ginac/indexed.cpp

   1 /** @file indexed.cpp
   2  *
   3  *  Implementation of GiNaC's indexed expressions. */
   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 <stdexcept>
  24 #include <algorithm>
  25
  26 #include "indexed.h"
  27 #include "idx.h"
  28 #include "add.h"
  29 #include "mul.h"
  30 #include "ncmul.h"
  31 #include "power.h"
  32 #include "lst.h"
  33 #include "inifcns.h" // for symmetrize()
  34 #include "print.h"
  35 #include "archive.h"
  36 #include "utils.h"
  37 #include "debugmsg.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(indexed, exprseq)
  42
  43 //////////
  44 // default constructor, destructor, copy constructor assignment operator and helpers
  45 //////////
  46
  47 indexed::indexed() : symmetry(unknown)
  48 {
  49         debugmsg("indexed default constructor", LOGLEVEL_CONSTRUCT);
  50         tinfo_key = TINFO_indexed;
  51 }
  52
  53 void indexed::copy(const indexed & other)
  54 {
  55         inherited::copy(other);
  56         symmetry = other.symmetry;
  57 }
  58
  59 DEFAULT_DESTROY(indexed)
  60
  61 //////////
  62 // other constructors
  63 //////////
  64
  65 indexed::indexed(const ex & b) : inherited(b), symmetry(unknown)
  66 {
  67         debugmsg("indexed constructor from ex", LOGLEVEL_CONSTRUCT);
  68         tinfo_key = TINFO_indexed;
  69         assert_all_indices_of_type_idx();
  70 }
  71
  72 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symmetry(unknown)
  73 {
  74         debugmsg("indexed constructor from ex,ex", LOGLEVEL_CONSTRUCT);
  75         tinfo_key = TINFO_indexed;
  76         assert_all_indices_of_type_idx();
  77 }
  78
  79 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symmetry(unknown)
  80 {
  81         debugmsg("indexed constructor from ex,ex,ex", LOGLEVEL_CONSTRUCT);
  82         tinfo_key = TINFO_indexed;
  83         assert_all_indices_of_type_idx();
  84 }
  85
  86 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symmetry(unknown)
  87 {
  88         debugmsg("indexed constructor from ex,ex,ex,ex", LOGLEVEL_CONSTRUCT);
  89         tinfo_key = TINFO_indexed;
  90         assert_all_indices_of_type_idx();
  91 }
  92
  93 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symmetry(unknown)
  94 {
  95         debugmsg("indexed constructor from ex,ex,ex,ex,ex", LOGLEVEL_CONSTRUCT);
  96         tinfo_key = TINFO_indexed;
  97         assert_all_indices_of_type_idx();
  98 }
  99
 100 indexed::indexed(const ex & b, symmetry_type symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symmetry(symm)
 101 {
 102         debugmsg("indexed constructor from ex,symmetry,ex,ex", LOGLEVEL_CONSTRUCT);
 103         tinfo_key = TINFO_indexed;
 104         assert_all_indices_of_type_idx();
 105 }
 106
 107 indexed::indexed(const ex & b, symmetry_type symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symmetry(symm)
 108 {
 109         debugmsg("indexed constructor from ex,symmetry,ex,ex,ex", LOGLEVEL_CONSTRUCT);
 110         tinfo_key = TINFO_indexed;
 111         assert_all_indices_of_type_idx();
 112 }
 113
 114 indexed::indexed(const ex & b, symmetry_type symm, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symmetry(symm)
 115 {
 116         debugmsg("indexed constructor from ex,symmetry,ex,ex,ex,ex", LOGLEVEL_CONSTRUCT);
 117         tinfo_key = TINFO_indexed;
 118         assert_all_indices_of_type_idx();
 119 }
 120
 121 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symmetry(unknown)
 122 {
 123         debugmsg("indexed constructor from ex,exvector", LOGLEVEL_CONSTRUCT);
 124         seq.insert(seq.end(), v.begin(), v.end());
 125         tinfo_key = TINFO_indexed;
 126         assert_all_indices_of_type_idx();
 127 }
 128
 129 indexed::indexed(const ex & b, symmetry_type symm, const exvector & v) : inherited(b), symmetry(symm)
 130 {
 131         debugmsg("indexed constructor from ex,symmetry,exvector", LOGLEVEL_CONSTRUCT);
 132         seq.insert(seq.end(), v.begin(), v.end());
 133         tinfo_key = TINFO_indexed;
 134         assert_all_indices_of_type_idx();
 135 }
 136
 137 indexed::indexed(symmetry_type symm, const exprseq & es) : inherited(es), symmetry(symm)
 138 {
 139         debugmsg("indexed constructor from symmetry,exprseq", LOGLEVEL_CONSTRUCT);
 140         tinfo_key = TINFO_indexed;
 141         assert_all_indices_of_type_idx();
 142 }
 143
 144 indexed::indexed(symmetry_type symm, const exvector & v, bool discardable) : inherited(v, discardable), symmetry(symm)
 145 {
 146         debugmsg("indexed constructor from symmetry,exvector", LOGLEVEL_CONSTRUCT);
 147         tinfo_key = TINFO_indexed;
 148         assert_all_indices_of_type_idx();
 149 }
 150
 151 indexed::indexed(symmetry_type symm, exvector * vp) : inherited(vp), symmetry(symm)
 152 {
 153         debugmsg("indexed constructor from symmetry,exvector *", LOGLEVEL_CONSTRUCT);
 154         tinfo_key = TINFO_indexed;
 155         assert_all_indices_of_type_idx();
 156 }
 157
 158 //////////
 159 // archiving
 160 //////////
 161
 162 indexed::indexed(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 163 {
 164         debugmsg("indexed constructor from archive_node", LOGLEVEL_CONSTRUCT);
 165         unsigned int symm;
 166         if (!(n.find_unsigned("symmetry", symm)))
 167                 throw (std::runtime_error("unknown indexed symmetry type in archive"));
 168 }
 169
 170 void indexed::archive(archive_node &n) const
 171 {
 172         inherited::archive(n);
 173         n.add_unsigned("symmetry", symmetry);
 174 }
 175
 176 DEFAULT_UNARCHIVE(indexed)
 177
 178 //////////
 179 // functions overriding virtual functions from bases classes
 180 //////////
 181
 182 void indexed::print(const print_context & c, unsigned level) const
 183 {
 184         debugmsg("indexed print", LOGLEVEL_PRINT);
 185         GINAC_ASSERT(seq.size() > 0);
 186
 187         if (is_of_type(c, print_tree)) {
 188
 189                 c.s << std::string(level, ' ') << class_name()
 190                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 191                     << ", " << seq.size()-1 << " indices";
 192                 switch (symmetry) {
 193                         case symmetric: c.s << ", symmetric"; break;
 194                         case antisymmetric: c.s << ", antisymmetric"; break;
 195                         default: break;
 196                 }
 197                 c.s << std::endl;
 198                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 199                 seq[0].print(c, level + delta_indent);
 200                 printindices(c, level + delta_indent);
 201
 202         } else {
 203
 204                 bool is_tex = is_of_type(c, print_latex);
 205                 const ex & base = seq[0];
 206                 bool need_parens = is_ex_exactly_of_type(base, add) || is_ex_exactly_of_type(base, mul)
 207                                 || is_ex_exactly_of_type(base, ncmul) || is_ex_exactly_of_type(base, power)
 208                                 || is_ex_of_type(base, indexed);
 209                 if (is_tex)
 210                         c.s << "{";
 211                 if (need_parens)
 212                         c.s << "(";
 213                 base.print(c);
 214                 if (need_parens)
 215                         c.s << ")";
 216                 if (is_tex)
 217                         c.s << "}";
 218                 printindices(c, level);
 219         }
 220 }
 221
 222 bool indexed::info(unsigned inf) const
 223 {
 224         if (inf == info_flags::indexed) return true;
 225         if (inf == info_flags::has_indices) return seq.size() > 1;
 226         return inherited::info(inf);
 227 }
 228
 229 struct idx_is_not : public binary_function<ex, unsigned, bool> {
 230         bool operator() (const ex & e, unsigned inf) const {
 231                 return !(ex_to_idx(e).get_value().info(inf));
 232         }
 233 };
 234
 235 bool indexed::all_index_values_are(unsigned inf) const
 236 {
 237         // No indices? Then no property can be fulfilled
 238         if (seq.size() < 2)
 239                 return false;
 240
 241         // Check all indices
 242         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 243 }
 244
 245 int indexed::compare_same_type(const basic & other) const
 246 {
 247         GINAC_ASSERT(is_of_type(other, indexed));
 248         return inherited::compare_same_type(other);
 249 }
 250
 251 // The main difference between sort_index_vector() and canonicalize_indices()
 252 // is that the latter takes the symmetry of the object into account. Once we
 253 // implement mixed symmetries, canonicalize_indices() will only be able to
 254 // reorder index pairs with known symmetry properties, while sort_index_vector()
 255 // always sorts the whole vector.
 256
 257 /** Bring a vector of indices into a canonic order. This operation only makes
 258  *  sense if the object carrying these indices is either symmetric or totally
 259  *  antisymmetric with respect to the indices.
 260  *
 261  *  @param itbegin Start of index vector
 262  *  @param itend End of index vector
 263  *  @param antisymm Whether the object is antisymmetric
 264  *  @return the sign introduced by the reordering of the indices if the object
 265  *          is antisymmetric (or 0 if two equal indices are encountered). For
 266  *          symmetric objects, this is always +1. If the index vector was
 267  *          already in a canonic order this function returns INT_MAX. */
 268 static int canonicalize_indices(exvector::iterator itbegin, exvector::iterator itend, bool antisymm)
 269 {
 270         bool something_changed = false;
 271         int sig = 1;
 272
 273         // Simple bubble sort algorithm should be sufficient for the small
 274         // number of indices expected
 275         exvector::iterator it1 = itbegin, next_to_last_idx = itend - 1;
 276         while (it1 != next_to_last_idx) {
 277                 exvector::iterator it2 = it1 + 1;
 278                 while (it2 != itend) {
 279                         int cmpval = it1->compare(*it2);
 280                         if (cmpval == 1) {
 281                                 it1->swap(*it2);
 282                                 something_changed = true;
 283                                 if (antisymm)
 284                                         sig = -sig;
 285                         } else if (cmpval == 0 && antisymm) {
 286                                 something_changed = true;
 287                                 sig = 0;
 288                         }
 289                         it2++;
 290                 }
 291                 it1++;
 292         }
 293
 294         return something_changed ? sig : INT_MAX;
 295 }
 296
 297 ex indexed::eval(int level) const
 298 {
 299         // First evaluate children, then we will end up here again
 300         if (level > 1)
 301                 return indexed(symmetry, evalchildren(level));
 302
 303         const ex &base = seq[0];
 304
 305         // If the base object is 0, the whole object is 0
 306         if (base.is_zero())
 307                 return _ex0();
 308
 309         // If the base object is a product, pull out the numeric factor
 310         if (is_ex_exactly_of_type(base, mul) && is_ex_exactly_of_type(base.op(base.nops() - 1), numeric)) {
 311                 exvector v(seq);
 312                 ex f = ex_to_numeric(base.op(base.nops() - 1));
 313                 v[0] = seq[0] / f;
 314                 return f * thisexprseq(v);
 315         }
 316
 317         // Canonicalize indices according to the symmetry properties
 318         if (seq.size() > 2 && (symmetry == symmetric || symmetry == antisymmetric)) {
 319                 exvector v(seq);
 320                 int sig = canonicalize_indices(v.begin() + 1, v.end(), symmetry == antisymmetric);
 321                 if (sig != INT_MAX) {
 322                         // Something has changed while sorting indices, more evaluations later
 323                         if (sig == 0)
 324                                 return _ex0();
 325                         return ex(sig) * thisexprseq(v);
 326                 }
 327         }
 328
 329         // Let the class of the base object perform additional evaluations
 330         return base.bp->eval_indexed(*this);
 331 }
 332
 333 int indexed::degree(const ex & s) const
 334 {
 335         return is_equal(*s.bp) ? 1 : 0;
 336 }
 337
 338 int indexed::ldegree(const ex & s) const
 339 {
 340         return is_equal(*s.bp) ? 1 : 0;
 341 }
 342
 343 ex indexed::coeff(const ex & s, int n) const
 344 {
 345         if (is_equal(*s.bp))
 346                 return n==1 ? _ex1() : _ex0();
 347         else
 348                 return n==0 ? ex(*this) : _ex0();
 349 }
 350
 351 ex indexed::thisexprseq(const exvector & v) const
 352 {
 353         return indexed(symmetry, v);
 354 }
 355
 356 ex indexed::thisexprseq(exvector * vp) const
 357 {
 358         return indexed(symmetry, vp);
 359 }
 360
 361 ex indexed::expand(unsigned options) const
 362 {
 363         GINAC_ASSERT(seq.size() > 0);
 364
 365         if ((options & expand_options::expand_indexed) && is_ex_exactly_of_type(seq[0], add)) {
 366
 367                 // expand_indexed expands (a+b).i -> a.i + b.i
 368                 const ex & base = seq[0];
 369                 ex sum = _ex0();
 370                 for (unsigned i=0; i<base.nops(); i++) {
 371                         exvector s = seq;
 372                         s[0] = base.op(i);
 373                         sum += thisexprseq(s).expand();
 374                 }
 375                 return sum;
 376
 377         } else
 378                 return inherited::expand(options);
 379 }
 380
 381 //////////
 382 // virtual functions which can be overridden by derived classes
 383 //////////
 384
 385 // none
 386
 387 //////////
 388 // non-virtual functions in this class
 389 //////////
 390
 391 void indexed::printindices(const print_context & c, unsigned level) const
 392 {
 393         if (seq.size() > 1) {
 394
 395                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 396
 397                 if (is_of_type(c, print_latex)) {
 398
 399                         // TeX output: group by variance
 400                         bool first = true;
 401                         bool covariant = true;
 402
 403                         while (it != itend) {
 404                                 bool cur_covariant = (is_ex_of_type(*it, varidx) ? ex_to_varidx(*it).is_covariant() : true);
 405                                 if (first || cur_covariant != covariant) {
 406                                         if (!first)
 407                                                 c.s << "}";
 408                                         covariant = cur_covariant;
 409                                         if (covariant)
 410                                                 c.s << "_{";
 411                                         else
 412                                                 c.s << "^{";
 413                                 }
 414                                 it->print(c, level);
 415                                 c.s << " ";
 416                                 first = false;
 417                                 it++;
 418                         }
 419                         c.s << "}";
 420
 421                 } else {
 422
 423                         // Ordinary output
 424                         while (it != itend) {
 425                                 it->print(c, level);
 426                                 it++;
 427                         }
 428                 }
 429         }
 430 }
 431
 432 /** Check whether all indices are of class idx. This function is used
 433  *  internally to make sure that all constructed indexed objects really
 434  *  carry indices and not some other classes. */
 435 void indexed::assert_all_indices_of_type_idx(void) const
 436 {
 437         GINAC_ASSERT(seq.size() > 0);
 438         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 439         while (it != itend) {
 440                 if (!is_ex_of_type(*it, idx))
 441                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 442                 it++;
 443         }
 444 }
 445
 446 //////////
 447 // global functions
 448 //////////
 449
 450 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 451 static bool indices_consistent(const exvector & v1, const exvector & v2)
 452 {
 453         // Number of indices must be the same
 454         if (v1.size() != v2.size())
 455                 return false;
 456
 457         return equal(v1.begin(), v1.end(), v2.begin(), ex_is_equal());
 458 }
 459
 460 exvector indexed::get_indices(void) const
 461 {
 462         GINAC_ASSERT(seq.size() >= 1);
 463         return exvector(seq.begin() + 1, seq.end());
 464 }
 465
 466 exvector indexed::get_dummy_indices(void) const
 467 {
 468         exvector free_indices, dummy_indices;
 469         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 470         return dummy_indices;
 471 }
 472
 473 exvector indexed::get_dummy_indices(const indexed & other) const
 474 {
 475         exvector indices = get_free_indices();
 476         exvector other_indices = other.get_free_indices();
 477         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 478         exvector dummy_indices;
 479         find_dummy_indices(indices, dummy_indices);
 480         return dummy_indices;
 481 }
 482
 483 bool indexed::has_dummy_index_for(const ex & i) const
 484 {
 485         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 486         while (it != itend) {
 487                 if (is_dummy_pair(*it, i))
 488                         return true;
 489                 it++;
 490         }
 491         return false;
 492 }
 493
 494 exvector indexed::get_free_indices(void) const
 495 {
 496         exvector free_indices, dummy_indices;
 497         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 498         return free_indices;
 499 }
 500
 501 exvector add::get_free_indices(void) const
 502 {
 503         exvector free_indices;
 504         for (unsigned i=0; i<nops(); i++) {
 505                 if (i == 0)
 506                         free_indices = op(i).get_free_indices();
 507                 else {
 508                         exvector free_indices_of_term = op(i).get_free_indices();
 509                         if (!indices_consistent(free_indices, free_indices_of_term))
 510                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 511                 }
 512         }
 513         return free_indices;
 514 }
 515
 516 exvector mul::get_free_indices(void) const
 517 {
 518         // Concatenate free indices of all factors
 519         exvector un;
 520         for (unsigned i=0; i<nops(); i++) {
 521                 exvector free_indices_of_factor = op(i).get_free_indices();
 522                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 523         }
 524
 525         // And remove the dummy indices
 526         exvector free_indices, dummy_indices;
 527         find_free_and_dummy(un, free_indices, dummy_indices);
 528         return free_indices;
 529 }
 530
 531 exvector ncmul::get_free_indices(void) const
 532 {
 533         // Concatenate free indices of all factors
 534         exvector un;
 535         for (unsigned i=0; i<nops(); i++) {
 536                 exvector free_indices_of_factor = op(i).get_free_indices();
 537                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 538         }
 539
 540         // And remove the dummy indices
 541         exvector free_indices, dummy_indices;
 542         find_free_and_dummy(un, free_indices, dummy_indices);
 543         return free_indices;
 544 }
 545
 546 exvector power::get_free_indices(void) const
 547 {
 548         // Return free indices of basis
 549         return basis.get_free_indices();
 550 }
 551
 552 /** Rename dummy indices in an expression.
 553  *
 554  *  @param e Expression to be worked on
 555  *  @param local_dummy_indices The set of dummy indices that appear in the
 556  *    expression "e"
 557  *  @param global_dummy_indices The set of dummy indices that have appeared
 558  *    before and which we would like to use in "e", too. This gets updated
 559  *    by the function */
 560 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 561 {
 562         int global_size = global_dummy_indices.size(),
 563             local_size = local_dummy_indices.size();
 564
 565         // Any local dummy indices at all?
 566         if (local_size == 0)
 567                 return e;
 568
 569         if (global_size < local_size) {
 570
 571                 // More local indices than we encountered before, add the new ones
 572                 // to the global set
 573                 int remaining = local_size - global_size;
 574                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 575                 while (it != itend && remaining > 0) {
 576                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(ex_is_equal(), *it)) == global_dummy_indices.end()) {
 577                                 global_dummy_indices.push_back(*it);
 578                                 global_size++;
 579                                 remaining--;
 580                         }
 581                         it++;
 582                 }
 583         }
 584
 585         // Replace index symbols in expression
 586         GINAC_ASSERT(local_size <= global_size);
 587         bool all_equal = true;
 588         lst local_syms, global_syms;
 589         for (unsigned i=0; i<local_size; i++) {
 590                 ex loc_sym = local_dummy_indices[i].op(0);
 591                 ex glob_sym = global_dummy_indices[i].op(0);
 592                 if (!loc_sym.is_equal(glob_sym)) {
 593                         all_equal = false;
 594                         local_syms.append(loc_sym);
 595                         global_syms.append(glob_sym);
 596                 }
 597         }
 598         if (all_equal)
 599                 return e;
 600         else
 601                 return e.subs(local_syms, global_syms);
 602 }
 603
 604 /** Simplify product of indexed expressions (commutative, noncommutative and
 605  *  simple squares), return list of free indices. */
 606 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 607 {
 608         // Remember whether the product was commutative or noncommutative
 609         // (because we chop it into factors and need to reassemble later)
 610         bool non_commutative = is_ex_exactly_of_type(e, ncmul);
 611
 612         // Collect factors in an exvector, store squares twice
 613         exvector v;
 614         v.reserve(e.nops() * 2);
 615
 616         if (is_ex_exactly_of_type(e, power)) {
 617                 // We only get called for simple squares, split a^2 -> a*a
 618                 GINAC_ASSERT(e.op(1).is_equal(_ex2()));
 619                 v.push_back(e.op(0));
 620                 v.push_back(e.op(0));
 621         } else {
 622                 for (int i=0; i<e.nops(); i++) {
 623                         ex f = e.op(i);
 624                         if (is_ex_exactly_of_type(f, power) && f.op(1).is_equal(_ex2())) {
 625                                 v.push_back(f.op(0));
 626                     v.push_back(f.op(0));
 627                         } else if (is_ex_exactly_of_type(f, ncmul)) {
 628                                 // Noncommutative factor found, split it as well
 629                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 630                                 for (int j=0; j<f.nops(); j++)
 631                                         v.push_back(f.op(j));
 632                         } else
 633                                 v.push_back(f);
 634                 }
 635         }
 636
 637         // Perform contractions
 638         bool something_changed = false;
 639         GINAC_ASSERT(v.size() > 1);
 640         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 641         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 642
 643 try_again:
 644                 if (!is_ex_of_type(*it1, indexed))
 645                         continue;
 646
 647                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 648
 649                 // Indexed factor found, get free indices and look for contraction
 650                 // candidates
 651                 exvector free1, dummy1;
 652                 find_free_and_dummy(ex_to_indexed(*it1).seq.begin() + 1, ex_to_indexed(*it1).seq.end(), free1, dummy1);
 653
 654                 exvector::iterator it2;
 655                 for (it2 = it1 + 1; it2 != itend; it2++) {
 656
 657                         if (!is_ex_of_type(*it2, indexed))
 658                                 continue;
 659
 660                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 661
 662                         // Find free indices of second factor and merge them with free
 663                         // indices of first factor
 664                         exvector un;
 665                         find_free_and_dummy(ex_to_indexed(*it2).seq.begin() + 1, ex_to_indexed(*it2).seq.end(), un, dummy1);
 666                         un.insert(un.end(), free1.begin(), free1.end());
 667
 668                         // Check whether the two factors share dummy indices
 669                         exvector free, dummy;
 670                         find_free_and_dummy(un, free, dummy);
 671                         if (dummy.size() == 0)
 672                                 continue;
 673
 674                         // At least one dummy index, is it a defined scalar product?
 675                         bool contracted = false;
 676                         if (free.size() == 0) {
 677                                 if (sp.is_defined(*it1, *it2)) {
 678                                         *it1 = sp.evaluate(*it1, *it2);
 679                                         *it2 = _ex1();
 680                                         goto contraction_done;
 681                                 }
 682                         }
 683
 684                         // Contraction of symmetric with antisymmetric object is zero
 685                         if ((ex_to_indexed(*it1).symmetry == indexed::symmetric &&
 686                              ex_to_indexed(*it2).symmetry == indexed::antisymmetric
 687                           || ex_to_indexed(*it1).symmetry == indexed::antisymmetric &&
 688                              ex_to_indexed(*it2).symmetry == indexed::symmetric)
 689                          && dummy.size() > 1) {
 690                                 free_indices.clear();
 691                                 return _ex0();
 692                         }
 693
 694                         // Try to contract the first one with the second one
 695                         contracted = it1->op(0).bp->contract_with(it1, it2, v);
 696                         if (!contracted) {
 697
 698                                 // That didn't work; maybe the second object knows how to
 699                                 // contract itself with the first one
 700                                 contracted = it2->op(0).bp->contract_with(it2, it1, v);
 701                         }
 702                         if (contracted) {
 703 contraction_done:
 704                                 if (first_noncommutative || second_noncommutative
 705                                  || is_ex_exactly_of_type(*it1, add) || is_ex_exactly_of_type(*it2, add)
 706                                  || is_ex_exactly_of_type(*it1, mul) || is_ex_exactly_of_type(*it2, mul)
 707                                  || is_ex_exactly_of_type(*it1, ncmul) || is_ex_exactly_of_type(*it2, ncmul)) {
 708
 709                                         // One of the factors became a sum or product:
 710                                         // re-expand expression and run again
 711                                         // Non-commutative products are always re-expanded to give
 712                                         // simplify_ncmul() the chance to re-order and canonicalize
 713                                         // the product
 714                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 715                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 716                                 }
 717
 718                                 // Both objects may have new indices now or they might
 719                                 // even not be indexed objects any more, so we have to
 720                                 // start over
 721                                 something_changed = true;
 722                                 goto try_again;
 723                         }
 724                 }
 725         }
 726
 727         // Find free indices (concatenate them all and call find_free_and_dummy())
 728         // and all dummy indices that appear
 729         exvector un, individual_dummy_indices;
 730         it1 = v.begin(); itend = v.end();
 731         while (it1 != itend) {
 732                 exvector free_indices_of_factor;
 733                 if (is_ex_of_type(*it1, indexed)) {
 734                         exvector dummy_indices_of_factor;
 735                         find_free_and_dummy(ex_to_indexed(*it1).seq.begin() + 1, ex_to_indexed(*it1).seq.end(), free_indices_of_factor, dummy_indices_of_factor);
 736                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 737                 } else
 738                         free_indices_of_factor = it1->get_free_indices();
 739                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 740                 it1++;
 741         }
 742         exvector local_dummy_indices;
 743         find_free_and_dummy(un, free_indices, local_dummy_indices);
 744         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 745
 746         ex r;
 747         if (something_changed)
 748                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 749         else
 750                 r = e;
 751
 752         // Dummy index renaming
 753         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 754
 755         // Product of indexed object with a scalar?
 756         if (is_ex_exactly_of_type(r, mul) && r.nops() == 2
 757          && is_ex_exactly_of_type(r.op(1), numeric) && is_ex_of_type(r.op(0), indexed))
 758                 return r.op(0).op(0).bp->scalar_mul_indexed(r.op(0), ex_to_numeric(r.op(1)));
 759         else
 760                 return r;
 761 }
 762
 763 /** Simplify indexed expression, return list of free indices. */
 764 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 765 {
 766         // Expand the expression
 767         ex e_expanded = e.expand();
 768
 769         // Simplification of single indexed object: just find the free indices
 770         // and perform dummy index renaming
 771         if (is_ex_of_type(e_expanded, indexed)) {
 772                 const indexed &i = ex_to_indexed(e_expanded);
 773                 exvector local_dummy_indices;
 774                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 775                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 776         }
 777
 778         // Simplification of sum = sum of simplifications, check consistency of
 779         // free indices in each term
 780         if (is_ex_exactly_of_type(e_expanded, add)) {
 781                 bool first = true;
 782                 ex sum = _ex0();
 783                 free_indices.clear();
 784
 785                 for (unsigned i=0; i<e_expanded.nops(); i++) {
 786                         exvector free_indices_of_term;
 787                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 788                         if (!term.is_zero()) {
 789                                 if (first) {
 790                                         free_indices = free_indices_of_term;
 791                                         sum = term;
 792                                         first = false;
 793                                 } else {
 794                                         if (!indices_consistent(free_indices, free_indices_of_term))
 795                                                 throw (std::runtime_error("simplify_indexed: inconsistent indices in sum"));
 796                                         if (is_ex_of_type(sum, indexed) && is_ex_of_type(term, indexed))
 797                                                 sum = sum.op(0).bp->add_indexed(sum, term);
 798                                         else
 799                                                 sum += term;
 800                                 }
 801                         }
 802                 }
 803
 804                 return sum;
 805         }
 806
 807         // Simplification of products
 808         if (is_ex_exactly_of_type(e_expanded, mul)
 809          || is_ex_exactly_of_type(e_expanded, ncmul)
 810          || (is_ex_exactly_of_type(e_expanded, power) && is_ex_of_type(e_expanded.op(0), indexed) && e_expanded.op(1).is_equal(_ex2())))
 811                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
 812
 813         // Cannot do anything
 814         free_indices.clear();
 815         return e_expanded;
 816 }
 817
 818 /** Simplify/canonicalize expression containing indexed objects. This
 819  *  performs contraction of dummy indices where possible and checks whether
 820  *  the free indices in sums are consistent.
 821  *
 822  *  @return simplified expression */
 823 ex ex::simplify_indexed(void) const
 824 {
 825         exvector free_indices, dummy_indices;
 826         scalar_products sp;
 827         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 828 }
 829
 830 /** Simplify/canonicalize expression containing indexed objects. This
 831  *  performs contraction of dummy indices where possible, checks whether
 832  *  the free indices in sums are consistent, and automatically replaces
 833  *  scalar products by known values if desired.
 834  *
 835  *  @param sp Scalar products to be replaced automatically
 836  *  @return simplified expression */
 837 ex ex::simplify_indexed(const scalar_products & sp) const
 838 {
 839         exvector free_indices, dummy_indices;
 840         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 841 }
 842
 843 /** Symmetrize expression over its free indices. */
 844 ex ex::symmetrize(void) const
 845 {
 846         return GiNaC::symmetrize(*this, get_free_indices());
 847 }
 848
 849 /** Antisymmetrize expression over its free indices. */
 850 ex ex::antisymmetrize(void) const
 851 {
 852         return GiNaC::antisymmetrize(*this, get_free_indices());
 853 }
 854
 855 //////////
 856 // helper classes
 857 //////////
 858
 859 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
 860 {
 861         spm[make_key(v1, v2)] = sp;
 862 }
 863
 864 void scalar_products::add_vectors(const lst & l)
 865 {
 866         // Add all possible pairs of products
 867         unsigned num = l.nops();
 868         for (unsigned i=0; i<num; i++) {
 869                 ex a = l.op(i);
 870                 for (unsigned j=0; j<num; j++) {
 871                         ex b = l.op(j);
 872                         add(a, b, a*b);
 873                 }
 874         }
 875 }
 876
 877 void scalar_products::clear(void)
 878 {
 879         spm.clear();
 880 }
 881
 882 /** Check whether scalar product pair is defined. */
 883 bool scalar_products::is_defined(const ex & v1, const ex & v2) const
 884 {
 885         return spm.find(make_key(v1, v2)) != spm.end();
 886 }
 887
 888 /** Return value of defined scalar product pair. */
 889 ex scalar_products::evaluate(const ex & v1, const ex & v2) const
 890 {
 891         return spm.find(make_key(v1, v2))->second;
 892 }
 893
 894 void scalar_products::debugprint(void) const
 895 {
 896         std::cerr << "map size=" << spm.size() << std::endl;
 897         for (spmap::const_iterator cit=spm.begin(); cit!=spm.end(); ++cit) {
 898                 const spmapkey & k = cit->first;
 899                 std::cerr << "item key=(" << k.first << "," << k.second;
 900                 std::cerr << "), value=" << cit->second << std::endl;
 901         }
 902 }
 903
 904 /** Make key from object pair. */
 905 spmapkey scalar_products::make_key(const ex & v1, const ex & v2)
 906 {
 907         // If indexed, extract base objects
 908         ex s1 = is_ex_of_type(v1, indexed) ? v1.op(0) : v1;
 909         ex s2 = is_ex_of_type(v2, indexed) ? v2.op(0) : v2;
 910
 911         // Enforce canonical order in pair
 912         if (s1.compare(s2) > 0)
 913                 return spmapkey(s2, s1);
 914         else
 915                 return spmapkey(s1, s2);
 916 }
 917
 918 } // namespace GiNaC