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 "inifcns.h"
  34 #include "symbol.h"
  35 #include "print.h"
  36 #include "archive.h"
  37 #include "debugmsg.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
  43
  44 typedef std::vector<int> intvector;
  45
  46 //////////
  47 // default ctor, dtor, copy ctor assignment operator and helpers
  48 //////////
  49
  50 power::power() : basic(TINFO_power)
  51 {
  52         debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
  53 }
  54
  55 void power::copy(const power & other)
  56 {
  57         inherited::copy(other);
  58         basis = other.basis;
  59         exponent = other.exponent;
  60 }
  61
  62 DEFAULT_DESTROY(power)
  63
  64 //////////
  65 // other ctors
  66 //////////
  67
  68 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  69 {
  70         debugmsg("power ctor from ex,ex",LOGLEVEL_CONSTRUCT);
  71 }
  72
  73 /** Ctor from an ex and a bare numeric.  This is somewhat more efficient than
  74  *  the normal ctor from two ex whenever it can be used. */
  75 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  76 {
  77         debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
  78 }
  79
  80 //////////
  81 // archiving
  82 //////////
  83
  84 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  85 {
  86         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
  87         n.find_ex("basis", basis, sym_lst);
  88         n.find_ex("exponent", exponent, sym_lst);
  89 }
  90
  91 void power::archive(archive_node &n) const
  92 {
  93         inherited::archive(n);
  94         n.add_ex("basis", basis);
  95         n.add_ex("exponent", exponent);
  96 }
  97
  98 DEFAULT_UNARCHIVE(power)
  99
 100 //////////
 101 // functions overriding virtual functions from bases classes
 102 //////////
 103
 104 // public
 105
 106 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 107 {
 108         // Optimal output of integer powers of symbols to aid compiler CSE.
 109         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 110         // to learn why such a hack is really necessary.
 111         if (exp == 1) {
 112                 x.print(c);
 113         } else if (exp == 2) {
 114                 x.print(c);
 115                 c.s << "*";
 116                 x.print(c);
 117         } else if (exp & 1) {
 118                 x.print(c);
 119                 c.s << "*";
 120                 print_sym_pow(c, x, exp-1);
 121         } else {
 122                 c.s << "(";
 123                 print_sym_pow(c, x, exp >> 1);
 124                 c.s << ")*(";
 125                 print_sym_pow(c, x, exp >> 1);
 126                 c.s << ")";
 127         }
 128 }
 129
 130 void power::print(const print_context & c, unsigned level) const
 131 {
 132         debugmsg("power print", LOGLEVEL_PRINT);
 133
 134         if (is_of_type(c, print_tree)) {
 135
 136                 inherited::print(c, level);
 137
 138         } else if (is_of_type(c, print_csrc)) {
 139
 140                 // Integer powers of symbols are printed in a special, optimized way
 141                 if (exponent.info(info_flags::integer)
 142                  && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
 143                         int exp = ex_to_numeric(exponent).to_int();
 144                         if (exp > 0)
 145                                 c.s << '(';
 146                         else {
 147                                 exp = -exp;
 148                                 if (is_of_type(c, print_csrc_cl_N))
 149                                         c.s << "recip(";
 150                                 else
 151                                         c.s << "1.0/(";
 152                         }
 153                         print_sym_pow(c, ex_to_symbol(basis), exp);
 154                         c.s << ')';
 155
 156                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 157                 } else if (exponent.compare(_num_1()) == 0) {
 158                         if (is_of_type(c, print_csrc_cl_N))
 159                                 c.s << "recip(";
 160                         else
 161                                 c.s << "1.0/(";
 162                         basis.print(c);
 163                         c.s << ')';
 164
 165                 // Otherwise, use the pow() or expt() (CLN) functions
 166                 } else {
 167                         if (is_of_type(c, print_csrc_cl_N))
 168                                 c.s << "expt(";
 169                         else
 170                                 c.s << "pow(";
 171                         basis.print(c);
 172                         c.s << ',';
 173                         exponent.print(c);
 174                         c.s << ')';
 175                 }
 176
 177         } else {
 178
 179                 if (exponent.is_equal(_ex1_2())) {
 180                         if (is_of_type(c, print_latex))
 181                                 c.s << "\\sqrt{";
 182                         else
 183                                 c.s << "sqrt(";
 184                         basis.print(c);
 185                         if (is_of_type(c, print_latex))
 186                                 c.s << '}';
 187                         else
 188                                 c.s << ')';
 189                 } else {
 190                         if (precedence() <= level) {
 191                                 if (is_of_type(c, print_latex))
 192                                         c.s << "{(";
 193                                 else
 194                                         c.s << "(";
 195                         }
 196                         basis.print(c, precedence());
 197                         c.s << '^';
 198                         if (is_of_type(c, print_latex))
 199                                 c.s << '{';
 200                         exponent.print(c, precedence());
 201                         if (is_of_type(c, print_latex))
 202                                 c.s << '}';
 203                         if (precedence() <= level) {
 204                                 if (is_of_type(c, print_latex))
 205                                         c.s << ")}";
 206                                 else
 207                                         c.s << ')';
 208                         }
 209                 }
 210         }
 211 }
 212
 213 bool power::info(unsigned inf) const
 214 {
 215         switch (inf) {
 216                 case info_flags::polynomial:
 217                 case info_flags::integer_polynomial:
 218                 case info_flags::cinteger_polynomial:
 219                 case info_flags::rational_polynomial:
 220                 case info_flags::crational_polynomial:
 221                         return exponent.info(info_flags::nonnegint);
 222                 case info_flags::rational_function:
 223                         return exponent.info(info_flags::integer);
 224                 case info_flags::algebraic:
 225                         return (!exponent.info(info_flags::integer) ||
 226                                         basis.info(inf));
 227         }
 228         return inherited::info(inf);
 229 }
 230
 231 unsigned power::nops() const
 232 {
 233         return 2;
 234 }
 235
 236 ex & power::let_op(int i)
 237 {
 238         GINAC_ASSERT(i>=0);
 239         GINAC_ASSERT(i<2);
 240
 241         return i==0 ? basis : exponent;
 242 }
 243
 244 ex power::map(map_func f) const
 245 {
 246         return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
 247 }
 248
 249 int power::degree(const ex & s) const
 250 {
 251         if (is_exactly_of_type(*exponent.bp,numeric)) {
 252                 if (basis.is_equal(s)) {
 253                         if (ex_to_numeric(exponent).is_integer())
 254                                 return ex_to_numeric(exponent).to_int();
 255                         else
 256                                 return 0;
 257                 } else
 258                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
 259         }
 260         return 0;
 261 }
 262
 263 int power::ldegree(const ex & s) const
 264 {
 265         if (is_exactly_of_type(*exponent.bp,numeric)) {
 266                 if (basis.is_equal(s)) {
 267                         if (ex_to_numeric(exponent).is_integer())
 268                                 return ex_to_numeric(exponent).to_int();
 269                         else
 270                                 return 0;
 271                 } else
 272                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
 273         }
 274         return 0;
 275 }
 276
 277 ex power::coeff(const ex & s, int n) const
 278 {
 279         if (!basis.is_equal(s)) {
 280                 // basis not equal to s
 281                 if (n == 0)
 282                         return *this;
 283                 else
 284                         return _ex0();
 285         } else {
 286                 // basis equal to s
 287                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
 288                         // integer exponent
 289                         int int_exp = ex_to_numeric(exponent).to_int();
 290                         if (n == int_exp)
 291                                 return _ex1();
 292                         else
 293                                 return _ex0();
 294                 } else {
 295                         // non-integer exponents are treated as zero
 296                         if (n == 0)
 297                                 return *this;
 298                         else
 299                                 return _ex0();
 300                 }
 301         }
 302 }
 303
 304 ex power::eval(int level) const
 305 {
 306         // simplifications: ^(x,0) -> 1 (0^0 handled here)
 307         //                  ^(x,1) -> x
 308         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
 309         //                  ^(1,x) -> 1
 310         //                  ^(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)
 311         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 312         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 313         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 314         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 315
 316         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 317
 318         if ((level==1) && (flags & status_flags::evaluated))
 319                 return *this;
 320         else if (level == -max_recursion_level)
 321                 throw(std::runtime_error("max recursion level reached"));
 322
 323         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 324         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 325
 326         bool basis_is_numerical = false;
 327         bool exponent_is_numerical = false;
 328         numeric * num_basis;
 329         numeric * num_exponent;
 330
 331         if (is_exactly_of_type(*ebasis.bp,numeric)) {
 332                 basis_is_numerical = true;
 333                 num_basis = static_cast<numeric *>(ebasis.bp);
 334         }
 335         if (is_exactly_of_type(*eexponent.bp,numeric)) {
 336                 exponent_is_numerical = true;
 337                 num_exponent = static_cast<numeric *>(eexponent.bp);
 338         }
 339
 340         // ^(x,0) -> 1 (0^0 also handled here)
 341         if (eexponent.is_zero()) {
 342                 if (ebasis.is_zero())
 343                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 344                 else
 345                         return _ex1();
 346         }
 347
 348         // ^(x,1) -> x
 349         if (eexponent.is_equal(_ex1()))
 350                 return ebasis;
 351
 352         // ^(0,c1) -> 0 or exception (depending on real value of c1)
 353         if (ebasis.is_zero() && exponent_is_numerical) {
 354                 if ((num_exponent->real()).is_zero())
 355                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 356                 else if ((num_exponent->real()).is_negative())
 357                         throw (pole_error("power::eval(): division by zero",1));
 358                 else
 359                         return _ex0();
 360         }
 361
 362         // ^(1,x) -> 1
 363         if (ebasis.is_equal(_ex1()))
 364                 return _ex1();
 365
 366         if (exponent_is_numerical) {
 367
 368                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
 369                 // except if c1,c2 are rational, but c1^c2 is not)
 370                 if (basis_is_numerical) {
 371                         bool basis_is_crational = num_basis->is_crational();
 372                         bool exponent_is_crational = num_exponent->is_crational();
 373                         numeric res = num_basis->power(*num_exponent);
 374
 375                         if ((!basis_is_crational || !exponent_is_crational)
 376                                 || res.is_crational()) {
 377                                 return res;
 378                         }
 379                         GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 380
 381                         // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 382                         if (basis_is_crational && exponent_is_crational
 383                                 && num_exponent->is_real()
 384                                 && !num_exponent->is_integer()) {
 385                                 numeric n = num_exponent->numer();
 386                                 numeric m = num_exponent->denom();
 387                                 numeric r;
 388                                 numeric q = iquo(n, m, r);
 389                                 if (r.is_negative()) {
 390                                         r = r.add(m);
 391                                         q = q.sub(_num1());
 392                                 }
 393                                 if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 394                                         return this->hold();
 395                                 else {
 396                                         epvector res;
 397                                         res.push_back(expair(ebasis,r.div(m)));
 398                                         return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
 399                                 }
 400                         }
 401                 }
 402
 403                 // ^(^(x,c1),c2) -> ^(x,c1*c2)
 404                 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 405                 // case c1==1 should not happen, see below!)
 406                 if (is_ex_exactly_of_type(ebasis,power)) {
 407                         const power & sub_power = ex_to_power(ebasis);
 408                         const ex & sub_basis = sub_power.basis;
 409                         const ex & sub_exponent = sub_power.exponent;
 410                         if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 411                                 const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
 412                                 GINAC_ASSERT(num_sub_exponent!=numeric(1));
 413                                 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1()).is_negative())
 414                                         return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 415                         }
 416                 }
 417
 418                 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 419                 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
 420                         return expand_mul(ex_to_mul(ebasis), *num_exponent);
 421                 }
 422
 423                 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 424                 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 425                 if (is_ex_exactly_of_type(ebasis,mul)) {
 426                         GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 427                         const mul & mulref = ex_to_mul(ebasis);
 428                         if (!mulref.overall_coeff.is_equal(_ex1())) {
 429                                 const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
 430                                 if (num_coeff.is_real()) {
 431                                         if (num_coeff.is_positive()) {
 432                                                 mul * mulp = new mul(mulref);
 433                                                 mulp->overall_coeff = _ex1();
 434                                                 mulp->clearflag(status_flags::evaluated);
 435                                                 mulp->clearflag(status_flags::hash_calculated);
 436                                                 return (new mul(power(*mulp,exponent),
 437                                                                 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 438                                         } else {
 439                                                 GINAC_ASSERT(num_coeff.compare(_num0())<0);
 440                                                 if (num_coeff.compare(_num_1())!=0) {
 441                                                         mul * mulp = new mul(mulref);
 442                                                         mulp->overall_coeff = _ex_1();
 443                                                         mulp->clearflag(status_flags::evaluated);
 444                                                         mulp->clearflag(status_flags::hash_calculated);
 445                                                         return (new mul(power(*mulp,exponent),
 446                                                                         power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 447                                                 }
 448                                         }
 449                                 }
 450                         }
 451                 }
 452
 453                 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer)
 454                 if (num_exponent->is_pos_integer() && ebasis.return_type() != return_types::commutative) {
 455                         return ncmul(exvector(num_exponent->to_int(), ebasis), true);
 456                 }
 457         }
 458
 459         if (are_ex_trivially_equal(ebasis,basis) &&
 460                 are_ex_trivially_equal(eexponent,exponent)) {
 461                 return this->hold();
 462         }
 463         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 464                                                                                                    status_flags::evaluated);
 465 }
 466
 467 ex power::evalf(int level) const
 468 {
 469         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 470
 471         ex ebasis;
 472         ex eexponent;
 473
 474         if (level==1) {
 475                 ebasis = basis;
 476                 eexponent = exponent;
 477         } else if (level == -max_recursion_level) {
 478                 throw(std::runtime_error("max recursion level reached"));
 479         } else {
 480                 ebasis = basis.evalf(level-1);
 481                 if (!is_ex_exactly_of_type(eexponent,numeric))
 482                         eexponent = exponent.evalf(level-1);
 483                 else
 484                         eexponent = exponent;
 485         }
 486
 487         return power(ebasis,eexponent);
 488 }
 489
 490 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
 491 {
 492         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
 493         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
 494
 495         if (are_ex_trivially_equal(basis, subsed_basis)
 496          && are_ex_trivially_equal(exponent, subsed_exponent))
 497                 return basic::subs(ls, lr, no_pattern);
 498         else
 499                 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
 500 }
 501
 502 ex power::simplify_ncmul(const exvector & v) const
 503 {
 504         return inherited::simplify_ncmul(v);
 505 }
 506
 507 // protected
 508
 509 /** Implementation of ex::diff() for a power.
 510  *  @see ex::diff */
 511 ex power::derivative(const symbol & s) const
 512 {
 513         if (exponent.info(info_flags::real)) {
 514                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 515                 epvector newseq;
 516                 newseq.reserve(2);
 517                 newseq.push_back(expair(basis, exponent - _ex1()));
 518                 newseq.push_back(expair(basis.diff(s), _ex1()));
 519                 return mul(newseq, exponent);
 520         } else {
 521                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 522                 return mul(*this,
 523                            add(mul(exponent.diff(s), log(basis)),
 524                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
 525         }
 526 }
 527
 528 int power::compare_same_type(const basic & other) const
 529 {
 530         GINAC_ASSERT(is_exactly_of_type(other, power));
 531         const power & o=static_cast<const power &>(const_cast<basic &>(other));
 532
 533         int cmpval;
 534         cmpval=basis.compare(o.basis);
 535         if (cmpval==0) {
 536                 return exponent.compare(o.exponent);
 537         }
 538         return cmpval;
 539 }
 540
 541 unsigned power::return_type(void) const
 542 {
 543         return basis.return_type();
 544 }
 545
 546 unsigned power::return_type_tinfo(void) const
 547 {
 548         return basis.return_type_tinfo();
 549 }
 550
 551 ex power::expand(unsigned options) const
 552 {
 553         if (flags & status_flags::expanded)
 554                 return *this;
 555
 556         ex expanded_basis = basis.expand(options);
 557         ex expanded_exponent = exponent.expand(options);
 558
 559         // x^(a+b) -> x^a * x^b
 560         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 561                 const add &a = ex_to_add(expanded_exponent);
 562                 exvector distrseq;
 563                 distrseq.reserve(a.seq.size() + 1);
 564                 epvector::const_iterator last = a.seq.end();
 565                 epvector::const_iterator cit = a.seq.begin();
 566                 while (cit!=last) {
 567                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 568                         cit++;
 569                 }
 570
 571                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 572                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
 573                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
 574                         int int_exponent = num_exponent.to_int();
 575                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 576                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
 577                         else
 578                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 579                 } else
 580                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 581
 582                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 583                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 584                 return r.expand();
 585         }
 586
 587         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 588                 !ex_to_numeric(expanded_exponent).is_integer()) {
 589                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 590                         return this->hold();
 591                 } else {
 592                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 593                 }
 594         }
 595
 596         // integer numeric exponent
 597         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
 598         int int_exponent = num_exponent.to_int();
 599
 600         // (x+y)^n, n>0
 601         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
 602                 return expand_add(ex_to_add(expanded_basis), int_exponent);
 603
 604         // (x*y)^n -> x^n * y^n
 605         if (is_ex_exactly_of_type(expanded_basis,mul))
 606                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
 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 // new virtual functions which can be overridden by derived classes
 617 //////////
 618
 619 // none
 620
 621 //////////
 622 // non-virtual functions in this class
 623 //////////
 624
 625 /** expand a^n where a is an add and n is an integer.
 626  *  @see power::expand */
 627 ex power::expand_add(const add & a, int n) const
 628 {
 629         if (n==2)
 630                 return expand_add_2(a);
 631
 632         int m = a.nops();
 633         exvector sum;
 634         sum.reserve((n+1)*(m-1));
 635         intvector k(m-1);
 636         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 637         intvector upper_limit(m-1);
 638         int l;
 639
 640         for (int l=0; l<m-1; l++) {
 641                 k[l] = 0;
 642                 k_cum[l] = 0;
 643                 upper_limit[l] = n;
 644         }
 645
 646         while (1) {
 647                 exvector term;
 648                 term.reserve(m+1);
 649                 for (l=0; l<m-1; l++) {
 650                         const ex & b = a.op(l);
 651                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 652                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 653                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 654                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 655                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 656                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 657                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 658                         if (is_ex_exactly_of_type(b,mul))
 659                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
 660                         else
 661                                 term.push_back(power(b,k[l]));
 662                 }
 663
 664                 const ex & b = a.op(l);
 665                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 666                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 667                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 668                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 669                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 670                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 671                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 672                 if (is_ex_exactly_of_type(b,mul))
 673                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
 674                 else
 675                         term.push_back(power(b,n-k_cum[m-2]));
 676
 677                 numeric f = binomial(numeric(n),numeric(k[0]));
 678                 for (l=1; l<m-1; l++)
 679                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 680
 681                 term.push_back(f);
 682
 683                 /*
 684                 cout << "begin term" << endl;
 685                 for (int i=0; i<m-1; i++) {
 686                         cout << "k[" << i << "]=" << k[i] << endl;
 687                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
 688                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
 689                 }
 690                 for_each(term.begin(), term.end(), ostream_iterator<ex>(cout, "\n"));
 691                 cout << "end term" << endl;
 692                 */
 693
 694                 // TODO: optimize this
 695                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 696
 697                 // increment k[]
 698                 l = m-2;
 699                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
 700                         k[l] = 0;
 701                         l--;
 702                 }
 703                 if (l<0) break;
 704
 705                 // recalc k_cum[] and upper_limit[]
 706                 if (l==0)
 707                         k_cum[0] = k[0];
 708                 else
 709                         k_cum[l] = k_cum[l-1]+k[l];
 710
 711                 for (int i=l+1; i<m-1; i++)
 712                         k_cum[i] = k_cum[i-1]+k[i];
 713
 714                 for (int i=l+1; i<m-1; i++)
 715                         upper_limit[i] = n-k_cum[i-1];
 716         }
 717         return (new add(sum))->setflag(status_flags::dynallocated |
 718                                                                    status_flags::expanded );
 719 }
 720
 721
 722 /** Special case of power::expand_add. Expands a^2 where a is an add.
 723  *  @see power::expand_add */
 724 ex power::expand_add_2(const add & a) const
 725 {
 726         epvector sum;
 727         unsigned a_nops = a.nops();
 728         sum.reserve((a_nops*(a_nops+1))/2);
 729         epvector::const_iterator last = a.seq.end();
 730
 731         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 732         // first part: ignore overall_coeff and expand other terms
 733         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 734                 const ex & r = (*cit0).rest;
 735                 const ex & c = (*cit0).coeff;
 736
 737                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
 738                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
 739                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
 740                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
 741                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
 742                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
 743                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
 744
 745                 if (are_ex_trivially_equal(c,_ex1())) {
 746                         if (is_ex_exactly_of_type(r,mul)) {
 747                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 748                                                      _ex1()));
 749                         } else {
 750                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 751                                                      _ex1()));
 752                         }
 753                 } else {
 754                         if (is_ex_exactly_of_type(r,mul)) {
 755                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 756                                                      ex_to_numeric(c).power_dyn(_num2())));
 757                         } else {
 758                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 759                                                      ex_to_numeric(c).power_dyn(_num2())));
 760                         }
 761                 }
 762
 763                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 764                         const ex & r1 = (*cit1).rest;
 765                         const ex & c1 = (*cit1).coeff;
 766                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 767                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
 768                 }
 769         }
 770
 771         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 772
 773         // second part: add terms coming from overall_factor (if != 0)
 774         if (!a.overall_coeff.is_zero()) {
 775                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
 776                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
 777                 }
 778                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
 779         }
 780
 781         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 782
 783         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 784 }
 785
 786 /** Expand factors of m in m^n where m is a mul and n is and integer
 787  *  @see power::expand */
 788 ex power::expand_mul(const mul & m, const numeric & n) const
 789 {
 790         if (n.is_zero())
 791                 return _ex1();
 792
 793         epvector distrseq;
 794         distrseq.reserve(m.seq.size());
 795         epvector::const_iterator last = m.seq.end();
 796         epvector::const_iterator cit = m.seq.begin();
 797         while (cit!=last) {
 798                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 799                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 800                 } else {
 801                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 802                         // since n is an integer
 803                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
 804                 }
 805                 ++cit;
 806         }
 807         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 808 }
 809
 810 /*
 811 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
 812                                                                 unsigned options) const
 813 {
 814         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
 815                         expand(options | expand_options::internal_do_not_expand_power_operands);
 816
 817         return ex(mul(rest_power,basis),0).
 818                expand(options | expand_options::internal_do_not_expand_mul_operands);
 819 }
 820 */
 821
 822 // helper function
 823
 824 ex sqrt(const ex & a)
 825 {
 826         return power(a,_ex1_2());
 827 }
 828
 829 } // namespace GiNaC