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 "ncmul.h"
  32 #include "numeric.h"
  33 #include "constant.h"
  34 #include "inifcns.h" // for log() in power::derivative()
  35 #include "matrix.h"
  36 #include "symbol.h"
  37 #include "print.h"
  38 #include "archive.h"
  39 #include "debugmsg.h"
  40 #include "utils.h"
  41
  42 namespace GiNaC {
  43
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
  45
  46 typedef std::vector<int> intvector;
  47
  48 //////////
  49 // default ctor, dtor, copy ctor assignment operator and helpers
  50 //////////
  51
  52 power::power() : inherited(TINFO_power)
  53 {
  54         debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
  55 }
  56
  57 void power::copy(const power & other)
  58 {
  59         inherited::copy(other);
  60         basis = other.basis;
  61         exponent = other.exponent;
  62 }
  63
  64 DEFAULT_DESTROY(power)
  65
  66 //////////
  67 // other ctors
  68 //////////
  69
  70 // all inlined
  71
  72 //////////
  73 // archiving
  74 //////////
  75
  76 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  77 {
  78         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
  79         n.find_ex("basis", basis, sym_lst);
  80         n.find_ex("exponent", exponent, sym_lst);
  81 }
  82
  83 void power::archive(archive_node &n) const
  84 {
  85         inherited::archive(n);
  86         n.add_ex("basis", basis);
  87         n.add_ex("exponent", exponent);
  88 }
  89
  90 DEFAULT_UNARCHIVE(power)
  91
  92 //////////
  93 // functions overriding virtual functions from base classes
  94 //////////
  95
  96 // public
  97
  98 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
  99 {
 100         // Optimal output of integer powers of symbols to aid compiler CSE.
 101         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 102         // to learn why such a hack is really necessary.
 103         if (exp == 1) {
 104                 x.print(c);
 105         } else if (exp == 2) {
 106                 x.print(c);
 107                 c.s << "*";
 108                 x.print(c);
 109         } else if (exp & 1) {
 110                 x.print(c);
 111                 c.s << "*";
 112                 print_sym_pow(c, x, exp-1);
 113         } else {
 114                 c.s << "(";
 115                 print_sym_pow(c, x, exp >> 1);
 116                 c.s << ")*(";
 117                 print_sym_pow(c, x, exp >> 1);
 118                 c.s << ")";
 119         }
 120 }
 121
 122 void power::print(const print_context & c, unsigned level) const
 123 {
 124         debugmsg("power print", LOGLEVEL_PRINT);
 125
 126         if (is_a<print_tree>(c)) {
 127
 128                 inherited::print(c, level);
 129
 130         } else if (is_a<print_csrc>(c)) {
 131
 132                 // Integer powers of symbols are printed in a special, optimized way
 133                 if (exponent.info(info_flags::integer)
 134                  && (is_exactly_a<symbol>(basis) || is_exactly_a<constant>(basis))) {
 135                         int exp = ex_to<numeric>(exponent).to_int();
 136                         if (exp > 0)
 137                                 c.s << '(';
 138                         else {
 139                                 exp = -exp;
 140                                 if (is_a<print_csrc_cl_N>(c))
 141                                         c.s << "recip(";
 142                                 else
 143                                         c.s << "1.0/(";
 144                         }
 145                         print_sym_pow(c, ex_to<symbol>(basis), exp);
 146                         c.s << ')';
 147
 148                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 149                 } else if (exponent.compare(_num_1()) == 0) {
 150                         if (is_a<print_csrc_cl_N>(c))
 151                                 c.s << "recip(";
 152                         else
 153                                 c.s << "1.0/(";
 154                         basis.print(c);
 155                         c.s << ')';
 156
 157                 // Otherwise, use the pow() or expt() (CLN) functions
 158                 } else {
 159                         if (is_a<print_csrc_cl_N>(c))
 160                                 c.s << "expt(";
 161                         else
 162                                 c.s << "pow(";
 163                         basis.print(c);
 164                         c.s << ',';
 165                         exponent.print(c);
 166                         c.s << ')';
 167                 }
 168
 169         } else {
 170
 171                 if (exponent.is_equal(_ex1_2())) {
 172                         if (is_a<print_latex>(c))
 173                                 c.s << "\\sqrt{";
 174                         else
 175                                 c.s << "sqrt(";
 176                         basis.print(c);
 177                         if (is_a<print_latex>(c))
 178                                 c.s << '}';
 179                         else
 180                                 c.s << ')';
 181                 } else {
 182                         if (precedence() <= level) {
 183                                 if (is_a<print_latex>(c))
 184                                         c.s << "{(";
 185                                 else
 186                                         c.s << "(";
 187                         }
 188                         basis.print(c, precedence());
 189                         c.s << '^';
 190                         if (is_a<print_latex>(c))
 191                                 c.s << '{';
 192                         exponent.print(c, precedence());
 193                         if (is_a<print_latex>(c))
 194                                 c.s << '}';
 195                         if (precedence() <= level) {
 196                                 if (is_a<print_latex>(c))
 197                                         c.s << ")}";
 198                                 else
 199                                         c.s << ')';
 200                         }
 201                 }
 202         }
 203 }
 204
 205 bool power::info(unsigned inf) const
 206 {
 207         switch (inf) {
 208                 case info_flags::polynomial:
 209                 case info_flags::integer_polynomial:
 210                 case info_flags::cinteger_polynomial:
 211                 case info_flags::rational_polynomial:
 212                 case info_flags::crational_polynomial:
 213                         return exponent.info(info_flags::nonnegint);
 214                 case info_flags::rational_function:
 215                         return exponent.info(info_flags::integer);
 216                 case info_flags::algebraic:
 217                         return (!exponent.info(info_flags::integer) ||
 218                                         basis.info(inf));
 219         }
 220         return inherited::info(inf);
 221 }
 222
 223 unsigned power::nops() const
 224 {
 225         return 2;
 226 }
 227
 228 ex & power::let_op(int i)
 229 {
 230         GINAC_ASSERT(i>=0);
 231         GINAC_ASSERT(i<2);
 232
 233         return i==0 ? basis : exponent;
 234 }
 235
 236 ex power::map(map_function & f) const
 237 {
 238         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
 239 }
 240
 241 int power::degree(const ex & s) const
 242 {
 243         if (is_exactly_of_type(*exponent.bp,numeric)) {
 244                 if (basis.is_equal(s)) {
 245                         if (ex_to<numeric>(exponent).is_integer())
 246                                 return ex_to<numeric>(exponent).to_int();
 247                         else
 248                                 return 0;
 249                 } else
 250                         return basis.degree(s) * ex_to<numeric>(exponent).to_int();
 251         }
 252         return 0;
 253 }
 254
 255 int power::ldegree(const ex & s) const
 256 {
 257         if (is_exactly_of_type(*exponent.bp,numeric)) {
 258                 if (basis.is_equal(s)) {
 259                         if (ex_to<numeric>(exponent).is_integer())
 260                                 return ex_to<numeric>(exponent).to_int();
 261                         else
 262                                 return 0;
 263                 } else
 264                         return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
 265         }
 266         return 0;
 267 }
 268
 269 ex power::coeff(const ex & s, int n) const
 270 {
 271         if (!basis.is_equal(s)) {
 272                 // basis not equal to s
 273                 if (n == 0)
 274                         return *this;
 275                 else
 276                         return _ex0();
 277         } else {
 278                 // basis equal to s
 279                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to<numeric>(exponent).is_integer()) {
 280                         // integer exponent
 281                         int int_exp = ex_to<numeric>(exponent).to_int();
 282                         if (n == int_exp)
 283                                 return _ex1();
 284                         else
 285                                 return _ex0();
 286                 } else {
 287                         // non-integer exponents are treated as zero
 288                         if (n == 0)
 289                                 return *this;
 290                         else
 291                                 return _ex0();
 292                 }
 293         }
 294 }
 295
 296 ex power::eval(int level) const
 297 {
 298         // simplifications: ^(x,0) -> 1 (0^0 handled here)
 299         //                  ^(x,1) -> x
 300         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
 301         //                  ^(1,x) -> 1
 302         //                  ^(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)
 303         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 304         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 305         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 306         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 307
 308         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 309
 310         if ((level==1) && (flags & status_flags::evaluated))
 311                 return *this;
 312         else if (level == -max_recursion_level)
 313                 throw(std::runtime_error("max recursion level reached"));
 314
 315         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 316         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 317
 318         bool basis_is_numerical = false;
 319         bool exponent_is_numerical = false;
 320         numeric * num_basis;
 321         numeric * num_exponent;
 322
 323         if (is_exactly_of_type(*ebasis.bp,numeric)) {
 324                 basis_is_numerical = true;
 325                 num_basis = static_cast<numeric *>(ebasis.bp);
 326         }
 327         if (is_exactly_of_type(*eexponent.bp,numeric)) {
 328                 exponent_is_numerical = true;
 329                 num_exponent = static_cast<numeric *>(eexponent.bp);
 330         }
 331
 332         // ^(x,0) -> 1 (0^0 also handled here)
 333         if (eexponent.is_zero()) {
 334                 if (ebasis.is_zero())
 335                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 336                 else
 337                         return _ex1();
 338         }
 339
 340         // ^(x,1) -> x
 341         if (eexponent.is_equal(_ex1()))
 342                 return ebasis;
 343
 344         // ^(0,c1) -> 0 or exception (depending on real value of c1)
 345         if (ebasis.is_zero() && exponent_is_numerical) {
 346                 if ((num_exponent->real()).is_zero())
 347                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 348                 else if ((num_exponent->real()).is_negative())
 349                         throw (pole_error("power::eval(): division by zero",1));
 350                 else
 351                         return _ex0();
 352         }
 353
 354         // ^(1,x) -> 1
 355         if (ebasis.is_equal(_ex1()))
 356                 return _ex1();
 357
 358         if (exponent_is_numerical) {
 359
 360                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
 361                 // except if c1,c2 are rational, but c1^c2 is not)
 362                 if (basis_is_numerical) {
 363                         bool basis_is_crational = num_basis->is_crational();
 364                         bool exponent_is_crational = num_exponent->is_crational();
 365                         numeric res = num_basis->power(*num_exponent);
 366
 367                         if ((!basis_is_crational || !exponent_is_crational)
 368                                 || res.is_crational()) {
 369                                 return res;
 370                         }
 371                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 372
 373                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 374                         if (basis_is_crational && exponent_is_crational
 375                                 && num_exponent->is_real()
 376                                 && !num_exponent->is_integer()) {
 377                                 numeric n = num_exponent->numer();
 378                                 numeric m = num_exponent->denom();
 379                                 numeric r;
 380                                 numeric q = iquo(n, m, r);
 381                                 if (r.is_negative()) {
 382                                         r = r.add(m);
 383                                         q = q.sub(_num1());
 384                                 }
 385                                 if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 386                                         return this->hold();
 387                                 else {
 388                                         epvector res;
 389                                         res.push_back(expair(ebasis,r.div(m)));
 390                                         return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
 391                                 }
 392                         }
 393                 }
 394
 395                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 396                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 397                 // case c1==1 should not happen, see below!)
 398                 if (is_ex_exactly_of_type(ebasis,power)) {
 399                         const power & sub_power = ex_to<power>(ebasis);
 400                         const ex & sub_basis = sub_power.basis;
 401                         const ex & sub_exponent = sub_power.exponent;
 402                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 403                                 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
 404                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 405                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1()).is_negative())
 406                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 407                         }
 408                 }
 409
 410                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 411                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
 412                         return expand_mul(ex_to<mul>(ebasis), *num_exponent);
 413                 }
 414
 415                 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 416                 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 417                 if (is_ex_exactly_of_type(ebasis,mul)) {
 418                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 419                         const mul & mulref = ex_to<mul>(ebasis);
 420                         if (!mulref.overall_coeff.is_equal(_ex1())) {
 421                                 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
 422                                 if (num_coeff.is_real()) {
 423                                         if (num_coeff.is_positive()) {
 424                                                 mul * mulp = new mul(mulref);
 425                                                 mulp->overall_coeff = _ex1();
 426                                                 mulp->clearflag(status_flags::evaluated);
 427                                                 mulp->clearflag(status_flags::hash_calculated);
 428                                                 return (new mul(power(*mulp,exponent),
 429                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 430                                         } else {
 431                                                 GINAC_ASSERT(num_coeff.compare(_num0())<0);
 432                                                 if (num_coeff.compare(_num_1())!=0) {
 433                                                         mul * mulp = new mul(mulref);
 434                                                         mulp->overall_coeff = _ex_1();
 435                                                         mulp->clearflag(status_flags::evaluated);
 436                                                         mulp->clearflag(status_flags::hash_calculated);
 437                                                         return (new mul(power(*mulp,exponent),
 438                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 439                                                 }
 440                                         }
 441                                 }
 442                         }
 443                 }
 444
 445                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
 446                 if (num_exponent->is_pos_integer() &&
 447                     ebasis.return_type() != return_types::commutative &&
 448                     !is_ex_of_type(ebasis,matrix)) {
 449                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 450                 }
 451         }
 452
 453         if (are_ex_trivially_equal(ebasis,basis) &&
 454                 are_ex_trivially_equal(eexponent,exponent)) {
 455                 return this->hold();
 456         }
 457         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 458                                                                                                    status_flags::evaluated);
 459 }
 460
 461 ex power::evalf(int level) const
 462 {
 463         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 464
 465         ex ebasis;
 466         ex eexponent;
 467
 468         if (level==1) {
 469                 ebasis = basis;
 470                 eexponent = exponent;
 471         } else if (level == -max_recursion_level) {
 472                 throw(std::runtime_error("max recursion level reached"));
 473         } else {
 474                 ebasis = basis.evalf(level-1);
 475                 if (!is_ex_exactly_of_type(eexponent,numeric))
 476                         eexponent = exponent.evalf(level-1);
 477                 else
 478                         eexponent = exponent;
 479         }
 480
 481         return power(ebasis,eexponent);
 482 }
 483
 484 ex power::evalm(void) const
 485 {
 486         const ex ebasis = basis.evalm();
 487         const ex eexponent = exponent.evalm();
 488         if (is_ex_of_type(ebasis,matrix)) {
 489                 if (is_ex_of_type(eexponent,numeric)) {
 490                         return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
 491                 }
 492         }
 493         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
 494 }
 495
 496 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
 497 {
 498         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
 499         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
 500
 501         if (are_ex_trivially_equal(basis, subsed_basis)
 502          && are_ex_trivially_equal(exponent, subsed_exponent))
 503                 return basic::subs(ls, lr, no_pattern);
 504         else
 505                 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
 506 }
 507
 508 ex power::simplify_ncmul(const exvector & v) const
 509 {
 510         return inherited::simplify_ncmul(v);
 511 }
 512
 513 // protected
 514
 515 /** Implementation of ex::diff() for a power.
 516  *  @see ex::diff */
 517 ex power::derivative(const symbol & s) const
 518 {
 519         if (exponent.info(info_flags::real)) {
 520                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 521                 epvector newseq;
 522                 newseq.reserve(2);
 523                 newseq.push_back(expair(basis, exponent - _ex1()));
 524                 newseq.push_back(expair(basis.diff(s), _ex1()));
 525                 return mul(newseq, exponent);
 526         } else {
 527                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 528                 return mul(*this,
 529                            add(mul(exponent.diff(s), log(basis)),
 530                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
 531         }
 532 }
 533
 534 int power::compare_same_type(const basic & other) const
 535 {
 536         GINAC_ASSERT(is_exactly_of_type(other, power));
 537         const power &o = static_cast<const power &>(other);
 538
 539         int cmpval = basis.compare(o.basis);
 540         if (cmpval)
 541                 return cmpval;
 542         else
 543                 return exponent.compare(o.exponent);
 544 }
 545
 546 unsigned power::return_type(void) const
 547 {
 548         return basis.return_type();
 549 }
 550
 551 unsigned power::return_type_tinfo(void) const
 552 {
 553         return basis.return_type_tinfo();
 554 }
 555
 556 ex power::expand(unsigned options) const
 557 {
 558         if (options == 0 && (flags & status_flags::expanded))
 559                 return *this;
 560
 561         ex expanded_basis = basis.expand(options);
 562         ex expanded_exponent = exponent.expand(options);
 563
 564         // x^(a+b) -> x^a * x^b
 565         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 566                 const add &a = ex_to<add>(expanded_exponent);
 567                 exvector distrseq;
 568                 distrseq.reserve(a.seq.size() + 1);
 569                 epvector::const_iterator last = a.seq.end();
 570                 epvector::const_iterator cit = a.seq.begin();
 571                 while (cit!=last) {
 572                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 573                         ++cit;
 574                 }
 575
 576                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 577                 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
 578                         const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
 579                         int int_exponent = num_exponent.to_int();
 580                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 581                                 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
 582                         else
 583                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 584                 } else
 585                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 586
 587                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 588                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 589                 return r.expand();
 590         }
 591
 592         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 593                 !ex_to<numeric>(expanded_exponent).is_integer()) {
 594                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 595                         return this->hold();
 596                 } else {
 597                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 598                 }
 599         }
 600
 601         // integer numeric exponent
 602         const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
 603         int int_exponent = num_exponent.to_int();
 604
 605         // (x+y)^n, n>0
 606         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
 607                 return expand_add(ex_to<add>(expanded_basis), int_exponent);
 608
 609         // (x*y)^n -> x^n * y^n
 610         if (is_ex_exactly_of_type(expanded_basis,mul))
 611                 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
 612
 613         // cannot expand further
 614         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 615                 return this->hold();
 616         else
 617                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
 618 }
 619
 620 //////////
 621 // new virtual functions which can be overridden by derived classes
 622 //////////
 623
 624 // none
 625
 626 //////////
 627 // non-virtual functions in this class
 628 //////////
 629
 630 /** expand a^n where a is an add and n is an integer.
 631  *  @see power::expand */
 632 ex power::expand_add(const add & a, int n) const
 633 {
 634         if (n==2)
 635                 return expand_add_2(a);
 636
 637         int m = a.nops();
 638         exvector sum;
 639         sum.reserve((n+1)*(m-1));
 640         intvector k(m-1);
 641         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 642         intvector upper_limit(m-1);
 643         int l;
 644
 645         for (int l=0; l<m-1; l++) {
 646                 k[l] = 0;
 647                 k_cum[l] = 0;
 648                 upper_limit[l] = n;
 649         }
 650
 651         while (true) {
 652                 exvector term;
 653                 term.reserve(m+1);
 654                 for (l=0; l<m-1; l++) {
 655                         const ex & b = a.op(l);
 656                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 657                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 658                                      !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
 659                                      !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 660                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
 661                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
 662                                      !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
 663                         if (is_ex_exactly_of_type(b,mul))
 664                                 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
 665                         else
 666                                 term.push_back(power(b,k[l]));
 667                 }
 668
 669                 const ex & b = a.op(l);
 670                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 671                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 672                              !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
 673                              !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
 674                              !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
 675                              !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
 676                              !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
 677                 if (is_ex_exactly_of_type(b,mul))
 678                         term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
 679                 else
 680                         term.push_back(power(b,n-k_cum[m-2]));
 681
 682                 numeric f = binomial(numeric(n),numeric(k[0]));
 683                 for (l=1; l<m-1; l++)
 684                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 685
 686                 term.push_back(f);
 687
 688                 // TODO: Can we optimize this?  Alex seemed to think so...
 689                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 690
 691                 // increment k[]
 692                 l = m-2;
 693                 while ((l>=0) && ((++k[l])>upper_limit[l])) {
 694                         k[l] = 0;
 695                         --l;
 696                 }
 697                 if (l<0) break;
 698
 699                 // recalc k_cum[] and upper_limit[]
 700                 if (l==0)
 701                         k_cum[0] = k[0];
 702                 else
 703                         k_cum[l] = k_cum[l-1]+k[l];
 704
 705                 for (int i=l+1; i<m-1; i++)
 706                         k_cum[i] = k_cum[i-1]+k[i];
 707
 708                 for (int i=l+1; i<m-1; i++)
 709                         upper_limit[i] = n-k_cum[i-1];
 710         }
 711         return (new add(sum))->setflag(status_flags::dynallocated |
 712                                        status_flags::expanded );
 713 }
 714
 715
 716 /** Special case of power::expand_add. Expands a^2 where a is an add.
 717  *  @see power::expand_add */
 718 ex power::expand_add_2(const add & a) const
 719 {
 720         epvector sum;
 721         unsigned a_nops = a.nops();
 722         sum.reserve((a_nops*(a_nops+1))/2);
 723         epvector::const_iterator last = a.seq.end();
 724
 725         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 726         // first part: ignore overall_coeff and expand other terms
 727         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 728                 const ex & r = cit0->rest;
 729                 const ex & c = cit0->coeff;
 730
 731                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
 732                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
 733                              !is_ex_exactly_of_type(ex_to<power>(r).exponent,numeric) ||
 734                              !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
 735                              !is_ex_exactly_of_type(ex_to<power>(r).basis,add) ||
 736                              !is_ex_exactly_of_type(ex_to<power>(r).basis,mul) ||
 737                              !is_ex_exactly_of_type(ex_to<power>(r).basis,power));
 738
 739                 if (are_ex_trivially_equal(c,_ex1())) {
 740                         if (is_ex_exactly_of_type(r,mul)) {
 741                                 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
 742                                                      _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_zero()) {
 769                 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
 770                 while (i != end) {
 771                         sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2())));
 772                         ++i;
 773                 }
 774                 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2()),_ex1()));
 775         }
 776
 777         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 778
 779         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 780 }
 781
 782 /** Expand factors of m in m^n where m is a mul and n is and integer
 783  *  @see power::expand */
 784 ex power::expand_mul(const mul & m, const numeric & n) const
 785 {
 786         if (n.is_zero())
 787                 return _ex1();
 788
 789         epvector distrseq;
 790         distrseq.reserve(m.seq.size());
 791         epvector::const_iterator last = m.seq.end();
 792         epvector::const_iterator cit = m.seq.begin();
 793         while (cit!=last) {
 794                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 795                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 796                 } else {
 797                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 798                         // since n is an integer
 799                         distrseq.push_back(expair((*cit).rest, 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)))->setflag(status_flags::dynallocated);
 804 }
 805
 806 // helper function
 807
 808 ex sqrt(const ex & a)
 809 {
 810         return power(a,_ex1_2());
 811 }
 812
 813 } // namespace GiNaC