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