ginac/indexed.cpp

   1 /** @file indexed.cpp
   2  *
   3  *  Implementation of GiNaC's indexed expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2001 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 <stdexcept>
  24 #include <algorithm>
  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 #include "debugmsg.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(indexed, exprseq)
  42
  43 //////////
  44 // default constructor, destructor, copy constructor assignment operator and helpers
  45 //////////
  46
  47 indexed::indexed() : symtree(sy_none())
  48 {
  49         debugmsg("indexed default constructor", LOGLEVEL_CONSTRUCT);
  50         tinfo_key = TINFO_indexed;
  51 }
  52
  53 void indexed::copy(const indexed & other)
  54 {
  55         inherited::copy(other);
  56         symtree = other.symtree;
  57 }
  58
  59 DEFAULT_DESTROY(indexed)
  60
  61 //////////
  62 // other constructors
  63 //////////
  64
  65 indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
  66 {
  67         debugmsg("indexed constructor from ex", LOGLEVEL_CONSTRUCT);
  68         tinfo_key = TINFO_indexed;
  69         validate();
  70 }
  71
  72 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(sy_none())
  73 {
  74         debugmsg("indexed constructor from ex,ex", LOGLEVEL_CONSTRUCT);
  75         tinfo_key = TINFO_indexed;
  76         validate();
  77 }
  78
  79 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(sy_none())
  80 {
  81         debugmsg("indexed constructor from ex,ex,ex", LOGLEVEL_CONSTRUCT);
  82         tinfo_key = TINFO_indexed;
  83         validate();
  84 }
  85
  86 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(sy_none())
  87 {
  88         debugmsg("indexed constructor from ex,ex,ex,ex", LOGLEVEL_CONSTRUCT);
  89         tinfo_key = TINFO_indexed;
  90         validate();
  91 }
  92
  93 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())
  94 {
  95         debugmsg("indexed constructor from ex,ex,ex,ex,ex", LOGLEVEL_CONSTRUCT);
  96         tinfo_key = TINFO_indexed;
  97         validate();
  98 }
  99
 100 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
 101 {
 102         debugmsg("indexed constructor from ex,symmetry,ex,ex", LOGLEVEL_CONSTRUCT);
 103         tinfo_key = TINFO_indexed;
 104         validate();
 105 }
 106
 107 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
 108 {
 109         debugmsg("indexed constructor from ex,symmetry,ex,ex,ex", LOGLEVEL_CONSTRUCT);
 110         tinfo_key = TINFO_indexed;
 111         validate();
 112 }
 113
 114 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)
 115 {
 116         debugmsg("indexed constructor from ex,symmetry,ex,ex,ex,ex", LOGLEVEL_CONSTRUCT);
 117         tinfo_key = TINFO_indexed;
 118         validate();
 119 }
 120
 121 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
 122 {
 123         debugmsg("indexed constructor from ex,exvector", LOGLEVEL_CONSTRUCT);
 124         seq.insert(seq.end(), v.begin(), v.end());
 125         tinfo_key = TINFO_indexed;
 126         validate();
 127 }
 128
 129 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 130 {
 131         debugmsg("indexed constructor from ex,symmetry,exvector", LOGLEVEL_CONSTRUCT);
 132         seq.insert(seq.end(), v.begin(), v.end());
 133         tinfo_key = TINFO_indexed;
 134         validate();
 135 }
 136
 137 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 138 {
 139         debugmsg("indexed constructor from symmetry,exprseq", LOGLEVEL_CONSTRUCT);
 140         tinfo_key = TINFO_indexed;
 141 }
 142
 143 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 144 {
 145         debugmsg("indexed constructor from symmetry,exvector", LOGLEVEL_CONSTRUCT);
 146         tinfo_key = TINFO_indexed;
 147 }
 148
 149 indexed::indexed(const symmetry & symm, exvector * vp) : inherited(vp), symtree(symm)
 150 {
 151         debugmsg("indexed constructor from symmetry,exvector *", LOGLEVEL_CONSTRUCT);
 152         tinfo_key = TINFO_indexed;
 153 }
 154
 155 //////////
 156 // archiving
 157 //////////
 158
 159 indexed::indexed(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 160 {
 161         debugmsg("indexed constructor from archive_node", LOGLEVEL_CONSTRUCT);
 162         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 163                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 164                 unsigned symm = 0;
 165                 n.find_unsigned("symmetry", symm);
 166                 switch (symm) {
 167                         case 1:
 168                                 symtree = sy_symm();
 169                                 break;
 170                         case 2:
 171                                 symtree = sy_anti();
 172                                 break;
 173                         default:
 174                                 symtree = sy_none();
 175                                 break;
 176                 }
 177                 ex_to_nonconst_symmetry(symtree).validate(seq.size() - 1);
 178         }
 179 }
 180
 181 void indexed::archive(archive_node &n) const
 182 {
 183         inherited::archive(n);
 184         n.add_ex("symmetry", symtree);
 185 }
 186
 187 DEFAULT_UNARCHIVE(indexed)
 188
 189 //////////
 190 // functions overriding virtual functions from base classes
 191 //////////
 192
 193 void indexed::print(const print_context & c, unsigned level) const
 194 {
 195         debugmsg("indexed print", LOGLEVEL_PRINT);
 196         GINAC_ASSERT(seq.size() > 0);
 197
 198         if (is_of_type(c, print_tree)) {
 199
 200                 c.s << std::string(level, ' ') << class_name()
 201                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 202                     << ", " << seq.size()-1 << " indices"
 203                     << ", symmetry=" << symtree << std::endl;
 204                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 205                 seq[0].print(c, level + delta_indent);
 206                 printindices(c, level + delta_indent);
 207
 208         } else {
 209
 210                 bool is_tex = is_of_type(c, print_latex);
 211                 const ex & base = seq[0];
 212                 bool need_parens = is_ex_exactly_of_type(base, add) || is_ex_exactly_of_type(base, mul)
 213                                 || is_ex_exactly_of_type(base, ncmul) || is_ex_exactly_of_type(base, power)
 214                                 || is_ex_of_type(base, indexed);
 215                 if (is_tex)
 216                         c.s << "{";
 217                 if (need_parens)
 218                         c.s << "(";
 219                 base.print(c);
 220                 if (need_parens)
 221                         c.s << ")";
 222                 if (is_tex)
 223                         c.s << "}";
 224                 printindices(c, level);
 225         }
 226 }
 227
 228 bool indexed::info(unsigned inf) const
 229 {
 230         if (inf == info_flags::indexed) return true;
 231         if (inf == info_flags::has_indices) return seq.size() > 1;
 232         return inherited::info(inf);
 233 }
 234
 235 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 236         bool operator() (const ex & e, unsigned inf) const {
 237                 return !(ex_to<idx>(e).get_value().info(inf));
 238         }
 239 };
 240
 241 bool indexed::all_index_values_are(unsigned inf) const
 242 {
 243         // No indices? Then no property can be fulfilled
 244         if (seq.size() < 2)
 245                 return false;
 246
 247         // Check all indices
 248         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 249 }
 250
 251 int indexed::compare_same_type(const basic & other) const
 252 {
 253         GINAC_ASSERT(is_of_type(other, indexed));
 254         return inherited::compare_same_type(other);
 255 }
 256
 257 ex indexed::eval(int level) const
 258 {
 259         // First evaluate children, then we will end up here again
 260         if (level > 1)
 261                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 262
 263         const ex &base = seq[0];
 264
 265         // If the base object is 0, the whole object is 0
 266         if (base.is_zero())
 267                 return _ex0();
 268
 269         // If the base object is a product, pull out the numeric factor
 270         if (is_ex_exactly_of_type(base, mul) && is_ex_exactly_of_type(base.op(base.nops() - 1), numeric)) {
 271                 exvector v(seq);
 272                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 273                 v[0] = seq[0] / f;
 274                 return f * thisexprseq(v);
 275         }
 276
 277         // Canonicalize indices according to the symmetry properties
 278         if (seq.size() > 2) {
 279                 exvector v = seq;
 280                 GINAC_ASSERT(is_ex_exactly_of_type(symtree, symmetry));
 281                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 282                 if (sig != INT_MAX) {
 283                         // Something has changed while sorting indices, more evaluations later
 284                         if (sig == 0)
 285                                 return _ex0();
 286                         return ex(sig) * thisexprseq(v);
 287                 }
 288         }
 289
 290         // Let the class of the base object perform additional evaluations
 291         return base.bp->eval_indexed(*this);
 292 }
 293
 294 int indexed::degree(const ex & s) const
 295 {
 296         return is_equal(*s.bp) ? 1 : 0;
 297 }
 298
 299 int indexed::ldegree(const ex & s) const
 300 {
 301         return is_equal(*s.bp) ? 1 : 0;
 302 }
 303
 304 ex indexed::coeff(const ex & s, int n) const
 305 {
 306         if (is_equal(*s.bp))
 307                 return n==1 ? _ex1() : _ex0();
 308         else
 309                 return n==0 ? ex(*this) : _ex0();
 310 }
 311
 312 ex indexed::thisexprseq(const exvector & v) const
 313 {
 314         return indexed(ex_to<symmetry>(symtree), v);
 315 }
 316
 317 ex indexed::thisexprseq(exvector * vp) const
 318 {
 319         return indexed(ex_to<symmetry>(symtree), vp);
 320 }
 321
 322 ex indexed::expand(unsigned options) const
 323 {
 324         GINAC_ASSERT(seq.size() > 0);
 325
 326         if ((options & expand_options::expand_indexed) && is_ex_exactly_of_type(seq[0], add)) {
 327
 328                 // expand_indexed expands (a+b).i -> a.i + b.i
 329                 const ex & base = seq[0];
 330                 ex sum = _ex0();
 331                 for (unsigned i=0; i<base.nops(); i++) {
 332                         exvector s = seq;
 333                         s[0] = base.op(i);
 334                         sum += thisexprseq(s).expand();
 335                 }
 336                 return sum;
 337
 338         } else
 339                 return inherited::expand(options);
 340 }
 341
 342 //////////
 343 // virtual functions which can be overridden by derived classes
 344 //////////
 345
 346 // none
 347
 348 //////////
 349 // non-virtual functions in this class
 350 //////////
 351
 352 void indexed::printindices(const print_context & c, unsigned level) const
 353 {
 354         if (seq.size() > 1) {
 355
 356                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 357
 358                 if (is_of_type(c, print_latex)) {
 359
 360                         // TeX output: group by variance
 361                         bool first = true;
 362                         bool covariant = true;
 363
 364                         while (it != itend) {
 365                                 bool cur_covariant = (is_ex_of_type(*it, varidx) ? ex_to<varidx>(*it).is_covariant() : true);
 366                                 if (first || cur_covariant != covariant) {
 367                                         if (!first)
 368                                                 c.s << "}";
 369                                         covariant = cur_covariant;
 370                                         if (covariant)
 371                                                 c.s << "_{";
 372                                         else
 373                                                 c.s << "^{";
 374                                 }
 375                                 it->print(c, level);
 376                                 c.s << " ";
 377                                 first = false;
 378                                 it++;
 379                         }
 380                         c.s << "}";
 381
 382                 } else {
 383
 384                         // Ordinary output
 385                         while (it != itend) {
 386                                 it->print(c, level);
 387                                 it++;
 388                         }
 389                 }
 390         }
 391 }
 392
 393 /** Check whether all indices are of class idx and validate the symmetry
 394  *  tree. This function is used internally to make sure that all constructed
 395  *  indexed objects really carry indices and not some other classes. */
 396 void indexed::validate(void) const
 397 {
 398         GINAC_ASSERT(seq.size() > 0);
 399         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 400         while (it != itend) {
 401                 if (!is_ex_of_type(*it, idx))
 402                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 403                 it++;
 404         }
 405
 406         if (!symtree.is_zero()) {
 407                 if (!is_ex_exactly_of_type(symtree, symmetry))
 408                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 409                 ex_to_nonconst_symmetry(symtree).validate(seq.size() - 1);
 410         }
 411 }
 412
 413 /** Implementation of ex::diff() for an indexed object always returns 0.
 414  *
 415  *  @see ex::diff */
 416 ex indexed::derivative(const symbol & s) const
 417 {
 418         return _ex0();
 419 }
 420
 421 //////////
 422 // global functions
 423 //////////
 424
 425 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 426 static bool indices_consistent(const exvector & v1, const exvector & v2)
 427 {
 428         // Number of indices must be the same
 429         if (v1.size() != v2.size())
 430                 return false;
 431
 432         return equal(v1.begin(), v1.end(), v2.begin(), ex_is_equal());
 433 }
 434
 435 exvector indexed::get_indices(void) const
 436 {
 437         GINAC_ASSERT(seq.size() >= 1);
 438         return exvector(seq.begin() + 1, seq.end());
 439 }
 440
 441 exvector indexed::get_dummy_indices(void) const
 442 {
 443         exvector free_indices, dummy_indices;
 444         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 445         return dummy_indices;
 446 }
 447
 448 exvector indexed::get_dummy_indices(const indexed & other) const
 449 {
 450         exvector indices = get_free_indices();
 451         exvector other_indices = other.get_free_indices();
 452         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 453         exvector dummy_indices;
 454         find_dummy_indices(indices, dummy_indices);
 455         return dummy_indices;
 456 }
 457
 458 bool indexed::has_dummy_index_for(const ex & i) const
 459 {
 460         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 461         while (it != itend) {
 462                 if (is_dummy_pair(*it, i))
 463                         return true;
 464                 it++;
 465         }
 466         return false;
 467 }
 468
 469 exvector indexed::get_free_indices(void) const
 470 {
 471         exvector free_indices, dummy_indices;
 472         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 473         return free_indices;
 474 }
 475
 476 exvector add::get_free_indices(void) const
 477 {
 478         exvector free_indices;
 479         for (unsigned i=0; i<nops(); i++) {
 480                 if (i == 0)
 481                         free_indices = op(i).get_free_indices();
 482                 else {
 483                         exvector free_indices_of_term = op(i).get_free_indices();
 484                         if (!indices_consistent(free_indices, free_indices_of_term))
 485                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 486                 }
 487         }
 488         return free_indices;
 489 }
 490
 491 exvector mul::get_free_indices(void) const
 492 {
 493         // Concatenate free indices of all factors
 494         exvector un;
 495         for (unsigned i=0; i<nops(); i++) {
 496                 exvector free_indices_of_factor = op(i).get_free_indices();
 497                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 498         }
 499
 500         // And remove the dummy indices
 501         exvector free_indices, dummy_indices;
 502         find_free_and_dummy(un, free_indices, dummy_indices);
 503         return free_indices;
 504 }
 505
 506 exvector ncmul::get_free_indices(void) const
 507 {
 508         // Concatenate free indices of all factors
 509         exvector un;
 510         for (unsigned i=0; i<nops(); i++) {
 511                 exvector free_indices_of_factor = op(i).get_free_indices();
 512                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 513         }
 514
 515         // And remove the dummy indices
 516         exvector free_indices, dummy_indices;
 517         find_free_and_dummy(un, free_indices, dummy_indices);
 518         return free_indices;
 519 }
 520
 521 exvector power::get_free_indices(void) const
 522 {
 523         // Return free indices of basis
 524         return basis.get_free_indices();
 525 }
 526
 527 /** Rename dummy indices in an expression.
 528  *
 529  *  @param e Expression to be worked on
 530  *  @param local_dummy_indices The set of dummy indices that appear in the
 531  *    expression "e"
 532  *  @param global_dummy_indices The set of dummy indices that have appeared
 533  *    before and which we would like to use in "e", too. This gets updated
 534  *    by the function */
 535 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 536 {
 537         unsigned global_size = global_dummy_indices.size(),
 538                  local_size = local_dummy_indices.size();
 539
 540         // Any local dummy indices at all?
 541         if (local_size == 0)
 542                 return e;
 543
 544         if (global_size < local_size) {
 545
 546                 // More local indices than we encountered before, add the new ones
 547                 // to the global set
 548                 int old_global_size = global_size;
 549                 int remaining = local_size - global_size;
 550                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 551                 while (it != itend && remaining > 0) {
 552                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(ex_is_equal(), *it)) == global_dummy_indices.end()) {
 553                                 global_dummy_indices.push_back(*it);
 554                                 global_size++;
 555                                 remaining--;
 556                         }
 557                         it++;
 558                 }
 559                 shaker_sort(global_dummy_indices.begin(), global_dummy_indices.end(), ex_is_less(), ex_swap());
 560
 561                 // If this is the first set of local indices, do nothing
 562                 if (old_global_size == 0)
 563                         return e;
 564         }
 565         GINAC_ASSERT(local_size <= global_size);
 566
 567         // Construct lists of index symbols
 568         exlist local_syms, global_syms;
 569         for (unsigned i=0; i<local_size; i++)
 570                 local_syms.push_back(local_dummy_indices[i].op(0));
 571         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 572         for (unsigned i=0; i<global_size; i++)
 573                 global_syms.push_back(global_dummy_indices[i].op(0));
 574
 575         // Remove common indices
 576         exlist local_uniq, global_uniq;
 577         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exlist>(local_uniq), ex_is_less());
 578         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exlist>(global_uniq), ex_is_less());
 579
 580         // Replace remaining non-common local index symbols by global ones
 581         if (local_uniq.empty())
 582                 return e;
 583         else {
 584                 while (global_uniq.size() > local_uniq.size())
 585                         global_uniq.pop_back();
 586                 return e.subs(lst(local_uniq), lst(global_uniq));
 587         }
 588 }
 589
 590 /** Simplify product of indexed expressions (commutative, noncommutative and
 591  *  simple squares), return list of free indices. */
 592 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 593 {
 594         // Remember whether the product was commutative or noncommutative
 595         // (because we chop it into factors and need to reassemble later)
 596         bool non_commutative = is_ex_exactly_of_type(e, ncmul);
 597
 598         // Collect factors in an exvector, store squares twice
 599         exvector v;
 600         v.reserve(e.nops() * 2);
 601
 602         if (is_ex_exactly_of_type(e, power)) {
 603                 // We only get called for simple squares, split a^2 -> a*a
 604                 GINAC_ASSERT(e.op(1).is_equal(_ex2()));
 605                 v.push_back(e.op(0));
 606                 v.push_back(e.op(0));
 607         } else {
 608                 for (unsigned i=0; i<e.nops(); i++) {
 609                         ex f = e.op(i);
 610                         if (is_ex_exactly_of_type(f, power) && f.op(1).is_equal(_ex2())) {
 611                                 v.push_back(f.op(0));
 612                     v.push_back(f.op(0));
 613                         } else if (is_ex_exactly_of_type(f, ncmul)) {
 614                                 // Noncommutative factor found, split it as well
 615                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 616                                 for (unsigned j=0; j<f.nops(); j++)
 617                                         v.push_back(f.op(j));
 618                         } else
 619                                 v.push_back(f);
 620                 }
 621         }
 622
 623         // Perform contractions
 624         bool something_changed = false;
 625         GINAC_ASSERT(v.size() > 1);
 626         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 627         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 628
 629 try_again:
 630                 if (!is_ex_of_type(*it1, indexed))
 631                         continue;
 632
 633                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 634
 635                 // Indexed factor found, get free indices and look for contraction
 636                 // candidates
 637                 exvector free1, dummy1;
 638                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 639
 640                 exvector::iterator it2;
 641                 for (it2 = it1 + 1; it2 != itend; it2++) {
 642
 643                         if (!is_ex_of_type(*it2, indexed))
 644                                 continue;
 645
 646                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 647
 648                         // Find free indices of second factor and merge them with free
 649                         // indices of first factor
 650                         exvector un;
 651                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 652                         un.insert(un.end(), free1.begin(), free1.end());
 653
 654                         // Check whether the two factors share dummy indices
 655                         exvector free, dummy;
 656                         find_free_and_dummy(un, free, dummy);
 657                         unsigned num_dummies = dummy.size();
 658                         if (num_dummies == 0)
 659                                 continue;
 660
 661                         // At least one dummy index, is it a defined scalar product?
 662                         bool contracted = false;
 663                         if (free.empty()) {
 664                                 if (sp.is_defined(*it1, *it2)) {
 665                                         *it1 = sp.evaluate(*it1, *it2);
 666                                         *it2 = _ex1();
 667                                         goto contraction_done;
 668                                 }
 669                         }
 670
 671                         // Try to contract the first one with the second one
 672                         contracted = it1->op(0).bp->contract_with(it1, it2, v);
 673                         if (!contracted) {
 674
 675                                 // That didn't work; maybe the second object knows how to
 676                                 // contract itself with the first one
 677                                 contracted = it2->op(0).bp->contract_with(it2, it1, v);
 678                         }
 679                         if (contracted) {
 680 contraction_done:
 681                                 if (first_noncommutative || second_noncommutative
 682                                  || is_ex_exactly_of_type(*it1, add) || is_ex_exactly_of_type(*it2, add)
 683                                  || is_ex_exactly_of_type(*it1, mul) || is_ex_exactly_of_type(*it2, mul)
 684                                  || is_ex_exactly_of_type(*it1, ncmul) || is_ex_exactly_of_type(*it2, ncmul)) {
 685
 686                                         // One of the factors became a sum or product:
 687                                         // re-expand expression and run again
 688                                         // Non-commutative products are always re-expanded to give
 689                                         // simplify_ncmul() the chance to re-order and canonicalize
 690                                         // the product
 691                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 692                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 693                                 }
 694
 695                                 // Both objects may have new indices now or they might
 696                                 // even not be indexed objects any more, so we have to
 697                                 // start over
 698                                 something_changed = true;
 699                                 goto try_again;
 700                         }
 701                 }
 702         }
 703
 704         // Find free indices (concatenate them all and call find_free_and_dummy())
 705         // and all dummy indices that appear
 706         exvector un, individual_dummy_indices;
 707         it1 = v.begin(); itend = v.end();
 708         while (it1 != itend) {
 709                 exvector free_indices_of_factor;
 710                 if (is_ex_of_type(*it1, indexed)) {
 711                         exvector dummy_indices_of_factor;
 712                         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);
 713                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 714                 } else
 715                         free_indices_of_factor = it1->get_free_indices();
 716                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 717                 it1++;
 718         }
 719         exvector local_dummy_indices;
 720         find_free_and_dummy(un, free_indices, local_dummy_indices);
 721         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 722
 723         ex r;
 724         if (something_changed)
 725                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 726         else
 727                 r = e;
 728
 729         // The result should be symmetric with respect to exchange of dummy
 730         // indices, so if the symmetrization vanishes, the whole expression is
 731         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 732         if (local_dummy_indices.size() >= 2) {
 733                 lst dummy_syms;
 734                 for (int i=0; i<local_dummy_indices.size(); i++)
 735                         dummy_syms.append(local_dummy_indices[i].op(0));
 736                 if (r.symmetrize(dummy_syms).is_zero()) {
 737                         free_indices.clear();
 738                         return _ex0();
 739                 }
 740         }
 741
 742         // Dummy index renaming
 743         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 744
 745         // Product of indexed object with a scalar?
 746         if (is_ex_exactly_of_type(r, mul) && r.nops() == 2
 747          && is_ex_exactly_of_type(r.op(1), numeric) && is_ex_of_type(r.op(0), indexed))
 748                 return r.op(0).op(0).bp->scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 749         else
 750                 return r;
 751 }
 752
 753 /** Simplify indexed expression, return list of free indices. */
 754 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 755 {
 756         // Expand the expression
 757         ex e_expanded = e.expand();
 758
 759         // Simplification of single indexed object: just find the free indices
 760         // and perform dummy index renaming
 761         if (is_ex_of_type(e_expanded, indexed)) {
 762                 const indexed &i = ex_to<indexed>(e_expanded);
 763                 exvector local_dummy_indices;
 764                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 765                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 766         }
 767
 768         // Simplification of sum = sum of simplifications, check consistency of
 769         // free indices in each term
 770         if (is_ex_exactly_of_type(e_expanded, add)) {
 771                 bool first = true;
 772                 ex sum = _ex0();
 773                 free_indices.clear();
 774
 775                 for (unsigned i=0; i<e_expanded.nops(); i++) {
 776                         exvector free_indices_of_term;
 777                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 778                         if (!term.is_zero()) {
 779                                 if (first) {
 780                                         free_indices = free_indices_of_term;
 781                                         sum = term;
 782                                         first = false;
 783                                 } else {
 784                                         if (!indices_consistent(free_indices, free_indices_of_term))
 785                                                 throw (std::runtime_error("simplify_indexed: inconsistent indices in sum"));
 786                                         if (is_ex_of_type(sum, indexed) && is_ex_of_type(term, indexed))
 787                                                 sum = sum.op(0).bp->add_indexed(sum, term);
 788                                         else
 789                                                 sum += term;
 790                                 }
 791                         }
 792                 }
 793
 794                 return sum;
 795         }
 796
 797         // Simplification of products
 798         if (is_ex_exactly_of_type(e_expanded, mul)
 799          || is_ex_exactly_of_type(e_expanded, ncmul)
 800          || (is_ex_exactly_of_type(e_expanded, power) && is_ex_of_type(e_expanded.op(0), indexed) && e_expanded.op(1).is_equal(_ex2())))
 801                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
 802
 803         // Cannot do anything
 804         free_indices.clear();
 805         return e_expanded;
 806 }
 807
 808 /** Simplify/canonicalize expression containing indexed objects. This
 809  *  performs contraction of dummy indices where possible and checks whether
 810  *  the free indices in sums are consistent.
 811  *
 812  *  @return simplified expression */
 813 ex ex::simplify_indexed(void) const
 814 {
 815         exvector free_indices, dummy_indices;
 816         scalar_products sp;
 817         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 818 }
 819
 820 /** Simplify/canonicalize expression containing indexed objects. This
 821  *  performs contraction of dummy indices where possible, checks whether
 822  *  the free indices in sums are consistent, and automatically replaces
 823  *  scalar products by known values if desired.
 824  *
 825  *  @param sp Scalar products to be replaced automatically
 826  *  @return simplified expression */
 827 ex ex::simplify_indexed(const scalar_products & sp) const
 828 {
 829         exvector free_indices, dummy_indices;
 830         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 831 }
 832
 833 /** Symmetrize expression over its free indices. */
 834 ex ex::symmetrize(void) const
 835 {
 836         return GiNaC::symmetrize(*this, get_free_indices());
 837 }
 838
 839 /** Antisymmetrize expression over its free indices. */
 840 ex ex::antisymmetrize(void) const
 841 {
 842         return GiNaC::antisymmetrize(*this, get_free_indices());
 843 }
 844
 845 /** Symmetrize expression by cyclic permutation over its free indices. */
 846 ex ex::symmetrize_cyclic(void) const
 847 {
 848         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
 849 }
 850
 851 //////////
 852 // helper classes
 853 //////////
 854
 855 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
 856 {
 857         spm[make_key(v1, v2)] = sp;
 858 }
 859
 860 void scalar_products::add_vectors(const lst & l)
 861 {
 862         // Add all possible pairs of products
 863         unsigned num = l.nops();
 864         for (unsigned i=0; i<num; i++) {
 865                 ex a = l.op(i);
 866                 for (unsigned j=0; j<num; j++) {
 867                         ex b = l.op(j);
 868                         add(a, b, a*b);
 869                 }
 870         }
 871 }
 872
 873 void scalar_products::clear(void)
 874 {
 875         spm.clear();
 876 }
 877
 878 /** Check whether scalar product pair is defined. */
 879 bool scalar_products::is_defined(const ex & v1, const ex & v2) const
 880 {
 881         return spm.find(make_key(v1, v2)) != spm.end();
 882 }
 883
 884 /** Return value of defined scalar product pair. */
 885 ex scalar_products::evaluate(const ex & v1, const ex & v2) const
 886 {
 887         return spm.find(make_key(v1, v2))->second;
 888 }
 889
 890 void scalar_products::debugprint(void) const
 891 {
 892         std::cerr << "map size=" << spm.size() << std::endl;
 893         spmap::const_iterator i = spm.begin(), end = spm.end();
 894         while (i != end) {
 895                 const spmapkey & k = i->first;
 896                 std::cerr << "item key=(" << k.first << "," << k.second;
 897                 std::cerr << "), value=" << i->second << std::endl;
 898                 ++i;
 899         }
 900 }
 901
 902 /** Make key from object pair. */
 903 spmapkey scalar_products::make_key(const ex & v1, const ex & v2)
 904 {
 905         // If indexed, extract base objects
 906         ex s1 = is_ex_of_type(v1, indexed) ? v1.op(0) : v1;
 907         ex s2 = is_ex_of_type(v2, indexed) ? v2.op(0) : v2;
 908
 909         // Enforce canonical order in pair
 910         if (s1.compare(s2) > 0)
 911                 return spmapkey(s2, s1);
 912         else
 913                 return spmapkey(s1, s2);
 914 }
 915
 916 } // namespace GiNaC