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 <iostream>
  24 #include <stdexcept>
  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 "symmetry.h"
  33 #include "lst.h"
  34 #include "print.h"
  35 #include "archive.h"
  36 #include "utils.h"
  37
  38 namespace GiNaC {
  39
  40 GINAC_IMPLEMENT_REGISTERED_CLASS(indexed, exprseq)
  41
  42 //////////
  43 // default ctor, dtor, copy ctor, assignment operator and helpers
  44 //////////
  45
  46 indexed::indexed() : symtree(sy_none())
  47 {
  48         tinfo_key = TINFO_indexed;
  49 }
  50
  51 void indexed::copy(const indexed & other)
  52 {
  53         inherited::copy(other);
  54         symtree = other.symtree;
  55 }
  56
  57 DEFAULT_DESTROY(indexed)
  58
  59 //////////
  60 // other constructors
  61 //////////
  62
  63 indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
  64 {
  65         tinfo_key = TINFO_indexed;
  66         validate();
  67 }
  68
  69 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(sy_none())
  70 {
  71         tinfo_key = TINFO_indexed;
  72         validate();
  73 }
  74
  75 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(sy_none())
  76 {
  77         tinfo_key = TINFO_indexed;
  78         validate();
  79 }
  80
  81 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(sy_none())
  82 {
  83         tinfo_key = TINFO_indexed;
  84         validate();
  85 }
  86
  87 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(sy_none())
  88 {
  89         tinfo_key = TINFO_indexed;
  90         validate();
  91 }
  92
  93 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
  94 {
  95         tinfo_key = TINFO_indexed;
  96         validate();
  97 }
  98
  99 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
 100 {
 101         tinfo_key = TINFO_indexed;
 102         validate();
 103 }
 104
 105 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(symm)
 106 {
 107         tinfo_key = TINFO_indexed;
 108         validate();
 109 }
 110
 111 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
 112 {
 113         seq.insert(seq.end(), v.begin(), v.end());
 114         tinfo_key = TINFO_indexed;
 115         validate();
 116 }
 117
 118 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 119 {
 120         seq.insert(seq.end(), v.begin(), v.end());
 121         tinfo_key = TINFO_indexed;
 122         validate();
 123 }
 124
 125 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 126 {
 127         tinfo_key = TINFO_indexed;
 128 }
 129
 130 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 131 {
 132         tinfo_key = TINFO_indexed;
 133 }
 134
 135 indexed::indexed(const symmetry & symm, exvector * vp) : inherited(vp), symtree(symm)
 136 {
 137         tinfo_key = TINFO_indexed;
 138 }
 139
 140 //////////
 141 // archiving
 142 //////////
 143
 144 indexed::indexed(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 145 {
 146         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 147                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 148                 unsigned symm = 0;
 149                 n.find_unsigned("symmetry", symm);
 150                 switch (symm) {
 151                         case 1:
 152                                 symtree = sy_symm();
 153                                 break;
 154                         case 2:
 155                                 symtree = sy_anti();
 156                                 break;
 157                         default:
 158                                 symtree = sy_none();
 159                                 break;
 160                 }
 161                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 162         }
 163 }
 164
 165 void indexed::archive(archive_node &n) const
 166 {
 167         inherited::archive(n);
 168         n.add_ex("symmetry", symtree);
 169 }
 170
 171 DEFAULT_UNARCHIVE(indexed)
 172
 173 //////////
 174 // functions overriding virtual functions from base classes
 175 //////////
 176
 177 void indexed::print(const print_context & c, unsigned level) const
 178 {
 179         GINAC_ASSERT(seq.size() > 0);
 180
 181         if (is_of_type(c, print_tree)) {
 182
 183                 c.s << std::string(level, ' ') << class_name()
 184                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 185                     << ", " << seq.size()-1 << " indices"
 186                     << ", symmetry=" << symtree << std::endl;
 187                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 188                 seq[0].print(c, level + delta_indent);
 189                 printindices(c, level + delta_indent);
 190
 191         } else {
 192
 193                 bool is_tex = is_of_type(c, print_latex);
 194                 const ex & base = seq[0];
 195                 bool need_parens = is_ex_exactly_of_type(base, add) || is_ex_exactly_of_type(base, mul)
 196                                 || is_ex_exactly_of_type(base, ncmul) || is_ex_exactly_of_type(base, power)
 197                                 || is_ex_of_type(base, indexed);
 198                 if (is_tex)
 199                         c.s << "{";
 200                 if (need_parens)
 201                         c.s << "(";
 202                 base.print(c);
 203                 if (need_parens)
 204                         c.s << ")";
 205                 if (is_tex)
 206                         c.s << "}";
 207                 printindices(c, level);
 208         }
 209 }
 210
 211 bool indexed::info(unsigned inf) const
 212 {
 213         if (inf == info_flags::indexed) return true;
 214         if (inf == info_flags::has_indices) return seq.size() > 1;
 215         return inherited::info(inf);
 216 }
 217
 218 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 219         bool operator() (const ex & e, unsigned inf) const {
 220                 return !(ex_to<idx>(e).get_value().info(inf));
 221         }
 222 };
 223
 224 bool indexed::all_index_values_are(unsigned inf) const
 225 {
 226         // No indices? Then no property can be fulfilled
 227         if (seq.size() < 2)
 228                 return false;
 229
 230         // Check all indices
 231         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 232 }
 233
 234 int indexed::compare_same_type(const basic & other) const
 235 {
 236         GINAC_ASSERT(is_a<indexed>(other));
 237         return inherited::compare_same_type(other);
 238 }
 239
 240 ex indexed::eval(int level) const
 241 {
 242         // First evaluate children, then we will end up here again
 243         if (level > 1)
 244                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 245
 246         const ex &base = seq[0];
 247
 248         // If the base object is 0, the whole object is 0
 249         if (base.is_zero())
 250                 return _ex0;
 251
 252         // If the base object is a product, pull out the numeric factor
 253         if (is_ex_exactly_of_type(base, mul) && is_ex_exactly_of_type(base.op(base.nops() - 1), numeric)) {
 254                 exvector v(seq);
 255                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 256                 v[0] = seq[0] / f;
 257                 return f * thisexprseq(v);
 258         }
 259
 260         // Canonicalize indices according to the symmetry properties
 261         if (seq.size() > 2) {
 262                 exvector v = seq;
 263                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 264                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 265                 if (sig != INT_MAX) {
 266                         // Something has changed while sorting indices, more evaluations later
 267                         if (sig == 0)
 268                                 return _ex0;
 269                         return ex(sig) * thisexprseq(v);
 270                 }
 271         }
 272
 273         // Let the class of the base object perform additional evaluations
 274         return ex_to<basic>(base).eval_indexed(*this);
 275 }
 276
 277 int indexed::degree(const ex & s) const
 278 {
 279         return is_equal(ex_to<basic>(s)) ? 1 : 0;
 280 }
 281
 282 int indexed::ldegree(const ex & s) const
 283 {
 284         return is_equal(ex_to<basic>(s)) ? 1 : 0;
 285 }
 286
 287 ex indexed::coeff(const ex & s, int n) const
 288 {
 289         if (is_equal(ex_to<basic>(s)))
 290                 return n==1 ? _ex1 : _ex0;
 291         else
 292                 return n==0 ? ex(*this) : _ex0;
 293 }
 294
 295 ex indexed::thisexprseq(const exvector & v) const
 296 {
 297         return indexed(ex_to<symmetry>(symtree), v);
 298 }
 299
 300 ex indexed::thisexprseq(exvector * vp) const
 301 {
 302         return indexed(ex_to<symmetry>(symtree), vp);
 303 }
 304
 305 ex indexed::expand(unsigned options) const
 306 {
 307         GINAC_ASSERT(seq.size() > 0);
 308
 309         if ((options & expand_options::expand_indexed) && is_ex_exactly_of_type(seq[0], add)) {
 310
 311                 // expand_indexed expands (a+b).i -> a.i + b.i
 312                 const ex & base = seq[0];
 313                 ex sum = _ex0;
 314                 for (unsigned i=0; i<base.nops(); i++) {
 315                         exvector s = seq;
 316                         s[0] = base.op(i);
 317                         sum += thisexprseq(s).expand();
 318                 }
 319                 return sum;
 320
 321         } else
 322                 return inherited::expand(options);
 323 }
 324
 325 //////////
 326 // virtual functions which can be overridden by derived classes
 327 //////////
 328
 329 // none
 330
 331 //////////
 332 // non-virtual functions in this class
 333 //////////
 334
 335 void indexed::printindices(const print_context & c, unsigned level) const
 336 {
 337         if (seq.size() > 1) {
 338
 339                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 340
 341                 if (is_of_type(c, print_latex)) {
 342
 343                         // TeX output: group by variance
 344                         bool first = true;
 345                         bool covariant = true;
 346
 347                         while (it != itend) {
 348                                 bool cur_covariant = (is_ex_of_type(*it, varidx) ? ex_to<varidx>(*it).is_covariant() : true);
 349                                 if (first || cur_covariant != covariant) {
 350                                         if (!first)
 351                                                 c.s << "}";
 352                                         covariant = cur_covariant;
 353                                         if (covariant)
 354                                                 c.s << "_{";
 355                                         else
 356                                                 c.s << "^{";
 357                                 }
 358                                 it->print(c, level);
 359                                 c.s << " ";
 360                                 first = false;
 361                                 it++;
 362                         }
 363                         c.s << "}";
 364
 365                 } else {
 366
 367                         // Ordinary output
 368                         while (it != itend) {
 369                                 it->print(c, level);
 370                                 it++;
 371                         }
 372                 }
 373         }
 374 }
 375
 376 /** Check whether all indices are of class idx and validate the symmetry
 377  *  tree. This function is used internally to make sure that all constructed
 378  *  indexed objects really carry indices and not some other classes. */
 379 void indexed::validate(void) const
 380 {
 381         GINAC_ASSERT(seq.size() > 0);
 382         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 383         while (it != itend) {
 384                 if (!is_ex_of_type(*it, idx))
 385                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 386                 it++;
 387         }
 388
 389         if (!symtree.is_zero()) {
 390                 if (!is_ex_exactly_of_type(symtree, symmetry))
 391                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 392                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 393         }
 394 }
 395
 396 /** Implementation of ex::diff() for an indexed object always returns 0.
 397  *
 398  *  @see ex::diff */
 399 ex indexed::derivative(const symbol & s) const
 400 {
 401         return _ex0;
 402 }
 403
 404 //////////
 405 // global functions
 406 //////////
 407
 408 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 409 static bool indices_consistent(const exvector & v1, const exvector & v2)
 410 {
 411         // Number of indices must be the same
 412         if (v1.size() != v2.size())
 413                 return false;
 414
 415         return equal(v1.begin(), v1.end(), v2.begin(), ex_is_equal());
 416 }
 417
 418 exvector indexed::get_indices(void) const
 419 {
 420         GINAC_ASSERT(seq.size() >= 1);
 421         return exvector(seq.begin() + 1, seq.end());
 422 }
 423
 424 exvector indexed::get_dummy_indices(void) const
 425 {
 426         exvector free_indices, dummy_indices;
 427         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 428         return dummy_indices;
 429 }
 430
 431 exvector indexed::get_dummy_indices(const indexed & other) const
 432 {
 433         exvector indices = get_free_indices();
 434         exvector other_indices = other.get_free_indices();
 435         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 436         exvector dummy_indices;
 437         find_dummy_indices(indices, dummy_indices);
 438         return dummy_indices;
 439 }
 440
 441 bool indexed::has_dummy_index_for(const ex & i) const
 442 {
 443         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 444         while (it != itend) {
 445                 if (is_dummy_pair(*it, i))
 446                         return true;
 447                 it++;
 448         }
 449         return false;
 450 }
 451
 452 exvector indexed::get_free_indices(void) const
 453 {
 454         exvector free_indices, dummy_indices;
 455         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 456         return free_indices;
 457 }
 458
 459 exvector add::get_free_indices(void) const
 460 {
 461         exvector free_indices;
 462         for (unsigned i=0; i<nops(); i++) {
 463                 if (i == 0)
 464                         free_indices = op(i).get_free_indices();
 465                 else {
 466                         exvector free_indices_of_term = op(i).get_free_indices();
 467                         if (!indices_consistent(free_indices, free_indices_of_term))
 468                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 469                 }
 470         }
 471         return free_indices;
 472 }
 473
 474 exvector mul::get_free_indices(void) const
 475 {
 476         // Concatenate free indices of all factors
 477         exvector un;
 478         for (unsigned i=0; i<nops(); i++) {
 479                 exvector free_indices_of_factor = op(i).get_free_indices();
 480                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 481         }
 482
 483         // And remove the dummy indices
 484         exvector free_indices, dummy_indices;
 485         find_free_and_dummy(un, free_indices, dummy_indices);
 486         return free_indices;
 487 }
 488
 489 exvector ncmul::get_free_indices(void) const
 490 {
 491         // Concatenate free indices of all factors
 492         exvector un;
 493         for (unsigned i=0; i<nops(); i++) {
 494                 exvector free_indices_of_factor = op(i).get_free_indices();
 495                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 496         }
 497
 498         // And remove the dummy indices
 499         exvector free_indices, dummy_indices;
 500         find_free_and_dummy(un, free_indices, dummy_indices);
 501         return free_indices;
 502 }
 503
 504 exvector power::get_free_indices(void) const
 505 {
 506         // Return free indices of basis
 507         return basis.get_free_indices();
 508 }
 509
 510 /** Rename dummy indices in an expression.
 511  *
 512  *  @param e Expression to be worked on
 513  *  @param local_dummy_indices The set of dummy indices that appear in the
 514  *    expression "e"
 515  *  @param global_dummy_indices The set of dummy indices that have appeared
 516  *    before and which we would like to use in "e", too. This gets updated
 517  *    by the function */
 518 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 519 {
 520         unsigned global_size = global_dummy_indices.size(),
 521                  local_size = local_dummy_indices.size();
 522
 523         // Any local dummy indices at all?
 524         if (local_size == 0)
 525                 return e;
 526
 527         if (global_size < local_size) {
 528
 529                 // More local indices than we encountered before, add the new ones
 530                 // to the global set
 531                 int old_global_size = global_size;
 532                 int remaining = local_size - global_size;
 533                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 534                 while (it != itend && remaining > 0) {
 535                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(ex_is_equal(), *it)) == global_dummy_indices.end()) {
 536                                 global_dummy_indices.push_back(*it);
 537                                 global_size++;
 538                                 remaining--;
 539                         }
 540                         it++;
 541                 }
 542                 shaker_sort(global_dummy_indices.begin(), global_dummy_indices.end(), ex_is_less(), ex_swap());
 543
 544                 // If this is the first set of local indices, do nothing
 545                 if (old_global_size == 0)
 546                         return e;
 547         }
 548         GINAC_ASSERT(local_size <= global_size);
 549
 550         // Construct lists of index symbols
 551         exlist local_syms, global_syms;
 552         for (unsigned i=0; i<local_size; i++)
 553                 local_syms.push_back(local_dummy_indices[i].op(0));
 554         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 555         for (unsigned i=0; i<global_size; i++)
 556                 global_syms.push_back(global_dummy_indices[i].op(0));
 557
 558         // Remove common indices
 559         exlist local_uniq, global_uniq;
 560         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exlist>(local_uniq), ex_is_less());
 561         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exlist>(global_uniq), ex_is_less());
 562
 563         // Replace remaining non-common local index symbols by global ones
 564         if (local_uniq.empty())
 565                 return e;
 566         else {
 567                 while (global_uniq.size() > local_uniq.size())
 568                         global_uniq.pop_back();
 569                 return e.subs(lst(local_uniq), lst(global_uniq));
 570         }
 571 }
 572
 573 /** Simplify product of indexed expressions (commutative, noncommutative and
 574  *  simple squares), return list of free indices. */
 575 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 576 {
 577         // Remember whether the product was commutative or noncommutative
 578         // (because we chop it into factors and need to reassemble later)
 579         bool non_commutative = is_ex_exactly_of_type(e, ncmul);
 580
 581         // Collect factors in an exvector, store squares twice
 582         exvector v;
 583         v.reserve(e.nops() * 2);
 584
 585         if (is_ex_exactly_of_type(e, power)) {
 586                 // We only get called for simple squares, split a^2 -> a*a
 587                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 588                 v.push_back(e.op(0));
 589                 v.push_back(e.op(0));
 590         } else {
 591                 for (unsigned i=0; i<e.nops(); i++) {
 592                         ex f = e.op(i);
 593                         if (is_ex_exactly_of_type(f, power) && f.op(1).is_equal(_ex2)) {
 594                                 v.push_back(f.op(0));
 595                     v.push_back(f.op(0));
 596                         } else if (is_ex_exactly_of_type(f, ncmul)) {
 597                                 // Noncommutative factor found, split it as well
 598                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 599                                 for (unsigned j=0; j<f.nops(); j++)
 600                                         v.push_back(f.op(j));
 601                         } else
 602                                 v.push_back(f);
 603                 }
 604         }
 605
 606         // Perform contractions
 607         bool something_changed = false;
 608         GINAC_ASSERT(v.size() > 1);
 609         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 610         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 611
 612 try_again:
 613                 if (!is_ex_of_type(*it1, indexed))
 614                         continue;
 615
 616                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 617
 618                 // Indexed factor found, get free indices and look for contraction
 619                 // candidates
 620                 exvector free1, dummy1;
 621                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 622
 623                 exvector::iterator it2;
 624                 for (it2 = it1 + 1; it2 != itend; it2++) {
 625
 626                         if (!is_ex_of_type(*it2, indexed))
 627                                 continue;
 628
 629                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 630
 631                         // Find free indices of second factor and merge them with free
 632                         // indices of first factor
 633                         exvector un;
 634                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 635                         un.insert(un.end(), free1.begin(), free1.end());
 636
 637                         // Check whether the two factors share dummy indices
 638                         exvector free, dummy;
 639                         find_free_and_dummy(un, free, dummy);
 640                         unsigned num_dummies = dummy.size();
 641                         if (num_dummies == 0)
 642                                 continue;
 643
 644                         // At least one dummy index, is it a defined scalar product?
 645                         bool contracted = false;
 646                         if (free.empty()) {
 647                                 if (sp.is_defined(*it1, *it2)) {
 648                                         *it1 = sp.evaluate(*it1, *it2);
 649                                         *it2 = _ex1;
 650                                         goto contraction_done;
 651                                 }
 652                         }
 653
 654                         // Try to contract the first one with the second one
 655                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 656                         if (!contracted) {
 657
 658                                 // That didn't work; maybe the second object knows how to
 659                                 // contract itself with the first one
 660                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 661                         }
 662                         if (contracted) {
 663 contraction_done:
 664                                 if (first_noncommutative || second_noncommutative
 665                                  || is_ex_exactly_of_type(*it1, add) || is_ex_exactly_of_type(*it2, add)
 666                                  || is_ex_exactly_of_type(*it1, mul) || is_ex_exactly_of_type(*it2, mul)
 667                                  || is_ex_exactly_of_type(*it1, ncmul) || is_ex_exactly_of_type(*it2, ncmul)) {
 668
 669                                         // One of the factors became a sum or product:
 670                                         // re-expand expression and run again
 671                                         // Non-commutative products are always re-expanded to give
 672                                         // simplify_ncmul() the chance to re-order and canonicalize
 673                                         // the product
 674                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 675                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 676                                 }
 677
 678                                 // Both objects may have new indices now or they might
 679                                 // even not be indexed objects any more, so we have to
 680                                 // start over
 681                                 something_changed = true;
 682                                 goto try_again;
 683                         }
 684                 }
 685         }
 686
 687         // Find free indices (concatenate them all and call find_free_and_dummy())
 688         // and all dummy indices that appear
 689         exvector un, individual_dummy_indices;
 690         it1 = v.begin(); itend = v.end();
 691         while (it1 != itend) {
 692                 exvector free_indices_of_factor;
 693                 if (is_ex_of_type(*it1, indexed)) {
 694                         exvector dummy_indices_of_factor;
 695                         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);
 696                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 697                 } else
 698                         free_indices_of_factor = it1->get_free_indices();
 699                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 700                 it1++;
 701         }
 702         exvector local_dummy_indices;
 703         find_free_and_dummy(un, free_indices, local_dummy_indices);
 704         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 705
 706         ex r;
 707         if (something_changed)
 708                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 709         else
 710                 r = e;
 711
 712         // The result should be symmetric with respect to exchange of dummy
 713         // indices, so if the symmetrization vanishes, the whole expression is
 714         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 715         if (local_dummy_indices.size() >= 2) {
 716                 lst dummy_syms;
 717                 for (int i=0; i<local_dummy_indices.size(); i++)
 718                         dummy_syms.append(local_dummy_indices[i].op(0));
 719                 if (r.symmetrize(dummy_syms).is_zero()) {
 720                         free_indices.clear();
 721                         return _ex0;
 722                 }
 723         }
 724
 725         // Dummy index renaming
 726         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 727
 728         // Product of indexed object with a scalar?
 729         if (is_ex_exactly_of_type(r, mul) && r.nops() == 2
 730          && is_ex_exactly_of_type(r.op(1), numeric) && is_ex_of_type(r.op(0), indexed))
 731                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 732         else
 733                 return r;
 734 }
 735
 736 /** Simplify indexed expression, return list of free indices. */
 737 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 738 {
 739         // Expand the expression
 740         ex e_expanded = e.expand();
 741
 742         // Simplification of single indexed object: just find the free indices
 743         // and perform dummy index renaming
 744         if (is_ex_of_type(e_expanded, indexed)) {
 745                 const indexed &i = ex_to<indexed>(e_expanded);
 746                 exvector local_dummy_indices;
 747                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 748                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 749         }
 750
 751         // Simplification of sum = sum of simplifications, check consistency of
 752         // free indices in each term
 753         if (is_ex_exactly_of_type(e_expanded, add)) {
 754                 bool first = true;
 755                 ex sum = _ex0;
 756                 free_indices.clear();
 757
 758                 for (unsigned i=0; i<e_expanded.nops(); i++) {
 759                         exvector free_indices_of_term;
 760                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 761                         if (!term.is_zero()) {
 762                                 if (first) {
 763                                         free_indices = free_indices_of_term;
 764                                         sum = term;
 765                                         first = false;
 766                                 } else {
 767                                         if (!indices_consistent(free_indices, free_indices_of_term))
 768                                                 throw (std::runtime_error("simplify_indexed: inconsistent indices in sum"));
 769                                         if (is_ex_of_type(sum, indexed) && is_ex_of_type(term, indexed))
 770                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
 771                                         else
 772                                                 sum += term;
 773                                 }
 774                         }
 775                 }
 776
 777                 return sum;
 778         }
 779
 780         // Simplification of products
 781         if (is_ex_exactly_of_type(e_expanded, mul)
 782          || is_ex_exactly_of_type(e_expanded, ncmul)
 783          || (is_ex_exactly_of_type(e_expanded, power) && is_ex_of_type(e_expanded.op(0), indexed) && e_expanded.op(1).is_equal(_ex2)))
 784                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
 785
 786         // Cannot do anything
 787         free_indices.clear();
 788         return e_expanded;
 789 }
 790
 791 /** Simplify/canonicalize expression containing indexed objects. This
 792  *  performs contraction of dummy indices where possible and checks whether
 793  *  the free indices in sums are consistent.
 794  *
 795  *  @return simplified expression */
 796 ex ex::simplify_indexed(void) const
 797 {
 798         exvector free_indices, dummy_indices;
 799         scalar_products sp;
 800         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 801 }
 802
 803 /** Simplify/canonicalize expression containing indexed objects. This
 804  *  performs contraction of dummy indices where possible, checks whether
 805  *  the free indices in sums are consistent, and automatically replaces
 806  *  scalar products by known values if desired.
 807  *
 808  *  @param sp Scalar products to be replaced automatically
 809  *  @return simplified expression */
 810 ex ex::simplify_indexed(const scalar_products & sp) const
 811 {
 812         exvector free_indices, dummy_indices;
 813         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 814 }
 815
 816 /** Symmetrize expression over its free indices. */
 817 ex ex::symmetrize(void) const
 818 {
 819         return GiNaC::symmetrize(*this, get_free_indices());
 820 }
 821
 822 /** Antisymmetrize expression over its free indices. */
 823 ex ex::antisymmetrize(void) const
 824 {
 825         return GiNaC::antisymmetrize(*this, get_free_indices());
 826 }
 827
 828 /** Symmetrize expression by cyclic permutation over its free indices. */
 829 ex ex::symmetrize_cyclic(void) const
 830 {
 831         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
 832 }
 833
 834 //////////
 835 // helper classes
 836 //////////
 837
 838 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
 839 {
 840         spm[make_key(v1, v2)] = sp;
 841 }
 842
 843 void scalar_products::add_vectors(const lst & l)
 844 {
 845         // Add all possible pairs of products
 846         unsigned num = l.nops();
 847         for (unsigned i=0; i<num; i++) {
 848                 ex a = l.op(i);
 849                 for (unsigned j=0; j<num; j++) {
 850                         ex b = l.op(j);
 851                         add(a, b, a*b);
 852                 }
 853         }
 854 }
 855
 856 void scalar_products::clear(void)
 857 {
 858         spm.clear();
 859 }
 860
 861 /** Check whether scalar product pair is defined. */
 862 bool scalar_products::is_defined(const ex & v1, const ex & v2) const
 863 {
 864         return spm.find(make_key(v1, v2)) != spm.end();
 865 }
 866
 867 /** Return value of defined scalar product pair. */
 868 ex scalar_products::evaluate(const ex & v1, const ex & v2) const
 869 {
 870         return spm.find(make_key(v1, v2))->second;
 871 }
 872
 873 void scalar_products::debugprint(void) const
 874 {
 875         std::cerr << "map size=" << spm.size() << std::endl;
 876         spmap::const_iterator i = spm.begin(), end = spm.end();
 877         while (i != end) {
 878                 const spmapkey & k = i->first;
 879                 std::cerr << "item key=(" << k.first << "," << k.second;
 880                 std::cerr << "), value=" << i->second << std::endl;
 881                 ++i;
 882         }
 883 }
 884
 885 /** Make key from object pair. */
 886 spmapkey scalar_products::make_key(const ex & v1, const ex & v2)
 887 {
 888         // If indexed, extract base objects
 889         ex s1 = is_ex_of_type(v1, indexed) ? v1.op(0) : v1;
 890         ex s2 = is_ex_of_type(v2, indexed) ? v2.op(0) : v2;
 891
 892         // Enforce canonical order in pair
 893         if (s1.compare(s2) > 0)
 894                 return spmapkey(s2, s1);
 895         else
 896                 return spmapkey(s1, s2);
 897 }
 898
 899 } // namespace GiNaC