ginac/indexed.cpp

   1 /** @file indexed.cpp
   2  *
   3  *  Implementation of GiNaC's indexed expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2002 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <iostream>
  24 #include <stdexcept>
  25
  26 #include "indexed.h"
  27 #include "idx.h"
  28 #include "add.h"
  29 #include "mul.h"
  30 #include "ncmul.h"
  31 #include "power.h"
  32 #include "relational.h"
  33 #include "symmetry.h"
  34 #include "lst.h"
  35 #include "print.h"
  36 #include "archive.h"
  37 #include "utils.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(indexed, exprseq)
  42
  43 //////////
  44 // default ctor, dtor, copy ctor, assignment operator and helpers
  45 //////////
  46
  47 indexed::indexed() : symtree(sy_none())
  48 {
  49         tinfo_key = TINFO_indexed;
  50 }
  51
  52 void indexed::copy(const indexed & other)
  53 {
  54         inherited::copy(other);
  55         symtree = other.symtree;
  56 }
  57
  58 DEFAULT_DESTROY(indexed)
  59
  60 //////////
  61 // other constructors
  62 //////////
  63
  64 indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
  65 {
  66         tinfo_key = TINFO_indexed;
  67         validate();
  68 }
  69
  70 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), 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) : inherited(b, i1, i2), 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) : inherited(b, i1, i2, i3), symtree(sy_none())
  83 {
  84         tinfo_key = TINFO_indexed;
  85         validate();
  86 }
  87
  88 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())
  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) : inherited(b, i1, i2), 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) : inherited(b, i1, i2, i3), symtree(symm)
 101 {
 102         tinfo_key = TINFO_indexed;
 103         validate();
 104 }
 105
 106 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)
 107 {
 108         tinfo_key = TINFO_indexed;
 109         validate();
 110 }
 111
 112 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
 113 {
 114         seq.insert(seq.end(), v.begin(), v.end());
 115         tinfo_key = TINFO_indexed;
 116         validate();
 117 }
 118
 119 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 120 {
 121         seq.insert(seq.end(), v.begin(), v.end());
 122         tinfo_key = TINFO_indexed;
 123         validate();
 124 }
 125
 126 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 127 {
 128         tinfo_key = TINFO_indexed;
 129 }
 130
 131 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 132 {
 133         tinfo_key = TINFO_indexed;
 134 }
 135
 136 indexed::indexed(const symmetry & symm, exvector * vp) : inherited(vp), symtree(symm)
 137 {
 138         tinfo_key = TINFO_indexed;
 139 }
 140
 141 //////////
 142 // archiving
 143 //////////
 144
 145 indexed::indexed(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 146 {
 147         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 148                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 149                 unsigned symm = 0;
 150                 n.find_unsigned("symmetry", symm);
 151                 switch (symm) {
 152                         case 1:
 153                                 symtree = sy_symm();
 154                                 break;
 155                         case 2:
 156                                 symtree = sy_anti();
 157                                 break;
 158                         default:
 159                                 symtree = sy_none();
 160                                 break;
 161                 }
 162                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 163         }
 164 }
 165
 166 void indexed::archive(archive_node &n) const
 167 {
 168         inherited::archive(n);
 169         n.add_ex("symmetry", symtree);
 170 }
 171
 172 DEFAULT_UNARCHIVE(indexed)
 173
 174 //////////
 175 // functions overriding virtual functions from base classes
 176 //////////
 177
 178 void indexed::print(const print_context & c, unsigned level) const
 179 {
 180         GINAC_ASSERT(seq.size() > 0);
 181
 182         if (is_a<print_tree>(c)) {
 183
 184                 c.s << std::string(level, ' ') << class_name()
 185                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 186                     << ", " << seq.size()-1 << " indices"
 187                     << ", symmetry=" << symtree << std::endl;
 188                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 189                 seq[0].print(c, level + delta_indent);
 190                 printindices(c, level + delta_indent);
 191
 192         } else {
 193
 194                 bool is_tex = is_a<print_latex>(c);
 195                 const ex & base = seq[0];
 196
 197                 if (precedence() <= level)
 198                         c.s << (is_tex ? "{(" : "(");
 199                 if (is_tex)
 200                         c.s << "{";
 201                 base.print(c, precedence());
 202                 if (is_tex)
 203                         c.s << "}";
 204                 printindices(c, level);
 205                 if (precedence() <= level)
 206                         c.s << (is_tex ? ")}" : ")");
 207         }
 208 }
 209
 210 bool indexed::info(unsigned inf) const
 211 {
 212         if (inf == info_flags::indexed) return true;
 213         if (inf == info_flags::has_indices) return seq.size() > 1;
 214         return inherited::info(inf);
 215 }
 216
 217 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 218         bool operator() (const ex & e, unsigned inf) const {
 219                 return !(ex_to<idx>(e).get_value().info(inf));
 220         }
 221 };
 222
 223 bool indexed::all_index_values_are(unsigned inf) const
 224 {
 225         // No indices? Then no property can be fulfilled
 226         if (seq.size() < 2)
 227                 return false;
 228
 229         // Check all indices
 230         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 231 }
 232
 233 int indexed::compare_same_type(const basic & other) const
 234 {
 235         GINAC_ASSERT(is_a<indexed>(other));
 236         return inherited::compare_same_type(other);
 237 }
 238
 239 ex indexed::eval(int level) const
 240 {
 241         // First evaluate children, then we will end up here again
 242         if (level > 1)
 243                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 244
 245         const ex &base = seq[0];
 246
 247         // If the base object is 0, the whole object is 0
 248         if (base.is_zero())
 249                 return _ex0;
 250
 251         // If the base object is a product, pull out the numeric factor
 252         if (is_ex_exactly_of_type(base, mul) && is_ex_exactly_of_type(base.op(base.nops() - 1), numeric)) {
 253                 exvector v(seq);
 254                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 255                 v[0] = seq[0] / f;
 256                 return f * thisexprseq(v);
 257         }
 258
 259         // Canonicalize indices according to the symmetry properties
 260         if (seq.size() > 2) {
 261                 exvector v = seq;
 262                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 263                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 264                 if (sig != INT_MAX) {
 265                         // Something has changed while sorting indices, more evaluations later
 266                         if (sig == 0)
 267                                 return _ex0;
 268                         return ex(sig) * thisexprseq(v);
 269                 }
 270         }
 271
 272         // Let the class of the base object perform additional evaluations
 273         return ex_to<basic>(base).eval_indexed(*this);
 274 }
 275
 276 ex indexed::thisexprseq(const exvector & v) const
 277 {
 278         return indexed(ex_to<symmetry>(symtree), v);
 279 }
 280
 281 ex indexed::thisexprseq(exvector * vp) const
 282 {
 283         return indexed(ex_to<symmetry>(symtree), vp);
 284 }
 285
 286 ex indexed::expand(unsigned options) const
 287 {
 288         GINAC_ASSERT(seq.size() > 0);
 289
 290         if ((options & expand_options::expand_indexed) && is_ex_exactly_of_type(seq[0], add)) {
 291
 292                 // expand_indexed expands (a+b).i -> a.i + b.i
 293                 const ex & base = seq[0];
 294                 ex sum = _ex0;
 295                 for (unsigned i=0; i<base.nops(); i++) {
 296                         exvector s = seq;
 297                         s[0] = base.op(i);
 298                         sum += thisexprseq(s).expand();
 299                 }
 300                 return sum;
 301
 302         } else
 303                 return inherited::expand(options);
 304 }
 305
 306 //////////
 307 // virtual functions which can be overridden by derived classes
 308 //////////
 309
 310 // none
 311
 312 //////////
 313 // non-virtual functions in this class
 314 //////////
 315
 316 void indexed::printindices(const print_context & c, unsigned level) const
 317 {
 318         if (seq.size() > 1) {
 319
 320                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 321
 322                 if (is_a<print_latex>(c)) {
 323
 324                         // TeX output: group by variance
 325                         bool first = true;
 326                         bool covariant = true;
 327
 328                         while (it != itend) {
 329                                 bool cur_covariant = (is_ex_of_type(*it, varidx) ? ex_to<varidx>(*it).is_covariant() : true);
 330                                 if (first || cur_covariant != covariant) { // Variance changed
 331                                         // The empty {} prevents indices from ending up on top of each other
 332                                         if (!first)
 333                                                 c.s << "}{}";
 334                                         covariant = cur_covariant;
 335                                         if (covariant)
 336                                                 c.s << "_{";
 337                                         else
 338                                                 c.s << "^{";
 339                                 }
 340                                 it->print(c, level);
 341                                 c.s << " ";
 342                                 first = false;
 343                                 it++;
 344                         }
 345                         c.s << "}";
 346
 347                 } else {
 348
 349                         // Ordinary output
 350                         while (it != itend) {
 351                                 it->print(c, level);
 352                                 it++;
 353                         }
 354                 }
 355         }
 356 }
 357
 358 /** Check whether all indices are of class idx and validate the symmetry
 359  *  tree. This function is used internally to make sure that all constructed
 360  *  indexed objects really carry indices and not some other classes. */
 361 void indexed::validate(void) const
 362 {
 363         GINAC_ASSERT(seq.size() > 0);
 364         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 365         while (it != itend) {
 366                 if (!is_ex_of_type(*it, idx))
 367                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 368                 it++;
 369         }
 370
 371         if (!symtree.is_zero()) {
 372                 if (!is_ex_exactly_of_type(symtree, symmetry))
 373                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 374                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 375         }
 376 }
 377
 378 /** Implementation of ex::diff() for an indexed object always returns 0.
 379  *
 380  *  @see ex::diff */
 381 ex indexed::derivative(const symbol & s) const
 382 {
 383         return _ex0;
 384 }
 385
 386 //////////
 387 // global functions
 388 //////////
 389
 390 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 391 static bool indices_consistent(const exvector & v1, const exvector & v2)
 392 {
 393         // Number of indices must be the same
 394         if (v1.size() != v2.size())
 395                 return false;
 396
 397         return equal(v1.begin(), v1.end(), v2.begin(), ex_is_equal());
 398 }
 399
 400 exvector indexed::get_indices(void) const
 401 {
 402         GINAC_ASSERT(seq.size() >= 1);
 403         return exvector(seq.begin() + 1, seq.end());
 404 }
 405
 406 exvector indexed::get_dummy_indices(void) const
 407 {
 408         exvector free_indices, dummy_indices;
 409         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 410         return dummy_indices;
 411 }
 412
 413 exvector indexed::get_dummy_indices(const indexed & other) const
 414 {
 415         exvector indices = get_free_indices();
 416         exvector other_indices = other.get_free_indices();
 417         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 418         exvector dummy_indices;
 419         find_dummy_indices(indices, dummy_indices);
 420         return dummy_indices;
 421 }
 422
 423 bool indexed::has_dummy_index_for(const ex & i) const
 424 {
 425         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 426         while (it != itend) {
 427                 if (is_dummy_pair(*it, i))
 428                         return true;
 429                 it++;
 430         }
 431         return false;
 432 }
 433
 434 exvector indexed::get_free_indices(void) const
 435 {
 436         exvector free_indices, dummy_indices;
 437         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 438         return free_indices;
 439 }
 440
 441 exvector add::get_free_indices(void) const
 442 {
 443         exvector free_indices;
 444         for (unsigned i=0; i<nops(); i++) {
 445                 if (i == 0)
 446                         free_indices = op(i).get_free_indices();
 447                 else {
 448                         exvector free_indices_of_term = op(i).get_free_indices();
 449                         if (!indices_consistent(free_indices, free_indices_of_term))
 450                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 451                 }
 452         }
 453         return free_indices;
 454 }
 455
 456 exvector mul::get_free_indices(void) const
 457 {
 458         // Concatenate free indices of all factors
 459         exvector un;
 460         for (unsigned i=0; i<nops(); i++) {
 461                 exvector free_indices_of_factor = op(i).get_free_indices();
 462                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 463         }
 464
 465         // And remove the dummy indices
 466         exvector free_indices, dummy_indices;
 467         find_free_and_dummy(un, free_indices, dummy_indices);
 468         return free_indices;
 469 }
 470
 471 exvector ncmul::get_free_indices(void) const
 472 {
 473         // Concatenate free indices of all factors
 474         exvector un;
 475         for (unsigned i=0; i<nops(); i++) {
 476                 exvector free_indices_of_factor = op(i).get_free_indices();
 477                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 478         }
 479
 480         // And remove the dummy indices
 481         exvector free_indices, dummy_indices;
 482         find_free_and_dummy(un, free_indices, dummy_indices);
 483         return free_indices;
 484 }
 485
 486 exvector power::get_free_indices(void) const
 487 {
 488         // Return free indices of basis
 489         return basis.get_free_indices();
 490 }
 491
 492 /** Rename dummy indices in an expression.
 493  *
 494  *  @param e Expression to work on
 495  *  @param local_dummy_indices The set of dummy indices that appear in the
 496  *    expression "e"
 497  *  @param global_dummy_indices The set of dummy indices that have appeared
 498  *    before and which we would like to use in "e", too. This gets updated
 499  *    by the function */
 500 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 501 {
 502         unsigned global_size = global_dummy_indices.size(),
 503                  local_size = local_dummy_indices.size();
 504
 505         // Any local dummy indices at all?
 506         if (local_size == 0)
 507                 return e;
 508
 509         if (global_size < local_size) {
 510
 511                 // More local indices than we encountered before, add the new ones
 512                 // to the global set
 513                 int old_global_size = global_size;
 514                 int remaining = local_size - global_size;
 515                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 516                 while (it != itend && remaining > 0) {
 517                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(ex_is_equal(), *it)) == global_dummy_indices.end()) {
 518                                 global_dummy_indices.push_back(*it);
 519                                 global_size++;
 520                                 remaining--;
 521                         }
 522                         it++;
 523                 }
 524
 525                 // If this is the first set of local indices, do nothing
 526                 if (old_global_size == 0)
 527                         return e;
 528         }
 529         GINAC_ASSERT(local_size <= global_size);
 530
 531         // Construct lists of index symbols
 532         exlist local_syms, global_syms;
 533         for (unsigned i=0; i<local_size; i++)
 534                 local_syms.push_back(local_dummy_indices[i].op(0));
 535         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 536         for (unsigned i=0; i<global_size; i++)
 537                 global_syms.push_back(global_dummy_indices[i].op(0));
 538         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 539
 540         // Remove common indices
 541         exlist local_uniq, global_uniq;
 542         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exlist>(local_uniq), ex_is_less());
 543         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exlist>(global_uniq), ex_is_less());
 544
 545         // Replace remaining non-common local index symbols by global ones
 546         if (local_uniq.empty())
 547                 return e;
 548         else {
 549                 while (global_uniq.size() > local_uniq.size())
 550                         global_uniq.pop_back();
 551                 return e.subs(lst(local_uniq), lst(global_uniq));
 552         }
 553 }
 554
 555 /** Given a set of indices, extract those of class varidx. */
 556 static void find_variant_indices(const exvector & v, exvector & variant_indices)
 557 {
 558         exvector::const_iterator it1, itend;
 559         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 560                 if (is_exactly_a<varidx>(*it1))
 561                         variant_indices.push_back(*it1);
 562         }
 563 }
 564
 565 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
 566  *  variance.
 567  *
 568  *  @param e Object to work on
 569  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
 570  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
 571  *  @return true if 'e' was changed */
 572 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
 573 {
 574         bool something_changed = false;
 575
 576         // If a dummy index is encountered for the first time in the
 577         // product, pull it up, otherwise, pull it down
 578         exvector::const_iterator it2, it2start, it2end;
 579         for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
 580                 if (!is_exactly_a<varidx>(*it2))
 581                         continue;
 582
 583                 exvector::iterator vit, vitend;
 584                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 585                         if (it2->op(0).is_equal(vit->op(0))) {
 586                                 if (ex_to<varidx>(*it2).is_covariant()) {
 587                                         e = e.subs(lst(
 588                                                 *it2 == ex_to<varidx>(*it2).toggle_variance(),
 589                                                 ex_to<varidx>(*it2).toggle_variance() == *it2
 590                                         ));
 591                                         something_changed = true;
 592                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 593                                         it2start = ex_to<indexed>(e).seq.begin();
 594                                         it2end = ex_to<indexed>(e).seq.end();
 595                                 }
 596                                 moved_indices.push_back(*vit);
 597                                 variant_dummy_indices.erase(vit);
 598                                 goto next_index;
 599                         }
 600                 }
 601
 602                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 603                         if (it2->op(0).is_equal(vit->op(0))) {
 604                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 605                                         e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance());
 606                                         something_changed = true;
 607                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 608                                         it2start = ex_to<indexed>(e).seq.begin();
 609                                         it2end = ex_to<indexed>(e).seq.end();
 610                                 }
 611                                 goto next_index;
 612                         }
 613                 }
 614
 615 next_index: ;
 616         }
 617
 618         return something_changed;
 619 }
 620
 621 /* Ordering that only compares the base expressions of indexed objects. */
 622 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
 623         bool operator() (const ex &lh, const ex &rh) const
 624         {
 625                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 626         }
 627 };
 628
 629 /** Simplify product of indexed expressions (commutative, noncommutative and
 630  *  simple squares), return list of free indices. */
 631 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 632 {
 633         // Remember whether the product was commutative or noncommutative
 634         // (because we chop it into factors and need to reassemble later)
 635         bool non_commutative = is_ex_exactly_of_type(e, ncmul);
 636
 637         // Collect factors in an exvector, store squares twice
 638         exvector v;
 639         v.reserve(e.nops() * 2);
 640
 641         if (is_ex_exactly_of_type(e, power)) {
 642                 // We only get called for simple squares, split a^2 -> a*a
 643                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 644                 v.push_back(e.op(0));
 645                 v.push_back(e.op(0));
 646         } else {
 647                 for (unsigned i=0; i<e.nops(); i++) {
 648                         ex f = e.op(i);
 649                         if (is_ex_exactly_of_type(f, power) && f.op(1).is_equal(_ex2)) {
 650                                 v.push_back(f.op(0));
 651                     v.push_back(f.op(0));
 652                         } else if (is_ex_exactly_of_type(f, ncmul)) {
 653                                 // Noncommutative factor found, split it as well
 654                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 655                                 for (unsigned j=0; j<f.nops(); j++)
 656                                         v.push_back(f.op(j));
 657                         } else
 658                                 v.push_back(f);
 659                 }
 660         }
 661
 662         // Perform contractions
 663         bool something_changed = false;
 664         GINAC_ASSERT(v.size() > 1);
 665         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 666         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 667
 668 try_again:
 669                 if (!is_ex_of_type(*it1, indexed))
 670                         continue;
 671
 672                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 673
 674                 // Indexed factor found, get free indices and look for contraction
 675                 // candidates
 676                 exvector free1, dummy1;
 677                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 678
 679                 exvector::iterator it2;
 680                 for (it2 = it1 + 1; it2 != itend; it2++) {
 681
 682                         if (!is_ex_of_type(*it2, indexed))
 683                                 continue;
 684
 685                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 686
 687                         // Find free indices of second factor and merge them with free
 688                         // indices of first factor
 689                         exvector un;
 690                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 691                         un.insert(un.end(), free1.begin(), free1.end());
 692
 693                         // Check whether the two factors share dummy indices
 694                         exvector free, dummy;
 695                         find_free_and_dummy(un, free, dummy);
 696                         unsigned num_dummies = dummy.size();
 697                         if (num_dummies == 0)
 698                                 continue;
 699
 700                         // At least one dummy index, is it a defined scalar product?
 701                         bool contracted = false;
 702                         if (free.empty()) {
 703                                 if (sp.is_defined(*it1, *it2)) {
 704                                         *it1 = sp.evaluate(*it1, *it2);
 705                                         *it2 = _ex1;
 706                                         goto contraction_done;
 707                                 }
 708                         }
 709
 710                         // Try to contract the first one with the second one
 711                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 712                         if (!contracted) {
 713
 714                                 // That didn't work; maybe the second object knows how to
 715                                 // contract itself with the first one
 716                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 717                         }
 718                         if (contracted) {
 719 contraction_done:
 720                                 if (first_noncommutative || second_noncommutative
 721                                  || is_ex_exactly_of_type(*it1, add) || is_ex_exactly_of_type(*it2, add)
 722                                  || is_ex_exactly_of_type(*it1, mul) || is_ex_exactly_of_type(*it2, mul)
 723                                  || is_ex_exactly_of_type(*it1, ncmul) || is_ex_exactly_of_type(*it2, ncmul)) {
 724
 725                                         // One of the factors became a sum or product:
 726                                         // re-expand expression and run again
 727                                         // Non-commutative products are always re-expanded to give
 728                                         // simplify_ncmul() the chance to re-order and canonicalize
 729                                         // the product
 730                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 731                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 732                                 }
 733
 734                                 // Both objects may have new indices now or they might
 735                                 // even not be indexed objects any more, so we have to
 736                                 // start over
 737                                 something_changed = true;
 738                                 goto try_again;
 739                         }
 740                 }
 741         }
 742
 743         // Find free indices (concatenate them all and call find_free_and_dummy())
 744         // and all dummy indices that appear
 745         exvector un, individual_dummy_indices;
 746         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 747                 exvector free_indices_of_factor;
 748                 if (is_ex_of_type(*it1, indexed)) {
 749                         exvector dummy_indices_of_factor;
 750                         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);
 751                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 752                 } else
 753                         free_indices_of_factor = it1->get_free_indices();
 754                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 755         }
 756         exvector local_dummy_indices;
 757         find_free_and_dummy(un, free_indices, local_dummy_indices);
 758         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 759
 760         // Filter out the dummy indices with variance
 761         exvector variant_dummy_indices;
 762         find_variant_indices(local_dummy_indices, variant_dummy_indices);
 763
 764         // Any indices with variance present at all?
 765         if (!variant_dummy_indices.empty()) {
 766
 767                 // Yes, bring the product into a canonical order that only depends on
 768                 // the base expressions of indexed objects
 769                 if (!non_commutative)
 770                         std::sort(v.begin(), v.end(), ex_base_is_less());
 771
 772                 exvector moved_indices;
 773
 774                 // Iterate over all indexed objects in the product
 775                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 776                         if (!is_ex_of_type(*it1, indexed))
 777                                 continue;
 778
 779                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
 780                                 something_changed = true;
 781                 }
 782         }
 783
 784         ex r;
 785         if (something_changed)
 786                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 787         else
 788                 r = e;
 789
 790         // The result should be symmetric with respect to exchange of dummy
 791         // indices, so if the symmetrization vanishes, the whole expression is
 792         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 793         if (local_dummy_indices.size() >= 2) {
 794                 lst dummy_syms;
 795                 for (int i=0; i<local_dummy_indices.size(); i++)
 796                         dummy_syms.append(local_dummy_indices[i].op(0));
 797                 if (r.symmetrize(dummy_syms).is_zero()) {
 798                         free_indices.clear();
 799                         return _ex0;
 800                 }
 801         }
 802
 803         // Dummy index renaming
 804         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 805
 806         // Product of indexed object with a scalar?
 807         if (is_ex_exactly_of_type(r, mul) && r.nops() == 2
 808          && is_ex_exactly_of_type(r.op(1), numeric) && is_ex_of_type(r.op(0), indexed))
 809                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 810         else
 811                 return r;
 812 }
 813
 814 /** Simplify indexed expression, return list of free indices. */
 815 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 816 {
 817         // Expand the expression
 818         ex e_expanded = e.expand();
 819
 820         // Simplification of single indexed object: just find the free indices
 821         // and perform dummy index renaming/repositioning
 822         if (is_ex_of_type(e_expanded, indexed)) {
 823
 824                 // Find the dummy indices
 825                 const indexed &i = ex_to<indexed>(e_expanded);
 826                 exvector local_dummy_indices;
 827                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 828
 829                 // Filter out the dummy indices with variance
 830                 exvector variant_dummy_indices;
 831                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
 832
 833                 // Any indices with variance present at all?
 834                 if (!variant_dummy_indices.empty()) {
 835
 836                         // Yes, reposition them
 837                         exvector moved_indices;
 838                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
 839                 }
 840
 841                 // Rename the dummy indices
 842                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 843         }
 844
 845         // Simplification of sum = sum of simplifications, check consistency of
 846         // free indices in each term
 847         if (is_ex_exactly_of_type(e_expanded, add)) {
 848                 bool first = true;
 849                 ex sum = _ex0;
 850                 free_indices.clear();
 851
 852                 for (unsigned i=0; i<e_expanded.nops(); i++) {
 853                         exvector free_indices_of_term;
 854                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 855                         if (!term.is_zero()) {
 856                                 if (first) {
 857                                         free_indices = free_indices_of_term;
 858                                         sum = term;
 859                                         first = false;
 860                                 } else {
 861                                         if (!indices_consistent(free_indices, free_indices_of_term))
 862                                                 throw (std::runtime_error("simplify_indexed: inconsistent indices in sum"));
 863                                         if (is_ex_of_type(sum, indexed) && is_ex_of_type(term, indexed))
 864                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
 865                                         else
 866                                                 sum += term;
 867                                 }
 868                         }
 869                 }
 870
 871                 if (sum.is_zero()) {
 872                         free_indices.clear();
 873                         return sum;
 874                 }
 875
 876                 // Symmetrizing over the dummy indices may cancel terms
 877                 int num_terms_orig = (is_a<add>(sum) ? sum.nops() : 1);
 878                 if (num_terms_orig > 1 && dummy_indices.size() >= 2) {
 879                         lst dummy_syms;
 880                         for (int i=0; i<dummy_indices.size(); i++)
 881                                 dummy_syms.append(dummy_indices[i].op(0));
 882                         ex sum_symm = sum.symmetrize(dummy_syms);
 883                         if (sum_symm.is_zero()) {
 884                                 free_indices.clear();
 885                                 return _ex0;
 886                         }
 887                         int num_terms = (is_a<add>(sum_symm) ? sum_symm.nops() : 1);
 888                         if (num_terms < num_terms_orig)
 889                                 return sum_symm;
 890                 }
 891
 892                 return sum;
 893         }
 894
 895         // Simplification of products
 896         if (is_ex_exactly_of_type(e_expanded, mul)
 897          || is_ex_exactly_of_type(e_expanded, ncmul)
 898          || (is_ex_exactly_of_type(e_expanded, power) && is_ex_of_type(e_expanded.op(0), indexed) && e_expanded.op(1).is_equal(_ex2)))
 899                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
 900
 901         // Cannot do anything
 902         free_indices.clear();
 903         return e_expanded;
 904 }
 905
 906 /** Simplify/canonicalize expression containing indexed objects. This
 907  *  performs contraction of dummy indices where possible and checks whether
 908  *  the free indices in sums are consistent.
 909  *
 910  *  @return simplified expression */
 911 ex ex::simplify_indexed(void) const
 912 {
 913         exvector free_indices, dummy_indices;
 914         scalar_products sp;
 915         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 916 }
 917
 918 /** Simplify/canonicalize expression containing indexed objects. This
 919  *  performs contraction of dummy indices where possible, checks whether
 920  *  the free indices in sums are consistent, and automatically replaces
 921  *  scalar products by known values if desired.
 922  *
 923  *  @param sp Scalar products to be replaced automatically
 924  *  @return simplified expression */
 925 ex ex::simplify_indexed(const scalar_products & sp) const
 926 {
 927         exvector free_indices, dummy_indices;
 928         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
 929 }
 930
 931 /** Symmetrize expression over its free indices. */
 932 ex ex::symmetrize(void) const
 933 {
 934         return GiNaC::symmetrize(*this, get_free_indices());
 935 }
 936
 937 /** Antisymmetrize expression over its free indices. */
 938 ex ex::antisymmetrize(void) const
 939 {
 940         return GiNaC::antisymmetrize(*this, get_free_indices());
 941 }
 942
 943 /** Symmetrize expression by cyclic permutation over its free indices. */
 944 ex ex::symmetrize_cyclic(void) const
 945 {
 946         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
 947 }
 948
 949 //////////
 950 // helper classes
 951 //////////
 952
 953 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
 954 {
 955         spm[make_key(v1, v2)] = sp;
 956 }
 957
 958 void scalar_products::add_vectors(const lst & l)
 959 {
 960         // Add all possible pairs of products
 961         unsigned num = l.nops();
 962         for (unsigned i=0; i<num; i++) {
 963                 ex a = l.op(i);
 964                 for (unsigned j=0; j<num; j++) {
 965                         ex b = l.op(j);
 966                         add(a, b, a*b);
 967                 }
 968         }
 969 }
 970
 971 void scalar_products::clear(void)
 972 {
 973         spm.clear();
 974 }
 975
 976 /** Check whether scalar product pair is defined. */
 977 bool scalar_products::is_defined(const ex & v1, const ex & v2) const
 978 {
 979         return spm.find(make_key(v1, v2)) != spm.end();
 980 }
 981
 982 /** Return value of defined scalar product pair. */
 983 ex scalar_products::evaluate(const ex & v1, const ex & v2) const
 984 {
 985         return spm.find(make_key(v1, v2))->second;
 986 }
 987
 988 void scalar_products::debugprint(void) const
 989 {
 990         std::cerr << "map size=" << spm.size() << std::endl;
 991         spmap::const_iterator i = spm.begin(), end = spm.end();
 992         while (i != end) {
 993                 const spmapkey & k = i->first;
 994                 std::cerr << "item key=(" << k.first << "," << k.second;
 995                 std::cerr << "), value=" << i->second << std::endl;
 996                 ++i;
 997         }
 998 }
 999
1000 /** Make key from object pair. */
1001 spmapkey scalar_products::make_key(const ex & v1, const ex & v2)
1002 {
1003         // If indexed, extract base objects
1004         ex s1 = is_ex_of_type(v1, indexed) ? v1.op(0) : v1;
1005         ex s2 = is_ex_of_type(v2, indexed) ? v2.op(0) : v2;
1006
1007         // Enforce canonical order in pair
1008         if (s1.compare(s2) > 0)
1009                 return spmapkey(s2, s1);
1010         else
1011                 return spmapkey(s1, s2);
1012 }
1013
1014 } // namespace GiNaC