ginac/indexed.cpp

   1 /** @file indexed.cpp
   2  *
   3  *  Implementation of GiNaC's indexed expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2002 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 "relational.h"
  33 #include "symmetry.h"
  34 #include "lst.h"
  35 #include "print.h"
  36 #include "archive.h"
  37 #include "utils.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(indexed, exprseq)
  42
  43 //////////
  44 // default ctor, dtor, copy ctor, assignment operator and helpers
  45 //////////
  46
  47 indexed::indexed() : symtree(sy_none())
  48 {
  49         tinfo_key = TINFO_indexed;
  50 }
  51
  52 void indexed::copy(const indexed & other)
  53 {
  54         inherited::copy(other);
  55         symtree = other.symtree;
  56 }
  57
  58 DEFAULT_DESTROY(indexed)
  59
  60 //////////
  61 // other constructors
  62 //////////
  63
  64 indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
  65 {
  66         tinfo_key = TINFO_indexed;
  67         validate();
  68 }
  69
  70 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(sy_none())
  71 {
  72         tinfo_key = TINFO_indexed;
  73         validate();
  74 }
  75
  76 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(sy_none())
  77 {
  78         tinfo_key = TINFO_indexed;
  79         validate();
  80 }
  81
  82 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(sy_none())
  83 {
  84         tinfo_key = TINFO_indexed;
  85         validate();
  86 }
  87
  88 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())
  89 {
  90         tinfo_key = TINFO_indexed;
  91         validate();
  92 }
  93
  94 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
  95 {
  96         tinfo_key = TINFO_indexed;
  97         validate();
  98 }
  99
 100 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
 101 {
 102         tinfo_key = TINFO_indexed;
 103         validate();
 104 }
 105
 106 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)
 107 {
 108         tinfo_key = TINFO_indexed;
 109         validate();
 110 }
 111
 112 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
 113 {
 114         seq.insert(seq.end(), v.begin(), v.end());
 115         tinfo_key = TINFO_indexed;
 116         validate();
 117 }
 118
 119 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 120 {
 121         seq.insert(seq.end(), v.begin(), v.end());
 122         tinfo_key = TINFO_indexed;
 123         validate();
 124 }
 125
 126 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 127 {
 128         tinfo_key = TINFO_indexed;
 129 }
 130
 131 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 132 {
 133         tinfo_key = TINFO_indexed;
 134 }
 135
 136 indexed::indexed(const symmetry & symm, exvector * vp) : inherited(vp), symtree(symm)
 137 {
 138         tinfo_key = TINFO_indexed;
 139 }
 140
 141 //////////
 142 // archiving
 143 //////////
 144
 145 indexed::indexed(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 146 {
 147         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 148                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 149                 unsigned symm = 0;
 150                 n.find_unsigned("symmetry", symm);
 151                 switch (symm) {
 152                         case 1:
 153                                 symtree = sy_symm();
 154                                 break;
 155                         case 2:
 156                                 symtree = sy_anti();
 157                                 break;
 158                         default:
 159                                 symtree = sy_none();
 160                                 break;
 161                 }
 162                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 163         }
 164 }
 165
 166 void indexed::archive(archive_node &n) const
 167 {
 168         inherited::archive(n);
 169         n.add_ex("symmetry", symtree);
 170 }
 171
 172 DEFAULT_UNARCHIVE(indexed)
 173
 174 //////////
 175 // functions overriding virtual functions from base classes
 176 //////////
 177
 178 void indexed::print(const print_context & c, unsigned level) const
 179 {
 180         GINAC_ASSERT(seq.size() > 0);
 181
 182         if (is_of_type(c, print_tree)) {
 183
 184                 c.s << std::string(level, ' ') << class_name()
 185                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 186                     << ", " << seq.size()-1 << " indices"
 187                     << ", symmetry=" << symtree << std::endl;
 188                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 189                 seq[0].print(c, level + delta_indent);
 190                 printindices(c, level + delta_indent);
 191
 192         } else {
 193
 194                 bool is_tex = is_of_type(c, print_latex);
 195                 const ex & base = seq[0];
 196                 bool need_parens = is_ex_exactly_of_type(base, add) || is_ex_exactly_of_type(base, mul)
 197                                 || is_ex_exactly_of_type(base, ncmul) || is_ex_exactly_of_type(base, power)
 198                                 || is_ex_of_type(base, indexed);
 199                 if (is_tex)
 200                         c.s << "{";
 201                 if (need_parens)
 202                         c.s << "(";
 203                 base.print(c);
 204                 if (need_parens)
 205                         c.s << ")";
 206                 if (is_tex)
 207                         c.s << "}";
 208                 printindices(c, level);
 209         }
 210 }
 211
 212 bool indexed::info(unsigned inf) const
 213 {
 214         if (inf == info_flags::indexed) return true;
 215         if (inf == info_flags::has_indices) return seq.size() > 1;
 216         return inherited::info(inf);
 217 }
 218
 219 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 220         bool operator() (const ex & e, unsigned inf) const {
 221                 return !(ex_to<idx>(e).get_value().info(inf));
 222         }
 223 };
 224
 225 bool indexed::all_index_values_are(unsigned inf) const
 226 {
 227         // No indices? Then no property can be fulfilled
 228         if (seq.size() < 2)
 229                 return false;
 230
 231         // Check all indices
 232         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 233 }
 234
 235 int indexed::compare_same_type(const basic & other) const
 236 {
 237         GINAC_ASSERT(is_a<indexed>(other));
 238         return inherited::compare_same_type(other);
 239 }
 240
 241 ex indexed::eval(int level) const
 242 {
 243         // First evaluate children, then we will end up here again
 244         if (level > 1)
 245                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 246
 247         const ex &base = seq[0];
 248
 249         // If the base object is 0, the whole object is 0
 250         if (base.is_zero())
 251                 return _ex0;
 252
 253         // If the base object is a product, pull out the numeric factor
 254         if (is_ex_exactly_of_type(base, mul) && is_ex_exactly_of_type(base.op(base.nops() - 1), numeric)) {
 255                 exvector v(seq);
 256                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 257                 v[0] = seq[0] / f;
 258                 return f * thisexprseq(v);
 259         }
 260
 261         // Canonicalize indices according to the symmetry properties
 262         if (seq.size() > 2) {
 263                 exvector v = seq;
 264                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 265                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 266                 if (sig != INT_MAX) {
 267                         // Something has changed while sorting indices, more evaluations later
 268                         if (sig == 0)
 269                                 return _ex0;
 270                         return ex(sig) * thisexprseq(v);
 271                 }
 272         }
 273
 274         // Let the class of the base object perform additional evaluations
 275         return ex_to<basic>(base).eval_indexed(*this);
 276 }
 277
 278 ex indexed::thisexprseq(const exvector & v) const
 279 {
 280         return indexed(ex_to<symmetry>(symtree), v);
 281 }
 282
 283 ex indexed::thisexprseq(exvector * vp) const
 284 {
 285         return indexed(ex_to<symmetry>(symtree), vp);
 286 }
 287
 288 ex indexed::expand(unsigned options) const
 289 {
 290         GINAC_ASSERT(seq.size() > 0);
 291
 292         if ((options & expand_options::expand_indexed) && is_ex_exactly_of_type(seq[0], add)) {
 293
 294                 // expand_indexed expands (a+b).i -> a.i + b.i
 295                 const ex & base = seq[0];
 296                 ex sum = _ex0;
 297                 for (unsigned i=0; i<base.nops(); i++) {
 298                         exvector s = seq;
 299                         s[0] = base.op(i);
 300                         sum += thisexprseq(s).expand();
 301                 }
 302                 return sum;
 303
 304         } else
 305                 return inherited::expand(options);
 306 }
 307
 308 //////////
 309 // virtual functions which can be overridden by derived classes
 310 //////////
 311
 312 // none
 313
 314 //////////
 315 // non-virtual functions in this class
 316 //////////
 317
 318 void indexed::printindices(const print_context & c, unsigned level) const
 319 {
 320         if (seq.size() > 1) {
 321
 322                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 323
 324                 if (is_of_type(c, print_latex)) {
 325
 326                         // TeX output: group by variance
 327                         bool first = true;
 328                         bool covariant = true;
 329
 330                         while (it != itend) {
 331                                 bool cur_covariant = (is_ex_of_type(*it, varidx) ? ex_to<varidx>(*it).is_covariant() : true);
 332                                 if (first || cur_covariant != covariant) { // Variance changed
 333                                         // The empty {} prevents indices from ending up on top of each other
 334                                         if (!first)
 335                                                 c.s << "}{}";
 336                                         covariant = cur_covariant;
 337                                         if (covariant)
 338                                                 c.s << "_{";
 339                                         else
 340                                                 c.s << "^{";
 341                                 }
 342                                 it->print(c, level);
 343                                 c.s << " ";
 344                                 first = false;
 345                                 it++;
 346                         }
 347                         c.s << "}";
 348
 349                 } else {
 350
 351                         // Ordinary output
 352                         while (it != itend) {
 353                                 it->print(c, level);
 354                                 it++;
 355                         }
 356                 }
 357         }
 358 }
 359
 360 /** Check whether all indices are of class idx and validate the symmetry
 361  *  tree. This function is used internally to make sure that all constructed
 362  *  indexed objects really carry indices and not some other classes. */
 363 void indexed::validate(void) const
 364 {
 365         GINAC_ASSERT(seq.size() > 0);
 366         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 367         while (it != itend) {
 368                 if (!is_ex_of_type(*it, idx))
 369                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 370                 it++;
 371         }
 372
 373         if (!symtree.is_zero()) {
 374                 if (!is_ex_exactly_of_type(symtree, symmetry))
 375                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 376                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 377         }
 378 }
 379
 380 /** Implementation of ex::diff() for an indexed object always returns 0.
 381  *
 382  *  @see ex::diff */
 383 ex indexed::derivative(const symbol & s) const
 384 {
 385         return _ex0;
 386 }
 387
 388 //////////
 389 // global functions
 390 //////////
 391
 392 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 393 static bool indices_consistent(const exvector & v1, const exvector & v2)
 394 {
 395         // Number of indices must be the same
 396         if (v1.size() != v2.size())
 397                 return false;
 398
 399         return equal(v1.begin(), v1.end(), v2.begin(), ex_is_equal());
 400 }
 401
 402 exvector indexed::get_indices(void) const
 403 {
 404         GINAC_ASSERT(seq.size() >= 1);
 405         return exvector(seq.begin() + 1, seq.end());
 406 }
 407
 408 exvector indexed::get_dummy_indices(void) const
 409 {
 410         exvector free_indices, dummy_indices;
 411         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 412         return dummy_indices;
 413 }
 414
 415 exvector indexed::get_dummy_indices(const indexed & other) const
 416 {
 417         exvector indices = get_free_indices();
 418         exvector other_indices = other.get_free_indices();
 419         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 420         exvector dummy_indices;
 421         find_dummy_indices(indices, dummy_indices);
 422         return dummy_indices;
 423 }
 424
 425 bool indexed::has_dummy_index_for(const ex & i) const
 426 {
 427         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 428         while (it != itend) {
 429                 if (is_dummy_pair(*it, i))
 430                         return true;
 431                 it++;
 432         }
 433         return false;
 434 }
 435
 436 exvector indexed::get_free_indices(void) const
 437 {
 438         exvector free_indices, dummy_indices;
 439         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 440         return free_indices;
 441 }
 442
 443 exvector add::get_free_indices(void) const
 444 {
 445         exvector free_indices;
 446         for (unsigned i=0; i<nops(); i++) {
 447                 if (i == 0)
 448                         free_indices = op(i).get_free_indices();
 449                 else {
 450                         exvector free_indices_of_term = op(i).get_free_indices();
 451                         if (!indices_consistent(free_indices, free_indices_of_term))
 452                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 453                 }
 454         }
 455         return free_indices;
 456 }
 457
 458 exvector mul::get_free_indices(void) const
 459 {
 460         // Concatenate free indices of all factors
 461         exvector un;
 462         for (unsigned i=0; i<nops(); i++) {
 463                 exvector free_indices_of_factor = op(i).get_free_indices();
 464                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 465         }
 466
 467         // And remove the dummy indices
 468         exvector free_indices, dummy_indices;
 469         find_free_and_dummy(un, free_indices, dummy_indices);
 470         return free_indices;
 471 }
 472
 473 exvector ncmul::get_free_indices(void) const
 474 {
 475         // Concatenate free indices of all factors
 476         exvector un;
 477         for (unsigned i=0; i<nops(); i++) {
 478                 exvector free_indices_of_factor = op(i).get_free_indices();
 479                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 480         }
 481
 482         // And remove the dummy indices
 483         exvector free_indices, dummy_indices;
 484         find_free_and_dummy(un, free_indices, dummy_indices);
 485         return free_indices;
 486 }
 487
 488 exvector power::get_free_indices(void) const
 489 {
 490         // Return free indices of basis
 491         return basis.get_free_indices();
 492 }
 493
 494 /** Rename dummy indices in an expression.
 495  *
 496  *  @param e Expression to be worked on
 497  *  @param local_dummy_indices The set of dummy indices that appear in the
 498  *    expression "e"
 499  *  @param global_dummy_indices The set of dummy indices that have appeared
 500  *    before and which we would like to use in "e", too. This gets updated
 501  *    by the function */
 502 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 503 {
 504         unsigned global_size = global_dummy_indices.size(),
 505                  local_size = local_dummy_indices.size();
 506
 507         // Any local dummy indices at all?
 508         if (local_size == 0)
 509                 return e;
 510
 511         if (global_size < local_size) {
 512
 513                 // More local indices than we encountered before, add the new ones
 514                 // to the global set
 515                 int old_global_size = global_size;
 516                 int remaining = local_size - global_size;
 517                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 518                 while (it != itend && remaining > 0) {
 519                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(ex_is_equal(), *it)) == global_dummy_indices.end()) {
 520                                 global_dummy_indices.push_back(*it);
 521                                 global_size++;
 522                                 remaining--;
 523                         }
 524                         it++;
 525                 }
 526
 527                 // If this is the first set of local indices, do nothing
 528                 if (old_global_size == 0)
 529                         return e;
 530         }
 531         GINAC_ASSERT(local_size <= global_size);
 532
 533         // Construct lists of index symbols
 534         exlist local_syms, global_syms;
 535         for (unsigned i=0; i<local_size; i++)
 536                 local_syms.push_back(local_dummy_indices[i].op(0));
 537         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 538         for (unsigned i=0; i<global_size; i++)
 539                 global_syms.push_back(global_dummy_indices[i].op(0));
 540         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 541
 542         // Remove common indices
 543         exlist local_uniq, global_uniq;
 544         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exlist>(local_uniq), ex_is_less());
 545         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exlist>(global_uniq), ex_is_less());
 546
 547         // Replace remaining non-common local index symbols by global ones
 548         if (local_uniq.empty())
 549                 return e;
 550         else {
 551                 while (global_uniq.size() > local_uniq.size())
 552                         global_uniq.pop_back();
 553                 return e.subs(lst(local_uniq), lst(global_uniq));
 554         }
 555 }
 556
 557 /* Ordering that only compares the base expressions of indexed objects. */
 558 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
 559         bool operator() (const ex &lh, const ex &rh) const
 560         {
 561                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 562         }
 563 };
 564
 565 /** Simplify product of indexed expressions (commutative, noncommutative and
 566  *  simple squares), return list of free indices. */
 567 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 568 {
 569         // Remember whether the product was commutative or noncommutative
 570         // (because we chop it into factors and need to reassemble later)
 571         bool non_commutative = is_ex_exactly_of_type(e, ncmul);
 572
 573         // Collect factors in an exvector, store squares twice
 574         exvector v;
 575         v.reserve(e.nops() * 2);
 576
 577         if (is_ex_exactly_of_type(e, power)) {
 578                 // We only get called for simple squares, split a^2 -> a*a
 579                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 580                 v.push_back(e.op(0));
 581                 v.push_back(e.op(0));
 582         } else {
 583                 for (unsigned i=0; i<e.nops(); i++) {
 584                         ex f = e.op(i);
 585                         if (is_ex_exactly_of_type(f, power) && f.op(1).is_equal(_ex2)) {
 586                                 v.push_back(f.op(0));
 587                     v.push_back(f.op(0));
 588                         } else if (is_ex_exactly_of_type(f, ncmul)) {
 589                                 // Noncommutative factor found, split it as well
 590                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 591                                 for (unsigned j=0; j<f.nops(); j++)
 592                                         v.push_back(f.op(j));
 593                         } else
 594                                 v.push_back(f);
 595                 }
 596         }
 597
 598         // Perform contractions
 599         bool something_changed = false;
 600         GINAC_ASSERT(v.size() > 1);
 601         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 602         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 603
 604 try_again:
 605                 if (!is_ex_of_type(*it1, indexed))
 606                         continue;
 607
 608                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 609
 610                 // Indexed factor found, get free indices and look for contraction
 611                 // candidates
 612                 exvector free1, dummy1;
 613                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 614
 615                 exvector::iterator it2;
 616                 for (it2 = it1 + 1; it2 != itend; it2++) {
 617
 618                         if (!is_ex_of_type(*it2, indexed))
 619                                 continue;
 620
 621                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 622
 623                         // Find free indices of second factor and merge them with free
 624                         // indices of first factor
 625                         exvector un;
 626                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 627                         un.insert(un.end(), free1.begin(), free1.end());
 628
 629                         // Check whether the two factors share dummy indices
 630                         exvector free, dummy;
 631                         find_free_and_dummy(un, free, dummy);
 632                         unsigned num_dummies = dummy.size();
 633                         if (num_dummies == 0)
 634                                 continue;
 635
 636                         // At least one dummy index, is it a defined scalar product?
 637                         bool contracted = false;
 638                         if (free.empty()) {
 639                                 if (sp.is_defined(*it1, *it2)) {
 640                                         *it1 = sp.evaluate(*it1, *it2);
 641                                         *it2 = _ex1;
 642                                         goto contraction_done;
 643                                 }
 644                         }
 645
 646                         // Try to contract the first one with the second one
 647                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 648                         if (!contracted) {
 649
 650                                 // That didn't work; maybe the second object knows how to
 651                                 // contract itself with the first one
 652                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 653                         }
 654                         if (contracted) {
 655 contraction_done:
 656                                 if (first_noncommutative || second_noncommutative
 657                                  || is_ex_exactly_of_type(*it1, add) || is_ex_exactly_of_type(*it2, add)
 658                                  || is_ex_exactly_of_type(*it1, mul) || is_ex_exactly_of_type(*it2, mul)
 659                                  || is_ex_exactly_of_type(*it1, ncmul) || is_ex_exactly_of_type(*it2, ncmul)) {
 660
 661                                         // One of the factors became a sum or product:
 662                                         // re-expand expression and run again
 663                                         // Non-commutative products are always re-expanded to give
 664                                         // simplify_ncmul() the chance to re-order and canonicalize
 665                                         // the product
 666                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 667                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 668                                 }
 669
 670                                 // Both objects may have new indices now or they might
 671                                 // even not be indexed objects any more, so we have to
 672                                 // start over
 673                                 something_changed = true;
 674                                 goto try_again;
 675                         }
 676                 }
 677         }
 678
 679         // Find free indices (concatenate them all and call find_free_and_dummy())
 680         // and all dummy indices that appear
 681         exvector un, individual_dummy_indices;
 682         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 683                 exvector free_indices_of_factor;
 684                 if (is_ex_of_type(*it1, indexed)) {
 685                         exvector dummy_indices_of_factor;
 686                         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);
 687                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 688                 } else
 689                         free_indices_of_factor = it1->get_free_indices();
 690                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 691         }
 692         exvector local_dummy_indices;
 693         find_free_and_dummy(un, free_indices, local_dummy_indices);
 694         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 695
 696         // Filter out the dummy indices with variance
 697         exvector variant_dummy_indices;
 698         for (it1 = local_dummy_indices.begin(), itend = local_dummy_indices.end(); it1 != itend; ++it1) {
 699                 if (is_exactly_a<varidx>(*it1))
 700                         variant_dummy_indices.push_back(*it1);
 701         }
 702
 703         // Any indices with variance present at all?
 704         if (!variant_dummy_indices.empty()) {
 705
 706                 // Yes, bring the product into a canonical order that only depends on
 707                 // the base expressions of indexed objects
 708                 if (!non_commutative)
 709                         std::sort(v.begin(), v.end(), ex_base_is_less());
 710
 711                 exvector moved_indices;
 712
 713                 // Iterate over all indexed objects in the product
 714                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 715                         if (!is_ex_of_type(*it1, indexed))
 716                                 continue;
 717
 718                         ex new_it1;
 719                         bool it1_dirty = false; // It this is true, then new_it1 holds a new value for *it1
 720
 721                         // If a dummy index is encountered for the first time in the
 722                         // product, pull it up, otherwise, pull it down
 723                         exvector::iterator it2, it2end;
 724                         for (it2 = const_cast<indexed &>(ex_to<indexed>(*it1)).seq.begin(), it2end = const_cast<indexed &>(ex_to<indexed>(*it1)).seq.end(); it2 != it2end; ++it2) {
 725                                 if (!is_exactly_a<varidx>(*it2))
 726                                         continue;
 727
 728                                 exvector::iterator vit, vitend;
 729                                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 730                                         if (it2->op(0).is_equal(vit->op(0))) {
 731                                                 if (ex_to<varidx>(*it2).is_covariant()) {
 732                                                         new_it1 = (it1_dirty ? new_it1 : *it1).subs(*it2 == ex_to<varidx>(*it2).toggle_variance());
 733                                                         it1_dirty = true;
 734                                                         something_changed = true;
 735                                                 }
 736                                                 moved_indices.push_back(*vit);
 737                                                 variant_dummy_indices.erase(vit);
 738                                                 goto next_index;
 739                                         }
 740                                 }
 741
 742                                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 743                                         if (it2->op(0).is_equal(vit->op(0))) {
 744                                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 745                                                         new_it1 = (it1_dirty ? new_it1 : *it1).subs(*it2 == ex_to<varidx>(*it2).toggle_variance());
 746                                                         it1_dirty = true;
 747                                                         something_changed = true;
 748                                                 }
 749                                                 goto next_index;
 750                                         }
 751                                 }
 752
 753 next_index:             ;
 754                         }
 755
 756                         if (it1_dirty)
 757                                 *it1 = new_it1;
 758                 }
 759         }
 760
 761         ex r;
 762         if (something_changed)
 763                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 764         else
 765                 r = e;
 766
 767         // The result should be symmetric with respect to exchange of dummy
 768         // indices, so if the symmetrization vanishes, the whole expression is
 769         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 770         if (local_dummy_indices.size() >= 2) {
 771                 lst dummy_syms;
 772                 for (int i=0; i<local_dummy_indices.size(); i++)
 773                         dummy_syms.append(local_dummy_indices[i].op(0));
 774                 if (r.symmetrize(dummy_syms).is_zero()) {
 775                         free_indices.clear();
 776                         return _ex0;
 777                 }
 778         }
 779
 780         // Dummy index renaming
 781         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 782
 783         // Product of indexed object with a scalar?
 784         if (is_ex_exactly_of_type(r, mul) && r.nops() == 2
 785          && is_ex_exactly_of_type(r.op(1), numeric) && is_ex_of_type(r.op(0), indexed))
 786                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 787         else
 788                 return r;
 789 }
 790
 791 /** Simplify indexed expression, return list of free indices. */
 792 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 793 {
 794         // Expand the expression
 795         ex e_expanded = e.expand();
 796
 797         // Simplification of single indexed object: just find the free indices
 798         // and perform dummy index renaming
 799         if (is_ex_of_type(e_expanded, indexed)) {
 800                 const indexed &i = ex_to<indexed>(e_expanded);
 801                 exvector local_dummy_indices;
 802                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 803                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 804         }
 805
 806         // Simplification of sum = sum of simplifications, check consistency of
 807         // free indices in each term
 808         if (is_ex_exactly_of_type(e_expanded, add)) {
 809                 bool first = true;
 810                 ex sum = _ex0;
 811                 free_indices.clear();
 812
 813                 for (unsigned i=0; i<e_expanded.nops(); i++) {
 814                         exvector free_indices_of_term;
 815                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 816                         if (!term.is_zero()) {
 817                                 if (first) {
 818                                         free_indices = free_indices_of_term;
 819                                         sum = term;
 820                                         first = false;
 821                                 } else {
 822                                         if (!indices_consistent(free_indices, free_indices_of_term))
 823                                                 throw (std::runtime_error("simplify_indexed: inconsistent indices in sum"));
 824                                         if (is_ex_of_type(sum, indexed) && is_ex_of_type(term, indexed))
 825                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
 826                                         else
 827                                                 sum += term;
 828                                 }
 829                         }
 830                 }
 831
 832                 return sum;
 833         }
 834
 835         // Simplification of products
 836         if (is_ex_exactly_of_type(e_expanded, mul)
 837          || is_ex_exactly_of_type(e_expanded, ncmul)
 838          || (is_ex_exactly_of_type(e_expanded, power) && is_ex_of_type(e_expanded.op(0), indexed) && e_expanded.op(1).is_equal(_ex2)))
 839                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
 840
 841         // Cannot do anything
 842         free_indices.clear();
 843         return e_expanded;
 844 }
 845
 846 /** Simplify/canonicalize expression containing indexed objects. This
 847  *  performs contraction of dummy indices where possible and checks whether
 848  *  the free indices in sums are consistent.
 849  *
 850  *  @return simplified expression */
 851 ex ex::simplify_indexed(void) const
 852 {
 853         exvector free_indices, dummy_indices;
 854         scalar_products sp;
 855         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 856 }
 857
 858 /** Simplify/canonicalize expression containing indexed objects. This
 859  *  performs contraction of dummy indices where possible, checks whether
 860  *  the free indices in sums are consistent, and automatically replaces
 861  *  scalar products by known values if desired.
 862  *
 863  *  @param sp Scalar products to be replaced automatically
 864  *  @return simplified expression */
 865 ex ex::simplify_indexed(const scalar_products & sp) const
 866 {
 867         exvector free_indices, dummy_indices;
 868         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 869 }
 870
 871 /** Symmetrize expression over its free indices. */
 872 ex ex::symmetrize(void) const
 873 {
 874         return GiNaC::symmetrize(*this, get_free_indices());
 875 }
 876
 877 /** Antisymmetrize expression over its free indices. */
 878 ex ex::antisymmetrize(void) const
 879 {
 880         return GiNaC::antisymmetrize(*this, get_free_indices());
 881 }
 882
 883 /** Symmetrize expression by cyclic permutation over its free indices. */
 884 ex ex::symmetrize_cyclic(void) const
 885 {
 886         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
 887 }
 888
 889 //////////
 890 // helper classes
 891 //////////
 892
 893 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
 894 {
 895         spm[make_key(v1, v2)] = sp;
 896 }
 897
 898 void scalar_products::add_vectors(const lst & l)
 899 {
 900         // Add all possible pairs of products
 901         unsigned num = l.nops();
 902         for (unsigned i=0; i<num; i++) {
 903                 ex a = l.op(i);
 904                 for (unsigned j=0; j<num; j++) {
 905                         ex b = l.op(j);
 906                         add(a, b, a*b);
 907                 }
 908         }
 909 }
 910
 911 void scalar_products::clear(void)
 912 {
 913         spm.clear();
 914 }
 915
 916 /** Check whether scalar product pair is defined. */
 917 bool scalar_products::is_defined(const ex & v1, const ex & v2) const
 918 {
 919         return spm.find(make_key(v1, v2)) != spm.end();
 920 }
 921
 922 /** Return value of defined scalar product pair. */
 923 ex scalar_products::evaluate(const ex & v1, const ex & v2) const
 924 {
 925         return spm.find(make_key(v1, v2))->second;
 926 }
 927
 928 void scalar_products::debugprint(void) const
 929 {
 930         std::cerr << "map size=" << spm.size() << std::endl;
 931         spmap::const_iterator i = spm.begin(), end = spm.end();
 932         while (i != end) {
 933                 const spmapkey & k = i->first;
 934                 std::cerr << "item key=(" << k.first << "," << k.second;
 935                 std::cerr << "), value=" << i->second << std::endl;
 936                 ++i;
 937         }
 938 }
 939
 940 /** Make key from object pair. */
 941 spmapkey scalar_products::make_key(const ex & v1, const ex & v2)
 942 {
 943         // If indexed, extract base objects
 944         ex s1 = is_ex_of_type(v1, indexed) ? v1.op(0) : v1;
 945         ex s2 = is_ex_of_type(v2, indexed) ? v2.op(0) : v2;
 946
 947         // Enforce canonical order in pair
 948         if (s1.compare(s2) > 0)
 949                 return spmapkey(s2, s1);
 950         else
 951                 return spmapkey(s1, s2);
 952 }
 953
 954 } // namespace GiNaC