ginac/indexed.cpp

   1 /** @file indexed.cpp
   2  *
   3  *  Implementation of GiNaC's indexed expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2002 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <iostream>
  24 #include <stdexcept>
  25
  26 #include "indexed.h"
  27 #include "idx.h"
  28 #include "add.h"
  29 #include "mul.h"
  30 #include "ncmul.h"
  31 #include "power.h"
  32 #include "relational.h"
  33 #include "symmetry.h"
  34 #include "lst.h"
  35 #include "print.h"
  36 #include "archive.h"
  37 #include "utils.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(indexed, exprseq)
  42
  43 //////////
  44 // default ctor, dtor, copy ctor, assignment operator and helpers
  45 //////////
  46
  47 indexed::indexed() : symtree(sy_none())
  48 {
  49         tinfo_key = TINFO_indexed;
  50 }
  51
  52 void indexed::copy(const indexed & other)
  53 {
  54         inherited::copy(other);
  55         symtree = other.symtree;
  56 }
  57
  58 DEFAULT_DESTROY(indexed)
  59
  60 //////////
  61 // other constructors
  62 //////////
  63
  64 indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
  65 {
  66         tinfo_key = TINFO_indexed;
  67         validate();
  68 }
  69
  70 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(sy_none())
  71 {
  72         tinfo_key = TINFO_indexed;
  73         validate();
  74 }
  75
  76 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(sy_none())
  77 {
  78         tinfo_key = TINFO_indexed;
  79         validate();
  80 }
  81
  82 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(sy_none())
  83 {
  84         tinfo_key = TINFO_indexed;
  85         validate();
  86 }
  87
  88 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(sy_none())
  89 {
  90         tinfo_key = TINFO_indexed;
  91         validate();
  92 }
  93
  94 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
  95 {
  96         tinfo_key = TINFO_indexed;
  97         validate();
  98 }
  99
 100 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
 101 {
 102         tinfo_key = TINFO_indexed;
 103         validate();
 104 }
 105
 106 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(symm)
 107 {
 108         tinfo_key = TINFO_indexed;
 109         validate();
 110 }
 111
 112 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
 113 {
 114         seq.insert(seq.end(), v.begin(), v.end());
 115         tinfo_key = TINFO_indexed;
 116         validate();
 117 }
 118
 119 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 120 {
 121         seq.insert(seq.end(), v.begin(), v.end());
 122         tinfo_key = TINFO_indexed;
 123         validate();
 124 }
 125
 126 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 127 {
 128         tinfo_key = TINFO_indexed;
 129 }
 130
 131 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 132 {
 133         tinfo_key = TINFO_indexed;
 134 }
 135
 136 indexed::indexed(const symmetry & symm, exvector * vp) : inherited(vp), symtree(symm)
 137 {
 138         tinfo_key = TINFO_indexed;
 139 }
 140
 141 //////////
 142 // archiving
 143 //////////
 144
 145 indexed::indexed(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 146 {
 147         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 148                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 149                 unsigned symm = 0;
 150                 n.find_unsigned("symmetry", symm);
 151                 switch (symm) {
 152                         case 1:
 153                                 symtree = sy_symm();
 154                                 break;
 155                         case 2:
 156                                 symtree = sy_anti();
 157                                 break;
 158                         default:
 159                                 symtree = sy_none();
 160                                 break;
 161                 }
 162                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 163         }
 164 }
 165
 166 void indexed::archive(archive_node &n) const
 167 {
 168         inherited::archive(n);
 169         n.add_ex("symmetry", symtree);
 170 }
 171
 172 DEFAULT_UNARCHIVE(indexed)
 173
 174 //////////
 175 // functions overriding virtual functions from base classes
 176 //////////
 177
 178 void indexed::print(const print_context & c, unsigned level) const
 179 {
 180         GINAC_ASSERT(seq.size() > 0);
 181
 182         if (is_of_type(c, print_tree)) {
 183
 184                 c.s << std::string(level, ' ') << class_name()
 185                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 186                     << ", " << seq.size()-1 << " indices"
 187                     << ", symmetry=" << symtree << std::endl;
 188                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 189                 seq[0].print(c, level + delta_indent);
 190                 printindices(c, level + delta_indent);
 191
 192         } else {
 193
 194                 bool is_tex = is_of_type(c, print_latex);
 195                 const ex & base = seq[0];
 196                 bool need_parens = is_ex_exactly_of_type(base, add) || is_ex_exactly_of_type(base, mul)
 197                                 || is_ex_exactly_of_type(base, ncmul) || is_ex_exactly_of_type(base, power)
 198                                 || is_ex_of_type(base, indexed);
 199                 if (is_tex)
 200                         c.s << "{";
 201                 if (need_parens)
 202                         c.s << "(";
 203                 base.print(c);
 204                 if (need_parens)
 205                         c.s << ")";
 206                 if (is_tex)
 207                         c.s << "}";
 208                 printindices(c, level);
 209         }
 210 }
 211
 212 bool indexed::info(unsigned inf) const
 213 {
 214         if (inf == info_flags::indexed) return true;
 215         if (inf == info_flags::has_indices) return seq.size() > 1;
 216         return inherited::info(inf);
 217 }
 218
 219 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 220         bool operator() (const ex & e, unsigned inf) const {
 221                 return !(ex_to<idx>(e).get_value().info(inf));
 222         }
 223 };
 224
 225 bool indexed::all_index_values_are(unsigned inf) const
 226 {
 227         // No indices? Then no property can be fulfilled
 228         if (seq.size() < 2)
 229                 return false;
 230
 231         // Check all indices
 232         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 233 }
 234
 235 int indexed::compare_same_type(const basic & other) const
 236 {
 237         GINAC_ASSERT(is_a<indexed>(other));
 238         return inherited::compare_same_type(other);
 239 }
 240
 241 ex indexed::eval(int level) const
 242 {
 243         // First evaluate children, then we will end up here again
 244         if (level > 1)
 245                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 246
 247         const ex &base = seq[0];
 248
 249         // If the base object is 0, the whole object is 0
 250         if (base.is_zero())
 251                 return _ex0;
 252
 253         // If the base object is a product, pull out the numeric factor
 254         if (is_ex_exactly_of_type(base, mul) && is_ex_exactly_of_type(base.op(base.nops() - 1), numeric)) {
 255                 exvector v(seq);
 256                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 257                 v[0] = seq[0] / f;
 258                 return f * thisexprseq(v);
 259         }
 260
 261         // Canonicalize indices according to the symmetry properties
 262         if (seq.size() > 2) {
 263                 exvector v = seq;
 264                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 265                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 266                 if (sig != INT_MAX) {
 267                         // Something has changed while sorting indices, more evaluations later
 268                         if (sig == 0)
 269                                 return _ex0;
 270                         return ex(sig) * thisexprseq(v);
 271                 }
 272         }
 273
 274         // Let the class of the base object perform additional evaluations
 275         return ex_to<basic>(base).eval_indexed(*this);
 276 }
 277
 278 ex indexed::thisexprseq(const exvector & v) const
 279 {
 280         return indexed(ex_to<symmetry>(symtree), v);
 281 }
 282
 283 ex indexed::thisexprseq(exvector * vp) const
 284 {
 285         return indexed(ex_to<symmetry>(symtree), vp);
 286 }
 287
 288 ex indexed::expand(unsigned options) const
 289 {
 290         GINAC_ASSERT(seq.size() > 0);
 291
 292         if ((options & expand_options::expand_indexed) && is_ex_exactly_of_type(seq[0], add)) {
 293
 294                 // expand_indexed expands (a+b).i -> a.i + b.i
 295                 const ex & base = seq[0];
 296                 ex sum = _ex0;
 297                 for (unsigned i=0; i<base.nops(); i++) {
 298                         exvector s = seq;
 299                         s[0] = base.op(i);
 300                         sum += thisexprseq(s).expand();
 301                 }
 302                 return sum;
 303
 304         } else
 305                 return inherited::expand(options);
 306 }
 307
 308 //////////
 309 // virtual functions which can be overridden by derived classes
 310 //////////
 311
 312 // none
 313
 314 //////////
 315 // non-virtual functions in this class
 316 //////////
 317
 318 void indexed::printindices(const print_context & c, unsigned level) const
 319 {
 320         if (seq.size() > 1) {
 321
 322                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 323
 324                 if (is_of_type(c, print_latex)) {
 325
 326                         // TeX output: group by variance
 327                         bool first = true;
 328                         bool covariant = true;
 329
 330                         while (it != itend) {
 331                                 bool cur_covariant = (is_ex_of_type(*it, varidx) ? ex_to<varidx>(*it).is_covariant() : true);
 332                                 if (first || cur_covariant != covariant) { // Variance changed
 333                                         // The empty {} prevents indices from ending up on top of each other
 334                                         if (!first)
 335                                                 c.s << "}{}";
 336                                         covariant = cur_covariant;
 337                                         if (covariant)
 338                                                 c.s << "_{";
 339                                         else
 340                                                 c.s << "^{";
 341                                 }
 342                                 it->print(c, level);
 343                                 c.s << " ";
 344                                 first = false;
 345                                 it++;
 346                         }
 347                         c.s << "}";
 348
 349                 } else {
 350
 351                         // Ordinary output
 352                         while (it != itend) {
 353                                 it->print(c, level);
 354                                 it++;
 355                         }
 356                 }
 357         }
 358 }
 359
 360 /** Check whether all indices are of class idx and validate the symmetry
 361  *  tree. This function is used internally to make sure that all constructed
 362  *  indexed objects really carry indices and not some other classes. */
 363 void indexed::validate(void) const
 364 {
 365         GINAC_ASSERT(seq.size() > 0);
 366         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 367         while (it != itend) {
 368                 if (!is_ex_of_type(*it, idx))
 369                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 370                 it++;
 371         }
 372
 373         if (!symtree.is_zero()) {
 374                 if (!is_ex_exactly_of_type(symtree, symmetry))
 375                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 376                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 377         }
 378 }
 379
 380 /** Implementation of ex::diff() for an indexed object always returns 0.
 381  *
 382  *  @see ex::diff */
 383 ex indexed::derivative(const symbol & s) const
 384 {
 385         return _ex0;
 386 }
 387
 388 //////////
 389 // global functions
 390 //////////
 391
 392 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 393 static bool indices_consistent(const exvector & v1, const exvector & v2)
 394 {
 395         // Number of indices must be the same
 396         if (v1.size() != v2.size())
 397                 return false;
 398
 399         return equal(v1.begin(), v1.end(), v2.begin(), ex_is_equal());
 400 }
 401
 402 exvector indexed::get_indices(void) const
 403 {
 404         GINAC_ASSERT(seq.size() >= 1);
 405         return exvector(seq.begin() + 1, seq.end());
 406 }
 407
 408 exvector indexed::get_dummy_indices(void) const
 409 {
 410         exvector free_indices, dummy_indices;
 411         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 412         return dummy_indices;
 413 }
 414
 415 exvector indexed::get_dummy_indices(const indexed & other) const
 416 {
 417         exvector indices = get_free_indices();
 418         exvector other_indices = other.get_free_indices();
 419         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 420         exvector dummy_indices;
 421         find_dummy_indices(indices, dummy_indices);
 422         return dummy_indices;
 423 }
 424
 425 bool indexed::has_dummy_index_for(const ex & i) const
 426 {
 427         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 428         while (it != itend) {
 429                 if (is_dummy_pair(*it, i))
 430                         return true;
 431                 it++;
 432         }
 433         return false;
 434 }
 435
 436 exvector indexed::get_free_indices(void) const
 437 {
 438         exvector free_indices, dummy_indices;
 439         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 440         return free_indices;
 441 }
 442
 443 exvector add::get_free_indices(void) const
 444 {
 445         exvector free_indices;
 446         for (unsigned i=0; i<nops(); i++) {
 447                 if (i == 0)
 448                         free_indices = op(i).get_free_indices();
 449                 else {
 450                         exvector free_indices_of_term = op(i).get_free_indices();
 451                         if (!indices_consistent(free_indices, free_indices_of_term))
 452                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 453                 }
 454         }
 455         return free_indices;
 456 }
 457
 458 exvector mul::get_free_indices(void) const
 459 {
 460         // Concatenate free indices of all factors
 461         exvector un;
 462         for (unsigned i=0; i<nops(); i++) {
 463                 exvector free_indices_of_factor = op(i).get_free_indices();
 464                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 465         }
 466
 467         // And remove the dummy indices
 468         exvector free_indices, dummy_indices;
 469         find_free_and_dummy(un, free_indices, dummy_indices);
 470         return free_indices;
 471 }
 472
 473 exvector ncmul::get_free_indices(void) const
 474 {
 475         // Concatenate free indices of all factors
 476         exvector un;
 477         for (unsigned i=0; i<nops(); i++) {
 478                 exvector free_indices_of_factor = op(i).get_free_indices();
 479                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 480         }
 481
 482         // And remove the dummy indices
 483         exvector free_indices, dummy_indices;
 484         find_free_and_dummy(un, free_indices, dummy_indices);
 485         return free_indices;
 486 }
 487
 488 exvector power::get_free_indices(void) const
 489 {
 490         // Return free indices of basis
 491         return basis.get_free_indices();
 492 }
 493
 494 /** Rename dummy indices in an expression.
 495  *
 496  *  @param e Expression to work on
 497  *  @param local_dummy_indices The set of dummy indices that appear in the
 498  *    expression "e"
 499  *  @param global_dummy_indices The set of dummy indices that have appeared
 500  *    before and which we would like to use in "e", too. This gets updated
 501  *    by the function */
 502 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 503 {
 504         unsigned global_size = global_dummy_indices.size(),
 505                  local_size = local_dummy_indices.size();
 506
 507         // Any local dummy indices at all?
 508         if (local_size == 0)
 509                 return e;
 510
 511         if (global_size < local_size) {
 512
 513                 // More local indices than we encountered before, add the new ones
 514                 // to the global set
 515                 int old_global_size = global_size;
 516                 int remaining = local_size - global_size;
 517                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 518                 while (it != itend && remaining > 0) {
 519                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(ex_is_equal(), *it)) == global_dummy_indices.end()) {
 520                                 global_dummy_indices.push_back(*it);
 521                                 global_size++;
 522                                 remaining--;
 523                         }
 524                         it++;
 525                 }
 526
 527                 // If this is the first set of local indices, do nothing
 528                 if (old_global_size == 0)
 529                         return e;
 530         }
 531         GINAC_ASSERT(local_size <= global_size);
 532
 533         // Construct lists of index symbols
 534         exlist local_syms, global_syms;
 535         for (unsigned i=0; i<local_size; i++)
 536                 local_syms.push_back(local_dummy_indices[i].op(0));
 537         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 538         for (unsigned i=0; i<global_size; i++)
 539                 global_syms.push_back(global_dummy_indices[i].op(0));
 540         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 541
 542         // Remove common indices
 543         exlist local_uniq, global_uniq;
 544         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exlist>(local_uniq), ex_is_less());
 545         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exlist>(global_uniq), ex_is_less());
 546
 547         // Replace remaining non-common local index symbols by global ones
 548         if (local_uniq.empty())
 549                 return e;
 550         else {
 551                 while (global_uniq.size() > local_uniq.size())
 552                         global_uniq.pop_back();
 553                 return e.subs(lst(local_uniq), lst(global_uniq));
 554         }
 555 }
 556
 557 /** Given a set of indices, extract those of class varidx. */
 558 static void find_variant_indices(const exvector & v, exvector & variant_indices)
 559 {
 560         exvector::const_iterator it1, itend;
 561         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 562                 if (is_exactly_a<varidx>(*it1))
 563                         variant_indices.push_back(*it1);
 564         }
 565 }
 566
 567 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
 568  *  variance.
 569  *
 570  *  @param e Object to work on
 571  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
 572  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
 573  *  @return true if 'e' was changed */
 574 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
 575 {
 576         bool something_changed = false;
 577
 578         // If a dummy index is encountered for the first time in the
 579         // product, pull it up, otherwise, pull it down
 580         exvector::const_iterator it2, it2start, it2end;
 581         for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
 582                 if (!is_exactly_a<varidx>(*it2))
 583                         continue;
 584
 585                 exvector::iterator vit, vitend;
 586                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 587                         if (it2->op(0).is_equal(vit->op(0))) {
 588                                 if (ex_to<varidx>(*it2).is_covariant()) {
 589                                         e = e.subs(lst(
 590                                                 *it2 == ex_to<varidx>(*it2).toggle_variance(),
 591                                                 ex_to<varidx>(*it2).toggle_variance() == *it2
 592                                         ));
 593                                         something_changed = true;
 594                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 595                                         it2start = ex_to<indexed>(e).seq.begin();
 596                                         it2end = ex_to<indexed>(e).seq.end();
 597                                 }
 598                                 moved_indices.push_back(*vit);
 599                                 variant_dummy_indices.erase(vit);
 600                                 goto next_index;
 601                         }
 602                 }
 603
 604                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 605                         if (it2->op(0).is_equal(vit->op(0))) {
 606                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 607                                         e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance());
 608                                         something_changed = true;
 609                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 610                                         it2start = ex_to<indexed>(e).seq.begin();
 611                                         it2end = ex_to<indexed>(e).seq.end();
 612                                 }
 613                                 goto next_index;
 614                         }
 615                 }
 616
 617 next_index: ;
 618         }
 619
 620         return something_changed;
 621 }
 622
 623 /* Ordering that only compares the base expressions of indexed objects. */
 624 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
 625         bool operator() (const ex &lh, const ex &rh) const
 626         {
 627                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 628         }
 629 };
 630
 631 /** Simplify product of indexed expressions (commutative, noncommutative and
 632  *  simple squares), return list of free indices. */
 633 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 634 {
 635         // Remember whether the product was commutative or noncommutative
 636         // (because we chop it into factors and need to reassemble later)
 637         bool non_commutative = is_ex_exactly_of_type(e, ncmul);
 638
 639         // Collect factors in an exvector, store squares twice
 640         exvector v;
 641         v.reserve(e.nops() * 2);
 642
 643         if (is_ex_exactly_of_type(e, power)) {
 644                 // We only get called for simple squares, split a^2 -> a*a
 645                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 646                 v.push_back(e.op(0));
 647                 v.push_back(e.op(0));
 648         } else {
 649                 for (unsigned i=0; i<e.nops(); i++) {
 650                         ex f = e.op(i);
 651                         if (is_ex_exactly_of_type(f, power) && f.op(1).is_equal(_ex2)) {
 652                                 v.push_back(f.op(0));
 653                     v.push_back(f.op(0));
 654                         } else if (is_ex_exactly_of_type(f, ncmul)) {
 655                                 // Noncommutative factor found, split it as well
 656                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 657                                 for (unsigned j=0; j<f.nops(); j++)
 658                                         v.push_back(f.op(j));
 659                         } else
 660                                 v.push_back(f);
 661                 }
 662         }
 663
 664         // Perform contractions
 665         bool something_changed = false;
 666         GINAC_ASSERT(v.size() > 1);
 667         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 668         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 669
 670 try_again:
 671                 if (!is_ex_of_type(*it1, indexed))
 672                         continue;
 673
 674                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 675
 676                 // Indexed factor found, get free indices and look for contraction
 677                 // candidates
 678                 exvector free1, dummy1;
 679                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 680
 681                 exvector::iterator it2;
 682                 for (it2 = it1 + 1; it2 != itend; it2++) {
 683
 684                         if (!is_ex_of_type(*it2, indexed))
 685                                 continue;
 686
 687                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 688
 689                         // Find free indices of second factor and merge them with free
 690                         // indices of first factor
 691                         exvector un;
 692                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 693                         un.insert(un.end(), free1.begin(), free1.end());
 694
 695                         // Check whether the two factors share dummy indices
 696                         exvector free, dummy;
 697                         find_free_and_dummy(un, free, dummy);
 698                         unsigned num_dummies = dummy.size();
 699                         if (num_dummies == 0)
 700                                 continue;
 701
 702                         // At least one dummy index, is it a defined scalar product?
 703                         bool contracted = false;
 704                         if (free.empty()) {
 705                                 if (sp.is_defined(*it1, *it2)) {
 706                                         *it1 = sp.evaluate(*it1, *it2);
 707                                         *it2 = _ex1;
 708                                         goto contraction_done;
 709                                 }
 710                         }
 711
 712                         // Try to contract the first one with the second one
 713                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 714                         if (!contracted) {
 715
 716                                 // That didn't work; maybe the second object knows how to
 717                                 // contract itself with the first one
 718                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 719                         }
 720                         if (contracted) {
 721 contraction_done:
 722                                 if (first_noncommutative || second_noncommutative
 723                                  || is_ex_exactly_of_type(*it1, add) || is_ex_exactly_of_type(*it2, add)
 724                                  || is_ex_exactly_of_type(*it1, mul) || is_ex_exactly_of_type(*it2, mul)
 725                                  || is_ex_exactly_of_type(*it1, ncmul) || is_ex_exactly_of_type(*it2, ncmul)) {
 726
 727                                         // One of the factors became a sum or product:
 728                                         // re-expand expression and run again
 729                                         // Non-commutative products are always re-expanded to give
 730                                         // simplify_ncmul() the chance to re-order and canonicalize
 731                                         // the product
 732                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 733                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 734                                 }
 735
 736                                 // Both objects may have new indices now or they might
 737                                 // even not be indexed objects any more, so we have to
 738                                 // start over
 739                                 something_changed = true;
 740                                 goto try_again;
 741                         }
 742                 }
 743         }
 744
 745         // Find free indices (concatenate them all and call find_free_and_dummy())
 746         // and all dummy indices that appear
 747         exvector un, individual_dummy_indices;
 748         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 749                 exvector free_indices_of_factor;
 750                 if (is_ex_of_type(*it1, indexed)) {
 751                         exvector dummy_indices_of_factor;
 752                         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);
 753                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 754                 } else
 755                         free_indices_of_factor = it1->get_free_indices();
 756                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 757         }
 758         exvector local_dummy_indices;
 759         find_free_and_dummy(un, free_indices, local_dummy_indices);
 760         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 761
 762         // Filter out the dummy indices with variance
 763         exvector variant_dummy_indices;
 764         find_variant_indices(local_dummy_indices, variant_dummy_indices);
 765
 766         // Any indices with variance present at all?
 767         if (!variant_dummy_indices.empty()) {
 768
 769                 // Yes, bring the product into a canonical order that only depends on
 770                 // the base expressions of indexed objects
 771                 if (!non_commutative)
 772                         std::sort(v.begin(), v.end(), ex_base_is_less());
 773
 774                 exvector moved_indices;
 775
 776                 // Iterate over all indexed objects in the product
 777                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 778                         if (!is_ex_of_type(*it1, indexed))
 779                                 continue;
 780
 781                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
 782                                 something_changed = true;
 783                 }
 784         }
 785
 786         ex r;
 787         if (something_changed)
 788                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 789         else
 790                 r = e;
 791
 792         // The result should be symmetric with respect to exchange of dummy
 793         // indices, so if the symmetrization vanishes, the whole expression is
 794         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 795         if (local_dummy_indices.size() >= 2) {
 796                 lst dummy_syms;
 797                 for (int i=0; i<local_dummy_indices.size(); i++)
 798                         dummy_syms.append(local_dummy_indices[i].op(0));
 799                 if (r.symmetrize(dummy_syms).is_zero()) {
 800                         free_indices.clear();
 801                         return _ex0;
 802                 }
 803         }
 804
 805         // Dummy index renaming
 806         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 807
 808         // Product of indexed object with a scalar?
 809         if (is_ex_exactly_of_type(r, mul) && r.nops() == 2
 810          && is_ex_exactly_of_type(r.op(1), numeric) && is_ex_of_type(r.op(0), indexed))
 811                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 812         else
 813                 return r;
 814 }
 815
 816 /** Simplify indexed expression, return list of free indices. */
 817 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 818 {
 819         // Expand the expression
 820         ex e_expanded = e.expand();
 821
 822         // Simplification of single indexed object: just find the free indices
 823         // and perform dummy index renaming/repositioning
 824         if (is_ex_of_type(e_expanded, indexed)) {
 825
 826                 // Find the dummy indices
 827                 const indexed &i = ex_to<indexed>(e_expanded);
 828                 exvector local_dummy_indices;
 829                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 830
 831                 // Filter out the dummy indices with variance
 832                 exvector variant_dummy_indices;
 833                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
 834
 835                 // Any indices with variance present at all?
 836                 if (!variant_dummy_indices.empty()) {
 837
 838                         // Yes, reposition them
 839                         exvector moved_indices;
 840                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
 841                 }
 842
 843                 // Rename the dummy indices
 844                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 845         }
 846
 847         // Simplification of sum = sum of simplifications, check consistency of
 848         // free indices in each term
 849         if (is_ex_exactly_of_type(e_expanded, add)) {
 850                 bool first = true;
 851                 ex sum = _ex0;
 852                 free_indices.clear();
 853
 854                 for (unsigned i=0; i<e_expanded.nops(); i++) {
 855                         exvector free_indices_of_term;
 856                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 857                         if (!term.is_zero()) {
 858                                 if (first) {
 859                                         free_indices = free_indices_of_term;
 860                                         sum = term;
 861                                         first = false;
 862                                 } else {
 863                                         if (!indices_consistent(free_indices, free_indices_of_term))
 864                                                 throw (std::runtime_error("simplify_indexed: inconsistent indices in sum"));
 865                                         if (is_ex_of_type(sum, indexed) && is_ex_of_type(term, indexed))
 866                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
 867                                         else
 868                                                 sum += term;
 869                                 }
 870                         }
 871                 }
 872
 873                 return sum;
 874         }
 875
 876         // Simplification of products
 877         if (is_ex_exactly_of_type(e_expanded, mul)
 878          || is_ex_exactly_of_type(e_expanded, ncmul)
 879          || (is_ex_exactly_of_type(e_expanded, power) && is_ex_of_type(e_expanded.op(0), indexed) && e_expanded.op(1).is_equal(_ex2)))
 880                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
 881
 882         // Cannot do anything
 883         free_indices.clear();
 884         return e_expanded;
 885 }
 886
 887 /** Simplify/canonicalize expression containing indexed objects. This
 888  *  performs contraction of dummy indices where possible and checks whether
 889  *  the free indices in sums are consistent.
 890  *
 891  *  @return simplified expression */
 892 ex ex::simplify_indexed(void) const
 893 {
 894         exvector free_indices, dummy_indices;
 895         scalar_products sp;
 896         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 897 }
 898
 899 /** Simplify/canonicalize expression containing indexed objects. This
 900  *  performs contraction of dummy indices where possible, checks whether
 901  *  the free indices in sums are consistent, and automatically replaces
 902  *  scalar products by known values if desired.
 903  *
 904  *  @param sp Scalar products to be replaced automatically
 905  *  @return simplified expression */
 906 ex ex::simplify_indexed(const scalar_products & sp) const
 907 {
 908         exvector free_indices, dummy_indices;
 909         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 910 }
 911
 912 /** Symmetrize expression over its free indices. */
 913 ex ex::symmetrize(void) const
 914 {
 915         return GiNaC::symmetrize(*this, get_free_indices());
 916 }
 917
 918 /** Antisymmetrize expression over its free indices. */
 919 ex ex::antisymmetrize(void) const
 920 {
 921         return GiNaC::antisymmetrize(*this, get_free_indices());
 922 }
 923
 924 /** Symmetrize expression by cyclic permutation over its free indices. */
 925 ex ex::symmetrize_cyclic(void) const
 926 {
 927         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
 928 }
 929
 930 //////////
 931 // helper classes
 932 //////////
 933
 934 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
 935 {
 936         spm[make_key(v1, v2)] = sp;
 937 }
 938
 939 void scalar_products::add_vectors(const lst & l)
 940 {
 941         // Add all possible pairs of products
 942         unsigned num = l.nops();
 943         for (unsigned i=0; i<num; i++) {
 944                 ex a = l.op(i);
 945                 for (unsigned j=0; j<num; j++) {
 946                         ex b = l.op(j);
 947                         add(a, b, a*b);
 948                 }
 949         }
 950 }
 951
 952 void scalar_products::clear(void)
 953 {
 954         spm.clear();
 955 }
 956
 957 /** Check whether scalar product pair is defined. */
 958 bool scalar_products::is_defined(const ex & v1, const ex & v2) const
 959 {
 960         return spm.find(make_key(v1, v2)) != spm.end();
 961 }
 962
 963 /** Return value of defined scalar product pair. */
 964 ex scalar_products::evaluate(const ex & v1, const ex & v2) const
 965 {
 966         return spm.find(make_key(v1, v2))->second;
 967 }
 968
 969 void scalar_products::debugprint(void) const
 970 {
 971         std::cerr << "map size=" << spm.size() << std::endl;
 972         spmap::const_iterator i = spm.begin(), end = spm.end();
 973         while (i != end) {
 974                 const spmapkey & k = i->first;
 975                 std::cerr << "item key=(" << k.first << "," << k.second;
 976                 std::cerr << "), value=" << i->second << std::endl;
 977                 ++i;
 978         }
 979 }
 980
 981 /** Make key from object pair. */
 982 spmapkey scalar_products::make_key(const ex & v1, const ex & v2)
 983 {
 984         // If indexed, extract base objects
 985         ex s1 = is_ex_of_type(v1, indexed) ? v1.op(0) : v1;
 986         ex s2 = is_ex_of_type(v2, indexed) ? v2.op(0) : v2;
 987
 988         // Enforce canonical order in pair
 989         if (s1.compare(s2) > 0)
 990                 return spmapkey(s2, s1);
 991         else
 992                 return spmapkey(s1, s2);
 993 }
 994
 995 } // namespace GiNaC