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 power::~power()
  59 {
  60         debugmsg("power destructor",LOGLEVEL_DESTRUCT);
  61         destroy(false);
  62 }
  63
  64 power::power(const power & other)
  65 {
  66         debugmsg("power copy constructor",LOGLEVEL_CONSTRUCT);
  67         copy(other);
  68 }
  69
  70 const power & power::operator=(const power & other)
  71 {
  72         debugmsg("power operator=",LOGLEVEL_ASSIGNMENT);
  73         if (this != &other) {
  74                 destroy(true);
  75                 copy(other);
  76         }
  77         return *this;
  78 }
  79
  80 // protected
  81
  82 void power::copy(const power & other)
  83 {
  84         inherited::copy(other);
  85         basis=other.basis;
  86         exponent=other.exponent;
  87 }
  88
  89 void power::destroy(bool call_parent)
  90 {
  91         if (call_parent) inherited::destroy(call_parent);
  92 }
  93
  94 //////////
  95 // other constructors
  96 //////////
  97
  98 // public
  99
 100 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
 101 {
 102         debugmsg("power constructor from ex,ex",LOGLEVEL_CONSTRUCT);
 103         GINAC_ASSERT(basis.return_type()==return_types::commutative);
 104 }
 105
 106 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
 107 {
 108         debugmsg("power constructor from ex,numeric",LOGLEVEL_CONSTRUCT);
 109         GINAC_ASSERT(basis.return_type()==return_types::commutative);
 110 }
 111
 112 //////////
 113 // archiving
 114 //////////
 115
 116 /** Construct object from archive_node. */
 117 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 118 {
 119         debugmsg("power constructor from archive_node", LOGLEVEL_CONSTRUCT);
 120         n.find_ex("basis", basis, sym_lst);
 121         n.find_ex("exponent", exponent, sym_lst);
 122 }
 123
 124 /** Unarchive the object. */
 125 ex power::unarchive(const archive_node &n, const lst &sym_lst)
 126 {
 127         return (new power(n, sym_lst))->setflag(status_flags::dynallocated);
 128 }
 129
 130 /** Archive the object. */
 131 void power::archive(archive_node &n) const
 132 {
 133         inherited::archive(n);
 134         n.add_ex("basis", basis);
 135         n.add_ex("exponent", exponent);
 136 }
 137
 138 //////////
 139 // functions overriding virtual functions from bases classes
 140 //////////
 141
 142 // public
 143
 144 basic * power::duplicate() const
 145 {
 146         debugmsg("power duplicate",LOGLEVEL_DUPLICATE);
 147         return new power(*this);
 148 }
 149
 150 void power::print(std::ostream & os, unsigned upper_precedence) const
 151 {
 152         debugmsg("power print",LOGLEVEL_PRINT);
 153         if (exponent.is_equal(_ex1_2())) {
 154                 os << "sqrt(" << basis << ")";
 155         } else {
 156                 if (precedence<=upper_precedence) os << "(";
 157                 basis.print(os,precedence);
 158                 os << "^";
 159                 exponent.print(os,precedence);
 160                 if (precedence<=upper_precedence) os << ")";
 161         }
 162 }
 163
 164 void power::printraw(std::ostream & os) const
 165 {
 166         debugmsg("power printraw",LOGLEVEL_PRINT);
 167
 168         os << "power(";
 169         basis.printraw(os);
 170         os << ",";
 171         exponent.printraw(os);
 172         os << ",hash=" << hashvalue << ",flags=" << flags << ")";
 173 }
 174
 175 void power::printtree(std::ostream & os, unsigned indent) const
 176 {
 177         debugmsg("power printtree",LOGLEVEL_PRINT);
 178
 179         os << std::string(indent,' ') << "power: "
 180            << "hash=" << hashvalue
 181            << " (0x" << std::hex << hashvalue << std::dec << ")"
 182            << ", flags=" << flags << std::endl;
 183         basis.printtree(os, indent+delta_indent);
 184         exponent.printtree(os, indent+delta_indent);
 185 }
 186
 187 static void print_sym_pow(std::ostream & os, unsigned type, const symbol &x, int exp)
 188 {
 189         // Optimal output of integer powers of symbols to aid compiler CSE
 190         if (exp == 1) {
 191                 x.printcsrc(os, type, 0);
 192         } else if (exp == 2) {
 193                 x.printcsrc(os, type, 0);
 194                 os << "*";
 195                 x.printcsrc(os, type, 0);
 196         } else if (exp & 1) {
 197                 x.printcsrc(os, 0);
 198                 os << "*";
 199                 print_sym_pow(os, type, x, exp-1);
 200         } else {
 201                 os << "(";
 202                 print_sym_pow(os, type, x, exp >> 1);
 203                 os << ")*(";
 204                 print_sym_pow(os, type, x, exp >> 1);
 205                 os << ")";
 206         }
 207 }
 208
 209 void power::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
 210 {
 211         debugmsg("power print csrc", LOGLEVEL_PRINT);
 212
 213         // Integer powers of symbols are printed in a special, optimized way
 214         if (exponent.info(info_flags::integer)
 215          && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
 216                 int exp = ex_to_numeric(exponent).to_int();
 217                 if (exp > 0)
 218                         os << "(";
 219                 else {
 220                         exp = -exp;
 221                         if (type == csrc_types::ctype_cl_N)
 222                                 os << "recip(";
 223                         else
 224                                 os << "1.0/(";
 225                 }
 226                 print_sym_pow(os, type, static_cast<const symbol &>(*basis.bp), exp);
 227                 os << ")";
 228
 229         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 230         } else if (exponent.compare(_num_1()) == 0) {
 231                 if (type == csrc_types::ctype_cl_N)
 232                         os << "recip(";
 233                 else
 234                         os << "1.0/(";
 235                 basis.bp->printcsrc(os, type, 0);
 236                 os << ")";
 237
 238         // Otherwise, use the pow() or expt() (CLN) functions
 239         } else {
 240                 if (type == csrc_types::ctype_cl_N)
 241                         os << "expt(";
 242                 else
 243                         os << "pow(";
 244                 basis.bp->printcsrc(os, type, 0);
 245                 os << ",";
 246                 exponent.bp->printcsrc(os, type, 0);
 247                 os << ")";
 248         }
 249 }
 250
 251 bool power::info(unsigned inf) const
 252 {
 253         switch (inf) {
 254                 case info_flags::polynomial:
 255                 case info_flags::integer_polynomial:
 256                 case info_flags::cinteger_polynomial:
 257                 case info_flags::rational_polynomial:
 258                 case info_flags::crational_polynomial:
 259                         return exponent.info(info_flags::nonnegint);
 260                 case info_flags::rational_function:
 261                         return exponent.info(info_flags::integer);
 262                 case info_flags::algebraic:
 263                         return (!exponent.info(info_flags::integer) ||
 264                                         basis.info(inf));
 265         }
 266         return inherited::info(inf);
 267 }
 268
 269 unsigned power::nops() const
 270 {
 271         return 2;
 272 }
 273
 274 ex & power::let_op(int i)
 275 {
 276         GINAC_ASSERT(i>=0);
 277         GINAC_ASSERT(i<2);
 278
 279         return i==0 ? basis : exponent;
 280 }
 281
 282 int power::degree(const symbol & s) const
 283 {
 284         if (is_exactly_of_type(*exponent.bp,numeric)) {
 285                 if ((*basis.bp).compare(s)==0) {
 286                         if (ex_to_numeric(exponent).is_integer())
 287                                 return ex_to_numeric(exponent).to_int();
 288                         else
 289                                 return 0;
 290                 } else
 291                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
 292         }
 293         return 0;
 294 }
 295
 296 int power::ldegree(const symbol & s) const
 297 {
 298         if (is_exactly_of_type(*exponent.bp,numeric)) {
 299                 if ((*basis.bp).compare(s)==0) {
 300                         if (ex_to_numeric(exponent).is_integer())
 301                                 return ex_to_numeric(exponent).to_int();
 302                         else
 303                                 return 0;
 304                 } else
 305                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
 306         }
 307         return 0;
 308 }
 309
 310 ex power::coeff(const symbol & s, int n) const
 311 {
 312         if ((*basis.bp).compare(s)!=0) {
 313                 // basis not equal to s
 314                 if (n == 0)
 315                         return *this;
 316                 else
 317                         return _ex0();
 318         } else {
 319                 // basis equal to s
 320                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
 321                         // integer exponent
 322                         int int_exp = ex_to_numeric(exponent).to_int();
 323                         if (n == int_exp)
 324                                 return _ex1();
 325                         else
 326                                 return _ex0();
 327                 } else {
 328                         // non-integer exponents are treated as zero
 329                         if (n == 0)
 330                                 return *this;
 331                         else
 332                                 return _ex0();
 333                 }
 334         }
 335 }
 336
 337 ex power::eval(int level) const
 338 {
 339         // simplifications: ^(x,0) -> 1 (0^0 handled here)
 340         //                  ^(x,1) -> x
 341         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
 342         //                  ^(1,x) -> 1
 343         //                  ^(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)
 344         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 345         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 346         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 347         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 348
 349         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 350
 351         if ((level==1) && (flags & status_flags::evaluated))
 352                 return *this;
 353         else if (level == -max_recursion_level)
 354                 throw(std::runtime_error("max recursion level reached"));
 355
 356         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 357         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 358
 359         bool basis_is_numerical = 0;
 360         bool exponent_is_numerical = 0;
 361         numeric * num_basis;
 362         numeric * num_exponent;
 363
 364         if (is_exactly_of_type(*ebasis.bp,numeric)) {
 365                 basis_is_numerical = 1;
 366                 num_basis = static_cast<numeric *>(ebasis.bp);
 367         }
 368         if (is_exactly_of_type(*eexponent.bp,numeric)) {
 369                 exponent_is_numerical = 1;
 370                 num_exponent = static_cast<numeric *>(eexponent.bp);
 371         }
 372
 373         // ^(x,0) -> 1 (0^0 also handled here)
 374         if (eexponent.is_zero())
 375                 if (ebasis.is_zero())
 376                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 377                 else
 378                         return _ex1();
 379
 380         // ^(x,1) -> x
 381         if (eexponent.is_equal(_ex1()))
 382                 return ebasis;
 383
 384         // ^(0,c1) -> 0 or exception (depending on real value of c1)
 385         if (ebasis.is_zero() && exponent_is_numerical) {
 386                 if ((num_exponent->real()).is_zero())
 387                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 388                 else if ((num_exponent->real()).is_negative())
 389                         throw (pole_error("power::eval(): division by zero",1));
 390                 else
 391                         return _ex0();
 392         }
 393
 394         // ^(1,x) -> 1
 395         if (ebasis.is_equal(_ex1()))
 396                 return _ex1();
 397
 398         if (basis_is_numerical && exponent_is_numerical) {
 399                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
 400                 // except if c1,c2 are rational, but c1^c2 is not)
 401                 bool basis_is_crational = num_basis->is_crational();
 402                 bool exponent_is_crational = num_exponent->is_crational();
 403                 numeric res = (*num_basis).power(*num_exponent);
 404
 405                 if ((!basis_is_crational || !exponent_is_crational)
 406                         || res.is_crational()) {
 407                         return res;
 408                 }
 409                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 410                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 411                 if (basis_is_crational && exponent_is_crational
 412                         && num_exponent->is_real()
 413                         && !num_exponent->is_integer()) {
 414                         numeric n = num_exponent->numer();
 415                         numeric m = num_exponent->denom();
 416                         numeric r;
 417                         numeric q = iquo(n, m, r);
 418                         if (r.is_negative()) {
 419                                 r = r.add(m);
 420                                 q = q.sub(_num1());
 421                         }
 422                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 423                                 return this->hold();
 424                         else {
 425                                 epvector res;
 426                                 res.push_back(expair(ebasis,r.div(m)));
 427                                 return (new mul(res,ex(num_basis->power(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
 428                         }
 429                 }
 430         }
 431
 432         // ^(^(x,c1),c2) -> ^(x,c1*c2)
 433         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 434         // case c1==1 should not happen, see below!)
 435         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
 436                 const power & sub_power = ex_to_power(ebasis);
 437                 const ex & sub_basis = sub_power.basis;
 438                 const ex & sub_exponent = sub_power.exponent;
 439                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 440                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
 441                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
 442                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
 443                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 444                         }
 445                 }
 446         }
 447
 448         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 449         if (exponent_is_numerical && num_exponent->is_integer() &&
 450                 is_ex_exactly_of_type(ebasis,mul)) {
 451                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
 452         }
 453
 454         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 455         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 456         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
 457                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 458                 const mul & mulref=ex_to_mul(ebasis);
 459                 if (!mulref.overall_coeff.is_equal(_ex1())) {
 460                         const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff);
 461                         if (num_coeff.is_real()) {
 462                                 if (num_coeff.is_positive()>0) {
 463                                         mul * mulp=new mul(mulref);
 464                                         mulp->overall_coeff=_ex1();
 465                                         mulp->clearflag(status_flags::evaluated);
 466                                         mulp->clearflag(status_flags::hash_calculated);
 467                                         return (new mul(power(*mulp,exponent),
 468                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 469                                 } else {
 470                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
 471                                         if (num_coeff.compare(_num_1())!=0) {
 472                                                 mul * mulp=new mul(mulref);
 473                                                 mulp->overall_coeff=_ex_1();
 474                                                 mulp->clearflag(status_flags::evaluated);
 475                                                 mulp->clearflag(status_flags::hash_calculated);
 476                                                 return (new mul(power(*mulp,exponent),
 477                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 478                                         }
 479                                 }
 480                         }
 481                 }
 482         }
 483
 484         if (are_ex_trivially_equal(ebasis,basis) &&
 485                 are_ex_trivially_equal(eexponent,exponent)) {
 486                 return this->hold();
 487         }
 488         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 489                                                                                                    status_flags::evaluated);
 490 }
 491
 492 ex power::evalf(int level) const
 493 {
 494         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 495
 496         ex ebasis;
 497         ex eexponent;
 498
 499         if (level==1) {
 500                 ebasis = basis;
 501                 eexponent = exponent;
 502         } else if (level == -max_recursion_level) {
 503                 throw(std::runtime_error("max recursion level reached"));
 504         } else {
 505                 ebasis = basis.evalf(level-1);
 506                 if (!is_ex_exactly_of_type(eexponent,numeric))
 507                         eexponent = exponent.evalf(level-1);
 508                 else
 509                         eexponent = exponent;
 510         }
 511
 512         return power(ebasis,eexponent);
 513 }
 514
 515 ex power::subs(const lst & ls, const lst & lr) const
 516 {
 517         const ex & subsed_basis=basis.subs(ls,lr);
 518         const ex & subsed_exponent=exponent.subs(ls,lr);
 519
 520         if (are_ex_trivially_equal(basis,subsed_basis)&&
 521                 are_ex_trivially_equal(exponent,subsed_exponent)) {
 522                 return *this;
 523         }
 524
 525         return power(subsed_basis, subsed_exponent);
 526 }
 527
 528 ex power::simplify_ncmul(const exvector & v) const
 529 {
 530         return inherited::simplify_ncmul(v);
 531 }
 532
 533 // protected
 534
 535 /** Implementation of ex::diff() for a power.
 536  *  @see ex::diff */
 537 ex power::derivative(const symbol & s) const
 538 {
 539         if (exponent.info(info_flags::real)) {
 540                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 541                 epvector newseq;
 542                 newseq.reserve(2);
 543                 newseq.push_back(expair(basis, exponent - _ex1()));
 544                 newseq.push_back(expair(basis.diff(s), _ex1()));
 545                 return mul(newseq, exponent);
 546         } else {
 547                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 548                 return mul(*this,
 549                            add(mul(exponent.diff(s), log(basis)),
 550                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
 551         }
 552 }
 553
 554 int power::compare_same_type(const basic & other) const
 555 {
 556         GINAC_ASSERT(is_exactly_of_type(other, power));
 557         const power & o=static_cast<const power &>(const_cast<basic &>(other));
 558
 559         int cmpval;
 560         cmpval=basis.compare(o.basis);
 561         if (cmpval==0) {
 562                 return exponent.compare(o.exponent);
 563         }
 564         return cmpval;
 565 }
 566
 567 unsigned power::return_type(void) const
 568 {
 569         return basis.return_type();
 570 }
 571
 572 unsigned power::return_type_tinfo(void) const
 573 {
 574         return basis.return_type_tinfo();
 575 }
 576
 577 ex power::expand(unsigned options) const
 578 {
 579         if (flags & status_flags::expanded)
 580                 return *this;
 581
 582         ex expanded_basis = basis.expand(options);
 583         ex expanded_exponent = exponent.expand(options);
 584
 585         // x^(a+b) -> x^a * x^b
 586         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 587                 const add &a = ex_to_add(expanded_exponent);
 588                 exvector distrseq;
 589                 distrseq.reserve(a.seq.size() + 1);
 590                 epvector::const_iterator last = a.seq.end();
 591                 epvector::const_iterator cit = a.seq.begin();
 592                 while (cit!=last) {
 593                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 594                         cit++;
 595                 }
 596
 597                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 598                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
 599                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
 600                         int int_exponent = num_exponent.to_int();
 601                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 602                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
 603                         else
 604                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 605                 } else
 606                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 607
 608                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 609                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 610                 return r.expand();
 611         }
 612
 613         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 614                 !ex_to_numeric(expanded_exponent).is_integer()) {
 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         // integer numeric exponent
 623         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
 624         int int_exponent = num_exponent.to_int();
 625
 626         // (x+y)^n, n>0
 627         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
 628                 return expand_add(ex_to_add(expanded_basis), int_exponent);
 629         }
 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
 636         // cannot expand further
 637         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 638                 return this->hold();
 639         } else {
 640                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 641         }
 642 }
 643
 644 //////////
 645 // new virtual functions which can be overridden by derived classes
 646 //////////
 647
 648 // none
 649
 650 //////////
 651 // non-virtual functions in this class
 652 //////////
 653
 654 /** expand a^n where a is an add and n is an integer.
 655  *  @see power::expand */
 656 ex power::expand_add(const add & a, int n) const
 657 {
 658         if (n==2)
 659                 return expand_add_2(a);
 660
 661         int m = a.nops();
 662         exvector sum;
 663         sum.reserve((n+1)*(m-1));
 664         intvector k(m-1);
 665         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 666         intvector upper_limit(m-1);
 667         int l;
 668
 669         for (int l=0; l<m-1; l++) {
 670                 k[l] = 0;
 671                 k_cum[l] = 0;
 672                 upper_limit[l] = n;
 673         }
 674
 675         while (1) {
 676                 exvector term;
 677                 term.reserve(m+1);
 678                 for (l=0; l<m-1; l++) {
 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(k[l])));
 689                         } else {
 690                                 term.push_back(power(b,k[l]));
 691                         }
 692                 }
 693
 694                 const ex & b = a.op(l);
 695                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 696                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 697                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 698                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 699                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 700                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 701                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 702                 if (is_ex_exactly_of_type(b,mul)) {
 703                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
 704                 } else {
 705                         term.push_back(power(b,n-k_cum[m-2]));
 706                 }
 707
 708                 numeric f = binomial(numeric(n),numeric(k[0]));
 709                 for (l=1; l<m-1; l++) {
 710                         f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 711                 }
 712                 term.push_back(f);
 713
 714                 /*
 715                 cout << "begin term" << endl;
 716                 for (int i=0; i<m-1; i++) {
 717                         cout << "k[" << i << "]=" << k[i] << endl;
 718                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
 719                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
 720                 }
 721                 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
 722                         cout << *cit << endl;
 723                 }
 724                 cout << "end term" << endl;
 725                 */
 726
 727                 // TODO: optimize this
 728                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 729
 730                 // increment k[]
 731                 l=m-2;
 732                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
 733                         k[l]=0;
 734                         l--;
 735                 }
 736                 if (l<0) break;
 737
 738                 // recalc k_cum[] and upper_limit[]
 739                 if (l==0) {
 740                         k_cum[0]=k[0];
 741                 } else {
 742                         k_cum[l]=k_cum[l-1]+k[l];
 743                 }
 744                 for (int i=l+1; i<m-1; i++) {
 745                         k_cum[i]=k_cum[i-1]+k[i];
 746                 }
 747
 748                 for (int i=l+1; i<m-1; i++) {
 749                         upper_limit[i]=n-k_cum[i-1];
 750                 }
 751         }
 752         return (new add(sum))->setflag(status_flags::dynallocated |
 753                                                                    status_flags::expanded );
 754 }
 755
 756
 757 /** Special case of power::expand_add. Expands a^2 where a is an add.
 758  *  @see power::expand_add */
 759 ex power::expand_add_2(const add & a) const
 760 {
 761         epvector sum;
 762         unsigned a_nops=a.nops();
 763         sum.reserve((a_nops*(a_nops+1))/2);
 764         epvector::const_iterator last=a.seq.end();
 765
 766         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 767         // first part: ignore overall_coeff and expand other terms
 768         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 769                 const ex & r=(*cit0).rest;
 770                 const ex & c=(*cit0).coeff;
 771
 772                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
 773                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
 774                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
 775                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
 776                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
 777                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
 778                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
 779
 780                 if (are_ex_trivially_equal(c,_ex1())) {
 781                         if (is_ex_exactly_of_type(r,mul)) {
 782                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 783                                                      _ex1()));
 784                         } else {
 785                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 786                                                      _ex1()));
 787                         }
 788                 } else {
 789                         if (is_ex_exactly_of_type(r,mul)) {
 790                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 791                                                      ex_to_numeric(c).power_dyn(_num2())));
 792                         } else {
 793                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 794                                                      ex_to_numeric(c).power_dyn(_num2())));
 795                         }
 796                 }
 797
 798                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 799                         const ex & r1=(*cit1).rest;
 800                         const ex & c1=(*cit1).coeff;
 801                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 802                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
 803                 }
 804         }
 805
 806         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 807
 808         // second part: add terms coming from overall_factor (if != 0)
 809         if (!a.overall_coeff.is_equal(_ex0())) {
 810                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
 811                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
 812                 }
 813                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
 814         }
 815
 816         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 817
 818         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 819 }
 820
 821 /** Expand factors of m in m^n where m is a mul and n is and integer
 822  *  @see power::expand */
 823 ex power::expand_mul(const mul & m, const numeric & n) const
 824 {
 825         if (n.is_equal(_num0()))
 826                 return _ex1();
 827
 828         epvector distrseq;
 829         distrseq.reserve(m.seq.size());
 830         epvector::const_iterator last = m.seq.end();
 831         epvector::const_iterator cit = m.seq.begin();
 832         while (cit!=last) {
 833                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 834                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 835                 } else {
 836                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 837                         // since n is an integer
 838                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
 839                 }
 840                 ++cit;
 841         }
 842         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 843 }
 844
 845 /*
 846 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
 847                                unsigned options) const
 848 {
 849         // obsolete
 850
 851         exvector distrseq;
 852         epvector splitseq;
 853
 854         const add & addref=static_cast<const add &>(*basis.bp);
 855
 856         splitseq=addref.seq;
 857         splitseq.pop_back();
 858         ex first_operands=add(splitseq);
 859         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
 860
 861         int n=exponent.to_int();
 862         for (int k=0; k<=n; k++) {
 863                 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
 864                                                  * power(last_operand,numeric(n-k)));
 865         }
 866         return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
 867 }
 868 */
 869
 870 /*
 871 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
 872                                                                 unsigned options) const
 873 {
 874         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
 875                         expand(options | expand_options::internal_do_not_expand_power_operands);
 876
 877         return ex(mul(rest_power,basis),0).
 878                expand(options | expand_options::internal_do_not_expand_mul_operands);
 879 }
 880 */
 881
 882 //////////
 883 // static member variables
 884 //////////
 885
 886 // protected
 887
 888 unsigned power::precedence = 60;
 889
 890 // helper function
 891
 892 ex sqrt(const ex & a)
 893 {
 894         return power(a,_ex1_2());
 895 }
 896
 897 #ifndef NO_NAMESPACE_GINAC
 898 } // namespace GiNaC
 899 #endif // ndef NO_NAMESPACE_GINAC