ginac/indexed.cpp

   1 /** @file indexed.cpp
   2  *
   3  *  Implementation of GiNaC's indexed expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <iostream>
  24 #include <sstream>
  25 #include <stdexcept>
  26
  27 #include "indexed.h"
  28 #include "idx.h"
  29 #include "add.h"
  30 #include "mul.h"
  31 #include "ncmul.h"
  32 #include "power.h"
  33 #include "relational.h"
  34 #include "symmetry.h"
  35 #include "operators.h"
  36 #include "lst.h"
  37 #include "print.h"
  38 #include "archive.h"
  39 #include "utils.h"
  40
  41 namespace GiNaC {
  42
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(indexed, exprseq)
  44
  45 //////////
  46 // default ctor, dtor, copy ctor, assignment operator and helpers
  47 //////////
  48
  49 indexed::indexed() : symtree(sy_none())
  50 {
  51         tinfo_key = TINFO_indexed;
  52 }
  53
  54 void indexed::copy(const indexed & other)
  55 {
  56         inherited::copy(other);
  57         symtree = other.symtree;
  58 }
  59
  60 DEFAULT_DESTROY(indexed)
  61
  62 //////////
  63 // other constructors
  64 //////////
  65
  66 indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
  67 {
  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         tinfo_key = TINFO_indexed;
  75         validate();
  76 }
  77
  78 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(sy_none())
  79 {
  80         tinfo_key = TINFO_indexed;
  81         validate();
  82 }
  83
  84 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(sy_none())
  85 {
  86         tinfo_key = TINFO_indexed;
  87         validate();
  88 }
  89
  90 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())
  91 {
  92         tinfo_key = TINFO_indexed;
  93         validate();
  94 }
  95
  96 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
  97 {
  98         tinfo_key = TINFO_indexed;
  99         validate();
 100 }
 101
 102 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
 103 {
 104         tinfo_key = TINFO_indexed;
 105         validate();
 106 }
 107
 108 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)
 109 {
 110         tinfo_key = TINFO_indexed;
 111         validate();
 112 }
 113
 114 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
 115 {
 116         seq.insert(seq.end(), v.begin(), v.end());
 117         tinfo_key = TINFO_indexed;
 118         validate();
 119 }
 120
 121 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 122 {
 123         seq.insert(seq.end(), v.begin(), v.end());
 124         tinfo_key = TINFO_indexed;
 125         validate();
 126 }
 127
 128 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 129 {
 130         tinfo_key = TINFO_indexed;
 131 }
 132
 133 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 134 {
 135         tinfo_key = TINFO_indexed;
 136 }
 137
 138 indexed::indexed(const symmetry & symm, exvector * vp) : inherited(vp), symtree(symm)
 139 {
 140         tinfo_key = TINFO_indexed;
 141 }
 142
 143 //////////
 144 // archiving
 145 //////////
 146
 147 indexed::indexed(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 148 {
 149         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 150                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 151                 unsigned symm = 0;
 152                 n.find_unsigned("symmetry", symm);
 153                 switch (symm) {
 154                         case 1:
 155                                 symtree = sy_symm();
 156                                 break;
 157                         case 2:
 158                                 symtree = sy_anti();
 159                                 break;
 160                         default:
 161                                 symtree = sy_none();
 162                                 break;
 163                 }
 164                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 165         }
 166 }
 167
 168 void indexed::archive(archive_node &n) const
 169 {
 170         inherited::archive(n);
 171         n.add_ex("symmetry", symtree);
 172 }
 173
 174 DEFAULT_UNARCHIVE(indexed)
 175
 176 //////////
 177 // functions overriding virtual functions from base classes
 178 //////////
 179
 180 void indexed::print(const print_context & c, unsigned level) const
 181 {
 182         GINAC_ASSERT(seq.size() > 0);
 183
 184         if (is_a<print_tree>(c)) {
 185
 186                 c.s << std::string(level, ' ') << class_name()
 187                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 188                     << ", " << seq.size()-1 << " indices"
 189                     << ", symmetry=" << symtree << std::endl;
 190                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 191                 seq[0].print(c, level + delta_indent);
 192                 printindices(c, level + delta_indent);
 193
 194         } else {
 195
 196                 bool is_tex = is_a<print_latex>(c);
 197                 const ex & base = seq[0];
 198
 199                 if (precedence() <= level)
 200                         c.s << (is_tex ? "{(" : "(");
 201                 if (is_tex)
 202                         c.s << "{";
 203                 base.print(c, precedence());
 204                 if (is_tex)
 205                         c.s << "}";
 206                 printindices(c, level);
 207                 if (precedence() <= level)
 208                         c.s << (is_tex ? ")}" : ")");
 209         }
 210 }
 211
 212 bool indexed::info(unsigned inf) const
 213 {
 214         if (inf == info_flags::indexed) return true;
 215         if (inf == info_flags::has_indices) return seq.size() > 1;
 216         return inherited::info(inf);
 217 }
 218
 219 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 220         bool operator() (const ex & e, unsigned inf) const {
 221                 return !(ex_to<idx>(e).get_value().info(inf));
 222         }
 223 };
 224
 225 bool indexed::all_index_values_are(unsigned inf) const
 226 {
 227         // No indices? Then no property can be fulfilled
 228         if (seq.size() < 2)
 229                 return false;
 230
 231         // Check all indices
 232         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 233 }
 234
 235 int indexed::compare_same_type(const basic & other) const
 236 {
 237         GINAC_ASSERT(is_a<indexed>(other));
 238         return inherited::compare_same_type(other);
 239 }
 240
 241 ex indexed::eval(int level) const
 242 {
 243         // First evaluate children, then we will end up here again
 244         if (level > 1)
 245                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 246
 247         const ex &base = seq[0];
 248
 249         // If the base object is 0, the whole object is 0
 250         if (base.is_zero())
 251                 return _ex0;
 252
 253         // If the base object is a product, pull out the numeric factor
 254         if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
 255                 exvector v(seq);
 256                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 257                 v[0] = seq[0] / f;
 258                 return f * thisexprseq(v);
 259         }
 260
 261         // Canonicalize indices according to the symmetry properties
 262         if (seq.size() > 2) {
 263                 exvector v = seq;
 264                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 265                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 266                 if (sig != INT_MAX) {
 267                         // Something has changed while sorting indices, more evaluations later
 268                         if (sig == 0)
 269                                 return _ex0;
 270                         return ex(sig) * thisexprseq(v);
 271                 }
 272         }
 273
 274         // Let the class of the base object perform additional evaluations
 275         return ex_to<basic>(base).eval_indexed(*this);
 276 }
 277
 278 ex indexed::thisexprseq(const exvector & v) const
 279 {
 280         return indexed(ex_to<symmetry>(symtree), v);
 281 }
 282
 283 ex indexed::thisexprseq(exvector * vp) const
 284 {
 285         return indexed(ex_to<symmetry>(symtree), vp);
 286 }
 287
 288 ex indexed::expand(unsigned options) const
 289 {
 290         GINAC_ASSERT(seq.size() > 0);
 291
 292         if ((options & expand_options::expand_indexed) && is_exactly_a<add>(seq[0])) {
 293
 294                 // expand_indexed expands (a+b).i -> a.i + b.i
 295                 const ex & base = seq[0];
 296                 ex sum = _ex0;
 297                 for (unsigned i=0; i<base.nops(); i++) {
 298                         exvector s = seq;
 299                         s[0] = base.op(i);
 300                         sum += thisexprseq(s).expand();
 301                 }
 302                 return sum;
 303
 304         } else
 305                 return inherited::expand(options);
 306 }
 307
 308 //////////
 309 // virtual functions which can be overridden by derived classes
 310 //////////
 311
 312 // none
 313
 314 //////////
 315 // non-virtual functions in this class
 316 //////////
 317
 318 void indexed::printindices(const print_context & c, unsigned level) const
 319 {
 320         if (seq.size() > 1) {
 321
 322                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 323
 324                 if (is_a<print_latex>(c)) {
 325
 326                         // TeX output: group by variance
 327                         bool first = true;
 328                         bool covariant = true;
 329
 330                         while (it != itend) {
 331                                 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
 332                                 if (first || cur_covariant != covariant) { // Variance changed
 333                                         // The empty {} prevents indices from ending up on top of each other
 334                                         if (!first)
 335                                                 c.s << "}{}";
 336                                         covariant = cur_covariant;
 337                                         if (covariant)
 338                                                 c.s << "_{";
 339                                         else
 340                                                 c.s << "^{";
 341                                 }
 342                                 it->print(c, level);
 343                                 c.s << " ";
 344                                 first = false;
 345                                 it++;
 346                         }
 347                         c.s << "}";
 348
 349                 } else {
 350
 351                         // Ordinary output
 352                         while (it != itend) {
 353                                 it->print(c, level);
 354                                 it++;
 355                         }
 356                 }
 357         }
 358 }
 359
 360 /** Check whether all indices are of class idx and validate the symmetry
 361  *  tree. This function is used internally to make sure that all constructed
 362  *  indexed objects really carry indices and not some other classes. */
 363 void indexed::validate(void) const
 364 {
 365         GINAC_ASSERT(seq.size() > 0);
 366         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 367         while (it != itend) {
 368                 if (!is_a<idx>(*it))
 369                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 370                 it++;
 371         }
 372
 373         if (!symtree.is_zero()) {
 374                 if (!is_exactly_a<symmetry>(symtree))
 375                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 376                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 377         }
 378 }
 379
 380 /** Implementation of ex::diff() for an indexed object always returns 0.
 381  *
 382  *  @see ex::diff */
 383 ex indexed::derivative(const symbol & s) const
 384 {
 385         return _ex0;
 386 }
 387
 388 //////////
 389 // global functions
 390 //////////
 391
 392 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
 393         bool operator() (const ex &lh, const ex &rh) const
 394         {
 395                 if (lh.is_equal(rh))
 396                         return true;
 397                 else
 398                         try {
 399                                 // Replacing the dimension might cause an error (e.g. with
 400                                 // index classes that only work in a fixed number of dimensions)
 401                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
 402                         } catch (...) {
 403                                 return false;
 404                         }
 405         }
 406 };
 407
 408 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 409 static bool indices_consistent(const exvector & v1, const exvector & v2)
 410 {
 411         // Number of indices must be the same
 412         if (v1.size() != v2.size())
 413                 return false;
 414
 415         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
 416 }
 417
 418 exvector indexed::get_indices(void) const
 419 {
 420         GINAC_ASSERT(seq.size() >= 1);
 421         return exvector(seq.begin() + 1, seq.end());
 422 }
 423
 424 exvector indexed::get_dummy_indices(void) const
 425 {
 426         exvector free_indices, dummy_indices;
 427         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 428         return dummy_indices;
 429 }
 430
 431 exvector indexed::get_dummy_indices(const indexed & other) const
 432 {
 433         exvector indices = get_free_indices();
 434         exvector other_indices = other.get_free_indices();
 435         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 436         exvector dummy_indices;
 437         find_dummy_indices(indices, dummy_indices);
 438         return dummy_indices;
 439 }
 440
 441 bool indexed::has_dummy_index_for(const ex & i) const
 442 {
 443         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 444         while (it != itend) {
 445                 if (is_dummy_pair(*it, i))
 446                         return true;
 447                 it++;
 448         }
 449         return false;
 450 }
 451
 452 exvector indexed::get_free_indices(void) const
 453 {
 454         exvector free_indices, dummy_indices;
 455         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 456         return free_indices;
 457 }
 458
 459 exvector add::get_free_indices(void) const
 460 {
 461         exvector free_indices;
 462         for (unsigned i=0; i<nops(); i++) {
 463                 if (i == 0)
 464                         free_indices = op(i).get_free_indices();
 465                 else {
 466                         exvector free_indices_of_term = op(i).get_free_indices();
 467                         if (!indices_consistent(free_indices, free_indices_of_term))
 468                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 469                 }
 470         }
 471         return free_indices;
 472 }
 473
 474 exvector mul::get_free_indices(void) const
 475 {
 476         // Concatenate free indices of all factors
 477         exvector un;
 478         for (unsigned i=0; i<nops(); i++) {
 479                 exvector free_indices_of_factor = op(i).get_free_indices();
 480                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 481         }
 482
 483         // And remove the dummy indices
 484         exvector free_indices, dummy_indices;
 485         find_free_and_dummy(un, free_indices, dummy_indices);
 486         return free_indices;
 487 }
 488
 489 exvector ncmul::get_free_indices(void) const
 490 {
 491         // Concatenate free indices of all factors
 492         exvector un;
 493         for (unsigned i=0; i<nops(); i++) {
 494                 exvector free_indices_of_factor = op(i).get_free_indices();
 495                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 496         }
 497
 498         // And remove the dummy indices
 499         exvector free_indices, dummy_indices;
 500         find_free_and_dummy(un, free_indices, dummy_indices);
 501         return free_indices;
 502 }
 503
 504 exvector power::get_free_indices(void) const
 505 {
 506         // Return free indices of basis
 507         return basis.get_free_indices();
 508 }
 509
 510 /** Rename dummy indices in an expression.
 511  *
 512  *  @param e Expression to work on
 513  *  @param local_dummy_indices The set of dummy indices that appear in the
 514  *    expression "e"
 515  *  @param global_dummy_indices The set of dummy indices that have appeared
 516  *    before and which we would like to use in "e", too. This gets updated
 517  *    by the function */
 518 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 519 {
 520         unsigned global_size = global_dummy_indices.size(),
 521                  local_size = local_dummy_indices.size();
 522
 523         // Any local dummy indices at all?
 524         if (local_size == 0)
 525                 return e;
 526
 527         if (global_size < local_size) {
 528
 529                 // More local indices than we encountered before, add the new ones
 530                 // to the global set
 531                 int old_global_size = global_size;
 532                 int remaining = local_size - global_size;
 533                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 534                 while (it != itend && remaining > 0) {
 535                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(ex_is_equal(), *it)) == global_dummy_indices.end()) {
 536                                 global_dummy_indices.push_back(*it);
 537                                 global_size++;
 538                                 remaining--;
 539                         }
 540                         it++;
 541                 }
 542
 543                 // If this is the first set of local indices, do nothing
 544                 if (old_global_size == 0)
 545                         return e;
 546         }
 547         GINAC_ASSERT(local_size <= global_size);
 548
 549         // Construct lists of index symbols
 550         exlist local_syms, global_syms;
 551         for (unsigned i=0; i<local_size; i++)
 552                 local_syms.push_back(local_dummy_indices[i].op(0));
 553         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 554         for (unsigned i=0; i<global_size; i++)
 555                 global_syms.push_back(global_dummy_indices[i].op(0));
 556         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 557
 558         // Remove common indices
 559         exlist local_uniq, global_uniq;
 560         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exlist>(local_uniq), ex_is_less());
 561         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exlist>(global_uniq), ex_is_less());
 562
 563         // Replace remaining non-common local index symbols by global ones
 564         if (local_uniq.empty())
 565                 return e;
 566         else {
 567                 while (global_uniq.size() > local_uniq.size())
 568                         global_uniq.pop_back();
 569                 return e.subs(lst(local_uniq), lst(global_uniq));
 570         }
 571 }
 572
 573 /** Given a set of indices, extract those of class varidx. */
 574 static void find_variant_indices(const exvector & v, exvector & variant_indices)
 575 {
 576         exvector::const_iterator it1, itend;
 577         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 578                 if (is_exactly_a<varidx>(*it1))
 579                         variant_indices.push_back(*it1);
 580         }
 581 }
 582
 583 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
 584  *  variance.
 585  *
 586  *  @param e Object to work on
 587  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
 588  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
 589  *  @return true if 'e' was changed */
 590 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
 591 {
 592         bool something_changed = false;
 593
 594         // If a dummy index is encountered for the first time in the
 595         // product, pull it up, otherwise, pull it down
 596         exvector::const_iterator it2, it2start, it2end;
 597         for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
 598                 if (!is_exactly_a<varidx>(*it2))
 599                         continue;
 600
 601                 exvector::iterator vit, vitend;
 602                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 603                         if (it2->op(0).is_equal(vit->op(0))) {
 604                                 if (ex_to<varidx>(*it2).is_covariant()) {
 605                                         e = e.subs(lst(
 606                                                 *it2 == ex_to<varidx>(*it2).toggle_variance(),
 607                                                 ex_to<varidx>(*it2).toggle_variance() == *it2
 608                                         ));
 609                                         something_changed = true;
 610                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 611                                         it2start = ex_to<indexed>(e).seq.begin();
 612                                         it2end = ex_to<indexed>(e).seq.end();
 613                                 }
 614                                 moved_indices.push_back(*vit);
 615                                 variant_dummy_indices.erase(vit);
 616                                 goto next_index;
 617                         }
 618                 }
 619
 620                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 621                         if (it2->op(0).is_equal(vit->op(0))) {
 622                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 623                                         e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance());
 624                                         something_changed = true;
 625                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 626                                         it2start = ex_to<indexed>(e).seq.begin();
 627                                         it2end = ex_to<indexed>(e).seq.end();
 628                                 }
 629                                 goto next_index;
 630                         }
 631                 }
 632
 633 next_index: ;
 634         }
 635
 636         return something_changed;
 637 }
 638
 639 /* Ordering that only compares the base expressions of indexed objects. */
 640 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
 641         bool operator() (const ex &lh, const ex &rh) const
 642         {
 643                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 644         }
 645 };
 646
 647 /** Simplify product of indexed expressions (commutative, noncommutative and
 648  *  simple squares), return list of free indices. */
 649 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 650 {
 651         // Remember whether the product was commutative or noncommutative
 652         // (because we chop it into factors and need to reassemble later)
 653         bool non_commutative = is_exactly_a<ncmul>(e);
 654
 655         // Collect factors in an exvector, store squares twice
 656         exvector v;
 657         v.reserve(e.nops() * 2);
 658
 659         if (is_exactly_a<power>(e)) {
 660                 // We only get called for simple squares, split a^2 -> a*a
 661                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 662                 v.push_back(e.op(0));
 663                 v.push_back(e.op(0));
 664         } else {
 665                 for (unsigned i=0; i<e.nops(); i++) {
 666                         ex f = e.op(i);
 667                         if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
 668                                 v.push_back(f.op(0));
 669                     v.push_back(f.op(0));
 670                         } else if (is_exactly_a<ncmul>(f)) {
 671                                 // Noncommutative factor found, split it as well
 672                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 673                                 for (unsigned j=0; j<f.nops(); j++)
 674                                         v.push_back(f.op(j));
 675                         } else
 676                                 v.push_back(f);
 677                 }
 678         }
 679
 680         // Perform contractions
 681         bool something_changed = false;
 682         GINAC_ASSERT(v.size() > 1);
 683         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 684         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 685
 686 try_again:
 687                 if (!is_a<indexed>(*it1))
 688                         continue;
 689
 690                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 691
 692                 // Indexed factor found, get free indices and look for contraction
 693                 // candidates
 694                 exvector free1, dummy1;
 695                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 696
 697                 exvector::iterator it2;
 698                 for (it2 = it1 + 1; it2 != itend; it2++) {
 699
 700                         if (!is_a<indexed>(*it2))
 701                                 continue;
 702
 703                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 704
 705                         // Find free indices of second factor and merge them with free
 706                         // indices of first factor
 707                         exvector un;
 708                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 709                         un.insert(un.end(), free1.begin(), free1.end());
 710
 711                         // Check whether the two factors share dummy indices
 712                         exvector free, dummy;
 713                         find_free_and_dummy(un, free, dummy);
 714                         unsigned num_dummies = dummy.size();
 715                         if (num_dummies == 0)
 716                                 continue;
 717
 718                         // At least one dummy index, is it a defined scalar product?
 719                         bool contracted = false;
 720                         if (free.empty()) {
 721
 722                                 // Find minimal dimension of all indices of both factors
 723                                 exvector::const_iterator dit = ex_to<indexed>(*it1).seq.begin() + 1, ditend = ex_to<indexed>(*it1).seq.end();
 724                                 ex dim = ex_to<idx>(*dit).get_dim();
 725                                 ++dit;
 726                                 for (; dit != ditend; ++dit) {
 727                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 728                                 }
 729                                 dit = ex_to<indexed>(*it2).seq.begin() + 1;
 730                                 ditend = ex_to<indexed>(*it2).seq.end();
 731                                 for (; dit != ditend; ++dit) {
 732                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 733                                 }
 734
 735                                 // User-defined scalar product?
 736                                 if (sp.is_defined(*it1, *it2, dim)) {
 737
 738                                         // Yes, substitute it
 739                                         *it1 = sp.evaluate(*it1, *it2, dim);
 740                                         *it2 = _ex1;
 741                                         goto contraction_done;
 742                                 }
 743                         }
 744
 745                         // Try to contract the first one with the second one
 746                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 747                         if (!contracted) {
 748
 749                                 // That didn't work; maybe the second object knows how to
 750                                 // contract itself with the first one
 751                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 752                         }
 753                         if (contracted) {
 754 contraction_done:
 755                                 if (first_noncommutative || second_noncommutative
 756                                  || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
 757                                  || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
 758                                  || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
 759
 760                                         // One of the factors became a sum or product:
 761                                         // re-expand expression and run again
 762                                         // Non-commutative products are always re-expanded to give
 763                                         // eval_ncmul() the chance to re-order and canonicalize
 764                                         // the product
 765                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 766                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 767                                 }
 768
 769                                 // Both objects may have new indices now or they might
 770                                 // even not be indexed objects any more, so we have to
 771                                 // start over
 772                                 something_changed = true;
 773                                 goto try_again;
 774                         }
 775                 }
 776         }
 777
 778         // Find free indices (concatenate them all and call find_free_and_dummy())
 779         // and all dummy indices that appear
 780         exvector un, individual_dummy_indices;
 781         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 782                 exvector free_indices_of_factor;
 783                 if (is_a<indexed>(*it1)) {
 784                         exvector dummy_indices_of_factor;
 785                         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);
 786                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 787                 } else
 788                         free_indices_of_factor = it1->get_free_indices();
 789                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 790         }
 791         exvector local_dummy_indices;
 792         find_free_and_dummy(un, free_indices, local_dummy_indices);
 793         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 794
 795         // Filter out the dummy indices with variance
 796         exvector variant_dummy_indices;
 797         find_variant_indices(local_dummy_indices, variant_dummy_indices);
 798
 799         // Any indices with variance present at all?
 800         if (!variant_dummy_indices.empty()) {
 801
 802                 // Yes, bring the product into a canonical order that only depends on
 803                 // the base expressions of indexed objects
 804                 if (!non_commutative)
 805                         std::sort(v.begin(), v.end(), ex_base_is_less());
 806
 807                 exvector moved_indices;
 808
 809                 // Iterate over all indexed objects in the product
 810                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 811                         if (!is_a<indexed>(*it1))
 812                                 continue;
 813
 814                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
 815                                 something_changed = true;
 816                 }
 817         }
 818
 819         ex r;
 820         if (something_changed)
 821                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 822         else
 823                 r = e;
 824
 825         // The result should be symmetric with respect to exchange of dummy
 826         // indices, so if the symmetrization vanishes, the whole expression is
 827         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 828         if (local_dummy_indices.size() >= 2) {
 829                 lst dummy_syms;
 830                 for (int i=0; i<local_dummy_indices.size(); i++)
 831                         dummy_syms.append(local_dummy_indices[i].op(0));
 832                 if (r.symmetrize(dummy_syms).is_zero()) {
 833                         free_indices.clear();
 834                         return _ex0;
 835                 }
 836         }
 837
 838         // Dummy index renaming
 839         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 840
 841         // Product of indexed object with a scalar?
 842         if (is_exactly_a<mul>(r) && r.nops() == 2
 843          && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
 844                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 845         else
 846                 return r;
 847 }
 848
 849 /** This structure stores the original and symmetrized versions of terms
 850  *  obtained during the simplification of sums. */
 851 class terminfo {
 852 public:
 853         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
 854
 855         ex orig; /**< original term */
 856         ex symm; /**< symmtrized term */
 857 };
 858
 859 class terminfo_is_less {
 860 public:
 861         bool operator() (const terminfo & ti1, const terminfo & ti2) const
 862         {
 863                 return (ti1.symm.compare(ti2.symm) < 0);
 864         }
 865 };
 866
 867 /** This structure stores the individual symmetrized terms obtained during
 868  *  the simplification of sums. */
 869 class symminfo {
 870 public:
 871         symminfo() : num(0) {}
 872
 873         symminfo(const ex & symmterm_, const ex & orig_, unsigned num_) : orig(orig_), num(num_)
 874         {
 875                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
 876                         coeff = symmterm_.op(symmterm_.nops()-1);
 877                         symmterm = symmterm_ / coeff;
 878                 } else {
 879                         coeff = 1;
 880                         symmterm = symmterm_;
 881                 }
 882         }
 883
 884         ex symmterm;  /**< symmetrized term */
 885         ex coeff;     /**< coefficient of symmetrized term */
 886         ex orig;      /**< original term */
 887         unsigned num; /**< how many symmetrized terms resulted from the original term */
 888 };
 889
 890 class symminfo_is_less_by_symmterm {
 891 public:
 892         bool operator() (const symminfo & si1, const symminfo & si2) const
 893         {
 894                 return (si1.symmterm.compare(si2.symmterm) < 0);
 895         }
 896 };
 897
 898 class symminfo_is_less_by_orig {
 899 public:
 900         bool operator() (const symminfo & si1, const symminfo & si2) const
 901         {
 902                 return (si1.orig.compare(si2.orig) < 0);
 903         }
 904 };
 905
 906 /** Simplify indexed expression, return list of free indices. */
 907 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 908 {
 909         // Expand the expression
 910         ex e_expanded = e.expand();
 911
 912         // Simplification of single indexed object: just find the free indices
 913         // and perform dummy index renaming/repositioning
 914         if (is_a<indexed>(e_expanded)) {
 915
 916                 // Find the dummy indices
 917                 const indexed &i = ex_to<indexed>(e_expanded);
 918                 exvector local_dummy_indices;
 919                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 920
 921                 // Filter out the dummy indices with variance
 922                 exvector variant_dummy_indices;
 923                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
 924
 925                 // Any indices with variance present at all?
 926                 if (!variant_dummy_indices.empty()) {
 927
 928                         // Yes, reposition them
 929                         exvector moved_indices;
 930                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
 931                 }
 932
 933                 // Rename the dummy indices
 934                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 935         }
 936
 937         // Simplification of sum = sum of simplifications, check consistency of
 938         // free indices in each term
 939         if (is_exactly_a<add>(e_expanded)) {
 940                 bool first = true;
 941                 ex sum;
 942                 free_indices.clear();
 943
 944                 for (unsigned i=0; i<e_expanded.nops(); i++) {
 945                         exvector free_indices_of_term;
 946                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 947                         if (!term.is_zero()) {
 948                                 if (first) {
 949                                         free_indices = free_indices_of_term;
 950                                         sum = term;
 951                                         first = false;
 952                                 } else {
 953                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
 954                                                 std::ostringstream s;
 955                                                 s << "simplify_indexed: inconsistent indices in sum: ";
 956                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
 957                                                 throw (std::runtime_error(s.str()));
 958                                         }
 959                                         if (is_a<indexed>(sum) && is_a<indexed>(term))
 960                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
 961                                         else
 962                                                 sum += term;
 963                                 }
 964                         }
 965                 }
 966
 967                 // If the sum turns out to be zero, we are finished
 968                 if (sum.is_zero()) {
 969                         free_indices.clear();
 970                         return sum;
 971                 }
 972
 973                 // More than one term and more than one dummy index?
 974                 int num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
 975                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
 976                         return sum;
 977
 978                 // Yes, construct list of all dummy index symbols
 979                 lst dummy_syms;
 980                 for (int i=0; i<dummy_indices.size(); i++)
 981                         dummy_syms.append(dummy_indices[i].op(0));
 982
 983                 // Chop the sum into terms and symmetrize each one over the dummy
 984                 // indices
 985                 std::vector<terminfo> terms;
 986                 for (unsigned i=0; i<sum.nops(); i++) {
 987                         const ex & term = sum.op(i);
 988                         ex term_symm = term.symmetrize(dummy_syms);
 989                         if (term_symm.is_zero())
 990                                 continue;
 991                         terms.push_back(terminfo(term, term_symm));
 992                 }
 993
 994                 // Sort by symmetrized terms
 995                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
 996
 997                 // Combine equal symmetrized terms
 998                 std::vector<terminfo> terms_pass2;
 999                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1000                         unsigned num = 1;
1001                         std::vector<terminfo>::const_iterator j = i + 1;
1002                         while (j != terms.end() && j->symm == i->symm) {
1003                                 num++;
1004                                 j++;
1005                         }
1006                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1007                         i = j;
1008                 }
1009
1010                 // If there is only one term left, we are finished
1011                 if (terms_pass2.size() == 1)
1012                         return terms_pass2[0].orig;
1013
1014                 // Chop the symmetrized terms into subterms
1015                 std::vector<symminfo> sy;
1016                 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1017                         if (is_exactly_a<add>(i->symm)) {
1018                                 unsigned num = i->symm.nops();
1019                                 for (unsigned j=0; j<num; j++)
1020                                         sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1021                         } else
1022                                 sy.push_back(symminfo(i->symm, i->orig, 1));
1023                 }
1024
1025                 // Sort by symmetrized subterms
1026                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1027
1028                 // Combine equal symmetrized subterms
1029                 std::vector<symminfo> sy_pass2;
1030                 exvector result;
1031                 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1032
1033                         // Combine equal terms
1034                         std::vector<symminfo>::const_iterator j = i + 1;
1035                         if (j != sy.end() && j->symmterm == i->symmterm) {
1036
1037                                 // More than one term, collect the coefficients
1038                                 ex coeff = i->coeff;
1039                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1040                                         coeff += j->coeff;
1041                                         j++;
1042                                 }
1043
1044                                 // Add combined term to result
1045                                 if (!coeff.is_zero())
1046                                         result.push_back(coeff * i->symmterm);
1047
1048                         } else {
1049
1050                                 // Single term, store for second pass
1051                                 sy_pass2.push_back(*i);
1052                         }
1053
1054                         i = j;
1055                 }
1056
1057                 // Were there any remaining terms that didn't get combined?
1058                 if (sy_pass2.size() > 0) {
1059
1060                         // Yes, sort by their original terms
1061                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1062
1063                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1064
1065                                 // How many symmetrized terms of this original term are left?
1066                                 unsigned num = 1;
1067                                 std::vector<symminfo>::const_iterator j = i + 1;
1068                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1069                                         num++;
1070                                         j++;
1071                                 }
1072
1073                                 if (num == i->num) {
1074
1075                                         // All terms left, then add the original term to the result
1076                                         result.push_back(i->orig);
1077
1078                                 } else {
1079
1080                                         // Some terms were combined with others, add up the remaining symmetrized terms
1081                                         std::vector<symminfo>::const_iterator k;
1082                                         for (k=i; k!=j; k++)
1083                                                 result.push_back(k->coeff * k->symmterm);
1084                                 }
1085
1086                                 i = j;
1087                         }
1088                 }
1089
1090                 // Add all resulting terms
1091                 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1092                 if (sum_symm.is_zero())
1093                         free_indices.clear();
1094                 return sum_symm;
1095         }
1096
1097         // Simplification of products
1098         if (is_exactly_a<mul>(e_expanded)
1099          || is_exactly_a<ncmul>(e_expanded)
1100          || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1101                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1102
1103         // Cannot do anything
1104         free_indices.clear();
1105         return e_expanded;
1106 }
1107
1108 /** Simplify/canonicalize expression containing indexed objects. This
1109  *  performs contraction of dummy indices where possible and checks whether
1110  *  the free indices in sums are consistent.
1111  *
1112  *  @return simplified expression */
1113 ex ex::simplify_indexed(void) const
1114 {
1115         exvector free_indices, dummy_indices;
1116         scalar_products sp;
1117         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1118 }
1119
1120 /** Simplify/canonicalize expression containing indexed objects. This
1121  *  performs contraction of dummy indices where possible, checks whether
1122  *  the free indices in sums are consistent, and automatically replaces
1123  *  scalar products by known values if desired.
1124  *
1125  *  @param sp Scalar products to be replaced automatically
1126  *  @return simplified expression */
1127 ex ex::simplify_indexed(const scalar_products & sp) const
1128 {
1129         exvector free_indices, dummy_indices;
1130         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1131 }
1132
1133 /** Symmetrize expression over its free indices. */
1134 ex ex::symmetrize(void) const
1135 {
1136         return GiNaC::symmetrize(*this, get_free_indices());
1137 }
1138
1139 /** Antisymmetrize expression over its free indices. */
1140 ex ex::antisymmetrize(void) const
1141 {
1142         return GiNaC::antisymmetrize(*this, get_free_indices());
1143 }
1144
1145 /** Symmetrize expression by cyclic permutation over its free indices. */
1146 ex ex::symmetrize_cyclic(void) const
1147 {
1148         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1149 }
1150
1151 //////////
1152 // helper classes
1153 //////////
1154
1155 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1156 {
1157         // If indexed, extract base objects
1158         ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1159         ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1160
1161         // Enforce canonical order in pair
1162         if (s1.compare(s2) > 0) {
1163                 v1 = s2;
1164                 v2 = s1;
1165         } else {
1166                 v1 = s1;
1167                 v2 = s2;
1168         }
1169 }
1170
1171 bool spmapkey::operator==(const spmapkey &other) const
1172 {
1173         if (!v1.is_equal(other.v1))
1174                 return false;
1175         if (!v2.is_equal(other.v2))
1176                 return false;
1177         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1178                 return true;
1179         else
1180                 return dim.is_equal(other.dim);
1181 }
1182
1183 bool spmapkey::operator<(const spmapkey &other) const
1184 {
1185         int cmp = v1.compare(other.v1);
1186         if (cmp)
1187                 return cmp < 0;
1188         cmp = v2.compare(other.v2);
1189         if (cmp)
1190                 return cmp < 0;
1191
1192         // Objects are equal, now check dimensions
1193         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1194                 return false;
1195         else
1196                 return dim.compare(other.dim) < 0;
1197 }
1198
1199 void spmapkey::debugprint(void) const
1200 {
1201         std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1202 }
1203
1204 scalar_products::scalar_products(const scalar_products & other) : spm(other.spm) {}
1205
1206 const scalar_products & scalar_products::operator=(const scalar_products & other)
1207 {
1208         if (this != &other) {
1209                 spm = other.spm;
1210         }
1211         return *this;
1212 }
1213
1214 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1215 {
1216         spm[spmapkey(v1, v2)] = sp;
1217 }
1218
1219 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1220 {
1221         spm[spmapkey(v1, v2, dim)] = sp;
1222 }
1223
1224 void scalar_products::add_vectors(const lst & l, const ex & dim)
1225 {
1226         // Add all possible pairs of products
1227         unsigned num = l.nops();
1228         for (unsigned i=0; i<num; i++) {
1229                 ex a = l.op(i);
1230                 for (unsigned j=0; j<num; j++) {
1231                         ex b = l.op(j);
1232                         add(a, b, dim, a*b);
1233                 }
1234         }
1235 }
1236
1237 void scalar_products::clear(void)
1238 {
1239         spm.clear();
1240 }
1241
1242 /** Check whether scalar product pair is defined. */
1243 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1244 {
1245         return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1246 }
1247
1248 /** Return value of defined scalar product pair. */
1249 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1250 {
1251         return spm.find(spmapkey(v1, v2, dim))->second;
1252 }
1253
1254 void scalar_products::debugprint(void) const
1255 {
1256         std::cerr << "map size=" << spm.size() << std::endl;
1257         spmap::const_iterator i = spm.begin(), end = spm.end();
1258         while (i != end) {
1259                 const spmapkey & k = i->first;
1260                 std::cerr << "item key=";
1261                 k.debugprint();
1262                 std::cerr << ", value=" << i->second << std::endl;
1263                 ++i;
1264         }
1265 }
1266
1267 } // namespace GiNaC