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