)) {
 521                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 522                 }
 523         }
 524         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 525 }
 526
 527 // from mul.cpp
 528 extern bool tryfactsubs(const ex &, const ex &, int &, lst &);
 529
 530 ex power::subs(const lst & ls, const lst & lr, unsigned options) const
 531 {
 532         const ex &subsed_basis = basis.subs(ls, lr, options);
 533         const ex &subsed_exponent = exponent.subs(ls, lr, options);
 534
 535         if (!are_ex_trivially_equal(basis, subsed_basis)
 536          || !are_ex_trivially_equal(exponent, subsed_exponent))
 537                 return power(subsed_basis, subsed_exponent).basic::subs(ls, lr, options);
 538
 539         if(!(options & subs_options::algebraic))
 540                 return basic::subs(ls, lr, options);
 541
 542         lst::const_iterator its, itr;
 543         for (its = ls.begin(), itr = lr.begin(); its != ls.end(); ++its, ++itr) {
 544                 int nummatches = std::numeric_limits<int>::max();
 545                 lst repls;
 546                 if (tryfactsubs(*this, *its, nummatches, repls))
 547                         return (ex_to<basic>((*this) * power(itr->subs(ex(repls), subs_options::no_pattern) / its->subs(ex(repls), subs_options::no_pattern), nummatches))).basic::subs(ls, lr, options);
 548         }
 549
 550         ex result=basic::subs(ls, lr, options);
 551         return result;
 552 }
 553
 554 ex power::eval_ncmul(const exvector & v) const
 555 {
 556         return inherited::eval_ncmul(v);
 557 }
 558
 559 // protected
 560
 561 /** Implementation of ex::diff() for a power.
 562  *  @see ex::diff */
 563 ex power::derivative(const symbol & s) const
 564 {
 565         if (exponent.info(info_flags::real)) {
 566                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 567                 epvector newseq;
 568                 newseq.reserve(2);
 569                 newseq.push_back(expair(basis, exponent - _ex1));
 570                 newseq.push_back(expair(basis.diff(s), _ex1));
 571                 return mul(newseq, exponent);
 572         } else {
 573                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 574                 return mul(*this,
 575                            add(mul(exponent.diff(s), log(basis)),
 576                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 577         }
 578 }
 579
 580 int power::compare_same_type(const basic & other) const
 581 {
 582         GINAC_ASSERT(is_exactly_a<power>(other));
 583         const power &o = static_cast<const power &>(other);
 584
 585         int cmpval = basis.compare(o.basis);
 586         if (cmpval)
 587                 return cmpval;
 588         else
 589                 return exponent.compare(o.exponent);
 590 }
 591
 592 unsigned power::return_type(void) const
 593 {
 594         return basis.return_type();
 595 }
 596
 597 unsigned power::return_type_tinfo(void) const
 598 {
 599         return basis.return_type_tinfo();
 600 }
 601
 602 ex power::expand(unsigned options) const
 603 {
 604         if (options == 0 && (flags & status_flags::expanded))
 605                 return *this;
 606
 607         const ex expanded_basis = basis.expand(options);
 608         const ex expanded_exponent = exponent.expand(options);
 609
 610         // x^(a+b) -> x^a * x^b
 611         if (is_exactly_a<add>(expanded_exponent)) {
 612                 const add &a = ex_to<add>(expanded_exponent);
 613                 exvector distrseq;
 614                 distrseq.reserve(a.seq.size() + 1);
 615                 epvector::const_iterator last = a.seq.end();
 616                 epvector::const_iterator cit = a.seq.begin();
 617                 while (cit!=last) {
 618                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 619                         ++cit;
 620                 }
 621
 622                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 623                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 624                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 625                         int int_exponent = num_exponent.to_int();
 626                         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 627                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
 628                         else
 629                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 630                 } else
 631                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 632
 633                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 634                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 635                 return r.expand();
 636         }
 637
 638         if (!is_exactly_a<numeric>(expanded_exponent) ||
 639                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 640                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 641                         return this->hold();
 642                 } else {
 643                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 644                 }
 645         }
 646
 647         // integer numeric exponent
 648         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 649         int int_exponent = num_exponent.to_int();
 650
 651         // (x+y)^n, n>0
 652         if (int_exponent > 0 && is_exactly_a<add>(expanded_basis))
 653                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
 654
 655         // (x*y)^n -> x^n * y^n
 656         if (is_exactly_a<mul>(expanded_basis))
 657                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
 658
 659         // cannot expand further
 660         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 661                 return this->hold();
 662         else
 663                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 664 }
 665
 666 //////////
 667 // new virtual functions which can be overridden by derived classes
 668 //////////
 669
 670 // none
 671
 672 //////////
 673 // non-virtual functions in this class
 674 //////////
 675
 676 /** expand a^n where a is an add and n is a positive integer.
 677  *  @see power::expand */
 678 ex power::expand_add(const add & a, int n) const
 679 {
 680         if (n==2)
 681                 return expand_add_2(a);
 682
 683         const size_t m = a.nops();
 684         exvector result;
 685         // The number of terms will be the number of combinatorial compositions,
 686         // i.e. the number of unordered arrangement of m nonnegative integers
 687         // which sum up to n.  It is frequently written as C_n(m) and directly
 688         // related with binomial coefficients:
 689         result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
 690         intvector k(m-1);
 691         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 692         intvector upper_limit(m-1);
 693         int l;
 694
 695         for (size_t l=0; l<m-1; ++l) {
 696                 k[l] = 0;
 697                 k_cum[l] = 0;
 698                 upper_limit[l] = n;
 699         }
 700
 701         while (true) {
 702                 exvector term;
 703                 term.reserve(m+1);
 704                 for (l=0; l<m-1; ++l) {
 705                         const ex & b = a.op(l);
 706                         GINAC_ASSERT(!is_exactly_a<add>(b));
 707                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 708                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 709                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 710                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 711                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 712                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 713                         if (is_exactly_a<mul>(b))
 714                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
 715                         else
 716                                 term.push_back(power(b,k[l]));
 717                 }
 718
 719                 const ex & b = a.op(l);
 720                 GINAC_ASSERT(!is_exactly_a<add>(b));
 721                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 722                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 723                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 724                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 725                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 726                              !is_exactly_a<power>(ex_to<power>(b).basis));
 727                 if (is_exactly_a<mul>(b))
 728                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
 729                 else
 730                         term.push_back(power(b,n-k_cum[m-2]));
 731
 732                 numeric f = binomial(numeric(n),numeric(k[0]));
 733                 for (l=1; l<m-1; ++l)
 734                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 735
 736                 term.push_back(f);
 737
 738                 result.push_back((new mul(term))->setflag(status_flags::dynallocated));
 739
 740                 // increment k[]
 741                 l = m-2;
 742                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 743                         k[l] = 0;
 744                         --l;
 745                 }
 746                 if (l<0) break;
 747
 748                 // recalc k_cum[] and upper_limit[]
 749                 k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
 750
 751                 for (size_t i=l+1; i<m-1; ++i)
 752                         k_cum[i] = k_cum[i-1]+k[i];
 753
 754                 for (size_t i=l+1; i<m-1; ++i)
 755                         upper_limit[i] = n-k_cum[i-1];
 756         }
 757
 758         return (new add(result))->setflag(status_flags::dynallocated |
 759                                           status_flags::expanded);
 760 }
 761
 762
 763 /** Special case of power::expand_add. Expands a^2 where a is an add.
 764  *  @see power::expand_add */
 765 ex power::expand_add_2(const add & a) const
 766 {
 767         epvector sum;
 768         size_t a_nops = a.nops();
 769         sum.reserve((a_nops*(a_nops+1))/2);
 770         epvector::const_iterator last = a.seq.end();
 771
 772         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 773         // first part: ignore overall_coeff and expand other terms
 774         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 775                 const ex & r = cit0->rest;
 776                 const ex & c = cit0->coeff;
 777
 778                 GINAC_ASSERT(!is_exactly_a<add>(r));
 779                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 780                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 781                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 782                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 783                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 784                              !is_exactly_a<power>(ex_to<power>(r).basis));
 785
 786                 if (c.is_equal(_ex1)) {
 787                         if (is_exactly_a<mul>(r)) {
 788                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
 789                                                      _ex1));
 790                         } else {
 791                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 792                                                      _ex1));
 793                         }
 794                 } else {
 795                         if (is_exactly_a<mul>(r)) {
 796                                 sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r),_num2),
 797                                                      ex_to<numeric>(c).power_dyn(_num2)));
 798                         } else {
 799                                 sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 800                                                      ex_to<numeric>(c).power_dyn(_num2)));
 801                         }
 802                 }
 803
 804                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 805                         const ex & r1 = cit1->rest;
 806                         const ex & c1 = cit1->coeff;
 807                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 808                                                                       _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 809                 }
 810         }
 811
 812         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 813
 814         // second part: add terms coming from overall_factor (if != 0)
 815         if (!a.overall_coeff.is_zero()) {
 816                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 817                 while (i != end) {
 818                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
 819                         ++i;
 820                 }
 821                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
 822         }
 823
 824         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 825
 826         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 827 }
 828
 829 /** Expand factors of m in m^n where m is a mul and n is and integer.
 830  *  @see power::expand */
 831 ex power::expand_mul(const mul & m, const numeric & n) const
 832 {
 833         GINAC_ASSERT(n.is_integer());
 834
 835         if (n.is_zero())
 836                 return _ex1;
 837
 838         epvector distrseq;
 839         distrseq.reserve(m.seq.size());
 840         epvector::const_iterator last = m.seq.end();
 841         epvector::const_iterator cit = m.seq.begin();
 842         while (cit!=last) {
 843                 if (is_exactly_a<numeric>(cit->rest)) {
 844                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit, n));
 845                 } else {
 846                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 847                         // since n is an integer
 848                         distrseq.push_back(expair(cit->rest, ex_to<numeric>(cit->coeff).mul(n)));
 849                 }
 850                 ++cit;
 851         }
 852         return (new mul(distrseq, ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 853 }
 854
 855 } // namespace GiNaC