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