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         // Canonicalize indices according to the symmetry properties
 301         if (seq.size() > 2) {
 302                 exvector v = seq;
 303                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 304                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 305                 if (sig != INT_MAX) {
 306                         // Something has changed while sorting indices, more evaluations later
 307                         if (sig == 0)
 308                                 return _ex0;
 309                         return ex(sig) * thiscontainer(v);
 310                 }
 311         }
 312
 313         // Let the class of the base object perform additional evaluations
 314         return ex_to<basic>(base).eval_indexed(*this);
 315 }
 316
 317 ex indexed::thiscontainer(const exvector & v) const
 318 {
 319         return indexed(ex_to<symmetry>(symtree), v);
 320 }
 321
 322 ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
 323 {
 324         return indexed(ex_to<symmetry>(symtree), vp);
 325 }
 326
 327 ex indexed::expand(unsigned options) const
 328 {
 329         GINAC_ASSERT(seq.size() > 0);
 330
 331         if (options & expand_options::expand_indexed) {
 332                 ex newbase = seq[0].expand(options);
 333                 if (is_exactly_a<add>(newbase)) {
 334                         ex sum = _ex0;
 335                         for (size_t i=0; i<newbase.nops(); i++) {
 336                                 exvector s = seq;
 337                                 s[0] = newbase.op(i);
 338                                 sum += thiscontainer(s).expand(options);
 339                         }
 340                         return sum;
 341                 }
 342                 if (!are_ex_trivially_equal(newbase, seq[0])) {
 343                         exvector s = seq;
 344                         s[0] = newbase;
 345                         return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
 346                 }
 347         }
 348         return inherited::expand(options);
 349 }
 350
 351 //////////
 352 // virtual functions which can be overridden by derived classes
 353 //////////
 354
 355 // none
 356
 357 //////////
 358 // non-virtual functions in this class
 359 //////////
 360
 361 /** Check whether all indices are of class idx and validate the symmetry
 362  *  tree. This function is used internally to make sure that all constructed
 363  *  indexed objects really carry indices and not some other classes. */
 364 void indexed::validate() const
 365 {
 366         GINAC_ASSERT(seq.size() > 0);
 367         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 368         while (it != itend) {
 369                 if (!is_a<idx>(*it))
 370                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 371                 it++;
 372         }
 373
 374         if (!symtree.is_zero()) {
 375                 if (!is_exactly_a<symmetry>(symtree))
 376                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 377                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 378         }
 379 }
 380
 381 /** Implementation of ex::diff() for an indexed object always returns 0.
 382  *
 383  *  @see ex::diff */
 384 ex indexed::derivative(const symbol & s) const
 385 {
 386         return _ex0;
 387 }
 388
 389 //////////
 390 // global functions
 391 //////////
 392
 393 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
 394         bool operator() (const ex &lh, const ex &rh) const
 395         {
 396                 if (lh.is_equal(rh))
 397                         return true;
 398                 else
 399                         try {
 400                                 // Replacing the dimension might cause an error (e.g. with
 401                                 // index classes that only work in a fixed number of dimensions)
 402                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
 403                         } catch (...) {
 404                                 return false;
 405                         }
 406         }
 407 };
 408
 409 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 410 static bool indices_consistent(const exvector & v1, const exvector & v2)
 411 {
 412         // Number of indices must be the same
 413         if (v1.size() != v2.size())
 414                 return false;
 415
 416         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
 417 }
 418
 419 exvector indexed::get_indices() const
 420 {
 421         GINAC_ASSERT(seq.size() >= 1);
 422         return exvector(seq.begin() + 1, seq.end());
 423 }
 424
 425 exvector indexed::get_dummy_indices() const
 426 {
 427         exvector free_indices, dummy_indices;
 428         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 429         return dummy_indices;
 430 }
 431
 432 exvector indexed::get_dummy_indices(const indexed & other) const
 433 {
 434         exvector indices = get_free_indices();
 435         exvector other_indices = other.get_free_indices();
 436         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 437         exvector dummy_indices;
 438         find_dummy_indices(indices, dummy_indices);
 439         return dummy_indices;
 440 }
 441
 442 bool indexed::has_dummy_index_for(const ex & i) const
 443 {
 444         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 445         while (it != itend) {
 446                 if (is_dummy_pair(*it, i))
 447                         return true;
 448                 it++;
 449         }
 450         return false;
 451 }
 452
 453 exvector indexed::get_free_indices() const
 454 {
 455         exvector free_indices, dummy_indices;
 456         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 457         return free_indices;
 458 }
 459
 460 exvector add::get_free_indices() const
 461 {
 462         exvector free_indices;
 463         for (size_t i=0; i<nops(); i++) {
 464                 if (i == 0)
 465                         free_indices = op(i).get_free_indices();
 466                 else {
 467                         exvector free_indices_of_term = op(i).get_free_indices();
 468                         if (!indices_consistent(free_indices, free_indices_of_term))
 469                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 470                 }
 471         }
 472         return free_indices;
 473 }
 474
 475 exvector mul::get_free_indices() const
 476 {
 477         // Concatenate free indices of all factors
 478         exvector un;
 479         for (size_t i=0; i<nops(); i++) {
 480                 exvector free_indices_of_factor = op(i).get_free_indices();
 481                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 482         }
 483
 484         // And remove the dummy indices
 485         exvector free_indices, dummy_indices;
 486         find_free_and_dummy(un, free_indices, dummy_indices);
 487         return free_indices;
 488 }
 489
 490 exvector ncmul::get_free_indices() const
 491 {
 492         // Concatenate free indices of all factors
 493         exvector un;
 494         for (size_t i=0; i<nops(); i++) {
 495                 exvector free_indices_of_factor = op(i).get_free_indices();
 496                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 497         }
 498
 499         // And remove the dummy indices
 500         exvector free_indices, dummy_indices;
 501         find_free_and_dummy(un, free_indices, dummy_indices);
 502         return free_indices;
 503 }
 504
 505 struct is_summation_idx : public std::unary_function<ex, bool> {
 506         bool operator()(const ex & e)
 507         {
 508                 return is_dummy_pair(e, e);
 509         }
 510 };
 511
 512 exvector power::get_free_indices() const
 513 {
 514         // Get free indices of basis
 515         exvector basis_indices = basis.get_free_indices();
 516
 517         if (exponent.info(info_flags::even)) {
 518                 // If the exponent is an even number, then any "free" index that
 519                 // forms a dummy pair with itself is actually a summation index
 520                 exvector really_free;
 521                 std::remove_copy_if(basis_indices.begin(), basis_indices.end(),
 522                                     std::back_inserter(really_free), is_summation_idx());
 523                 return really_free;
 524         } else
 525                 return basis_indices;
 526 }
 527
 528 exvector integral::get_free_indices() const
 529 {
 530         if (a.get_free_indices().size() || b.get_free_indices().size())
 531                 throw (std::runtime_error("integral::get_free_indices: boundary values should not have free indices"));
 532         return f.get_free_indices();
 533 }
 534
 535 /** Rename dummy indices in an expression.
 536  *
 537  *  @param e Expression to work on
 538  *  @param local_dummy_indices The set of dummy indices that appear in the
 539  *    expression "e"
 540  *  @param global_dummy_indices The set of dummy indices that have appeared
 541  *    before and which we would like to use in "e", too. This gets updated
 542  *    by the function */
 543 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 544 {
 545         size_t global_size = global_dummy_indices.size(),
 546                local_size = local_dummy_indices.size();
 547
 548         // Any local dummy indices at all?
 549         if (local_size == 0)
 550                 return e;
 551
 552         if (global_size < local_size) {
 553
 554                 // More local indices than we encountered before, add the new ones
 555                 // to the global set
 556                 size_t old_global_size = global_size;
 557                 int remaining = local_size - global_size;
 558                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 559                 while (it != itend && remaining > 0) {
 560                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(op0_is_equal(), *it)) == global_dummy_indices.end()) {
 561                                 global_dummy_indices.push_back(*it);
 562                                 global_size++;
 563                                 remaining--;
 564                         }
 565                         it++;
 566                 }
 567
 568                 // If this is the first set of local indices, do nothing
 569                 if (old_global_size == 0)
 570                         return e;
 571         }
 572         GINAC_ASSERT(local_size <= global_size);
 573
 574         // Construct vectors of index symbols
 575         exvector local_syms, global_syms;
 576         local_syms.reserve(local_size);
 577         global_syms.reserve(local_size);
 578         for (size_t i=0; i<local_size; i++)
 579                 local_syms.push_back(local_dummy_indices[i].op(0));
 580         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 581         for (size_t i=0; i<local_size; i++) // don't use more global symbols than necessary
 582                 global_syms.push_back(global_dummy_indices[i].op(0));
 583         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 584
 585         // Remove common indices
 586         exvector local_uniq, global_uniq;
 587         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
 588         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
 589
 590         // Replace remaining non-common local index symbols by global ones
 591         if (local_uniq.empty())
 592                 return e;
 593         else {
 594                 while (global_uniq.size() > local_uniq.size())
 595                         global_uniq.pop_back();
 596                 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
 597         }
 598 }
 599
 600 /** Given a set of indices, extract those of class varidx. */
 601 static void find_variant_indices(const exvector & v, exvector & variant_indices)
 602 {
 603         exvector::const_iterator it1, itend;
 604         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 605                 if (is_exactly_a<varidx>(*it1))
 606                         variant_indices.push_back(*it1);
 607         }
 608 }
 609
 610 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
 611  *  variance.
 612  *
 613  *  @param e Object to work on
 614  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
 615  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
 616  *  @return true if 'e' was changed */
 617 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
 618 {
 619         bool something_changed = false;
 620
 621         // If a dummy index is encountered for the first time in the
 622         // product, pull it up, otherwise, pull it down
 623         exvector::const_iterator it2, it2start, it2end;
 624         for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
 625                 if (!is_exactly_a<varidx>(*it2))
 626                         continue;
 627
 628                 exvector::iterator vit, vitend;
 629                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 630                         if (it2->op(0).is_equal(vit->op(0))) {
 631                                 if (ex_to<varidx>(*it2).is_covariant()) {
 632                                         e = e.subs(lst(
 633                                                 *it2 == ex_to<varidx>(*it2).toggle_variance(),
 634                                                 ex_to<varidx>(*it2).toggle_variance() == *it2
 635                                         ), subs_options::no_pattern);
 636                                         something_changed = true;
 637                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 638                                         it2start = ex_to<indexed>(e).seq.begin();
 639                                         it2end = ex_to<indexed>(e).seq.end();
 640                                 }
 641                                 moved_indices.push_back(*vit);
 642                                 variant_dummy_indices.erase(vit);
 643                                 goto next_index;
 644                         }
 645                 }
 646
 647                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 648                         if (it2->op(0).is_equal(vit->op(0))) {
 649                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 650                                         e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance(), subs_options::no_pattern);
 651                                         something_changed = true;
 652                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 653                                         it2start = ex_to<indexed>(e).seq.begin();
 654                                         it2end = ex_to<indexed>(e).seq.end();
 655                                 }
 656                                 goto next_index;
 657                         }
 658                 }
 659
 660 next_index: ;
 661         }
 662
 663         return something_changed;
 664 }
 665
 666 /* Ordering that only compares the base expressions of indexed objects. */
 667 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
 668         bool operator() (const ex &lh, const ex &rh) const
 669         {
 670                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 671         }
 672 };
 673
 674 /* An auxiliary function used by simplify_indexed() and expand_dummy_sum()
 675  * It returns an exvector of factors from the supplied product */
 676 static void product_to_exvector(const ex & e, exvector & v, bool & non_commutative)
 677 {
 678         // Remember whether the product was commutative or noncommutative
 679         // (because we chop it into factors and need to reassemble later)
 680         non_commutative = is_exactly_a<ncmul>(e);
 681
 682         // Collect factors in an exvector, store squares twice
 683         v.reserve(e.nops() * 2);
 684
 685         if (is_exactly_a<power>(e)) {
 686                 // We only get called for simple squares, split a^2 -> a*a
 687                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 688                 v.push_back(e.op(0));
 689                 v.push_back(e.op(0));
 690         } else {
 691                 for (size_t i=0; i<e.nops(); i++) {
 692                         ex f = e.op(i);
 693                         if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
 694                                 v.push_back(f.op(0));
 695                                 v.push_back(f.op(0));
 696                         } else if (is_exactly_a<ncmul>(f)) {
 697                                 // Noncommutative factor found, split it as well
 698                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 699                                 for (size_t j=0; j<f.nops(); j++)
 700                                         v.push_back(f.op(j));
 701                         } else
 702                                 v.push_back(f);
 703                 }
 704         }
 705 }
 706
 707 /** Simplify product of indexed expressions (commutative, noncommutative and
 708  *  simple squares), return list of free indices. */
 709 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 710 {
 711         // Collect factors in an exvector
 712         exvector v;
 713
 714         // Remember whether the product was commutative or noncommutative
 715         // (because we chop it into factors and need to reassemble later)
 716         bool non_commutative;
 717         product_to_exvector(e, v, non_commutative);
 718
 719         // Perform contractions
 720         bool something_changed = false;
 721         GINAC_ASSERT(v.size() > 1);
 722         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 723         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 724
 725 try_again:
 726                 if (!is_a<indexed>(*it1))
 727                         continue;
 728
 729                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 730
 731                 // Indexed factor found, get free indices and look for contraction
 732                 // candidates
 733                 exvector free1, dummy1;
 734                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 735
 736                 exvector::iterator it2;
 737                 for (it2 = it1 + 1; it2 != itend; it2++) {
 738
 739                         if (!is_a<indexed>(*it2))
 740                                 continue;
 741
 742                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 743
 744                         // Find free indices of second factor and merge them with free
 745                         // indices of first factor
 746                         exvector un;
 747                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 748                         un.insert(un.end(), free1.begin(), free1.end());
 749
 750                         // Check whether the two factors share dummy indices
 751                         exvector free, dummy;
 752                         find_free_and_dummy(un, free, dummy);
 753                         size_t num_dummies = dummy.size();
 754                         if (num_dummies == 0)
 755                                 continue;
 756
 757                         // At least one dummy index, is it a defined scalar product?
 758                         bool contracted = false;
 759                         if (free.empty()) {
 760
 761                                 // Find minimal dimension of all indices of both factors
 762                                 exvector::const_iterator dit = ex_to<indexed>(*it1).seq.begin() + 1, ditend = ex_to<indexed>(*it1).seq.end();
 763                                 ex dim = ex_to<idx>(*dit).get_dim();
 764                                 ++dit;
 765                                 for (; dit != ditend; ++dit) {
 766                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 767                                 }
 768                                 dit = ex_to<indexed>(*it2).seq.begin() + 1;
 769                                 ditend = ex_to<indexed>(*it2).seq.end();
 770                                 for (; dit != ditend; ++dit) {
 771                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 772                                 }
 773
 774                                 // User-defined scalar product?
 775                                 if (sp.is_defined(*it1, *it2, dim)) {
 776
 777                                         // Yes, substitute it
 778                                         *it1 = sp.evaluate(*it1, *it2, dim);
 779                                         *it2 = _ex1;
 780                                         goto contraction_done;
 781                                 }
 782                         }
 783
 784                         // Try to contract the first one with the second one
 785                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 786                         if (!contracted) {
 787
 788                                 // That didn't work; maybe the second object knows how to
 789                                 // contract itself with the first one
 790                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 791                         }
 792                         if (contracted) {
 793 contraction_done:
 794                                 if (first_noncommutative || second_noncommutative
 795                                  || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
 796                                  || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
 797                                  || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
 798
 799                                         // One of the factors became a sum or product:
 800                                         // re-expand expression and run again
 801                                         // Non-commutative products are always re-expanded to give
 802                                         // eval_ncmul() the chance to re-order and canonicalize
 803                                         // the product
 804                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 805                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 806                                 }
 807
 808                                 // Both objects may have new indices now or they might
 809                                 // even not be indexed objects any more, so we have to
 810                                 // start over
 811                                 something_changed = true;
 812                                 goto try_again;
 813                         }
 814                 }
 815         }
 816
 817         // Find free indices (concatenate them all and call find_free_and_dummy())
 818         // and all dummy indices that appear
 819         exvector un, individual_dummy_indices;
 820         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 821                 exvector free_indices_of_factor;
 822                 if (is_a<indexed>(*it1)) {
 823                         exvector dummy_indices_of_factor;
 824                         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);
 825                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 826                 } else
 827                         free_indices_of_factor = it1->get_free_indices();
 828                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 829         }
 830         exvector local_dummy_indices;
 831         find_free_and_dummy(un, free_indices, local_dummy_indices);
 832         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 833
 834         // Filter out the dummy indices with variance
 835         exvector variant_dummy_indices;
 836         find_variant_indices(local_dummy_indices, variant_dummy_indices);
 837
 838         // Any indices with variance present at all?
 839         if (!variant_dummy_indices.empty()) {
 840
 841                 // Yes, bring the product into a canonical order that only depends on
 842                 // the base expressions of indexed objects
 843                 if (!non_commutative)
 844                         std::sort(v.begin(), v.end(), ex_base_is_less());
 845
 846                 exvector moved_indices;
 847
 848                 // Iterate over all indexed objects in the product
 849                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 850                         if (!is_a<indexed>(*it1))
 851                                 continue;
 852
 853                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
 854                                 something_changed = true;
 855                 }
 856         }
 857
 858         ex r;
 859         if (something_changed)
 860                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 861         else
 862                 r = e;
 863
 864         // The result should be symmetric with respect to exchange of dummy
 865         // indices, so if the symmetrization vanishes, the whole expression is
 866         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 867         if (local_dummy_indices.size() >= 2) {
 868                 exvector dummy_syms;
 869                 dummy_syms.reserve(local_dummy_indices.size());
 870                 for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
 871                         dummy_syms.push_back(it->op(0));
 872                 if (symmetrize(r, dummy_syms).is_zero()) {
 873                         free_indices.clear();
 874                         return _ex0;
 875                 }
 876         }
 877
 878         // Dummy index renaming
 879         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 880
 881         // Product of indexed object with a scalar?
 882         if (is_exactly_a<mul>(r) && r.nops() == 2
 883          && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
 884                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 885         else
 886                 return r;
 887 }
 888
 889 /** This structure stores the original and symmetrized versions of terms
 890  *  obtained during the simplification of sums. */
 891 class terminfo {
 892 public:
 893         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
 894
 895         ex orig; /**< original term */
 896         ex symm; /**< symmtrized term */
 897 };
 898
 899 class terminfo_is_less {
 900 public:
 901         bool operator() (const terminfo & ti1, const terminfo & ti2) const
 902         {
 903                 return (ti1.symm.compare(ti2.symm) < 0);
 904         }
 905 };
 906
 907 /** This structure stores the individual symmetrized terms obtained during
 908  *  the simplification of sums. */
 909 class symminfo {
 910 public:
 911         symminfo() : num(0) {}
 912
 913         symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
 914         {
 915                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
 916                         coeff = symmterm_.op(symmterm_.nops()-1);
 917                         symmterm = symmterm_ / coeff;
 918                 } else {
 919                         coeff = 1;
 920                         symmterm = symmterm_;
 921                 }
 922         }
 923
 924         ex symmterm;  /**< symmetrized term */
 925         ex coeff;     /**< coefficient of symmetrized term */
 926         ex orig;      /**< original term */
 927         size_t num; /**< how many symmetrized terms resulted from the original term */
 928 };
 929
 930 class symminfo_is_less_by_symmterm {
 931 public:
 932         bool operator() (const symminfo & si1, const symminfo & si2) const
 933         {
 934                 return (si1.symmterm.compare(si2.symmterm) < 0);
 935         }
 936 };
 937
 938 class symminfo_is_less_by_orig {
 939 public:
 940         bool operator() (const symminfo & si1, const symminfo & si2) const
 941         {
 942                 return (si1.orig.compare(si2.orig) < 0);
 943         }
 944 };
 945
 946 /** Simplify indexed expression, return list of free indices. */
 947 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 948 {
 949         // Expand the expression
 950         ex e_expanded = e.expand();
 951
 952         // Simplification of single indexed object: just find the free indices
 953         // and perform dummy index renaming/repositioning
 954         if (is_a<indexed>(e_expanded)) {
 955
 956                 // Find the dummy indices
 957                 const indexed &i = ex_to<indexed>(e_expanded);
 958                 exvector local_dummy_indices;
 959                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 960
 961                 // Filter out the dummy indices with variance
 962                 exvector variant_dummy_indices;
 963                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
 964
 965                 // Any indices with variance present at all?
 966                 if (!variant_dummy_indices.empty()) {
 967
 968                         // Yes, reposition them
 969                         exvector moved_indices;
 970                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
 971                 }
 972
 973                 // Rename the dummy indices
 974                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 975         }
 976
 977         // Simplification of sum = sum of simplifications, check consistency of
 978         // free indices in each term
 979         if (is_exactly_a<add>(e_expanded)) {
 980                 bool first = true;
 981                 ex sum;
 982                 free_indices.clear();
 983
 984                 for (size_t i=0; i<e_expanded.nops(); i++) {
 985                         exvector free_indices_of_term;
 986                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 987                         if (!term.is_zero()) {
 988                                 if (first) {
 989                                         free_indices = free_indices_of_term;
 990                                         sum = term;
 991                                         first = false;
 992                                 } else {
 993                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
 994                                                 std::ostringstream s;
 995                                                 s << "simplify_indexed: inconsistent indices in sum: ";
 996                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
 997                                                 throw (std::runtime_error(s.str()));
 998                                         }
 999                                         if (is_a<indexed>(sum) && is_a<indexed>(term))
1000                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
1001                                         else
1002                                                 sum += term;
1003                                 }
1004                         }
1005                 }
1006
1007                 // If the sum turns out to be zero, we are finished
1008                 if (sum.is_zero()) {
1009                         free_indices.clear();
1010                         return sum;
1011                 }
1012
1013                 // More than one term and more than one dummy index?
1014                 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
1015                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
1016                         return sum;
1017
1018                 // Yes, construct vector of all dummy index symbols
1019                 exvector dummy_syms;
1020                 dummy_syms.reserve(dummy_indices.size());
1021                 for (exvector::const_iterator it = dummy_indices.begin(); it != dummy_indices.end(); ++it)
1022                         dummy_syms.push_back(it->op(0));
1023
1024                 // Chop the sum into terms and symmetrize each one over the dummy
1025                 // indices
1026                 std::vector<terminfo> terms;
1027                 for (size_t i=0; i<sum.nops(); i++) {
1028                         const ex & term = sum.op(i);
1029                         ex term_symm = symmetrize(term, dummy_syms);
1030                         if (term_symm.is_zero())
1031                                 continue;
1032                         terms.push_back(terminfo(term, term_symm));
1033                 }
1034
1035                 // Sort by symmetrized terms
1036                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
1037
1038                 // Combine equal symmetrized terms
1039                 std::vector<terminfo> terms_pass2;
1040                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1041                         size_t num = 1;
1042                         std::vector<terminfo>::const_iterator j = i + 1;
1043                         while (j != terms.end() && j->symm == i->symm) {
1044                                 num++;
1045                                 j++;
1046                         }
1047                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1048                         i = j;
1049                 }
1050
1051                 // If there is only one term left, we are finished
1052                 if (terms_pass2.size() == 1)
1053                         return terms_pass2[0].orig;
1054
1055                 // Chop the symmetrized terms into subterms
1056                 std::vector<symminfo> sy;
1057                 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1058                         if (is_exactly_a<add>(i->symm)) {
1059                                 size_t num = i->symm.nops();
1060                                 for (size_t j=0; j<num; j++)
1061                                         sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1062                         } else
1063                                 sy.push_back(symminfo(i->symm, i->orig, 1));
1064                 }
1065
1066                 // Sort by symmetrized subterms
1067                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1068
1069                 // Combine equal symmetrized subterms
1070                 std::vector<symminfo> sy_pass2;
1071                 exvector result;
1072                 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1073
1074                         // Combine equal terms
1075                         std::vector<symminfo>::const_iterator j = i + 1;
1076                         if (j != sy.end() && j->symmterm == i->symmterm) {
1077
1078                                 // More than one term, collect the coefficients
1079                                 ex coeff = i->coeff;
1080                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1081                                         coeff += j->coeff;
1082                                         j++;
1083                                 }
1084
1085                                 // Add combined term to result
1086                                 if (!coeff.is_zero())
1087                                         result.push_back(coeff * i->symmterm);
1088
1089                         } else {
1090
1091                                 // Single term, store for second pass
1092                                 sy_pass2.push_back(*i);
1093                         }
1094
1095                         i = j;
1096                 }
1097
1098                 // Were there any remaining terms that didn't get combined?
1099                 if (sy_pass2.size() > 0) {
1100
1101                         // Yes, sort by their original terms
1102                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1103
1104                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1105
1106                                 // How many symmetrized terms of this original term are left?
1107                                 size_t num = 1;
1108                                 std::vector<symminfo>::const_iterator j = i + 1;
1109                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1110                                         num++;
1111                                         j++;
1112                                 }
1113
1114                                 if (num == i->num) {
1115
1116                                         // All terms left, then add the original term to the result
1117                                         result.push_back(i->orig);
1118
1119                                 } else {
1120
1121                                         // Some terms were combined with others, add up the remaining symmetrized terms
1122                                         std::vector<symminfo>::const_iterator k;
1123                                         for (k=i; k!=j; k++)
1124                                                 result.push_back(k->coeff * k->symmterm);
1125                                 }
1126
1127                                 i = j;
1128                         }
1129                 }
1130
1131                 // Add all resulting terms
1132                 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1133                 if (sum_symm.is_zero())
1134                         free_indices.clear();
1135                 return sum_symm;
1136         }
1137
1138         // Simplification of products
1139         if (is_exactly_a<mul>(e_expanded)
1140          || is_exactly_a<ncmul>(e_expanded)
1141          || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1142                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1143
1144         // Cannot do anything
1145         free_indices.clear();
1146         return e_expanded;
1147 }
1148
1149 /** Simplify/canonicalize expression containing indexed objects. This
1150  *  performs contraction of dummy indices where possible and checks whether
1151  *  the free indices in sums are consistent.
1152  *
1153  *  @param options Simplification options (currently unused)
1154  *  @return simplified expression */
1155 ex ex::simplify_indexed(unsigned options) const
1156 {
1157         exvector free_indices, dummy_indices;
1158         scalar_products sp;
1159         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1160 }
1161
1162 /** Simplify/canonicalize expression containing indexed objects. This
1163  *  performs contraction of dummy indices where possible, checks whether
1164  *  the free indices in sums are consistent, and automatically replaces
1165  *  scalar products by known values if desired.
1166  *
1167  *  @param sp Scalar products to be replaced automatically
1168  *  @param options Simplification options (currently unused)
1169  *  @return simplified expression */
1170 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1171 {
1172         exvector free_indices, dummy_indices;
1173         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1174 }
1175
1176 /** Symmetrize expression over its free indices. */
1177 ex ex::symmetrize() const
1178 {
1179         return GiNaC::symmetrize(*this, get_free_indices());
1180 }
1181
1182 /** Antisymmetrize expression over its free indices. */
1183 ex ex::antisymmetrize() const
1184 {
1185         return GiNaC::antisymmetrize(*this, get_free_indices());
1186 }
1187
1188 /** Symmetrize expression by cyclic permutation over its free indices. */
1189 ex ex::symmetrize_cyclic() const
1190 {
1191         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1192 }
1193
1194 //////////
1195 // helper classes
1196 //////////
1197
1198 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1199 {
1200         // If indexed, extract base objects
1201         ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1202         ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1203
1204         // Enforce canonical order in pair
1205         if (s1.compare(s2) > 0) {
1206                 v1 = s2;
1207                 v2 = s1;
1208         } else {
1209                 v1 = s1;
1210                 v2 = s2;
1211         }
1212 }
1213
1214 bool spmapkey::operator==(const spmapkey &other) const
1215 {
1216         if (!v1.is_equal(other.v1))
1217                 return false;
1218         if (!v2.is_equal(other.v2))
1219                 return false;
1220         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1221                 return true;
1222         else
1223                 return dim.is_equal(other.dim);
1224 }
1225
1226 bool spmapkey::operator<(const spmapkey &other) const
1227 {
1228         int cmp = v1.compare(other.v1);
1229         if (cmp)
1230                 return cmp < 0;
1231         cmp = v2.compare(other.v2);
1232         if (cmp)
1233                 return cmp < 0;
1234
1235         // Objects are equal, now check dimensions
1236         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1237                 return false;
1238         else
1239                 return dim.compare(other.dim) < 0;
1240 }
1241
1242 void spmapkey::debugprint() const
1243 {
1244         std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1245 }
1246
1247 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1248 {
1249         spm[spmapkey(v1, v2)] = sp;
1250 }
1251
1252 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1253 {
1254         spm[spmapkey(v1, v2, dim)] = sp;
1255 }
1256
1257 void scalar_products::add_vectors(const lst & l, const ex & dim)
1258 {
1259         // Add all possible pairs of products
1260         for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1261                 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1262                         add(*it1, *it2, *it1 * *it2);
1263 }
1264
1265 void scalar_products::clear()
1266 {
1267         spm.clear();
1268 }
1269
1270 /** Check whether scalar product pair is defined. */
1271 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1272 {
1273         return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1274 }
1275
1276 /** Return value of defined scalar product pair. */
1277 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1278 {
1279         return spm.find(spmapkey(v1, v2, dim))->second;
1280 }
1281
1282 void scalar_products::debugprint() const
1283 {
1284         std::cerr << "map size=" << spm.size() << std::endl;
1285         spmap::const_iterator i = spm.begin(), end = spm.end();
1286         while (i != end) {
1287                 const spmapkey & k = i->first;
1288                 std::cerr << "item key=";
1289                 k.debugprint();
1290                 std::cerr << ", value=" << i->second << std::endl;
1291                 ++i;
1292         }
1293 }
1294
1295 /** Returns all dummy indices from the exvector */
1296 exvector get_all_dummy_indices(const ex & e)
1297 {
1298         exvector p;
1299         bool nc;
1300         product_to_exvector(e, p, nc);
1301         exvector::const_iterator ip = p.begin(), ipend = p.end();
1302         exvector v, v1;
1303         while (ip != ipend) {
1304                 if (is_a<indexed>(*ip)) {
1305                         v1 = ex_to<indexed>(*ip).get_dummy_indices();
1306                         v.insert(v.end(), v1.begin(), v1.end());
1307                         exvector::const_iterator ip1 = ip+1;
1308                         while (ip1 != ipend) {
1309                                 if (is_a<indexed>(*ip1)) {
1310                                         v1 = ex_to<indexed>(*ip).get_dummy_indices(ex_to<indexed>(*ip1));
1311                                         v.insert(v.end(), v1.begin(), v1.end());
1312                                 }
1313                                 ++ip1;
1314                         }
1315                 }
1316                 ++ip;
1317         }
1318         return v;
1319 }
1320
1321 ex rename_dummy_indices_uniquely(const ex & a, const ex & b)
1322 {
1323         exvector va = get_all_dummy_indices(a), vb = get_all_dummy_indices(b), common_indices;
1324         set_intersection(va.begin(), va.end(), vb.begin(), vb.end(), std::back_insert_iterator<exvector>(common_indices), ex_is_less());
1325         if (common_indices.empty()) {
1326                 return b;
1327         } else {
1328                 exvector new_indices, old_indices;
1329                 old_indices.reserve(2*common_indices.size());
1330                 new_indices.reserve(2*common_indices.size());
1331                 exvector::const_iterator ip = common_indices.begin(), ipend = common_indices.end();
1332                 while (ip != ipend) {
1333                         if (is_a<varidx>(*ip)) {
1334                                 varidx mu((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(*ip).get_dim(), ex_to<varidx>(*ip).is_covariant());
1335                                 old_indices.push_back(*ip);
1336                                 new_indices.push_back(mu);
1337                                 old_indices.push_back(ex_to<varidx>(*ip).toggle_variance());
1338                                 new_indices.push_back(mu.toggle_variance());
1339                         } else {
1340                                 old_indices.push_back(*ip);
1341                                 new_indices.push_back(idx((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(*ip).get_dim()));
1342                         }
1343                         ++ip;
1344                 }
1345                 return b.subs(lst(old_indices.begin(), old_indices.end()), lst(new_indices.begin(), new_indices.end()), subs_options::no_pattern);
1346         }
1347 }
1348
1349 ex expand_dummy_sum(const ex & e, bool subs_idx)
1350 {
1351         ex e_expanded = e.expand();
1352         pointer_to_map_function_1arg<bool> fcn(expand_dummy_sum, subs_idx);
1353         if (is_a<add>(e_expanded) || is_a<lst>(e_expanded) || is_a<matrix>(e_expanded)) {
1354                 return e_expanded.map(fcn);
1355         } else if (is_a<ncmul>(e_expanded) || is_a<mul>(e_expanded) || is_a<power>(e_expanded)) {
1356                 exvector v = get_all_dummy_indices(e_expanded);
1357                 exvector::const_iterator it = v.begin(), itend = v.end();
1358                 while (it != itend) {
1359                         varidx nu = ex_to<varidx>(*it);
1360                         if (nu.is_dim_numeric()) {
1361                                 ex en = 0;
1362                                 for (int i=0; i < ex_to<numeric>(nu.get_dim()).to_int(); i++) {
1363                                         if (is_a<varidx>(nu) && !subs_idx) {
1364                                                 en += e_expanded.subs(lst(nu == varidx(i, nu.get_dim(), true), nu.toggle_variance() == varidx(i, nu.get_dim())));
1365                                         } else {
1366                                                 en += e_expanded.subs(lst(nu == idx(i, nu.get_dim()), nu.toggle_variance() == idx(i, nu.get_dim())));
1367                                         }
1368                                 }
1369                                 return expand_dummy_sum(en, subs_idx);
1370                         }
1371                         ++it;
1372                 }
1373                 return e;
1374         } else if (is_a<indexed>(e_expanded)) {
1375                 exvector v = ex_to<indexed>(e_expanded).get_dummy_indices();
1376                 exvector::const_iterator it = v.begin(), itend = v.end();
1377                 while (it != itend) {
1378                         varidx nu = ex_to<varidx>(*it);
1379                         if (nu.is_dim_numeric()) {
1380                                 ex en = 0;
1381                                 for (int i=0; i < ex_to<numeric>(nu.get_dim()).to_int(); i++) {
1382                                         if (is_a<varidx>(nu) && !subs_idx) {
1383                                                 en += e_expanded.subs(lst(nu == varidx(i, nu.get_dim(), true), nu.toggle_variance() == varidx(i, nu.get_dim())));
1384                                         } else {
1385                                                 en += e_expanded.subs(lst(nu == idx(i, nu.get_dim()), nu.toggle_variance() == idx(i, nu.get_dim())));
1386                                         }
1387                                 }
1388                                 return expand_dummy_sum(en, subs_idx);
1389                         }
1390                         ++it;
1391                 }
1392                 return e;
1393         } else {
1394                 return e;
1395         }
1396 }
1397
1398 } // namespace GiNaC