ginac/indexed.cpp

   1 /** @file indexed.cpp
   2  *
   3  *  Implementation of GiNaC's indexed expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <iostream>
  24 #include <sstream>
  25 #include <stdexcept>
  26
  27 #include "indexed.h"
  28 #include "idx.h"
  29 #include "add.h"
  30 #include "mul.h"
  31 #include "ncmul.h"
  32 #include "power.h"
  33 #include "relational.h"
  34 #include "symmetry.h"
  35 #include "operators.h"
  36 #include "lst.h"
  37 #include "archive.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(indexed, exprseq,
  43   print_func<print_context>(&indexed::do_print).
  44   print_func<print_latex>(&indexed::do_print_latex).
  45   print_func<print_tree>(&indexed::do_print_tree))
  46
  47 //////////
  48 // default constructor
  49 //////////
  50
  51 indexed::indexed() : symtree(sy_none())
  52 {
  53         tinfo_key = TINFO_indexed;
  54 }
  55
  56 //////////
  57 // other constructors
  58 //////////
  59
  60 indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
  61 {
  62         tinfo_key = TINFO_indexed;
  63         validate();
  64 }
  65
  66 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(sy_none())
  67 {
  68         tinfo_key = TINFO_indexed;
  69         validate();
  70 }
  71
  72 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(sy_none())
  73 {
  74         tinfo_key = TINFO_indexed;
  75         validate();
  76 }
  77
  78 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(sy_none())
  79 {
  80         tinfo_key = TINFO_indexed;
  81         validate();
  82 }
  83
  84 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(sy_none())
  85 {
  86         tinfo_key = TINFO_indexed;
  87         validate();
  88 }
  89
  90 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
  91 {
  92         tinfo_key = TINFO_indexed;
  93         validate();
  94 }
  95
  96 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
  97 {
  98         tinfo_key = TINFO_indexed;
  99         validate();
 100 }
 101
 102 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3, const ex & i4) : inherited(b, i1, i2, i3, i4), symtree(symm)
 103 {
 104         tinfo_key = TINFO_indexed;
 105         validate();
 106 }
 107
 108 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
 109 {
 110         seq.insert(seq.end(), v.begin(), v.end());
 111         tinfo_key = TINFO_indexed;
 112         validate();
 113 }
 114
 115 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 116 {
 117         seq.insert(seq.end(), v.begin(), v.end());
 118         tinfo_key = TINFO_indexed;
 119         validate();
 120 }
 121
 122 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 123 {
 124         tinfo_key = TINFO_indexed;
 125 }
 126
 127 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 128 {
 129         tinfo_key = TINFO_indexed;
 130 }
 131
 132 indexed::indexed(const symmetry & symm, std::auto_ptr<exvector> vp) : inherited(vp), symtree(symm)
 133 {
 134         tinfo_key = TINFO_indexed;
 135 }
 136
 137 //////////
 138 // archiving
 139 //////////
 140
 141 indexed::indexed(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 142 {
 143         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 144                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 145                 unsigned symm = 0;
 146                 n.find_unsigned("symmetry", symm);
 147                 switch (symm) {
 148                         case 1:
 149                                 symtree = sy_symm();
 150                                 break;
 151                         case 2:
 152                                 symtree = sy_anti();
 153                                 break;
 154                         default:
 155                                 symtree = sy_none();
 156                                 break;
 157                 }
 158                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 159         }
 160 }
 161
 162 void indexed::archive(archive_node &n) const
 163 {
 164         inherited::archive(n);
 165         n.add_ex("symmetry", symtree);
 166 }
 167
 168 DEFAULT_UNARCHIVE(indexed)
 169
 170 //////////
 171 // functions overriding virtual functions from base classes
 172 //////////
 173
 174 void indexed::printindices(const print_context & c, unsigned level) const
 175 {
 176         if (seq.size() > 1) {
 177
 178                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 179
 180                 if (is_a<print_latex>(c)) {
 181
 182                         // TeX output: group by variance
 183                         bool first = true;
 184                         bool covariant = true;
 185
 186                         while (it != itend) {
 187                                 bool cur_covariant = (is_a<varidx>(*it) ? ex_to<varidx>(*it).is_covariant() : true);
 188                                 if (first || cur_covariant != covariant) { // Variance changed
 189                                         // The empty {} prevents indices from ending up on top of each other
 190                                         if (!first)
 191                                                 c.s << "}{}";
 192                                         covariant = cur_covariant;
 193                                         if (covariant)
 194                                                 c.s << "_{";
 195                                         else
 196                                                 c.s << "^{";
 197                                 }
 198                                 it->print(c, level);
 199                                 c.s << " ";
 200                                 first = false;
 201                                 it++;
 202                         }
 203                         c.s << "}";
 204
 205                 } else {
 206
 207                         // Ordinary output
 208                         while (it != itend) {
 209                                 it->print(c, level);
 210                                 it++;
 211                         }
 212                 }
 213         }
 214 }
 215
 216 void indexed::print_indexed(const print_context & c, const char *openbrace, const char *closebrace, unsigned level) const
 217 {
 218         if (precedence() <= level)
 219                 c.s << openbrace << '(';
 220         c.s << openbrace;
 221         seq[0].print(c, precedence());
 222         c.s << closebrace;
 223         printindices(c, level);
 224         if (precedence() <= level)
 225                 c.s << ')' << closebrace;
 226 }
 227
 228 void indexed::do_print(const print_context & c, unsigned level) const
 229 {
 230         print_indexed(c, "", "", level);
 231 }
 232
 233 void indexed::do_print_latex(const print_latex & c, unsigned level) const
 234 {
 235         print_indexed(c, "{", "}", level);
 236 }
 237
 238 void indexed::do_print_tree(const print_tree & c, unsigned level) const
 239 {
 240         c.s << std::string(level, ' ') << class_name() << " @" << this
 241             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 242             << ", " << seq.size()-1 << " indices"
 243             << ", symmetry=" << symtree << std::endl;
 244         seq[0].print(c, level + c.delta_indent);
 245         printindices(c, level + c.delta_indent);
 246 }
 247
 248 bool indexed::info(unsigned inf) const
 249 {
 250         if (inf == info_flags::indexed) return true;
 251         if (inf == info_flags::has_indices) return seq.size() > 1;
 252         return inherited::info(inf);
 253 }
 254
 255 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 256         bool operator() (const ex & e, unsigned inf) const {
 257                 return !(ex_to<idx>(e).get_value().info(inf));
 258         }
 259 };
 260
 261 bool indexed::all_index_values_are(unsigned inf) const
 262 {
 263         // No indices? Then no property can be fulfilled
 264         if (seq.size() < 2)
 265                 return false;
 266
 267         // Check all indices
 268         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 269 }
 270
 271 int indexed::compare_same_type(const basic & other) const
 272 {
 273         GINAC_ASSERT(is_a<indexed>(other));
 274         return inherited::compare_same_type(other);
 275 }
 276
 277 ex indexed::eval(int level) const
 278 {
 279         // First evaluate children, then we will end up here again
 280         if (level > 1)
 281                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 282
 283         const ex &base = seq[0];
 284
 285         // If the base object is 0, the whole object is 0
 286         if (base.is_zero())
 287                 return _ex0;
 288
 289         // If the base object is a product, pull out the numeric factor
 290         if (is_exactly_a<mul>(base) && is_exactly_a<numeric>(base.op(base.nops() - 1))) {
 291                 exvector v(seq);
 292                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 293                 v[0] = seq[0] / f;
 294                 return f * thiscontainer(v);
 295         }
 296
 297         // Canonicalize indices according to the symmetry properties
 298         if (seq.size() > 2) {
 299                 exvector v = seq;
 300                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 301                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 302                 if (sig != INT_MAX) {
 303                         // Something has changed while sorting indices, more evaluations later
 304                         if (sig == 0)
 305                                 return _ex0;
 306                         return ex(sig) * thiscontainer(v);
 307                 }
 308         }
 309
 310         // Let the class of the base object perform additional evaluations
 311         return ex_to<basic>(base).eval_indexed(*this);
 312 }
 313
 314 ex indexed::thiscontainer(const exvector & v) const
 315 {
 316         return indexed(ex_to<symmetry>(symtree), v);
 317 }
 318
 319 ex indexed::thiscontainer(std::auto_ptr<exvector> vp) const
 320 {
 321         return indexed(ex_to<symmetry>(symtree), vp);
 322 }
 323
 324 ex indexed::expand(unsigned options) const
 325 {
 326         GINAC_ASSERT(seq.size() > 0);
 327
 328         if (options & expand_options::expand_indexed) {
 329                 ex newbase = seq[0].expand(options);
 330                 if (is_exactly_a<add>(newbase)) {
 331                         ex sum = _ex0;
 332                         for (size_t i=0; i<newbase.nops(); i++) {
 333                                 exvector s = seq;
 334                                 s[0] = newbase.op(i);
 335                                 sum += thiscontainer(s).expand(options);
 336                         }
 337                         return sum;
 338                 }
 339                 if (!are_ex_trivially_equal(newbase, seq[0])) {
 340                         exvector s = seq;
 341                         s[0] = newbase;
 342                         return ex_to<indexed>(thiscontainer(s)).inherited::expand(options);
 343                 }
 344         }
 345         return inherited::expand(options);
 346 }
 347
 348 //////////
 349 // virtual functions which can be overridden by derived classes
 350 //////////
 351
 352 // none
 353
 354 //////////
 355 // non-virtual functions in this class
 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() 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_a<idx>(*it))
 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_exactly_a<symmetry>(symtree))
 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 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
 391         bool operator() (const ex &lh, const ex &rh) const
 392         {
 393                 if (lh.is_equal(rh))
 394                         return true;
 395                 else
 396                         try {
 397                                 // Replacing the dimension might cause an error (e.g. with
 398                                 // index classes that only work in a fixed number of dimensions)
 399                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
 400                         } catch (...) {
 401                                 return false;
 402                         }
 403         }
 404 };
 405
 406 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 407 static bool indices_consistent(const exvector & v1, const exvector & v2)
 408 {
 409         // Number of indices must be the same
 410         if (v1.size() != v2.size())
 411                 return false;
 412
 413         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
 414 }
 415
 416 exvector indexed::get_indices() const
 417 {
 418         GINAC_ASSERT(seq.size() >= 1);
 419         return exvector(seq.begin() + 1, seq.end());
 420 }
 421
 422 exvector indexed::get_dummy_indices() const
 423 {
 424         exvector free_indices, dummy_indices;
 425         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 426         return dummy_indices;
 427 }
 428
 429 exvector indexed::get_dummy_indices(const indexed & other) const
 430 {
 431         exvector indices = get_free_indices();
 432         exvector other_indices = other.get_free_indices();
 433         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 434         exvector dummy_indices;
 435         find_dummy_indices(indices, dummy_indices);
 436         return dummy_indices;
 437 }
 438
 439 bool indexed::has_dummy_index_for(const ex & i) const
 440 {
 441         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 442         while (it != itend) {
 443                 if (is_dummy_pair(*it, i))
 444                         return true;
 445                 it++;
 446         }
 447         return false;
 448 }
 449
 450 exvector indexed::get_free_indices() const
 451 {
 452         exvector free_indices, dummy_indices;
 453         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 454         return free_indices;
 455 }
 456
 457 exvector add::get_free_indices() const
 458 {
 459         exvector free_indices;
 460         for (size_t i=0; i<nops(); i++) {
 461                 if (i == 0)
 462                         free_indices = op(i).get_free_indices();
 463                 else {
 464                         exvector free_indices_of_term = op(i).get_free_indices();
 465                         if (!indices_consistent(free_indices, free_indices_of_term))
 466                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 467                 }
 468         }
 469         return free_indices;
 470 }
 471
 472 exvector mul::get_free_indices() const
 473 {
 474         // Concatenate free indices of all factors
 475         exvector un;
 476         for (size_t i=0; i<nops(); i++) {
 477                 exvector free_indices_of_factor = op(i).get_free_indices();
 478                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 479         }
 480
 481         // And remove the dummy indices
 482         exvector free_indices, dummy_indices;
 483         find_free_and_dummy(un, free_indices, dummy_indices);
 484         return free_indices;
 485 }
 486
 487 exvector ncmul::get_free_indices() const
 488 {
 489         // Concatenate free indices of all factors
 490         exvector un;
 491         for (size_t i=0; i<nops(); i++) {
 492                 exvector free_indices_of_factor = op(i).get_free_indices();
 493                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 494         }
 495
 496         // And remove the dummy indices
 497         exvector free_indices, dummy_indices;
 498         find_free_and_dummy(un, free_indices, dummy_indices);
 499         return free_indices;
 500 }
 501
 502 exvector power::get_free_indices() const
 503 {
 504         // Return free indices of basis
 505         return basis.get_free_indices();
 506 }
 507
 508 /** Rename dummy indices in an expression.
 509  *
 510  *  @param e Expression to work on
 511  *  @param local_dummy_indices The set of dummy indices that appear in the
 512  *    expression "e"
 513  *  @param global_dummy_indices The set of dummy indices that have appeared
 514  *    before and which we would like to use in "e", too. This gets updated
 515  *    by the function */
 516 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 517 {
 518         size_t global_size = global_dummy_indices.size(),
 519                local_size = local_dummy_indices.size();
 520
 521         // Any local dummy indices at all?
 522         if (local_size == 0)
 523                 return e;
 524
 525         if (global_size < local_size) {
 526
 527                 // More local indices than we encountered before, add the new ones
 528                 // to the global set
 529                 size_t old_global_size = global_size;
 530                 int remaining = local_size - global_size;
 531                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 532                 while (it != itend && remaining > 0) {
 533                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(op0_is_equal(), *it)) == global_dummy_indices.end()) {
 534                                 global_dummy_indices.push_back(*it);
 535                                 global_size++;
 536                                 remaining--;
 537                         }
 538                         it++;
 539                 }
 540
 541                 // If this is the first set of local indices, do nothing
 542                 if (old_global_size == 0)
 543                         return e;
 544         }
 545         GINAC_ASSERT(local_size <= global_size);
 546
 547         // Construct vectors of index symbols
 548         exvector local_syms, global_syms;
 549         local_syms.reserve(local_size);
 550         global_syms.reserve(local_size);
 551         for (size_t i=0; i<local_size; i++)
 552                 local_syms.push_back(local_dummy_indices[i].op(0));
 553         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 554         for (size_t i=0; i<local_size; i++) // don't use more global symbols than necessary
 555                 global_syms.push_back(global_dummy_indices[i].op(0));
 556         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 557
 558         // Remove common indices
 559         exvector local_uniq, global_uniq;
 560         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exvector>(local_uniq), ex_is_less());
 561         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exvector>(global_uniq), ex_is_less());
 562
 563         // Replace remaining non-common local index symbols by global ones
 564         if (local_uniq.empty())
 565                 return e;
 566         else {
 567                 while (global_uniq.size() > local_uniq.size())
 568                         global_uniq.pop_back();
 569                 return e.subs(lst(local_uniq.begin(), local_uniq.end()), lst(global_uniq.begin(), global_uniq.end()), subs_options::no_pattern);
 570         }
 571 }
 572
 573 /** Given a set of indices, extract those of class varidx. */
 574 static void find_variant_indices(const exvector & v, exvector & variant_indices)
 575 {
 576         exvector::const_iterator it1, itend;
 577         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 578                 if (is_exactly_a<varidx>(*it1))
 579                         variant_indices.push_back(*it1);
 580         }
 581 }
 582
 583 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
 584  *  variance.
 585  *
 586  *  @param e Object to work on
 587  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
 588  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
 589  *  @return true if 'e' was changed */
 590 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
 591 {
 592         bool something_changed = false;
 593
 594         // If a dummy index is encountered for the first time in the
 595         // product, pull it up, otherwise, pull it down
 596         exvector::const_iterator it2, it2start, it2end;
 597         for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
 598                 if (!is_exactly_a<varidx>(*it2))
 599                         continue;
 600
 601                 exvector::iterator vit, vitend;
 602                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 603                         if (it2->op(0).is_equal(vit->op(0))) {
 604                                 if (ex_to<varidx>(*it2).is_covariant()) {
 605                                         e = e.subs(lst(
 606                                                 *it2 == ex_to<varidx>(*it2).toggle_variance(),
 607                                                 ex_to<varidx>(*it2).toggle_variance() == *it2
 608                                         ), subs_options::no_pattern);
 609                                         something_changed = true;
 610                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 611                                         it2start = ex_to<indexed>(e).seq.begin();
 612                                         it2end = ex_to<indexed>(e).seq.end();
 613                                 }
 614                                 moved_indices.push_back(*vit);
 615                                 variant_dummy_indices.erase(vit);
 616                                 goto next_index;
 617                         }
 618                 }
 619
 620                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 621                         if (it2->op(0).is_equal(vit->op(0))) {
 622                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 623                                         e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance(), subs_options::no_pattern);
 624                                         something_changed = true;
 625                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 626                                         it2start = ex_to<indexed>(e).seq.begin();
 627                                         it2end = ex_to<indexed>(e).seq.end();
 628                                 }
 629                                 goto next_index;
 630                         }
 631                 }
 632
 633 next_index: ;
 634         }
 635
 636         return something_changed;
 637 }
 638
 639 /* Ordering that only compares the base expressions of indexed objects. */
 640 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
 641         bool operator() (const ex &lh, const ex &rh) const
 642         {
 643                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 644         }
 645 };
 646
 647 /** Simplify product of indexed expressions (commutative, noncommutative and
 648  *  simple squares), return list of free indices. */
 649 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 650 {
 651         // Remember whether the product was commutative or noncommutative
 652         // (because we chop it into factors and need to reassemble later)
 653         bool non_commutative = is_exactly_a<ncmul>(e);
 654
 655         // Collect factors in an exvector, store squares twice
 656         exvector v;
 657         v.reserve(e.nops() * 2);
 658
 659         if (is_exactly_a<power>(e)) {
 660                 // We only get called for simple squares, split a^2 -> a*a
 661                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 662                 v.push_back(e.op(0));
 663                 v.push_back(e.op(0));
 664         } else {
 665                 for (size_t i=0; i<e.nops(); i++) {
 666                         ex f = e.op(i);
 667                         if (is_exactly_a<power>(f) && f.op(1).is_equal(_ex2)) {
 668                                 v.push_back(f.op(0));
 669                     v.push_back(f.op(0));
 670                         } else if (is_exactly_a<ncmul>(f)) {
 671                                 // Noncommutative factor found, split it as well
 672                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 673                                 for (size_t j=0; j<f.nops(); j++)
 674                                         v.push_back(f.op(j));
 675                         } else
 676                                 v.push_back(f);
 677                 }
 678         }
 679
 680         // Perform contractions
 681         bool something_changed = false;
 682         GINAC_ASSERT(v.size() > 1);
 683         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 684         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 685
 686 try_again:
 687                 if (!is_a<indexed>(*it1))
 688                         continue;
 689
 690                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 691
 692                 // Indexed factor found, get free indices and look for contraction
 693                 // candidates
 694                 exvector free1, dummy1;
 695                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 696
 697                 exvector::iterator it2;
 698                 for (it2 = it1 + 1; it2 != itend; it2++) {
 699
 700                         if (!is_a<indexed>(*it2))
 701                                 continue;
 702
 703                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 704
 705                         // Find free indices of second factor and merge them with free
 706                         // indices of first factor
 707                         exvector un;
 708                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 709                         un.insert(un.end(), free1.begin(), free1.end());
 710
 711                         // Check whether the two factors share dummy indices
 712                         exvector free, dummy;
 713                         find_free_and_dummy(un, free, dummy);
 714                         size_t num_dummies = dummy.size();
 715                         if (num_dummies == 0)
 716                                 continue;
 717
 718                         // At least one dummy index, is it a defined scalar product?
 719                         bool contracted = false;
 720                         if (free.empty()) {
 721
 722                                 // Find minimal dimension of all indices of both factors
 723                                 exvector::const_iterator dit = ex_to<indexed>(*it1).seq.begin() + 1, ditend = ex_to<indexed>(*it1).seq.end();
 724                                 ex dim = ex_to<idx>(*dit).get_dim();
 725                                 ++dit;
 726                                 for (; dit != ditend; ++dit) {
 727                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 728                                 }
 729                                 dit = ex_to<indexed>(*it2).seq.begin() + 1;
 730                                 ditend = ex_to<indexed>(*it2).seq.end();
 731                                 for (; dit != ditend; ++dit) {
 732                                         dim = minimal_dim(dim, ex_to<idx>(*dit).get_dim());
 733                                 }
 734
 735                                 // User-defined scalar product?
 736                                 if (sp.is_defined(*it1, *it2, dim)) {
 737
 738                                         // Yes, substitute it
 739                                         *it1 = sp.evaluate(*it1, *it2, dim);
 740                                         *it2 = _ex1;
 741                                         goto contraction_done;
 742                                 }
 743                         }
 744
 745                         // Try to contract the first one with the second one
 746                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 747                         if (!contracted) {
 748
 749                                 // That didn't work; maybe the second object knows how to
 750                                 // contract itself with the first one
 751                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 752                         }
 753                         if (contracted) {
 754 contraction_done:
 755                                 if (first_noncommutative || second_noncommutative
 756                                  || is_exactly_a<add>(*it1) || is_exactly_a<add>(*it2)
 757                                  || is_exactly_a<mul>(*it1) || is_exactly_a<mul>(*it2)
 758                                  || is_exactly_a<ncmul>(*it1) || is_exactly_a<ncmul>(*it2)) {
 759
 760                                         // One of the factors became a sum or product:
 761                                         // re-expand expression and run again
 762                                         // Non-commutative products are always re-expanded to give
 763                                         // eval_ncmul() the chance to re-order and canonicalize
 764                                         // the product
 765                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 766                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 767                                 }
 768
 769                                 // Both objects may have new indices now or they might
 770                                 // even not be indexed objects any more, so we have to
 771                                 // start over
 772                                 something_changed = true;
 773                                 goto try_again;
 774                         }
 775                 }
 776         }
 777
 778         // Find free indices (concatenate them all and call find_free_and_dummy())
 779         // and all dummy indices that appear
 780         exvector un, individual_dummy_indices;
 781         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 782                 exvector free_indices_of_factor;
 783                 if (is_a<indexed>(*it1)) {
 784                         exvector dummy_indices_of_factor;
 785                         find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free_indices_of_factor, dummy_indices_of_factor);
 786                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 787                 } else
 788                         free_indices_of_factor = it1->get_free_indices();
 789                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 790         }
 791         exvector local_dummy_indices;
 792         find_free_and_dummy(un, free_indices, local_dummy_indices);
 793         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 794
 795         // Filter out the dummy indices with variance
 796         exvector variant_dummy_indices;
 797         find_variant_indices(local_dummy_indices, variant_dummy_indices);
 798
 799         // Any indices with variance present at all?
 800         if (!variant_dummy_indices.empty()) {
 801
 802                 // Yes, bring the product into a canonical order that only depends on
 803                 // the base expressions of indexed objects
 804                 if (!non_commutative)
 805                         std::sort(v.begin(), v.end(), ex_base_is_less());
 806
 807                 exvector moved_indices;
 808
 809                 // Iterate over all indexed objects in the product
 810                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 811                         if (!is_a<indexed>(*it1))
 812                                 continue;
 813
 814                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
 815                                 something_changed = true;
 816                 }
 817         }
 818
 819         ex r;
 820         if (something_changed)
 821                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 822         else
 823                 r = e;
 824
 825         // The result should be symmetric with respect to exchange of dummy
 826         // indices, so if the symmetrization vanishes, the whole expression is
 827         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 828         if (local_dummy_indices.size() >= 2) {
 829                 exvector dummy_syms;
 830                 dummy_syms.reserve(local_dummy_indices.size());
 831                 for (exvector::const_iterator it = local_dummy_indices.begin(); it != local_dummy_indices.end(); ++it)
 832                         dummy_syms.push_back(it->op(0));
 833                 if (symmetrize(r, dummy_syms).is_zero()) {
 834                         free_indices.clear();
 835                         return _ex0;
 836                 }
 837         }
 838
 839         // Dummy index renaming
 840         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 841
 842         // Product of indexed object with a scalar?
 843         if (is_exactly_a<mul>(r) && r.nops() == 2
 844          && is_exactly_a<numeric>(r.op(1)) && is_a<indexed>(r.op(0)))
 845                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 846         else
 847                 return r;
 848 }
 849
 850 /** This structure stores the original and symmetrized versions of terms
 851  *  obtained during the simplification of sums. */
 852 class terminfo {
 853 public:
 854         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
 855
 856         ex orig; /**< original term */
 857         ex symm; /**< symmtrized term */
 858 };
 859
 860 class terminfo_is_less {
 861 public:
 862         bool operator() (const terminfo & ti1, const terminfo & ti2) const
 863         {
 864                 return (ti1.symm.compare(ti2.symm) < 0);
 865         }
 866 };
 867
 868 /** This structure stores the individual symmetrized terms obtained during
 869  *  the simplification of sums. */
 870 class symminfo {
 871 public:
 872         symminfo() : num(0) {}
 873
 874         symminfo(const ex & symmterm_, const ex & orig_, size_t num_) : orig(orig_), num(num_)
 875         {
 876                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
 877                         coeff = symmterm_.op(symmterm_.nops()-1);
 878                         symmterm = symmterm_ / coeff;
 879                 } else {
 880                         coeff = 1;
 881                         symmterm = symmterm_;
 882                 }
 883         }
 884
 885         ex symmterm;  /**< symmetrized term */
 886         ex coeff;     /**< coefficient of symmetrized term */
 887         ex orig;      /**< original term */
 888         size_t num; /**< how many symmetrized terms resulted from the original term */
 889 };
 890
 891 class symminfo_is_less_by_symmterm {
 892 public:
 893         bool operator() (const symminfo & si1, const symminfo & si2) const
 894         {
 895                 return (si1.symmterm.compare(si2.symmterm) < 0);
 896         }
 897 };
 898
 899 class symminfo_is_less_by_orig {
 900 public:
 901         bool operator() (const symminfo & si1, const symminfo & si2) const
 902         {
 903                 return (si1.orig.compare(si2.orig) < 0);
 904         }
 905 };
 906
 907 /** Simplify indexed expression, return list of free indices. */
 908 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 909 {
 910         // Expand the expression
 911         ex e_expanded = e.expand();
 912
 913         // Simplification of single indexed object: just find the free indices
 914         // and perform dummy index renaming/repositioning
 915         if (is_a<indexed>(e_expanded)) {
 916
 917                 // Find the dummy indices
 918                 const indexed &i = ex_to<indexed>(e_expanded);
 919                 exvector local_dummy_indices;
 920                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 921
 922                 // Filter out the dummy indices with variance
 923                 exvector variant_dummy_indices;
 924                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
 925
 926                 // Any indices with variance present at all?
 927                 if (!variant_dummy_indices.empty()) {
 928
 929                         // Yes, reposition them
 930                         exvector moved_indices;
 931                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
 932                 }
 933
 934                 // Rename the dummy indices
 935                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 936         }
 937
 938         // Simplification of sum = sum of simplifications, check consistency of
 939         // free indices in each term
 940         if (is_exactly_a<add>(e_expanded)) {
 941                 bool first = true;
 942                 ex sum;
 943                 free_indices.clear();
 944
 945                 for (size_t i=0; i<e_expanded.nops(); i++) {
 946                         exvector free_indices_of_term;
 947                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 948                         if (!term.is_zero()) {
 949                                 if (first) {
 950                                         free_indices = free_indices_of_term;
 951                                         sum = term;
 952                                         first = false;
 953                                 } else {
 954                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
 955                                                 std::ostringstream s;
 956                                                 s << "simplify_indexed: inconsistent indices in sum: ";
 957                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
 958                                                 throw (std::runtime_error(s.str()));
 959                                         }
 960                                         if (is_a<indexed>(sum) && is_a<indexed>(term))
 961                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
 962                                         else
 963                                                 sum += term;
 964                                 }
 965                         }
 966                 }
 967
 968                 // If the sum turns out to be zero, we are finished
 969                 if (sum.is_zero()) {
 970                         free_indices.clear();
 971                         return sum;
 972                 }
 973
 974                 // More than one term and more than one dummy index?
 975                 size_t num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
 976                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
 977                         return sum;
 978
 979                 // Yes, construct vector of all dummy index symbols
 980                 exvector dummy_syms;
 981                 dummy_syms.reserve(dummy_indices.size());
 982                 for (exvector::const_iterator it = dummy_indices.begin(); it != dummy_indices.end(); ++it)
 983                         dummy_syms.push_back(it->op(0));
 984
 985                 // Chop the sum into terms and symmetrize each one over the dummy
 986                 // indices
 987                 std::vector<terminfo> terms;
 988                 for (size_t i=0; i<sum.nops(); i++) {
 989                         const ex & term = sum.op(i);
 990                         ex term_symm = symmetrize(term, dummy_syms);
 991                         if (term_symm.is_zero())
 992                                 continue;
 993                         terms.push_back(terminfo(term, term_symm));
 994                 }
 995
 996                 // Sort by symmetrized terms
 997                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
 998
 999                 // Combine equal symmetrized terms
1000                 std::vector<terminfo> terms_pass2;
1001                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
1002                         size_t num = 1;
1003                         std::vector<terminfo>::const_iterator j = i + 1;
1004                         while (j != terms.end() && j->symm == i->symm) {
1005                                 num++;
1006                                 j++;
1007                         }
1008                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
1009                         i = j;
1010                 }
1011
1012                 // If there is only one term left, we are finished
1013                 if (terms_pass2.size() == 1)
1014                         return terms_pass2[0].orig;
1015
1016                 // Chop the symmetrized terms into subterms
1017                 std::vector<symminfo> sy;
1018                 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
1019                         if (is_exactly_a<add>(i->symm)) {
1020                                 size_t num = i->symm.nops();
1021                                 for (size_t j=0; j<num; j++)
1022                                         sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1023                         } else
1024                                 sy.push_back(symminfo(i->symm, i->orig, 1));
1025                 }
1026
1027                 // Sort by symmetrized subterms
1028                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1029
1030                 // Combine equal symmetrized subterms
1031                 std::vector<symminfo> sy_pass2;
1032                 exvector result;
1033                 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1034
1035                         // Combine equal terms
1036                         std::vector<symminfo>::const_iterator j = i + 1;
1037                         if (j != sy.end() && j->symmterm == i->symmterm) {
1038
1039                                 // More than one term, collect the coefficients
1040                                 ex coeff = i->coeff;
1041                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1042                                         coeff += j->coeff;
1043                                         j++;
1044                                 }
1045
1046                                 // Add combined term to result
1047                                 if (!coeff.is_zero())
1048                                         result.push_back(coeff * i->symmterm);
1049
1050                         } else {
1051
1052                                 // Single term, store for second pass
1053                                 sy_pass2.push_back(*i);
1054                         }
1055
1056                         i = j;
1057                 }
1058
1059                 // Were there any remaining terms that didn't get combined?
1060                 if (sy_pass2.size() > 0) {
1061
1062                         // Yes, sort by their original terms
1063                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1064
1065                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1066
1067                                 // How many symmetrized terms of this original term are left?
1068                                 size_t num = 1;
1069                                 std::vector<symminfo>::const_iterator j = i + 1;
1070                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1071                                         num++;
1072                                         j++;
1073                                 }
1074
1075                                 if (num == i->num) {
1076
1077                                         // All terms left, then add the original term to the result
1078                                         result.push_back(i->orig);
1079
1080                                 } else {
1081
1082                                         // Some terms were combined with others, add up the remaining symmetrized terms
1083                                         std::vector<symminfo>::const_iterator k;
1084                                         for (k=i; k!=j; k++)
1085                                                 result.push_back(k->coeff * k->symmterm);
1086                                 }
1087
1088                                 i = j;
1089                         }
1090                 }
1091
1092                 // Add all resulting terms
1093                 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1094                 if (sum_symm.is_zero())
1095                         free_indices.clear();
1096                 return sum_symm;
1097         }
1098
1099         // Simplification of products
1100         if (is_exactly_a<mul>(e_expanded)
1101          || is_exactly_a<ncmul>(e_expanded)
1102          || (is_exactly_a<power>(e_expanded) && is_a<indexed>(e_expanded.op(0)) && e_expanded.op(1).is_equal(_ex2)))
1103                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1104
1105         // Cannot do anything
1106         free_indices.clear();
1107         return e_expanded;
1108 }
1109
1110 /** Simplify/canonicalize expression containing indexed objects. This
1111  *  performs contraction of dummy indices where possible and checks whether
1112  *  the free indices in sums are consistent.
1113  *
1114  *  @return simplified expression */
1115 ex ex::simplify_indexed(unsigned options) const
1116 {
1117         exvector free_indices, dummy_indices;
1118         scalar_products sp;
1119         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1120 }
1121
1122 /** Simplify/canonicalize expression containing indexed objects. This
1123  *  performs contraction of dummy indices where possible, checks whether
1124  *  the free indices in sums are consistent, and automatically replaces
1125  *  scalar products by known values if desired.
1126  *
1127  *  @param sp Scalar products to be replaced automatically
1128  *  @return simplified expression */
1129 ex ex::simplify_indexed(const scalar_products & sp, unsigned options) const
1130 {
1131         exvector free_indices, dummy_indices;
1132         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1133 }
1134
1135 /** Symmetrize expression over its free indices. */
1136 ex ex::symmetrize() const
1137 {
1138         return GiNaC::symmetrize(*this, get_free_indices());
1139 }
1140
1141 /** Antisymmetrize expression over its free indices. */
1142 ex ex::antisymmetrize() const
1143 {
1144         return GiNaC::antisymmetrize(*this, get_free_indices());
1145 }
1146
1147 /** Symmetrize expression by cyclic permutation over its free indices. */
1148 ex ex::symmetrize_cyclic() const
1149 {
1150         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1151 }
1152
1153 //////////
1154 // helper classes
1155 //////////
1156
1157 spmapkey::spmapkey(const ex & v1_, const ex & v2_, const ex & dim_) : dim(dim_)
1158 {
1159         // If indexed, extract base objects
1160         ex s1 = is_a<indexed>(v1_) ? v1_.op(0) : v1_;
1161         ex s2 = is_a<indexed>(v2_) ? v2_.op(0) : v2_;
1162
1163         // Enforce canonical order in pair
1164         if (s1.compare(s2) > 0) {
1165                 v1 = s2;
1166                 v2 = s1;
1167         } else {
1168                 v1 = s1;
1169                 v2 = s2;
1170         }
1171 }
1172
1173 bool spmapkey::operator==(const spmapkey &other) const
1174 {
1175         if (!v1.is_equal(other.v1))
1176                 return false;
1177         if (!v2.is_equal(other.v2))
1178                 return false;
1179         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1180                 return true;
1181         else
1182                 return dim.is_equal(other.dim);
1183 }
1184
1185 bool spmapkey::operator<(const spmapkey &other) const
1186 {
1187         int cmp = v1.compare(other.v1);
1188         if (cmp)
1189                 return cmp < 0;
1190         cmp = v2.compare(other.v2);
1191         if (cmp)
1192                 return cmp < 0;
1193
1194         // Objects are equal, now check dimensions
1195         if (is_a<wildcard>(dim) || is_a<wildcard>(other.dim))
1196                 return false;
1197         else
1198                 return dim.compare(other.dim) < 0;
1199 }
1200
1201 void spmapkey::debugprint() const
1202 {
1203         std::cerr << "(" << v1 << "," << v2 << "," << dim << ")";
1204 }
1205
1206 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1207 {
1208         spm[spmapkey(v1, v2)] = sp;
1209 }
1210
1211 void scalar_products::add(const ex & v1, const ex & v2, const ex & dim, const ex & sp)
1212 {
1213         spm[spmapkey(v1, v2, dim)] = sp;
1214 }
1215
1216 void scalar_products::add_vectors(const lst & l, const ex & dim)
1217 {
1218         // Add all possible pairs of products
1219         for (lst::const_iterator it1 = l.begin(); it1 != l.end(); ++it1)
1220                 for (lst::const_iterator it2 = l.begin(); it2 != l.end(); ++it2)
1221                         add(*it1, *it2, *it1 * *it2);
1222 }
1223
1224 void scalar_products::clear()
1225 {
1226         spm.clear();
1227 }
1228
1229 /** Check whether scalar product pair is defined. */
1230 bool scalar_products::is_defined(const ex & v1, const ex & v2, const ex & dim) const
1231 {
1232         return spm.find(spmapkey(v1, v2, dim)) != spm.end();
1233 }
1234
1235 /** Return value of defined scalar product pair. */
1236 ex scalar_products::evaluate(const ex & v1, const ex & v2, const ex & dim) const
1237 {
1238         return spm.find(spmapkey(v1, v2, dim))->second;
1239 }
1240
1241 void scalar_products::debugprint() const
1242 {
1243         std::cerr << "map size=" << spm.size() << std::endl;
1244         spmap::const_iterator i = spm.begin(), end = spm.end();
1245         while (i != end) {
1246                 const spmapkey & k = i->first;
1247                 std::cerr << "item key=";
1248                 k.debugprint();
1249                 std::cerr << ", value=" << i->second << std::endl;
1250                 ++i;
1251         }
1252 }
1253
1254 } // namespace GiNaC