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 bool indexed::all_index_values_are(unsigned inf) const
 229 {
 230         // No indices? Then no property can be fulfilled
 231         if (seq.size() < 2)
 232                 return false;
 233
 234         // Check all indices
 235         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 236         while (it != itend) {
 237                 GINAC_ASSERT(is_ex_of_type(*it, idx));
 238                 if (!ex_to_idx(*it).get_value().info(inf))
 239                         return false;
 240                 it++;
 241         }
 242         return true;
 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 != unknown && symmetry != mixed)) {
 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         // And also the indices themselves
 458         exvector::const_iterator ait = v1.begin(), aitend = v1.end(),
 459                                  bit = v2.begin(), bitend = v2.end();
 460         while (ait != aitend) {
 461                 if (!ait->is_equal(*bit))
 462                         return false;
 463                 ait++; bit++;
 464         }
 465         return true;
 466 }
 467
 468 exvector indexed::get_indices(void) const
 469 {
 470         GINAC_ASSERT(seq.size() >= 1);
 471         return exvector(seq.begin() + 1, seq.end());
 472 }
 473
 474 exvector indexed::get_dummy_indices(void) const
 475 {
 476         exvector free_indices, dummy_indices;
 477         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 478         return dummy_indices;
 479 }
 480
 481 exvector indexed::get_dummy_indices(const indexed & other) const
 482 {
 483         exvector indices = get_free_indices();
 484         exvector other_indices = other.get_free_indices();
 485         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 486         exvector dummy_indices;
 487         find_dummy_indices(indices, dummy_indices);
 488         return dummy_indices;
 489 }
 490
 491 exvector indexed::get_free_indices(void) const
 492 {
 493         exvector free_indices, dummy_indices;
 494         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 495         return free_indices;
 496 }
 497
 498 exvector add::get_free_indices(void) const
 499 {
 500         exvector free_indices;
 501         for (unsigned i=0; i<nops(); i++) {
 502                 if (i == 0)
 503                         free_indices = op(i).get_free_indices();
 504                 else {
 505                         exvector free_indices_of_term = op(i).get_free_indices();
 506                         if (!indices_consistent(free_indices, free_indices_of_term))
 507                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 508                 }
 509         }
 510         return free_indices;
 511 }
 512
 513 exvector mul::get_free_indices(void) const
 514 {
 515         // Concatenate free indices of all factors
 516         exvector un;
 517         for (unsigned i=0; i<nops(); i++) {
 518                 exvector free_indices_of_factor = op(i).get_free_indices();
 519                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 520         }
 521
 522         // And remove the dummy indices
 523         exvector free_indices, dummy_indices;
 524         find_free_and_dummy(un, free_indices, dummy_indices);
 525         return free_indices;
 526 }
 527
 528 exvector ncmul::get_free_indices(void) const
 529 {
 530         // Concatenate free indices of all factors
 531         exvector un;
 532         for (unsigned i=0; i<nops(); i++) {
 533                 exvector free_indices_of_factor = op(i).get_free_indices();
 534                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 535         }
 536
 537         // And remove the dummy indices
 538         exvector free_indices, dummy_indices;
 539         find_free_and_dummy(un, free_indices, dummy_indices);
 540         return free_indices;
 541 }
 542
 543 exvector power::get_free_indices(void) const
 544 {
 545         // Return free indices of basis
 546         return basis.get_free_indices();
 547 }
 548
 549 /* Function object for STL sort() */
 550 struct ex_is_less {
 551         bool operator() (const ex &lh, const ex &rh) const
 552         {
 553                 return lh.compare(rh) < 0;
 554         }
 555 };
 556
 557 /** Rename dummy indices in an expression.
 558  *
 559  *  @param e Expression to be worked on
 560  *  @param local_dummy_indices The set of dummy indices that appear in the
 561  *    expression "e"
 562  *  @param global_dummy_indices The set of dummy indices that have appeared
 563  *    before and which we would like to use in "e", too. This gets updated
 564  *    by the function */
 565 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 566 {
 567         int global_size = global_dummy_indices.size(),
 568             local_size = local_dummy_indices.size();
 569
 570         // Any local dummy indices at all?
 571         if (local_size == 0)
 572                 return e;
 573
 574         sort(local_dummy_indices.begin(), local_dummy_indices.end(), ex_is_less());
 575
 576         if (global_size < local_size) {
 577
 578                 // More local indices than we encountered before, add the new ones
 579                 // to the global set
 580                 int remaining = local_size - global_size;
 581                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 582                 while (it != itend && remaining > 0) {
 583                         exvector::const_iterator git = global_dummy_indices.begin(), gitend = global_dummy_indices.end();
 584                         while (git != gitend) {
 585                                 if (it->is_equal(*git))
 586                                         goto found;
 587                                 git++;
 588                         }
 589                         global_dummy_indices.push_back(*it);
 590                         global_size++;
 591                         remaining--;
 592 found:          it++;
 593                 }
 594                 sort(global_dummy_indices.begin(), global_dummy_indices.end(), ex_is_less());
 595         }
 596
 597         // Replace index symbols in expression
 598         GINAC_ASSERT(local_size <= global_size);
 599         bool all_equal = true;
 600         lst local_syms, global_syms;
 601         for (unsigned i=0; i<local_size; i++) {
 602                 ex loc_sym = local_dummy_indices[i].op(0);
 603                 ex glob_sym = global_dummy_indices[i].op(0);
 604                 if (!loc_sym.is_equal(glob_sym))
 605                         all_equal = false;
 606                 local_syms.append(loc_sym);
 607                 global_syms.append(glob_sym);
 608         }
 609         if (all_equal)
 610                 return e;
 611         else
 612                 return e.subs(local_syms, global_syms);
 613 }
 614
 615 /** Simplify product of indexed expressions (commutative, noncommutative and
 616  *  simple squares), return list of free indices. */
 617 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 618 {
 619         // Remember whether the product was commutative or noncommutative
 620         // (because we chop it into factors and need to reassemble later)
 621         bool non_commutative = is_ex_exactly_of_type(e, ncmul);
 622
 623         // Collect factors in an exvector, store squares twice
 624         exvector v;
 625         v.reserve(e.nops() * 2);
 626
 627         if (is_ex_exactly_of_type(e, power)) {
 628                 // We only get called for simple squares, split a^2 -> a*a
 629                 GINAC_ASSERT(e.op(1).is_equal(_ex2()));
 630                 v.push_back(e.op(0));
 631                 v.push_back(e.op(0));
 632         } else {
 633                 for (int i=0; i<e.nops(); i++) {
 634                         ex f = e.op(i);
 635                         if (is_ex_exactly_of_type(f, power) && f.op(1).is_equal(_ex2())) {
 636                                 v.push_back(f.op(0));
 637                     v.push_back(f.op(0));
 638                         } else if (is_ex_exactly_of_type(f, ncmul)) {
 639                                 // Noncommutative factor found, split it as well
 640                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 641                                 for (int j=0; j<f.nops(); j++)
 642                                         v.push_back(f.op(j));
 643                         } else
 644                                 v.push_back(f);
 645                 }
 646         }
 647
 648         // Perform contractions
 649         bool something_changed = false;
 650         GINAC_ASSERT(v.size() > 1);
 651         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 652         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 653
 654 try_again:
 655                 if (!is_ex_of_type(*it1, indexed))
 656                         continue;
 657
 658                 // Indexed factor found, get free indices and look for contraction
 659                 // candidates
 660                 exvector free1, dummy1;
 661                 find_free_and_dummy(ex_to_indexed(*it1).seq.begin() + 1, ex_to_indexed(*it1).seq.end(), free1, dummy1);
 662
 663                 exvector::iterator it2;
 664                 for (it2 = it1 + 1; it2 != itend; it2++) {
 665
 666                         if (!is_ex_of_type(*it2, indexed))
 667                                 continue;
 668
 669                         // Find free indices of second factor and merge them with free
 670                         // indices of first factor
 671                         exvector un;
 672                         find_free_and_dummy(ex_to_indexed(*it2).seq.begin() + 1, ex_to_indexed(*it2).seq.end(), un, dummy1);
 673                         un.insert(un.end(), free1.begin(), free1.end());
 674
 675                         // Check whether the two factors share dummy indices
 676                         exvector free, dummy;
 677                         find_free_and_dummy(un, free, dummy);
 678                         if (dummy.size() == 0)
 679                                 continue;
 680
 681                         // At least one dummy index, is it a defined scalar product?
 682                         bool contracted = false;
 683                         if (free.size() == 0) {
 684                                 if (sp.is_defined(*it1, *it2)) {
 685                                         *it1 = sp.evaluate(*it1, *it2);
 686                                         *it2 = _ex1();
 687                                         goto contraction_done;
 688                                 }
 689                         }
 690
 691                         // Contraction of symmetric with antisymmetric object is zero
 692                         if ((ex_to_indexed(*it1).symmetry == indexed::symmetric &&
 693                              ex_to_indexed(*it2).symmetry == indexed::antisymmetric
 694                           || ex_to_indexed(*it1).symmetry == indexed::antisymmetric &&
 695                              ex_to_indexed(*it2).symmetry == indexed::symmetric)
 696                          && dummy.size() > 1) {
 697                                 free_indices.clear();
 698                                 return _ex0();
 699                         }
 700
 701                         // Try to contract the first one with the second one
 702                         contracted = it1->op(0).bp->contract_with(it1, it2, v);
 703                         if (!contracted) {
 704
 705                                 // That didn't work; maybe the second object knows how to
 706                                 // contract itself with the first one
 707                                 contracted = it2->op(0).bp->contract_with(it2, it1, v);
 708                         }
 709                         if (contracted) {
 710 contraction_done:
 711                                 if (non_commutative
 712                                  || is_ex_exactly_of_type(*it1, add) || is_ex_exactly_of_type(*it2, add)
 713                                  || is_ex_exactly_of_type(*it1, mul) || is_ex_exactly_of_type(*it2, mul)
 714                                  || is_ex_exactly_of_type(*it1, ncmul) || is_ex_exactly_of_type(*it2, ncmul)) {
 715
 716                                         // One of the factors became a sum or product:
 717                                         // re-expand expression and run again
 718                                         // Non-commutative products are always re-expanded to give
 719                                         // simplify_ncmul() the chance to re-order and canonicalize
 720                                         // the product
 721                                         ex r = (non_commutative ? ex(ncmul(v)) : ex(mul(v)));
 722                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 723                                 }
 724
 725                                 // Both objects may have new indices now or they might
 726                                 // even not be indexed objects any more, so we have to
 727                                 // start over
 728                                 something_changed = true;
 729                                 goto try_again;
 730                         }
 731                 }
 732         }
 733
 734         // Find free indices (concatenate them all and call find_free_and_dummy())
 735         exvector un, local_dummy_indices;
 736         it1 = v.begin(); itend = v.end();
 737         while (it1 != itend) {
 738                 exvector 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         find_free_and_dummy(un, free_indices, local_dummy_indices);
 743
 744         ex r;
 745         if (something_changed)
 746                 r = non_commutative ? ex(ncmul(v)) : ex(mul(v));
 747         else
 748                 r = e;
 749
 750         // Dummy index renaming
 751         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 752
 753         // Product of indexed object with a scalar?
 754         if (is_ex_exactly_of_type(r, mul) && r.nops() == 2
 755          && is_ex_exactly_of_type(r.op(1), numeric) && is_ex_of_type(r.op(0), indexed))
 756                 return r.op(0).op(0).bp->scalar_mul_indexed(r.op(0), ex_to_numeric(r.op(1)));
 757         else
 758                 return r;
 759 }
 760
 761 /** Simplify indexed expression, return list of free indices. */
 762 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 763 {
 764         // Expand the expression
 765         ex e_expanded = e.expand();
 766
 767         // Simplification of single indexed object: just find the free indices
 768         // (and perform dummy index renaming if
 769         if (is_ex_of_type(e_expanded, indexed)) {
 770                 const indexed &i = ex_to_indexed(e_expanded);
 771                 exvector local_dummy_indices;
 772                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 773                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 774         }
 775
 776         // Simplification of sum = sum of simplifications, check consistency of
 777         // free indices in each term
 778         if (is_ex_exactly_of_type(e_expanded, add)) {
 779                 bool first = true;
 780                 ex sum = _ex0();
 781                 free_indices.clear();
 782
 783                 for (unsigned i=0; i<e_expanded.nops(); i++) {
 784                         exvector free_indices_of_term;
 785                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 786                         if (!term.is_zero()) {
 787                                 if (first) {
 788                                         free_indices = free_indices_of_term;
 789                                         sum = term;
 790                                         first = false;
 791                                 } else {
 792                                         if (!indices_consistent(free_indices, free_indices_of_term))
 793                                                 throw (std::runtime_error("simplify_indexed: inconsistent indices in sum"));
 794                                         if (is_ex_of_type(sum, indexed) && is_ex_of_type(term, indexed))
 795                                                 sum = sum.op(0).bp->add_indexed(sum, term);
 796                                         else
 797                                                 sum += term;
 798                                 }
 799                         }
 800                 }
 801
 802                 return sum;
 803         }
 804
 805         // Simplification of products
 806         if (is_ex_exactly_of_type(e_expanded, mul)
 807          || is_ex_exactly_of_type(e_expanded, ncmul)
 808          || (is_ex_exactly_of_type(e_expanded, power) && is_ex_of_type(e_expanded.op(0), indexed) && e_expanded.op(1).is_equal(_ex2())))
 809                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
 810
 811         // Cannot do anything
 812         free_indices.clear();
 813         return e_expanded;
 814 }
 815
 816 ex simplify_indexed(const ex & e)
 817 {
 818         exvector free_indices, dummy_indices;
 819         scalar_products sp;
 820         return simplify_indexed(e, free_indices, dummy_indices, sp);
 821 }
 822
 823 ex simplify_indexed(const ex & e, const scalar_products & sp)
 824 {
 825         exvector free_indices, dummy_indices;
 826         return simplify_indexed(e, free_indices, dummy_indices, sp);
 827 }
 828
 829 //////////
 830 // helper classes
 831 //////////
 832
 833 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
 834 {
 835         spm[make_key(v1, v2)] = sp;
 836 }
 837
 838 void scalar_products::add_vectors(const lst & l)
 839 {
 840         // Add all possible pairs of products
 841         unsigned num = l.nops();
 842         for (unsigned i=0; i<num; i++) {
 843                 ex a = l.op(i);
 844                 for (unsigned j=0; j<num; j++) {
 845                         ex b = l.op(j);
 846                         add(a, b, a*b);
 847                 }
 848         }
 849 }
 850
 851 void scalar_products::clear(void)
 852 {
 853         spm.clear();
 854 }
 855
 856 /** Check whether scalar product pair is defined. */
 857 bool scalar_products::is_defined(const ex & v1, const ex & v2) const
 858 {
 859         return spm.find(make_key(v1, v2)) != spm.end();
 860 }
 861
 862 /** Return value of defined scalar product pair. */
 863 ex scalar_products::evaluate(const ex & v1, const ex & v2) const
 864 {
 865         return spm.find(make_key(v1, v2))->second;
 866 }
 867
 868 void scalar_products::debugprint(void) const
 869 {
 870         std::cerr << "map size=" << spm.size() << std::endl;
 871         for (spmap::const_iterator cit=spm.begin(); cit!=spm.end(); ++cit) {
 872                 const spmapkey & k = cit->first;
 873                 std::cerr << "item key=(" << k.first << "," << k.second;
 874                 std::cerr << "), value=" << cit->second << std::endl;
 875         }
 876 }
 877
 878 /** Make key from object pair. */
 879 spmapkey scalar_products::make_key(const ex & v1, const ex & v2)
 880 {
 881         // If indexed, extract base objects
 882         ex s1 = is_ex_of_type(v1, indexed) ? v1.op(0) : v1;
 883         ex s2 = is_ex_of_type(v2, indexed) ? v2.op(0) : v2;
 884
 885         // Enforce canonical order in pair
 886         if (s1.compare(s2) > 0)
 887                 return spmapkey(s2, s1);
 888         else
 889                 return spmapkey(s1, s2);
 890 }
 891
 892 } // namespace GiNaC