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 "debugmsg.h"
  40 #include "utils.h"
  41
  42 namespace GiNaC {
  43
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
  45
  46 typedef std::vector<int> intvector;
  47
  48 //////////
  49 // default ctor, dtor, copy ctor assignment operator and helpers
  50 //////////
  51
  52 power::power() : inherited(TINFO_power)
  53 {
  54         debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
  55 }
  56
  57 void power::copy(const power & other)
  58 {
  59         inherited::copy(other);
  60         basis = other.basis;
  61         exponent = other.exponent;
  62 }
  63
  64 DEFAULT_DESTROY(power)
  65
  66 //////////
  67 // other ctors
  68 //////////
  69
  70 power::power(const ex & lh, const ex & rh) : inherited(TINFO_power), basis(lh), exponent(rh)
  71 {
  72         debugmsg("power ctor from ex,ex",LOGLEVEL_CONSTRUCT);
  73 }
  74
  75 /** Ctor from an ex and a bare numeric.  This is somewhat more efficient than
  76  *  the normal ctor from two ex whenever it can be used. */
  77 power::power(const ex & lh, const numeric & rh) : inherited(TINFO_power), basis(lh), exponent(rh)
  78 {
  79         debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
  80 }
  81
  82 //////////
  83 // archiving
  84 //////////
  85
  86 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  87 {
  88         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
  89         n.find_ex("basis", basis, sym_lst);
  90         n.find_ex("exponent", exponent, sym_lst);
  91 }
  92
  93 void power::archive(archive_node &n) const
  94 {
  95         inherited::archive(n);
  96         n.add_ex("basis", basis);
  97         n.add_ex("exponent", exponent);
  98 }
  99
 100 DEFAULT_UNARCHIVE(power)
 101
 102 //////////
 103 // functions overriding virtual functions from bases classes
 104 //////////
 105
 106 // public
 107
 108 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 109 {
 110         // Optimal output of integer powers of symbols to aid compiler CSE.
 111         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 112         // to learn why such a hack is really necessary.
 113         if (exp == 1) {
 114                 x.print(c);
 115         } else if (exp == 2) {
 116                 x.print(c);
 117                 c.s << "*";
 118                 x.print(c);
 119         } else if (exp & 1) {
 120                 x.print(c);
 121                 c.s << "*";
 122                 print_sym_pow(c, x, exp-1);
 123         } else {
 124                 c.s << "(";
 125                 print_sym_pow(c, x, exp >> 1);
 126                 c.s << ")*(";
 127                 print_sym_pow(c, x, exp >> 1);
 128                 c.s << ")";
 129         }
 130 }
 131
 132 void power::print(const print_context & c, unsigned level) const
 133 {
 134         debugmsg("power print", LOGLEVEL_PRINT);
 135
 136         if (is_a<print_tree>(c)) {
 137
 138                 inherited::print(c, level);
 139
 140         } else if (is_a<print_csrc>(c)) {
 141
 142                 // Integer powers of symbols are printed in a special, optimized way
 143                 if (exponent.info(info_flags::integer)
 144                  && (is_exactly_a<symbol>(basis) || is_exactly_a<constant>(basis))) {
 145                         int exp = ex_to<numeric>(exponent).to_int();
 146                         if (exp > 0)
 147                                 c.s << '(';
 148                         else {
 149                                 exp = -exp;
 150                                 if (is_a<print_csrc_cl_N>(c))
 151                                         c.s << "recip(";
 152                                 else
 153                                         c.s << "1.0/(";
 154                         }
 155                         print_sym_pow(c, ex_to<symbol>(basis), exp);
 156                         c.s << ')';
 157
 158                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 159                 } else if (exponent.compare(_num_1()) == 0) {
 160                         if (is_a<print_csrc_cl_N>(c))
 161                                 c.s << "recip(";
 162                         else
 163                                 c.s << "1.0/(";
 164                         basis.print(c);
 165                         c.s << ')';
 166
 167                 // Otherwise, use the pow() or expt() (CLN) functions
 168                 } else {
 169                         if (is_a<print_csrc_cl_N>(c))
 170                                 c.s << "expt(";
 171                         else
 172                                 c.s << "pow(";
 173                         basis.print(c);
 174                         c.s << ',';
 175                         exponent.print(c);
 176                         c.s << ')';
 177                 }
 178
 179         } else {
 180
 181                 if (exponent.is_equal(_ex1_2())) {
 182                         if (is_a<print_latex>(c))
 183                                 c.s << "\\sqrt{";
 184                         else
 185                                 c.s << "sqrt(";
 186                         basis.print(c);
 187                         if (is_a<print_latex>(c))
 188                                 c.s << '}';
 189                         else
 190                                 c.s << ')';
 191                 } else {
 192                         if (precedence() <= level) {
 193                                 if (is_a<print_latex>(c))
 194                                         c.s << "{(";
 195                                 else
 196                                         c.s << "(";
 197                         }
 198                         basis.print(c, precedence());
 199                         c.s << '^';
 200                         if (is_a<print_latex>(c))
 201                                 c.s << '{';
 202                         exponent.print(c, precedence());
 203                         if (is_a<print_latex>(c))
 204                                 c.s << '}';
 205                         if (precedence() <= level) {
 206                                 if (is_a<print_latex>(c))
 207                                         c.s << ")}";
 208                                 else
 209                                         c.s << ')';
 210                         }
 211                 }
 212         }
 213 }
 214
 215 bool power::info(unsigned inf) const
 216 {
 217         switch (inf) {
 218                 case info_flags::polynomial:
 219                 case info_flags::integer_polynomial:
 220                 case info_flags::cinteger_polynomial:
 221                 case info_flags::rational_polynomial:
 222                 case info_flags::crational_polynomial:
 223                         return exponent.info(info_flags::nonnegint);
 224                 case info_flags::rational_function:
 225                         return exponent.info(info_flags::integer);
 226                 case info_flags::algebraic:
 227                         return (!exponent.info(info_flags::integer) ||
 228                                         basis.info(inf));
 229         }
 230         return inherited::info(inf);
 231 }
 232
 233 unsigned power::nops() const
 234 {
 235         return 2;
 236 }
 237
 238 ex & power::let_op(int i)
 239 {
 240         GINAC_ASSERT(i>=0);
 241         GINAC_ASSERT(i<2);
 242
 243         return i==0 ? basis : exponent;
 244 }
 245
 246 ex power::map(map_function & f) const
 247 {
 248         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
 249 }
 250
 251 int power::degree(const ex & s) const
 252 {
 253         if (is_exactly_of_type(*exponent.bp,numeric)) {
 254                 if (basis.is_equal(s)) {
 255                         if (ex_to<numeric>(exponent).is_integer())
 256                                 return ex_to<numeric>(exponent).to_int();
 257                         else
 258                                 return 0;
 259                 } else
 260                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 261         }
 262         return 0;
 263 }
 264
 265 int power::ldegree(const ex & s) const
 266 {
 267         if (is_exactly_of_type(*exponent.bp,numeric)) {
 268                 if (basis.is_equal(s)) {
 269                         if (ex_to<numeric>(exponent).is_integer())
 270                                 return ex_to<numeric>(exponent).to_int();
 271                         else
 272                                 return 0;
 273                 } else
 274                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 275         }
 276         return 0;
 277 }
 278
 279 ex power::coeff(const ex & s, int n) const
 280 {
 281         if (!basis.is_equal(s)) {
 282                 // basis not equal to s
 283                 if (n == 0)
 284                         return *this;
 285                 else
 286                         return _ex0();
 287         } else {
 288                 // basis equal to s
 289                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to<numeric>(exponent).is_integer()) {
 290                         // integer exponent
 291                         int int_exp = ex_to<numeric>(exponent).to_int();
 292                         if (n == int_exp)
 293                                 return _ex1();
 294                         else
 295                                 return _ex0();
 296                 } else {
 297                         // non-integer exponents are treated as zero
 298                         if (n == 0)
 299                                 return *this;
 300                         else
 301                                 return _ex0();
 302                 }
 303         }
 304 }
 305
 306 ex power::eval(int level) const
 307 {
 308         // simplifications: ^(x,0) -> 1 (0^0 handled here)
 309         //                  ^(x,1) -> x
 310         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
 311         //                  ^(1,x) -> 1
 312         //                  ^(c1,c2) -> *(c1^n,c1^(c2-n)) (c1, c2 numeric(), 0<(c2-n)<1 except if c1,c2 are rational, but c1^c2 is not)
 313         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 314         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 315         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 316         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 317
 318         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 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         numeric * num_basis;
 331         numeric * num_exponent;
 332
 333         if (is_exactly_of_type(*ebasis.bp,numeric)) {
 334                 basis_is_numerical = true;
 335                 num_basis = static_cast<numeric *>(ebasis.bp);
 336         }
 337         if (is_exactly_of_type(*eexponent.bp,numeric)) {
 338                 exponent_is_numerical = true;
 339                 num_exponent = static_cast<numeric *>(eexponent.bp);
 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                         bool basis_is_crational = num_basis->is_crational();
 374                         bool exponent_is_crational = num_exponent->is_crational();
 375                         numeric res = num_basis->power(*num_exponent);
 376
 377                         if ((!basis_is_crational || !exponent_is_crational)
 378                                 || res.is_crational()) {
 379                                 return res;
 380                         }
 381                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 382
 383                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 384                         if (basis_is_crational && exponent_is_crational
 385                                 && num_exponent->is_real()
 386                                 && !num_exponent->is_integer()) {
 387                                 numeric n = num_exponent->numer();
 388                                 numeric m = num_exponent->denom();
 389                                 numeric r;
 390                                 numeric q = iquo(n, m, r);
 391                                 if (r.is_negative()) {
 392                                         r = r.add(m);
 393                                         q = q.sub(_num1());
 394                                 }
 395                                 if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 396                                         return this->hold();
 397                                 else {
 398                                         epvector res;
 399                                         res.push_back(expair(ebasis,r.div(m)));
 400                                         return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
 401                                 }
 402                         }
 403                 }
 404
 405                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 406                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 407                 // case c1==1 should not happen, see below!)
 408                 if (is_ex_exactly_of_type(ebasis,power)) {
 409                         const power & sub_power = ex_to<power>(ebasis);
 410                         const ex & sub_basis = sub_power.basis;
 411                         const ex & sub_exponent = sub_power.exponent;
 412                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 413                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 414                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 415                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1()).is_negative())
 416                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 417                         }
 418                 }
 419
 420                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 421                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
 422                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
 423                 }
 424
 425                 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 426                 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 427                 if (is_ex_exactly_of_type(ebasis,mul)) {
 428                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 429                         const mul & mulref = ex_to<mul>(ebasis);
 430                         if (!mulref.overall_coeff.is_equal(_ex1())) {
 431                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 432                                 if (num_coeff.is_real()) {
 433                                         if (num_coeff.is_positive()) {
 434                                                 mul * mulp = new mul(mulref);
 435                                                 mulp->overall_coeff = _ex1();
 436                                                 mulp->clearflag(status_flags::evaluated);
 437                                                 mulp->clearflag(status_flags::hash_calculated);
 438                                                 return (new mul(power(*mulp,exponent),
 439                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 440                                         } else {
 441                                                 GINAC_ASSERT(num_coeff.compare(_num0())<0);
 442                                                 if (num_coeff.compare(_num_1())!=0) {
 443                                                         mul * mulp = new mul(mulref);
 444                                                         mulp->overall_coeff = _ex_1();
 445                                                         mulp->clearflag(status_flags::evaluated);
 446                                                         mulp->clearflag(status_flags::hash_calculated);
 447                                                         return (new mul(power(*mulp,exponent),
 448                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 449                                                 }
 450                                         }
 451                                 }
 452                         }
 453                 }
 454
 455                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 456                 if (num_exponent->is_pos_integer() &&
 457                     ebasis.return_type() != return_types::commutative &&
 458                     !is_ex_of_type(ebasis,matrix)) {
 459                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 460                 }
 461         }
 462
 463         if (are_ex_trivially_equal(ebasis,basis) &&
 464                 are_ex_trivially_equal(eexponent,exponent)) {
 465                 return this->hold();
 466         }
 467         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 468                                                                                                    status_flags::evaluated);
 469 }
 470
 471 ex power::evalf(int level) const
 472 {
 473         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 474
 475         ex ebasis;
 476         ex eexponent;
 477
 478         if (level==1) {
 479                 ebasis = basis;
 480                 eexponent = exponent;
 481         } else if (level == -max_recursion_level) {
 482                 throw(std::runtime_error("max recursion level reached"));
 483         } else {
 484                 ebasis = basis.evalf(level-1);
 485                 if (!is_ex_exactly_of_type(eexponent,numeric))
 486                         eexponent = exponent.evalf(level-1);
 487                 else
 488                         eexponent = exponent;
 489         }
 490
 491         return power(ebasis,eexponent);
 492 }
 493
 494 ex power::evalm(void) const
 495 {
 496         ex ebasis = basis.evalm();
 497         ex eexponent = exponent.evalm();
 498         if (is_ex_of_type(ebasis,matrix)) {
 499                 if (is_ex_of_type(eexponent,numeric)) {
 500                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 501                 }
 502         }
 503         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 504 }
 505
 506 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
 507 {
 508         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
 509         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
 510
 511         if (are_ex_trivially_equal(basis, subsed_basis)
 512          && are_ex_trivially_equal(exponent, subsed_exponent))
 513                 return basic::subs(ls, lr, no_pattern);
 514         else
 515                 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
 516 }
 517
 518 ex power::simplify_ncmul(const exvector & v) const
 519 {
 520         return inherited::simplify_ncmul(v);
 521 }
 522
 523 // protected
 524
 525 /** Implementation of ex::diff() for a power.
 526  *  @see ex::diff */
 527 ex power::derivative(const symbol & s) const
 528 {
 529         if (exponent.info(info_flags::real)) {
 530                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 531                 epvector newseq;
 532                 newseq.reserve(2);
 533                 newseq.push_back(expair(basis, exponent - _ex1()));
 534                 newseq.push_back(expair(basis.diff(s), _ex1()));
 535                 return mul(newseq, exponent);
 536         } else {
 537                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 538                 return mul(*this,
 539                            add(mul(exponent.diff(s), log(basis)),
 540                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
 541         }
 542 }
 543
 544 int power::compare_same_type(const basic & other) const
 545 {
 546         GINAC_ASSERT(is_exactly_of_type(other, power));
 547         const power &o = static_cast<const power &>(other);
 548
 549         int cmpval = basis.compare(o.basis);
 550         if (cmpval)
 551                 return cmpval;
 552         else
 553                 return exponent.compare(o.exponent);
 554 }
 555
 556 unsigned power::return_type(void) const
 557 {
 558         return basis.return_type();
 559 }
 560
 561 unsigned power::return_type_tinfo(void) const
 562 {
 563         return basis.return_type_tinfo();
 564 }
 565
 566 ex power::expand(unsigned options) const
 567 {
 568         if (flags & status_flags::expanded)
 569                 return *this;
 570
 571         ex expanded_basis = basis.expand(options);
 572         ex expanded_exponent = exponent.expand(options);
 573
 574         // x^(a+b) -> x^a * x^b
 575         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 576                 const add &a = ex_to<add>(expanded_exponent);
 577                 exvector distrseq;
 578                 distrseq.reserve(a.seq.size() + 1);
 579                 epvector::const_iterator last = a.seq.end();
 580                 epvector::const_iterator cit = a.seq.begin();
 581                 while (cit!=last) {
 582                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 583                         cit++;
 584                 }
 585
 586                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 587                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 588                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 589                         int int_exponent = num_exponent.to_int();
 590                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 591                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
 592                         else
 593                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 594                 } else
 595                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 596
 597                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 598                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 599                 return r.expand();
 600         }
 601
 602         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 603                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 604                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 605                         return this->hold();
 606                 } else {
 607                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 608                 }
 609         }
 610
 611         // integer numeric exponent
 612         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 613         int int_exponent = num_exponent.to_int();
 614
 615         // (x+y)^n, n>0
 616         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
 617                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
 618
 619         // (x*y)^n -> x^n * y^n
 620         if (is_ex_exactly_of_type(expanded_basis,mul))
 621                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
 622
 623         // cannot expand further
 624         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 625                 return this->hold();
 626         else
 627                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 628 }
 629
 630 //////////
 631 // new virtual functions which can be overridden by derived classes
 632 //////////
 633
 634 // none
 635
 636 //////////
 637 // non-virtual functions in this class
 638 //////////
 639
 640 /** expand a^n where a is an add and n is an integer.
 641  *  @see power::expand */
 642 ex power::expand_add(const add & a, int n) const
 643 {
 644         if (n==2)
 645                 return expand_add_2(a);
 646
 647         int m = a.nops();
 648         exvector sum;
 649         sum.reserve((n+1)*(m-1));
 650         intvector k(m-1);
 651         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 652         intvector upper_limit(m-1);
 653         int l;
 654
 655         for (int l=0; l<m-1; l++) {
 656                 k[l] = 0;
 657                 k_cum[l] = 0;
 658                 upper_limit[l] = n;
 659         }
 660
 661         while (1) {
 662                 exvector term;
 663                 term.reserve(m+1);
 664                 for (l=0; l<m-1; l++) {
 665                         const ex & b = a.op(l);
 666                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 667                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 668                                      !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
 669                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 670                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
 671                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
 672                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
 673                         if (is_ex_exactly_of_type(b,mul))
 674                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
 675                         else
 676                                 term.push_back(power(b,k[l]));
 677                 }
 678
 679                 const ex & b = a.op(l);
 680                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 681                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 682                              !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
 683                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 684                              !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
 685                              !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
 686                              !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
 687                 if (is_ex_exactly_of_type(b,mul))
 688                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
 689                 else
 690                         term.push_back(power(b,n-k_cum[m-2]));
 691
 692                 numeric f = binomial(numeric(n),numeric(k[0]));
 693                 for (l=1; l<m-1; l++)
 694                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 695
 696                 term.push_back(f);
 697
 698                 /*
 699                 cout << "begin term" << endl;
 700                 for (int i=0; i<m-1; i++) {
 701                         cout << "k[" << i << "]=" << k[i] << endl;
 702                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
 703                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
 704                 }
 705                 for_each(term.begin(), term.end(), ostream_iterator<ex>(cout, "\n"));
 706                 cout << "end term" << endl;
 707                 */
 708
 709                 // TODO: optimize this
 710                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 711
 712                 // increment k[]
 713                 l = m-2;
 714                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
 715                         k[l] = 0;
 716                         l--;
 717                 }
 718                 if (l<0) break;
 719
 720                 // recalc k_cum[] and upper_limit[]
 721                 if (l==0)
 722                         k_cum[0] = k[0];
 723                 else
 724                         k_cum[l] = k_cum[l-1]+k[l];
 725
 726                 for (int i=l+1; i<m-1; i++)
 727                         k_cum[i] = k_cum[i-1]+k[i];
 728
 729                 for (int i=l+1; i<m-1; i++)
 730                         upper_limit[i] = n-k_cum[i-1];
 731         }
 732         return (new add(sum))->setflag(status_flags::dynallocated |
 733                                                                    status_flags::expanded );
 734 }
 735
 736
 737 /** Special case of power::expand_add. Expands a^2 where a is an add.
 738  *  @see power::expand_add */
 739 ex power::expand_add_2(const add & a) const
 740 {
 741         epvector sum;
 742         unsigned a_nops = a.nops();
 743         sum.reserve((a_nops*(a_nops+1))/2);
 744         epvector::const_iterator last = a.seq.end();
 745
 746         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 747         // first part: ignore overall_coeff and expand other terms
 748         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 749                 const ex & r = (*cit0).rest;
 750                 const ex & c = (*cit0).coeff;
 751
 752                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
 753                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
 754                              !is_ex_exactly_of_type(ex_to<power>(r).exponent,numeric) ||
 755                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 756                              !is_ex_exactly_of_type(ex_to<power>(r).basis,add) ||
 757                              !is_ex_exactly_of_type(ex_to<power>(r).basis,mul) ||
 758                              !is_ex_exactly_of_type(ex_to<power>(r).basis,power));
 759
 760                 if (are_ex_trivially_equal(c,_ex1())) {
 761                         if (is_ex_exactly_of_type(r,mul)) {
 762                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
 763                                                      _ex1()));
 764                         } else {
 765                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 766                                                      _ex1()));
 767                         }
 768                 } else {
 769                         if (is_ex_exactly_of_type(r,mul)) {
 770                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
 771                                                      ex_to<numeric>(c).power_dyn(_num2())));
 772                         } else {
 773                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 774                                                      ex_to<numeric>(c).power_dyn(_num2())));
 775                         }
 776                 }
 777
 778                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 779                         const ex & r1 = (*cit1).rest;
 780                         const ex & c1 = (*cit1).coeff;
 781                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 782                                                                       _num2().mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
 783                 }
 784         }
 785
 786         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 787
 788         // second part: add terms coming from overall_factor (if != 0)
 789         if (!a.overall_coeff.is_zero()) {
 790                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
 791                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to<numeric>(a.overall_coeff).mul_dyn(_num2())));
 792                 }
 793                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2()),_ex1()));
 794         }
 795
 796         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 797
 798         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 799 }
 800
 801 /** Expand factors of m in m^n where m is a mul and n is and integer
 802  *  @see power::expand */
 803 ex power::expand_mul(const mul & m, const numeric & n) const
 804 {
 805         if (n.is_zero())
 806                 return _ex1();
 807
 808         epvector distrseq;
 809         distrseq.reserve(m.seq.size());
 810         epvector::const_iterator last = m.seq.end();
 811         epvector::const_iterator cit = m.seq.begin();
 812         while (cit!=last) {
 813                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 814                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 815                 } else {
 816                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 817                         // since n is an integer
 818                         distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
 819                 }
 820                 ++cit;
 821         }
 822         return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 823 }
 824
 825 /*
 826 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
 827                                                                 unsigned options) const
 828 {
 829         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
 830                         expand(options | expand_options::internal_do_not_expand_power_operands);
 831
 832         return ex(mul(rest_power,basis),0).
 833                expand(options | expand_options::internal_do_not_expand_mul_operands);
 834 }
 835 */
 836
 837 // helper function
 838
 839 ex sqrt(const ex & a)
 840 {
 841         return power(a,_ex1_2());
 842 }
 843
 844 } // namespace GiNaC