ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2001 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
  27 #include "power.h"
  28 #include "expairseq.h"
  29 #include "add.h"
  30 #include "mul.h"
  31 #include "ncmul.h"
  32 #include "numeric.h"
  33 #include "constant.h"
  34 #include "inifcns.h" // for log() in power::derivative()
  35 #include "matrix.h"
  36 #include "symbol.h"
  37 #include "print.h"
  38 #include "archive.h"
  39 #include "utils.h"
  40
  41 namespace GiNaC {
  42
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
  44
  45 typedef std::vector<int> intvector;
  46
  47 //////////
  48 // default ctor, dtor, copy ctor, assignment operator and helpers
  49 //////////
  50
  51 power::power() : inherited(TINFO_power) { }
  52
  53 void power::copy(const power & other)
  54 {
  55         inherited::copy(other);
  56         basis = other.basis;
  57         exponent = other.exponent;
  58 }
  59
  60 DEFAULT_DESTROY(power)
  61
  62 //////////
  63 // other ctors
  64 //////////
  65
  66 // all inlined
  67
  68 //////////
  69 // archiving
  70 //////////
  71
  72 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  73 {
  74         n.find_ex("basis", basis, sym_lst);
  75         n.find_ex("exponent", exponent, sym_lst);
  76 }
  77
  78 void power::archive(archive_node &n) const
  79 {
  80         inherited::archive(n);
  81         n.add_ex("basis", basis);
  82         n.add_ex("exponent", exponent);
  83 }
  84
  85 DEFAULT_UNARCHIVE(power)
  86
  87 //////////
  88 // functions overriding virtual functions from base classes
  89 //////////
  90
  91 // public
  92
  93 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
  94 {
  95         // Optimal output of integer powers of symbols to aid compiler CSE.
  96         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
  97         // to learn why such a parenthisation is really necessary.
  98         if (exp == 1) {
  99                 x.print(c);
 100         } else if (exp == 2) {
 101                 x.print(c);
 102                 c.s << "*";
 103                 x.print(c);
 104         } else if (exp & 1) {
 105                 x.print(c);
 106                 c.s << "*";
 107                 print_sym_pow(c, x, exp-1);
 108         } else {
 109                 c.s << "(";
 110                 print_sym_pow(c, x, exp >> 1);
 111                 c.s << ")*(";
 112                 print_sym_pow(c, x, exp >> 1);
 113                 c.s << ")";
 114         }
 115 }
 116
 117 void power::print(const print_context & c, unsigned level) const
 118 {
 119         if (is_a<print_tree>(c)) {
 120
 121                 inherited::print(c, level);
 122
 123         } else if (is_a<print_csrc>(c)) {
 124
 125                 // Integer powers of symbols are printed in a special, optimized way
 126                 if (exponent.info(info_flags::integer)
 127                  && (is_exactly_a<symbol>(basis) || is_exactly_a<constant>(basis))) {
 128                         int exp = ex_to<numeric>(exponent).to_int();
 129                         if (exp > 0)
 130                                 c.s << '(';
 131                         else {
 132                                 exp = -exp;
 133                                 if (is_a<print_csrc_cl_N>(c))
 134                                         c.s << "recip(";
 135                                 else
 136                                         c.s << "1.0/(";
 137                         }
 138                         print_sym_pow(c, ex_to<symbol>(basis), exp);
 139                         c.s << ')';
 140
 141                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 142                 } else if (exponent.compare(_num_1) == 0) {
 143                         if (is_a<print_csrc_cl_N>(c))
 144                                 c.s << "recip(";
 145                         else
 146                                 c.s << "1.0/(";
 147                         basis.print(c);
 148                         c.s << ')';
 149
 150                 // Otherwise, use the pow() or expt() (CLN) functions
 151                 } else {
 152                         if (is_a<print_csrc_cl_N>(c))
 153                                 c.s << "expt(";
 154                         else
 155                                 c.s << "pow(";
 156                         basis.print(c);
 157                         c.s << ',';
 158                         exponent.print(c);
 159                         c.s << ')';
 160                 }
 161
 162         } else {
 163
 164                 if (exponent.is_equal(_ex1_2)) {
 165                         if (is_a<print_latex>(c))
 166                                 c.s << "\\sqrt{";
 167                         else
 168                                 c.s << "sqrt(";
 169                         basis.print(c);
 170                         if (is_a<print_latex>(c))
 171                                 c.s << '}';
 172                         else
 173                                 c.s << ')';
 174                 } else {
 175                         if (precedence() <= level) {
 176                                 if (is_a<print_latex>(c))
 177                                         c.s << "{(";
 178                                 else
 179                                         c.s << "(";
 180                         }
 181                         basis.print(c, precedence());
 182                         c.s << '^';
 183                         if (is_a<print_latex>(c))
 184                                 c.s << '{';
 185                         exponent.print(c, precedence());
 186                         if (is_a<print_latex>(c))
 187                                 c.s << '}';
 188                         if (precedence() <= level) {
 189                                 if (is_a<print_latex>(c))
 190                                         c.s << ")}";
 191                                 else
 192                                         c.s << ')';
 193                         }
 194                 }
 195         }
 196 }
 197
 198 bool power::info(unsigned inf) const
 199 {
 200         switch (inf) {
 201                 case info_flags::polynomial:
 202                 case info_flags::integer_polynomial:
 203                 case info_flags::cinteger_polynomial:
 204                 case info_flags::rational_polynomial:
 205                 case info_flags::crational_polynomial:
 206                         return exponent.info(info_flags::nonnegint);
 207                 case info_flags::rational_function:
 208                         return exponent.info(info_flags::integer);
 209                 case info_flags::algebraic:
 210                         return (!exponent.info(info_flags::integer) ||
 211                                         basis.info(inf));
 212         }
 213         return inherited::info(inf);
 214 }
 215
 216 unsigned power::nops() const
 217 {
 218         return 2;
 219 }
 220
 221 ex & power::let_op(int i)
 222 {
 223         GINAC_ASSERT(i>=0);
 224         GINAC_ASSERT(i<2);
 225
 226         return i==0 ? basis : exponent;
 227 }
 228
 229 ex power::map(map_function & f) const
 230 {
 231         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
 232 }
 233
 234 int power::degree(const ex & s) const
 235 {
 236         if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
 237                 if (basis.is_equal(s))
 238                         return ex_to<numeric>(exponent).to_int();
 239                 else
 240                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 241         } else if (basis.has(s))
 242                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 243         else
 244                 return 0;
 245 }
 246
 247 int power::ldegree(const ex & s) const
 248 {
 249         if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
 250                 if (basis.is_equal(s))
 251                         return ex_to<numeric>(exponent).to_int();
 252                 else
 253                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 254         } else if (basis.has(s))
 255                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 256         else
 257                 return 0;
 258 }
 259
 260 ex power::coeff(const ex & s, int n) const
 261 {
 262         if (!basis.is_equal(s)) {
 263                 // basis not equal to s
 264                 if (n == 0)
 265                         return *this;
 266                 else
 267                         return _ex0;
 268         } else {
 269                 // basis equal to s
 270                 if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
 271                         // integer exponent
 272                         int int_exp = ex_to<numeric>(exponent).to_int();
 273                         if (n == int_exp)
 274                                 return _ex1;
 275                         else
 276                                 return _ex0;
 277                 } else {
 278                         // non-integer exponents are treated as zero
 279                         if (n == 0)
 280                                 return *this;
 281                         else
 282                                 return _ex0;
 283                 }
 284         }
 285 }
 286
 287 /** Perform automatic term rewriting rules in this class.  In the following
 288  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 289  *  stand for such expressions that contain a plain number.
 290  *  - ^(x,0) -> 1  (also handles ^(0,0))
 291  *  - ^(x,1) -> x
 292  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 293  *  - ^(1,x) -> 1
 294  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 295  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 296  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 297  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 298  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 299  *
 300  *  @param level cut-off in recursive evaluation */
 301 ex power::eval(int level) const
 302 {
 303         if ((level==1) && (flags & status_flags::evaluated))
 304                 return *this;
 305         else if (level == -max_recursion_level)
 306                 throw(std::runtime_error("max recursion level reached"));
 307
 308         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 309         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 310
 311         bool basis_is_numerical = false;
 312         bool exponent_is_numerical = false;
 313         const numeric *num_basis;
 314         const numeric *num_exponent;
 315
 316         if (is_ex_exactly_of_type(ebasis, numeric)) {
 317                 basis_is_numerical = true;
 318                 num_basis = &ex_to<numeric>(ebasis);
 319         }
 320         if (is_ex_exactly_of_type(eexponent, numeric)) {
 321                 exponent_is_numerical = true;
 322                 num_exponent = &ex_to<numeric>(eexponent);
 323         }
 324
 325         // ^(x,0) -> 1  (0^0 also handled here)
 326         if (eexponent.is_zero()) {
 327                 if (ebasis.is_zero())
 328                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 329                 else
 330                         return _ex1;
 331         }
 332
 333         // ^(x,1) -> x
 334         if (eexponent.is_equal(_ex1))
 335                 return ebasis;
 336
 337         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 338         if (ebasis.is_zero() && exponent_is_numerical) {
 339                 if ((num_exponent->real()).is_zero())
 340                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 341                 else if ((num_exponent->real()).is_negative())
 342                         throw (pole_error("power::eval(): division by zero",1));
 343                 else
 344                         return _ex0;
 345         }
 346
 347         // ^(1,x) -> 1
 348         if (ebasis.is_equal(_ex1))
 349                 return _ex1;
 350
 351         if (exponent_is_numerical) {
 352
 353                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 354                 // except if c1,c2 are rational, but c1^c2 is not)
 355                 if (basis_is_numerical) {
 356                         const bool basis_is_crational = num_basis->is_crational();
 357                         const bool exponent_is_crational = num_exponent->is_crational();
 358                         if (!basis_is_crational || !exponent_is_crational) {
 359                                 // return a plain float
 360                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 361                                                                                                status_flags::evaluated |
 362                                                                                                status_flags::expanded);
 363                         }
 364
 365                         const numeric res = num_basis->power(*num_exponent);
 366                         if (res.is_crational()) {
 367                                 return res;
 368                         }
 369                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 370
 371                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 372                         if (basis_is_crational && exponent_is_crational
 373                             && num_exponent->is_real()
 374                             && !num_exponent->is_integer()) {
 375                                 const numeric n = num_exponent->numer();
 376                                 const numeric m = num_exponent->denom();
 377                                 numeric r;
 378                                 numeric q = iquo(n, m, r);
 379                                 if (r.is_negative()) {
 380                                         r += m;
 381                                         --q;
 382                                 }
 383                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 384                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 385                                                 // try it for numerator and denominator separately, in order to
 386                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 387                                                 const numeric bnum = num_basis->numer();
 388                                                 const numeric bden = num_basis->denom();
 389                                                 const numeric res_bnum = bnum.power(*num_exponent);
 390                                                 const numeric res_bden = bden.power(*num_exponent);
 391                                                 if (res_bnum.is_integer())
 392                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 393                                                 if (res_bden.is_integer())
 394                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 395                                         }
 396                                         return this->hold();
 397                                 } else {
 398                                         // assemble resulting product, but allowing for a re-evaluation,
 399                                         // because otherwise we'll end up with something like
 400                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 401                                         // instead of 7/16*7^(1/3).
 402                                         ex prod = power(*num_basis,r.div(m));
 403                                         return prod*power(*num_basis,q);
 404                                 }
 405                         }
 406                 }
 407
 408                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 409                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 410                 // case c1==1 should not happen, see below!)
 411                 if (is_ex_exactly_of_type(ebasis,power)) {
 412                         const power & sub_power = ex_to<power>(ebasis);
 413                         const ex & sub_basis = sub_power.basis;
 414                         const ex & sub_exponent = sub_power.exponent;
 415                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 416                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 417                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 418                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1).is_negative())
 419                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 420                         }
 421                 }
 422
 423                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 424                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
 425                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
 426                 }
 427
 428                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 429                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 430                 if (is_ex_exactly_of_type(ebasis,mul)) {
 431                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 432                         const mul & mulref = ex_to<mul>(ebasis);
 433                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 434                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 435                                 if (num_coeff.is_real()) {
 436                                         if (num_coeff.is_positive()) {
 437                                                 mul *mulp = new mul(mulref);
 438                                                 mulp->overall_coeff = _ex1;
 439                                                 mulp->clearflag(status_flags::evaluated);
 440                                                 mulp->clearflag(status_flags::hash_calculated);
 441                                                 return (new mul(power(*mulp,exponent),
 442                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 443                                         } else {
 444                                                 GINAC_ASSERT(num_coeff.compare(_num0)<0);
 445                                                 if (!num_coeff.is_equal(_num_1)) {
 446                                                         mul *mulp = new mul(mulref);
 447                                                         mulp->overall_coeff = _ex_1;
 448                                                         mulp->clearflag(status_flags::evaluated);
 449                                                         mulp->clearflag(status_flags::hash_calculated);
 450                                                         return (new mul(power(*mulp,exponent),
 451                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 452                                                 }
 453                                         }
 454                                 }
 455                         }
 456                 }
 457
 458                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 459                 if (num_exponent->is_pos_integer() &&
 460                     ebasis.return_type() != return_types::commutative &&
 461                     !is_ex_of_type(ebasis,matrix)) {
 462                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 463                 }
 464         }
 465
 466         if (are_ex_trivially_equal(ebasis,basis) &&
 467             are_ex_trivially_equal(eexponent,exponent)) {
 468                 return this->hold();
 469         }
 470         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 471                                                        status_flags::evaluated);
 472 }
 473
 474 ex power::evalf(int level) const
 475 {
 476         ex ebasis;
 477         ex eexponent;
 478
 479         if (level==1) {
 480                 ebasis = basis;
 481                 eexponent = exponent;
 482         } else if (level == -max_recursion_level) {
 483                 throw(std::runtime_error("max recursion level reached"));
 484         } else {
 485                 ebasis = basis.evalf(level-1);
 486                 if (!is_exactly_a<numeric>(exponent))
 487                         eexponent = exponent.evalf(level-1);
 488                 else
 489                         eexponent = exponent;
 490         }
 491
 492         return power(ebasis,eexponent);
 493 }
 494
 495 ex power::evalm(void) const
 496 {
 497         const ex ebasis = basis.evalm();
 498         const ex eexponent = exponent.evalm();
 499         if (is_ex_of_type(ebasis,matrix)) {
 500                 if (is_ex_of_type(eexponent,numeric)) {
 501                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 502                 }
 503         }
 504         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 505 }
 506
 507 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
 508 {
 509         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
 510         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
 511
 512         if (are_ex_trivially_equal(basis, subsed_basis)
 513          && are_ex_trivially_equal(exponent, subsed_exponent))
 514                 return basic::subs(ls, lr, no_pattern);
 515         else
 516                 return power(subsed_basis, subsed_exponent).basic::subs(ls, lr, no_pattern);
 517 }
 518
 519 ex power::simplify_ncmul(const exvector & v) const
 520 {
 521         return inherited::simplify_ncmul(v);
 522 }
 523
 524 // protected
 525
 526 /** Implementation of ex::diff() for a power.
 527  *  @see ex::diff */
 528 ex power::derivative(const symbol & s) const
 529 {
 530         if (exponent.info(info_flags::real)) {
 531                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 532                 epvector newseq;
 533                 newseq.reserve(2);
 534                 newseq.push_back(expair(basis, exponent - _ex1));
 535                 newseq.push_back(expair(basis.diff(s), _ex1));
 536                 return mul(newseq, exponent);
 537         } else {
 538                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 539                 return mul(*this,
 540                            add(mul(exponent.diff(s), log(basis)),
 541                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 542         }
 543 }
 544
 545 int power::compare_same_type(const basic & other) const
 546 {
 547         GINAC_ASSERT(is_exactly_a<power>(other));
 548         const power &o = static_cast<const power &>(other);
 549
 550         int cmpval = basis.compare(o.basis);
 551         if (cmpval)
 552                 return cmpval;
 553         else
 554                 return exponent.compare(o.exponent);
 555 }
 556
 557 unsigned power::return_type(void) const
 558 {
 559         return basis.return_type();
 560 }
 561
 562 unsigned power::return_type_tinfo(void) const
 563 {
 564         return basis.return_type_tinfo();
 565 }
 566
 567 ex power::expand(unsigned options) const
 568 {
 569         if (options == 0 && (flags & status_flags::expanded))
 570                 return *this;
 571
 572         const ex expanded_basis = basis.expand(options);
 573         const ex expanded_exponent = exponent.expand(options);
 574
 575         // x^(a+b) -> x^a * x^b
 576         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 577                 const add &a = ex_to<add>(expanded_exponent);
 578                 exvector distrseq;
 579                 distrseq.reserve(a.seq.size() + 1);
 580                 epvector::const_iterator last = a.seq.end();
 581                 epvector::const_iterator cit = a.seq.begin();
 582                 while (cit!=last) {
 583                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 584                         ++cit;
 585                 }
 586
 587                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 588                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 589                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 590                         int int_exponent = num_exponent.to_int();
 591                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 592                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
 593                         else
 594                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 595                 } else
 596                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 597
 598                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 599                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 600                 return r.expand();
 601         }
 602
 603         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 604                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 605                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 606                         return this->hold();
 607                 } else {
 608                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 609                 }
 610         }
 611
 612         // integer numeric exponent
 613         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 614         int int_exponent = num_exponent.to_int();
 615
 616         // (x+y)^n, n>0
 617         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
 618                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
 619
 620         // (x*y)^n -> x^n * y^n
 621         if (is_ex_exactly_of_type(expanded_basis,mul))
 622                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
 623
 624         // cannot expand further
 625         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 626                 return this->hold();
 627         else
 628                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 629 }
 630
 631 //////////
 632 // new virtual functions which can be overridden by derived classes
 633 //////////
 634
 635 // none
 636
 637 //////////
 638 // non-virtual functions in this class
 639 //////////
 640
 641 /** expand a^n where a is an add and n is an integer.
 642  *  @see power::expand */
 643 ex power::expand_add(const add & a, int n) const
 644 {
 645         if (n==2)
 646                 return expand_add_2(a);
 647
 648         int m = a.nops();
 649         exvector sum;
 650         sum.reserve((n+1)*(m-1));
 651         intvector k(m-1);
 652         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 653         intvector upper_limit(m-1);
 654         int l;
 655
 656         for (int l=0; l<m-1; l++) {
 657                 k[l] = 0;
 658                 k_cum[l] = 0;
 659                 upper_limit[l] = n;
 660         }
 661
 662         while (true) {
 663                 exvector term;
 664                 term.reserve(m+1);
 665                 for (l=0; l<m-1; l++) {
 666                         const ex & b = a.op(l);
 667                         GINAC_ASSERT(!is_exactly_a<add>(b));
 668                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 669                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 670                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 671                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 672                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 673                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 674                         if (is_ex_exactly_of_type(b,mul))
 675                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
 676                         else
 677                                 term.push_back(power(b,k[l]));
 678                 }
 679
 680                 const ex & b = a.op(l);
 681                 GINAC_ASSERT(!is_exactly_a<add>(b));
 682                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 683                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 684                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 685                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 686                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 687                              !is_exactly_a<power>(ex_to<power>(b).basis));
 688                 if (is_ex_exactly_of_type(b,mul))
 689                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
 690                 else
 691                         term.push_back(power(b,n-k_cum[m-2]));
 692
 693                 numeric f = binomial(numeric(n),numeric(k[0]));
 694                 for (l=1; l<m-1; l++)
 695                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 696
 697                 term.push_back(f);
 698
 699                 // TODO: Can we optimize this?  Alex seemed to think so...
 700                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 701
 702                 // increment k[]
 703                 l = m-2;
 704                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 705                         k[l] = 0;
 706                         --l;
 707                 }
 708                 if (l<0) break;
 709
 710                 // recalc k_cum[] and upper_limit[]
 711                 if (l==0)
 712                         k_cum[0] = k[0];
 713                 else
 714                         k_cum[l] = k_cum[l-1]+k[l];
 715
 716                 for (int i=l+1; i<m-1; i++)
 717                         k_cum[i] = k_cum[i-1]+k[i];
 718
 719                 for (int i=l+1; i<m-1; i++)
 720                         upper_limit[i] = n-k_cum[i-1];
 721         }
 722         return (new add(sum))->setflag(status_flags::dynallocated |
 723                                        status_flags::expanded );
 724 }
 725
 726
 727 /** Special case of power::expand_add. Expands a^2 where a is an add.
 728  *  @see power::expand_add */
 729 ex power::expand_add_2(const add & a) const
 730 {
 731         epvector sum;
 732         unsigned a_nops = a.nops();
 733         sum.reserve((a_nops*(a_nops+1))/2);
 734         epvector::const_iterator last = a.seq.end();
 735
 736         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 737         // first part: ignore overall_coeff and expand other terms
 738         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 739                 const ex & r = cit0->rest;
 740                 const ex & c = cit0->coeff;
 741
 742                 GINAC_ASSERT(!is_exactly_a<add>(r));
 743                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 744                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 745                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 746                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 747                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 748                              !is_exactly_a<power>(ex_to<power>(r).basis));
 749
 750                 if (are_ex_trivially_equal(c,_ex1)) {
 751                         if (is_ex_exactly_of_type(r,mul)) {
 752                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
 753                                                      _ex1));
 754                         } else {
 755                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 756                                                      _ex1));
 757                         }
 758                 } else {
 759                         if (is_ex_exactly_of_type(r,mul)) {
 760                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
 761                                                      ex_to<numeric>(c).power_dyn(_num2)));
 762                         } else {
 763                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 764                                                      ex_to<numeric>(c).power_dyn(_num2)));
 765                         }
 766                 }
 767
 768                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 769                         const ex & r1 = cit1->rest;
 770                         const ex & c1 = cit1->coeff;
 771                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 772                                                                       _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 773                 }
 774         }
 775
 776         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 777
 778         // second part: add terms coming from overall_factor (if != 0)
 779         if (!a.overall_coeff.is_zero()) {
 780                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 781                 while (i != end) {
 782                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
 783                         ++i;
 784                 }
 785                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
 786         }
 787
 788         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 789
 790         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 791 }
 792
 793 /** Expand factors of m in m^n where m is a mul and n is and integer
 794  *  @see power::expand */
 795 ex power::expand_mul(const mul & m, const numeric & n) const
 796 {
 797         if (n.is_zero())
 798                 return _ex1;
 799
 800         epvector distrseq;
 801         distrseq.reserve(m.seq.size());
 802         epvector::const_iterator last = m.seq.end();
 803         epvector::const_iterator cit = m.seq.begin();
 804         while (cit!=last) {
 805                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 806                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 807                 } else {
 808                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 809                         // since n is an integer
 810                         distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
 811                 }
 812                 ++cit;
 813         }
 814         return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 815 }
 816
 817 } // namespace GiNaC