ginac/indexed.cpp

   1 /** @file indexed.cpp
   2  *
   3  *  Implementation of GiNaC's indexed expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2005 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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include <iostream>
  24 #include <sstream>
  25 #include <stdexcept>
  26
  27 #include "indexed.h"
  28 #include "idx.h"
  29 #include "add.h"
  30 #include "mul.h"
  31 #include "ncmul.h"
  32 #include "power.h"
  33 #include "relational.h"
  34 #include "symmetry.h"
  35 #include "operators.h"
  36 #include "lst.h"
  37 #include "archive.h"
  38 #include "symbol.h"
  39 #include "utils.h"
  40 #include "integral.h"
  41 #include "matrix.h"
  42
  43 namespace GiNaC {
  44
  45 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(indexed, exprseq,
  46   print_func<print_context>(&indexed::do_print).
  47   print_func<print_latex>(&indexed::do_print_latex).
  48   print_func<print_tree>(&indexed::do_print_tree))
  49
  50 //////////
  51 // default constructor
  52 //////////
  53
  54 indexed::indexed() : symtree(not_symmetric())
  55 {
  56         tinfo_key = TINFO_indexed;
  57 }
  58
  59 //////////
  60 // other constructors
  61 //////////
  62
  63 indexed::indexed(const ex & b) : inherited(b), symtree(not_symmetric())
  64 {
  65         tinfo_key = TINFO_indexed;
  66         validate();
  67 }
  68
  69 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(not_symmetric())
  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(not_symmetric())
  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(not_symmetric())
  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(not_symmetric())
  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(not_symmetric())
 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, std::auto_ptr<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, 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 = not_symmetric();
 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::printindices(const print_context & c, unsigned level) const
 178 {
 179         if (seq.size() > 1) {
 180
 181                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 182
 183                 if (is_a<print_latex>(c)) {
 184
 185                         // TeX output: group by variance
 186                         bool first = true;
 187                         bool covariant = true;
 188
 189                         while (it != itend) {
 190                                 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
 191                                 if (first || cur_covariant != covariant) { // Variance changed
 192                                         // The empty {} prevents indices from ending up on top of each other
 193                                         if (!first)
 194                                                 c.s << "}{}";
 195                                         covariant = cur_covariant;
 196                                         if (covariant)
 197                                                 c.s << "_{";
 198                                         else
 199                                                 c.s << "^{";
 200                                 }
 201                                 it->print(c, level);
 202                                 c.s << " ";
 203                                 first = false;
 204                                 it++;
 205                         }
 206                         c.s << "}";
 207
 208                 } else {
 209
 210                         // Ordinary output
 211                         while (it != itend) {
 212                                 it->print(c, level);
 213                                 it++;
 214                         }
 215                 }
 216         }
 217 }
 218
 219 void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
 220 {
 221         if (precedence() <= level)
 222                 c.s << openbrace << '(';
 223         c.s << openbrace;
 224         seq[0].print(c, precedence());
 225         c.s << closebrace;
 226         printindices(c, level);
 227         if (precedence() <= level)
 228                 c.s << ')' << closebrace;
 229 }
 230
 231 void indexed::do_print(const print_context & c, unsigned level) const
 232 {
 233         print_indexed(c, "", "", level);
 234 }
 235
 236 void indexed::do_print_latex(const print_latex & c, unsigned level) const
 237 {
 238         print_indexed(c, "{", "}", level);
 239 }
 240
 241 void indexed::do_print_tree(const print_tree & c, unsigned level) const
 242 {
 243         c.s << std::string(level, ' ') << class_name() << " @" << this
 244             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 245             << ", " << seq.size()-1 << " indices"
 246             << ", symmetry=" << symtree << std::endl;
 247         seq[0].print(c, level + c.delta_indent);
 248         printindices(c, level + c.delta_indent);
 249 }
 250
 251 bool indexed::info(unsigned inf) const
 252 {
 253         if (inf == info_flags::indexed) return true;
 254         if (inf == info_flags::has_indices) return seq.size() > 1;
 255         return inherited::info(inf);
 256 }
 257
 258 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 259         bool operator() (const ex & e, unsigned inf) const {
 260                 return !(ex_to<idx>(e).get_value().info(inf));
 261         }
 262 };
 263
 264 bool indexed::all_index_values_are(unsigned inf) const
 265 {
 266         // No indices? Then no property can be fulfilled
 267         if (seq.size() < 2)
 268                 return false;
 269
 270         // Check all indices
 271         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 272 }
 273
 274 int indexed::compare_same_type(const basic & other) const
 275 {
 276         GINAC_ASSERT(is_a<indexed>(other));
 277         return inherited::compare_same_type(other);
 278 }
 279
 280 ex indexed::eval(int level) const
 281 {
 282         // First evaluate children, then we will end up here again
 283         if (level > 1)
 284                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 285
 286         const ex &base = seq[0];
 287
 288         // If the base object is 0, the whole object is 0
 289         if (base.is_zero())
 290                 return _ex0;
 291
 292         // If the base object is a product, pull out the numeric factor
 293         if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
 294                 exvector v(seq);
 295                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 296                 v[0] = seq[0] / f;
 297                 return f * thiscontainer(v);
 298         }
 299
 300         if(this->tinfo()==TINFO_indexed && seq.size()==1)
 301                 return base;
 302
 303         // Canonicalize indices according to the symmetry properties
 304         if (seq.size() > 2) {
 305                 exvector v = seq;
 306                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 307                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 308                 if (sig != INT_MAX) {
 309                         // Something has changed while sorting indices, more evaluations later
 310                         if (sig == 0)
 311                                 return _ex0;
 312                         return ex(sig) * thiscontainer(v);
 313                 }
 314         }
 315
 316         // Let the class of the base object perform additional evaluations
 317         return ex_to<basic>(base).eval_indexed(*this);
 318 }
 319
 320 ex indexed::thiscontainer(const exvector & v) const
 321 {
 322         return indexed(ex_to<symmetry>(symtree), v);
 323 }
 324
 325 ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
 326 {
 327         return indexed(ex_to<symmetry>(symtree), vp);
 328 }
 329
 330 unsigned indexed::return_type() const
 331 {
 332         if(is_a<matrix>(op(0)))
 333                 return return_types::commutative;
 334         else
 335                 return op(0).return_type();
 336 }
 337
 338 ex indexed::expand(unsigned options) const
 339 {
 340         GINAC_ASSERT(seq.size() > 0);
 341
 342         if (options & expand_options::expand_indexed) {
 343                 ex newbase = seq[0].expand(options);
 344                 if (is_exactly_a<add>(newbase)) {
 345                         ex sum = _ex0;
 346                         for (size_t i=0; i<newbase.nops(); i++) {
 347                                 exvector s = seq;
 348                                 s[0] = newbase.op(i);
 349                                 sum += thiscontainer(s).expand(options);
 350                         }
 351                         return sum;
 352                 }
 353                 if (!are_ex_trivially_equal(newbase, seq[0])) {
 354                         exvector s = seq;
 355                         s[0] = newbase;
 356                         return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
 357                 }
 358         }
 359         return inherited::expand(options);
 360 }
 361
 362 //////////
 363 // virtual functions which can be overridden by derived classes
 364 //////////
 365
 366 // none
 367
 368 //////////
 369 // non-virtual functions in this class
 370 //////////
 371
 372 /** Check whether all indices are of class idx and validate the symmetry
 373  *  tree. This function is used internally to make sure that all constructed
 374  *  indexed objects really carry indices and not some other classes. */
 375 void indexed::validate() const
 376 {
 377         GINAC_ASSERT(seq.size() > 0);
 378         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 379         while (it != itend) {
 380                 if (!is_a<idx>(*it))
 381                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 382                 it++;
 383         }
 384
 385         if (!symtree.is_zero()) {
 386                 if (!is_exactly_a<symmetry>(symtree))
 387                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 388                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 389         }
 390 }
 391
 392 /** Implementation of ex::diff() for an indexed object always returns 0.
 393  *
 394  *  @see ex::diff */
 395 ex indexed::derivative(const symbol & s) const
 396 {
 397         return _ex0;
 398 }
 399
 400 //////////
 401 // global functions
 402 //////////
 403
 404 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
 405         bool operator() (const ex &lh, const ex &rh) const
 406         {
 407                 if (lh.is_equal(rh))
 408                         return true;
 409                 else
 410                         try {
 411                                 // Replacing the dimension might cause an error (e.g. with
 412                                 // index classes that only work in a fixed number of dimensions)
 413                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
 414                         } catch (...) {
 415                                 return false;
 416                         }
 417         }
 418 };
 419
 420 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 421 static bool indices_consistent(const exvector & v1, const exvector & v2)
 422 {
 423         // Number of indices must be the same
 424         if (v1.size() != v2.size())
 425                 return false;
 426
 427         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
 428 }
 429
 430 exvector indexed::get_indices() const
 431 {
 432         GINAC_ASSERT(seq.size() >= 1);
 433         return exvector(seq.begin() + 1, seq.end());
 434 }
 435
 436 exvector indexed::get_dummy_indices() const
 437 {
 438         exvector free_indices, dummy_indices;
 439         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 440         return dummy_indices;
 441 }
 442
 443 exvector indexed::get_dummy_indices(const indexed & other) const
 444 {
 445         exvector indices = get_free_indices();
 446         exvector other_indices = other.get_free_indices();
 447         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 448         exvector dummy_indices;
 449         find_dummy_indices(indices, dummy_indices);
 450         return dummy_indices;
 451 }
 452
 453 bool indexed::has_dummy_index_for(const ex & i) const
 454 {
 455         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 456         while (it != itend) {
 457                 if (is_dummy_pair(*it, i))
 458                         return true;
 459                 it++;
 460         }
 461         return false;
 462 }
 463
 464 exvector indexed::get_free_indices() const
 465 {
 466         exvector free_indices, dummy_indices;
 467         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 468         return free_indices;
 469 }
 470
 471 exvector add::get_free_indices() const
 472 {
 473         exvector free_indices;
 474         for (size_t i=0; i<nops(); i++) {
 475                 if (i == 0)
 476                         free_indices = op(i).get_free_indices();
 477                 else {
 478                         exvector free_indices_of_term = op(i).get_free_indices();
 479                         if (!indices_consistent(free_indices, free_indices_of_term))
 480                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 481                 }
 482         }
 483         return free_indices;
 484 }
 485
 486 exvector mul::get_free_indices() const
 487 {
 488         // Concatenate free indices of all factors
 489         exvector un;
 490         for (size_t i=0; i<nops(); i++) {
 491                 exvector free_indices_of_factor = op(i).get_free_indices();
 492                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 493         }
 494
 495         // And remove the dummy indices
 496         exvector free_indices, dummy_indices;
 497         find_free_and_dummy(un, free_indices, dummy_indices);
 498         return free_indices;
 499 }
 500
 501 exvector ncmul::get_free_indices() const
 502 {
 503         // Concatenate free indices of all factors
 504         exvector un;
 505         for (size_t i=0; i<nops(); i++) {
 506                 exvector free_indices_of_factor = op(i).get_free_indices();
 507                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 508         }
 509
 510         // And remove the dummy indices
 511         exvector free_indices, dummy_indices;
 512         find_free_and_dummy(un, free_indices, dummy_indices);
 513         return free_indices;
 514 }
 515
 516 struct is_summation_idx : public std::unary_function<ex, bool> {
 517         bool operator()(const ex & e)
 518         {
 519                 return is_dummy_pair(e, e);
 520         }
 521 };
 522
 523 exvector power::get_free_indices() const
 524 {
 525         // Get free indices of basis
 526         exvector basis_indices = basis.get_free_indices();
 527
 528         if (exponent.info(info_flags::even)) {
 529                 // If the exponent is an even number, then any "free" index that
 530                 // forms a dummy pair with itself is actually a summation index
 531                 exvector really_free;
 532                 std::remove_copy_if(basis_indices.begin(), basis_indices.end(),
 533                                     std::back_inserter(really_free), is_summation_idx());
 534                 return really_free;
 535         } else
 536                 return basis_indices;
 537 }
 538
 539 exvector integral::get_free_indices() const
 540 {
 541         if (a.get_free_indices().size() || b.get_free_indices().size())
 542                 throw (std::runtime_error("integral::get_free_indices: boundary values should not have free indices"));
 543         return f.get_free_indices();
 544 }
 545
 546 template<class T> size_t number_of_type(const exvector&v)
 547 {
 548         size_t number = 0;
 549         for(exvector::const_iterator i=v.begin(); i!=v.end(); ++i)
 550                 if(is_exactly_a<T>(*i))
 551                         ++number;
 552         return number;
 553 }
 554
 555 /** Rename dummy indices in an expression.
 556  *
 557  *  @param e Expression to work on
 558  *  @param local_dummy_indices The set of dummy indices that appear in the
 559  *    expression "e"
 560  *  @param global_dummy_indices The set of dummy indices that have appeared
 561  *    before and which we would like to use in "e", too. This gets updated
 562  *    by the function */
 563 template<class T> static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 564 {
 565         size_t global_size = number_of_type<T>(global_dummy_indices),
 566                local_size = number_of_type<T>(local_dummy_indices);
 567
 568         // Any local dummy indices at all?
 569         if (local_size == 0)
 570                 return e;
 571
 572         if (global_size < local_size) {
 573
 574                 // More local indices than we encountered before, add the new ones
 575                 // to the global set
 576                 size_t old_global_size = global_size;
 577                 int remaining = local_size - global_size;
 578                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 579                 while (it != itend && remaining > 0) {
 580                         if (is_exactly_a<T>(*it) && find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(idx_is_equal_ignore_dim(), *it)) == global_dummy_indices.end()) {
 581                                 global_dummy_indices.push_back(*it);
 582                                 global_size++;
 583                                 remaining--;
 584                         }
 585                         it++;
 586                 }
 587
 588                 // If this is the first set of local indices, do nothing
 589                 if (old_global_size == 0)
 590                         return e;
 591         }
 592         GINAC_ASSERT(local_size <= global_size);
 593
 594         // Construct vectors of index symbols
 595         exvector local_syms, global_syms;
 596         local_syms.reserve(local_size);
 597         global_syms.reserve(local_size);
 598         for (size_t i=0; local_syms.size()!=local_size; i++)
 599                 if(is_exactly_a<T>(local_dummy_indices[i]))
 600                         local_syms.push_back(local_dummy_indices[i].op(0));
 601         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 602         for (size_t i=0; global_syms.size()!=local_size; i++) // don't use more global symbols than necessary
 603                 if(is_exactly_a<T>(global_dummy_indices[i]))
 604                         global_syms.push_back(global_dummy_indices[i].op(0));
 605         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 606
 607         // Remove common indices
 608         exvector local_uniq, global_uniq;
 609         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
 610         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
 611
 612         // Replace remaining non-common local index symbols by global ones
 613         if (local_uniq.empty())
 614                 return e;
 615         else {
 616                 while (global_uniq.size() > local_uniq.size())
 617                         global_uniq.pop_back();
 618                 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
 619         }
 620 }
 621
 622 /** Given a set of indices, extract those of class varidx. */
 623 static void find_variant_indices(const exvector & v, exvector & variant_indices)
 624 {
 625         exvector::const_iterator it1, itend;
 626         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 627                 if (is_exactly_a<varidx>(*it1))
 628                         variant_indices.push_back(*it1);
 629         }
 630 }
 631
 632 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
 633  *  variance.
 634  *
 635  *  @param e Object to work on
 636  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
 637  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
 638  *  @return true if 'e' was changed */
 639 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
 640 {
 641         bool something_changed = false;
 642
 643         // If a dummy index is encountered for the first time in the
 644         // product, pull it up, otherwise, pull it down
 645         exvector::const_iterator it2, it2start, it2end;
 646         for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
 647                 if (!is_exactly_a<varidx>(*it2))
 648                         continue;
 649
 650                 exvector::iterator vit, vitend;
 651                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 652                         if (it2->op(0).is_equal(vit->op(0))) {
 653                                 if (ex_to<varidx>(*it2).is_covariant()) {
 654                                         e = e.subs(lst(
 655                                                 *it2 == ex_to<varidx>(*it2).toggle_variance(),
 656                                                 ex_to<varidx>(*it2).toggle_variance() == *it2
 657                                         ), subs_options::no_pattern);
 658                                         something_changed = true;
 659                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 660                                         it2start = ex_to<indexed>(e).seq.begin();
 661                                         it2end = ex_to<indexed>(e).seq.end();
 662                                 }
 663                                 moved_indices.push_back(*vit);
 664                                 variant_dummy_indices.erase(vit);
 665                                 goto next_index;
 666                         }
 667                 }
 668
 669                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 670                         if (it2->op(0).is_equal(vit->op(0))) {
 671                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 672                                         e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance(), subs_options::no_pattern);
 673                                         something_changed = true;
 674                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 675                                         it2start = ex_to<indexed>(e).seq.begin();
 676                                         it2end = ex_to<indexed>(e).seq.end();
 677                                 }
 678                                 goto next_index;
 679                         }
 680                 }
 681
 682 next_index: ;
 683         }
 684
 685         return something_changed;
 686 }
 687
 688 /* Ordering that only compares the base expressions of indexed objects. */
 689 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
 690         bool operator() (const ex &lh, const ex &rh) const
 691         {
 692                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 693         }
 694 };
 695
 696 /* An auxiliary function used by simplify_indexed() and expand_dummy_sum()
 697  * It returns an exvector of factors from the supplied product */
 698 static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
 699 {
 700         // Remember whether the product was commutative or noncommutative
 701         // (because we chop it into factors and need to reassemble later)
 702         non_commutative = is_exactly_a<ncmul>(e);
 703
 704         // Collect factors in an exvector, store squares twice
 705         v.reserve(e.nops() * 2);
 706
 707         if (is_exactly_a<power>(e)) {
 708                 // We only get called for simple squares, split a^2 -> a*a
 709                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 710                 v.push_back(e.op(0));
 711                 v.push_back(e.op(0));
 712         } else {
 713                 for (size_t i=0; i<e.nops(); i++) {
 714                         ex f = e.op(i);
 715                         if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
 716                                 v.push_back(f.op(0));
 717                                 v.push_back(f.op(0));
 718                         } else if (is_exactly_a<ncmul>(f)) {
 719                                 // Noncommutative factor found, split it as well
 720                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 721                                 for (size_t j=0; j<f.nops(); j++)
 722                                         v.push_back(f.op(j));
 723                         } else
 724                                 v.push_back(f);
 725                 }
 726         }
 727 }
 728
 729 template<class T> ex idx_symmetrization(const ex& r,const exvector& local_dummy_indices)
 730 {       exvector dummy_syms;
 731         dummy_syms.reserve(r.nops());
 732         for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
 733                         if(is_exactly_a<T>(*it))
 734                                 dummy_syms.push_back(it->op(0));
 735         if(dummy_syms.size() < 2)
 736                 return r;
 737         ex q=symmetrize(r, dummy_syms);
 738         return q;
 739 }
 740
 741 // Forward declaration needed in absence of friend injection, C.f. [namespace.memdef]:
 742 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp);
 743
 744 /** Simplify product of indexed expressions (commutative, noncommutative and
 745  *  simple squares), return list of free indices. */
 746 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 747 {
 748         // Collect factors in an exvector
 749         exvector v;
 750
 751         // Remember whether the product was commutative or noncommutative
 752         // (because we chop it into factors and need to reassemble later)
 753         bool non_commutative;
 754         product_to_exvector(e, v, non_commutative);
 755
 756         // Perform contractions
 757         bool something_changed = false;
 758         GINAC_ASSERT(v.size() > 1);
 759         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 760         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 761
 762 try_again:
 763                 if (!is_a<indexed>(*it1))
 764                         continue;
 765
 766                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 767
 768                 // Indexed factor found, get free indices and look for contraction
 769                 // candidates
 770                 exvector free1, dummy1;
 771                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 772
 773                 exvector::iterator it2;
 774                 for (it2 = it1 + 1; it2 != itend; it2++) {
 775
 776                         if (!is_a<indexed>(*it2))
 777                                 continue;
 778
 779                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 780
 781                         // Find free indices of second factor and merge them with free
 782                         // indices of first factor
 783                         exvector un;
 784                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 785                         un.insert(un.end(), free1.begin(), free1.end());
 786
 787                         // Check whether the two factors share dummy indices
 788                         exvector free, dummy;
 789                         find_free_and_dummy(un, free, dummy);
 790                         size_t num_dummies = dummy.size();
 791                         if (num_dummies == 0)
 792                                 continue;
 793
 794                         // At least one dummy index, is it a defined scalar product?
 795                         bool contracted = false;
 796                         if (free.empty()) {
 797
 798                                 try {
 799                                         // Find minimal dimension of all indices of both factors
 800                                         exvector::const_iterator dit = ex_to<indexed>(*it1).seq.begin() + 1, ditend = ex_to<indexed>(*it1).seq.end();
 801                                         ex dim = ex_to<idx>(*dit).get_dim();
 802                                         ++dit;
 803                                         for (; dit != ditend; ++dit) {
 804                                                 dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 805                                         }
 806                                         dit = ex_to<indexed>(*it2).seq.begin() + 1;
 807                                         ditend = ex_to<indexed>(*it2).seq.end();
 808                                         for (; dit != ditend; ++dit) {
 809                                                 dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 810                                         }
 811
 812                                         // User-defined scalar product?
 813                                         if (sp.is_defined(*it1, *it2, dim)) {
 814
 815                                                 // Yes, substitute it
 816                                                 *it1 = sp.evaluate(*it1, *it2, dim);
 817                                                 *it2 = _ex1;
 818                                                 goto contraction_done;
 819                                         }
 820                                 } catch (const std::runtime_error&) {}
 821                         }
 822
 823                         // Try to contract the first one with the second one
 824                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 825                         if (!contracted) {
 826
 827                                 // That didn't work; maybe the second object knows how to
 828                                 // contract itself with the first one
 829                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 830                         }
 831                         if (contracted) {
 832 contraction_done:
 833                                 if (first_noncommutative || second_noncommutative
 834                                  || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
 835                                  || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
 836                                  || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
 837
 838                                         // One of the factors became a sum or product:
 839                                         // re-expand expression and run again
 840                                         // Non-commutative products are always re-expanded to give
 841                                         // eval_ncmul() the chance to re-order and canonicalize
 842                                         // the product
 843                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 844                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 845                                 }
 846
 847                                 // Both objects may have new indices now or they might
 848                                 // even not be indexed objects any more, so we have to
 849                                 // start over
 850                                 something_changed = true;
 851                                 goto try_again;
 852                         }
 853                 }
 854         }
 855
 856         // Find free indices (concatenate them all and call find_free_and_dummy())
 857         // and all dummy indices that appear
 858         exvector un, individual_dummy_indices;
 859         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 860                 exvector free_indices_of_factor;
 861                 if (is_a<indexed>(*it1)) {
 862                         exvector dummy_indices_of_factor;
 863                         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);
 864                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 865                 } else
 866                         free_indices_of_factor = it1->get_free_indices();
 867                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 868         }
 869         exvector local_dummy_indices;
 870         find_free_and_dummy(un, free_indices, local_dummy_indices);
 871         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 872
 873         // Filter out the dummy indices with variance
 874         exvector variant_dummy_indices;
 875         find_variant_indices(local_dummy_indices, variant_dummy_indices);
 876
 877         // Any indices with variance present at all?
 878         if (!variant_dummy_indices.empty()) {
 879
 880                 // Yes, bring the product into a canonical order that only depends on
 881                 // the base expressions of indexed objects
 882                 if (!non_commutative)
 883                         std::sort(v.begin(), v.end(), ex_base_is_less());
 884
 885                 exvector moved_indices;
 886
 887                 // Iterate over all indexed objects in the product
 888                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 889                         if (!is_a<indexed>(*it1))
 890                                 continue;
 891
 892                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
 893                                 something_changed = true;
 894                 }
 895         }
 896
 897         ex r;
 898         if (something_changed)
 899                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 900         else
 901                 r = e;
 902
 903         // The result should be symmetric with respect to exchange of dummy
 904         // indices, so if the symmetrization vanishes, the whole expression is
 905         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 906         ex q = idx_symmetrization<idx>(r, local_dummy_indices);
 907         if (q.is_zero()) {
 908                 free_indices.clear();
 909                 return _ex0;
 910         }
 911         q = idx_symmetrization<varidx>(q, local_dummy_indices);
 912         if (q.is_zero()) {
 913                 free_indices.clear();
 914                 return _ex0;
 915         }
 916         q = idx_symmetrization<spinidx>(q, local_dummy_indices);
 917         if (q.is_zero()) {
 918                 free_indices.clear();
 919                 return _ex0;
 920         }
 921
 922         // Dummy index renaming
 923         r = rename_dummy_indices<idx>(r, dummy_indices, local_dummy_indices);
 924         r = rename_dummy_indices<varidx>(r, dummy_indices, local_dummy_indices);
 925         r = rename_dummy_indices<spinidx>(r, dummy_indices, local_dummy_indices);
 926
 927         // Product of indexed object with a scalar?
 928         if (is_exactly_a<mul>(r) && r.nops() == 2
 929          && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
 930                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 931         else
 932                 return r;
 933 }
 934
 935 /** This structure stores the original and symmetrized versions of terms
 936  *  obtained during the simplification of sums. */
 937 class terminfo {
 938 public:
 939         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
 940
 941         ex orig; /**< original term */
 942         ex symm; /**< symmtrized term */
 943 };
 944
 945 class terminfo_is_less {
 946 public:
 947         bool operator() (const terminfo & ti1, const terminfo & ti2) const
 948         {
 949                 return (ti1.symm.compare(ti2.symm) < 0);
 950         }
 951 };
 952
 953 /** This structure stores the individual symmetrized terms obtained during
 954  *  the simplification of sums. */
 955 class symminfo {
 956 public:
 957         symminfo() : num(0) {}
 958
 959         symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
 960         {
 961                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
 962                         coeff = symmterm_.op(symmterm_.nops()-1);
 963                         symmterm = symmterm_ / coeff;
 964                 } else {
 965                         coeff = 1;
 966                         symmterm = symmterm_;
 967                 }
 968         }
 969
 970         ex symmterm;  /**< symmetrized term */
 971         ex coeff;     /**< coefficient of symmetrized term */
 972         ex orig;      /**< original term */
 973         size_t num; /**< how many symmetrized terms resulted from the original term */
 974 };
 975
 976 class symminfo_is_less_by_symmterm {
 977 public:
 978         bool operator() (const symminfo & si1, const symminfo & si2) const
 979         {
 980                 return (si1.symmterm.compare(si2.symmterm) < 0);
 981         }
 982 };
 983
 984 class symminfo_is_less_by_orig {
 985 public:
 986         bool operator() (const symminfo & si1, const symminfo & si2) const
 987         {
 988                 return (si1.orig.compare(si2.orig) < 0);
 989         }
 990 };
 991
 992 bool hasindex(const ex &x, const ex &sym)
 993 {
 994         if(is_a<idx>(x) && x.op(0)==sym)
 995                 return true;
 996         else
 997                 for(size_t i=0; i<x.nops(); ++i)
 998                         if(hasindex(x.op(i), sym))
 999                                 return true;
1000         return false;
1001 }
1002
1003 /** Simplify indexed expression, return list of free indices. */
1004 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
1005 {
1006         // Expand the expression
1007         ex e_expanded = e.expand();
1008
1009         // Simplification of single indexed object: just find the free indices
1010         // and perform dummy index renaming/repositioning
1011         if (is_a<indexed>(e_expanded)) {
1012
1013                 // Find the dummy indices
1014                 const indexed &i = ex_to<indexed>(e_expanded);
1015                 exvector local_dummy_indices;
1016                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
1017
1018                 // Filter out the dummy indices with variance
1019                 exvector variant_dummy_indices;
1020                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
1021
1022                 // Any indices with variance present at all?
1023                 if (!variant_dummy_indices.empty()) {
1024
1025                         // Yes, reposition them
1026                         exvector moved_indices;
1027                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
1028                 }
1029
1030                 // Rename the dummy indices
1031                 e_expanded = rename_dummy_indices<idx>(e_expanded, dummy_indices, local_dummy_indices);
1032                 e_expanded = rename_dummy_indices<varidx>(e_expanded, dummy_indices, local_dummy_indices);
1033                 e_expanded = rename_dummy_indices<spinidx>(e_expanded, dummy_indices, local_dummy_indices);
1034                 return e_expanded;
1035         }
1036
1037         // Simplification of sum = sum of simplifications, check consistency of
1038         // free indices in each term
1039         if (is_exactly_a<add>(e_expanded)) {
1040                 bool first = true;
1041                 ex sum;
1042                 free_indices.clear();
1043
1044                 for (size_t i=0; i<e_expanded.nops(); i++) {
1045                         exvector free_indices_of_term;
1046                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
1047                         if (!term.is_zero()) {
1048                                 if (first) {
1049                                         free_indices = free_indices_of_term;
1050                                         sum = term;
1051                                         first = false;
1052                                 } else {
1053                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
1054                                                 std::ostringstream s;
1055                                                 s << "simplify_indexed: inconsistent indices in sum: ";
1056                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
1057                                                 throw (std::runtime_error(s.str()));
1058                                         }
1059                                         if (is_a<indexed>(sum) && is_a<indexed>(term))
1060                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
1061                                         else
1062                                                 sum += term;
1063                                 }
1064                         }
1065                 }
1066
1067                 // If the sum turns out to be zero, we are finished
1068                 if (sum.is_zero()) {
1069                         free_indices.clear();
1070                         return sum;
1071                 }
1072
1073                 // More than one term and more than one dummy index?
1074                 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
1075                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
1076                         return sum;
1077
1078                 // Chop the sum into terms and symmetrize each one over the dummy
1079                 // indices
1080                 std::vector<terminfo> terms;
1081                 for (size_t i=0; i<sum.nops(); i++) {
1082                         const ex & term = sum.op(i);
1083                         exvector dummy_indices_of_term;
1084                         dummy_indices_of_term.reserve(dummy_indices.size());
1085                         for(exvector::iterator i=dummy_indices.begin(); i!=dummy_indices.end(); ++i)
1086                                 if(hasindex(term,i->op(0)))
1087                                         dummy_indices_of_term.push_back(*i);
1088                         ex term_symm = idx_symmetrization<idx>(term, dummy_indices_of_term);
1089                         term_symm = idx_symmetrization<varidx>(term_symm, dummy_indices_of_term);
1090                         term_symm = idx_symmetrization<spinidx>(term_symm, dummy_indices_of_term);
1091                         if (term_symm.is_zero())
1092                                 continue;
1093                         terms.push_back(terminfo(term, term_symm));
1094                 }
1095
1096                 // Sort by symmetrized terms
1097                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
1098
1099                 // Combine equal symmetrized terms
1100                 std::vector<terminfo> terms_pass2;
1101                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1102                         size_t num = 1;
1103                         std::vector<terminfo>::const_iterator j = i + 1;
1104                         while (j != terms.end() && j->symm == i->symm) {
1105                                 num++;
1106                                 j++;
1107                         }
1108                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1109                         i = j;
1110                 }
1111
1112                 // If there is only one term left, we are finished
1113                 if (terms_pass2.size() == 1)
1114                         return terms_pass2[0].orig;
1115
1116                 // Chop the symmetrized terms into subterms
1117                 std::vector<symminfo> sy;
1118                 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1119                         if (is_exactly_a<add>(i->symm)) {
1120                                 size_t num = i->symm.nops();
1121                                 for (size_t j=0; j<num; j++)
1122                                         sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1123                         } else
1124                                 sy.push_back(symminfo(i->symm, i->orig, 1));
1125                 }
1126
1127                 // Sort by symmetrized subterms
1128                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1129
1130                 // Combine equal symmetrized subterms
1131                 std::vector<symminfo> sy_pass2;
1132                 exvector result;
1133                 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1134
1135                         // Combine equal terms
1136                         std::vector<symminfo>::const_iterator j = i + 1;
1137                         if (j != sy.end() && j->symmterm == i->symmterm) {
1138
1139                                 // More than one term, collect the coefficients
1140                                 ex coeff = i->coeff;
1141                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1142                                         coeff += j->coeff;
1143                                         j++;
1144                                 }
1145
1146                                 // Add combined term to result
1147                                 if (!coeff.is_zero())
1148                                         result.push_back(coeff * i->symmterm);
1149
1150                         } else {
1151
1152                                 // Single term, store for second pass
1153                                 sy_pass2.push_back(*i);
1154                         }
1155
1156                         i = j;
1157                 }
1158
1159                 // Were there any remaining terms that didn't get combined?
1160                 if (sy_pass2.size() > 0) {
1161
1162                         // Yes, sort by their original terms
1163                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1164
1165                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1166
1167                                 // How many symmetrized terms of this original term are left?
1168                                 size_t num = 1;
1169                                 std::vector<symminfo>::const_iterator j = i + 1;
1170                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1171                                         num++;
1172                                         j++;
1173                                 }
1174
1175                                 if (num == i->num) {
1176
1177                                         // All terms left, then add the original term to the result
1178                                         result.push_back(i->orig);
1179
1180                                 } else {
1181
1182                                         // Some terms were combined with others, add up the remaining symmetrized terms
1183                                         std::vector<symminfo>::const_iterator k;
1184                                         for (k=i; k!=j; k++)
1185                                                 result.push_back(k->coeff * k->symmterm);
1186                                 }
1187
1188                                 i = j;
1189                         }
1190                 }
1191
1192                 // Add all resulting terms
1193                 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1194                 if (sum_symm.is_zero())
1195                         free_indices.clear();
1196                 return sum_symm;
1197         }
1198
1199         // Simplification of products
1200         if (is_exactly_a<mul>(e_expanded)
1201          || is_exactly_a<ncmul>(e_expanded)
1202          || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1203                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1204
1205         // Cannot do anything
1206         free_indices.clear();
1207         return e_expanded;
1208 }
1209
1210 /** Simplify/canonicalize expression containing indexed objects. This
1211  *  performs contraction of dummy indices where possible and checks whether
1212  *  the free indices in sums are consistent.
1213  *
1214  *  @param options Simplification options (currently unused)
1215  *  @return simplified expression */
1216 ex ex::simplify_indexed(unsigned options) const
1217 {
1218         exvector free_indices, dummy_indices;
1219         scalar_products sp;
1220         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1221 }
1222
1223 /** Simplify/canonicalize expression containing indexed objects. This
1224  *  performs contraction of dummy indices where possible, checks whether
1225  *  the free indices in sums are consistent, and automatically replaces
1226  *  scalar products by known values if desired.
1227  *
1228  *  @param sp Scalar products to be replaced automatically
1229  *  @param options Simplification options (currently unused)
1230  *  @return simplified expression */
1231 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1232 {
1233         exvector free_indices, dummy_indices;
1234         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1235 }
1236
1237 /** Symmetrize expression over its free indices. */
1238 ex ex::symmetrize() const
1239 {
1240         return GiNaC::symmetrize(*this, get_free_indices());
1241 }
1242
1243 /** Antisymmetrize expression over its free indices. */
1244 ex ex::antisymmetrize() const
1245 {
1246         return GiNaC::antisymmetrize(*this, get_free_indices());
1247 }
1248
1249 /** Symmetrize expression by cyclic permutation over its free indices. */
1250 ex ex::symmetrize_cyclic() const
1251 {
1252         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1253 }
1254
1255 //////////
1256 // helper classes
1257 //////////
1258
1259 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1260 {
1261         // If indexed, extract base objects
1262         ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1263         ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1264
1265         // Enforce canonical order in pair
1266         if (s1.compare(s2) > 0) {
1267                 v1 = s2;
1268                 v2 = s1;
1269         } else {
1270                 v1 = s1;
1271                 v2 = s2;
1272         }
1273 }
1274
1275 bool spmapkey::operator==(const spmapkey &other) const
1276 {
1277         if (!v1.is_equal(other.v1))
1278                 return false;
1279         if (!v2.is_equal(other.v2))
1280                 return false;
1281         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1282                 return true;
1283         else
1284                 return dim.is_equal(other.dim);
1285 }
1286
1287 bool spmapkey::operator<(const spmapkey &other) const
1288 {
1289         int cmp = v1.compare(other.v1);
1290         if (cmp)
1291                 return cmp < 0;
1292         cmp = v2.compare(other.v2);
1293         if (cmp)
1294                 return cmp < 0;
1295
1296         // Objects are equal, now check dimensions
1297         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1298                 return false;
1299         else
1300                 return dim.compare(other.dim) < 0;
1301 }
1302
1303 void spmapkey::debugprint() const
1304 {
1305         std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1306 }
1307
1308 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1309 {
1310         spm[spmapkey(v1, v2)] = sp;
1311 }
1312
1313 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1314 {
1315         spm[spmapkey(v1, v2, dim)] = sp;
1316 }
1317
1318 void scalar_products::add_vectors(const lst & l, const ex & dim)
1319 {
1320         // Add all possible pairs of products
1321         for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1322                 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1323                         add(*it1, *it2, *it1 * *it2);
1324 }
1325
1326 void scalar_products::clear()
1327 {
1328         spm.clear();
1329 }
1330
1331 /** Check whether scalar product pair is defined. */
1332 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1333 {
1334         return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1335 }
1336
1337 /** Return value of defined scalar product pair. */
1338 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1339 {
1340         return spm.find(spmapkey(v1, v2, dim))->second;
1341 }
1342
1343 void scalar_products::debugprint() const
1344 {
1345         std::cerr << "map size=" << spm.size() << std::endl;
1346         spmap::const_iterator i = spm.begin(), end = spm.end();
1347         while (i != end) {
1348                 const spmapkey & k = i->first;
1349                 std::cerr << "item key=";
1350                 k.debugprint();
1351                 std::cerr << ", value=" << i->second << std::endl;
1352                 ++i;
1353         }
1354 }
1355
1356 /** Returns all dummy indices from the exvector */
1357 exvector get_all_dummy_indices(const ex & e)
1358 {
1359         exvector p;
1360         bool nc;
1361         product_to_exvector(e, p, nc);
1362         exvector::const_iterator ip = p.begin(), ipend = p.end();
1363         exvector v, v1;
1364         while (ip != ipend) {
1365                 if (is_a<indexed>(*ip)) {
1366                         v1 = ex_to<indexed>(*ip).get_dummy_indices();
1367                         v.insert(v.end(), v1.begin(), v1.end());
1368                         exvector::const_iterator ip1 = ip+1;
1369                         while (ip1 != ipend) {
1370                                 if (is_a<indexed>(*ip1)) {
1371                                         v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
1372                                         v.insert(v.end(), v1.begin(), v1.end());
1373                                 }
1374                                 ++ip1;
1375                         }
1376                 }
1377                 ++ip;
1378         }
1379         return v;
1380 }
1381
1382 lst rename_dummy_indices_uniquely(const exvector & va, const exvector & vb)
1383 {
1384         exvector common_indices;
1385         set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
1386         if (common_indices.empty()) {
1387                 return lst(lst(), lst());
1388         } else {
1389                 exvector new_indices, old_indices;
1390                 old_indices.reserve(2*common_indices.size());
1391                 new_indices.reserve(2*common_indices.size());
1392                 exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
1393                 while (ip != ipend) {
1394                         ex newsym=(new symbol)->setflag(status_flags::dynallocated);
1395                         ex newidx;
1396                         if(is_exactly_a<spinidx>(*ip))
1397                                 newidx = (new spinidx(newsym, ex_to<spinidx>(*ip).get_dim(),
1398                                                 ex_to<spinidx>(*ip).is_covariant(),
1399                                                 ex_to<spinidx>(*ip).is_dotted()))
1400                                         -> setflag(status_flags::dynallocated);
1401                         else if (is_exactly_a<varidx>(*ip))
1402                                 newidx = (new varidx(newsym, ex_to<varidx>(*ip).get_dim(),
1403                                                 ex_to<varidx>(*ip).is_covariant()))
1404                                         -> setflag(status_flags::dynallocated);
1405                         else
1406                                 newidx = (new idx(newsym, ex_to<idx>(*ip).get_dim()))
1407                                         -> setflag(status_flags::dynallocated);
1408                         old_indices.push_back(*ip);
1409                         new_indices.push_back(newidx);
1410                         if(is_a<varidx>(*ip)) {
1411                                 old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
1412                                 new_indices.push_back(ex_to<varidx>(newidx).toggle_variance());
1413                         }
1414                         ++ip;
1415                 }
1416                 return lst(lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end()));
1417         }
1418 }
1419
1420 ex rename_dummy_indices_uniquely(const exvector & va, const exvector & vb, const ex & b)
1421 {
1422         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1423         return (indices_subs.op(0).nops()>0 ? b.subs((lst)indices_subs.op(0), (lst)indices_subs.op(1), subs_options::no_pattern) : b);
1424 }
1425
1426 ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
1427 {
1428         exvector va = get_all_dummy_indices(a);
1429         if (va.size() > 0) {
1430                 exvector vb = get_all_dummy_indices(b);
1431                 if (vb.size() > 0) {
1432                         sort(va.begin(), va.end(), ex_is_less());
1433                         sort(vb.begin(), vb.end(), ex_is_less());
1434                         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1435                         if (indices_subs.op(0).nops() > 0)
1436                                 return b.subs((lst)indices_subs.op(0), (lst)indices_subs.op(1), subs_options::no_pattern);
1437                 }
1438         }
1439         return b;
1440 }
1441
1442 ex rename_dummy_indices_uniquely(exvector & va, const ex & b, bool modify_va)
1443 {
1444         if (va.size() > 0) {
1445                 exvector vb = get_all_dummy_indices(b);
1446                 if (vb.size() > 0) {
1447                         sort(vb.begin(), vb.end(), ex_is_less());
1448                         lst indices_subs = rename_dummy_indices_uniquely(va, vb);
1449                         if (indices_subs.op(0).nops() > 0) {
1450                                 if (modify_va) {
1451                                         for (lst::const_iterator i = ex_to<lst>(indices_subs.op(1)).begin(); i != ex_to<lst>(indices_subs.op(1)).end(); ++i)
1452                                                 va.push_back(*i);
1453                                         exvector uncommon_indices;
1454                                         set_difference(vb.begin(), vb.end(), indices_subs.op(0).begin(), indices_subs.op(0).end(), std::back_insert_iterator<exvector>(uncommon_indices), ex_is_less());
1455                                         exvector::const_iterator ip = uncommon_indices.begin(), ipend = uncommon_indices.end();
1456                                         while (ip != ipend) {
1457                                                 va.push_back(*ip);
1458                                                 ++ip;
1459                                         }
1460                                         sort(va.begin(), va.end(), ex_is_less());
1461                                 }
1462                                 return b.subs((lst)indices_subs.op(0), (lst)indices_subs.op(1), subs_options::no_pattern);
1463                         }
1464                 }
1465         }
1466         return b;
1467 }
1468
1469 ex expand_dummy_sum(const ex & e, bool subs_idx)
1470 {
1471         ex e_expanded = e.expand();
1472         pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
1473         if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
1474                 return e_expanded.map(fcn);
1475         } else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded)) {
1476                 exvector v = get_all_dummy_indices(e_expanded);
1477                 exvector::const_iterator it = v.begin(), itend = v.end();
1478                 while (it != itend) {
1479                         varidx nu = ex_to<varidx>(*it);
1480                         if (nu.is_dim_numeric()) {
1481                                 ex en = 0;
1482                                 for (int i=0; i < ex_to<numeric>(nu.get_dim()).to_int(); i++) {
1483                                         if (is_a<varidx>(nu) && !subs_idx) {
1484                                                 en += e_expanded.subs(lst(nu == varidx(i, nu.get_dim(), true), nu.toggle_variance() == varidx(i, nu.get_dim())));
1485                                         } else {
1486                                                 en += e_expanded.subs(lst(nu == idx(i, nu.get_dim()), nu.toggle_variance() == idx(i, nu.get_dim())));
1487                                         }
1488                                 }
1489                                 return expand_dummy_sum(en, subs_idx);
1490                         }
1491                         ++it;
1492                 }
1493                 return e;
1494         } else if (is_a<indexed>(e_expanded)) {
1495                 exvector v = ex_to<indexed>(e_expanded).get_dummy_indices();
1496                 exvector::const_iterator it = v.begin(), itend = v.end();
1497                 while (it != itend) {
1498                         varidx nu = ex_to<varidx>(*it);
1499                         if (nu.is_dim_numeric()) {
1500                                 ex en = 0;
1501                                 for (int i=0; i < ex_to<numeric>(nu.get_dim()).to_int(); i++) {
1502                                         if (is_a<varidx>(nu) && !subs_idx) {
1503                                                 en += e_expanded.subs(lst(nu == varidx(i, nu.get_dim(), true), nu.toggle_variance() == varidx(i, nu.get_dim())));
1504                                         } else {
1505                                                 en += e_expanded.subs(lst(nu == idx(i, nu.get_dim()), nu.toggle_variance() == idx(i, nu.get_dim())));
1506                                         }
1507                                 }
1508                                 return expand_dummy_sum(en, subs_idx);
1509                         }
1510                         ++it;
1511                 }
1512                 return e;
1513         } else {
1514                 return e;
1515         }
1516 }
1517
1518 } // namespace GiNaC