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 "numeric.h"
  32 #include "inifcns.h"
  33 #include "relational.h"
  34 #include "symbol.h"
  35 #include "archive.h"
  36 #include "debugmsg.h"
  37 #include "utils.h"
  38
  39 #ifndef NO_NAMESPACE_GINAC
  40 namespace GiNaC {
  41 #endif // ndef NO_NAMESPACE_GINAC
  42
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
  44
  45 typedef std::vector<int> intvector;
  46
  47 //////////
  48 // default constructor, destructor, copy constructor assignment operator and helpers
  49 //////////
  50
  51 // public
  52
  53 power::power() : basic(TINFO_power)
  54 {
  55         debugmsg("power default constructor",LOGLEVEL_CONSTRUCT);
  56 }
  57
  58 // protected
  59
  60 void power::copy(const power & other)
  61 {
  62         inherited::copy(other);
  63         basis=other.basis;
  64         exponent=other.exponent;
  65 }
  66
  67 void power::destroy(bool call_parent)
  68 {
  69         if (call_parent) inherited::destroy(call_parent);
  70 }
  71
  72 //////////
  73 // other constructors
  74 //////////
  75
  76 // public
  77
  78 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  79 {
  80         debugmsg("power constructor from ex,ex",LOGLEVEL_CONSTRUCT);
  81         GINAC_ASSERT(basis.return_type()==return_types::commutative);
  82 }
  83
  84 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  85 {
  86         debugmsg("power constructor from ex,numeric",LOGLEVEL_CONSTRUCT);
  87         GINAC_ASSERT(basis.return_type()==return_types::commutative);
  88 }
  89
  90 //////////
  91 // archiving
  92 //////////
  93
  94 /** Construct object from archive_node. */
  95 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  96 {
  97         debugmsg("power constructor from archive_node", LOGLEVEL_CONSTRUCT);
  98         n.find_ex("basis", basis, sym_lst);
  99         n.find_ex("exponent", exponent, sym_lst);
 100 }
 101
 102 /** Unarchive the object. */
 103 ex power::unarchive(const archive_node &n, const lst &sym_lst)
 104 {
 105         return (new power(n, sym_lst))->setflag(status_flags::dynallocated);
 106 }
 107
 108 /** Archive the object. */
 109 void power::archive(archive_node &n) const
 110 {
 111         inherited::archive(n);
 112         n.add_ex("basis", basis);
 113         n.add_ex("exponent", exponent);
 114 }
 115
 116 //////////
 117 // functions overriding virtual functions from bases classes
 118 //////////
 119
 120 // public
 121
 122 basic * power::duplicate() const
 123 {
 124         debugmsg("power duplicate",LOGLEVEL_DUPLICATE);
 125         return new power(*this);
 126 }
 127
 128 void power::print(std::ostream & os, unsigned upper_precedence) const
 129 {
 130         debugmsg("power print",LOGLEVEL_PRINT);
 131         if (exponent.is_equal(_ex1_2())) {
 132                 os << "sqrt(" << basis << ")";
 133         } else {
 134                 if (precedence<=upper_precedence) os << "(";
 135                 basis.print(os,precedence);
 136                 os << "^";
 137                 exponent.print(os,precedence);
 138                 if (precedence<=upper_precedence) os << ")";
 139         }
 140 }
 141
 142 void power::printraw(std::ostream & os) const
 143 {
 144         debugmsg("power printraw",LOGLEVEL_PRINT);
 145
 146         os << "power(";
 147         basis.printraw(os);
 148         os << ",";
 149         exponent.printraw(os);
 150         os << ",hash=" << hashvalue << ",flags=" << flags << ")";
 151 }
 152
 153 void power::printtree(std::ostream & os, unsigned indent) const
 154 {
 155         debugmsg("power printtree",LOGLEVEL_PRINT);
 156
 157         os << std::string(indent,' ') << "power: "
 158            << "hash=" << hashvalue
 159            << " (0x" << std::hex << hashvalue << std::dec << ")"
 160            << ", flags=" << flags << std::endl;
 161         basis.printtree(os, indent+delta_indent);
 162         exponent.printtree(os, indent+delta_indent);
 163 }
 164
 165 static void print_sym_pow(std::ostream & os, unsigned type, const symbol &x, int exp)
 166 {
 167         // Optimal output of integer powers of symbols to aid compiler CSE
 168         if (exp == 1) {
 169                 x.printcsrc(os, type, 0);
 170         } else if (exp == 2) {
 171                 x.printcsrc(os, type, 0);
 172                 os << "*";
 173                 x.printcsrc(os, type, 0);
 174         } else if (exp & 1) {
 175                 x.printcsrc(os, 0);
 176                 os << "*";
 177                 print_sym_pow(os, type, x, exp-1);
 178         } else {
 179                 os << "(";
 180                 print_sym_pow(os, type, x, exp >> 1);
 181                 os << ")*(";
 182                 print_sym_pow(os, type, x, exp >> 1);
 183                 os << ")";
 184         }
 185 }
 186
 187 void power::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
 188 {
 189         debugmsg("power print csrc", LOGLEVEL_PRINT);
 190
 191         // Integer powers of symbols are printed in a special, optimized way
 192         if (exponent.info(info_flags::integer)
 193          && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
 194                 int exp = ex_to_numeric(exponent).to_int();
 195                 if (exp > 0)
 196                         os << "(";
 197                 else {
 198                         exp = -exp;
 199                         if (type == csrc_types::ctype_cl_N)
 200                                 os << "recip(";
 201                         else
 202                                 os << "1.0/(";
 203                 }
 204                 print_sym_pow(os, type, static_cast<const symbol &>(*basis.bp), exp);
 205                 os << ")";
 206
 207         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 208         } else if (exponent.compare(_num_1()) == 0) {
 209                 if (type == csrc_types::ctype_cl_N)
 210                         os << "recip(";
 211                 else
 212                         os << "1.0/(";
 213                 basis.bp->printcsrc(os, type, 0);
 214                 os << ")";
 215
 216         // Otherwise, use the pow() or expt() (CLN) functions
 217         } else {
 218                 if (type == csrc_types::ctype_cl_N)
 219                         os << "expt(";
 220                 else
 221                         os << "pow(";
 222                 basis.bp->printcsrc(os, type, 0);
 223                 os << ",";
 224                 exponent.bp->printcsrc(os, type, 0);
 225                 os << ")";
 226         }
 227 }
 228
 229 bool power::info(unsigned inf) const
 230 {
 231         switch (inf) {
 232                 case info_flags::polynomial:
 233                 case info_flags::integer_polynomial:
 234                 case info_flags::cinteger_polynomial:
 235                 case info_flags::rational_polynomial:
 236                 case info_flags::crational_polynomial:
 237                         return exponent.info(info_flags::nonnegint);
 238                 case info_flags::rational_function:
 239                         return exponent.info(info_flags::integer);
 240                 case info_flags::algebraic:
 241                         return (!exponent.info(info_flags::integer) ||
 242                                         basis.info(inf));
 243         }
 244         return inherited::info(inf);
 245 }
 246
 247 unsigned power::nops() const
 248 {
 249         return 2;
 250 }
 251
 252 ex & power::let_op(int i)
 253 {
 254         GINAC_ASSERT(i>=0);
 255         GINAC_ASSERT(i<2);
 256
 257         return i==0 ? basis : exponent;
 258 }
 259
 260 int power::degree(const symbol & s) const
 261 {
 262         if (is_exactly_of_type(*exponent.bp,numeric)) {
 263                 if ((*basis.bp).compare(s)==0) {
 264                         if (ex_to_numeric(exponent).is_integer())
 265                                 return ex_to_numeric(exponent).to_int();
 266                         else
 267                                 return 0;
 268                 } else
 269                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
 270         }
 271         return 0;
 272 }
 273
 274 int power::ldegree(const symbol & s) const
 275 {
 276         if (is_exactly_of_type(*exponent.bp,numeric)) {
 277                 if ((*basis.bp).compare(s)==0) {
 278                         if (ex_to_numeric(exponent).is_integer())
 279                                 return ex_to_numeric(exponent).to_int();
 280                         else
 281                                 return 0;
 282                 } else
 283                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
 284         }
 285         return 0;
 286 }
 287
 288 ex power::coeff(const symbol & s, int n) const
 289 {
 290         if ((*basis.bp).compare(s)!=0) {
 291                 // basis not equal to s
 292                 if (n == 0)
 293                         return *this;
 294                 else
 295                         return _ex0();
 296         } else {
 297                 // basis equal to s
 298                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
 299                         // integer exponent
 300                         int int_exp = ex_to_numeric(exponent).to_int();
 301                         if (n == int_exp)
 302                                 return _ex1();
 303                         else
 304                                 return _ex0();
 305                 } else {
 306                         // non-integer exponents are treated as zero
 307                         if (n == 0)
 308                                 return *this;
 309                         else
 310                                 return _ex0();
 311                 }
 312         }
 313 }
 314
 315 ex power::eval(int level) const
 316 {
 317         // simplifications: ^(x,0) -> 1 (0^0 handled here)
 318         //                  ^(x,1) -> x
 319         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
 320         //                  ^(1,x) -> 1
 321         //                  ^(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)
 322         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 323         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 324         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 325         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 326
 327         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 328
 329         if ((level==1) && (flags & status_flags::evaluated))
 330                 return *this;
 331         else if (level == -max_recursion_level)
 332                 throw(std::runtime_error("max recursion level reached"));
 333
 334         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 335         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 336
 337         bool basis_is_numerical = 0;
 338         bool exponent_is_numerical = 0;
 339         numeric * num_basis;
 340         numeric * num_exponent;
 341
 342         if (is_exactly_of_type(*ebasis.bp,numeric)) {
 343                 basis_is_numerical = 1;
 344                 num_basis = static_cast<numeric *>(ebasis.bp);
 345         }
 346         if (is_exactly_of_type(*eexponent.bp,numeric)) {
 347                 exponent_is_numerical = 1;
 348                 num_exponent = static_cast<numeric *>(eexponent.bp);
 349         }
 350
 351         // ^(x,0) -> 1 (0^0 also handled here)
 352         if (eexponent.is_zero())
 353                 if (ebasis.is_zero())
 354                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 355                 else
 356                         return _ex1();
 357
 358         // ^(x,1) -> x
 359         if (eexponent.is_equal(_ex1()))
 360                 return ebasis;
 361
 362         // ^(0,c1) -> 0 or exception (depending on real value of c1)
 363         if (ebasis.is_zero() && exponent_is_numerical) {
 364                 if ((num_exponent->real()).is_zero())
 365                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 366                 else if ((num_exponent->real()).is_negative())
 367                         throw (pole_error("power::eval(): division by zero",1));
 368                 else
 369                         return _ex0();
 370         }
 371
 372         // ^(1,x) -> 1
 373         if (ebasis.is_equal(_ex1()))
 374                 return _ex1();
 375
 376         if (basis_is_numerical && exponent_is_numerical) {
 377                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
 378                 // except if c1,c2 are rational, but c1^c2 is not)
 379                 bool basis_is_crational = num_basis->is_crational();
 380                 bool exponent_is_crational = num_exponent->is_crational();
 381                 numeric res = (*num_basis).power(*num_exponent);
 382
 383                 if ((!basis_is_crational || !exponent_is_crational)
 384                         || res.is_crational()) {
 385                         return res;
 386                 }
 387                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 388                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 389                 if (basis_is_crational && exponent_is_crational
 390                         && num_exponent->is_real()
 391                         && !num_exponent->is_integer()) {
 392                         numeric n = num_exponent->numer();
 393                         numeric m = num_exponent->denom();
 394                         numeric r;
 395                         numeric q = iquo(n, m, r);
 396                         if (r.is_negative()) {
 397                                 r = r.add(m);
 398                                 q = q.sub(_num1());
 399                         }
 400                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 401                                 return this->hold();
 402                         else {
 403                                 epvector res;
 404                                 res.push_back(expair(ebasis,r.div(m)));
 405                                 return (new mul(res,ex(num_basis->power(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
 406                         }
 407                 }
 408         }
 409
 410         // ^(^(x,c1),c2) -> ^(x,c1*c2)
 411         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 412         // case c1==1 should not happen, see below!)
 413         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
 414                 const power & sub_power = ex_to_power(ebasis);
 415                 const ex & sub_basis = sub_power.basis;
 416                 const ex & sub_exponent = sub_power.exponent;
 417                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 418                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
 419                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
 420                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
 421                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 422                         }
 423                 }
 424         }
 425
 426         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 427         if (exponent_is_numerical && num_exponent->is_integer() &&
 428                 is_ex_exactly_of_type(ebasis,mul)) {
 429                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
 430         }
 431
 432         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 433         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 434         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
 435                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 436                 const mul & mulref=ex_to_mul(ebasis);
 437                 if (!mulref.overall_coeff.is_equal(_ex1())) {
 438                         const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff);
 439                         if (num_coeff.is_real()) {
 440                                 if (num_coeff.is_positive()>0) {
 441                                         mul * mulp=new mul(mulref);
 442                                         mulp->overall_coeff=_ex1();
 443                                         mulp->clearflag(status_flags::evaluated);
 444                                         mulp->clearflag(status_flags::hash_calculated);
 445                                         return (new mul(power(*mulp,exponent),
 446                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 447                                 } else {
 448                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
 449                                         if (num_coeff.compare(_num_1())!=0) {
 450                                                 mul * mulp=new mul(mulref);
 451                                                 mulp->overall_coeff=_ex_1();
 452                                                 mulp->clearflag(status_flags::evaluated);
 453                                                 mulp->clearflag(status_flags::hash_calculated);
 454                                                 return (new mul(power(*mulp,exponent),
 455                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 456                                         }
 457                                 }
 458                         }
 459                 }
 460         }
 461
 462         if (are_ex_trivially_equal(ebasis,basis) &&
 463                 are_ex_trivially_equal(eexponent,exponent)) {
 464                 return this->hold();
 465         }
 466         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 467                                                                                                    status_flags::evaluated);
 468 }
 469
 470 ex power::evalf(int level) const
 471 {
 472         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 473
 474         ex ebasis;
 475         ex eexponent;
 476
 477         if (level==1) {
 478                 ebasis = basis;
 479                 eexponent = exponent;
 480         } else if (level == -max_recursion_level) {
 481                 throw(std::runtime_error("max recursion level reached"));
 482         } else {
 483                 ebasis = basis.evalf(level-1);
 484                 if (!is_ex_exactly_of_type(eexponent,numeric))
 485                         eexponent = exponent.evalf(level-1);
 486                 else
 487                         eexponent = exponent;
 488         }
 489
 490         return power(ebasis,eexponent);
 491 }
 492
 493 ex power::subs(const lst & ls, const lst & lr) const
 494 {
 495         const ex & subsed_basis=basis.subs(ls,lr);
 496         const ex & subsed_exponent=exponent.subs(ls,lr);
 497
 498         if (are_ex_trivially_equal(basis,subsed_basis)&&
 499                 are_ex_trivially_equal(exponent,subsed_exponent)) {
 500                 return *this;
 501         }
 502
 503         return power(subsed_basis, subsed_exponent);
 504 }
 505
 506 ex power::simplify_ncmul(const exvector & v) const
 507 {
 508         return inherited::simplify_ncmul(v);
 509 }
 510
 511 // protected
 512
 513 /** Implementation of ex::diff() for a power.
 514  *  @see ex::diff */
 515 ex power::derivative(const symbol & s) const
 516 {
 517         if (exponent.info(info_flags::real)) {
 518                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 519                 epvector newseq;
 520                 newseq.reserve(2);
 521                 newseq.push_back(expair(basis, exponent - _ex1()));
 522                 newseq.push_back(expair(basis.diff(s), _ex1()));
 523                 return mul(newseq, exponent);
 524         } else {
 525                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 526                 return mul(*this,
 527                            add(mul(exponent.diff(s), log(basis)),
 528                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
 529         }
 530 }
 531
 532 int power::compare_same_type(const basic & other) const
 533 {
 534         GINAC_ASSERT(is_exactly_of_type(other, power));
 535         const power & o=static_cast<const power &>(const_cast<basic &>(other));
 536
 537         int cmpval;
 538         cmpval=basis.compare(o.basis);
 539         if (cmpval==0) {
 540                 return exponent.compare(o.exponent);
 541         }
 542         return cmpval;
 543 }
 544
 545 unsigned power::return_type(void) const
 546 {
 547         return basis.return_type();
 548 }
 549
 550 unsigned power::return_type_tinfo(void) const
 551 {
 552         return basis.return_type_tinfo();
 553 }
 554
 555 ex power::expand(unsigned options) const
 556 {
 557         if (flags & status_flags::expanded)
 558                 return *this;
 559
 560         ex expanded_basis = basis.expand(options);
 561         ex expanded_exponent = exponent.expand(options);
 562
 563         // x^(a+b) -> x^a * x^b
 564         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 565                 const add &a = ex_to_add(expanded_exponent);
 566                 exvector distrseq;
 567                 distrseq.reserve(a.seq.size() + 1);
 568                 epvector::const_iterator last = a.seq.end();
 569                 epvector::const_iterator cit = a.seq.begin();
 570                 while (cit!=last) {
 571                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 572                         cit++;
 573                 }
 574
 575                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 576                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
 577                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
 578                         int int_exponent = num_exponent.to_int();
 579                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 580                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
 581                         else
 582                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 583                 } else
 584                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 585
 586                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 587                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 588                 return r.expand();
 589         }
 590
 591         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 592                 !ex_to_numeric(expanded_exponent).is_integer()) {
 593                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 594                         return this->hold();
 595                 } else {
 596                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 597                 }
 598         }
 599
 600         // integer numeric exponent
 601         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
 602         int int_exponent = num_exponent.to_int();
 603
 604         // (x+y)^n, n>0
 605         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
 606                 return expand_add(ex_to_add(expanded_basis), int_exponent);
 607         }
 608
 609         // (x*y)^n -> x^n * y^n
 610         if (is_ex_exactly_of_type(expanded_basis,mul)) {
 611                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
 612         }
 613
 614         // cannot expand further
 615         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 616                 return this->hold();
 617         } else {
 618                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 619         }
 620 }
 621
 622 //////////
 623 // new virtual functions which can be overridden by derived classes
 624 //////////
 625
 626 // none
 627
 628 //////////
 629 // non-virtual functions in this class
 630 //////////
 631
 632 /** expand a^n where a is an add and n is an integer.
 633  *  @see power::expand */
 634 ex power::expand_add(const add & a, int n) const
 635 {
 636         if (n==2)
 637                 return expand_add_2(a);
 638
 639         int m = a.nops();
 640         exvector sum;
 641         sum.reserve((n+1)*(m-1));
 642         intvector k(m-1);
 643         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 644         intvector upper_limit(m-1);
 645         int l;
 646
 647         for (int l=0; l<m-1; l++) {
 648                 k[l] = 0;
 649                 k_cum[l] = 0;
 650                 upper_limit[l] = n;
 651         }
 652
 653         while (1) {
 654                 exvector term;
 655                 term.reserve(m+1);
 656                 for (l=0; l<m-1; l++) {
 657                         const ex & b = a.op(l);
 658                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 659                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 660                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 661                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 662                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 663                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 664                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 665                         if (is_ex_exactly_of_type(b,mul)) {
 666                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
 667                         } else {
 668                                 term.push_back(power(b,k[l]));
 669                         }
 670                 }
 671
 672                 const ex & b = a.op(l);
 673                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 674                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 675                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 676                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 677                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 678                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 679                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 680                 if (is_ex_exactly_of_type(b,mul)) {
 681                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
 682                 } else {
 683                         term.push_back(power(b,n-k_cum[m-2]));
 684                 }
 685
 686                 numeric f = binomial(numeric(n),numeric(k[0]));
 687                 for (l=1; l<m-1; l++) {
 688                         f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 689                 }
 690                 term.push_back(f);
 691
 692                 /*
 693                 cout << "begin term" << endl;
 694                 for (int i=0; i<m-1; i++) {
 695                         cout << "k[" << i << "]=" << k[i] << endl;
 696                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
 697                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
 698                 }
 699                 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
 700                         cout << *cit << endl;
 701                 }
 702                 cout << "end term" << endl;
 703                 */
 704
 705                 // TODO: optimize this
 706                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 707
 708                 // increment k[]
 709                 l=m-2;
 710                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
 711                         k[l]=0;
 712                         l--;
 713                 }
 714                 if (l<0) break;
 715
 716                 // recalc k_cum[] and upper_limit[]
 717                 if (l==0) {
 718                         k_cum[0]=k[0];
 719                 } else {
 720                         k_cum[l]=k_cum[l-1]+k[l];
 721                 }
 722                 for (int i=l+1; i<m-1; i++) {
 723                         k_cum[i]=k_cum[i-1]+k[i];
 724                 }
 725
 726                 for (int i=l+1; i<m-1; i++) {
 727                         upper_limit[i]=n-k_cum[i-1];
 728                 }
 729         }
 730         return (new add(sum))->setflag(status_flags::dynallocated |
 731                                                                    status_flags::expanded );
 732 }
 733
 734
 735 /** Special case of power::expand_add. Expands a^2 where a is an add.
 736  *  @see power::expand_add */
 737 ex power::expand_add_2(const add & a) const
 738 {
 739         epvector sum;
 740         unsigned a_nops=a.nops();
 741         sum.reserve((a_nops*(a_nops+1))/2);
 742         epvector::const_iterator last=a.seq.end();
 743
 744         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 745         // first part: ignore overall_coeff and expand other terms
 746         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 747                 const ex & r=(*cit0).rest;
 748                 const ex & c=(*cit0).coeff;
 749
 750                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
 751                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
 752                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
 753                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
 754                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
 755                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
 756                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
 757
 758                 if (are_ex_trivially_equal(c,_ex1())) {
 759                         if (is_ex_exactly_of_type(r,mul)) {
 760                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 761                                                      _ex1()));
 762                         } else {
 763                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 764                                                      _ex1()));
 765                         }
 766                 } else {
 767                         if (is_ex_exactly_of_type(r,mul)) {
 768                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 769                                                      ex_to_numeric(c).power_dyn(_num2())));
 770                         } else {
 771                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 772                                                      ex_to_numeric(c).power_dyn(_num2())));
 773                         }
 774                 }
 775
 776                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 777                         const ex & r1=(*cit1).rest;
 778                         const ex & c1=(*cit1).coeff;
 779                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 780                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
 781                 }
 782         }
 783
 784         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 785
 786         // second part: add terms coming from overall_factor (if != 0)
 787         if (!a.overall_coeff.is_equal(_ex0())) {
 788                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
 789                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
 790                 }
 791                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
 792         }
 793
 794         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 795
 796         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 797 }
 798
 799 /** Expand factors of m in m^n where m is a mul and n is and integer
 800  *  @see power::expand */
 801 ex power::expand_mul(const mul & m, const numeric & n) const
 802 {
 803         if (n.is_equal(_num0()))
 804                 return _ex1();
 805
 806         epvector distrseq;
 807         distrseq.reserve(m.seq.size());
 808         epvector::const_iterator last = m.seq.end();
 809         epvector::const_iterator cit = m.seq.begin();
 810         while (cit!=last) {
 811                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 812                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 813                 } else {
 814                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 815                         // since n is an integer
 816                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
 817                 }
 818                 ++cit;
 819         }
 820         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 821 }
 822
 823 /*
 824 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
 825                                unsigned options) const
 826 {
 827         // obsolete
 828
 829         exvector distrseq;
 830         epvector splitseq;
 831
 832         const add & addref=static_cast<const add &>(*basis.bp);
 833
 834         splitseq=addref.seq;
 835         splitseq.pop_back();
 836         ex first_operands=add(splitseq);
 837         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
 838
 839         int n=exponent.to_int();
 840         for (int k=0; k<=n; k++) {
 841                 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
 842                                                  * power(last_operand,numeric(n-k)));
 843         }
 844         return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
 845 }
 846 */
 847
 848 /*
 849 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
 850                                                                 unsigned options) const
 851 {
 852         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
 853                         expand(options | expand_options::internal_do_not_expand_power_operands);
 854
 855         return ex(mul(rest_power,basis),0).
 856                expand(options | expand_options::internal_do_not_expand_mul_operands);
 857 }
 858 */
 859
 860 //////////
 861 // static member variables
 862 //////////
 863
 864 // protected
 865
 866 unsigned power::precedence = 60;
 867
 868 // helper function
 869
 870 ex sqrt(const ex & a)
 871 {
 872         return power(a,_ex1_2());
 873 }
 874
 875 #ifndef NO_NAMESPACE_GINAC
 876 } // namespace GiNaC
 877 #endif // ndef NO_NAMESPACE_GINAC