ginac/indexed.cpp

   1 /** @file indexed.cpp
   2  *
   3  *  Implementation of GiNaC's indexed expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <iostream>
  24 #include <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 "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(indexed, exprseq,
  43   print_func<print_context>(&indexed::do_print).
  44   print_func<print_latex>(&indexed::do_print_latex).
  45   print_func<print_tree>(&indexed::do_print_tree))
  46
  47 //////////
  48 // default constructor
  49 //////////
  50
  51 indexed::indexed() : symtree(sy_none())
  52 {
  53         tinfo_key = TINFO_indexed;
  54 }
  55
  56 //////////
  57 // other constructors
  58 //////////
  59
  60 indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
  61 {
  62         tinfo_key = TINFO_indexed;
  63         validate();
  64 }
  65
  66 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(sy_none())
  67 {
  68         tinfo_key = TINFO_indexed;
  69         validate();
  70 }
  71
  72 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(sy_none())
  73 {
  74         tinfo_key = TINFO_indexed;
  75         validate();
  76 }
  77
  78 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(sy_none())
  79 {
  80         tinfo_key = TINFO_indexed;
  81         validate();
  82 }
  83
  84 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(sy_none())
  85 {
  86         tinfo_key = TINFO_indexed;
  87         validate();
  88 }
  89
  90 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
  91 {
  92         tinfo_key = TINFO_indexed;
  93         validate();
  94 }
  95
  96 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
  97 {
  98         tinfo_key = TINFO_indexed;
  99         validate();
 100 }
 101
 102 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)
 103 {
 104         tinfo_key = TINFO_indexed;
 105         validate();
 106 }
 107
 108 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
 109 {
 110         seq.insert(seq.end(), v.begin(), v.end());
 111         tinfo_key = TINFO_indexed;
 112         validate();
 113 }
 114
 115 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 116 {
 117         seq.insert(seq.end(), v.begin(), v.end());
 118         tinfo_key = TINFO_indexed;
 119         validate();
 120 }
 121
 122 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 123 {
 124         tinfo_key = TINFO_indexed;
 125 }
 126
 127 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 128 {
 129         tinfo_key = TINFO_indexed;
 130 }
 131
 132 indexed::indexed(const symmetry & symm, std::auto_ptr<exvector> vp) : inherited(vp), symtree(symm)
 133 {
 134         tinfo_key = TINFO_indexed;
 135 }
 136
 137 //////////
 138 // archiving
 139 //////////
 140
 141 indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 142 {
 143         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 144                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 145                 unsigned symm = 0;
 146                 n.find_unsigned("symmetry", symm);
 147                 switch (symm) {
 148                         case 1:
 149                                 symtree = sy_symm();
 150                                 break;
 151                         case 2:
 152                                 symtree = sy_anti();
 153                                 break;
 154                         default:
 155                                 symtree = sy_none();
 156                                 break;
 157                 }
 158                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 159         }
 160 }
 161
 162 void indexed::archive(archive_node &n) const
 163 {
 164         inherited::archive(n);
 165         n.add_ex("symmetry", symtree);
 166 }
 167
 168 DEFAULT_UNARCHIVE(indexed)
 169
 170 //////////
 171 // functions overriding virtual functions from base classes
 172 //////////
 173
 174 void indexed::printindices(const print_context & c, unsigned level) const
 175 {
 176         if (seq.size() > 1) {
 177
 178                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 179
 180                 if (is_a<print_latex>(c)) {
 181
 182                         // TeX output: group by variance
 183                         bool first = true;
 184                         bool covariant = true;
 185
 186                         while (it != itend) {
 187                                 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
 188                                 if (first || cur_covariant != covariant) { // Variance changed
 189                                         // The empty {} prevents indices from ending up on top of each other
 190                                         if (!first)
 191                                                 c.s << "}{}";
 192                                         covariant = cur_covariant;
 193                                         if (covariant)
 194                                                 c.s << "_{";
 195                                         else
 196                                                 c.s << "^{";
 197                                 }
 198                                 it->print(c, level);
 199                                 c.s << " ";
 200                                 first = false;
 201                                 it++;
 202                         }
 203                         c.s << "}";
 204
 205                 } else {
 206
 207                         // Ordinary output
 208                         while (it != itend) {
 209                                 it->print(c, level);
 210                                 it++;
 211                         }
 212                 }
 213         }
 214 }
 215
 216 void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
 217 {
 218         if (precedence() <= level)
 219                 c.s << openbrace << '(';
 220         c.s << openbrace;
 221         seq[0].print(c, precedence());
 222         c.s << closebrace;
 223         printindices(c, level);
 224         if (precedence() <= level)
 225                 c.s << ')' << closebrace;
 226 }
 227
 228 void indexed::do_print(const print_context & c, unsigned level) const
 229 {
 230         print_indexed(c, "", "", level);
 231 }
 232
 233 void indexed::do_print_latex(const print_latex & c, unsigned level) const
 234 {
 235         print_indexed(c, "{", "}", level);
 236 }
 237
 238 void indexed::do_print_tree(const print_tree & c, unsigned level) const
 239 {
 240         c.s << std::string(level, ' ') << class_name() << " @" << this
 241             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 242             << ", " << seq.size()-1 << " indices"
 243             << ", symmetry=" << symtree << std::endl;
 244         seq[0].print(c, level + c.delta_indent);
 245         printindices(c, level + c.delta_indent);
 246 }
 247
 248 bool indexed::info(unsigned inf) const
 249 {
 250         if (inf == info_flags::indexed) return true;
 251         if (inf == info_flags::has_indices) return seq.size() > 1;
 252         return inherited::info(inf);
 253 }
 254
 255 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 256         bool operator() (const ex & e, unsigned inf) const {
 257                 return !(ex_to<idx>(e).get_value().info(inf));
 258         }
 259 };
 260
 261 bool indexed::all_index_values_are(unsigned inf) const
 262 {
 263         // No indices? Then no property can be fulfilled
 264         if (seq.size() < 2)
 265                 return false;
 266
 267         // Check all indices
 268         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 269 }
 270
 271 int indexed::compare_same_type(const basic & other) const
 272 {
 273         GINAC_ASSERT(is_a<indexed>(other));
 274         return inherited::compare_same_type(other);
 275 }
 276
 277 ex indexed::eval(int level) const
 278 {
 279         // First evaluate children, then we will end up here again
 280         if (level > 1)
 281                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 282
 283         const ex &base = seq[0];
 284
 285         // If the base object is 0, the whole object is 0
 286         if (base.is_zero())
 287                 return _ex0;
 288
 289         // If the base object is a product, pull out the numeric factor
 290         if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
 291                 exvector v(seq);
 292                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 293                 v[0] = seq[0] / f;
 294                 return f * thiscontainer(v);
 295         }
 296
 297         // Canonicalize indices according to the symmetry properties
 298         if (seq.size() > 2) {
 299                 exvector v = seq;
 300                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 301                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 302                 if (sig != INT_MAX) {
 303                         // Something has changed while sorting indices, more evaluations later
 304                         if (sig == 0)
 305                                 return _ex0;
 306                         return ex(sig) * thiscontainer(v);
 307                 }
 308         }
 309
 310         // Let the class of the base object perform additional evaluations
 311         return ex_to<basic>(base).eval_indexed(*this);
 312 }
 313
 314 ex indexed::thiscontainer(const exvector & v) const
 315 {
 316         return indexed(ex_to<symmetry>(symtree), v);
 317 }
 318
 319 ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
 320 {
 321         return indexed(ex_to<symmetry>(symtree), vp);
 322 }
 323
 324 ex indexed::expand(unsigned options) const
 325 {
 326         GINAC_ASSERT(seq.size() > 0);
 327
 328         if ((options & expand_options::expand_indexed) && is_exactly_a<add>(seq[0])) {
 329
 330                 // expand_indexed expands (a+b).i -> a.i + b.i
 331                 const ex & base = seq[0];
 332                 ex sum = _ex0;
 333                 for (size_t i=0; i<base.nops(); i++) {
 334                         exvector s = seq;
 335                         s[0] = base.op(i);
 336                         sum += thiscontainer(s).expand();
 337                 }
 338                 return sum;
 339
 340         } else
 341                 return inherited::expand(options);
 342 }
 343
 344 //////////
 345 // virtual functions which can be overridden by derived classes
 346 //////////
 347
 348 // none
 349
 350 //////////
 351 // non-virtual functions in this class
 352 //////////
 353
 354 /** Check whether all indices are of class idx and validate the symmetry
 355  *  tree. This function is used internally to make sure that all constructed
 356  *  indexed objects really carry indices and not some other classes. */
 357 void indexed::validate() const
 358 {
 359         GINAC_ASSERT(seq.size() > 0);
 360         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 361         while (it != itend) {
 362                 if (!is_a<idx>(*it))
 363                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 364                 it++;
 365         }
 366
 367         if (!symtree.is_zero()) {
 368                 if (!is_exactly_a<symmetry>(symtree))
 369                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 370                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 371         }
 372 }
 373
 374 /** Implementation of ex::diff() for an indexed object always returns 0.
 375  *
 376  *  @see ex::diff */
 377 ex indexed::derivative(const symbol & s) const
 378 {
 379         return _ex0;
 380 }
 381
 382 //////////
 383 // global functions
 384 //////////
 385
 386 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
 387         bool operator() (const ex &lh, const ex &rh) const
 388         {
 389                 if (lh.is_equal(rh))
 390                         return true;
 391                 else
 392                         try {
 393                                 // Replacing the dimension might cause an error (e.g. with
 394                                 // index classes that only work in a fixed number of dimensions)
 395                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
 396                         } catch (...) {
 397                                 return false;
 398                         }
 399         }
 400 };
 401
 402 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 403 static bool indices_consistent(const exvector & v1, const exvector & v2)
 404 {
 405         // Number of indices must be the same
 406         if (v1.size() != v2.size())
 407                 return false;
 408
 409         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
 410 }
 411
 412 exvector indexed::get_indices() const
 413 {
 414         GINAC_ASSERT(seq.size() >= 1);
 415         return exvector(seq.begin() + 1, seq.end());
 416 }
 417
 418 exvector indexed::get_dummy_indices() const
 419 {
 420         exvector free_indices, dummy_indices;
 421         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 422         return dummy_indices;
 423 }
 424
 425 exvector indexed::get_dummy_indices(const indexed & other) const
 426 {
 427         exvector indices = get_free_indices();
 428         exvector other_indices = other.get_free_indices();
 429         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 430         exvector dummy_indices;
 431         find_dummy_indices(indices, dummy_indices);
 432         return dummy_indices;
 433 }
 434
 435 bool indexed::has_dummy_index_for(const ex & i) const
 436 {
 437         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 438         while (it != itend) {
 439                 if (is_dummy_pair(*it, i))
 440                         return true;
 441                 it++;
 442         }
 443         return false;
 444 }
 445
 446 exvector indexed::get_free_indices() const
 447 {
 448         exvector free_indices, dummy_indices;
 449         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 450         return free_indices;
 451 }
 452
 453 exvector add::get_free_indices() const
 454 {
 455         exvector free_indices;
 456         for (size_t i=0; i<nops(); i++) {
 457                 if (i == 0)
 458                         free_indices = op(i).get_free_indices();
 459                 else {
 460                         exvector free_indices_of_term = op(i).get_free_indices();
 461                         if (!indices_consistent(free_indices, free_indices_of_term))
 462                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 463                 }
 464         }
 465         return free_indices;
 466 }
 467
 468 exvector mul::get_free_indices() const
 469 {
 470         // Concatenate free indices of all factors
 471         exvector un;
 472         for (size_t i=0; i<nops(); i++) {
 473                 exvector free_indices_of_factor = op(i).get_free_indices();
 474                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 475         }
 476
 477         // And remove the dummy indices
 478         exvector free_indices, dummy_indices;
 479         find_free_and_dummy(un, free_indices, dummy_indices);
 480         return free_indices;
 481 }
 482
 483 exvector ncmul::get_free_indices() const
 484 {
 485         // Concatenate free indices of all factors
 486         exvector un;
 487         for (size_t i=0; i<nops(); i++) {
 488                 exvector free_indices_of_factor = op(i).get_free_indices();
 489                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 490         }
 491
 492         // And remove the dummy indices
 493         exvector free_indices, dummy_indices;
 494         find_free_and_dummy(un, free_indices, dummy_indices);
 495         return free_indices;
 496 }
 497
 498 exvector power::get_free_indices() const
 499 {
 500         // Return free indices of basis
 501         return basis.get_free_indices();
 502 }
 503
 504 /** Rename dummy indices in an expression.
 505  *
 506  *  @param e Expression to work on
 507  *  @param local_dummy_indices The set of dummy indices that appear in the
 508  *    expression "e"
 509  *  @param global_dummy_indices The set of dummy indices that have appeared
 510  *    before and which we would like to use in "e", too. This gets updated
 511  *    by the function */
 512 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 513 {
 514         size_t global_size = global_dummy_indices.size(),
 515                local_size = local_dummy_indices.size();
 516
 517         // Any local dummy indices at all?
 518         if (local_size == 0)
 519                 return e;
 520
 521         if (global_size < local_size) {
 522
 523                 // More local indices than we encountered before, add the new ones
 524                 // to the global set
 525                 size_t old_global_size = global_size;
 526                 int remaining = local_size - global_size;
 527                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 528                 while (it != itend && remaining > 0) {
 529                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(op0_is_equal(), *it)) == global_dummy_indices.end()) {
 530                                 global_dummy_indices.push_back(*it);
 531                                 global_size++;
 532                                 remaining--;
 533                         }
 534                         it++;
 535                 }
 536
 537                 // If this is the first set of local indices, do nothing
 538                 if (old_global_size == 0)
 539                         return e;
 540         }
 541         GINAC_ASSERT(local_size <= global_size);
 542
 543         // Construct vectors of index symbols
 544         exvector local_syms, global_syms;
 545         local_syms.reserve(local_size);
 546         global_syms.reserve(local_size);
 547         for (size_t i=0; i<local_size; i++)
 548                 local_syms.push_back(local_dummy_indices[i].op(0));
 549         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 550         for (size_t i=0; i<local_size; i++) // don't use more global symbols than necessary
 551                 global_syms.push_back(global_dummy_indices[i].op(0));
 552         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 553
 554         // Remove common indices
 555         exvector local_uniq, global_uniq;
 556         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
 557         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
 558
 559         // Replace remaining non-common local index symbols by global ones
 560         if (local_uniq.empty())
 561                 return e;
 562         else {
 563                 while (global_uniq.size() > local_uniq.size())
 564                         global_uniq.pop_back();
 565                 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
 566         }
 567 }
 568
 569 /** Given a set of indices, extract those of class varidx. */
 570 static void find_variant_indices(const exvector & v, exvector & variant_indices)
 571 {
 572         exvector::const_iterator it1, itend;
 573         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 574                 if (is_exactly_a<varidx>(*it1))
 575                         variant_indices.push_back(*it1);
 576         }
 577 }
 578
 579 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
 580  *  variance.
 581  *
 582  *  @param e Object to work on
 583  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
 584  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
 585  *  @return true if 'e' was changed */
 586 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
 587 {
 588         bool something_changed = false;
 589
 590         // If a dummy index is encountered for the first time in the
 591         // product, pull it up, otherwise, pull it down
 592         exvector::const_iterator it2, it2start, it2end;
 593         for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
 594                 if (!is_exactly_a<varidx>(*it2))
 595                         continue;
 596
 597                 exvector::iterator vit, vitend;
 598                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 599                         if (it2->op(0).is_equal(vit->op(0))) {
 600                                 if (ex_to<varidx>(*it2).is_covariant()) {
 601                                         e = e.subs(lst(
 602                                                 *it2 == ex_to<varidx>(*it2).toggle_variance(),
 603                                                 ex_to<varidx>(*it2).toggle_variance() == *it2
 604                                         ), subs_options::no_pattern);
 605                                         something_changed = true;
 606                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 607                                         it2start = ex_to<indexed>(e).seq.begin();
 608                                         it2end = ex_to<indexed>(e).seq.end();
 609                                 }
 610                                 moved_indices.push_back(*vit);
 611                                 variant_dummy_indices.erase(vit);
 612                                 goto next_index;
 613                         }
 614                 }
 615
 616                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 617                         if (it2->op(0).is_equal(vit->op(0))) {
 618                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 619                                         e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance(), subs_options::no_pattern);
 620                                         something_changed = true;
 621                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 622                                         it2start = ex_to<indexed>(e).seq.begin();
 623                                         it2end = ex_to<indexed>(e).seq.end();
 624                                 }
 625                                 goto next_index;
 626                         }
 627                 }
 628
 629 next_index: ;
 630         }
 631
 632         return something_changed;
 633 }
 634
 635 /* Ordering that only compares the base expressions of indexed objects. */
 636 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
 637         bool operator() (const ex &lh, const ex &rh) const
 638         {
 639                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 640         }
 641 };
 642
 643 /** Simplify product of indexed expressions (commutative, noncommutative and
 644  *  simple squares), return list of free indices. */
 645 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 646 {
 647         // Remember whether the product was commutative or noncommutative
 648         // (because we chop it into factors and need to reassemble later)
 649         bool non_commutative = is_exactly_a<ncmul>(e);
 650
 651         // Collect factors in an exvector, store squares twice
 652         exvector v;
 653         v.reserve(e.nops() * 2);
 654
 655         if (is_exactly_a<power>(e)) {
 656                 // We only get called for simple squares, split a^2 -> a*a
 657                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 658                 v.push_back(e.op(0));
 659                 v.push_back(e.op(0));
 660         } else {
 661                 for (size_t i=0; i<e.nops(); i++) {
 662                         ex f = e.op(i);
 663                         if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
 664                                 v.push_back(f.op(0));
 665                     v.push_back(f.op(0));
 666                         } else if (is_exactly_a<ncmul>(f)) {
 667                                 // Noncommutative factor found, split it as well
 668                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 669                                 for (size_t j=0; j<f.nops(); j++)
 670                                         v.push_back(f.op(j));
 671                         } else
 672                                 v.push_back(f);
 673                 }
 674         }
 675
 676         // Perform contractions
 677         bool something_changed = false;
 678         GINAC_ASSERT(v.size() > 1);
 679         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 680         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 681
 682 try_again:
 683                 if (!is_a<indexed>(*it1))
 684                         continue;
 685
 686                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 687
 688                 // Indexed factor found, get free indices and look for contraction
 689                 // candidates
 690                 exvector free1, dummy1;
 691                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 692
 693                 exvector::iterator it2;
 694                 for (it2 = it1 + 1; it2 != itend; it2++) {
 695
 696                         if (!is_a<indexed>(*it2))
 697                                 continue;
 698
 699                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 700
 701                         // Find free indices of second factor and merge them with free
 702                         // indices of first factor
 703                         exvector un;
 704                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 705                         un.insert(un.end(), free1.begin(), free1.end());
 706
 707                         // Check whether the two factors share dummy indices
 708                         exvector free, dummy;
 709                         find_free_and_dummy(un, free, dummy);
 710                         size_t num_dummies = dummy.size();
 711                         if (num_dummies == 0)
 712                                 continue;
 713
 714                         // At least one dummy index, is it a defined scalar product?
 715                         bool contracted = false;
 716                         if (free.empty()) {
 717
 718                                 // Find minimal dimension of all indices of both factors
 719                                 exvector::const_iterator dit = ex_to<indexed>(*it1).seq.begin() + 1, ditend = ex_to<indexed>(*it1).seq.end();
 720                                 ex dim = ex_to<idx>(*dit).get_dim();
 721                                 ++dit;
 722                                 for (; dit != ditend; ++dit) {
 723                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 724                                 }
 725                                 dit = ex_to<indexed>(*it2).seq.begin() + 1;
 726                                 ditend = ex_to<indexed>(*it2).seq.end();
 727                                 for (; dit != ditend; ++dit) {
 728                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 729                                 }
 730
 731                                 // User-defined scalar product?
 732                                 if (sp.is_defined(*it1, *it2, dim)) {
 733
 734                                         // Yes, substitute it
 735                                         *it1 = sp.evaluate(*it1, *it2, dim);
 736                                         *it2 = _ex1;
 737                                         goto contraction_done;
 738                                 }
 739                         }
 740
 741                         // Try to contract the first one with the second one
 742                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 743                         if (!contracted) {
 744
 745                                 // That didn't work; maybe the second object knows how to
 746                                 // contract itself with the first one
 747                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 748                         }
 749                         if (contracted) {
 750 contraction_done:
 751                                 if (first_noncommutative || second_noncommutative
 752                                  || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
 753                                  || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
 754                                  || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
 755
 756                                         // One of the factors became a sum or product:
 757                                         // re-expand expression and run again
 758                                         // Non-commutative products are always re-expanded to give
 759                                         // eval_ncmul() the chance to re-order and canonicalize
 760                                         // the product
 761                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 762                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 763                                 }
 764
 765                                 // Both objects may have new indices now or they might
 766                                 // even not be indexed objects any more, so we have to
 767                                 // start over
 768                                 something_changed = true;
 769                                 goto try_again;
 770                         }
 771                 }
 772         }
 773
 774         // Find free indices (concatenate them all and call find_free_and_dummy())
 775         // and all dummy indices that appear
 776         exvector un, individual_dummy_indices;
 777         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 778                 exvector free_indices_of_factor;
 779                 if (is_a<indexed>(*it1)) {
 780                         exvector dummy_indices_of_factor;
 781                         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);
 782                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 783                 } else
 784                         free_indices_of_factor = it1->get_free_indices();
 785                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 786         }
 787         exvector local_dummy_indices;
 788         find_free_and_dummy(un, free_indices, local_dummy_indices);
 789         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 790
 791         // Filter out the dummy indices with variance
 792         exvector variant_dummy_indices;
 793         find_variant_indices(local_dummy_indices, variant_dummy_indices);
 794
 795         // Any indices with variance present at all?
 796         if (!variant_dummy_indices.empty()) {
 797
 798                 // Yes, bring the product into a canonical order that only depends on
 799                 // the base expressions of indexed objects
 800                 if (!non_commutative)
 801                         std::sort(v.begin(), v.end(), ex_base_is_less());
 802
 803                 exvector moved_indices;
 804
 805                 // Iterate over all indexed objects in the product
 806                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 807                         if (!is_a<indexed>(*it1))
 808                                 continue;
 809
 810                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
 811                                 something_changed = true;
 812                 }
 813         }
 814
 815         ex r;
 816         if (something_changed)
 817                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 818         else
 819                 r = e;
 820
 821         // The result should be symmetric with respect to exchange of dummy
 822         // indices, so if the symmetrization vanishes, the whole expression is
 823         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 824         if (local_dummy_indices.size() >= 2) {
 825                 exvector dummy_syms;
 826                 dummy_syms.reserve(local_dummy_indices.size());
 827                 for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
 828                         dummy_syms.push_back(it->op(0));
 829                 if (symmetrize(r, dummy_syms).is_zero()) {
 830                         free_indices.clear();
 831                         return _ex0;
 832                 }
 833         }
 834
 835         // Dummy index renaming
 836         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 837
 838         // Product of indexed object with a scalar?
 839         if (is_exactly_a<mul>(r) && r.nops() == 2
 840          && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
 841                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 842         else
 843                 return r;
 844 }
 845
 846 /** This structure stores the original and symmetrized versions of terms
 847  *  obtained during the simplification of sums. */
 848 class terminfo {
 849 public:
 850         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
 851
 852         ex orig; /**< original term */
 853         ex symm; /**< symmtrized term */
 854 };
 855
 856 class terminfo_is_less {
 857 public:
 858         bool operator() (const terminfo & ti1, const terminfo & ti2) const
 859         {
 860                 return (ti1.symm.compare(ti2.symm) < 0);
 861         }
 862 };
 863
 864 /** This structure stores the individual symmetrized terms obtained during
 865  *  the simplification of sums. */
 866 class symminfo {
 867 public:
 868         symminfo() : num(0) {}
 869
 870         symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
 871         {
 872                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
 873                         coeff = symmterm_.op(symmterm_.nops()-1);
 874                         symmterm = symmterm_ / coeff;
 875                 } else {
 876                         coeff = 1;
 877                         symmterm = symmterm_;
 878                 }
 879         }
 880
 881         ex symmterm;  /**< symmetrized term */
 882         ex coeff;     /**< coefficient of symmetrized term */
 883         ex orig;      /**< original term */
 884         size_t num; /**< how many symmetrized terms resulted from the original term */
 885 };
 886
 887 class symminfo_is_less_by_symmterm {
 888 public:
 889         bool operator() (const symminfo & si1, const symminfo & si2) const
 890         {
 891                 return (si1.symmterm.compare(si2.symmterm) < 0);
 892         }
 893 };
 894
 895 class symminfo_is_less_by_orig {
 896 public:
 897         bool operator() (const symminfo & si1, const symminfo & si2) const
 898         {
 899                 return (si1.orig.compare(si2.orig) < 0);
 900         }
 901 };
 902
 903 /** Simplify indexed expression, return list of free indices. */
 904 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 905 {
 906         // Expand the expression
 907         ex e_expanded = e.expand();
 908
 909         // Simplification of single indexed object: just find the free indices
 910         // and perform dummy index renaming/repositioning
 911         if (is_a<indexed>(e_expanded)) {
 912
 913                 // Find the dummy indices
 914                 const indexed &i = ex_to<indexed>(e_expanded);
 915                 exvector local_dummy_indices;
 916                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 917
 918                 // Filter out the dummy indices with variance
 919                 exvector variant_dummy_indices;
 920                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
 921
 922                 // Any indices with variance present at all?
 923                 if (!variant_dummy_indices.empty()) {
 924
 925                         // Yes, reposition them
 926                         exvector moved_indices;
 927                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
 928                 }
 929
 930                 // Rename the dummy indices
 931                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 932         }
 933
 934         // Simplification of sum = sum of simplifications, check consistency of
 935         // free indices in each term
 936         if (is_exactly_a<add>(e_expanded)) {
 937                 bool first = true;
 938                 ex sum;
 939                 free_indices.clear();
 940
 941                 for (size_t i=0; i<e_expanded.nops(); i++) {
 942                         exvector free_indices_of_term;
 943                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 944                         if (!term.is_zero()) {
 945                                 if (first) {
 946                                         free_indices = free_indices_of_term;
 947                                         sum = term;
 948                                         first = false;
 949                                 } else {
 950                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
 951                                                 std::ostringstream s;
 952                                                 s << "simplify_indexed: inconsistent indices in sum: ";
 953                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
 954                                                 throw (std::runtime_error(s.str()));
 955                                         }
 956                                         if (is_a<indexed>(sum) && is_a<indexed>(term))
 957                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
 958                                         else
 959                                                 sum += term;
 960                                 }
 961                         }
 962                 }
 963
 964                 // If the sum turns out to be zero, we are finished
 965                 if (sum.is_zero()) {
 966                         free_indices.clear();
 967                         return sum;
 968                 }
 969
 970                 // More than one term and more than one dummy index?
 971                 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
 972                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
 973                         return sum;
 974
 975                 // Yes, construct vector of all dummy index symbols
 976                 exvector dummy_syms;
 977                 dummy_syms.reserve(dummy_indices.size());
 978                 for (exvector::const_iterator it = dummy_indices.begin(); it != dummy_indices.end(); ++it)
 979                         dummy_syms.push_back(it->op(0));
 980
 981                 // Chop the sum into terms and symmetrize each one over the dummy
 982                 // indices
 983                 std::vector<terminfo> terms;
 984                 for (size_t i=0; i<sum.nops(); i++) {
 985                         const ex & term = sum.op(i);
 986                         ex term_symm = symmetrize(term, dummy_syms);
 987                         if (term_symm.is_zero())
 988                                 continue;
 989                         terms.push_back(terminfo(term, term_symm));
 990                 }
 991
 992                 // Sort by symmetrized terms
 993                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
 994
 995                 // Combine equal symmetrized terms
 996                 std::vector<terminfo> terms_pass2;
 997                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
 998                         size_t num = 1;
 999                         std::vector<terminfo>::const_iterator j = i + 1;
1000                         while (j != terms.end() && j->symm == i->symm) {
1001                                 num++;
1002                                 j++;
1003                         }
1004                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1005                         i = j;
1006                 }
1007
1008                 // If there is only one term left, we are finished
1009                 if (terms_pass2.size() == 1)
1010                         return terms_pass2[0].orig;
1011
1012                 // Chop the symmetrized terms into subterms
1013                 std::vector<symminfo> sy;
1014                 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1015                         if (is_exactly_a<add>(i->symm)) {
1016                                 size_t num = i->symm.nops();
1017                                 for (size_t j=0; j<num; j++)
1018                                         sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1019                         } else
1020                                 sy.push_back(symminfo(i->symm, i->orig, 1));
1021                 }
1022
1023                 // Sort by symmetrized subterms
1024                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1025
1026                 // Combine equal symmetrized subterms
1027                 std::vector<symminfo> sy_pass2;
1028                 exvector result;
1029                 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1030
1031                         // Combine equal terms
1032                         std::vector<symminfo>::const_iterator j = i + 1;
1033                         if (j != sy.end() && j->symmterm == i->symmterm) {
1034
1035                                 // More than one term, collect the coefficients
1036                                 ex coeff = i->coeff;
1037                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1038                                         coeff += j->coeff;
1039                                         j++;
1040                                 }
1041
1042                                 // Add combined term to result
1043                                 if (!coeff.is_zero())
1044                                         result.push_back(coeff * i->symmterm);
1045
1046                         } else {
1047
1048                                 // Single term, store for second pass
1049                                 sy_pass2.push_back(*i);
1050                         }
1051
1052                         i = j;
1053                 }
1054
1055                 // Were there any remaining terms that didn't get combined?
1056                 if (sy_pass2.size() > 0) {
1057
1058                         // Yes, sort by their original terms
1059                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1060
1061                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1062
1063                                 // How many symmetrized terms of this original term are left?
1064                                 size_t num = 1;
1065                                 std::vector<symminfo>::const_iterator j = i + 1;
1066                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1067                                         num++;
1068                                         j++;
1069                                 }
1070
1071                                 if (num == i->num) {
1072
1073                                         // All terms left, then add the original term to the result
1074                                         result.push_back(i->orig);
1075
1076                                 } else {
1077
1078                                         // Some terms were combined with others, add up the remaining symmetrized terms
1079                                         std::vector<symminfo>::const_iterator k;
1080                                         for (k=i; k!=j; k++)
1081                                                 result.push_back(k->coeff * k->symmterm);
1082                                 }
1083
1084                                 i = j;
1085                         }
1086                 }
1087
1088                 // Add all resulting terms
1089                 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1090                 if (sum_symm.is_zero())
1091                         free_indices.clear();
1092                 return sum_symm;
1093         }
1094
1095         // Simplification of products
1096         if (is_exactly_a<mul>(e_expanded)
1097          || is_exactly_a<ncmul>(e_expanded)
1098          || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1099                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1100
1101         // Cannot do anything
1102         free_indices.clear();
1103         return e_expanded;
1104 }
1105
1106 /** Simplify/canonicalize expression containing indexed objects. This
1107  *  performs contraction of dummy indices where possible and checks whether
1108  *  the free indices in sums are consistent.
1109  *
1110  *  @return simplified expression */
1111 ex ex::simplify_indexed() const
1112 {
1113         exvector free_indices, dummy_indices;
1114         scalar_products sp;
1115         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1116 }
1117
1118 /** Simplify/canonicalize expression containing indexed objects. This
1119  *  performs contraction of dummy indices where possible, checks whether
1120  *  the free indices in sums are consistent, and automatically replaces
1121  *  scalar products by known values if desired.
1122  *
1123  *  @param sp Scalar products to be replaced automatically
1124  *  @return simplified expression */
1125 ex ex::simplify_indexed(const scalar_products & sp) const
1126 {
1127         exvector free_indices, dummy_indices;
1128         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1129 }
1130
1131 /** Symmetrize expression over its free indices. */
1132 ex ex::symmetrize() const
1133 {
1134         return GiNaC::symmetrize(*this, get_free_indices());
1135 }
1136
1137 /** Antisymmetrize expression over its free indices. */
1138 ex ex::antisymmetrize() const
1139 {
1140         return GiNaC::antisymmetrize(*this, get_free_indices());
1141 }
1142
1143 /** Symmetrize expression by cyclic permutation over its free indices. */
1144 ex ex::symmetrize_cyclic() const
1145 {
1146         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1147 }
1148
1149 //////////
1150 // helper classes
1151 //////////
1152
1153 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1154 {
1155         // If indexed, extract base objects
1156         ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1157         ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1158
1159         // Enforce canonical order in pair
1160         if (s1.compare(s2) > 0) {
1161                 v1 = s2;
1162                 v2 = s1;
1163         } else {
1164                 v1 = s1;
1165                 v2 = s2;
1166         }
1167 }
1168
1169 bool spmapkey::operator==(const spmapkey &other) const
1170 {
1171         if (!v1.is_equal(other.v1))
1172                 return false;
1173         if (!v2.is_equal(other.v2))
1174                 return false;
1175         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1176                 return true;
1177         else
1178                 return dim.is_equal(other.dim);
1179 }
1180
1181 bool spmapkey::operator<(const spmapkey &other) const
1182 {
1183         int cmp = v1.compare(other.v1);
1184         if (cmp)
1185                 return cmp < 0;
1186         cmp = v2.compare(other.v2);
1187         if (cmp)
1188                 return cmp < 0;
1189
1190         // Objects are equal, now check dimensions
1191         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1192                 return false;
1193         else
1194                 return dim.compare(other.dim) < 0;
1195 }
1196
1197 void spmapkey::debugprint() const
1198 {
1199         std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1200 }
1201
1202 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1203 {
1204         spm[spmapkey(v1, v2)] = sp;
1205 }
1206
1207 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1208 {
1209         spm[spmapkey(v1, v2, dim)] = sp;
1210 }
1211
1212 void scalar_products::add_vectors(const lst & l, const ex & dim)
1213 {
1214         // Add all possible pairs of products
1215         for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1216                 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1217                         add(*it1, *it2, *it1 * *it2);
1218 }
1219
1220 void scalar_products::clear()
1221 {
1222         spm.clear();
1223 }
1224
1225 /** Check whether scalar product pair is defined. */
1226 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1227 {
1228         return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1229 }
1230
1231 /** Return value of defined scalar product pair. */
1232 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1233 {
1234         return spm.find(spmapkey(v1, v2, dim))->second;
1235 }
1236
1237 void scalar_products::debugprint() const
1238 {
1239         std::cerr << "map size=" << spm.size() << std::endl;
1240         spmap::const_iterator i = spm.begin(), end = spm.end();
1241         while (i != end) {
1242                 const spmapkey & k = i->first;
1243                 std::cerr << "item key=";
1244                 k.debugprint();
1245                 std::cerr << ", value=" << i->second << std::endl;
1246                 ++i;
1247         }
1248 }
1249
1250 } // namespace GiNaC