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 /** Ctor from an ex and a bare numeric.  This is somewhat more efficient than
  75  *  the normal ctor from two ex whenever it can be used. */
  76 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
  77 {
  78         debugmsg("power ctor from ex,numeric",LOGLEVEL_CONSTRUCT);
  79         GINAC_ASSERT(basis.return_type()==return_types::commutative);
  80 }
  81
  82 //////////
  83 // archiving
  84 //////////
  85
  86 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
  87 {
  88         debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
  89         n.find_ex("basis", basis, sym_lst);
  90         n.find_ex("exponent", exponent, sym_lst);
  91 }
  92
  93 void power::archive(archive_node &n) const
  94 {
  95         inherited::archive(n);
  96         n.add_ex("basis", basis);
  97         n.add_ex("exponent", exponent);
  98 }
  99
 100 DEFAULT_UNARCHIVE(power)
 101
 102 //////////
 103 // functions overriding virtual functions from bases classes
 104 //////////
 105
 106 // public
 107
 108 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
 109 {
 110         // Optimal output of integer powers of symbols to aid compiler CSE.
 111         // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
 112         // to learn why such a hack is really necessary.
 113         if (exp == 1) {
 114                 x.print(c);
 115         } else if (exp == 2) {
 116                 x.print(c);
 117                 c.s << "*";
 118                 x.print(c);
 119         } else if (exp & 1) {
 120                 x.print(c);
 121                 c.s << "*";
 122                 print_sym_pow(c, x, exp-1);
 123         } else {
 124                 c.s << "(";
 125                 print_sym_pow(c, x, exp >> 1);
 126                 c.s << ")*(";
 127                 print_sym_pow(c, x, exp >> 1);
 128                 c.s << ")";
 129         }
 130 }
 131
 132 void power::print(const print_context & c, unsigned level) const
 133 {
 134         debugmsg("power print", LOGLEVEL_PRINT);
 135
 136         if (is_of_type(c, print_tree)) {
 137
 138                 inherited::print(c, level);
 139
 140         } else if (is_of_type(c, print_csrc)) {
 141
 142                 // Integer powers of symbols are printed in a special, optimized way
 143                 if (exponent.info(info_flags::integer)
 144                  && (is_ex_exactly_of_type(basis, symbol) || is_ex_exactly_of_type(basis, constant))) {
 145                         int exp = ex_to_numeric(exponent).to_int();
 146                         if (exp > 0)
 147                                 c.s << "(";
 148                         else {
 149                                 exp = -exp;
 150                                 if (is_of_type(c, print_csrc_cl_N))
 151                                         c.s << "recip(";
 152                                 else
 153                                         c.s << "1.0/(";
 154                         }
 155                         print_sym_pow(c, ex_to_symbol(basis), exp);
 156                         c.s << ")";
 157
 158                 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
 159                 } else if (exponent.compare(_num_1()) == 0) {
 160                         if (is_of_type(c, print_csrc_cl_N))
 161                                 c.s << "recip(";
 162                         else
 163                                 c.s << "1.0/(";
 164                         basis.print(c);
 165                         c.s << ")";
 166
 167                 // Otherwise, use the pow() or expt() (CLN) functions
 168                 } else {
 169                         if (is_of_type(c, print_csrc_cl_N))
 170                                 c.s << "expt(";
 171                         else
 172                                 c.s << "pow(";
 173                         basis.print(c);
 174                         c.s << ",";
 175                         exponent.print(c);
 176                         c.s << ")";
 177                 }
 178
 179         } else {
 180
 181                 if (exponent.is_equal(_ex1_2())) {
 182                         if (is_of_type(c, print_latex))
 183                                 c.s << "\\sqrt{";
 184                         else
 185                                 c.s << "sqrt(";
 186                         basis.print(c);
 187                         if (is_of_type(c, print_latex))
 188                                 c.s << "}";
 189                         else
 190                                 c.s << ")";
 191                 } else {
 192                         if (precedence() <= level) {
 193                                 if (is_of_type(c, print_latex))
 194                                         c.s << "{(";
 195                                 else
 196                                         c.s << "(";
 197                         }
 198                         basis.print(c, precedence());
 199                         c.s << "^";
 200                         exponent.print(c, precedence());
 201                         if (precedence() <= level) {
 202                                 if (is_of_type(c, print_latex))
 203                                         c.s << ")}";
 204                                 else
 205                                         c.s << ")";
 206                         }
 207                 }
 208         }
 209 }
 210
 211 bool power::info(unsigned inf) const
 212 {
 213         switch (inf) {
 214                 case info_flags::polynomial:
 215                 case info_flags::integer_polynomial:
 216                 case info_flags::cinteger_polynomial:
 217                 case info_flags::rational_polynomial:
 218                 case info_flags::crational_polynomial:
 219                         return exponent.info(info_flags::nonnegint);
 220                 case info_flags::rational_function:
 221                         return exponent.info(info_flags::integer);
 222                 case info_flags::algebraic:
 223                         return (!exponent.info(info_flags::integer) ||
 224                                         basis.info(inf));
 225         }
 226         return inherited::info(inf);
 227 }
 228
 229 unsigned power::nops() const
 230 {
 231         return 2;
 232 }
 233
 234 ex & power::let_op(int i)
 235 {
 236         GINAC_ASSERT(i>=0);
 237         GINAC_ASSERT(i<2);
 238
 239         return i==0 ? basis : exponent;
 240 }
 241
 242 int power::degree(const ex & s) const
 243 {
 244         if (is_exactly_of_type(*exponent.bp,numeric)) {
 245                 if (basis.is_equal(s)) {
 246                         if (ex_to_numeric(exponent).is_integer())
 247                                 return ex_to_numeric(exponent).to_int();
 248                         else
 249                                 return 0;
 250                 } else
 251                         return basis.degree(s) * ex_to_numeric(exponent).to_int();
 252         }
 253         return 0;
 254 }
 255
 256 int power::ldegree(const ex & s) const
 257 {
 258         if (is_exactly_of_type(*exponent.bp,numeric)) {
 259                 if (basis.is_equal(s)) {
 260                         if (ex_to_numeric(exponent).is_integer())
 261                                 return ex_to_numeric(exponent).to_int();
 262                         else
 263                                 return 0;
 264                 } else
 265                         return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
 266         }
 267         return 0;
 268 }
 269
 270 ex power::coeff(const ex & s, int n) const
 271 {
 272         if (!basis.is_equal(s)) {
 273                 // basis not equal to s
 274                 if (n == 0)
 275                         return *this;
 276                 else
 277                         return _ex0();
 278         } else {
 279                 // basis equal to s
 280                 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to_numeric(exponent).is_integer()) {
 281                         // integer exponent
 282                         int int_exp = ex_to_numeric(exponent).to_int();
 283                         if (n == int_exp)
 284                                 return _ex1();
 285                         else
 286                                 return _ex0();
 287                 } else {
 288                         // non-integer exponents are treated as zero
 289                         if (n == 0)
 290                                 return *this;
 291                         else
 292                                 return _ex0();
 293                 }
 294         }
 295 }
 296
 297 ex power::eval(int level) const
 298 {
 299         // simplifications: ^(x,0) -> 1 (0^0 handled here)
 300         //                  ^(x,1) -> x
 301         //                  ^(0,c1) -> 0 or exception (depending on real value of c1)
 302         //                  ^(1,x) -> 1
 303         //                  ^(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)
 304         //                  ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
 305         //                  ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 306         //                  ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
 307         //                  ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
 308
 309         debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
 310
 311         if ((level==1) && (flags & status_flags::evaluated))
 312                 return *this;
 313         else if (level == -max_recursion_level)
 314                 throw(std::runtime_error("max recursion level reached"));
 315
 316         const ex & ebasis    = level==1 ? basis    : basis.eval(level-1);
 317         const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
 318
 319         bool basis_is_numerical = 0;
 320         bool exponent_is_numerical = 0;
 321         numeric * num_basis;
 322         numeric * num_exponent;
 323
 324         if (is_exactly_of_type(*ebasis.bp,numeric)) {
 325                 basis_is_numerical = 1;
 326                 num_basis = static_cast<numeric *>(ebasis.bp);
 327         }
 328         if (is_exactly_of_type(*eexponent.bp,numeric)) {
 329                 exponent_is_numerical = 1;
 330                 num_exponent = static_cast<numeric *>(eexponent.bp);
 331         }
 332
 333         // ^(x,0) -> 1 (0^0 also handled here)
 334         if (eexponent.is_zero()) {
 335                 if (ebasis.is_zero())
 336                         throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
 337                 else
 338                         return _ex1();
 339         }
 340
 341         // ^(x,1) -> x
 342         if (eexponent.is_equal(_ex1()))
 343                 return ebasis;
 344
 345         // ^(0,c1) -> 0 or exception (depending on real value of c1)
 346         if (ebasis.is_zero() && exponent_is_numerical) {
 347                 if ((num_exponent->real()).is_zero())
 348                         throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
 349                 else if ((num_exponent->real()).is_negative())
 350                         throw (pole_error("power::eval(): division by zero",1));
 351                 else
 352                         return _ex0();
 353         }
 354
 355         // ^(1,x) -> 1
 356         if (ebasis.is_equal(_ex1()))
 357                 return _ex1();
 358
 359         if (basis_is_numerical && exponent_is_numerical) {
 360                 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
 361                 // except if c1,c2 are rational, but c1^c2 is not)
 362                 bool basis_is_crational = num_basis->is_crational();
 363                 bool exponent_is_crational = num_exponent->is_crational();
 364                 numeric res = num_basis->power(*num_exponent);
 365
 366                 if ((!basis_is_crational || !exponent_is_crational)
 367                         || res.is_crational()) {
 368                         return res;
 369                 }
 370                 GINAC_ASSERT(!num_exponent->is_integer());  // has been handled by now
 371                 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
 372                 if (basis_is_crational && exponent_is_crational
 373                         && num_exponent->is_real()
 374                         && !num_exponent->is_integer()) {
 375                         numeric n = num_exponent->numer();
 376                         numeric m = num_exponent->denom();
 377                         numeric r;
 378                         numeric q = iquo(n, m, r);
 379                         if (r.is_negative()) {
 380                                 r = r.add(m);
 381                                 q = q.sub(_num1());
 382                         }
 383                         if (q.is_zero())  // the exponent was in the allowed range 0<(n/m)<1
 384                                 return this->hold();
 385                         else {
 386                                 epvector res;
 387                                 res.push_back(expair(ebasis,r.div(m)));
 388                                 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
 389                         }
 390                 }
 391         }
 392
 393         // ^(^(x,c1),c2) -> ^(x,c1*c2)
 394         // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
 395         // case c1==1 should not happen, see below!)
 396         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
 397                 const power & sub_power = ex_to_power(ebasis);
 398                 const ex & sub_basis = sub_power.basis;
 399                 const ex & sub_exponent = sub_power.exponent;
 400                 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
 401                         const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
 402                         GINAC_ASSERT(num_sub_exponent!=numeric(1));
 403                         if (num_exponent->is_integer() || abs(num_sub_exponent)<1)
 404                                 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
 405                 }
 406         }
 407
 408         // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
 409         if (exponent_is_numerical && num_exponent->is_integer() &&
 410                 is_ex_exactly_of_type(ebasis,mul)) {
 411                 return expand_mul(ex_to_mul(ebasis), *num_exponent);
 412         }
 413
 414         // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
 415         // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
 416         if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
 417                 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
 418                 const mul & mulref = ex_to_mul(ebasis);
 419                 if (!mulref.overall_coeff.is_equal(_ex1())) {
 420                         const numeric & num_coeff = ex_to_numeric(mulref.overall_coeff);
 421                         if (num_coeff.is_real()) {
 422                                 if (num_coeff.is_positive()) {
 423                                         mul * mulp = new mul(mulref);
 424                                         mulp->overall_coeff = _ex1();
 425                                         mulp->clearflag(status_flags::evaluated);
 426                                         mulp->clearflag(status_flags::hash_calculated);
 427                                         return (new mul(power(*mulp,exponent),
 428                                                         power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
 429                                 } else {
 430                                         GINAC_ASSERT(num_coeff.compare(_num0())<0);
 431                                         if (num_coeff.compare(_num_1())!=0) {
 432                                                 mul * mulp = new mul(mulref);
 433                                                 mulp->overall_coeff = _ex_1();
 434                                                 mulp->clearflag(status_flags::evaluated);
 435                                                 mulp->clearflag(status_flags::hash_calculated);
 436                                                 return (new mul(power(*mulp,exponent),
 437                                                                 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
 438                                         }
 439                                 }
 440                         }
 441                 }
 442         }
 443
 444         if (are_ex_trivially_equal(ebasis,basis) &&
 445                 are_ex_trivially_equal(eexponent,exponent)) {
 446                 return this->hold();
 447         }
 448         return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
 449                                                                                                    status_flags::evaluated);
 450 }
 451
 452 ex power::evalf(int level) const
 453 {
 454         debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
 455
 456         ex ebasis;
 457         ex eexponent;
 458
 459         if (level==1) {
 460                 ebasis = basis;
 461                 eexponent = exponent;
 462         } else if (level == -max_recursion_level) {
 463                 throw(std::runtime_error("max recursion level reached"));
 464         } else {
 465                 ebasis = basis.evalf(level-1);
 466                 if (!is_ex_exactly_of_type(eexponent,numeric))
 467                         eexponent = exponent.evalf(level-1);
 468                 else
 469                         eexponent = exponent;
 470         }
 471
 472         return power(ebasis,eexponent);
 473 }
 474
 475 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
 476 {
 477         const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
 478         const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
 479
 480         if (are_ex_trivially_equal(basis, subsed_basis)
 481          && are_ex_trivially_equal(exponent, subsed_exponent))
 482                 return basic::subs(ls, lr, no_pattern);
 483         else
 484                 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
 485 }
 486
 487 ex power::simplify_ncmul(const exvector & v) const
 488 {
 489         return inherited::simplify_ncmul(v);
 490 }
 491
 492 // protected
 493
 494 /** Implementation of ex::diff() for a power.
 495  *  @see ex::diff */
 496 ex power::derivative(const symbol & s) const
 497 {
 498         if (exponent.info(info_flags::real)) {
 499                 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
 500                 epvector newseq;
 501                 newseq.reserve(2);
 502                 newseq.push_back(expair(basis, exponent - _ex1()));
 503                 newseq.push_back(expair(basis.diff(s), _ex1()));
 504                 return mul(newseq, exponent);
 505         } else {
 506                 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
 507                 return mul(*this,
 508                            add(mul(exponent.diff(s), log(basis)),
 509                            mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
 510         }
 511 }
 512
 513 int power::compare_same_type(const basic & other) const
 514 {
 515         GINAC_ASSERT(is_exactly_of_type(other, power));
 516         const power & o=static_cast<const power &>(const_cast<basic &>(other));
 517
 518         int cmpval;
 519         cmpval=basis.compare(o.basis);
 520         if (cmpval==0) {
 521                 return exponent.compare(o.exponent);
 522         }
 523         return cmpval;
 524 }
 525
 526 unsigned power::return_type(void) const
 527 {
 528         return basis.return_type();
 529 }
 530
 531 unsigned power::return_type_tinfo(void) const
 532 {
 533         return basis.return_type_tinfo();
 534 }
 535
 536 ex power::expand(unsigned options) const
 537 {
 538         if (flags & status_flags::expanded)
 539                 return *this;
 540
 541         ex expanded_basis = basis.expand(options);
 542         ex expanded_exponent = exponent.expand(options);
 543
 544         // x^(a+b) -> x^a * x^b
 545         if (is_ex_exactly_of_type(expanded_exponent, add)) {
 546                 const add &a = ex_to_add(expanded_exponent);
 547                 exvector distrseq;
 548                 distrseq.reserve(a.seq.size() + 1);
 549                 epvector::const_iterator last = a.seq.end();
 550                 epvector::const_iterator cit = a.seq.begin();
 551                 while (cit!=last) {
 552                         distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
 553                         cit++;
 554                 }
 555
 556                 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
 557                 if (ex_to_numeric(a.overall_coeff).is_integer()) {
 558                         const numeric &num_exponent = ex_to_numeric(a.overall_coeff);
 559                         int int_exponent = num_exponent.to_int();
 560                         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
 561                                 distrseq.push_back(expand_add(ex_to_add(expanded_basis), int_exponent));
 562                         else
 563                                 distrseq.push_back(power(expanded_basis, a.overall_coeff));
 564                 } else
 565                         distrseq.push_back(power(expanded_basis, a.overall_coeff));
 566
 567                 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
 568                 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
 569                 return r.expand();
 570         }
 571
 572         if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
 573                 !ex_to_numeric(expanded_exponent).is_integer()) {
 574                 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
 575                         return this->hold();
 576                 } else {
 577                         return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | status_flags::expanded);
 578                 }
 579         }
 580
 581         // integer numeric exponent
 582         const numeric & num_exponent = ex_to_numeric(expanded_exponent);
 583         int int_exponent = num_exponent.to_int();
 584
 585         // (x+y)^n, n>0
 586         if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
 587                 return expand_add(ex_to_add(expanded_basis), int_exponent);
 588
 589         // (x*y)^n -> x^n * y^n
 590         if (is_ex_exactly_of_type(expanded_basis,mul))
 591                 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
 592
 593         // cannot expand further
 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 | status_flags::expanded);
 598 }
 599
 600 //////////
 601 // new virtual functions which can be overridden by derived classes
 602 //////////
 603
 604 // none
 605
 606 //////////
 607 // non-virtual functions in this class
 608 //////////
 609
 610 /** expand a^n where a is an add and n is an integer.
 611  *  @see power::expand */
 612 ex power::expand_add(const add & a, int n) const
 613 {
 614         if (n==2)
 615                 return expand_add_2(a);
 616
 617         int m = a.nops();
 618         exvector sum;
 619         sum.reserve((n+1)*(m-1));
 620         intvector k(m-1);
 621         intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
 622         intvector upper_limit(m-1);
 623         int l;
 624
 625         for (int l=0; l<m-1; l++) {
 626                 k[l] = 0;
 627                 k_cum[l] = 0;
 628                 upper_limit[l] = n;
 629         }
 630
 631         while (1) {
 632                 exvector term;
 633                 term.reserve(m+1);
 634                 for (l=0; l<m-1; l++) {
 635                         const ex & b = a.op(l);
 636                         GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 637                         GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 638                                      !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 639                                      !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 640                                      !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 641                                      !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 642                                      !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 643                         if (is_ex_exactly_of_type(b,mul))
 644                                 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
 645                         else
 646                                 term.push_back(power(b,k[l]));
 647                 }
 648
 649                 const ex & b = a.op(l);
 650                 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
 651                 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
 652                              !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric) ||
 653                              !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer() ||
 654                              !is_ex_exactly_of_type(ex_to_power(b).basis,add) ||
 655                              !is_ex_exactly_of_type(ex_to_power(b).basis,mul) ||
 656                              !is_ex_exactly_of_type(ex_to_power(b).basis,power));
 657                 if (is_ex_exactly_of_type(b,mul))
 658                         term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
 659                 else
 660                         term.push_back(power(b,n-k_cum[m-2]));
 661
 662                 numeric f = binomial(numeric(n),numeric(k[0]));
 663                 for (l=1; l<m-1; l++)
 664                         f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
 665
 666                 term.push_back(f);
 667
 668                 /*
 669                 cout << "begin term" << endl;
 670                 for (int i=0; i<m-1; i++) {
 671                         cout << "k[" << i << "]=" << k[i] << endl;
 672                         cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
 673                         cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
 674                 }
 675                 for_each(term.begin(), term.end(), ostream_iterator<ex>(cout, "\n"));
 676                 cout << "end term" << endl;
 677                 */
 678
 679                 // TODO: optimize this
 680                 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
 681
 682                 // increment k[]
 683                 l = m-2;
 684                 while ((l>=0)&&((++k[l])>upper_limit[l])) {
 685                         k[l] = 0;
 686                         l--;
 687                 }
 688                 if (l<0) break;
 689
 690                 // recalc k_cum[] and upper_limit[]
 691                 if (l==0)
 692                         k_cum[0] = k[0];
 693                 else
 694                         k_cum[l] = k_cum[l-1]+k[l];
 695
 696                 for (int i=l+1; i<m-1; i++)
 697                         k_cum[i] = k_cum[i-1]+k[i];
 698
 699                 for (int i=l+1; i<m-1; i++)
 700                         upper_limit[i] = n-k_cum[i-1];
 701         }
 702         return (new add(sum))->setflag(status_flags::dynallocated |
 703                                                                    status_flags::expanded );
 704 }
 705
 706
 707 /** Special case of power::expand_add. Expands a^2 where a is an add.
 708  *  @see power::expand_add */
 709 ex power::expand_add_2(const add & a) const
 710 {
 711         epvector sum;
 712         unsigned a_nops = a.nops();
 713         sum.reserve((a_nops*(a_nops+1))/2);
 714         epvector::const_iterator last = a.seq.end();
 715
 716         // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
 717         // first part: ignore overall_coeff and expand other terms
 718         for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
 719                 const ex & r = (*cit0).rest;
 720                 const ex & c = (*cit0).coeff;
 721
 722                 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
 723                 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
 724                              !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric) ||
 725                              !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer() ||
 726                              !is_ex_exactly_of_type(ex_to_power(r).basis,add) ||
 727                              !is_ex_exactly_of_type(ex_to_power(r).basis,mul) ||
 728                              !is_ex_exactly_of_type(ex_to_power(r).basis,power));
 729
 730                 if (are_ex_trivially_equal(c,_ex1())) {
 731                         if (is_ex_exactly_of_type(r,mul)) {
 732                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 733                                                      _ex1()));
 734                         } else {
 735                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 736                                                      _ex1()));
 737                         }
 738                 } else {
 739                         if (is_ex_exactly_of_type(r,mul)) {
 740                                 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
 741                                                      ex_to_numeric(c).power_dyn(_num2())));
 742                         } else {
 743                                 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
 744                                                      ex_to_numeric(c).power_dyn(_num2())));
 745                         }
 746                 }
 747
 748                 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
 749                         const ex & r1 = (*cit1).rest;
 750                         const ex & c1 = (*cit1).coeff;
 751                         sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
 752                                                                       _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
 753                 }
 754         }
 755
 756         GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
 757
 758         // second part: add terms coming from overall_factor (if != 0)
 759         if (!a.overall_coeff.is_zero()) {
 760                 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
 761                         sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
 762                 }
 763                 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
 764         }
 765
 766         GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
 767
 768         return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
 769 }
 770
 771 /** Expand factors of m in m^n where m is a mul and n is and integer
 772  *  @see power::expand */
 773 ex power::expand_mul(const mul & m, const numeric & n) const
 774 {
 775         if (n.is_zero())
 776                 return _ex1();
 777
 778         epvector distrseq;
 779         distrseq.reserve(m.seq.size());
 780         epvector::const_iterator last = m.seq.end();
 781         epvector::const_iterator cit = m.seq.begin();
 782         while (cit!=last) {
 783                 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
 784                         distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
 785                 } else {
 786                         // it is safe not to call mul::combine_pair_with_coeff_to_pair()
 787                         // since n is an integer
 788                         distrseq.push_back(expair((*cit).rest, ex_to_numeric((*cit).coeff).mul(n)));
 789                 }
 790                 ++cit;
 791         }
 792         return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
 793 }
 794
 795 /*
 796 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
 797                                unsigned options) const
 798 {
 799         // obsolete
 800
 801         exvector distrseq;
 802         epvector splitseq;
 803
 804         const add & addref=static_cast<const add &>(*basis.bp);
 805
 806         splitseq=addref.seq;
 807         splitseq.pop_back();
 808         ex first_operands=add(splitseq);
 809         ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
 810
 811         int n=exponent.to_int();
 812         for (int k=0; k<=n; k++) {
 813                 distrseq.push_back(binomial(n,k) * power(first_operands,numeric(k))
 814                                                  * power(last_operand,numeric(n-k)));
 815         }
 816         return ex((new add(distrseq))->setflag(status_flags::expanded | status_flags::dynallocated)).expand(options);
 817 }
 818 */
 819
 820 /*
 821 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
 822                                                                 unsigned options) const
 823 {
 824         ex rest_power = ex(power(basis,exponent.add(_num_1()))).
 825                         expand(options | expand_options::internal_do_not_expand_power_operands);
 826
 827         return ex(mul(rest_power,basis),0).
 828                expand(options | expand_options::internal_do_not_expand_mul_operands);
 829 }
 830 */
 831
 832 // helper function
 833
 834 ex sqrt(const ex & a)
 835 {
 836         return power(a,_ex1_2());
 837 }
 838
 839 } // namespace GiNaC