ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2002 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.is_equal(_ex_1)) {
 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 if (is_a<print_python_repr>(c)) {
 163
 164                 c.s << class_name() << '(';
 165                 basis.print(c);
 166                 c.s << ',';
 167                 exponent.print(c);
 168                 c.s << ')';
 169
 170         } else {
 171
 172                 if (exponent.is_equal(_ex1_2)) {
 173                         if (is_a<print_latex>(c))
 174                                 c.s << "\\sqrt{";
 175                         else
 176                                 c.s << "sqrt(";
 177                         basis.print(c);
 178                         if (is_a<print_latex>(c))
 179                                 c.s << '}';
 180                         else
 181                                 c.s << ')';
 182                 } else {
 183                         if (precedence() <= level) {
 184                                 if (is_a<print_latex>(c))
 185                                         c.s << "{(";
 186                                 else
 187                                         c.s << "(";
 188                         }
 189                         basis.print(c, precedence());
 190                         if (is_a<print_python>(c))
 191                                 c.s << "**";
 192                         else
 193                                 c.s << '^';
 194                         if (is_a<print_latex>(c))
 195                                 c.s << '{';
 196                         exponent.print(c, precedence());
 197                         if (is_a<print_latex>(c))
 198                                 c.s << '}';
 199                         if (precedence() <= level) {
 200                                 if (is_a<print_latex>(c))
 201                                         c.s << ")}";
 202                                 else
 203                                         c.s << ')';
 204                         }
 205                 }
 206         }
 207 }
 208
 209 bool power::info(unsigned inf) const
 210 {
 211         switch (inf) {
 212                 case info_flags::polynomial:
 213                 case info_flags::integer_polynomial:
 214                 case info_flags::cinteger_polynomial:
 215                 case info_flags::rational_polynomial:
 216                 case info_flags::crational_polynomial:
 217                         return exponent.info(info_flags::nonnegint);
 218                 case info_flags::rational_function:
 219                         return exponent.info(info_flags::integer);
 220                 case info_flags::algebraic:
 221                         return (!exponent.info(info_flags::integer) ||
 222                                         basis.info(inf));
 223         }
 224         return inherited::info(inf);
 225 }
 226
 227 unsigned power::nops() const
 228 {
 229         return 2;
 230 }
 231
 232 ex & power::let_op(int i)
 233 {
 234         GINAC_ASSERT(i>=0);
 235         GINAC_ASSERT(i<2);
 236
 237         return i==0 ? basis : exponent;
 238 }
 239
 240 ex power::map(map_function & f) const
 241 {
 242         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
 243 }
 244
 245 int power::degree(const ex & s) const
 246 {
 247         if (is_equal(ex_to<basic>(s)))
 248                 return 1;
 249         else 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.degree(s) * ex_to<numeric>(exponent).to_int();
 254         } else if (basis.has(s))
 255                 throw(std::runtime_error("power::degree(): undefined degree because of non-integer exponent"));
 256         else
 257                 return 0;
 258 }
 259
 260 int power::ldegree(const ex & s) const
 261 {
 262         if (is_equal(ex_to<basic>(s)))
 263                 return 1;
 264         else if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
 265                 if (basis.is_equal(s))
 266                         return ex_to<numeric>(exponent).to_int();
 267                 else
 268                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 269         } else if (basis.has(s))
 270                 throw(std::runtime_error("power::ldegree(): undefined degree because of non-integer exponent"));
 271         else
 272                 return 0;
 273 }
 274
 275 ex power::coeff(const ex & s, int n) const
 276 {
 277         if (is_equal(ex_to<basic>(s)))
 278                 return n==1 ? _ex1 : _ex0;
 279         else if (!basis.is_equal(s)) {
 280                 // basis not equal to s
 281                 if (n == 0)
 282                         return *this;
 283                 else
 284                         return _ex0;
 285         } else {
 286                 // basis equal to s
 287                 if (is_ex_exactly_of_type(exponent, numeric) && ex_to<numeric>(exponent).is_integer()) {
 288                         // integer exponent
 289                         int int_exp = ex_to<numeric>(exponent).to_int();
 290                         if (n == int_exp)
 291                                 return _ex1;
 292                         else
 293                                 return _ex0;
 294                 } else {
 295                         // non-integer exponents are treated as zero
 296                         if (n == 0)
 297                                 return *this;
 298                         else
 299                                 return _ex0;
 300                 }
 301         }
 302 }
 303
 304 /** Perform automatic term rewriting rules in this class.  In the following
 305  *  x, x1, x2,... stand for a symbolic variables of type ex and c, c1, c2...
 306  *  stand for such expressions that contain a plain number.
 307  *  - ^(x,0) -> 1  (also handles ^(0,0))
 308  *  - ^(x,1) -> x
 309  *  - ^(0,c) -> 0 or exception  (depending on the real part of c)
 310  *  - ^(1,x) -> 1
 311  *  - ^(c1,c2) -> *(c1^n,c1^(c2-n))  (so that 0<(c2-n)<1, try to evaluate roots, possibly in numerator and denominator of c1)
 312  *  - ^(^(x,c1),c2) -> ^(x,c1*c2)  (c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 313  *  - ^(*(x,y,z),c) -> *(x^c,y^c,z^c)  (if c integer)
 314  *  - ^(*(x,c1),c2) -> ^(x,c2)*c1^c2  (c1>0)
 315  *  - ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2  (c1<0)
 316  *
 317  *  @param level cut-off in recursive evaluation */
 318 ex power::eval(int level) const
 319 {
 320         if ((level==1) && (flags & status_flags::evaluated))
 321                 return *this;
 322         else if (level == -max_recursion_level)
 323                 throw(std::runtime_error("max recursion level reached"));
 324
 325         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 326         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 327
 328         bool basis_is_numerical = false;
 329         bool exponent_is_numerical = false;
 330         const numeric *num_basis;
 331         const numeric *num_exponent;
 332
 333         if (is_ex_exactly_of_type(ebasis, numeric)) {
 334                 basis_is_numerical = true;
 335                 num_basis = &ex_to<numeric>(ebasis);
 336         }
 337         if (is_ex_exactly_of_type(eexponent, numeric)) {
 338                 exponent_is_numerical = true;
 339                 num_exponent = &ex_to<numeric>(eexponent);
 340         }
 341
 342         // ^(x,0) -> 1  (0^0 also handled here)
 343         if (eexponent.is_zero()) {
 344                 if (ebasis.is_zero())
 345                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 346                 else
 347                         return _ex1;
 348         }
 349
 350         // ^(x,1) -> x
 351         if (eexponent.is_equal(_ex1))
 352                 return ebasis;
 353
 354         // ^(0,c1) -> 0 or exception  (depending on real value of c1)
 355         if (ebasis.is_zero() && exponent_is_numerical) {
 356                 if ((num_exponent->real()).is_zero())
 357                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 358                 else if ((num_exponent->real()).is_negative())
 359                         throw (pole_error("power::eval(): division by zero",1));
 360                 else
 361                         return _ex0;
 362         }
 363
 364         // ^(1,x) -> 1
 365         if (ebasis.is_equal(_ex1))
 366                 return _ex1;
 367
 368         if (exponent_is_numerical) {
 369
 370                 // ^(c1,c2) -> c1^c2  (c1, c2 numeric(),
 371                 // except if c1,c2 are rational, but c1^c2 is not)
 372                 if (basis_is_numerical) {
 373                         const bool basis_is_crational = num_basis->is_crational();
 374                         const bool exponent_is_crational = num_exponent->is_crational();
 375                         if (!basis_is_crational || !exponent_is_crational) {
 376                                 // return a plain float
 377                                 return (new numeric(num_basis->power(*num_exponent)))->setflag(status_flags::dynallocated |
 378                                                                                                status_flags::evaluated |
 379                                                                                                status_flags::expanded);
 380                         }
 381
 382                         const numeric res = num_basis->power(*num_exponent);
 383                         if (res.is_crational()) {
 384                                 return res;
 385                         }
 386                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 387
 388                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-q)<1, q integer
 389                         if (basis_is_crational && exponent_is_crational
 390                             && num_exponent->is_real()
 391                             && !num_exponent->is_integer()) {
 392                                 const numeric n = num_exponent->numer();
 393                                 const numeric m = num_exponent->denom();
 394                                 numeric r;
 395                                 numeric q = iquo(n, m, r);
 396                                 if (r.is_negative()) {
 397                                         r += m;
 398                                         --q;
 399                                 }
 400                                 if (q.is_zero()) {  // the exponent was in the allowed range 0<(n/m)<1
 401                                         if (num_basis->is_rational() && !num_basis->is_integer()) {
 402                                                 // try it for numerator and denominator separately, in order to
 403                                                 // partially simplify things like (5/8)^(1/3) -> 1/2*5^(1/3)
 404                                                 const numeric bnum = num_basis->numer();
 405                                                 const numeric bden = num_basis->denom();
 406                                                 const numeric res_bnum = bnum.power(*num_exponent);
 407                                                 const numeric res_bden = bden.power(*num_exponent);
 408                                                 if (res_bnum.is_integer())
 409                                                         return (new mul(power(bden,-*num_exponent),res_bnum))->setflag(status_flags::dynallocated | status_flags::evaluated);
 410                                                 if (res_bden.is_integer())
 411                                                         return (new mul(power(bnum,*num_exponent),res_bden.inverse()))->setflag(status_flags::dynallocated | status_flags::evaluated);
 412                                         }
 413                                         return this->hold();
 414                                 } else {
 415                                         // assemble resulting product, but allowing for a re-evaluation,
 416                                         // because otherwise we'll end up with something like
 417                                         //    (7/8)^(4/3)  ->  7/8*(1/2*7^(1/3))
 418                                         // instead of 7/16*7^(1/3).
 419                                         ex prod = power(*num_basis,r.div(m));
 420                                         return prod*power(*num_basis,q);
 421                                 }
 422                         }
 423                 }
 424
 425                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 426                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 427                 // case c1==1 should not happen, see below!)
 428                 if (is_ex_exactly_of_type(ebasis,power)) {
 429                         const power & sub_power = ex_to<power>(ebasis);
 430                         const ex & sub_basis = sub_power.basis;
 431                         const ex & sub_exponent = sub_power.exponent;
 432                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 433                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 434                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 435                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1).is_negative())
 436                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 437                         }
 438                 }
 439
 440                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 441                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
 442                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
 443                 }
 444
 445                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;1),c2),c1^c2)  (c1, c2 numeric(), c1>0)
 446                 // ^(*(...,x;c1),c2) -> *(^(*(...,x;-1),c2),(-c1)^c2)  (c1, c2 numeric(), c1<0)
 447                 if (is_ex_exactly_of_type(ebasis,mul)) {
 448                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 449                         const mul & mulref = ex_to<mul>(ebasis);
 450                         if (!mulref.overall_coeff.is_equal(_ex1)) {
 451                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 452                                 if (num_coeff.is_real()) {
 453                                         if (num_coeff.is_positive()) {
 454                                                 mul *mulp = new mul(mulref);
 455                                                 mulp->overall_coeff = _ex1;
 456                                                 mulp->clearflag(status_flags::evaluated);
 457                                                 mulp->clearflag(status_flags::hash_calculated);
 458                                                 return (new mul(power(*mulp,exponent),
 459                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 460                                         } else {
 461                                                 GINAC_ASSERT(num_coeff.compare(_num0)<0);
 462                                                 if (!num_coeff.is_equal(_num_1)) {
 463                                                         mul *mulp = new mul(mulref);
 464                                                         mulp->overall_coeff = _ex_1;
 465                                                         mulp->clearflag(status_flags::evaluated);
 466                                                         mulp->clearflag(status_flags::hash_calculated);
 467                                                         return (new mul(power(*mulp,exponent),
 468                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 469                                                 }
 470                                         }
 471                                 }
 472                         }
 473                 }
 474
 475                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 476                 if (num_exponent->is_pos_integer() &&
 477                     ebasis.return_type() != return_types::commutative &&
 478                     !is_ex_of_type(ebasis,matrix)) {
 479                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 480                 }
 481         }
 482
 483         if (are_ex_trivially_equal(ebasis,basis) &&
 484             are_ex_trivially_equal(eexponent,exponent)) {
 485                 return this->hold();
 486         }
 487         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 488                                                        status_flags::evaluated);
 489 }
 490
 491 ex power::evalf(int level) const
 492 {
 493         ex ebasis;
 494         ex eexponent;
 495
 496         if (level==1) {
 497                 ebasis = basis;
 498                 eexponent = exponent;
 499         } else if (level == -max_recursion_level) {
 500                 throw(std::runtime_error("max recursion level reached"));
 501         } else {
 502                 ebasis = basis.evalf(level-1);
 503                 if (!is_exactly_a<numeric>(exponent))
 504                         eexponent = exponent.evalf(level-1);
 505                 else
 506                         eexponent = exponent;
 507         }
 508
 509         return power(ebasis,eexponent);
 510 }
 511
 512 ex power::evalm(void) const
 513 {
 514         const ex ebasis = basis.evalm();
 515         const ex eexponent = exponent.evalm();
 516         if (is_ex_of_type(ebasis,matrix)) {
 517                 if (is_ex_of_type(eexponent,numeric)) {
 518                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 519                 }
 520         }
 521         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 522 }
 523
 524 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
 525 {
 526         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
 527         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
 528
 529         if (are_ex_trivially_equal(basis, subsed_basis)
 530          && are_ex_trivially_equal(exponent, subsed_exponent))
 531                 return basic::subs(ls, lr, no_pattern);
 532         else
 533                 return power(subsed_basis, subsed_exponent).basic::subs(ls, lr, no_pattern);
 534 }
 535
 536 ex power::simplify_ncmul(const exvector & v) const
 537 {
 538         return inherited::simplify_ncmul(v);
 539 }
 540
 541 // protected
 542
 543 /** Implementation of ex::diff() for a power.
 544  *  @see ex::diff */
 545 ex power::derivative(const symbol & s) const
 546 {
 547         if (exponent.info(info_flags::real)) {
 548                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 549                 epvector newseq;
 550                 newseq.reserve(2);
 551                 newseq.push_back(expair(basis, exponent - _ex1));
 552                 newseq.push_back(expair(basis.diff(s), _ex1));
 553                 return mul(newseq, exponent);
 554         } else {
 555                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 556                 return mul(*this,
 557                            add(mul(exponent.diff(s), log(basis)),
 558                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1))));
 559         }
 560 }
 561
 562 int power::compare_same_type(const basic & other) const
 563 {
 564         GINAC_ASSERT(is_exactly_a<power>(other));
 565         const power &o = static_cast<const power &>(other);
 566
 567         int cmpval = basis.compare(o.basis);
 568         if (cmpval)
 569                 return cmpval;
 570         else
 571                 return exponent.compare(o.exponent);
 572 }
 573
 574 unsigned power::return_type(void) const
 575 {
 576         return basis.return_type();
 577 }
 578
 579 unsigned power::return_type_tinfo(void) const
 580 {
 581         return basis.return_type_tinfo();
 582 }
 583
 584 ex power::expand(unsigned options) const
 585 {
 586         if (options == 0 && (flags & status_flags::expanded))
 587                 return *this;
 588
 589         const ex expanded_basis = basis.expand(options);
 590         const ex expanded_exponent = exponent.expand(options);
 591
 592         // x^(a+b) -> x^a * x^b
 593         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 594                 const add &a = ex_to<add>(expanded_exponent);
 595                 exvector distrseq;
 596                 distrseq.reserve(a.seq.size() + 1);
 597                 epvector::const_iterator last = a.seq.end();
 598                 epvector::const_iterator cit = a.seq.begin();
 599                 while (cit!=last) {
 600                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 601                         ++cit;
 602                 }
 603
 604                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 605                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 606                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 607                         int int_exponent = num_exponent.to_int();
 608                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 609                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
 610                         else
 611                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 612                 } else
 613                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 614
 615                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 616                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 617                 return r.expand();
 618         }
 619
 620         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 621                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 622                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 623                         return this->hold();
 624                 } else {
 625                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 626                 }
 627         }
 628
 629         // integer numeric exponent
 630         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 631         int int_exponent = num_exponent.to_int();
 632
 633         // (x+y)^n, n>0
 634         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
 635                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
 636
 637         // (x*y)^n -> x^n * y^n
 638         if (is_ex_exactly_of_type(expanded_basis,mul))
 639                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
 640
 641         // cannot expand further
 642         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 643                 return this->hold();
 644         else
 645                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 646 }
 647
 648 //////////
 649 // new virtual functions which can be overridden by derived classes
 650 //////////
 651
 652 // none
 653
 654 //////////
 655 // non-virtual functions in this class
 656 //////////
 657
 658 /** expand a^n where a is an add and n is an integer.
 659  *  @see power::expand */
 660 ex power::expand_add(const add & a, int n) const
 661 {
 662         if (n==2)
 663                 return expand_add_2(a);
 664
 665         int m = a.nops();
 666         exvector sum;
 667         sum.reserve((n+1)*(m-1));
 668         intvector k(m-1);
 669         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 670         intvector upper_limit(m-1);
 671         int l;
 672
 673         for (int l=0; l<m-1; l++) {
 674                 k[l] = 0;
 675                 k_cum[l] = 0;
 676                 upper_limit[l] = n;
 677         }
 678
 679         while (true) {
 680                 exvector term;
 681                 term.reserve(m+1);
 682                 for (l=0; l<m-1; l++) {
 683                         const ex & b = a.op(l);
 684                         GINAC_ASSERT(!is_exactly_a<add>(b));
 685                         GINAC_ASSERT(!is_exactly_a<power>(b) ||
 686                                      !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 687                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 688                                      !is_exactly_a<add>(ex_to<power>(b).basis) ||
 689                                      !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 690                                      !is_exactly_a<power>(ex_to<power>(b).basis));
 691                         if (is_ex_exactly_of_type(b,mul))
 692                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
 693                         else
 694                                 term.push_back(power(b,k[l]));
 695                 }
 696
 697                 const ex & b = a.op(l);
 698                 GINAC_ASSERT(!is_exactly_a<add>(b));
 699                 GINAC_ASSERT(!is_exactly_a<power>(b) ||
 700                              !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
 701                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 702                              !is_exactly_a<add>(ex_to<power>(b).basis) ||
 703                              !is_exactly_a<mul>(ex_to<power>(b).basis) ||
 704                              !is_exactly_a<power>(ex_to<power>(b).basis));
 705                 if (is_ex_exactly_of_type(b,mul))
 706                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
 707                 else
 708                         term.push_back(power(b,n-k_cum[m-2]));
 709
 710                 numeric f = binomial(numeric(n),numeric(k[0]));
 711                 for (l=1; l<m-1; l++)
 712                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 713
 714                 term.push_back(f);
 715
 716                 // TODO: Can we optimize this?  Alex seemed to think so...
 717                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 718
 719                 // increment k[]
 720                 l = m-2;
 721                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 722                         k[l] = 0;
 723                         --l;
 724                 }
 725                 if (l<0) break;
 726
 727                 // recalc k_cum[] and upper_limit[]
 728                 if (l==0)
 729                         k_cum[0] = k[0];
 730                 else
 731                         k_cum[l] = k_cum[l-1]+k[l];
 732
 733                 for (int i=l+1; i<m-1; i++)
 734                         k_cum[i] = k_cum[i-1]+k[i];
 735
 736                 for (int i=l+1; i<m-1; i++)
 737                         upper_limit[i] = n-k_cum[i-1];
 738         }
 739         return (new add(sum))->setflag(status_flags::dynallocated |
 740                                        status_flags::expanded );
 741 }
 742
 743
 744 /** Special case of power::expand_add. Expands a^2 where a is an add.
 745  *  @see power::expand_add */
 746 ex power::expand_add_2(const add & a) const
 747 {
 748         epvector sum;
 749         unsigned a_nops = a.nops();
 750         sum.reserve((a_nops*(a_nops+1))/2);
 751         epvector::const_iterator last = a.seq.end();
 752
 753         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 754         // first part: ignore overall_coeff and expand other terms
 755         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 756                 const ex & r = cit0->rest;
 757                 const ex & c = cit0->coeff;
 758
 759                 GINAC_ASSERT(!is_exactly_a<add>(r));
 760                 GINAC_ASSERT(!is_exactly_a<power>(r) ||
 761                              !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
 762                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 763                              !is_exactly_a<add>(ex_to<power>(r).basis) ||
 764                              !is_exactly_a<mul>(ex_to<power>(r).basis) ||
 765                              !is_exactly_a<power>(ex_to<power>(r).basis));
 766
 767                 if (are_ex_trivially_equal(c,_ex1)) {
 768                         if (is_ex_exactly_of_type(r,mul)) {
 769                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
 770                                                      _ex1));
 771                         } else {
 772                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 773                                                      _ex1));
 774                         }
 775                 } else {
 776                         if (is_ex_exactly_of_type(r,mul)) {
 777                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2),
 778                                                      ex_to<numeric>(c).power_dyn(_num2)));
 779                         } else {
 780                                 sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
 781                                                      ex_to<numeric>(c).power_dyn(_num2)));
 782                         }
 783                 }
 784
 785                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 786                         const ex & r1 = cit1->rest;
 787                         const ex & c1 = cit1->coeff;
 788                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 789                                                                       _num2.mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 790                 }
 791         }
 792
 793         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 794
 795         // second part: add terms coming from overall_factor (if != 0)
 796         if (!a.overall_coeff.is_zero()) {
 797                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 798                 while (i != end) {
 799                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2)));
 800                         ++i;
 801                 }
 802                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2),_ex1));
 803         }
 804
 805         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 806
 807         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 808 }
 809
 810 /** Expand factors of m in m^n where m is a mul and n is and integer
 811  *  @see power::expand */
 812 ex power::expand_mul(const mul & m, const numeric & n) const
 813 {
 814         if (n.is_zero())
 815                 return _ex1;
 816
 817         epvector distrseq;
 818         distrseq.reserve(m.seq.size());
 819         epvector::const_iterator last = m.seq.end();
 820         epvector::const_iterator cit = m.seq.begin();
 821         while (cit!=last) {
 822                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 823                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 824                 } else {
 825                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 826                         // since n is an integer
 827                         distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
 828                 }
 829                 ++cit;
 830         }
 831         return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 832 }
 833
 834 } // namespace GiNaC