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 "archive.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(indexed, exprseq)
  43
  44 //////////
  45 // default constructor
  46 //////////
  47
  48 indexed::indexed() : symtree(sy_none())
  49 {
  50         tinfo_key = TINFO_indexed;
  51 }
  52
  53 //////////
  54 // other constructors
  55 //////////
  56
  57 indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
  58 {
  59         tinfo_key = TINFO_indexed;
  60         validate();
  61 }
  62
  63 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(sy_none())
  64 {
  65         tinfo_key = TINFO_indexed;
  66         validate();
  67 }
  68
  69 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(sy_none())
  70 {
  71         tinfo_key = TINFO_indexed;
  72         validate();
  73 }
  74
  75 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(sy_none())
  76 {
  77         tinfo_key = TINFO_indexed;
  78         validate();
  79 }
  80
  81 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(sy_none())
  82 {
  83         tinfo_key = TINFO_indexed;
  84         validate();
  85 }
  86
  87 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
  88 {
  89         tinfo_key = TINFO_indexed;
  90         validate();
  91 }
  92
  93 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
  94 {
  95         tinfo_key = TINFO_indexed;
  96         validate();
  97 }
  98
  99 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(symm)
 100 {
 101         tinfo_key = TINFO_indexed;
 102         validate();
 103 }
 104
 105 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
 106 {
 107         seq.insert(seq.end(), v.begin(), v.end());
 108         tinfo_key = TINFO_indexed;
 109         validate();
 110 }
 111
 112 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 113 {
 114         seq.insert(seq.end(), v.begin(), v.end());
 115         tinfo_key = TINFO_indexed;
 116         validate();
 117 }
 118
 119 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 120 {
 121         tinfo_key = TINFO_indexed;
 122 }
 123
 124 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 125 {
 126         tinfo_key = TINFO_indexed;
 127 }
 128
 129 indexed::indexed(const symmetry & symm, exvector * vp) : inherited(vp), symtree(symm)
 130 {
 131         tinfo_key = TINFO_indexed;
 132 }
 133
 134 //////////
 135 // archiving
 136 //////////
 137
 138 indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 139 {
 140         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 141                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 142                 unsigned symm = 0;
 143                 n.find_unsigned("symmetry", symm);
 144                 switch (symm) {
 145                         case 1:
 146                                 symtree = sy_symm();
 147                                 break;
 148                         case 2:
 149                                 symtree = sy_anti();
 150                                 break;
 151                         default:
 152                                 symtree = sy_none();
 153                                 break;
 154                 }
 155                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 156         }
 157 }
 158
 159 void indexed::archive(archive_node &n) const
 160 {
 161         inherited::archive(n);
 162         n.add_ex("symmetry", symtree);
 163 }
 164
 165 DEFAULT_UNARCHIVE(indexed)
 166
 167 //////////
 168 // functions overriding virtual functions from base classes
 169 //////////
 170
 171 void indexed::print(const print_context & c, unsigned level) const
 172 {
 173         GINAC_ASSERT(seq.size() > 0);
 174
 175         if (is_a<print_tree>(c)) {
 176
 177                 c.s << std::string(level, ' ') << class_name()
 178                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 179                     << ", " << seq.size()-1 << " indices"
 180                     << ", symmetry=" << symtree << std::endl;
 181                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 182                 seq[0].print(c, level + delta_indent);
 183                 printindices(c, level + delta_indent);
 184
 185         } else {
 186
 187                 bool is_tex = is_a<print_latex>(c);
 188                 const ex & base = seq[0];
 189
 190                 if (precedence() <= level)
 191                         c.s << (is_tex ? "{(" : "(");
 192                 if (is_tex)
 193                         c.s << "{";
 194                 base.print(c, precedence());
 195                 if (is_tex)
 196                         c.s << "}";
 197                 printindices(c, level);
 198                 if (precedence() <= level)
 199                         c.s << (is_tex ? ")}" : ")");
 200         }
 201 }
 202
 203 bool indexed::info(unsigned inf) const
 204 {
 205         if (inf == info_flags::indexed) return true;
 206         if (inf == info_flags::has_indices) return seq.size() > 1;
 207         return inherited::info(inf);
 208 }
 209
 210 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 211         bool operator() (const ex & e, unsigned inf) const {
 212                 return !(ex_to<idx>(e).get_value().info(inf));
 213         }
 214 };
 215
 216 bool indexed::all_index_values_are(unsigned inf) const
 217 {
 218         // No indices? Then no property can be fulfilled
 219         if (seq.size() < 2)
 220                 return false;
 221
 222         // Check all indices
 223         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 224 }
 225
 226 int indexed::compare_same_type(const basic & other) const
 227 {
 228         GINAC_ASSERT(is_a<indexed>(other));
 229         return inherited::compare_same_type(other);
 230 }
 231
 232 ex indexed::eval(int level) const
 233 {
 234         // First evaluate children, then we will end up here again
 235         if (level > 1)
 236                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 237
 238         const ex &base = seq[0];
 239
 240         // If the base object is 0, the whole object is 0
 241         if (base.is_zero())
 242                 return _ex0;
 243
 244         // If the base object is a product, pull out the numeric factor
 245         if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
 246                 exvector v(seq);
 247                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 248                 v[0] = seq[0] / f;
 249                 return f * thiscontainer(v);
 250         }
 251
 252         // Canonicalize indices according to the symmetry properties
 253         if (seq.size() > 2) {
 254                 exvector v = seq;
 255                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 256                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 257                 if (sig != INT_MAX) {
 258                         // Something has changed while sorting indices, more evaluations later
 259                         if (sig == 0)
 260                                 return _ex0;
 261                         return ex(sig) * thiscontainer(v);
 262                 }
 263         }
 264
 265         // Let the class of the base object perform additional evaluations
 266         return ex_to<basic>(base).eval_indexed(*this);
 267 }
 268
 269 ex indexed::thiscontainer(const exvector & v) const
 270 {
 271         return indexed(ex_to<symmetry>(symtree), v);
 272 }
 273
 274 ex indexed::thiscontainer(exvector * vp) const
 275 {
 276         return indexed(ex_to<symmetry>(symtree), vp);
 277 }
 278
 279 ex indexed::expand(unsigned options) const
 280 {
 281         GINAC_ASSERT(seq.size() > 0);
 282
 283         if ((options & expand_options::expand_indexed) && is_exactly_a<add>(seq[0])) {
 284
 285                 // expand_indexed expands (a+b).i -> a.i + b.i
 286                 const ex & base = seq[0];
 287                 ex sum = _ex0;
 288                 for (size_t i=0; i<base.nops(); i++) {
 289                         exvector s = seq;
 290                         s[0] = base.op(i);
 291                         sum += thiscontainer(s).expand();
 292                 }
 293                 return sum;
 294
 295         } else
 296                 return inherited::expand(options);
 297 }
 298
 299 //////////
 300 // virtual functions which can be overridden by derived classes
 301 //////////
 302
 303 // none
 304
 305 //////////
 306 // non-virtual functions in this class
 307 //////////
 308
 309 void indexed::printindices(const print_context & c, unsigned level) const
 310 {
 311         if (seq.size() > 1) {
 312
 313                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 314
 315                 if (is_a<print_latex>(c)) {
 316
 317                         // TeX output: group by variance
 318                         bool first = true;
 319                         bool covariant = true;
 320
 321                         while (it != itend) {
 322                                 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
 323                                 if (first || cur_covariant != covariant) { // Variance changed
 324                                         // The empty {} prevents indices from ending up on top of each other
 325                                         if (!first)
 326                                                 c.s << "}{}";
 327                                         covariant = cur_covariant;
 328                                         if (covariant)
 329                                                 c.s << "_{";
 330                                         else
 331                                                 c.s << "^{";
 332                                 }
 333                                 it->print(c, level);
 334                                 c.s << " ";
 335                                 first = false;
 336                                 it++;
 337                         }
 338                         c.s << "}";
 339
 340                 } else {
 341
 342                         // Ordinary output
 343                         while (it != itend) {
 344                                 it->print(c, level);
 345                                 it++;
 346                         }
 347                 }
 348         }
 349 }
 350
 351 /** Check whether all indices are of class idx and validate the symmetry
 352  *  tree. This function is used internally to make sure that all constructed
 353  *  indexed objects really carry indices and not some other classes. */
 354 void indexed::validate() const
 355 {
 356         GINAC_ASSERT(seq.size() > 0);
 357         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 358         while (it != itend) {
 359                 if (!is_a<idx>(*it))
 360                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 361                 it++;
 362         }
 363
 364         if (!symtree.is_zero()) {
 365                 if (!is_exactly_a<symmetry>(symtree))
 366                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 367                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 368         }
 369 }
 370
 371 /** Implementation of ex::diff() for an indexed object always returns 0.
 372  *
 373  *  @see ex::diff */
 374 ex indexed::derivative(const symbol & s) const
 375 {
 376         return _ex0;
 377 }
 378
 379 //////////
 380 // global functions
 381 //////////
 382
 383 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
 384         bool operator() (const ex &lh, const ex &rh) const
 385         {
 386                 if (lh.is_equal(rh))
 387                         return true;
 388                 else
 389                         try {
 390                                 // Replacing the dimension might cause an error (e.g. with
 391                                 // index classes that only work in a fixed number of dimensions)
 392                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
 393                         } catch (...) {
 394                                 return false;
 395                         }
 396         }
 397 };
 398
 399 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 400 static bool indices_consistent(const exvector & v1, const exvector & v2)
 401 {
 402         // Number of indices must be the same
 403         if (v1.size() != v2.size())
 404                 return false;
 405
 406         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
 407 }
 408
 409 exvector indexed::get_indices() const
 410 {
 411         GINAC_ASSERT(seq.size() >= 1);
 412         return exvector(seq.begin() + 1, seq.end());
 413 }
 414
 415 exvector indexed::get_dummy_indices() const
 416 {
 417         exvector free_indices, dummy_indices;
 418         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 419         return dummy_indices;
 420 }
 421
 422 exvector indexed::get_dummy_indices(const indexed & other) const
 423 {
 424         exvector indices = get_free_indices();
 425         exvector other_indices = other.get_free_indices();
 426         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 427         exvector dummy_indices;
 428         find_dummy_indices(indices, dummy_indices);
 429         return dummy_indices;
 430 }
 431
 432 bool indexed::has_dummy_index_for(const ex & i) const
 433 {
 434         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 435         while (it != itend) {
 436                 if (is_dummy_pair(*it, i))
 437                         return true;
 438                 it++;
 439         }
 440         return false;
 441 }
 442
 443 exvector indexed::get_free_indices() const
 444 {
 445         exvector free_indices, dummy_indices;
 446         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 447         return free_indices;
 448 }
 449
 450 exvector add::get_free_indices() const
 451 {
 452         exvector free_indices;
 453         for (size_t i=0; i<nops(); i++) {
 454                 if (i == 0)
 455                         free_indices = op(i).get_free_indices();
 456                 else {
 457                         exvector free_indices_of_term = op(i).get_free_indices();
 458                         if (!indices_consistent(free_indices, free_indices_of_term))
 459                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 460                 }
 461         }
 462         return free_indices;
 463 }
 464
 465 exvector mul::get_free_indices() const
 466 {
 467         // Concatenate free indices of all factors
 468         exvector un;
 469         for (size_t i=0; i<nops(); i++) {
 470                 exvector free_indices_of_factor = op(i).get_free_indices();
 471                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 472         }
 473
 474         // And remove the dummy indices
 475         exvector free_indices, dummy_indices;
 476         find_free_and_dummy(un, free_indices, dummy_indices);
 477         return free_indices;
 478 }
 479
 480 exvector ncmul::get_free_indices() const
 481 {
 482         // Concatenate free indices of all factors
 483         exvector un;
 484         for (size_t i=0; i<nops(); i++) {
 485                 exvector free_indices_of_factor = op(i).get_free_indices();
 486                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 487         }
 488
 489         // And remove the dummy indices
 490         exvector free_indices, dummy_indices;
 491         find_free_and_dummy(un, free_indices, dummy_indices);
 492         return free_indices;
 493 }
 494
 495 exvector power::get_free_indices() const
 496 {
 497         // Return free indices of basis
 498         return basis.get_free_indices();
 499 }
 500
 501 /** Rename dummy indices in an expression.
 502  *
 503  *  @param e Expression to work on
 504  *  @param local_dummy_indices The set of dummy indices that appear in the
 505  *    expression "e"
 506  *  @param global_dummy_indices The set of dummy indices that have appeared
 507  *    before and which we would like to use in "e", too. This gets updated
 508  *    by the function */
 509 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 510 {
 511         size_t global_size = global_dummy_indices.size(),
 512                local_size = local_dummy_indices.size();
 513
 514         // Any local dummy indices at all?
 515         if (local_size == 0)
 516                 return e;
 517
 518         if (global_size < local_size) {
 519
 520                 // More local indices than we encountered before, add the new ones
 521                 // to the global set
 522                 size_t old_global_size = global_size;
 523                 int remaining = local_size - global_size;
 524                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 525                 while (it != itend && remaining > 0) {
 526                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(op0_is_equal(), *it)) == global_dummy_indices.end()) {
 527                                 global_dummy_indices.push_back(*it);
 528                                 global_size++;
 529                                 remaining--;
 530                         }
 531                         it++;
 532                 }
 533
 534                 // If this is the first set of local indices, do nothing
 535                 if (old_global_size == 0)
 536                         return e;
 537         }
 538         GINAC_ASSERT(local_size <= global_size);
 539
 540         // Construct vectors of index symbols
 541         exvector local_syms, global_syms;
 542         local_syms.reserve(local_size);
 543         global_syms.reserve(local_size);
 544         for (size_t i=0; i<local_size; i++)
 545                 local_syms.push_back(local_dummy_indices[i].op(0));
 546         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 547         for (size_t i=0; i<local_size; i++) // don't use more global symbols than necessary
 548                 global_syms.push_back(global_dummy_indices[i].op(0));
 549         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 550
 551         // Remove common indices
 552         exvector local_uniq, global_uniq;
 553         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
 554         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
 555
 556         // Replace remaining non-common local index symbols by global ones
 557         if (local_uniq.empty())
 558                 return e;
 559         else {
 560                 while (global_uniq.size() > local_uniq.size())
 561                         global_uniq.pop_back();
 562                 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
 563         }
 564 }
 565
 566 /** Given a set of indices, extract those of class varidx. */
 567 static void find_variant_indices(const exvector & v, exvector & variant_indices)
 568 {
 569         exvector::const_iterator it1, itend;
 570         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 571                 if (is_exactly_a<varidx>(*it1))
 572                         variant_indices.push_back(*it1);
 573         }
 574 }
 575
 576 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
 577  *  variance.
 578  *
 579  *  @param e Object to work on
 580  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
 581  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
 582  *  @return true if 'e' was changed */
 583 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
 584 {
 585         bool something_changed = false;
 586
 587         // If a dummy index is encountered for the first time in the
 588         // product, pull it up, otherwise, pull it down
 589         exvector::const_iterator it2, it2start, it2end;
 590         for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
 591                 if (!is_exactly_a<varidx>(*it2))
 592                         continue;
 593
 594                 exvector::iterator vit, vitend;
 595                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 596                         if (it2->op(0).is_equal(vit->op(0))) {
 597                                 if (ex_to<varidx>(*it2).is_covariant()) {
 598                                         e = e.subs(lst(
 599                                                 *it2 == ex_to<varidx>(*it2).toggle_variance(),
 600                                                 ex_to<varidx>(*it2).toggle_variance() == *it2
 601                                         ), subs_options::no_pattern);
 602                                         something_changed = true;
 603                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 604                                         it2start = ex_to<indexed>(e).seq.begin();
 605                                         it2end = ex_to<indexed>(e).seq.end();
 606                                 }
 607                                 moved_indices.push_back(*vit);
 608                                 variant_dummy_indices.erase(vit);
 609                                 goto next_index;
 610                         }
 611                 }
 612
 613                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 614                         if (it2->op(0).is_equal(vit->op(0))) {
 615                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 616                                         e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance(), subs_options::no_pattern);
 617                                         something_changed = true;
 618                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 619                                         it2start = ex_to<indexed>(e).seq.begin();
 620                                         it2end = ex_to<indexed>(e).seq.end();
 621                                 }
 622                                 goto next_index;
 623                         }
 624                 }
 625
 626 next_index: ;
 627         }
 628
 629         return something_changed;
 630 }
 631
 632 /* Ordering that only compares the base expressions of indexed objects. */
 633 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
 634         bool operator() (const ex &lh, const ex &rh) const
 635         {
 636                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 637         }
 638 };
 639
 640 /** Simplify product of indexed expressions (commutative, noncommutative and
 641  *  simple squares), return list of free indices. */
 642 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 643 {
 644         // Remember whether the product was commutative or noncommutative
 645         // (because we chop it into factors and need to reassemble later)
 646         bool non_commutative = is_exactly_a<ncmul>(e);
 647
 648         // Collect factors in an exvector, store squares twice
 649         exvector v;
 650         v.reserve(e.nops() * 2);
 651
 652         if (is_exactly_a<power>(e)) {
 653                 // We only get called for simple squares, split a^2 -> a*a
 654                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 655                 v.push_back(e.op(0));
 656                 v.push_back(e.op(0));
 657         } else {
 658                 for (size_t i=0; i<e.nops(); i++) {
 659                         ex f = e.op(i);
 660                         if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
 661                                 v.push_back(f.op(0));
 662                     v.push_back(f.op(0));
 663                         } else if (is_exactly_a<ncmul>(f)) {
 664                                 // Noncommutative factor found, split it as well
 665                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 666                                 for (size_t j=0; j<f.nops(); j++)
 667                                         v.push_back(f.op(j));
 668                         } else
 669                                 v.push_back(f);
 670                 }
 671         }
 672
 673         // Perform contractions
 674         bool something_changed = false;
 675         GINAC_ASSERT(v.size() > 1);
 676         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 677         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 678
 679 try_again:
 680                 if (!is_a<indexed>(*it1))
 681                         continue;
 682
 683                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 684
 685                 // Indexed factor found, get free indices and look for contraction
 686                 // candidates
 687                 exvector free1, dummy1;
 688                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 689
 690                 exvector::iterator it2;
 691                 for (it2 = it1 + 1; it2 != itend; it2++) {
 692
 693                         if (!is_a<indexed>(*it2))
 694                                 continue;
 695
 696                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 697
 698                         // Find free indices of second factor and merge them with free
 699                         // indices of first factor
 700                         exvector un;
 701                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 702                         un.insert(un.end(), free1.begin(), free1.end());
 703
 704                         // Check whether the two factors share dummy indices
 705                         exvector free, dummy;
 706                         find_free_and_dummy(un, free, dummy);
 707                         size_t num_dummies = dummy.size();
 708                         if (num_dummies == 0)
 709                                 continue;
 710
 711                         // At least one dummy index, is it a defined scalar product?
 712                         bool contracted = false;
 713                         if (free.empty()) {
 714
 715                                 // Find minimal dimension of all indices of both factors
 716                                 exvector::const_iterator dit = ex_to<indexed>(*it1).seq.begin() + 1, ditend = ex_to<indexed>(*it1).seq.end();
 717                                 ex dim = ex_to<idx>(*dit).get_dim();
 718                                 ++dit;
 719                                 for (; dit != ditend; ++dit) {
 720                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 721                                 }
 722                                 dit = ex_to<indexed>(*it2).seq.begin() + 1;
 723                                 ditend = ex_to<indexed>(*it2).seq.end();
 724                                 for (; dit != ditend; ++dit) {
 725                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 726                                 }
 727
 728                                 // User-defined scalar product?
 729                                 if (sp.is_defined(*it1, *it2, dim)) {
 730
 731                                         // Yes, substitute it
 732                                         *it1 = sp.evaluate(*it1, *it2, dim);
 733                                         *it2 = _ex1;
 734                                         goto contraction_done;
 735                                 }
 736                         }
 737
 738                         // Try to contract the first one with the second one
 739                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 740                         if (!contracted) {
 741
 742                                 // That didn't work; maybe the second object knows how to
 743                                 // contract itself with the first one
 744                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 745                         }
 746                         if (contracted) {
 747 contraction_done:
 748                                 if (first_noncommutative || second_noncommutative
 749                                  || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
 750                                  || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
 751                                  || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
 752
 753                                         // One of the factors became a sum or product:
 754                                         // re-expand expression and run again
 755                                         // Non-commutative products are always re-expanded to give
 756                                         // eval_ncmul() the chance to re-order and canonicalize
 757                                         // the product
 758                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 759                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 760                                 }
 761
 762                                 // Both objects may have new indices now or they might
 763                                 // even not be indexed objects any more, so we have to
 764                                 // start over
 765                                 something_changed = true;
 766                                 goto try_again;
 767                         }
 768                 }
 769         }
 770
 771         // Find free indices (concatenate them all and call find_free_and_dummy())
 772         // and all dummy indices that appear
 773         exvector un, individual_dummy_indices;
 774         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 775                 exvector free_indices_of_factor;
 776                 if (is_a<indexed>(*it1)) {
 777                         exvector dummy_indices_of_factor;
 778                         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);
 779                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 780                 } else
 781                         free_indices_of_factor = it1->get_free_indices();
 782                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 783         }
 784         exvector local_dummy_indices;
 785         find_free_and_dummy(un, free_indices, local_dummy_indices);
 786         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 787
 788         // Filter out the dummy indices with variance
 789         exvector variant_dummy_indices;
 790         find_variant_indices(local_dummy_indices, variant_dummy_indices);
 791
 792         // Any indices with variance present at all?
 793         if (!variant_dummy_indices.empty()) {
 794
 795                 // Yes, bring the product into a canonical order that only depends on
 796                 // the base expressions of indexed objects
 797                 if (!non_commutative)
 798                         std::sort(v.begin(), v.end(), ex_base_is_less());
 799
 800                 exvector moved_indices;
 801
 802                 // Iterate over all indexed objects in the product
 803                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 804                         if (!is_a<indexed>(*it1))
 805                                 continue;
 806
 807                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
 808                                 something_changed = true;
 809                 }
 810         }
 811
 812         ex r;
 813         if (something_changed)
 814                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 815         else
 816                 r = e;
 817
 818         // The result should be symmetric with respect to exchange of dummy
 819         // indices, so if the symmetrization vanishes, the whole expression is
 820         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 821         if (local_dummy_indices.size() >= 2) {
 822                 exvector dummy_syms;
 823                 dummy_syms.reserve(local_dummy_indices.size());
 824                 for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
 825                         dummy_syms.push_back(it->op(0));
 826                 if (symmetrize(r, dummy_syms).is_zero()) {
 827                         free_indices.clear();
 828                         return _ex0;
 829                 }
 830         }
 831
 832         // Dummy index renaming
 833         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 834
 835         // Product of indexed object with a scalar?
 836         if (is_exactly_a<mul>(r) && r.nops() == 2
 837          && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
 838                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 839         else
 840                 return r;
 841 }
 842
 843 /** This structure stores the original and symmetrized versions of terms
 844  *  obtained during the simplification of sums. */
 845 class terminfo {
 846 public:
 847         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
 848
 849         ex orig; /**< original term */
 850         ex symm; /**< symmtrized term */
 851 };
 852
 853 class terminfo_is_less {
 854 public:
 855         bool operator() (const terminfo & ti1, const terminfo & ti2) const
 856         {
 857                 return (ti1.symm.compare(ti2.symm) < 0);
 858         }
 859 };
 860
 861 /** This structure stores the individual symmetrized terms obtained during
 862  *  the simplification of sums. */
 863 class symminfo {
 864 public:
 865         symminfo() : num(0) {}
 866
 867         symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
 868         {
 869                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
 870                         coeff = symmterm_.op(symmterm_.nops()-1);
 871                         symmterm = symmterm_ / coeff;
 872                 } else {
 873                         coeff = 1;
 874                         symmterm = symmterm_;
 875                 }
 876         }
 877
 878         ex symmterm;  /**< symmetrized term */
 879         ex coeff;     /**< coefficient of symmetrized term */
 880         ex orig;      /**< original term */
 881         size_t num; /**< how many symmetrized terms resulted from the original term */
 882 };
 883
 884 class symminfo_is_less_by_symmterm {
 885 public:
 886         bool operator() (const symminfo & si1, const symminfo & si2) const
 887         {
 888                 return (si1.symmterm.compare(si2.symmterm) < 0);
 889         }
 890 };
 891
 892 class symminfo_is_less_by_orig {
 893 public:
 894         bool operator() (const symminfo & si1, const symminfo & si2) const
 895         {
 896                 return (si1.orig.compare(si2.orig) < 0);
 897         }
 898 };
 899
 900 /** Simplify indexed expression, return list of free indices. */
 901 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 902 {
 903         // Expand the expression
 904         ex e_expanded = e.expand();
 905
 906         // Simplification of single indexed object: just find the free indices
 907         // and perform dummy index renaming/repositioning
 908         if (is_a<indexed>(e_expanded)) {
 909
 910                 // Find the dummy indices
 911                 const indexed &i = ex_to<indexed>(e_expanded);
 912                 exvector local_dummy_indices;
 913                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 914
 915                 // Filter out the dummy indices with variance
 916                 exvector variant_dummy_indices;
 917                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
 918
 919                 // Any indices with variance present at all?
 920                 if (!variant_dummy_indices.empty()) {
 921
 922                         // Yes, reposition them
 923                         exvector moved_indices;
 924                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
 925                 }
 926
 927                 // Rename the dummy indices
 928                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 929         }
 930
 931         // Simplification of sum = sum of simplifications, check consistency of
 932         // free indices in each term
 933         if (is_exactly_a<add>(e_expanded)) {
 934                 bool first = true;
 935                 ex sum;
 936                 free_indices.clear();
 937
 938                 for (size_t i=0; i<e_expanded.nops(); i++) {
 939                         exvector free_indices_of_term;
 940                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 941                         if (!term.is_zero()) {
 942                                 if (first) {
 943                                         free_indices = free_indices_of_term;
 944                                         sum = term;
 945                                         first = false;
 946                                 } else {
 947                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
 948                                                 std::ostringstream s;
 949                                                 s << "simplify_indexed: inconsistent indices in sum: ";
 950                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
 951                                                 throw (std::runtime_error(s.str()));
 952                                         }
 953                                         if (is_a<indexed>(sum) && is_a<indexed>(term))
 954                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
 955                                         else
 956                                                 sum += term;
 957                                 }
 958                         }
 959                 }
 960
 961                 // If the sum turns out to be zero, we are finished
 962                 if (sum.is_zero()) {
 963                         free_indices.clear();
 964                         return sum;
 965                 }
 966
 967                 // More than one term and more than one dummy index?
 968                 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
 969                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
 970                         return sum;
 971
 972                 // Yes, construct vector of all dummy index symbols
 973                 exvector dummy_syms;
 974                 dummy_syms.reserve(dummy_indices.size());
 975                 for (exvector::const_iterator it = dummy_indices.begin(); it != dummy_indices.end(); ++it)
 976                         dummy_syms.push_back(it->op(0));
 977
 978                 // Chop the sum into terms and symmetrize each one over the dummy
 979                 // indices
 980                 std::vector<terminfo> terms;
 981                 for (size_t i=0; i<sum.nops(); i++) {
 982                         const ex & term = sum.op(i);
 983                         ex term_symm = symmetrize(term, dummy_syms);
 984                         if (term_symm.is_zero())
 985                                 continue;
 986                         terms.push_back(terminfo(term, term_symm));
 987                 }
 988
 989                 // Sort by symmetrized terms
 990                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
 991
 992                 // Combine equal symmetrized terms
 993                 std::vector<terminfo> terms_pass2;
 994                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
 995                         size_t num = 1;
 996                         std::vector<terminfo>::const_iterator j = i + 1;
 997                         while (j != terms.end() && j->symm == i->symm) {
 998                                 num++;
 999                                 j++;
1000                         }
1001                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1002                         i = j;
1003                 }
1004
1005                 // If there is only one term left, we are finished
1006                 if (terms_pass2.size() == 1)
1007                         return terms_pass2[0].orig;
1008
1009                 // Chop the symmetrized terms into subterms
1010                 std::vector<symminfo> sy;
1011                 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1012                         if (is_exactly_a<add>(i->symm)) {
1013                                 size_t num = i->symm.nops();
1014                                 for (size_t j=0; j<num; j++)
1015                                         sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1016                         } else
1017                                 sy.push_back(symminfo(i->symm, i->orig, 1));
1018                 }
1019
1020                 // Sort by symmetrized subterms
1021                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1022
1023                 // Combine equal symmetrized subterms
1024                 std::vector<symminfo> sy_pass2;
1025                 exvector result;
1026                 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1027
1028                         // Combine equal terms
1029                         std::vector<symminfo>::const_iterator j = i + 1;
1030                         if (j != sy.end() && j->symmterm == i->symmterm) {
1031
1032                                 // More than one term, collect the coefficients
1033                                 ex coeff = i->coeff;
1034                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1035                                         coeff += j->coeff;
1036                                         j++;
1037                                 }
1038
1039                                 // Add combined term to result
1040                                 if (!coeff.is_zero())
1041                                         result.push_back(coeff * i->symmterm);
1042
1043                         } else {
1044
1045                                 // Single term, store for second pass
1046                                 sy_pass2.push_back(*i);
1047                         }
1048
1049                         i = j;
1050                 }
1051
1052                 // Were there any remaining terms that didn't get combined?
1053                 if (sy_pass2.size() > 0) {
1054
1055                         // Yes, sort by their original terms
1056                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1057
1058                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1059
1060                                 // How many symmetrized terms of this original term are left?
1061                                 size_t num = 1;
1062                                 std::vector<symminfo>::const_iterator j = i + 1;
1063                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1064                                         num++;
1065                                         j++;
1066                                 }
1067
1068                                 if (num == i->num) {
1069
1070                                         // All terms left, then add the original term to the result
1071                                         result.push_back(i->orig);
1072
1073                                 } else {
1074
1075                                         // Some terms were combined with others, add up the remaining symmetrized terms
1076                                         std::vector<symminfo>::const_iterator k;
1077                                         for (k=i; k!=j; k++)
1078                                                 result.push_back(k->coeff * k->symmterm);
1079                                 }
1080
1081                                 i = j;
1082                         }
1083                 }
1084
1085                 // Add all resulting terms
1086                 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1087                 if (sum_symm.is_zero())
1088                         free_indices.clear();
1089                 return sum_symm;
1090         }
1091
1092         // Simplification of products
1093         if (is_exactly_a<mul>(e_expanded)
1094          || is_exactly_a<ncmul>(e_expanded)
1095          || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1096                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1097
1098         // Cannot do anything
1099         free_indices.clear();
1100         return e_expanded;
1101 }
1102
1103 /** Simplify/canonicalize expression containing indexed objects. This
1104  *  performs contraction of dummy indices where possible and checks whether
1105  *  the free indices in sums are consistent.
1106  *
1107  *  @return simplified expression */
1108 ex ex::simplify_indexed() const
1109 {
1110         exvector free_indices, dummy_indices;
1111         scalar_products sp;
1112         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1113 }
1114
1115 /** Simplify/canonicalize expression containing indexed objects. This
1116  *  performs contraction of dummy indices where possible, checks whether
1117  *  the free indices in sums are consistent, and automatically replaces
1118  *  scalar products by known values if desired.
1119  *
1120  *  @param sp Scalar products to be replaced automatically
1121  *  @return simplified expression */
1122 ex ex::simplify_indexed(const scalar_products & sp) const
1123 {
1124         exvector free_indices, dummy_indices;
1125         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1126 }
1127
1128 /** Symmetrize expression over its free indices. */
1129 ex ex::symmetrize() const
1130 {
1131         return GiNaC::symmetrize(*this, get_free_indices());
1132 }
1133
1134 /** Antisymmetrize expression over its free indices. */
1135 ex ex::antisymmetrize() const
1136 {
1137         return GiNaC::antisymmetrize(*this, get_free_indices());
1138 }
1139
1140 /** Symmetrize expression by cyclic permutation over its free indices. */
1141 ex ex::symmetrize_cyclic() const
1142 {
1143         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1144 }
1145
1146 //////////
1147 // helper classes
1148 //////////
1149
1150 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1151 {
1152         // If indexed, extract base objects
1153         ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1154         ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1155
1156         // Enforce canonical order in pair
1157         if (s1.compare(s2) > 0) {
1158                 v1 = s2;
1159                 v2 = s1;
1160         } else {
1161                 v1 = s1;
1162                 v2 = s2;
1163         }
1164 }
1165
1166 bool spmapkey::operator==(const spmapkey &other) const
1167 {
1168         if (!v1.is_equal(other.v1))
1169                 return false;
1170         if (!v2.is_equal(other.v2))
1171                 return false;
1172         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1173                 return true;
1174         else
1175                 return dim.is_equal(other.dim);
1176 }
1177
1178 bool spmapkey::operator<(const spmapkey &other) const
1179 {
1180         int cmp = v1.compare(other.v1);
1181         if (cmp)
1182                 return cmp < 0;
1183         cmp = v2.compare(other.v2);
1184         if (cmp)
1185                 return cmp < 0;
1186
1187         // Objects are equal, now check dimensions
1188         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1189                 return false;
1190         else
1191                 return dim.compare(other.dim) < 0;
1192 }
1193
1194 void spmapkey::debugprint() const
1195 {
1196         std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1197 }
1198
1199 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1200 {
1201         spm[spmapkey(v1, v2)] = sp;
1202 }
1203
1204 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1205 {
1206         spm[spmapkey(v1, v2, dim)] = sp;
1207 }
1208
1209 void scalar_products::add_vectors(const lst & l, const ex & dim)
1210 {
1211         // Add all possible pairs of products
1212         for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1213                 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1214                         add(*it1, *it2, *it1 * *it2);
1215 }
1216
1217 void scalar_products::clear()
1218 {
1219         spm.clear();
1220 }
1221
1222 /** Check whether scalar product pair is defined. */
1223 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1224 {
1225         return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1226 }
1227
1228 /** Return value of defined scalar product pair. */
1229 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1230 {
1231         return spm.find(spmapkey(v1, v2, dim))->second;
1232 }
1233
1234 void scalar_products::debugprint() const
1235 {
1236         std::cerr << "map size=" << spm.size() << std::endl;
1237         spmap::const_iterator i = spm.begin(), end = spm.end();
1238         while (i != end) {
1239                 const spmapkey & k = i->first;
1240                 std::cerr << "item key=";
1241                 k.debugprint();
1242                 std::cerr << ", value=" << i->second << std::endl;
1243                 ++i;
1244         }
1245 }
1246
1247 } // namespace GiNaC