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