ginac/indexed.cpp

   1 /** @file indexed.cpp
   2  *
   3  *  Implementation of GiNaC's indexed expressions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <iostream>
  24 #include <sstream>
  25 #include <stdexcept>
  26
  27 #include "indexed.h"
  28 #include "idx.h"
  29 #include "add.h"
  30 #include "mul.h"
  31 #include "ncmul.h"
  32 #include "power.h"
  33 #include "relational.h"
  34 #include "symmetry.h"
  35 #include "lst.h"
  36 #include "print.h"
  37 #include "archive.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(indexed, exprseq)
  43
  44 //////////
  45 // default ctor, dtor, copy ctor, assignment operator and helpers
  46 //////////
  47
  48 indexed::indexed() : symtree(sy_none())
  49 {
  50         tinfo_key = TINFO_indexed;
  51 }
  52
  53 void indexed::copy(const indexed & other)
  54 {
  55         inherited::copy(other);
  56         symtree = other.symtree;
  57 }
  58
  59 DEFAULT_DESTROY(indexed)
  60
  61 //////////
  62 // other constructors
  63 //////////
  64
  65 indexed::indexed(const ex & b) : inherited(b), symtree(sy_none())
  66 {
  67         tinfo_key = TINFO_indexed;
  68         validate();
  69 }
  70
  71 indexed::indexed(const ex & b, const ex & i1) : inherited(b, i1), symtree(sy_none())
  72 {
  73         tinfo_key = TINFO_indexed;
  74         validate();
  75 }
  76
  77 indexed::indexed(const ex & b, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(sy_none())
  78 {
  79         tinfo_key = TINFO_indexed;
  80         validate();
  81 }
  82
  83 indexed::indexed(const ex & b, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(sy_none())
  84 {
  85         tinfo_key = TINFO_indexed;
  86         validate();
  87 }
  88
  89 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())
  90 {
  91         tinfo_key = TINFO_indexed;
  92         validate();
  93 }
  94
  95 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2) : inherited(b, i1, i2), symtree(symm)
  96 {
  97         tinfo_key = TINFO_indexed;
  98         validate();
  99 }
 100
 101 indexed::indexed(const ex & b, const symmetry & symm, const ex & i1, const ex & i2, const ex & i3) : inherited(b, i1, i2, i3), symtree(symm)
 102 {
 103         tinfo_key = TINFO_indexed;
 104         validate();
 105 }
 106
 107 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)
 108 {
 109         tinfo_key = TINFO_indexed;
 110         validate();
 111 }
 112
 113 indexed::indexed(const ex & b, const exvector & v) : inherited(b), symtree(sy_none())
 114 {
 115         seq.insert(seq.end(), v.begin(), v.end());
 116         tinfo_key = TINFO_indexed;
 117         validate();
 118 }
 119
 120 indexed::indexed(const ex & b, const symmetry & symm, const exvector & v) : inherited(b), symtree(symm)
 121 {
 122         seq.insert(seq.end(), v.begin(), v.end());
 123         tinfo_key = TINFO_indexed;
 124         validate();
 125 }
 126
 127 indexed::indexed(const symmetry & symm, const exprseq & es) : inherited(es), symtree(symm)
 128 {
 129         tinfo_key = TINFO_indexed;
 130 }
 131
 132 indexed::indexed(const symmetry & symm, const exvector & v, bool discardable) : inherited(v, discardable), symtree(symm)
 133 {
 134         tinfo_key = TINFO_indexed;
 135 }
 136
 137 indexed::indexed(const symmetry & symm, exvector * vp) : inherited(vp), symtree(symm)
 138 {
 139         tinfo_key = TINFO_indexed;
 140 }
 141
 142 //////////
 143 // archiving
 144 //////////
 145
 146 indexed::indexed(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 147 {
 148         if (!n.find_ex("symmetry", symtree, sym_lst)) {
 149                 // GiNaC versions <= 0.9.0 had an unsigned "symmetry" property
 150                 unsigned symm = 0;
 151                 n.find_unsigned("symmetry", symm);
 152                 switch (symm) {
 153                         case 1:
 154                                 symtree = sy_symm();
 155                                 break;
 156                         case 2:
 157                                 symtree = sy_anti();
 158                                 break;
 159                         default:
 160                                 symtree = sy_none();
 161                                 break;
 162                 }
 163                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 164         }
 165 }
 166
 167 void indexed::archive(archive_node &n) const
 168 {
 169         inherited::archive(n);
 170         n.add_ex("symmetry", symtree);
 171 }
 172
 173 DEFAULT_UNARCHIVE(indexed)
 174
 175 //////////
 176 // functions overriding virtual functions from base classes
 177 //////////
 178
 179 void indexed::print(const print_context & c, unsigned level) const
 180 {
 181         GINAC_ASSERT(seq.size() > 0);
 182
 183         if (is_a<print_tree>(c)) {
 184
 185                 c.s << std::string(level, ' ') << class_name()
 186                     << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
 187                     << ", " << seq.size()-1 << " indices"
 188                     << ", symmetry=" << symtree << std::endl;
 189                 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
 190                 seq[0].print(c, level + delta_indent);
 191                 printindices(c, level + delta_indent);
 192
 193         } else {
 194
 195                 bool is_tex = is_a<print_latex>(c);
 196                 const ex & base = seq[0];
 197
 198                 if (precedence() <= level)
 199                         c.s << (is_tex ? "{(" : "(");
 200                 if (is_tex)
 201                         c.s << "{";
 202                 base.print(c, precedence());
 203                 if (is_tex)
 204                         c.s << "}";
 205                 printindices(c, level);
 206                 if (precedence() <= level)
 207                         c.s << (is_tex ? ")}" : ")");
 208         }
 209 }
 210
 211 bool indexed::info(unsigned inf) const
 212 {
 213         if (inf == info_flags::indexed) return true;
 214         if (inf == info_flags::has_indices) return seq.size() > 1;
 215         return inherited::info(inf);
 216 }
 217
 218 struct idx_is_not : public std::binary_function<ex, unsigned, bool> {
 219         bool operator() (const ex & e, unsigned inf) const {
 220                 return !(ex_to<idx>(e).get_value().info(inf));
 221         }
 222 };
 223
 224 bool indexed::all_index_values_are(unsigned inf) const
 225 {
 226         // No indices? Then no property can be fulfilled
 227         if (seq.size() < 2)
 228                 return false;
 229
 230         // Check all indices
 231         return find_if(seq.begin() + 1, seq.end(), bind2nd(idx_is_not(), inf)) == seq.end();
 232 }
 233
 234 int indexed::compare_same_type(const basic & other) const
 235 {
 236         GINAC_ASSERT(is_a<indexed>(other));
 237         return inherited::compare_same_type(other);
 238 }
 239
 240 ex indexed::eval(int level) const
 241 {
 242         // First evaluate children, then we will end up here again
 243         if (level > 1)
 244                 return indexed(ex_to<symmetry>(symtree), evalchildren(level));
 245
 246         const ex &base = seq[0];
 247
 248         // If the base object is 0, the whole object is 0
 249         if (base.is_zero())
 250                 return _ex0;
 251
 252         // If the base object is a product, pull out the numeric factor
 253         if (is_ex_exactly_of_type(base, mul) && is_ex_exactly_of_type(base.op(base.nops() - 1), numeric)) {
 254                 exvector v(seq);
 255                 ex f = ex_to<numeric>(base.op(base.nops() - 1));
 256                 v[0] = seq[0] / f;
 257                 return f * thisexprseq(v);
 258         }
 259
 260         // Canonicalize indices according to the symmetry properties
 261         if (seq.size() > 2) {
 262                 exvector v = seq;
 263                 GINAC_ASSERT(is_exactly_a<symmetry>(symtree));
 264                 int sig = canonicalize(v.begin() + 1, ex_to<symmetry>(symtree));
 265                 if (sig != INT_MAX) {
 266                         // Something has changed while sorting indices, more evaluations later
 267                         if (sig == 0)
 268                                 return _ex0;
 269                         return ex(sig) * thisexprseq(v);
 270                 }
 271         }
 272
 273         // Let the class of the base object perform additional evaluations
 274         return ex_to<basic>(base).eval_indexed(*this);
 275 }
 276
 277 ex indexed::thisexprseq(const exvector & v) const
 278 {
 279         return indexed(ex_to<symmetry>(symtree), v);
 280 }
 281
 282 ex indexed::thisexprseq(exvector * vp) const
 283 {
 284         return indexed(ex_to<symmetry>(symtree), vp);
 285 }
 286
 287 ex indexed::expand(unsigned options) const
 288 {
 289         GINAC_ASSERT(seq.size() > 0);
 290
 291         if ((options & expand_options::expand_indexed) && is_ex_exactly_of_type(seq[0], add)) {
 292
 293                 // expand_indexed expands (a+b).i -> a.i + b.i
 294                 const ex & base = seq[0];
 295                 ex sum = _ex0;
 296                 for (unsigned i=0; i<base.nops(); i++) {
 297                         exvector s = seq;
 298                         s[0] = base.op(i);
 299                         sum += thisexprseq(s).expand();
 300                 }
 301                 return sum;
 302
 303         } else
 304                 return inherited::expand(options);
 305 }
 306
 307 //////////
 308 // virtual functions which can be overridden by derived classes
 309 //////////
 310
 311 // none
 312
 313 //////////
 314 // non-virtual functions in this class
 315 //////////
 316
 317 void indexed::printindices(const print_context & c, unsigned level) const
 318 {
 319         if (seq.size() > 1) {
 320
 321                 exvector::const_iterator it=seq.begin() + 1, itend = seq.end();
 322
 323                 if (is_a<print_latex>(c)) {
 324
 325                         // TeX output: group by variance
 326                         bool first = true;
 327                         bool covariant = true;
 328
 329                         while (it != itend) {
 330                                 bool cur_covariant = (is_ex_of_type(*it, varidx) ? ex_to<varidx>(*it).is_covariant() : true);
 331                                 if (first || cur_covariant != covariant) { // Variance changed
 332                                         // The empty {} prevents indices from ending up on top of each other
 333                                         if (!first)
 334                                                 c.s << "}{}";
 335                                         covariant = cur_covariant;
 336                                         if (covariant)
 337                                                 c.s << "_{";
 338                                         else
 339                                                 c.s << "^{";
 340                                 }
 341                                 it->print(c, level);
 342                                 c.s << " ";
 343                                 first = false;
 344                                 it++;
 345                         }
 346                         c.s << "}";
 347
 348                 } else {
 349
 350                         // Ordinary output
 351                         while (it != itend) {
 352                                 it->print(c, level);
 353                                 it++;
 354                         }
 355                 }
 356         }
 357 }
 358
 359 /** Check whether all indices are of class idx and validate the symmetry
 360  *  tree. This function is used internally to make sure that all constructed
 361  *  indexed objects really carry indices and not some other classes. */
 362 void indexed::validate(void) const
 363 {
 364         GINAC_ASSERT(seq.size() > 0);
 365         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 366         while (it != itend) {
 367                 if (!is_ex_of_type(*it, idx))
 368                         throw(std::invalid_argument("indices of indexed object must be of type idx"));
 369                 it++;
 370         }
 371
 372         if (!symtree.is_zero()) {
 373                 if (!is_ex_exactly_of_type(symtree, symmetry))
 374                         throw(std::invalid_argument("symmetry of indexed object must be of type symmetry"));
 375                 const_cast<symmetry &>(ex_to<symmetry>(symtree)).validate(seq.size() - 1);
 376         }
 377 }
 378
 379 /** Implementation of ex::diff() for an indexed object always returns 0.
 380  *
 381  *  @see ex::diff */
 382 ex indexed::derivative(const symbol & s) const
 383 {
 384         return _ex0;
 385 }
 386
 387 //////////
 388 // global functions
 389 //////////
 390
 391 struct idx_is_equal_ignore_dim : public std::binary_function<ex, ex, bool> {
 392         bool operator() (const ex &lh, const ex &rh) const
 393         {
 394                 if (lh.is_equal(rh))
 395                         return true;
 396                 else
 397                         try {
 398                                 // Replacing the dimension might cause an error (e.g. with
 399                                 // index classes that only work in a fixed number of dimensions)
 400                                 return lh.is_equal(ex_to<idx>(rh).replace_dim(ex_to<idx>(lh).get_dim()));
 401                         } catch (...) {
 402                                 return false;
 403                         }
 404         }
 405 };
 406
 407 /** Check whether two sorted index vectors are consistent (i.e. equal). */
 408 static bool indices_consistent(const exvector & v1, const exvector & v2)
 409 {
 410         // Number of indices must be the same
 411         if (v1.size() != v2.size())
 412                 return false;
 413
 414         return equal(v1.begin(), v1.end(), v2.begin(), idx_is_equal_ignore_dim());
 415 }
 416
 417 exvector indexed::get_indices(void) const
 418 {
 419         GINAC_ASSERT(seq.size() >= 1);
 420         return exvector(seq.begin() + 1, seq.end());
 421 }
 422
 423 exvector indexed::get_dummy_indices(void) const
 424 {
 425         exvector free_indices, dummy_indices;
 426         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 427         return dummy_indices;
 428 }
 429
 430 exvector indexed::get_dummy_indices(const indexed & other) const
 431 {
 432         exvector indices = get_free_indices();
 433         exvector other_indices = other.get_free_indices();
 434         indices.insert(indices.end(), other_indices.begin(), other_indices.end());
 435         exvector dummy_indices;
 436         find_dummy_indices(indices, dummy_indices);
 437         return dummy_indices;
 438 }
 439
 440 bool indexed::has_dummy_index_for(const ex & i) const
 441 {
 442         exvector::const_iterator it = seq.begin() + 1, itend = seq.end();
 443         while (it != itend) {
 444                 if (is_dummy_pair(*it, i))
 445                         return true;
 446                 it++;
 447         }
 448         return false;
 449 }
 450
 451 exvector indexed::get_free_indices(void) const
 452 {
 453         exvector free_indices, dummy_indices;
 454         find_free_and_dummy(seq.begin() + 1, seq.end(), free_indices, dummy_indices);
 455         return free_indices;
 456 }
 457
 458 exvector add::get_free_indices(void) const
 459 {
 460         exvector free_indices;
 461         for (unsigned i=0; i<nops(); i++) {
 462                 if (i == 0)
 463                         free_indices = op(i).get_free_indices();
 464                 else {
 465                         exvector free_indices_of_term = op(i).get_free_indices();
 466                         if (!indices_consistent(free_indices, free_indices_of_term))
 467                                 throw (std::runtime_error("add::get_free_indices: inconsistent indices in sum"));
 468                 }
 469         }
 470         return free_indices;
 471 }
 472
 473 exvector mul::get_free_indices(void) const
 474 {
 475         // Concatenate free indices of all factors
 476         exvector un;
 477         for (unsigned i=0; i<nops(); i++) {
 478                 exvector free_indices_of_factor = op(i).get_free_indices();
 479                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 480         }
 481
 482         // And remove the dummy indices
 483         exvector free_indices, dummy_indices;
 484         find_free_and_dummy(un, free_indices, dummy_indices);
 485         return free_indices;
 486 }
 487
 488 exvector ncmul::get_free_indices(void) const
 489 {
 490         // Concatenate free indices of all factors
 491         exvector un;
 492         for (unsigned i=0; i<nops(); i++) {
 493                 exvector free_indices_of_factor = op(i).get_free_indices();
 494                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 495         }
 496
 497         // And remove the dummy indices
 498         exvector free_indices, dummy_indices;
 499         find_free_and_dummy(un, free_indices, dummy_indices);
 500         return free_indices;
 501 }
 502
 503 exvector power::get_free_indices(void) const
 504 {
 505         // Return free indices of basis
 506         return basis.get_free_indices();
 507 }
 508
 509 /** Rename dummy indices in an expression.
 510  *
 511  *  @param e Expression to work on
 512  *  @param local_dummy_indices The set of dummy indices that appear in the
 513  *    expression "e"
 514  *  @param global_dummy_indices The set of dummy indices that have appeared
 515  *    before and which we would like to use in "e", too. This gets updated
 516  *    by the function */
 517 static ex rename_dummy_indices(const ex & e, exvector & global_dummy_indices, exvector & local_dummy_indices)
 518 {
 519         unsigned global_size = global_dummy_indices.size(),
 520                  local_size = local_dummy_indices.size();
 521
 522         // Any local dummy indices at all?
 523         if (local_size == 0)
 524                 return e;
 525
 526         if (global_size < local_size) {
 527
 528                 // More local indices than we encountered before, add the new ones
 529                 // to the global set
 530                 int old_global_size = global_size;
 531                 int remaining = local_size - global_size;
 532                 exvector::const_iterator it = local_dummy_indices.begin(), itend = local_dummy_indices.end();
 533                 while (it != itend && remaining > 0) {
 534                         if (find_if(global_dummy_indices.begin(), global_dummy_indices.end(), bind2nd(ex_is_equal(), *it)) == global_dummy_indices.end()) {
 535                                 global_dummy_indices.push_back(*it);
 536                                 global_size++;
 537                                 remaining--;
 538                         }
 539                         it++;
 540                 }
 541
 542                 // If this is the first set of local indices, do nothing
 543                 if (old_global_size == 0)
 544                         return e;
 545         }
 546         GINAC_ASSERT(local_size <= global_size);
 547
 548         // Construct lists of index symbols
 549         exlist local_syms, global_syms;
 550         for (unsigned i=0; i<local_size; i++)
 551                 local_syms.push_back(local_dummy_indices[i].op(0));
 552         shaker_sort(local_syms.begin(), local_syms.end(), ex_is_less(), ex_swap());
 553         for (unsigned i=0; i<global_size; i++)
 554                 global_syms.push_back(global_dummy_indices[i].op(0));
 555         shaker_sort(global_syms.begin(), global_syms.end(), ex_is_less(), ex_swap());
 556
 557         // Remove common indices
 558         exlist local_uniq, global_uniq;
 559         set_difference(local_syms.begin(), local_syms.end(), global_syms.begin(), global_syms.end(), std::back_insert_iterator<exlist>(local_uniq), ex_is_less());
 560         set_difference(global_syms.begin(), global_syms.end(), local_syms.begin(), local_syms.end(), std::back_insert_iterator<exlist>(global_uniq), ex_is_less());
 561
 562         // Replace remaining non-common local index symbols by global ones
 563         if (local_uniq.empty())
 564                 return e;
 565         else {
 566                 while (global_uniq.size() > local_uniq.size())
 567                         global_uniq.pop_back();
 568                 return e.subs(lst(local_uniq), lst(global_uniq));
 569         }
 570 }
 571
 572 /** Given a set of indices, extract those of class varidx. */
 573 static void find_variant_indices(const exvector & v, exvector & variant_indices)
 574 {
 575         exvector::const_iterator it1, itend;
 576         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 577                 if (is_exactly_a<varidx>(*it1))
 578                         variant_indices.push_back(*it1);
 579         }
 580 }
 581
 582 /** Raise/lower dummy indices in a single indexed objects to canonicalize their
 583  *  variance.
 584  *
 585  *  @param e Object to work on
 586  *  @param variant_dummy_indices The set of indices that might need repositioning (will be changed by this function)
 587  *  @param moved_indices The set of indices that have been repositioned (will be changed by this function)
 588  *  @return true if 'e' was changed */
 589 bool reposition_dummy_indices(ex & e, exvector & variant_dummy_indices, exvector & moved_indices)
 590 {
 591         bool something_changed = false;
 592
 593         // If a dummy index is encountered for the first time in the
 594         // product, pull it up, otherwise, pull it down
 595         exvector::const_iterator it2, it2start, it2end;
 596         for (it2start = ex_to<indexed>(e).seq.begin(), it2end = ex_to<indexed>(e).seq.end(), it2 = it2start + 1; it2 != it2end; ++it2) {
 597                 if (!is_exactly_a<varidx>(*it2))
 598                         continue;
 599
 600                 exvector::iterator vit, vitend;
 601                 for (vit = variant_dummy_indices.begin(), vitend = variant_dummy_indices.end(); vit != vitend; ++vit) {
 602                         if (it2->op(0).is_equal(vit->op(0))) {
 603                                 if (ex_to<varidx>(*it2).is_covariant()) {
 604                                         e = e.subs(lst(
 605                                                 *it2 == ex_to<varidx>(*it2).toggle_variance(),
 606                                                 ex_to<varidx>(*it2).toggle_variance() == *it2
 607                                         ));
 608                                         something_changed = true;
 609                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 610                                         it2start = ex_to<indexed>(e).seq.begin();
 611                                         it2end = ex_to<indexed>(e).seq.end();
 612                                 }
 613                                 moved_indices.push_back(*vit);
 614                                 variant_dummy_indices.erase(vit);
 615                                 goto next_index;
 616                         }
 617                 }
 618
 619                 for (vit = moved_indices.begin(), vitend = moved_indices.end(); vit != vitend; ++vit) {
 620                         if (it2->op(0).is_equal(vit->op(0))) {
 621                                 if (ex_to<varidx>(*it2).is_contravariant()) {
 622                                         e = e.subs(*it2 == ex_to<varidx>(*it2).toggle_variance());
 623                                         something_changed = true;
 624                                         it2 = ex_to<indexed>(e).seq.begin() + (it2 - it2start);
 625                                         it2start = ex_to<indexed>(e).seq.begin();
 626                                         it2end = ex_to<indexed>(e).seq.end();
 627                                 }
 628                                 goto next_index;
 629                         }
 630                 }
 631
 632 next_index: ;
 633         }
 634
 635         return something_changed;
 636 }
 637
 638 /* Ordering that only compares the base expressions of indexed objects. */
 639 struct ex_base_is_less : public std::binary_function<ex, ex, bool> {
 640         bool operator() (const ex &lh, const ex &rh) const
 641         {
 642                 return (is_a<indexed>(lh) ? lh.op(0) : lh).compare(is_a<indexed>(rh) ? rh.op(0) : rh) < 0;
 643         }
 644 };
 645
 646 /** Simplify product of indexed expressions (commutative, noncommutative and
 647  *  simple squares), return list of free indices. */
 648 ex simplify_indexed_product(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 649 {
 650         // Remember whether the product was commutative or noncommutative
 651         // (because we chop it into factors and need to reassemble later)
 652         bool non_commutative = is_ex_exactly_of_type(e, ncmul);
 653
 654         // Collect factors in an exvector, store squares twice
 655         exvector v;
 656         v.reserve(e.nops() * 2);
 657
 658         if (is_ex_exactly_of_type(e, power)) {
 659                 // We only get called for simple squares, split a^2 -> a*a
 660                 GINAC_ASSERT(e.op(1).is_equal(_ex2));
 661                 v.push_back(e.op(0));
 662                 v.push_back(e.op(0));
 663         } else {
 664                 for (unsigned i=0; i<e.nops(); i++) {
 665                         ex f = e.op(i);
 666                         if (is_ex_exactly_of_type(f, power) && f.op(1).is_equal(_ex2)) {
 667                                 v.push_back(f.op(0));
 668                     v.push_back(f.op(0));
 669                         } else if (is_ex_exactly_of_type(f, ncmul)) {
 670                                 // Noncommutative factor found, split it as well
 671                                 non_commutative = true; // everything becomes noncommutative, ncmul will sort out the commutative factors later
 672                                 for (unsigned j=0; j<f.nops(); j++)
 673                                         v.push_back(f.op(j));
 674                         } else
 675                                 v.push_back(f);
 676                 }
 677         }
 678
 679         // Perform contractions
 680         bool something_changed = false;
 681         GINAC_ASSERT(v.size() > 1);
 682         exvector::iterator it1, itend = v.end(), next_to_last = itend - 1;
 683         for (it1 = v.begin(); it1 != next_to_last; it1++) {
 684
 685 try_again:
 686                 if (!is_ex_of_type(*it1, indexed))
 687                         continue;
 688
 689                 bool first_noncommutative = (it1->return_type() != return_types::commutative);
 690
 691                 // Indexed factor found, get free indices and look for contraction
 692                 // candidates
 693                 exvector free1, dummy1;
 694                 find_free_and_dummy(ex_to<indexed>(*it1).seq.begin() + 1, ex_to<indexed>(*it1).seq.end(), free1, dummy1);
 695
 696                 exvector::iterator it2;
 697                 for (it2 = it1 + 1; it2 != itend; it2++) {
 698
 699                         if (!is_ex_of_type(*it2, indexed))
 700                                 continue;
 701
 702                         bool second_noncommutative = (it2->return_type() != return_types::commutative);
 703
 704                         // Find free indices of second factor and merge them with free
 705                         // indices of first factor
 706                         exvector un;
 707                         find_free_and_dummy(ex_to<indexed>(*it2).seq.begin() + 1, ex_to<indexed>(*it2).seq.end(), un, dummy1);
 708                         un.insert(un.end(), free1.begin(), free1.end());
 709
 710                         // Check whether the two factors share dummy indices
 711                         exvector free, dummy;
 712                         find_free_and_dummy(un, free, dummy);
 713                         unsigned num_dummies = dummy.size();
 714                         if (num_dummies == 0)
 715                                 continue;
 716
 717                         // At least one dummy index, is it a defined scalar product?
 718                         bool contracted = false;
 719                         if (free.empty()) {
 720                                 if (sp.is_defined(*it1, *it2)) {
 721                                         *it1 = sp.evaluate(*it1, *it2);
 722                                         *it2 = _ex1;
 723                                         goto contraction_done;
 724                                 }
 725                         }
 726
 727                         // Try to contract the first one with the second one
 728                         contracted = ex_to<basic>(it1->op(0)).contract_with(it1, it2, v);
 729                         if (!contracted) {
 730
 731                                 // That didn't work; maybe the second object knows how to
 732                                 // contract itself with the first one
 733                                 contracted = ex_to<basic>(it2->op(0)).contract_with(it2, it1, v);
 734                         }
 735                         if (contracted) {
 736 contraction_done:
 737                                 if (first_noncommutative || second_noncommutative
 738                                  || is_ex_exactly_of_type(*it1, add) || is_ex_exactly_of_type(*it2, add)
 739                                  || is_ex_exactly_of_type(*it1, mul) || is_ex_exactly_of_type(*it2, mul)
 740                                  || is_ex_exactly_of_type(*it1, ncmul) || is_ex_exactly_of_type(*it2, ncmul)) {
 741
 742                                         // One of the factors became a sum or product:
 743                                         // re-expand expression and run again
 744                                         // Non-commutative products are always re-expanded to give
 745                                         // simplify_ncmul() the chance to re-order and canonicalize
 746                                         // the product
 747                                         ex r = (non_commutative ? ex(ncmul(v, true)) : ex(mul(v)));
 748                                         return simplify_indexed(r, free_indices, dummy_indices, sp);
 749                                 }
 750
 751                                 // Both objects may have new indices now or they might
 752                                 // even not be indexed objects any more, so we have to
 753                                 // start over
 754                                 something_changed = true;
 755                                 goto try_again;
 756                         }
 757                 }
 758         }
 759
 760         // Find free indices (concatenate them all and call find_free_and_dummy())
 761         // and all dummy indices that appear
 762         exvector un, individual_dummy_indices;
 763         for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 764                 exvector free_indices_of_factor;
 765                 if (is_ex_of_type(*it1, indexed)) {
 766                         exvector dummy_indices_of_factor;
 767                         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);
 768                         individual_dummy_indices.insert(individual_dummy_indices.end(), dummy_indices_of_factor.begin(), dummy_indices_of_factor.end());
 769                 } else
 770                         free_indices_of_factor = it1->get_free_indices();
 771                 un.insert(un.end(), free_indices_of_factor.begin(), free_indices_of_factor.end());
 772         }
 773         exvector local_dummy_indices;
 774         find_free_and_dummy(un, free_indices, local_dummy_indices);
 775         local_dummy_indices.insert(local_dummy_indices.end(), individual_dummy_indices.begin(), individual_dummy_indices.end());
 776
 777         // Filter out the dummy indices with variance
 778         exvector variant_dummy_indices;
 779         find_variant_indices(local_dummy_indices, variant_dummy_indices);
 780
 781         // Any indices with variance present at all?
 782         if (!variant_dummy_indices.empty()) {
 783
 784                 // Yes, bring the product into a canonical order that only depends on
 785                 // the base expressions of indexed objects
 786                 if (!non_commutative)
 787                         std::sort(v.begin(), v.end(), ex_base_is_less());
 788
 789                 exvector moved_indices;
 790
 791                 // Iterate over all indexed objects in the product
 792                 for (it1 = v.begin(), itend = v.end(); it1 != itend; ++it1) {
 793                         if (!is_ex_of_type(*it1, indexed))
 794                                 continue;
 795
 796                         if (reposition_dummy_indices(*it1, variant_dummy_indices, moved_indices))
 797                                 something_changed = true;
 798                 }
 799         }
 800
 801         ex r;
 802         if (something_changed)
 803                 r = non_commutative ? ex(ncmul(v, true)) : ex(mul(v));
 804         else
 805                 r = e;
 806
 807         // The result should be symmetric with respect to exchange of dummy
 808         // indices, so if the symmetrization vanishes, the whole expression is
 809         // zero. This detects things like eps.i.j.k * p.j * p.k = 0.
 810         if (local_dummy_indices.size() >= 2) {
 811                 lst dummy_syms;
 812                 for (int i=0; i<local_dummy_indices.size(); i++)
 813                         dummy_syms.append(local_dummy_indices[i].op(0));
 814                 if (r.symmetrize(dummy_syms).is_zero()) {
 815                         free_indices.clear();
 816                         return _ex0;
 817                 }
 818         }
 819
 820         // Dummy index renaming
 821         r = rename_dummy_indices(r, dummy_indices, local_dummy_indices);
 822
 823         // Product of indexed object with a scalar?
 824         if (is_ex_exactly_of_type(r, mul) && r.nops() == 2
 825          && is_ex_exactly_of_type(r.op(1), numeric) && is_ex_of_type(r.op(0), indexed))
 826                 return ex_to<basic>(r.op(0).op(0)).scalar_mul_indexed(r.op(0), ex_to<numeric>(r.op(1)));
 827         else
 828                 return r;
 829 }
 830
 831 /** This structure stores the original and symmetrized versions of terms
 832  *  obtained during the simplification of sums. */
 833 class terminfo {
 834 public:
 835         terminfo(const ex & orig_, const ex & symm_) : orig(orig_), symm(symm_) {}
 836
 837         ex orig; /**< original term */
 838         ex symm; /**< symmtrized term */
 839 };
 840
 841 class terminfo_is_less {
 842 public:
 843         bool operator() (const terminfo & ti1, const terminfo & ti2) const
 844         {
 845                 return (ti1.symm.compare(ti2.symm) < 0);
 846         }
 847 };
 848
 849 /** This structure stores the individual symmetrized terms obtained during
 850  *  the simplification of sums. */
 851 class symminfo {
 852 public:
 853         symminfo() : num(0) {}
 854
 855         symminfo(const ex & symmterm_, const ex & orig_, unsigned num_) : orig(orig_), num(num_)
 856         {
 857                 if (is_exactly_a<mul>(symmterm_) && is_exactly_a<numeric>(symmterm_.op(symmterm_.nops()-1))) {
 858                         coeff = symmterm_.op(symmterm_.nops()-1);
 859                         symmterm = symmterm_ / coeff;
 860                 } else {
 861                         coeff = 1;
 862                         symmterm = symmterm_;
 863                 }
 864         }
 865
 866         ex symmterm;  /**< symmetrized term */
 867         ex coeff;     /**< coefficient of symmetrized term */
 868         ex orig;      /**< original term */
 869         unsigned num; /**< how many symmetrized terms resulted from the original term */
 870 };
 871
 872 class symminfo_is_less_by_symmterm {
 873 public:
 874         bool operator() (const symminfo & si1, const symminfo & si2) const
 875         {
 876                 return (si1.symmterm.compare(si2.symmterm) < 0);
 877         }
 878 };
 879
 880 class symminfo_is_less_by_orig {
 881 public:
 882         bool operator() (const symminfo & si1, const symminfo & si2) const
 883         {
 884                 return (si1.orig.compare(si2.orig) < 0);
 885         }
 886 };
 887
 888 /** Simplify indexed expression, return list of free indices. */
 889 ex simplify_indexed(const ex & e, exvector & free_indices, exvector & dummy_indices, const scalar_products & sp)
 890 {
 891         // Expand the expression
 892         ex e_expanded = e.expand();
 893
 894         // Simplification of single indexed object: just find the free indices
 895         // and perform dummy index renaming/repositioning
 896         if (is_ex_of_type(e_expanded, indexed)) {
 897
 898                 // Find the dummy indices
 899                 const indexed &i = ex_to<indexed>(e_expanded);
 900                 exvector local_dummy_indices;
 901                 find_free_and_dummy(i.seq.begin() + 1, i.seq.end(), free_indices, local_dummy_indices);
 902
 903                 // Filter out the dummy indices with variance
 904                 exvector variant_dummy_indices;
 905                 find_variant_indices(local_dummy_indices, variant_dummy_indices);
 906
 907                 // Any indices with variance present at all?
 908                 if (!variant_dummy_indices.empty()) {
 909
 910                         // Yes, reposition them
 911                         exvector moved_indices;
 912                         reposition_dummy_indices(e_expanded, variant_dummy_indices, moved_indices);
 913                 }
 914
 915                 // Rename the dummy indices
 916                 return rename_dummy_indices(e_expanded, dummy_indices, local_dummy_indices);
 917         }
 918
 919         // Simplification of sum = sum of simplifications, check consistency of
 920         // free indices in each term
 921         if (is_ex_exactly_of_type(e_expanded, add)) {
 922                 bool first = true;
 923                 ex sum;
 924                 free_indices.clear();
 925
 926                 for (unsigned i=0; i<e_expanded.nops(); i++) {
 927                         exvector free_indices_of_term;
 928                         ex term = simplify_indexed(e_expanded.op(i), free_indices_of_term, dummy_indices, sp);
 929                         if (!term.is_zero()) {
 930                                 if (first) {
 931                                         free_indices = free_indices_of_term;
 932                                         sum = term;
 933                                         first = false;
 934                                 } else {
 935                                         if (!indices_consistent(free_indices, free_indices_of_term)) {
 936                                                 std::ostringstream s;
 937                                                 s << "simplify_indexed: inconsistent indices in sum: ";
 938                                                 s << exprseq(free_indices) << " vs. " << exprseq(free_indices_of_term);
 939                                                 throw (std::runtime_error(s.str()));
 940                                         }
 941                                         if (is_ex_of_type(sum, indexed) && is_ex_of_type(term, indexed))
 942                                                 sum = ex_to<basic>(sum.op(0)).add_indexed(sum, term);
 943                                         else
 944                                                 sum += term;
 945                                 }
 946                         }
 947                 }
 948
 949                 // If the sum turns out to be zero, we are finished
 950                 if (sum.is_zero()) {
 951                         free_indices.clear();
 952                         return sum;
 953                 }
 954
 955                 // More than one term and more than one dummy index?
 956                 int num_terms_orig = (is_exactly_a<add>(sum) ? sum.nops() : 1);
 957                 if (num_terms_orig < 2 || dummy_indices.size() < 2)
 958                         return sum;
 959
 960                 // Yes, construct list of all dummy index symbols
 961                 lst dummy_syms;
 962                 for (int i=0; i<dummy_indices.size(); i++)
 963                         dummy_syms.append(dummy_indices[i].op(0));
 964
 965                 // Chop the sum into terms and symmetrize each one over the dummy
 966                 // indices
 967                 std::vector<terminfo> terms;
 968                 for (unsigned i=0; i<sum.nops(); i++) {
 969                         const ex & term = sum.op(i);
 970                         ex term_symm = term.symmetrize(dummy_syms);
 971                         if (term_symm.is_zero())
 972                                 continue;
 973                         terms.push_back(terminfo(term, term_symm));
 974                 }
 975
 976                 // Sort by symmetrized terms
 977                 std::sort(terms.begin(), terms.end(), terminfo_is_less());
 978
 979                 // Combine equal symmetrized terms
 980                 std::vector<terminfo> terms_pass2;
 981                 for (std::vector<terminfo>::const_iterator i=terms.begin(); i!=terms.end(); ) {
 982                         unsigned num = 1;
 983                         std::vector<terminfo>::const_iterator j = i + 1;
 984                         while (j != terms.end() && j->symm == i->symm) {
 985                                 num++;
 986                                 j++;
 987                         }
 988                         terms_pass2.push_back(terminfo(i->orig * num, i->symm * num));
 989                         i = j;
 990                 }
 991
 992                 // If there is only one term left, we are finished
 993                 if (terms_pass2.size() == 1)
 994                         return terms_pass2[0].orig;
 995
 996                 // Chop the symmetrized terms into subterms
 997                 std::vector<symminfo> sy;
 998                 for (std::vector<terminfo>::const_iterator i=terms_pass2.begin(); i!=terms_pass2.end(); ++i) {
 999                         if (is_exactly_a<add>(i->symm)) {
1000                                 unsigned num = i->symm.nops();
1001                                 for (unsigned j=0; j<num; j++)
1002                                         sy.push_back(symminfo(i->symm.op(j), i->orig, num));
1003                         } else
1004                                 sy.push_back(symminfo(i->symm, i->orig, 1));
1005                 }
1006
1007                 // Sort by symmetrized subterms
1008                 std::sort(sy.begin(), sy.end(), symminfo_is_less_by_symmterm());
1009
1010                 // Combine equal symmetrized subterms
1011                 std::vector<symminfo> sy_pass2;
1012                 exvector result;
1013                 for (std::vector<symminfo>::const_iterator i=sy.begin(); i!=sy.end(); ) {
1014
1015                         // Combine equal terms
1016                         std::vector<symminfo>::const_iterator j = i + 1;
1017                         if (j != sy.end() && j->symmterm == i->symmterm) {
1018
1019                                 // More than one term, collect the coefficients
1020                                 ex coeff = i->coeff;
1021                                 while (j != sy.end() && j->symmterm == i->symmterm) {
1022                                         coeff += j->coeff;
1023                                         j++;
1024                                 }
1025
1026                                 // Add combined term to result
1027                                 if (!coeff.is_zero())
1028                                         result.push_back(coeff * i->symmterm);
1029
1030                         } else {
1031
1032                                 // Single term, store for second pass
1033                                 sy_pass2.push_back(*i);
1034                         }
1035
1036                         i = j;
1037                 }
1038
1039                 // Were there any remaining terms that didn't get combined?
1040                 if (sy_pass2.size() > 0) {
1041
1042                         // Yes, sort by their original terms
1043                         std::sort(sy_pass2.begin(), sy_pass2.end(), symminfo_is_less_by_orig());
1044
1045                         for (std::vector<symminfo>::const_iterator i=sy_pass2.begin(); i!=sy_pass2.end(); ) {
1046
1047                                 // How many symmetrized terms of this original term are left?
1048                                 unsigned num = 1;
1049                                 std::vector<symminfo>::const_iterator j = i + 1;
1050                                 while (j != sy_pass2.end() && j->orig == i->orig) {
1051                                         num++;
1052                                         j++;
1053                                 }
1054
1055                                 if (num == i->num) {
1056
1057                                         // All terms left, then add the original term to the result
1058                                         result.push_back(i->orig);
1059
1060                                 } else {
1061
1062                                         // Some terms were combined with others, add up the remaining symmetrized terms
1063                                         std::vector<symminfo>::const_iterator k;
1064                                         for (k=i; k!=j; k++)
1065                                                 result.push_back(k->coeff * k->symmterm);
1066                                 }
1067
1068                                 i = j;
1069                         }
1070                 }
1071
1072                 // Add all resulting terms
1073                 ex sum_symm = (new add(result))->setflag(status_flags::dynallocated);
1074                 if (sum_symm.is_zero())
1075                         free_indices.clear();
1076                 return sum_symm;
1077         }
1078
1079         // Simplification of products
1080         if (is_ex_exactly_of_type(e_expanded, mul)
1081          || is_ex_exactly_of_type(e_expanded, ncmul)
1082          || (is_ex_exactly_of_type(e_expanded, power) && is_ex_of_type(e_expanded.op(0), indexed) && e_expanded.op(1).is_equal(_ex2)))
1083                 return simplify_indexed_product(e_expanded, free_indices, dummy_indices, sp);
1084
1085         // Cannot do anything
1086         free_indices.clear();
1087         return e_expanded;
1088 }
1089
1090 /** Simplify/canonicalize expression containing indexed objects. This
1091  *  performs contraction of dummy indices where possible and checks whether
1092  *  the free indices in sums are consistent.
1093  *
1094  *  @return simplified expression */
1095 ex ex::simplify_indexed(void) const
1096 {
1097         exvector free_indices, dummy_indices;
1098         scalar_products sp;
1099         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1100 }
1101
1102 /** Simplify/canonicalize expression containing indexed objects. This
1103  *  performs contraction of dummy indices where possible, checks whether
1104  *  the free indices in sums are consistent, and automatically replaces
1105  *  scalar products by known values if desired.
1106  *
1107  *  @param sp Scalar products to be replaced automatically
1108  *  @return simplified expression */
1109 ex ex::simplify_indexed(const scalar_products & sp) const
1110 {
1111         exvector free_indices, dummy_indices;
1112         return GiNaC::simplify_indexed(*this, free_indices, dummy_indices, sp);
1113 }
1114
1115 /** Symmetrize expression over its free indices. */
1116 ex ex::symmetrize(void) const
1117 {
1118         return GiNaC::symmetrize(*this, get_free_indices());
1119 }
1120
1121 /** Antisymmetrize expression over its free indices. */
1122 ex ex::antisymmetrize(void) const
1123 {
1124         return GiNaC::antisymmetrize(*this, get_free_indices());
1125 }
1126
1127 /** Symmetrize expression by cyclic permutation over its free indices. */
1128 ex ex::symmetrize_cyclic(void) const
1129 {
1130         return GiNaC::symmetrize_cyclic(*this, get_free_indices());
1131 }
1132
1133 //////////
1134 // helper classes
1135 //////////
1136
1137 void scalar_products::add(const ex & v1, const ex & v2, const ex & sp)
1138 {
1139         spm[make_key(v1, v2)] = sp;
1140 }
1141
1142 void scalar_products::add_vectors(const lst & l)
1143 {
1144         // Add all possible pairs of products
1145         unsigned num = l.nops();
1146         for (unsigned i=0; i<num; i++) {
1147                 ex a = l.op(i);
1148                 for (unsigned j=0; j<num; j++) {
1149                         ex b = l.op(j);
1150                         add(a, b, a*b);
1151                 }
1152         }
1153 }
1154
1155 void scalar_products::clear(void)
1156 {
1157         spm.clear();
1158 }
1159
1160 /** Check whether scalar product pair is defined. */
1161 bool scalar_products::is_defined(const ex & v1, const ex & v2) const
1162 {
1163         return spm.find(make_key(v1, v2)) != spm.end();
1164 }
1165
1166 /** Return value of defined scalar product pair. */
1167 ex scalar_products::evaluate(const ex & v1, const ex & v2) const
1168 {
1169         return spm.find(make_key(v1, v2))->second;
1170 }
1171
1172 void scalar_products::debugprint(void) const
1173 {
1174         std::cerr << "map size=" << spm.size() << std::endl;
1175         spmap::const_iterator i = spm.begin(), end = spm.end();
1176         while (i != end) {
1177                 const spmapkey & k = i->first;
1178                 std::cerr << "item key=(" << k.first << "," << k.second;
1179                 std::cerr << "), value=" << i->second << std::endl;
1180                 ++i;
1181         }
1182 }
1183
1184 /** Make key from object pair. */
1185 spmapkey scalar_products::make_key(const ex & v1, const ex & v2)
1186 {
1187         // If indexed, extract base objects
1188         ex s1 = is_ex_of_type(v1, indexed) ? v1.op(0) : v1;
1189         ex s2 = is_ex_of_type(v2, indexed) ? v2.op(0) : v2;
1190
1191         // Enforce canonical order in pair
1192         if (s1.compare(s2) > 0)
1193                 return spmapkey(s2, s1);
1194         else
1195                 return spmapkey(s1, s2);
1196 }
1197
1198 } // namespace GiNaC