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(0);
  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(1);
  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) ||
 216                  is_ex_exactly_of_type(basis, constant))) {
 217                 int exp = ex_to_numeric(exponent).to_int();
 218                 if (exp > 0)
 219                         os << "(";
 220                 else {
 221                         exp = -exp;
 222                         if (type == csrc_types::ctype_cl_N)
 223                                 os << "recip(";
 224                         else
 225                                 os << "1.0/(";
 226                 }
 227                 print_sym_pow(os, type, static_cast<const symbol &>(*basis.bp), exp);
 228                 os << ")";
 229
 230         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 231         } else if (exponent.compare(_num_1()) == 0) {
 232                 if (type == csrc_types::ctype_cl_N)
 233                         os << "recip(";
 234                 else
 235                         os << "1.0/(";
 236                 basis.bp->printcsrc(os, type, 0);
 237                 os << ")";
 238
 239         // Otherwise, use the pow() or expt() (CLN) functions
 240         } else {
 241                 if (type == csrc_types::ctype_cl_N)
 242                         os << "expt(";
 243                 else
 244                         os << "pow(";
 245                 basis.bp->printcsrc(os, type, 0);
 246                 os << ",";
 247                 exponent.bp->printcsrc(os, type, 0);
 248                 os << ")";
 249         }
 250 }
 251
 252 bool power::info(unsigned inf) const
 253 {
 254         switch (inf) {
 255                 case info_flags::polynomial:
 256                 case info_flags::integer_polynomial:
 257                 case info_flags::cinteger_polynomial:
 258                 case info_flags::rational_polynomial:
 259                 case info_flags::crational_polynomial:
 260                         return exponent.info(info_flags::nonnegint);
 261                 case info_flags::rational_function:
 262                         return exponent.info(info_flags::integer);
 263                 case info_flags::algebraic:
 264                         return (!exponent.info(info_flags::integer) ||
 265                                         basis.info(inf));
 266         }
 267         return inherited::info(inf);
 268 }
 269
 270 unsigned power::nops() const
 271 {
 272         return 2;
 273 }
 274
 275 ex & power::let_op(int i)
 276 {
 277         GINAC_ASSERT(i>=0);
 278         GINAC_ASSERT(i<2);
 279
 280         return i==0 ? basis : exponent;
 281 }
 282
 283 int power::degree(const symbol & s) const
 284 {
 285         if (is_exactly_of_type(*exponent.bp,numeric)) {
 286                 if ((*basis.bp).compare(s)==0)
 287                         return ex_to_numeric(exponent).to_int();
 288                 else
 289                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
 290         }
 291         return 0;
 292 }
 293
 294 int power::ldegree(const symbol & s) const
 295 {
 296         if (is_exactly_of_type(*exponent.bp,numeric)) {
 297                 if ((*basis.bp).compare(s)==0)
 298                         return ex_to_numeric(exponent).to_int();
 299                 else
 300                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
 301         }
 302         return 0;
 303 }
 304
 305 ex power::coeff(const symbol & s, int n) const
 306 {
 307         if ((*basis.bp).compare(s)!=0) {
 308                 // basis not equal to s
 309                 if (n==0) {
 310                         return *this;
 311                 } else {
 312                         return _ex0();
 313                 }
 314         } else if (is_exactly_of_type(*exponent.bp,numeric)&&
 315                            (static_cast<const numeric &>(*exponent.bp).compare(numeric(n))==0)) {
 316                 return _ex1();
 317         }
 318
 319         return _ex0();
 320 }
 321
 322 ex power::eval(int level) const
 323 {
 324         // simplifications: ^(x,0) -> 1 (0^0 handled here)
 325         //                  ^(x,1) -> x
 326         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
 327         //                  ^(1,x) -> 1
 328         //                  ^(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)
 329         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 330         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 331         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 332         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 333
 334         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 335
 336         if ((level==1) && (flags & status_flags::evaluated))
 337                 return *this;
 338         else if (level == -max_recursion_level)
 339                 throw(std::runtime_error("max recursion level reached"));
 340
 341         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 342         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 343
 344         bool basis_is_numerical = 0;
 345         bool exponent_is_numerical = 0;
 346         numeric * num_basis;
 347         numeric * num_exponent;
 348
 349         if (is_exactly_of_type(*ebasis.bp,numeric)) {
 350                 basis_is_numerical = 1;
 351                 num_basis = static_cast<numeric *>(ebasis.bp);
 352         }
 353         if (is_exactly_of_type(*eexponent.bp,numeric)) {
 354                 exponent_is_numerical = 1;
 355                 num_exponent = static_cast<numeric *>(eexponent.bp);
 356         }
 357
 358         // ^(x,0) -> 1 (0^0 also handled here)
 359         if (eexponent.is_zero())
 360                 if (ebasis.is_zero())
 361                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 362                 else
 363                         return _ex1();
 364
 365         // ^(x,1) -> x
 366         if (eexponent.is_equal(_ex1()))
 367                 return ebasis;
 368
 369         // ^(0,c1) -> 0 or exception (depending on real value of c1)
 370         if (ebasis.is_zero() && exponent_is_numerical) {
 371                 if ((num_exponent->real()).is_zero())
 372                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 373                 else if ((num_exponent->real()).is_negative())
 374                         throw (pole_error("power::eval(): division by zero",1));
 375                 else
 376                         return _ex0();
 377         }
 378
 379         // ^(1,x) -> 1
 380         if (ebasis.is_equal(_ex1()))
 381                 return _ex1();
 382
 383         if (basis_is_numerical && exponent_is_numerical) {
 384                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
 385                 // except if c1,c2 are rational, but c1^c2 is not)
 386                 bool basis_is_crational = num_basis->is_crational();
 387                 bool exponent_is_crational = num_exponent->is_crational();
 388                 numeric res = (*num_basis).power(*num_exponent);
 389
 390                 if ((!basis_is_crational || !exponent_is_crational)
 391                         || res.is_crational()) {
 392                         return res;
 393                 }
 394                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 395                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 396                 if (basis_is_crational && exponent_is_crational
 397                         && num_exponent->is_real()
 398                         && !num_exponent->is_integer()) {
 399                         numeric n = num_exponent->numer();
 400                         numeric m = num_exponent->denom();
 401                         numeric r;
 402                         numeric q = iquo(n, m, r);
 403                         if (r.is_negative()) {
 404                                 r = r.add(m);
 405                                 q = q.sub(_num1());
 406                         }
 407                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 408                                 return this->hold();
 409                         else {
 410                                 epvector res;
 411                                 res.push_back(expair(ebasis,r.div(m)));
 412                                 return (new mul(res,ex(num_basis->power(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
 413                         }
 414                 }
 415         }
 416
 417         // ^(^(x,c1),c2) -> ^(x,c1*c2)
 418         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 419         // case c1==1 should not happen, see below!)
 420         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
 421                 const power & sub_power = ex_to_power(ebasis);
 422                 const ex & sub_basis = sub_power.basis;
 423                 const ex & sub_exponent = sub_power.exponent;
 424                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 425                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
 426                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
 427                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
 428                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 429                         }
 430                 }
 431         }
 432
 433         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 434         if (exponent_is_numerical && num_exponent->is_integer() &&
 435                 is_ex_exactly_of_type(ebasis,mul)) {
 436                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
 437         }
 438
 439         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 440         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 441         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
 442                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 443                 const mul & mulref=ex_to_mul(ebasis);
 444                 if (!mulref.overall_coeff.is_equal(_ex1())) {
 445                         const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff);
 446                         if (num_coeff.is_real()) {
 447                                 if (num_coeff.is_positive()>0) {
 448                                         mul * mulp=new mul(mulref);
 449                                         mulp->overall_coeff=_ex1();
 450                                         mulp->clearflag(status_flags::evaluated);
 451                                         mulp->clearflag(status_flags::hash_calculated);
 452                                         return (new mul(power(*mulp,exponent),
 453                                                                         power(num_coeff,*num_exponent)))->
 454                                                 setflag(status_flags::dynallocated);
 455                                 } else {
 456                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
 457                                         if (num_coeff.compare(_num_1())!=0) {
 458                                                 mul * mulp=new mul(mulref);
 459                                                 mulp->overall_coeff=_ex_1();
 460                                                 mulp->clearflag(status_flags::evaluated);
 461                                                 mulp->clearflag(status_flags::hash_calculated);
 462                                                 return (new mul(power(*mulp,exponent),
 463                                                                                 power(abs(num_coeff),*num_exponent)))->
 464                                                         setflag(status_flags::dynallocated);
 465                                         }
 466                                 }
 467                         }
 468                 }
 469         }
 470
 471         if (are_ex_trivially_equal(ebasis,basis) &&
 472                 are_ex_trivially_equal(eexponent,exponent)) {
 473                 return this->hold();
 474         }
 475         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 476                                                                                                    status_flags::evaluated);
 477 }
 478
 479 ex power::evalf(int level) const
 480 {
 481         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 482
 483         ex ebasis;
 484         ex eexponent;
 485
 486         if (level==1) {
 487                 ebasis = basis;
 488                 eexponent = exponent;
 489         } else if (level == -max_recursion_level) {
 490                 throw(std::runtime_error("max recursion level reached"));
 491         } else {
 492                 ebasis = basis.evalf(level-1);
 493                 if (!is_ex_exactly_of_type(eexponent,numeric))
 494                         eexponent = exponent.evalf(level-1);
 495                 else
 496                         eexponent = exponent;
 497         }
 498
 499         return power(ebasis,eexponent);
 500 }
 501
 502 ex power::subs(const lst & ls, const lst & lr) const
 503 {
 504         const ex & subsed_basis=basis.subs(ls,lr);
 505         const ex & subsed_exponent=exponent.subs(ls,lr);
 506
 507         if (are_ex_trivially_equal(basis,subsed_basis)&&
 508                 are_ex_trivially_equal(exponent,subsed_exponent)) {
 509                 return *this;
 510         }
 511
 512         return power(subsed_basis, subsed_exponent);
 513 }
 514
 515 ex power::simplify_ncmul(const exvector & v) const
 516 {
 517         return inherited::simplify_ncmul(v);
 518 }
 519
 520 // protected
 521
 522 /** Implementation of ex::diff() for a power.
 523  *  @see ex::diff */
 524 ex power::derivative(const symbol & s) const
 525 {
 526         if (exponent.info(info_flags::real)) {
 527                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 528                 epvector newseq;
 529                 newseq.reserve(2);
 530                 newseq.push_back(expair(basis, exponent - _ex1()));
 531                 newseq.push_back(expair(basis.diff(s), _ex1()));
 532                 return mul(newseq, exponent);
 533         } else {
 534                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 535                 return mul(power(basis, exponent),
 536                                    add(mul(exponent.diff(s), log(basis)),
 537                                            mul(mul(exponent, basis.diff(s)), power(basis, -1))));
 538         }
 539 }
 540
 541 int power::compare_same_type(const basic & other) const
 542 {
 543         GINAC_ASSERT(is_exactly_of_type(other, power));
 544         const power & o=static_cast<const power &>(const_cast<basic &>(other));
 545
 546         int cmpval;
 547         cmpval=basis.compare(o.basis);
 548         if (cmpval==0) {
 549                 return exponent.compare(o.exponent);
 550         }
 551         return cmpval;
 552 }
 553
 554 unsigned power::return_type(void) const
 555 {
 556         return basis.return_type();
 557 }
 558
 559 unsigned power::return_type_tinfo(void) const
 560 {
 561         return basis.return_type_tinfo();
 562 }
 563
 564 ex power::expand(unsigned options) const
 565 {
 566         if (flags & status_flags::expanded)
 567                 return *this;
 568
 569         ex expanded_basis = basis.expand(options);
 570
 571         if (!is_ex_exactly_of_type(exponent,numeric) ||
 572                 !ex_to_numeric(exponent).is_integer()) {
 573                 if (are_ex_trivially_equal(basis,expanded_basis)) {
 574                         return this->hold();
 575                 } else {
 576                         return (new power(expanded_basis,exponent))->
 577                                 setflag(status_flags::dynallocated |
 578                                                 status_flags::expanded);
 579                 }
 580         }
 581
 582         // integer numeric exponent
 583         const numeric & num_exponent = ex_to_numeric(exponent);
 584         int int_exponent = num_exponent.to_int();
 585
 586         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
 587                 return expand_add(ex_to_add(expanded_basis), int_exponent);
 588         }
 589
 590         if (is_ex_exactly_of_type(expanded_basis,mul)) {
 591                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
 592         }
 593
 594         // cannot expand further
 595         if (are_ex_trivially_equal(basis,expanded_basis)) {
 596                 return this->hold();
 597         } else {
 598                 return (new power(expanded_basis,exponent))->
 599                            setflag(status_flags::dynallocated |
 600                                            status_flags::expanded);
 601         }
 602 }
 603
 604 //////////
 605 // new virtual functions which can be overridden by derived classes
 606 //////////
 607
 608 // none
 609
 610 //////////
 611 // non-virtual functions in this class
 612 //////////
 613
 614 /** expand a^n where a is an add and n is an integer.
 615  *  @see power::expand */
 616 ex power::expand_add(const add & a, int n) const
 617 {
 618         if (n==2)
 619                 return expand_add_2(a);
 620
 621         int m = a.nops();
 622         exvector sum;
 623         sum.reserve((n+1)*(m-1));
 624         intvector k(m-1);
 625         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 626         intvector upper_limit(m-1);
 627         int l;
 628
 629         for (int l=0; l<m-1; l++) {
 630                 k[l] = 0;
 631                 k_cum[l] = 0;
 632                 upper_limit[l] = n;
 633         }
 634
 635         while (1) {
 636                 exvector term;
 637                 term.reserve(m+1);
 638                 for (l=0; l<m-1; l++) {
 639                         const ex & b = a.op(l);
 640                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 641                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power)||
 642                                                  !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
 643                                                  !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer()||
 644                                                  !is_ex_exactly_of_type(ex_to_power(b).basis,add)||
 645                                                  !is_ex_exactly_of_type(ex_to_power(b).basis,mul)||
 646                                                  !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 647                         if (is_ex_exactly_of_type(b,mul)) {
 648                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
 649                         } else {
 650                                 term.push_back(power(b,k[l]));
 651                         }
 652                 }
 653
 654                 const ex & b = a.op(l);
 655                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 656                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power)||
 657                                          !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
 658                                          !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer()||
 659                                          !is_ex_exactly_of_type(ex_to_power(b).basis,add)||
 660                                          !is_ex_exactly_of_type(ex_to_power(b).basis,mul)||
 661                                          !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 662                 if (is_ex_exactly_of_type(b,mul)) {
 663                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
 664                 } else {
 665                         term.push_back(power(b,n-k_cum[m-2]));
 666                 }
 667
 668                 numeric f = binomial(numeric(n),numeric(k[0]));
 669                 for (l=1; l<m-1; l++) {
 670                         f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 671                 }
 672                 term.push_back(f);
 673
 674                 /*
 675                 cout << "begin term" << endl;
 676                 for (int i=0; i<m-1; i++) {
 677                         cout << "k[" << i << "]=" << k[i] << endl;
 678                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
 679                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
 680                 }
 681                 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
 682                         cout << *cit << endl;
 683                 }
 684                 cout << "end term" << endl;
 685                 */
 686
 687                 // TODO: optimize this
 688                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 689
 690                 // increment k[]
 691                 l=m-2;
 692                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
 693                         k[l]=0;
 694                         l--;
 695                 }
 696                 if (l<0) break;
 697
 698                 // recalc k_cum[] and upper_limit[]
 699                 if (l==0) {
 700                         k_cum[0]=k[0];
 701                 } else {
 702                         k_cum[l]=k_cum[l-1]+k[l];
 703                 }
 704                 for (int i=l+1; i<m-1; i++) {
 705                         k_cum[i]=k_cum[i-1]+k[i];
 706                 }
 707
 708                 for (int i=l+1; i<m-1; i++) {
 709                         upper_limit[i]=n-k_cum[i-1];
 710                 }
 711         }
 712         return (new add(sum))->setflag(status_flags::dynallocated |
 713                                                                    status_flags::expanded );
 714 }
 715
 716
 717 /** Special case of power::expand_add. Expands a^2 where a is an add.
 718  *  @see power::expand_add */
 719 ex power::expand_add_2(const add & a) const
 720 {
 721         epvector sum;
 722         unsigned a_nops=a.nops();
 723         sum.reserve((a_nops*(a_nops+1))/2);
 724         epvector::const_iterator last=a.seq.end();
 725
 726         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 727         // first part: ignore overall_coeff and expand other terms
 728         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 729                 const ex & r=(*cit0).rest;
 730                 const ex & c=(*cit0).coeff;
 731
 732                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
 733                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power)||
 734                            !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric)||
 735                            !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer()||
 736                            !is_ex_exactly_of_type(ex_to_power(r).basis,add)||
 737                            !is_ex_exactly_of_type(ex_to_power(r).basis,mul)||
 738                            !is_ex_exactly_of_type(ex_to_power(r).basis,power));
 739
 740                 if (are_ex_trivially_equal(c,_ex1())) {
 741                         if (is_ex_exactly_of_type(r,mul)) {
 742                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),_ex1()));
 743                         } else {
 744                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 745                                                                          _ex1()));
 746                         }
 747                 } else {
 748                         if (is_ex_exactly_of_type(r,mul)) {
 749                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 750                                                                          ex_to_numeric(c).power_dyn(_num2())));
 751                         } else {
 752                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 753                                                                          ex_to_numeric(c).power_dyn(_num2())));
 754                         }
 755                 }
 756
 757                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 758                         const ex & r1=(*cit1).rest;
 759                         const ex & c1=(*cit1).coeff;
 760                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 761                                                                                                                   _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
 762                 }
 763         }
 764
 765         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 766
 767         // second part: add terms coming from overall_factor (if != 0)
 768         if (!a.overall_coeff.is_equal(_ex0())) {
 769                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
 770                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
 771                 }
 772                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
 773         }
 774
 775         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 776
 777         return (new add(sum))->setflag(status_flags::dynallocated |
 778                                                                    status_flags::expanded );
 779 }
 780
 781 /** Expand factors of m in m^n where m is a mul and n is and integer
 782  *  @see power::expand */
 783 ex power::expand_mul(const mul & m, const numeric & n) const
 784 {
 785         if (n.is_equal(_num0()))
 786                 return _ex1();
 787
 788         epvector distrseq;
 789         distrseq.reserve(m.seq.size());
 790         epvector::const_iterator last = m.seq.end();
 791         epvector::const_iterator cit = m.seq.begin();
 792         while (cit!=last) {
 793                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 794                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 795                 } else {
 796                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 797                         // since n is an integer
 798                         distrseq.push_back(expair((*cit).rest,
 799                                                                           ex_to_numeric((*cit).coeff).mul(n)));
 800                 }
 801                 ++cit;
 802         }
 803         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))
 804                 ->setflag(status_flags::dynallocated);
 805 }
 806
 807 /*
 808 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
 809                                                          unsigned options) const
 810 {
 811         // obsolete
 812
 813         exvector distrseq;
 814         epvector splitseq;
 815
 816         const add & addref=static_cast<const add &>(*basis.bp);
 817
 818         splitseq=addref.seq;
 819         splitseq.pop_back();
 820         ex first_operands=add(splitseq);
 821         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
 822
 823         int n=exponent.to_int();
 824         for (int k=0; k<=n; k++) {
 825                 distrseq.push_back(binomial(n,k)*power(first_operands,numeric(k))*
 826                                                    power(last_operand,numeric(n-k)));
 827         }
 828         return ex((new add(distrseq))->setflag(status_flags::expanded |
 829                                                                                    status_flags::dynallocated )).
 830                    expand(options);
 831 }
 832 */
 833
 834 /*
 835 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
 836                                                                 unsigned options) const
 837 {
 838         ex rest_power=ex(power(basis,exponent.add(_num_1()))).
 839                                   expand(options | expand_options::internal_do_not_expand_power_operands);
 840
 841         return ex(mul(rest_power,basis),0).
 842                    expand(options | expand_options::internal_do_not_expand_mul_operands);
 843 }
 844 */
 845
 846 //////////
 847 // static member variables
 848 //////////
 849
 850 // protected
 851
 852 unsigned power::precedence = 60;
 853
 854 //////////
 855 // global constants
 856 //////////
 857
 858 const power some_power;
 859 const type_info & typeid_power=typeid(some_power);
 860
 861 // helper function
 862
 863 ex sqrt(const ex & a)
 864 {
 865         return power(a,_ex1_2());
 866 }
 867
 868 #ifndef NO_NAMESPACE_GINAC
 869 } // namespace GiNaC
 870 #endif // ndef NO_NAMESPACE_GINAC