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 "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         GINAC_ASSERT(basis.return_type()==return_types::commutative);
  72 }
  73
  74 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  75 {
  76         debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
  77         GINAC_ASSERT(basis.return_type()==return_types::commutative);
  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                         c.s << "sqrt(";
 181                         basis.print(c);
 182                         c.s << ")";
 183                 } else {
 184                         if (precedence <= level)
 185                                 c.s << "(";
 186                         basis.print(c, precedence);
 187                         c.s << "^";
 188                         exponent.print(c, precedence);
 189                         if (precedence <= level)
 190                                 c.s << ")";
 191                 }
 192         }
 193 }
 194
 195 bool power::info(unsigned inf) const
 196 {
 197         switch (inf) {
 198                 case info_flags::polynomial:
 199                 case info_flags::integer_polynomial:
 200                 case info_flags::cinteger_polynomial:
 201                 case info_flags::rational_polynomial:
 202                 case info_flags::crational_polynomial:
 203                         return exponent.info(info_flags::nonnegint);
 204                 case info_flags::rational_function:
 205                         return exponent.info(info_flags::integer);
 206                 case info_flags::algebraic:
 207                         return (!exponent.info(info_flags::integer) ||
 208                                         basis.info(inf));
 209         }
 210         return inherited::info(inf);
 211 }
 212
 213 unsigned power::nops() const
 214 {
 215         return 2;
 216 }
 217
 218 ex & power::let_op(int i)
 219 {
 220         GINAC_ASSERT(i>=0);
 221         GINAC_ASSERT(i<2);
 222
 223         return i==0 ? basis : exponent;
 224 }
 225
 226 int power::degree(const ex & s) const
 227 {
 228         if (is_exactly_of_type(*exponent.bp,numeric)) {
 229                 if (basis.is_equal(s)) {
 230                         if (ex_to_numeric(exponent).is_integer())
 231                                 return ex_to_numeric(exponent).to_int();
 232                         else
 233                                 return 0;
 234                 } else
 235                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
 236         }
 237         return 0;
 238 }
 239
 240 int power::ldegree(const ex & s) const
 241 {
 242         if (is_exactly_of_type(*exponent.bp,numeric)) {
 243                 if (basis.is_equal(s)) {
 244                         if (ex_to_numeric(exponent).is_integer())
 245                                 return ex_to_numeric(exponent).to_int();
 246                         else
 247                                 return 0;
 248                 } else
 249                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
 250         }
 251         return 0;
 252 }
 253
 254 ex power::coeff(const ex & s, int n) const
 255 {
 256         if (!basis.is_equal(s)) {
 257                 // basis not equal to s
 258                 if (n == 0)
 259                         return *this;
 260                 else
 261                         return _ex0();
 262         } else {
 263                 // basis equal to s
 264                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
 265                         // integer exponent
 266                         int int_exp = ex_to_numeric(exponent).to_int();
 267                         if (n == int_exp)
 268                                 return _ex1();
 269                         else
 270                                 return _ex0();
 271                 } else {
 272                         // non-integer exponents are treated as zero
 273                         if (n == 0)
 274                                 return *this;
 275                         else
 276                                 return _ex0();
 277                 }
 278         }
 279 }
 280
 281 ex power::eval(int level) const
 282 {
 283         // simplifications: ^(x,0) -> 1 (0^0 handled here)
 284         //                  ^(x,1) -> x
 285         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
 286         //                  ^(1,x) -> 1
 287         //                  ^(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)
 288         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 289         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 290         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 291         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 292
 293         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 294
 295         if ((level==1) && (flags & status_flags::evaluated))
 296                 return *this;
 297         else if (level == -max_recursion_level)
 298                 throw(std::runtime_error("max recursion level reached"));
 299
 300         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 301         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 302
 303         bool basis_is_numerical = 0;
 304         bool exponent_is_numerical = 0;
 305         numeric * num_basis;
 306         numeric * num_exponent;
 307
 308         if (is_exactly_of_type(*ebasis.bp,numeric)) {
 309                 basis_is_numerical = 1;
 310                 num_basis = static_cast<numeric *>(ebasis.bp);
 311         }
 312         if (is_exactly_of_type(*eexponent.bp,numeric)) {
 313                 exponent_is_numerical = 1;
 314                 num_exponent = static_cast<numeric *>(eexponent.bp);
 315         }
 316
 317         // ^(x,0) -> 1 (0^0 also handled here)
 318         if (eexponent.is_zero()) {
 319                 if (ebasis.is_zero())
 320                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 321                 else
 322                         return _ex1();
 323         }
 324
 325         // ^(x,1) -> x
 326         if (eexponent.is_equal(_ex1()))
 327                 return ebasis;
 328
 329         // ^(0,c1) -> 0 or exception (depending on real value of c1)
 330         if (ebasis.is_zero() && exponent_is_numerical) {
 331                 if ((num_exponent->real()).is_zero())
 332                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 333                 else if ((num_exponent->real()).is_negative())
 334                         throw (pole_error("power::eval(): division by zero",1));
 335                 else
 336                         return _ex0();
 337         }
 338
 339         // ^(1,x) -> 1
 340         if (ebasis.is_equal(_ex1()))
 341                 return _ex1();
 342
 343         if (basis_is_numerical && exponent_is_numerical) {
 344                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
 345                 // except if c1,c2 are rational, but c1^c2 is not)
 346                 bool basis_is_crational = num_basis->is_crational();
 347                 bool exponent_is_crational = num_exponent->is_crational();
 348                 numeric res = num_basis->power(*num_exponent);
 349
 350                 if ((!basis_is_crational || !exponent_is_crational)
 351                         || res.is_crational()) {
 352                         return res;
 353                 }
 354                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 355                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 356                 if (basis_is_crational && exponent_is_crational
 357                         && num_exponent->is_real()
 358                         && !num_exponent->is_integer()) {
 359                         numeric n = num_exponent->numer();
 360                         numeric m = num_exponent->denom();
 361                         numeric r;
 362                         numeric q = iquo(n, m, r);
 363                         if (r.is_negative()) {
 364                                 r = r.add(m);
 365                                 q = q.sub(_num1());
 366                         }
 367                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 368                                 return this->hold();
 369                         else {
 370                                 epvector res;
 371                                 res.push_back(expair(ebasis,r.div(m)));
 372                                 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
 373                         }
 374                 }
 375         }
 376
 377         // ^(^(x,c1),c2) -> ^(x,c1*c2)
 378         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 379         // case c1==1 should not happen, see below!)
 380         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
 381                 const power & sub_power = ex_to_power(ebasis);
 382                 const ex & sub_basis = sub_power.basis;
 383                 const ex & sub_exponent = sub_power.exponent;
 384                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 385                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
 386                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
 387                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1)
 388                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 389                 }
 390         }
 391
 392         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 393         if (exponent_is_numerical && num_exponent->is_integer() &&
 394                 is_ex_exactly_of_type(ebasis,mul)) {
 395                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
 396         }
 397
 398         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 399         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 400         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
 401                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 402                 const mul & mulref = ex_to_mul(ebasis);
 403                 if (!mulref.overall_coeff.is_equal(_ex1())) {
 404                         const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
 405                         if (num_coeff.is_real()) {
 406                                 if (num_coeff.is_positive()) {
 407                                         mul * mulp = new mul(mulref);
 408                                         mulp->overall_coeff = _ex1();
 409                                         mulp->clearflag(status_flags::evaluated);
 410                                         mulp->clearflag(status_flags::hash_calculated);
 411                                         return (new mul(power(*mulp,exponent),
 412                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 413                                 } else {
 414                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
 415                                         if (num_coeff.compare(_num_1())!=0) {
 416                                                 mul * mulp = new mul(mulref);
 417                                                 mulp->overall_coeff = _ex_1();
 418                                                 mulp->clearflag(status_flags::evaluated);
 419                                                 mulp->clearflag(status_flags::hash_calculated);
 420                                                 return (new mul(power(*mulp,exponent),
 421                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 422                                         }
 423                                 }
 424                         }
 425                 }
 426         }
 427
 428         if (are_ex_trivially_equal(ebasis,basis) &&
 429                 are_ex_trivially_equal(eexponent,exponent)) {
 430                 return this->hold();
 431         }
 432         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 433                                                                                                    status_flags::evaluated);
 434 }
 435
 436 ex power::evalf(int level) const
 437 {
 438         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 439
 440         ex ebasis;
 441         ex eexponent;
 442
 443         if (level==1) {
 444                 ebasis = basis;
 445                 eexponent = exponent;
 446         } else if (level == -max_recursion_level) {
 447                 throw(std::runtime_error("max recursion level reached"));
 448         } else {
 449                 ebasis = basis.evalf(level-1);
 450                 if (!is_ex_exactly_of_type(eexponent,numeric))
 451                         eexponent = exponent.evalf(level-1);
 452                 else
 453                         eexponent = exponent;
 454         }
 455
 456         return power(ebasis,eexponent);
 457 }
 458
 459 ex power::subs(const lst & ls, const lst & lr) const
 460 {
 461         const ex & subsed_basis=basis.subs(ls,lr);
 462         const ex & subsed_exponent=exponent.subs(ls,lr);
 463
 464         if (are_ex_trivially_equal(basis,subsed_basis)&&
 465                 are_ex_trivially_equal(exponent,subsed_exponent)) {
 466                 return inherited::subs(ls, lr);
 467         }
 468
 469         return power(subsed_basis, subsed_exponent);
 470 }
 471
 472 ex power::simplify_ncmul(const exvector & v) const
 473 {
 474         return inherited::simplify_ncmul(v);
 475 }
 476
 477 // protected
 478
 479 /** Implementation of ex::diff() for a power.
 480  *  @see ex::diff */
 481 ex power::derivative(const symbol & s) const
 482 {
 483         if (exponent.info(info_flags::real)) {
 484                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 485                 epvector newseq;
 486                 newseq.reserve(2);
 487                 newseq.push_back(expair(basis, exponent - _ex1()));
 488                 newseq.push_back(expair(basis.diff(s), _ex1()));
 489                 return mul(newseq, exponent);
 490         } else {
 491                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 492                 return mul(*this,
 493                            add(mul(exponent.diff(s), log(basis)),
 494                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
 495         }
 496 }
 497
 498 int power::compare_same_type(const basic & other) const
 499 {
 500         GINAC_ASSERT(is_exactly_of_type(other, power));
 501         const power & o=static_cast<const power &>(const_cast<basic &>(other));
 502
 503         int cmpval;
 504         cmpval=basis.compare(o.basis);
 505         if (cmpval==0) {
 506                 return exponent.compare(o.exponent);
 507         }
 508         return cmpval;
 509 }
 510
 511 unsigned power::return_type(void) const
 512 {
 513         return basis.return_type();
 514 }
 515
 516 unsigned power::return_type_tinfo(void) const
 517 {
 518         return basis.return_type_tinfo();
 519 }
 520
 521 ex power::expand(unsigned options) const
 522 {
 523         if (flags & status_flags::expanded)
 524                 return *this;
 525
 526         ex expanded_basis = basis.expand(options);
 527         ex expanded_exponent = exponent.expand(options);
 528
 529         // x^(a+b) -> x^a * x^b
 530         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 531                 const add &a = ex_to_add(expanded_exponent);
 532                 exvector distrseq;
 533                 distrseq.reserve(a.seq.size() + 1);
 534                 epvector::const_iterator last = a.seq.end();
 535                 epvector::const_iterator cit = a.seq.begin();
 536                 while (cit!=last) {
 537                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 538                         cit++;
 539                 }
 540
 541                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 542                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
 543                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
 544                         int int_exponent = num_exponent.to_int();
 545                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 546                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
 547                         else
 548                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 549                 } else
 550                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 551
 552                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 553                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 554                 return r.expand();
 555         }
 556
 557         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 558                 !ex_to_numeric(expanded_exponent).is_integer()) {
 559                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 560                         return this->hold();
 561                 } else {
 562                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 563                 }
 564         }
 565
 566         // integer numeric exponent
 567         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
 568         int int_exponent = num_exponent.to_int();
 569
 570         // (x+y)^n, n>0
 571         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
 572                 return expand_add(ex_to_add(expanded_basis), int_exponent);
 573
 574         // (x*y)^n -> x^n * y^n
 575         if (is_ex_exactly_of_type(expanded_basis,mul))
 576                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
 577
 578         // cannot expand further
 579         if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
 580                 return this->hold();
 581         else
 582                 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 583 }
 584
 585 //////////
 586 // new virtual functions which can be overridden by derived classes
 587 //////////
 588
 589 // none
 590
 591 //////////
 592 // non-virtual functions in this class
 593 //////////
 594
 595 /** expand a^n where a is an add and n is an integer.
 596  *  @see power::expand */
 597 ex power::expand_add(const add & a, int n) const
 598 {
 599         if (n==2)
 600                 return expand_add_2(a);
 601
 602         int m = a.nops();
 603         exvector sum;
 604         sum.reserve((n+1)*(m-1));
 605         intvector k(m-1);
 606         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 607         intvector upper_limit(m-1);
 608         int l;
 609
 610         for (int l=0; l<m-1; l++) {
 611                 k[l] = 0;
 612                 k_cum[l] = 0;
 613                 upper_limit[l] = n;
 614         }
 615
 616         while (1) {
 617                 exvector term;
 618                 term.reserve(m+1);
 619                 for (l=0; l<m-1; l++) {
 620                         const ex & b = a.op(l);
 621                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 622                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 623                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 624                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 625                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 626                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 627                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 628                         if (is_ex_exactly_of_type(b,mul))
 629                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
 630                         else
 631                                 term.push_back(power(b,k[l]));
 632                 }
 633
 634                 const ex & b = a.op(l);
 635                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 636                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 637                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 638                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 639                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 640                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 641                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 642                 if (is_ex_exactly_of_type(b,mul))
 643                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
 644                 else
 645                         term.push_back(power(b,n-k_cum[m-2]));
 646
 647                 numeric f = binomial(numeric(n),numeric(k[0]));
 648                 for (l=1; l<m-1; l++)
 649                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 650
 651                 term.push_back(f);
 652
 653                 /*
 654                 cout << "begin term" << endl;
 655                 for (int i=0; i<m-1; i++) {
 656                         cout << "k[" << i << "]=" << k[i] << endl;
 657                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
 658                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
 659                 }
 660                 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
 661                         cout << *cit << endl;
 662                 }
 663                 cout << "end term" << endl;
 664                 */
 665
 666                 // TODO: optimize this
 667                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 668
 669                 // increment k[]
 670                 l = m-2;
 671                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
 672                         k[l] = 0;
 673                         l--;
 674                 }
 675                 if (l<0) break;
 676
 677                 // recalc k_cum[] and upper_limit[]
 678                 if (l==0)
 679                         k_cum[0] = k[0];
 680                 else
 681                         k_cum[l] = k_cum[l-1]+k[l];
 682
 683                 for (int i=l+1; i<m-1; i++)
 684                         k_cum[i] = k_cum[i-1]+k[i];
 685
 686                 for (int i=l+1; i<m-1; i++)
 687                         upper_limit[i] = n-k_cum[i-1];
 688         }
 689         return (new add(sum))->setflag(status_flags::dynallocated |
 690                                                                    status_flags::expanded );
 691 }
 692
 693
 694 /** Special case of power::expand_add. Expands a^2 where a is an add.
 695  *  @see power::expand_add */
 696 ex power::expand_add_2(const add & a) const
 697 {
 698         epvector sum;
 699         unsigned a_nops = a.nops();
 700         sum.reserve((a_nops*(a_nops+1))/2);
 701         epvector::const_iterator last = a.seq.end();
 702
 703         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 704         // first part: ignore overall_coeff and expand other terms
 705         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 706                 const ex & r = (*cit0).rest;
 707                 const ex & c = (*cit0).coeff;
 708
 709                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
 710                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
 711                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
 712                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
 713                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
 714                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
 715                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
 716
 717                 if (are_ex_trivially_equal(c,_ex1())) {
 718                         if (is_ex_exactly_of_type(r,mul)) {
 719                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 720                                                      _ex1()));
 721                         } else {
 722                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 723                                                      _ex1()));
 724                         }
 725                 } else {
 726                         if (is_ex_exactly_of_type(r,mul)) {
 727                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 728                                                      ex_to_numeric(c).power_dyn(_num2())));
 729                         } else {
 730                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 731                                                      ex_to_numeric(c).power_dyn(_num2())));
 732                         }
 733                 }
 734
 735                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 736                         const ex & r1 = (*cit1).rest;
 737                         const ex & c1 = (*cit1).coeff;
 738                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 739                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
 740                 }
 741         }
 742
 743         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 744
 745         // second part: add terms coming from overall_factor (if != 0)
 746         if (!a.overall_coeff.is_zero()) {
 747                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
 748                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
 749                 }
 750                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
 751         }
 752
 753         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 754
 755         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 756 }
 757
 758 /** Expand factors of m in m^n where m is a mul and n is and integer
 759  *  @see power::expand */
 760 ex power::expand_mul(const mul & m, const numeric & n) const
 761 {
 762         if (n.is_zero())
 763                 return _ex1();
 764
 765         epvector distrseq;
 766         distrseq.reserve(m.seq.size());
 767         epvector::const_iterator last = m.seq.end();
 768         epvector::const_iterator cit = m.seq.begin();
 769         while (cit!=last) {
 770                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 771                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 772                 } else {
 773                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 774                         // since n is an integer
 775                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
 776                 }
 777                 ++cit;
 778         }
 779         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 780 }
 781
 782 /*
 783 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
 784                                unsigned options) const
 785 {
 786         // obsolete
 787
 788         exvector distrseq;
 789         epvector splitseq;
 790
 791         const add & addref=static_cast<const add &>(*basis.bp);
 792
 793         splitseq=addref.seq;
 794         splitseq.pop_back();
 795         ex first_operands=add(splitseq);
 796         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
 797
 798         int n=exponent.to_int();
 799         for (int k=0; k<=n; k++) {
 800                 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
 801                                                  * power(last_operand,numeric(n-k)));
 802         }
 803         return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
 804 }
 805 */
 806
 807 /*
 808 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
 809                                                                 unsigned options) const
 810 {
 811         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
 812                         expand(options | expand_options::internal_do_not_expand_power_operands);
 813
 814         return ex(mul(rest_power,basis),0).
 815                expand(options | expand_options::internal_do_not_expand_mul_operands);
 816 }
 817 */
 818
 819 //////////
 820 // static member variables
 821 //////////
 822
 823 // protected
 824
 825 unsigned power::precedence = 60;
 826
 827 // helper function
 828
 829 ex sqrt(const ex & a)
 830 {
 831         return power(a,_ex1_2());
 832 }
 833
 834 } // namespace GiNaC