ginac/power.cpp

   1 /** @file power.cpp
   2  *
   3  *  Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <vector>
  24 #include <iostream>
  25 #include <stdexcept>
  26
  27 #include "power.h"
  28 #include "expairseq.h"
  29 #include "add.h"
  30 #include "mul.h"
  31 #include "numeric.h"
  32 #include "inifcns.h"
  33 #include "relational.h"
  34 #include "symbol.h"
  35 #include "archive.h"
  36 #include "debugmsg.h"
  37 #include "utils.h"
  38
  39 #ifndef NO_NAMESPACE_GINAC
  40 namespace GiNaC {
  41 #endif // ndef NO_NAMESPACE_GINAC
  42
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
  44
  45 typedef std::vector<int> intvector;
  46
  47 //////////
  48 // default constructor, destructor, copy constructor assignment operator and helpers
  49 //////////
  50
  51 // public
  52
  53 power::power() : basic(TINFO_power)
  54 {
  55         debugmsg("power default constructor",LOGLEVEL_CONSTRUCT);
  56 }
  57
  58 // protected
  59
  60 void power::copy(const power & other)
  61 {
  62         inherited::copy(other);
  63         basis=other.basis;
  64         exponent=other.exponent;
  65 }
  66
  67 void power::destroy(bool call_parent)
  68 {
  69         if (call_parent) inherited::destroy(call_parent);
  70 }
  71
  72 //////////
  73 // other constructors
  74 //////////
  75
  76 // public
  77
  78 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  79 {
  80         debugmsg("power constructor from ex,ex",LOGLEVEL_CONSTRUCT);
  81         GINAC_ASSERT(basis.return_type()==return_types::commutative);
  82 }
  83
  84 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  85 {
  86         debugmsg("power constructor from ex,numeric",LOGLEVEL_CONSTRUCT);
  87         GINAC_ASSERT(basis.return_type()==return_types::commutative);
  88 }
  89
  90 //////////
  91 // archiving
  92 //////////
  93
  94 /** Construct object from archive_node. */
  95 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  96 {
  97         debugmsg("power constructor from archive_node", LOGLEVEL_CONSTRUCT);
  98         n.find_ex("basis", basis, sym_lst);
  99         n.find_ex("exponent", exponent, sym_lst);
 100 }
 101
 102 /** Unarchive the object. */
 103 ex power::unarchive(const archive_node &n, const lst &sym_lst)
 104 {
 105         return (new power(n, sym_lst))->setflag(status_flags::dynallocated);
 106 }
 107
 108 /** Archive the object. */
 109 void power::archive(archive_node &n) const
 110 {
 111         inherited::archive(n);
 112         n.add_ex("basis", basis);
 113         n.add_ex("exponent", exponent);
 114 }
 115
 116 //////////
 117 // functions overriding virtual functions from bases classes
 118 //////////
 119
 120 // public
 121
 122 void power::print(std::ostream & os, unsigned upper_precedence) const
 123 {
 124         debugmsg("power print",LOGLEVEL_PRINT);
 125         if (exponent.is_equal(_ex1_2())) {
 126                 os << "sqrt(" << basis << ")";
 127         } else {
 128                 if (precedence<=upper_precedence) os << "(";
 129                 basis.print(os,precedence);
 130                 os << "^";
 131                 exponent.print(os,precedence);
 132                 if (precedence<=upper_precedence) os << ")";
 133         }
 134 }
 135
 136 void power::printraw(std::ostream & os) const
 137 {
 138         debugmsg("power printraw",LOGLEVEL_PRINT);
 139
 140         os << "power(";
 141         basis.printraw(os);
 142         os << ",";
 143         exponent.printraw(os);
 144         os << ",hash=" << hashvalue << ",flags=" << flags << ")";
 145 }
 146
 147 void power::printtree(std::ostream & os, unsigned indent) const
 148 {
 149         debugmsg("power printtree",LOGLEVEL_PRINT);
 150
 151         os << std::string(indent,' ') << "power: "
 152            << "hash=" << hashvalue
 153            << " (0x" << std::hex << hashvalue << std::dec << ")"
 154            << ", flags=" << flags << std::endl;
 155         basis.printtree(os, indent+delta_indent);
 156         exponent.printtree(os, indent+delta_indent);
 157 }
 158
 159 static void print_sym_pow(std::ostream & os, unsigned type, const symbol &x, int exp)
 160 {
 161         // Optimal output of integer powers of symbols to aid compiler CSE
 162         if (exp == 1) {
 163                 x.printcsrc(os, type, 0);
 164         } else if (exp == 2) {
 165                 x.printcsrc(os, type, 0);
 166                 os << "*";
 167                 x.printcsrc(os, type, 0);
 168         } else if (exp & 1) {
 169                 x.printcsrc(os, 0);
 170                 os << "*";
 171                 print_sym_pow(os, type, x, exp-1);
 172         } else {
 173                 os << "(";
 174                 print_sym_pow(os, type, x, exp >> 1);
 175                 os << ")*(";
 176                 print_sym_pow(os, type, x, exp >> 1);
 177                 os << ")";
 178         }
 179 }
 180
 181 void power::printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence) const
 182 {
 183         debugmsg("power print csrc", LOGLEVEL_PRINT);
 184
 185         // Integer powers of symbols are printed in a special, optimized way
 186         if (exponent.info(info_flags::integer)
 187          && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
 188                 int exp = ex_to_numeric(exponent).to_int();
 189                 if (exp > 0)
 190                         os << "(";
 191                 else {
 192                         exp = -exp;
 193                         if (type == csrc_types::ctype_cl_N)
 194                                 os << "recip(";
 195                         else
 196                                 os << "1.0/(";
 197                 }
 198                 print_sym_pow(os, type, static_cast<const symbol &>(*basis.bp), exp);
 199                 os << ")";
 200
 201         // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 202         } else if (exponent.compare(_num_1()) == 0) {
 203                 if (type == csrc_types::ctype_cl_N)
 204                         os << "recip(";
 205                 else
 206                         os << "1.0/(";
 207                 basis.bp->printcsrc(os, type, 0);
 208                 os << ")";
 209
 210         // Otherwise, use the pow() or expt() (CLN) functions
 211         } else {
 212                 if (type == csrc_types::ctype_cl_N)
 213                         os << "expt(";
 214                 else
 215                         os << "pow(";
 216                 basis.bp->printcsrc(os, type, 0);
 217                 os << ",";
 218                 exponent.bp->printcsrc(os, type, 0);
 219                 os << ")";
 220         }
 221 }
 222
 223 bool power::info(unsigned inf) const
 224 {
 225         switch (inf) {
 226                 case info_flags::polynomial:
 227                 case info_flags::integer_polynomial:
 228                 case info_flags::cinteger_polynomial:
 229                 case info_flags::rational_polynomial:
 230                 case info_flags::crational_polynomial:
 231                         return exponent.info(info_flags::nonnegint);
 232                 case info_flags::rational_function:
 233                         return exponent.info(info_flags::integer);
 234                 case info_flags::algebraic:
 235                         return (!exponent.info(info_flags::integer) ||
 236                                         basis.info(inf));
 237         }
 238         return inherited::info(inf);
 239 }
 240
 241 unsigned power::nops() const
 242 {
 243         return 2;
 244 }
 245
 246 ex & power::let_op(int i)
 247 {
 248         GINAC_ASSERT(i>=0);
 249         GINAC_ASSERT(i<2);
 250
 251         return i==0 ? basis : exponent;
 252 }
 253
 254 int power::degree(const symbol & s) const
 255 {
 256         if (is_exactly_of_type(*exponent.bp,numeric)) {
 257                 if ((*basis.bp).compare(s)==0) {
 258                         if (ex_to_numeric(exponent).is_integer())
 259                                 return ex_to_numeric(exponent).to_int();
 260                         else
 261                                 return 0;
 262                 } else
 263                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
 264         }
 265         return 0;
 266 }
 267
 268 int power::ldegree(const symbol & s) const
 269 {
 270         if (is_exactly_of_type(*exponent.bp,numeric)) {
 271                 if ((*basis.bp).compare(s)==0) {
 272                         if (ex_to_numeric(exponent).is_integer())
 273                                 return ex_to_numeric(exponent).to_int();
 274                         else
 275                                 return 0;
 276                 } else
 277                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
 278         }
 279         return 0;
 280 }
 281
 282 ex power::coeff(const symbol & s, int n) const
 283 {
 284         if ((*basis.bp).compare(s)!=0) {
 285                 // basis not equal to s
 286                 if (n == 0)
 287                         return *this;
 288                 else
 289                         return _ex0();
 290         } else {
 291                 // basis equal to s
 292                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
 293                         // integer exponent
 294                         int int_exp = ex_to_numeric(exponent).to_int();
 295                         if (n == int_exp)
 296                                 return _ex1();
 297                         else
 298                                 return _ex0();
 299                 } else {
 300                         // non-integer exponents are treated as zero
 301                         if (n == 0)
 302                                 return *this;
 303                         else
 304                                 return _ex0();
 305                 }
 306         }
 307 }
 308
 309 ex power::eval(int level) const
 310 {
 311         // simplifications: ^(x,0) -> 1 (0^0 handled here)
 312         //                  ^(x,1) -> x
 313         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
 314         //                  ^(1,x) -> 1
 315         //                  ^(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)
 316         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 317         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 318         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 319         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 320
 321         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 322
 323         if ((level==1) && (flags & status_flags::evaluated))
 324                 return *this;
 325         else if (level == -max_recursion_level)
 326                 throw(std::runtime_error("max recursion level reached"));
 327
 328         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 329         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 330
 331         bool basis_is_numerical = 0;
 332         bool exponent_is_numerical = 0;
 333         numeric * num_basis;
 334         numeric * num_exponent;
 335
 336         if (is_exactly_of_type(*ebasis.bp,numeric)) {
 337                 basis_is_numerical = 1;
 338                 num_basis = static_cast<numeric *>(ebasis.bp);
 339         }
 340         if (is_exactly_of_type(*eexponent.bp,numeric)) {
 341                 exponent_is_numerical = 1;
 342                 num_exponent = static_cast<numeric *>(eexponent.bp);
 343         }
 344
 345         // ^(x,0) -> 1 (0^0 also handled here)
 346         if (eexponent.is_zero())
 347                 if (ebasis.is_zero())
 348                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 349                 else
 350                         return _ex1();
 351
 352         // ^(x,1) -> x
 353         if (eexponent.is_equal(_ex1()))
 354                 return ebasis;
 355
 356         // ^(0,c1) -> 0 or exception (depending on real value of c1)
 357         if (ebasis.is_zero() && exponent_is_numerical) {
 358                 if ((num_exponent->real()).is_zero())
 359                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 360                 else if ((num_exponent->real()).is_negative())
 361                         throw (pole_error("power::eval(): division by zero",1));
 362                 else
 363                         return _ex0();
 364         }
 365
 366         // ^(1,x) -> 1
 367         if (ebasis.is_equal(_ex1()))
 368                 return _ex1();
 369
 370         if (basis_is_numerical && exponent_is_numerical) {
 371                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
 372                 // except if c1,c2 are rational, but c1^c2 is not)
 373                 bool basis_is_crational = num_basis->is_crational();
 374                 bool exponent_is_crational = num_exponent->is_crational();
 375                 numeric res = (*num_basis).power(*num_exponent);
 376
 377                 if ((!basis_is_crational || !exponent_is_crational)
 378                         || res.is_crational()) {
 379                         return res;
 380                 }
 381                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 382                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 383                 if (basis_is_crational && exponent_is_crational
 384                         && num_exponent->is_real()
 385                         && !num_exponent->is_integer()) {
 386                         numeric n = num_exponent->numer();
 387                         numeric m = num_exponent->denom();
 388                         numeric r;
 389                         numeric q = iquo(n, m, r);
 390                         if (r.is_negative()) {
 391                                 r = r.add(m);
 392                                 q = q.sub(_num1());
 393                         }
 394                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 395                                 return this->hold();
 396                         else {
 397                                 epvector res;
 398                                 res.push_back(expair(ebasis,r.div(m)));
 399                                 return (new mul(res,ex(num_basis->power(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
 400                         }
 401                 }
 402         }
 403
 404         // ^(^(x,c1),c2) -> ^(x,c1*c2)
 405         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 406         // case c1==1 should not happen, see below!)
 407         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
 408                 const power & sub_power = ex_to_power(ebasis);
 409                 const ex & sub_basis = sub_power.basis;
 410                 const ex & sub_exponent = sub_power.exponent;
 411                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 412                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
 413                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
 414                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
 415                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 416                         }
 417                 }
 418         }
 419
 420         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 421         if (exponent_is_numerical && num_exponent->is_integer() &&
 422                 is_ex_exactly_of_type(ebasis,mul)) {
 423                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
 424         }
 425
 426         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 427         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 428         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
 429                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 430                 const mul & mulref=ex_to_mul(ebasis);
 431                 if (!mulref.overall_coeff.is_equal(_ex1())) {
 432                         const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff);
 433                         if (num_coeff.is_real()) {
 434                                 if (num_coeff.is_positive()>0) {
 435                                         mul * mulp=new mul(mulref);
 436                                         mulp->overall_coeff=_ex1();
 437                                         mulp->clearflag(status_flags::evaluated);
 438                                         mulp->clearflag(status_flags::hash_calculated);
 439                                         return (new mul(power(*mulp,exponent),
 440                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 441                                 } else {
 442                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
 443                                         if (num_coeff.compare(_num_1())!=0) {
 444                                                 mul * mulp=new mul(mulref);
 445                                                 mulp->overall_coeff=_ex_1();
 446                                                 mulp->clearflag(status_flags::evaluated);
 447                                                 mulp->clearflag(status_flags::hash_calculated);
 448                                                 return (new mul(power(*mulp,exponent),
 449                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 450                                         }
 451                                 }
 452                         }
 453                 }
 454         }
 455
 456         if (are_ex_trivially_equal(ebasis,basis) &&
 457                 are_ex_trivially_equal(eexponent,exponent)) {
 458                 return this->hold();
 459         }
 460         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 461                                                                                                    status_flags::evaluated);
 462 }
 463
 464 ex power::evalf(int level) const
 465 {
 466         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 467
 468         ex ebasis;
 469         ex eexponent;
 470
 471         if (level==1) {
 472                 ebasis = basis;
 473                 eexponent = exponent;
 474         } else if (level == -max_recursion_level) {
 475                 throw(std::runtime_error("max recursion level reached"));
 476         } else {
 477                 ebasis = basis.evalf(level-1);
 478                 if (!is_ex_exactly_of_type(eexponent,numeric))
 479                         eexponent = exponent.evalf(level-1);
 480                 else
 481                         eexponent = exponent;
 482         }
 483
 484         return power(ebasis,eexponent);
 485 }
 486
 487 ex power::subs(const lst & ls, const lst & lr) const
 488 {
 489         const ex & subsed_basis=basis.subs(ls,lr);
 490         const ex & subsed_exponent=exponent.subs(ls,lr);
 491
 492         if (are_ex_trivially_equal(basis,subsed_basis)&&
 493                 are_ex_trivially_equal(exponent,subsed_exponent)) {
 494                 return *this;
 495         }
 496
 497         return power(subsed_basis, subsed_exponent);
 498 }
 499
 500 ex power::simplify_ncmul(const exvector & v) const
 501 {
 502         return inherited::simplify_ncmul(v);
 503 }
 504
 505 // protected
 506
 507 /** Implementation of ex::diff() for a power.
 508  *  @see ex::diff */
 509 ex power::derivative(const symbol & s) const
 510 {
 511         if (exponent.info(info_flags::real)) {
 512                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 513                 epvector newseq;
 514                 newseq.reserve(2);
 515                 newseq.push_back(expair(basis, exponent - _ex1()));
 516                 newseq.push_back(expair(basis.diff(s), _ex1()));
 517                 return mul(newseq, exponent);
 518         } else {
 519                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 520                 return mul(*this,
 521                            add(mul(exponent.diff(s), log(basis)),
 522                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
 523         }
 524 }
 525
 526 int power::compare_same_type(const basic & other) const
 527 {
 528         GINAC_ASSERT(is_exactly_of_type(other, power));
 529         const power & o=static_cast<const power &>(const_cast<basic &>(other));
 530
 531         int cmpval;
 532         cmpval=basis.compare(o.basis);
 533         if (cmpval==0) {
 534                 return exponent.compare(o.exponent);
 535         }
 536         return cmpval;
 537 }
 538
 539 unsigned power::return_type(void) const
 540 {
 541         return basis.return_type();
 542 }
 543
 544 unsigned power::return_type_tinfo(void) const
 545 {
 546         return basis.return_type_tinfo();
 547 }
 548
 549 ex power::expand(unsigned options) const
 550 {
 551         if (flags & status_flags::expanded)
 552                 return *this;
 553
 554         ex expanded_basis = basis.expand(options);
 555         ex expanded_exponent = exponent.expand(options);
 556
 557         // x^(a+b) -> x^a * x^b
 558         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 559                 const add &a = ex_to_add(expanded_exponent);
 560                 exvector distrseq;
 561                 distrseq.reserve(a.seq.size() + 1);
 562                 epvector::const_iterator last = a.seq.end();
 563                 epvector::const_iterator cit = a.seq.begin();
 564                 while (cit!=last) {
 565                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 566                         cit++;
 567                 }
 568
 569                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 570                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
 571                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
 572                         int int_exponent = num_exponent.to_int();
 573                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 574                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
 575                         else
 576                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 577                 } else
 578                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 579
 580                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 581                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 582                 return r.expand();
 583         }
 584
 585         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 586                 !ex_to_numeric(expanded_exponent).is_integer()) {
 587                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 588                         return this->hold();
 589                 } else {
 590                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 591                 }
 592         }
 593
 594         // integer numeric exponent
 595         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
 596         int int_exponent = num_exponent.to_int();
 597
 598         // (x+y)^n, n>0
 599         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
 600                 return expand_add(ex_to_add(expanded_basis), int_exponent);
 601         }
 602
 603         // (x*y)^n -> x^n * y^n
 604         if (is_ex_exactly_of_type(expanded_basis,mul)) {
 605                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
 606         }
 607
 608         // cannot expand further
 609         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 610                 return this->hold();
 611         } else {
 612                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 613         }
 614 }
 615
 616 //////////
 617 // new virtual functions which can be overridden by derived classes
 618 //////////
 619
 620 // none
 621
 622 //////////
 623 // non-virtual functions in this class
 624 //////////
 625
 626 /** expand a^n where a is an add and n is an integer.
 627  *  @see power::expand */
 628 ex power::expand_add(const add & a, int n) const
 629 {
 630         if (n==2)
 631                 return expand_add_2(a);
 632
 633         int m = a.nops();
 634         exvector sum;
 635         sum.reserve((n+1)*(m-1));
 636         intvector k(m-1);
 637         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 638         intvector upper_limit(m-1);
 639         int l;
 640
 641         for (int l=0; l<m-1; l++) {
 642                 k[l] = 0;
 643                 k_cum[l] = 0;
 644                 upper_limit[l] = n;
 645         }
 646
 647         while (1) {
 648                 exvector term;
 649                 term.reserve(m+1);
 650                 for (l=0; l<m-1; l++) {
 651                         const ex & b = a.op(l);
 652                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 653                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 654                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 655                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 656                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 657                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 658                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 659                         if (is_ex_exactly_of_type(b,mul)) {
 660                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
 661                         } else {
 662                                 term.push_back(power(b,k[l]));
 663                         }
 664                 }
 665
 666                 const ex & b = a.op(l);
 667                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 668                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 669                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 670                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 671                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 672                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 673                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 674                 if (is_ex_exactly_of_type(b,mul)) {
 675                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
 676                 } else {
 677                         term.push_back(power(b,n-k_cum[m-2]));
 678                 }
 679
 680                 numeric f = binomial(numeric(n),numeric(k[0]));
 681                 for (l=1; l<m-1; l++) {
 682                         f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 683                 }
 684                 term.push_back(f);
 685
 686                 /*
 687                 cout << "begin term" << endl;
 688                 for (int i=0; i<m-1; i++) {
 689                         cout << "k[" << i << "]=" << k[i] << endl;
 690                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
 691                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
 692                 }
 693                 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
 694                         cout << *cit << endl;
 695                 }
 696                 cout << "end term" << endl;
 697                 */
 698
 699                 // TODO: optimize this
 700                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 701
 702                 // increment k[]
 703                 l=m-2;
 704                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
 705                         k[l]=0;
 706                         l--;
 707                 }
 708                 if (l<0) break;
 709
 710                 // recalc k_cum[] and upper_limit[]
 711                 if (l==0) {
 712                         k_cum[0]=k[0];
 713                 } else {
 714                         k_cum[l]=k_cum[l-1]+k[l];
 715                 }
 716                 for (int i=l+1; i<m-1; i++) {
 717                         k_cum[i]=k_cum[i-1]+k[i];
 718                 }
 719
 720                 for (int i=l+1; i<m-1; i++) {
 721                         upper_limit[i]=n-k_cum[i-1];
 722                 }
 723         }
 724         return (new add(sum))->setflag(status_flags::dynallocated |
 725                                                                    status_flags::expanded );
 726 }
 727
 728
 729 /** Special case of power::expand_add. Expands a^2 where a is an add.
 730  *  @see power::expand_add */
 731 ex power::expand_add_2(const add & a) const
 732 {
 733         epvector sum;
 734         unsigned a_nops=a.nops();
 735         sum.reserve((a_nops*(a_nops+1))/2);
 736         epvector::const_iterator last=a.seq.end();
 737
 738         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 739         // first part: ignore overall_coeff and expand other terms
 740         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 741                 const ex & r=(*cit0).rest;
 742                 const ex & c=(*cit0).coeff;
 743
 744                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
 745                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
 746                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
 747                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
 748                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
 749                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
 750                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
 751
 752                 if (are_ex_trivially_equal(c,_ex1())) {
 753                         if (is_ex_exactly_of_type(r,mul)) {
 754                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 755                                                      _ex1()));
 756                         } else {
 757                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 758                                                      _ex1()));
 759                         }
 760                 } else {
 761                         if (is_ex_exactly_of_type(r,mul)) {
 762                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 763                                                      ex_to_numeric(c).power_dyn(_num2())));
 764                         } else {
 765                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 766                                                      ex_to_numeric(c).power_dyn(_num2())));
 767                         }
 768                 }
 769
 770                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 771                         const ex & r1=(*cit1).rest;
 772                         const ex & c1=(*cit1).coeff;
 773                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 774                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
 775                 }
 776         }
 777
 778         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 779
 780         // second part: add terms coming from overall_factor (if != 0)
 781         if (!a.overall_coeff.is_equal(_ex0())) {
 782                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
 783                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
 784                 }
 785                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
 786         }
 787
 788         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 789
 790         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 791 }
 792
 793 /** Expand factors of m in m^n where m is a mul and n is and integer
 794  *  @see power::expand */
 795 ex power::expand_mul(const mul & m, const numeric & n) const
 796 {
 797         if (n.is_equal(_num0()))
 798                 return _ex1();
 799
 800         epvector distrseq;
 801         distrseq.reserve(m.seq.size());
 802         epvector::const_iterator last = m.seq.end();
 803         epvector::const_iterator cit = m.seq.begin();
 804         while (cit!=last) {
 805                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 806                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 807                 } else {
 808                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 809                         // since n is an integer
 810                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
 811                 }
 812                 ++cit;
 813         }
 814         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 815 }
 816
 817 /*
 818 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
 819                                unsigned options) const
 820 {
 821         // obsolete
 822
 823         exvector distrseq;
 824         epvector splitseq;
 825
 826         const add & addref=static_cast<const add &>(*basis.bp);
 827
 828         splitseq=addref.seq;
 829         splitseq.pop_back();
 830         ex first_operands=add(splitseq);
 831         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
 832
 833         int n=exponent.to_int();
 834         for (int k=0; k<=n; k++) {
 835                 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
 836                                                  * power(last_operand,numeric(n-k)));
 837         }
 838         return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
 839 }
 840 */
 841
 842 /*
 843 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
 844                                                                 unsigned options) const
 845 {
 846         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
 847                         expand(options | expand_options::internal_do_not_expand_power_operands);
 848
 849         return ex(mul(rest_power,basis),0).
 850                expand(options | expand_options::internal_do_not_expand_mul_operands);
 851 }
 852 */
 853
 854 //////////
 855 // static member variables
 856 //////////
 857
 858 // protected
 859
 860 unsigned power::precedence = 60;
 861
 862 // helper function
 863
 864 ex sqrt(const ex & a)
 865 {
 866         return power(a,_ex1_2());
 867 }
 868
 869 #ifndef NO_NAMESPACE_GINAC
 870 } // namespace GiNaC
 871 #endif // ndef NO_NAMESPACE_GINAC