ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2000 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                         return ex_to_numeric(exponent).to_int();
 287                 else
 288                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
 289         }
 290         return 0;
 291 }
 292
 293 int power::ldegree(const symbol & s) const
 294 {
 295         if (is_exactly_of_type(*exponent.bp,numeric)) {
 296                 if ((*basis.bp).compare(s)==0)
 297                         return ex_to_numeric(exponent).to_int();
 298                 else
 299                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
 300         }
 301         return 0;
 302 }
 303
 304 ex power::coeff(const symbol & s, int n) const
 305 {
 306         if ((*basis.bp).compare(s)!=0) {
 307                 // basis not equal to s
 308                 if (n==0) {
 309                         return *this;
 310                 } else {
 311                         return _ex0();
 312                 }
 313         } else if (is_exactly_of_type(*exponent.bp,numeric)&&
 314                            (static_cast<const numeric &>(*exponent.bp).compare(numeric(n))==0)) {
 315                 return _ex1();
 316         }
 317
 318         return _ex0();
 319 }
 320
 321 ex power::eval(int level) const
 322 {
 323         // simplifications: ^(x,0) -> 1 (0^0 handled here)
 324         //                  ^(x,1) -> x
 325         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
 326         //                  ^(1,x) -> 1
 327         //                  ^(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)
 328         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 329         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 330         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 331         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 332
 333         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 334
 335         if ((level==1) && (flags & status_flags::evaluated))
 336                 return *this;
 337         else if (level == -max_recursion_level)
 338                 throw(std::runtime_error("max recursion level reached"));
 339
 340         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 341         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 342
 343         bool basis_is_numerical = 0;
 344         bool exponent_is_numerical = 0;
 345         numeric * num_basis;
 346         numeric * num_exponent;
 347
 348         if (is_exactly_of_type(*ebasis.bp,numeric)) {
 349                 basis_is_numerical = 1;
 350                 num_basis = static_cast<numeric *>(ebasis.bp);
 351         }
 352         if (is_exactly_of_type(*eexponent.bp,numeric)) {
 353                 exponent_is_numerical = 1;
 354                 num_exponent = static_cast<numeric *>(eexponent.bp);
 355         }
 356
 357         // ^(x,0) -> 1 (0^0 also handled here)
 358         if (eexponent.is_zero())
 359                 if (ebasis.is_zero())
 360                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 361                 else
 362                         return _ex1();
 363
 364         // ^(x,1) -> x
 365         if (eexponent.is_equal(_ex1()))
 366                 return ebasis;
 367
 368         // ^(0,c1) -> 0 or exception (depending on real value of c1)
 369         if (ebasis.is_zero() && exponent_is_numerical) {
 370                 if ((num_exponent->real()).is_zero())
 371                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 372                 else if ((num_exponent->real()).is_negative())
 373                         throw (pole_error("power::eval(): division by zero",1));
 374                 else
 375                         return _ex0();
 376         }
 377
 378         // ^(1,x) -> 1
 379         if (ebasis.is_equal(_ex1()))
 380                 return _ex1();
 381
 382         if (basis_is_numerical && exponent_is_numerical) {
 383                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
 384                 // except if c1,c2 are rational, but c1^c2 is not)
 385                 bool basis_is_crational = num_basis->is_crational();
 386                 bool exponent_is_crational = num_exponent->is_crational();
 387                 numeric res = (*num_basis).power(*num_exponent);
 388
 389                 if ((!basis_is_crational || !exponent_is_crational)
 390                         || res.is_crational()) {
 391                         return res;
 392                 }
 393                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 394                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 395                 if (basis_is_crational && exponent_is_crational
 396                         && num_exponent->is_real()
 397                         && !num_exponent->is_integer()) {
 398                         numeric n = num_exponent->numer();
 399                         numeric m = num_exponent->denom();
 400                         numeric r;
 401                         numeric q = iquo(n, m, r);
 402                         if (r.is_negative()) {
 403                                 r = r.add(m);
 404                                 q = q.sub(_num1());
 405                         }
 406                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 407                                 return this->hold();
 408                         else {
 409                                 epvector res;
 410                                 res.push_back(expair(ebasis,r.div(m)));
 411                                 return (new mul(res,ex(num_basis->power(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
 412                         }
 413                 }
 414         }
 415
 416         // ^(^(x,c1),c2) -> ^(x,c1*c2)
 417         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 418         // case c1==1 should not happen, see below!)
 419         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
 420                 const power & sub_power = ex_to_power(ebasis);
 421                 const ex & sub_basis = sub_power.basis;
 422                 const ex & sub_exponent = sub_power.exponent;
 423                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 424                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
 425                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
 426                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
 427                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 428                         }
 429                 }
 430         }
 431
 432         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 433         if (exponent_is_numerical && num_exponent->is_integer() &&
 434                 is_ex_exactly_of_type(ebasis,mul)) {
 435                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
 436         }
 437
 438         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 439         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 440         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
 441                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 442                 const mul & mulref=ex_to_mul(ebasis);
 443                 if (!mulref.overall_coeff.is_equal(_ex1())) {
 444                         const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff);
 445                         if (num_coeff.is_real()) {
 446                                 if (num_coeff.is_positive()>0) {
 447                                         mul * mulp=new mul(mulref);
 448                                         mulp->overall_coeff=_ex1();
 449                                         mulp->clearflag(status_flags::evaluated);
 450                                         mulp->clearflag(status_flags::hash_calculated);
 451                                         return (new mul(power(*mulp,exponent),
 452                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 453                                 } else {
 454                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
 455                                         if (num_coeff.compare(_num_1())!=0) {
 456                                                 mul * mulp=new mul(mulref);
 457                                                 mulp->overall_coeff=_ex_1();
 458                                                 mulp->clearflag(status_flags::evaluated);
 459                                                 mulp->clearflag(status_flags::hash_calculated);
 460                                                 return (new mul(power(*mulp,exponent),
 461                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 462                                         }
 463                                 }
 464                         }
 465                 }
 466         }
 467
 468         if (are_ex_trivially_equal(ebasis,basis) &&
 469                 are_ex_trivially_equal(eexponent,exponent)) {
 470                 return this->hold();
 471         }
 472         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 473                                                                                                    status_flags::evaluated);
 474 }
 475
 476 ex power::evalf(int level) const
 477 {
 478         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 479
 480         ex ebasis;
 481         ex eexponent;
 482
 483         if (level==1) {
 484                 ebasis = basis;
 485                 eexponent = exponent;
 486         } else if (level == -max_recursion_level) {
 487                 throw(std::runtime_error("max recursion level reached"));
 488         } else {
 489                 ebasis = basis.evalf(level-1);
 490                 if (!is_ex_exactly_of_type(eexponent,numeric))
 491                         eexponent = exponent.evalf(level-1);
 492                 else
 493                         eexponent = exponent;
 494         }
 495
 496         return power(ebasis,eexponent);
 497 }
 498
 499 ex power::subs(const lst & ls, const lst & lr) const
 500 {
 501         const ex & subsed_basis=basis.subs(ls,lr);
 502         const ex & subsed_exponent=exponent.subs(ls,lr);
 503
 504         if (are_ex_trivially_equal(basis,subsed_basis)&&
 505                 are_ex_trivially_equal(exponent,subsed_exponent)) {
 506                 return *this;
 507         }
 508
 509         return power(subsed_basis, subsed_exponent);
 510 }
 511
 512 ex power::simplify_ncmul(const exvector & v) const
 513 {
 514         return inherited::simplify_ncmul(v);
 515 }
 516
 517 // protected
 518
 519 /** Implementation of ex::diff() for a power.
 520  *  @see ex::diff */
 521 ex power::derivative(const symbol & s) const
 522 {
 523         if (exponent.info(info_flags::real)) {
 524                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 525                 epvector newseq;
 526                 newseq.reserve(2);
 527                 newseq.push_back(expair(basis, exponent - _ex1()));
 528                 newseq.push_back(expair(basis.diff(s), _ex1()));
 529                 return mul(newseq, exponent);
 530         } else {
 531                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 532                 return mul(power(basis, exponent),
 533                            add(mul(exponent.diff(s), log(basis)),
 534                            mul(mul(exponent, basis.diff(s)), power(basis, -1))));
 535         }
 536 }
 537
 538 int power::compare_same_type(const basic & other) const
 539 {
 540         GINAC_ASSERT(is_exactly_of_type(other, power));
 541         const power & o=static_cast<const power &>(const_cast<basic &>(other));
 542
 543         int cmpval;
 544         cmpval=basis.compare(o.basis);
 545         if (cmpval==0) {
 546                 return exponent.compare(o.exponent);
 547         }
 548         return cmpval;
 549 }
 550
 551 unsigned power::return_type(void) const
 552 {
 553         return basis.return_type();
 554 }
 555
 556 unsigned power::return_type_tinfo(void) const
 557 {
 558         return basis.return_type_tinfo();
 559 }
 560
 561 ex power::expand(unsigned options) const
 562 {
 563         if (flags & status_flags::expanded)
 564                 return *this;
 565
 566         ex expanded_basis = basis.expand(options);
 567
 568         if (!is_ex_exactly_of_type(exponent,numeric) ||
 569                 !ex_to_numeric(exponent).is_integer()) {
 570                 if (are_ex_trivially_equal(basis,expanded_basis)) {
 571                         return this->hold();
 572                 } else {
 573                         return (new power(expanded_basis,exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 574                 }
 575         }
 576
 577         // integer numeric exponent
 578         const numeric & num_exponent = ex_to_numeric(exponent);
 579         int int_exponent = num_exponent.to_int();
 580
 581         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
 582                 return expand_add(ex_to_add(expanded_basis), int_exponent);
 583         }
 584
 585         if (is_ex_exactly_of_type(expanded_basis,mul)) {
 586                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
 587         }
 588
 589         // cannot expand further
 590         if (are_ex_trivially_equal(basis,expanded_basis)) {
 591                 return this->hold();
 592         } else {
 593                 return (new power(expanded_basis,exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 594         }
 595 }
 596
 597 //////////
 598 // new virtual functions which can be overridden by derived classes
 599 //////////
 600
 601 // none
 602
 603 //////////
 604 // non-virtual functions in this class
 605 //////////
 606
 607 /** expand a^n where a is an add and n is an integer.
 608  *  @see power::expand */
 609 ex power::expand_add(const add & a, int n) const
 610 {
 611         if (n==2)
 612                 return expand_add_2(a);
 613
 614         int m = a.nops();
 615         exvector sum;
 616         sum.reserve((n+1)*(m-1));
 617         intvector k(m-1);
 618         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 619         intvector upper_limit(m-1);
 620         int l;
 621
 622         for (int l=0; l<m-1; l++) {
 623                 k[l] = 0;
 624                 k_cum[l] = 0;
 625                 upper_limit[l] = n;
 626         }
 627
 628         while (1) {
 629                 exvector term;
 630                 term.reserve(m+1);
 631                 for (l=0; l<m-1; l++) {
 632                         const ex & b = a.op(l);
 633                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 634                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 635                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 636                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 637                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 638                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 639                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 640                         if (is_ex_exactly_of_type(b,mul)) {
 641                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
 642                         } else {
 643                                 term.push_back(power(b,k[l]));
 644                         }
 645                 }
 646
 647                 const ex & b = a.op(l);
 648                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 649                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 650                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 651                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 652                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 653                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 654                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 655                 if (is_ex_exactly_of_type(b,mul)) {
 656                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
 657                 } else {
 658                         term.push_back(power(b,n-k_cum[m-2]));
 659                 }
 660
 661                 numeric f = binomial(numeric(n),numeric(k[0]));
 662                 for (l=1; l<m-1; l++) {
 663                         f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 664                 }
 665                 term.push_back(f);
 666
 667                 /*
 668                 cout << "begin term" << endl;
 669                 for (int i=0; i<m-1; i++) {
 670                         cout << "k[" << i << "]=" << k[i] << endl;
 671                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
 672                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
 673                 }
 674                 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
 675                         cout << *cit << endl;
 676                 }
 677                 cout << "end term" << endl;
 678                 */
 679
 680                 // TODO: optimize this
 681                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 682
 683                 // increment k[]
 684                 l=m-2;
 685                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
 686                         k[l]=0;
 687                         l--;
 688                 }
 689                 if (l<0) break;
 690
 691                 // recalc k_cum[] and upper_limit[]
 692                 if (l==0) {
 693                         k_cum[0]=k[0];
 694                 } else {
 695                         k_cum[l]=k_cum[l-1]+k[l];
 696                 }
 697                 for (int i=l+1; i<m-1; i++) {
 698                         k_cum[i]=k_cum[i-1]+k[i];
 699                 }
 700
 701                 for (int i=l+1; i<m-1; i++) {
 702                         upper_limit[i]=n-k_cum[i-1];
 703                 }
 704         }
 705         return (new add(sum))->setflag(status_flags::dynallocated |
 706                                                                    status_flags::expanded );
 707 }
 708
 709
 710 /** Special case of power::expand_add. Expands a^2 where a is an add.
 711  *  @see power::expand_add */
 712 ex power::expand_add_2(const add & a) const
 713 {
 714         epvector sum;
 715         unsigned a_nops=a.nops();
 716         sum.reserve((a_nops*(a_nops+1))/2);
 717         epvector::const_iterator last=a.seq.end();
 718
 719         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 720         // first part: ignore overall_coeff and expand other terms
 721         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 722                 const ex & r=(*cit0).rest;
 723                 const ex & c=(*cit0).coeff;
 724
 725                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
 726                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
 727                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
 728                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
 729                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
 730                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
 731                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
 732
 733                 if (are_ex_trivially_equal(c,_ex1())) {
 734                         if (is_ex_exactly_of_type(r,mul)) {
 735                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 736                                                      _ex1()));
 737                         } else {
 738                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 739                                                      _ex1()));
 740                         }
 741                 } else {
 742                         if (is_ex_exactly_of_type(r,mul)) {
 743                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 744                                                      ex_to_numeric(c).power_dyn(_num2())));
 745                         } else {
 746                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 747                                                      ex_to_numeric(c).power_dyn(_num2())));
 748                         }
 749                 }
 750
 751                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 752                         const ex & r1=(*cit1).rest;
 753                         const ex & c1=(*cit1).coeff;
 754                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 755                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
 756                 }
 757         }
 758
 759         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 760
 761         // second part: add terms coming from overall_factor (if != 0)
 762         if (!a.overall_coeff.is_equal(_ex0())) {
 763                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
 764                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
 765                 }
 766                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
 767         }
 768
 769         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 770
 771         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 772 }
 773
 774 /** Expand factors of m in m^n where m is a mul and n is and integer
 775  *  @see power::expand */
 776 ex power::expand_mul(const mul & m, const numeric & n) const
 777 {
 778         if (n.is_equal(_num0()))
 779                 return _ex1();
 780
 781         epvector distrseq;
 782         distrseq.reserve(m.seq.size());
 783         epvector::const_iterator last = m.seq.end();
 784         epvector::const_iterator cit = m.seq.begin();
 785         while (cit!=last) {
 786                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 787                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 788                 } else {
 789                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 790                         // since n is an integer
 791                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
 792                 }
 793                 ++cit;
 794         }
 795         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 796 }
 797
 798 /*
 799 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
 800                                unsigned options) const
 801 {
 802         // obsolete
 803
 804         exvector distrseq;
 805         epvector splitseq;
 806
 807         const add & addref=static_cast<const add &>(*basis.bp);
 808
 809         splitseq=addref.seq;
 810         splitseq.pop_back();
 811         ex first_operands=add(splitseq);
 812         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
 813
 814         int n=exponent.to_int();
 815         for (int k=0; k<=n; k++) {
 816                 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
 817                                                  * power(last_operand,numeric(n-k)));
 818         }
 819         return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
 820 }
 821 */
 822
 823 /*
 824 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
 825                                                                 unsigned options) const
 826 {
 827         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
 828                         expand(options | expand_options::internal_do_not_expand_power_operands);
 829
 830         return ex(mul(rest_power,basis),0).
 831                expand(options | expand_options::internal_do_not_expand_mul_operands);
 832 }
 833 */
 834
 835 //////////
 836 // static member variables
 837 //////////
 838
 839 // protected
 840
 841 unsigned power::precedence = 60;
 842
 843 //////////
 844 // global constants
 845 //////////
 846
 847 const power some_power;
 848 const std::type_info & typeid_power=typeid(some_power);
 849
 850 // helper function
 851
 852 ex sqrt(const ex & a)
 853 {
 854         return power(a,_ex1_2());
 855 }
 856
 857 #ifndef NO_NAMESPACE_GINAC
 858 } // namespace GiNaC
 859 #endif // ndef NO_NAMESPACE_GINAC