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