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 constructor
  47 //////////
  48
  49 indexed::indexed() : symtree(sy_none())
  50 {
  51         tinfo_key = TINFO_indexed;
  52 }
  53
  54 //////////
  55 // other constructors
  56 //////////
  57
  58 indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
  59 {
  60         tinfo_key = TINFO_indexed;
  61         validate();
  62 }
  63
  64 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(sy_none())
  65 {
  66         tinfo_key = TINFO_indexed;
  67         validate();
  68 }
  69
  70 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(sy_none())
  71 {
  72         tinfo_key = TINFO_indexed;
  73         validate();
  74 }
  75
  76 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(sy_none())
  77 {
  78         tinfo_key = TINFO_indexed;
  79         validate();
  80 }
  81
  82 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())
  83 {
  84         tinfo_key = TINFO_indexed;
  85         validate();
  86 }
  87
  88 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
  89 {
  90         tinfo_key = TINFO_indexed;
  91         validate();
  92 }
  93
  94 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
  95 {
  96         tinfo_key = TINFO_indexed;
  97         validate();
  98 }
  99
 100 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)
 101 {
 102         tinfo_key = TINFO_indexed;
 103         validate();
 104 }
 105
 106 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
 107 {
 108         seq.insert(seq.end(), v.begin(), v.end());
 109         tinfo_key = TINFO_indexed;
 110         validate();
 111 }
 112
 113 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 114 {
 115         seq.insert(seq.end(), v.begin(), v.end());
 116         tinfo_key = TINFO_indexed;
 117         validate();
 118 }
 119
 120 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 121 {
 122         tinfo_key = TINFO_indexed;
 123 }
 124
 125 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 126 {
 127         tinfo_key = TINFO_indexed;
 128 }
 129
 130 indexed::indexed(const symmetry & symm, exvector * vp) : inherited(vp), symtree(symm)
 131 {
 132         tinfo_key = TINFO_indexed;
 133 }
 134
 135 //////////
 136 // archiving
 137 //////////
 138
 139 indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 140 {
 141         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 142                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 143                 unsigned symm = 0;
 144                 n.find_unsigned("symmetry", symm);
 145                 switch (symm) {
 146                         case 1:
 147                                 symtree = sy_symm();
 148                                 break;
 149                         case 2:
 150                                 symtree = sy_anti();
 151                                 break;
 152                         default:
 153                                 symtree = sy_none();
 154                                 break;
 155                 }
 156                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 157         }
 158 }
 159
 160 void indexed::archive(archive_node &n) const
 161 {
 162         inherited::archive(n);
 163         n.add_ex("symmetry", symtree);
 164 }
 165
 166 DEFAULT_UNARCHIVE(indexed)
 167
 168 //////////
 169 // functions overriding virtual functions from base classes
 170 //////////
 171
 172 void indexed::print(const print_context & c, unsigned level) const
 173 {
 174         GINAC_ASSERT(seq.size() > 0);
 175
 176         if (is_a<print_tree>(c)) {
 177
 178                 c.s << std::string(level, ' ') << class_name()
 179                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 180                     << ", " << seq.size()-1 << " indices"
 181                     << ", symmetry=" << symtree << std::endl;
 182                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 183                 seq[0].print(c, level + delta_indent);
 184                 printindices(c, level + delta_indent);
 185
 186         } else {
 187
 188                 bool is_tex = is_a<print_latex>(c);
 189                 const ex & base = seq[0];
 190
 191                 if (precedence() <= level)
 192                         c.s << (is_tex ? "{(" : "(");
 193                 if (is_tex)
 194                         c.s << "{";
 195                 base.print(c, precedence());
 196                 if (is_tex)
 197                         c.s << "}";
 198                 printindices(c, level);
 199                 if (precedence() <= level)
 200                         c.s << (is_tex ? ")}" : ")");
 201         }
 202 }
 203
 204 bool indexed::info(unsigned inf) const
 205 {
 206         if (inf == info_flags::indexed) return true;
 207         if (inf == info_flags::has_indices) return seq.size() > 1;
 208         return inherited::info(inf);
 209 }
 210
 211 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 212         bool operator() (const ex & e, unsigned inf) const {
 213                 return !(ex_to<idx>(e).get_value().info(inf));
 214         }
 215 };
 216
 217 bool indexed::all_index_values_are(unsigned inf) const
 218 {
 219         // No indices? Then no property can be fulfilled
 220         if (seq.size() < 2)
 221                 return false;
 222
 223         // Check all indices
 224         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 225 }
 226
 227 int indexed::compare_same_type(const basic & other) const
 228 {
 229         GINAC_ASSERT(is_a<indexed>(other));
 230         return inherited::compare_same_type(other);
 231 }
 232
 233 ex indexed::eval(int level) const
 234 {
 235         // First evaluate children, then we will end up here again
 236         if (level > 1)
 237                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 238
 239         const ex &base = seq[0];
 240
 241         // If the base object is 0, the whole object is 0
 242         if (base.is_zero())
 243                 return _ex0;
 244
 245         // If the base object is a product, pull out the numeric factor
 246         if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
 247                 exvector v(seq);
 248                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 249                 v[0] = seq[0] / f;
 250                 return f * thiscontainer(v);
 251         }
 252
 253         // Canonicalize indices according to the symmetry properties
 254         if (seq.size() > 2) {
 255                 exvector v = seq;
 256                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 257                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 258                 if (sig != INT_MAX) {
 259                         // Something has changed while sorting indices, more evaluations later
 260                         if (sig == 0)
 261                                 return _ex0;
 262                         return ex(sig) * thiscontainer(v);
 263                 }
 264         }
 265
 266         // Let the class of the base object perform additional evaluations
 267         return ex_to<basic>(base).eval_indexed(*this);
 268 }
 269
 270 ex indexed::thiscontainer(const exvector & v) const
 271 {
 272         return indexed(ex_to<symmetry>(symtree), v);
 273 }
 274
 275 ex indexed::thiscontainer(exvector * vp) const
 276 {
 277         return indexed(ex_to<symmetry>(symtree), vp);
 278 }
 279
 280 ex indexed::expand(unsigned options) const
 281 {
 282         GINAC_ASSERT(seq.size() > 0);
 283
 284         if ((options & expand_options::expand_indexed) && is_exactly_a<add>(seq[0])) {
 285
 286                 // expand_indexed expands (a+b).i -> a.i + b.i
 287                 const ex & base = seq[0];
 288                 ex sum = _ex0;
 289                 for (size_t i=0; i<base.nops(); i++) {
 290                         exvector s = seq;
 291                         s[0] = base.op(i);
 292                         sum += thiscontainer(s).expand();
 293                 }
 294                 return sum;
 295
 296         } else
 297                 return inherited::expand(options);
 298 }
 299
 300 //////////
 301 // virtual functions which can be overridden by derived classes
 302 //////////
 303
 304 // none
 305
 306 //////////
 307 // non-virtual functions in this class
 308 //////////
 309
 310 void indexed::printindices(const print_context & c, unsigned level) const
 311 {
 312         if (seq.size() > 1) {
 313
 314                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 315
 316                 if (is_a<print_latex>(c)) {
 317
 318                         // TeX output: group by variance
 319                         bool first = true;
 320                         bool covariant = true;
 321
 322                         while (it != itend) {
 323                                 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
 324                                 if (first || cur_covariant != covariant) { // Variance changed
 325                                         // The empty {} prevents indices from ending up on top of each other
 326                                         if (!first)
 327                                                 c.s << "}{}";
 328                                         covariant = cur_covariant;
 329                                         if (covariant)
 330                                                 c.s << "_{";
 331                                         else
 332                                                 c.s << "^{";
 333                                 }
 334                                 it->print(c, level);
 335                                 c.s << " ";
 336                                 first = false;
 337                                 it++;
 338                         }
 339                         c.s << "}";
 340
 341                 } else {
 342
 343                         // Ordinary output
 344                         while (it != itend) {
 345                                 it->print(c, level);
 346                                 it++;
 347                         }
 348                 }
 349         }
 350 }
 351
 352 /** Check whether all indices are of class idx and validate the symmetry
 353  *  tree. This function is used internally to make sure that all constructed
 354  *  indexed objects really carry indices and not some other classes. */
 355 void indexed::validate() const
 356 {
 357         GINAC_ASSERT(seq.size() > 0);
 358         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 359         while (it != itend) {
 360                 if (!is_a<idx>(*it))
 361                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 362                 it++;
 363         }
 364
 365         if (!symtree.is_zero()) {
 366                 if (!is_exactly_a<symmetry>(symtree))
 367                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 368                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 369         }
 370 }
 371
 372 /** Implementation of ex::diff() for an indexed object always returns 0.
 373  *
 374  *  @see ex::diff */
 375 ex indexed::derivative(const symbol & s) const
 376 {
 377         return _ex0;
 378 }
 379
 380 //////////
 381 // global functions
 382 //////////
 383
 384 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
 385         bool operator() (const ex &lh, const ex &rh) const
 386         {
 387                 if (lh.is_equal(rh))
 388                         return true;
 389                 else
 390                         try {
 391                                 // Replacing the dimension might cause an error (e.g. with
 392                                 // index classes that only work in a fixed number of dimensions)
 393                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
 394                         } catch (...) {
 395                                 return false;
 396                         }
 397         }
 398 };
 399
 400 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 401 static bool indices_consistent(const exvector & v1, const exvector & v2)
 402 {
 403         // Number of indices must be the same
 404         if (v1.size() != v2.size())
 405                 return false;
 406
 407         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
 408 }
 409
 410 exvector indexed::get_indices() const
 411 {
 412         GINAC_ASSERT(seq.size() >= 1);
 413         return exvector(seq.begin() + 1, seq.end());
 414 }
 415
 416 exvector indexed::get_dummy_indices() const
 417 {
 418         exvector free_indices, dummy_indices;
 419         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 420         return dummy_indices;
 421 }
 422
 423 exvector indexed::get_dummy_indices(const indexed & other) const
 424 {
 425         exvector indices = get_free_indices();
 426         exvector other_indices = other.get_free_indices();
 427         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 428         exvector dummy_indices;
 429         find_dummy_indices(indices, dummy_indices);
 430         return dummy_indices;
 431 }
 432
 433 bool indexed::has_dummy_index_for(const ex & i) const
 434 {
 435         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 436         while (it != itend) {
 437                 if (is_dummy_pair(*it, i))
 438                         return true;
 439                 it++;
 440         }
 441         return false;
 442 }
 443
 444 exvector indexed::get_free_indices() const
 445 {
 446         exvector free_indices, dummy_indices;
 447         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 448         return free_indices;
 449 }
 450
 451 exvector add::get_free_indices() const
 452 {
 453         exvector free_indices;
 454         for (size_t i=0; i<nops(); i++) {
 455                 if (i == 0)
 456                         free_indices = op(i).get_free_indices();
 457                 else {
 458                         exvector free_indices_of_term = op(i).get_free_indices();
 459                         if (!indices_consistent(free_indices, free_indices_of_term))
 460                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 461                 }
 462         }
 463         return free_indices;
 464 }
 465
 466 exvector mul::get_free_indices() const
 467 {
 468         // Concatenate free indices of all factors
 469         exvector un;
 470         for (size_t i=0; i<nops(); i++) {
 471                 exvector free_indices_of_factor = op(i).get_free_indices();
 472                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 473         }
 474
 475         // And remove the dummy indices
 476         exvector free_indices, dummy_indices;
 477         find_free_and_dummy(un, free_indices, dummy_indices);
 478         return free_indices;
 479 }
 480
 481 exvector ncmul::get_free_indices() const
 482 {
 483         // Concatenate free indices of all factors
 484         exvector un;
 485         for (size_t i=0; i<nops(); i++) {
 486                 exvector free_indices_of_factor = op(i).get_free_indices();
 487                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 488         }
 489
 490         // And remove the dummy indices
 491         exvector free_indices, dummy_indices;
 492         find_free_and_dummy(un, free_indices, dummy_indices);
 493         return free_indices;
 494 }
 495
 496 exvector power::get_free_indices() const
 497 {
 498         // Return free indices of basis
 499         return basis.get_free_indices();
 500 }
 501
 502 /** Rename dummy indices in an expression.
 503  *
 504  *  @param e Expression to work on
 505  *  @param local_dummy_indices The set of dummy indices that appear in the
 506  *    expression "e"
 507  *  @param global_dummy_indices The set of dummy indices that have appeared
 508  *    before and which we would like to use in "e", too. This gets updated
 509  *    by the function */
 510 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 511 {
 512         size_t global_size = global_dummy_indices.size(),
 513                local_size = local_dummy_indices.size();
 514
 515         // Any local dummy indices at all?
 516         if (local_size == 0)
 517                 return e;
 518
 519         if (global_size < local_size) {
 520
 521                 // More local indices than we encountered before, add the new ones
 522                 // to the global set
 523                 size_t old_global_size = global_size;
 524                 int remaining = local_size - global_size;
 525                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 526                 while (it != itend && remaining > 0) {
 527                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(op0_is_equal(), *it)) == global_dummy_indices.end()) {
 528                                 global_dummy_indices.push_back(*it);
 529                                 global_size++;
 530                                 remaining--;
 531                         }
 532                         it++;
 533                 }
 534
 535                 // If this is the first set of local indices, do nothing
 536                 if (old_global_size == 0)
 537                         return e;
 538         }
 539         GINAC_ASSERT(local_size <= global_size);
 540
 541         // Construct vectors of index symbols
 542         exvector local_syms, global_syms;
 543         local_syms.reserve(local_size);
 544         global_syms.reserve(local_size);
 545         for (size_t i=0; i<local_size; i++)
 546                 local_syms.push_back(local_dummy_indices[i].op(0));
 547         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 548         for (size_t i=0; i<local_size; i++) // don't use more global symbols than necessary
 549                 global_syms.push_back(global_dummy_indices[i].op(0));
 550         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 551
 552         // Remove common indices
 553         exvector local_uniq, global_uniq;
 554         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
 555         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
 556
 557         // Replace remaining non-common local index symbols by global ones
 558         if (local_uniq.empty())
 559                 return e;
 560         else {
 561                 while (global_uniq.size() > local_uniq.size())
 562                         global_uniq.pop_back();
 563                 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()));
 564         }
 565 }
 566
 567 /** Given a set of indices, extract those of class varidx. */
 568 static void find_variant_indices(const exvector & v, exvector & variant_indices)
 569 {
 570         exvector::const_iterator it1, itend;
 571         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 572                 if (is_exactly_a<varidx>(*it1))
 573                         variant_indices.push_back(*it1);
 574         }
 575 }
 576
 577 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
 578  *  variance.
 579  *
 580  *  @param e Object to work on
 581  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
 582  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
 583  *  @return true if 'e' was changed */
 584 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
 585 {
 586         bool something_changed = false;
 587
 588         // If a dummy index is encountered for the first time in the
 589         // product, pull it up, otherwise, pull it down
 590         exvector::const_iterator it2, it2start, it2end;
 591         for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
 592                 if (!is_exactly_a<varidx>(*it2))
 593                         continue;
 594
 595                 exvector::iterator vit, vitend;
 596                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 597                         if (it2->op(0).is_equal(vit->op(0))) {
 598                                 if (ex_to<varidx>(*it2).is_covariant()) {
 599                                         e = e.subs(lst(
 600                                                 *it2 == ex_to<varidx>(*it2).toggle_variance(),
 601                                                 ex_to<varidx>(*it2).toggle_variance() == *it2
 602                                         ));
 603                                         something_changed = true;
 604                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 605                                         it2start = ex_to<indexed>(e).seq.begin();
 606                                         it2end = ex_to<indexed>(e).seq.end();
 607                                 }
 608                                 moved_indices.push_back(*vit);
 609                                 variant_dummy_indices.erase(vit);
 610                                 goto next_index;
 611                         }
 612                 }
 613
 614                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 615                         if (it2->op(0).is_equal(vit->op(0))) {
 616                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 617                                         e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance());
 618                                         something_changed = true;
 619                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 620                                         it2start = ex_to<indexed>(e).seq.begin();
 621                                         it2end = ex_to<indexed>(e).seq.end();
 622                                 }
 623                                 goto next_index;
 624                         }
 625                 }
 626
 627 next_index: ;
 628         }
 629
 630         return something_changed;
 631 }
 632
 633 /* Ordering that only compares the base expressions of indexed objects. */
 634 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
 635         bool operator() (const ex &lh, const ex &rh) const
 636         {
 637                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 638         }
 639 };
 640
 641 /** Simplify product of indexed expressions (commutative, noncommutative and
 642  *  simple squares), return list of free indices. */
 643 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 644 {
 645         // Remember whether the product was commutative or noncommutative
 646         // (because we chop it into factors and need to reassemble later)
 647         bool non_commutative = is_exactly_a<ncmul>(e);
 648
 649         // Collect factors in an exvector, store squares twice
 650         exvector v;
 651         v.reserve(e.nops() * 2);
 652
 653         if (is_exactly_a<power>(e)) {
 654                 // We only get called for simple squares, split a^2 -> a*a
 655                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 656                 v.push_back(e.op(0));
 657                 v.push_back(e.op(0));
 658         } else {
 659                 for (size_t i=0; i<e.nops(); i++) {
 660                         ex f = e.op(i);
 661                         if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
 662                                 v.push_back(f.op(0));
 663                     v.push_back(f.op(0));
 664                         } else if (is_exactly_a<ncmul>(f)) {
 665                                 // Noncommutative factor found, split it as well
 666                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 667                                 for (size_t j=0; j<f.nops(); j++)
 668                                         v.push_back(f.op(j));
 669                         } else
 670                                 v.push_back(f);
 671                 }
 672         }
 673
 674         // Perform contractions
 675         bool something_changed = false;
 676         GINAC_ASSERT(v.size() > 1);
 677         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 678         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 679
 680 try_again:
 681                 if (!is_a<indexed>(*it1))
 682                         continue;
 683
 684                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 685
 686                 // Indexed factor found, get free indices and look for contraction
 687                 // candidates
 688                 exvector free1, dummy1;
 689                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 690
 691                 exvector::iterator it2;
 692                 for (it2 = it1 + 1; it2 != itend; it2++) {
 693
 694                         if (!is_a<indexed>(*it2))
 695                                 continue;
 696
 697                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 698
 699                         // Find free indices of second factor and merge them with free
 700                         // indices of first factor
 701                         exvector un;
 702                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 703                         un.insert(un.end(), free1.begin(), free1.end());
 704
 705                         // Check whether the two factors share dummy indices
 706                         exvector free, dummy;
 707                         find_free_and_dummy(un, free, dummy);
 708                         size_t num_dummies = dummy.size();
 709                         if (num_dummies == 0)
 710                                 continue;
 711
 712                         // At least one dummy index, is it a defined scalar product?
 713                         bool contracted = false;
 714                         if (free.empty()) {
 715
 716                                 // Find minimal dimension of all indices of both factors
 717                                 exvector::const_iterator dit = ex_to<indexed>(*it1).seq.begin() + 1, ditend = ex_to<indexed>(*it1).seq.end();
 718                                 ex dim = ex_to<idx>(*dit).get_dim();
 719                                 ++dit;
 720                                 for (; dit != ditend; ++dit) {
 721                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 722                                 }
 723                                 dit = ex_to<indexed>(*it2).seq.begin() + 1;
 724                                 ditend = ex_to<indexed>(*it2).seq.end();
 725                                 for (; dit != ditend; ++dit) {
 726                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 727                                 }
 728
 729                                 // User-defined scalar product?
 730                                 if (sp.is_defined(*it1, *it2, dim)) {
 731
 732                                         // Yes, substitute it
 733                                         *it1 = sp.evaluate(*it1, *it2, dim);
 734                                         *it2 = _ex1;
 735                                         goto contraction_done;
 736                                 }
 737                         }
 738
 739                         // Try to contract the first one with the second one
 740                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 741                         if (!contracted) {
 742
 743                                 // That didn't work; maybe the second object knows how to
 744                                 // contract itself with the first one
 745                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 746                         }
 747                         if (contracted) {
 748 contraction_done:
 749                                 if (first_noncommutative || second_noncommutative
 750                                  || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
 751                                  || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
 752                                  || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
 753
 754                                         // One of the factors became a sum or product:
 755                                         // re-expand expression and run again
 756                                         // Non-commutative products are always re-expanded to give
 757                                         // eval_ncmul() the chance to re-order and canonicalize
 758                                         // the product
 759                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 760                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 761                                 }
 762
 763                                 // Both objects may have new indices now or they might
 764                                 // even not be indexed objects any more, so we have to
 765                                 // start over
 766                                 something_changed = true;
 767                                 goto try_again;
 768                         }
 769                 }
 770         }
 771
 772         // Find free indices (concatenate them all and call find_free_and_dummy())
 773         // and all dummy indices that appear
 774         exvector un, individual_dummy_indices;
 775         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 776                 exvector free_indices_of_factor;
 777                 if (is_a<indexed>(*it1)) {
 778                         exvector dummy_indices_of_factor;
 779                         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);
 780                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 781                 } else
 782                         free_indices_of_factor = it1->get_free_indices();
 783                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 784         }
 785         exvector local_dummy_indices;
 786         find_free_and_dummy(un, free_indices, local_dummy_indices);
 787         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 788
 789         // Filter out the dummy indices with variance
 790         exvector variant_dummy_indices;
 791         find_variant_indices(local_dummy_indices, variant_dummy_indices);
 792
 793         // Any indices with variance present at all?
 794         if (!variant_dummy_indices.empty()) {
 795
 796                 // Yes, bring the product into a canonical order that only depends on
 797                 // the base expressions of indexed objects
 798                 if (!non_commutative)
 799                         std::sort(v.begin(), v.end(), ex_base_is_less());
 800
 801                 exvector moved_indices;
 802
 803                 // Iterate over all indexed objects in the product
 804                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 805                         if (!is_a<indexed>(*it1))
 806                                 continue;
 807
 808                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
 809                                 something_changed = true;
 810                 }
 811         }
 812
 813         ex r;
 814         if (something_changed)
 815                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 816         else
 817                 r = e;
 818
 819         // The result should be symmetric with respect to exchange of dummy
 820         // indices, so if the symmetrization vanishes, the whole expression is
 821         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 822         if (local_dummy_indices.size() >= 2) {
 823                 exvector dummy_syms;
 824                 dummy_syms.reserve(local_dummy_indices.size());
 825                 for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
 826                         dummy_syms.push_back(it->op(0));
 827                 if (symmetrize(r, dummy_syms).is_zero()) {
 828                         free_indices.clear();
 829                         return _ex0;
 830                 }
 831         }
 832
 833         // Dummy index renaming
 834         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 835
 836         // Product of indexed object with a scalar?
 837         if (is_exactly_a<mul>(r) && r.nops() == 2
 838          && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
 839                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 840         else
 841                 return r;
 842 }
 843
 844 /** This structure stores the original and symmetrized versions of terms
 845  *  obtained during the simplification of sums. */
 846 class terminfo {
 847 public:
 848         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
 849
 850         ex orig; /**< original term */
 851         ex symm; /**< symmtrized term */
 852 };
 853
 854 class terminfo_is_less {
 855 public:
 856         bool operator() (const terminfo & ti1, const terminfo & ti2) const
 857         {
 858                 return (ti1.symm.compare(ti2.symm) < 0);
 859         }
 860 };
 861
 862 /** This structure stores the individual symmetrized terms obtained during
 863  *  the simplification of sums. */
 864 class symminfo {
 865 public:
 866         symminfo() : num(0) {}
 867
 868         symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
 869         {
 870                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
 871                         coeff = symmterm_.op(symmterm_.nops()-1);
 872                         symmterm = symmterm_ / coeff;
 873                 } else {
 874                         coeff = 1;
 875                         symmterm = symmterm_;
 876                 }
 877         }
 878
 879         ex symmterm;  /**< symmetrized term */
 880         ex coeff;     /**< coefficient of symmetrized term */
 881         ex orig;      /**< original term */
 882         size_t num; /**< how many symmetrized terms resulted from the original term */
 883 };
 884
 885 class symminfo_is_less_by_symmterm {
 886 public:
 887         bool operator() (const symminfo & si1, const symminfo & si2) const
 888         {
 889                 return (si1.symmterm.compare(si2.symmterm) < 0);
 890         }
 891 };
 892
 893 class symminfo_is_less_by_orig {
 894 public:
 895         bool operator() (const symminfo & si1, const symminfo & si2) const
 896         {
 897                 return (si1.orig.compare(si2.orig) < 0);
 898         }
 899 };
 900
 901 /** Simplify indexed expression, return list of free indices. */
 902 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 903 {
 904         // Expand the expression
 905         ex e_expanded = e.expand();
 906
 907         // Simplification of single indexed object: just find the free indices
 908         // and perform dummy index renaming/repositioning
 909         if (is_a<indexed>(e_expanded)) {
 910
 911                 // Find the dummy indices
 912                 const indexed &i = ex_to<indexed>(e_expanded);
 913                 exvector local_dummy_indices;
 914                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 915
 916                 // Filter out the dummy indices with variance
 917                 exvector variant_dummy_indices;
 918                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
 919
 920                 // Any indices with variance present at all?
 921                 if (!variant_dummy_indices.empty()) {
 922
 923                         // Yes, reposition them
 924                         exvector moved_indices;
 925                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
 926                 }
 927
 928                 // Rename the dummy indices
 929                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 930         }
 931
 932         // Simplification of sum = sum of simplifications, check consistency of
 933         // free indices in each term
 934         if (is_exactly_a<add>(e_expanded)) {
 935                 bool first = true;
 936                 ex sum;
 937                 free_indices.clear();
 938
 939                 for (size_t i=0; i<e_expanded.nops(); i++) {
 940                         exvector free_indices_of_term;
 941                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 942                         if (!term.is_zero()) {
 943                                 if (first) {
 944                                         free_indices = free_indices_of_term;
 945                                         sum = term;
 946                                         first = false;
 947                                 } else {
 948                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
 949                                                 std::ostringstream s;
 950                                                 s << "simplify_indexed: inconsistent indices in sum: ";
 951                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
 952                                                 throw (std::runtime_error(s.str()));
 953                                         }
 954                                         if (is_a<indexed>(sum) && is_a<indexed>(term))
 955                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
 956                                         else
 957                                                 sum += term;
 958                                 }
 959                         }
 960                 }
 961
 962                 // If the sum turns out to be zero, we are finished
 963                 if (sum.is_zero()) {
 964                         free_indices.clear();
 965                         return sum;
 966                 }
 967
 968                 // More than one term and more than one dummy index?
 969                 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
 970                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
 971                         return sum;
 972
 973                 // Yes, construct vector of all dummy index symbols
 974                 exvector dummy_syms;
 975                 dummy_syms.reserve(dummy_indices.size());
 976                 for (exvector::const_iterator it = dummy_indices.begin(); it != dummy_indices.end(); ++it)
 977                         dummy_syms.push_back(it->op(0));
 978
 979                 // Chop the sum into terms and symmetrize each one over the dummy
 980                 // indices
 981                 std::vector<terminfo> terms;
 982                 for (size_t i=0; i<sum.nops(); i++) {
 983                         const ex & term = sum.op(i);
 984                         ex term_symm = symmetrize(term, dummy_syms);
 985                         if (term_symm.is_zero())
 986                                 continue;
 987                         terms.push_back(terminfo(term, term_symm));
 988                 }
 989
 990                 // Sort by symmetrized terms
 991                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
 992
 993                 // Combine equal symmetrized terms
 994                 std::vector<terminfo> terms_pass2;
 995                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
 996                         size_t num = 1;
 997                         std::vector<terminfo>::const_iterator j = i + 1;
 998                         while (j != terms.end() && j->symm == i->symm) {
 999                                 num++;
1000                                 j++;
1001                         }
1002                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1003                         i = j;
1004                 }
1005
1006                 // If there is only one term left, we are finished
1007                 if (terms_pass2.size() == 1)
1008                         return terms_pass2[0].orig;
1009
1010                 // Chop the symmetrized terms into subterms
1011                 std::vector<symminfo> sy;
1012                 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1013                         if (is_exactly_a<add>(i->symm)) {
1014                                 size_t num = i->symm.nops();
1015                                 for (size_t j=0; j<num; j++)
1016                                         sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1017                         } else
1018                                 sy.push_back(symminfo(i->symm, i->orig, 1));
1019                 }
1020
1021                 // Sort by symmetrized subterms
1022                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1023
1024                 // Combine equal symmetrized subterms
1025                 std::vector<symminfo> sy_pass2;
1026                 exvector result;
1027                 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1028
1029                         // Combine equal terms
1030                         std::vector<symminfo>::const_iterator j = i + 1;
1031                         if (j != sy.end() && j->symmterm == i->symmterm) {
1032
1033                                 // More than one term, collect the coefficients
1034                                 ex coeff = i->coeff;
1035                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1036                                         coeff += j->coeff;
1037                                         j++;
1038                                 }
1039
1040                                 // Add combined term to result
1041                                 if (!coeff.is_zero())
1042                                         result.push_back(coeff * i->symmterm);
1043
1044                         } else {
1045
1046                                 // Single term, store for second pass
1047                                 sy_pass2.push_back(*i);
1048                         }
1049
1050                         i = j;
1051                 }
1052
1053                 // Were there any remaining terms that didn't get combined?
1054                 if (sy_pass2.size() > 0) {
1055
1056                         // Yes, sort by their original terms
1057                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1058
1059                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1060
1061                                 // How many symmetrized terms of this original term are left?
1062                                 size_t num = 1;
1063                                 std::vector<symminfo>::const_iterator j = i + 1;
1064                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1065                                         num++;
1066                                         j++;
1067                                 }
1068
1069                                 if (num == i->num) {
1070
1071                                         // All terms left, then add the original term to the result
1072                                         result.push_back(i->orig);
1073
1074                                 } else {
1075
1076                                         // Some terms were combined with others, add up the remaining symmetrized terms
1077                                         std::vector<symminfo>::const_iterator k;
1078                                         for (k=i; k!=j; k++)
1079                                                 result.push_back(k->coeff * k->symmterm);
1080                                 }
1081
1082                                 i = j;
1083                         }
1084                 }
1085
1086                 // Add all resulting terms
1087                 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1088                 if (sum_symm.is_zero())
1089                         free_indices.clear();
1090                 return sum_symm;
1091         }
1092
1093         // Simplification of products
1094         if (is_exactly_a<mul>(e_expanded)
1095          || is_exactly_a<ncmul>(e_expanded)
1096          || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1097                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1098
1099         // Cannot do anything
1100         free_indices.clear();
1101         return e_expanded;
1102 }
1103
1104 /** Simplify/canonicalize expression containing indexed objects. This
1105  *  performs contraction of dummy indices where possible and checks whether
1106  *  the free indices in sums are consistent.
1107  *
1108  *  @return simplified expression */
1109 ex ex::simplify_indexed() const
1110 {
1111         exvector free_indices, dummy_indices;
1112         scalar_products sp;
1113         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1114 }
1115
1116 /** Simplify/canonicalize expression containing indexed objects. This
1117  *  performs contraction of dummy indices where possible, checks whether
1118  *  the free indices in sums are consistent, and automatically replaces
1119  *  scalar products by known values if desired.
1120  *
1121  *  @param sp Scalar products to be replaced automatically
1122  *  @return simplified expression */
1123 ex ex::simplify_indexed(const scalar_products & sp) const
1124 {
1125         exvector free_indices, dummy_indices;
1126         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1127 }
1128
1129 /** Symmetrize expression over its free indices. */
1130 ex ex::symmetrize() const
1131 {
1132         return GiNaC::symmetrize(*this, get_free_indices());
1133 }
1134
1135 /** Antisymmetrize expression over its free indices. */
1136 ex ex::antisymmetrize() const
1137 {
1138         return GiNaC::antisymmetrize(*this, get_free_indices());
1139 }
1140
1141 /** Symmetrize expression by cyclic permutation over its free indices. */
1142 ex ex::symmetrize_cyclic() const
1143 {
1144         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1145 }
1146
1147 //////////
1148 // helper classes
1149 //////////
1150
1151 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1152 {
1153         // If indexed, extract base objects
1154         ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1155         ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1156
1157         // Enforce canonical order in pair
1158         if (s1.compare(s2) > 0) {
1159                 v1 = s2;
1160                 v2 = s1;
1161         } else {
1162                 v1 = s1;
1163                 v2 = s2;
1164         }
1165 }
1166
1167 bool spmapkey::operator==(const spmapkey &other) const
1168 {
1169         if (!v1.is_equal(other.v1))
1170                 return false;
1171         if (!v2.is_equal(other.v2))
1172                 return false;
1173         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1174                 return true;
1175         else
1176                 return dim.is_equal(other.dim);
1177 }
1178
1179 bool spmapkey::operator<(const spmapkey &other) const
1180 {
1181         int cmp = v1.compare(other.v1);
1182         if (cmp)
1183                 return cmp < 0;
1184         cmp = v2.compare(other.v2);
1185         if (cmp)
1186                 return cmp < 0;
1187
1188         // Objects are equal, now check dimensions
1189         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1190                 return false;
1191         else
1192                 return dim.compare(other.dim) < 0;
1193 }
1194
1195 void spmapkey::debugprint() const
1196 {
1197         std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1198 }
1199
1200 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1201 {
1202         spm[spmapkey(v1, v2)] = sp;
1203 }
1204
1205 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1206 {
1207         spm[spmapkey(v1, v2, dim)] = sp;
1208 }
1209
1210 void scalar_products::add_vectors(const lst & l, const ex & dim)
1211 {
1212         // Add all possible pairs of products
1213         for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1214                 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1215                         add(*it1, *it2, *it1 * *it2);
1216 }
1217
1218 void scalar_products::clear()
1219 {
1220         spm.clear();
1221 }
1222
1223 /** Check whether scalar product pair is defined. */
1224 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1225 {
1226         return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1227 }
1228
1229 /** Return value of defined scalar product pair. */
1230 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1231 {
1232         return spm.find(spmapkey(v1, v2, dim))->second;
1233 }
1234
1235 void scalar_products::debugprint() const
1236 {
1237         std::cerr << "map size=" << spm.size() << std::endl;
1238         spmap::const_iterator i = spm.begin(), end = spm.end();
1239         while (i != end) {
1240                 const spmapkey & k = i->first;
1241                 std::cerr << "item key=";
1242                 k.debugprint();
1243                 std::cerr << ", value=" << i->second << std::endl;
1244                 ++i;
1245         }
1246 }
1247
1248 } // namespace GiNaC