ginac/color.cpp

   1 /** @file color.cpp
   2  *
   3  *  Implementation of GiNaC's color (SU(3) Lie algebra) objects. */
   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 <stdexcept>
  25
  26 #include "color.h"
  27 #include "idx.h"
  28 #include "ncmul.h"
  29 #include "symmetry.h"
  30 #include "operators.h"
  31 #include "numeric.h"
  32 #include "mul.h"
  33 #include "power.h" // for sqrt()
  34 #include "symbol.h"
  35 #include "print.h"
  36 #include "archive.h"
  37 #include "utils.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
  42
  43 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3one, tensor,
  44   print_func<print_dflt>(&su3one::do_print).
  45   print_func<print_latex>(&su3one::do_print_latex))
  46
  47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3t, tensor,
  48   print_func<print_dflt>(&su3t::do_print).
  49   print_func<print_latex>(&su3t::do_print))
  50
  51 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3f, tensor,
  52   print_func<print_dflt>(&su3f::do_print).
  53   print_func<print_latex>(&su3f::do_print))
  54
  55 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3d, tensor,
  56   print_func<print_dflt>(&su3d::do_print).
  57   print_func<print_latex>(&su3d::do_print))
  58
  59 //////////
  60 // default constructors
  61 //////////
  62
  63 color::color() : representation_label(0)
  64 {
  65         tinfo_key = TINFO_color;
  66 }
  67
  68 DEFAULT_CTOR(su3one)
  69 DEFAULT_CTOR(su3t)
  70 DEFAULT_CTOR(su3f)
  71 DEFAULT_CTOR(su3d)
  72
  73 //////////
  74 // other constructors
  75 //////////
  76
  77 /** Construct object without any color index. This constructor is for
  78  *  internal use only. Use the color_ONE() function instead.
  79  *  @see color_ONE */
  80 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  81 {
  82         tinfo_key = TINFO_color;
  83 }
  84
  85 /** Construct object with one color index. This constructor is for internal
  86  *  use only. Use the color_T() function instead.
  87  *  @see color_T */
  88 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
  89 {
  90         tinfo_key = TINFO_color;
  91 }
  92
  93 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
  94 {
  95         tinfo_key = TINFO_color;
  96 }
  97
  98 color::color(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
  99 {
 100         tinfo_key = TINFO_color;
 101 }
 102
 103 //////////
 104 // archiving
 105 //////////
 106
 107 color::color(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 108 {
 109         unsigned rl;
 110         n.find_unsigned("label", rl);
 111         representation_label = rl;
 112 }
 113
 114 void color::archive(archive_node &n) const
 115 {
 116         inherited::archive(n);
 117         n.add_unsigned("label", representation_label);
 118 }
 119
 120 DEFAULT_UNARCHIVE(color)
 121 DEFAULT_ARCHIVING(su3one)
 122 DEFAULT_ARCHIVING(su3t)
 123 DEFAULT_ARCHIVING(su3f)
 124 DEFAULT_ARCHIVING(su3d)
 125
 126 //////////
 127 // functions overriding virtual functions from base classes
 128 //////////
 129
 130 int color::compare_same_type(const basic & other) const
 131 {
 132         GINAC_ASSERT(is_a<color>(other));
 133         const color &o = static_cast<const color &>(other);
 134
 135         if (representation_label != o.representation_label) {
 136                 // different representation label
 137                 return representation_label < o.representation_label ? -1 : 1;
 138         }
 139
 140         return inherited::compare_same_type(other);
 141 }
 142
 143 bool color::match_same_type(const basic & other) const
 144 {
 145         GINAC_ASSERT(is_a<color>(other));
 146         const color &o = static_cast<const color &>(other);
 147
 148         return representation_label == o.representation_label;
 149 }
 150
 151 DEFAULT_COMPARE(su3one)
 152 DEFAULT_COMPARE(su3t)
 153 DEFAULT_COMPARE(su3f)
 154 DEFAULT_COMPARE(su3d)
 155
 156 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
 157 DEFAULT_PRINT(su3t, "T")
 158 DEFAULT_PRINT(su3f, "f")
 159 DEFAULT_PRINT(su3d, "d")
 160
 161 /** Perform automatic simplification on noncommutative product of color
 162  *  objects. This removes superfluous ONEs. */
 163 ex color::eval_ncmul(const exvector & v) const
 164 {
 165         exvector s;
 166         s.reserve(v.size());
 167
 168         // Remove superfluous ONEs
 169         exvector::const_iterator it = v.begin(), itend = v.end();
 170         while (it != itend) {
 171                 if (!is_a<su3one>(it->op(0)))
 172                         s.push_back(*it);
 173                 it++;
 174         }
 175
 176         if (s.empty())
 177                 return color(su3one(), representation_label);
 178         else
 179                 return hold_ncmul(s);
 180 }
 181
 182 ex color::thiscontainer(const exvector & v) const
 183 {
 184         return color(representation_label, v);
 185 }
 186
 187 ex color::thiscontainer(exvector * vp) const
 188 {
 189         return color(representation_label, vp);
 190 }
 191
 192 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
 193  *  is a subset of iv3, return the (free) index that is in iv3 but not in
 194  *  iv2 and the sign introduced by permuting that index to the front.
 195  *
 196  *  @param iv3 Vector of 3 indices
 197  *  @param iv2 Vector of 2 indices, must be a subset of iv3
 198  *  @param sig Returs sign introduced by index permutation
 199  *  @return the free index (the one that is in iv3 but not in iv2) */
 200 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
 201 {
 202         GINAC_ASSERT(iv3.size() == 3);
 203         GINAC_ASSERT(iv2.size() == 2);
 204
 205         sig = 1;
 206
 207 #define TEST_PERMUTATION(A,B,C,P) \
 208         if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
 209                 sig = P; \
 210                 return iv3[A]; \
 211         }
 212
 213         TEST_PERMUTATION(0,1,2,  1);
 214         TEST_PERMUTATION(0,2,1, -1);
 215         TEST_PERMUTATION(1,0,2, -1);
 216         TEST_PERMUTATION(1,2,0,  1);
 217         TEST_PERMUTATION(2,0,1,  1);
 218         TEST_PERMUTATION(2,1,0, -1);
 219
 220         throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
 221 }
 222
 223 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
 224 ex su3d::eval_indexed(const basic & i) const
 225 {
 226         GINAC_ASSERT(is_a<indexed>(i));
 227         GINAC_ASSERT(i.nops() == 4);
 228         GINAC_ASSERT(is_a<su3d>(i.op(0)));
 229
 230         // Convolutions are zero
 231         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 232                 return _ex0;
 233
 234         // Numeric evaluation
 235         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 236
 237                 // Sort indices
 238                 int v[3];
 239                 for (unsigned j=0; j<3; j++)
 240                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 241                 if (v[0] > v[1]) std::swap(v[0], v[1]);
 242                 if (v[0] > v[2]) std::swap(v[0], v[2]);
 243                 if (v[1] > v[2]) std::swap(v[1], v[2]);
 244
 245 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
 246
 247                 // Check for non-zero elements
 248                 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
 249                  || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
 250                         return _ex1_2;
 251                 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
 252                         return _ex_1_2;
 253                 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
 254                         return sqrt(_ex3)*_ex1_3;
 255                 else if (CMPINDICES(8,8,8))
 256                         return sqrt(_ex3)*_ex_1_3;
 257                 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
 258                       || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
 259                         return sqrt(_ex3)/_ex_6;
 260                 else
 261                         return _ex0;
 262         }
 263
 264         // No further simplifications
 265         return i.hold();
 266 }
 267
 268 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
 269 ex su3f::eval_indexed(const basic & i) const
 270 {
 271         GINAC_ASSERT(is_a<indexed>(i));
 272         GINAC_ASSERT(i.nops() == 4);
 273         GINAC_ASSERT(is_a<su3f>(i.op(0)));
 274
 275         // Numeric evaluation
 276         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 277
 278                 // Sort indices, remember permutation sign
 279                 int v[3];
 280                 for (unsigned j=0; j<3; j++)
 281                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 282                 int sign = 1;
 283                 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
 284                 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
 285                 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
 286
 287                 // Check for non-zero elements
 288                 if (CMPINDICES(1,2,3))
 289                         return sign;
 290                 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
 291                       || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
 292                         return _ex1_2 * sign;
 293                 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
 294                         return _ex_1_2 * sign;
 295                 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
 296                         return sqrt(_ex3)/2 * sign;
 297                 else
 298                         return _ex0;
 299         }
 300
 301         // No further simplifications
 302         return i.hold();
 303 }
 304
 305
 306 /** Contraction of generator with something else. */
 307 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 308 {
 309         GINAC_ASSERT(is_a<indexed>(*self));
 310         GINAC_ASSERT(is_a<indexed>(*other));
 311         GINAC_ASSERT(self->nops() == 2);
 312         GINAC_ASSERT(is_a<su3t>(self->op(0)));
 313         unsigned char rl = ex_to<color>(*self).get_representation_label();
 314
 315         if (is_exactly_a<su3t>(other->op(0))) {
 316
 317                 // Contraction only makes sense if the represenation labels are equal
 318                 GINAC_ASSERT(is_a<color>(*other));
 319                 if (ex_to<color>(*other).get_representation_label() != rl)
 320                         return false;
 321
 322                 // T.a T.a = 4/3 ONE
 323                 if (other - self == 1) {
 324                         *self = numeric(4, 3);
 325                         *other = color_ONE(rl);
 326                         return true;
 327
 328                 // T.a T.b T.a = -1/6 T.b
 329                 } else if (other - self == 2
 330                         && is_a<color>(self[1])) {
 331                         *self = numeric(-1, 6);
 332                         *other = _ex1;
 333                         return true;
 334
 335                 // T.a S T.a = 1/2 Tr(S) - 1/6 S
 336                 } else {
 337                         exvector::iterator it = self + 1;
 338                         while (it != other) {
 339                                 if (!is_a<color>(*it)) {
 340                                         return false;
 341                                 }
 342                                 it++;
 343                         }
 344
 345                         it = self + 1;
 346                         ex S = _ex1;
 347                         while (it != other) {
 348                                 S *= *it;
 349                                 *it++ = _ex1;
 350                         }
 351
 352                         *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
 353                         *other = _ex1;
 354                         return true;
 355                 }
 356         }
 357
 358         return false;
 359 }
 360
 361 /** Contraction of an indexed symmetric structure constant with something else. */
 362 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 363 {
 364         GINAC_ASSERT(is_a<indexed>(*self));
 365         GINAC_ASSERT(is_a<indexed>(*other));
 366         GINAC_ASSERT(self->nops() == 4);
 367         GINAC_ASSERT(is_a<su3d>(self->op(0)));
 368
 369         if (is_exactly_a<su3d>(other->op(0))) {
 370
 371                 // Find the dummy indices of the contraction
 372                 exvector self_indices = ex_to<indexed>(*self).get_indices();
 373                 exvector other_indices = ex_to<indexed>(*other).get_indices();
 374                 exvector all_indices = self_indices;
 375                 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
 376                 exvector free_indices, dummy_indices;
 377                 find_free_and_dummy(all_indices, free_indices, dummy_indices);
 378
 379                 // d.abc d.abc = 40/3
 380                 if (dummy_indices.size() == 3) {
 381                         *self = numeric(40, 3);
 382                         *other = _ex1;
 383                         return true;
 384
 385                 // d.akl d.bkl = 5/3 delta.ab
 386                 } else if (dummy_indices.size() == 2) {
 387                         exvector a;
 388                         std::back_insert_iterator<exvector> ita(a);
 389                         ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 390                         ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 391                         GINAC_ASSERT(a.size() == 2);
 392                         *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
 393                         *other = _ex1;
 394                         return true;
 395                 }
 396
 397         } else if (is_exactly_a<su3t>(other->op(0))) {
 398
 399                 // d.abc T.b T.c = 5/6 T.a
 400                 if (other+1 != v.end()
 401                  && is_exactly_a<su3t>(other[1].op(0))
 402                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 403
 404                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 405                         exvector dummy_indices;
 406                         dummy_indices.push_back(other[0].op(1));
 407                         dummy_indices.push_back(other[1].op(1));
 408                         int sig;
 409                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 410                         *self = numeric(5, 6);
 411                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 412                         other[1] = _ex1;
 413                         return true;
 414                 }
 415         }
 416
 417         return false;
 418 }
 419
 420 /** Contraction of an indexed antisymmetric structure constant with something else. */
 421 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 422 {
 423         GINAC_ASSERT(is_a<indexed>(*self));
 424         GINAC_ASSERT(is_a<indexed>(*other));
 425         GINAC_ASSERT(self->nops() == 4);
 426         GINAC_ASSERT(is_a<su3f>(self->op(0)));
 427
 428         if (is_exactly_a<su3f>(other->op(0))) { // f*d is handled by su3d class
 429
 430                 // Find the dummy indices of the contraction
 431                 exvector dummy_indices;
 432                 dummy_indices = ex_to<indexed>(*self).get_dummy_indices(ex_to<indexed>(*other));
 433
 434                 // f.abc f.abc = 24
 435                 if (dummy_indices.size() == 3) {
 436                         *self = 24;
 437                         *other = _ex1;
 438                         return true;
 439
 440                 // f.akl f.bkl = 3 delta.ab
 441                 } else if (dummy_indices.size() == 2) {
 442                         int sign1, sign2;
 443                         ex a = permute_free_index_to_front(ex_to<indexed>(*self).get_indices(), dummy_indices, sign1);
 444                         ex b = permute_free_index_to_front(ex_to<indexed>(*other).get_indices(), dummy_indices, sign2);
 445                         *self = sign1 * sign2 * 3 * delta_tensor(a, b);
 446                         *other = _ex1;
 447                         return true;
 448                 }
 449
 450         } else if (is_exactly_a<su3t>(other->op(0))) {
 451
 452                 // f.abc T.b T.c = 3/2 I T.a
 453                 if (other+1 != v.end()
 454                  && is_exactly_a<su3t>(other[1].op(0))
 455                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 456
 457                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 458                         exvector dummy_indices;
 459                         dummy_indices.push_back(other[0].op(1));
 460                         dummy_indices.push_back(other[1].op(1));
 461                         int sig;
 462                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 463                         *self = numeric(3, 2) * sig * I;
 464                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 465                         other[1] = _ex1;
 466                         return true;
 467                 }
 468         }
 469
 470         return false;
 471 }
 472
 473 //////////
 474 // global functions
 475 //////////
 476
 477 ex color_ONE(unsigned char rl)
 478 {
 479         return color(su3one(), rl);
 480 }
 481
 482 ex color_T(const ex & a, unsigned char rl)
 483 {
 484         if (!is_a<idx>(a))
 485                 throw(std::invalid_argument("indices of color_T must be of type idx"));
 486         if (!ex_to<idx>(a).get_dim().is_equal(8))
 487                 throw(std::invalid_argument("index dimension for color_T must be 8"));
 488
 489         return color(su3t(), a, rl);
 490 }
 491
 492 ex color_f(const ex & a, const ex & b, const ex & c)
 493 {
 494         if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
 495                 throw(std::invalid_argument("indices of color_f must be of type idx"));
 496         if (!ex_to<idx>(a).get_dim().is_equal(8) || !ex_to<idx>(b).get_dim().is_equal(8) || !ex_to<idx>(c).get_dim().is_equal(8))
 497                 throw(std::invalid_argument("index dimension for color_f must be 8"));
 498
 499         return indexed(su3f(), sy_anti(), a, b, c);
 500 }
 501
 502 ex color_d(const ex & a, const ex & b, const ex & c)
 503 {
 504         if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
 505                 throw(std::invalid_argument("indices of color_d must be of type idx"));
 506         if (!ex_to<idx>(a).get_dim().is_equal(8) || !ex_to<idx>(b).get_dim().is_equal(8) || !ex_to<idx>(c).get_dim().is_equal(8))
 507                 throw(std::invalid_argument("index dimension for color_d must be 8"));
 508
 509         return indexed(su3d(), sy_symm(), a, b, c);
 510 }
 511
 512 ex color_h(const ex & a, const ex & b, const ex & c)
 513 {
 514         return color_d(a, b, c) + I * color_f(a, b, c);
 515 }
 516
 517 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 518  *  is that of a color object with the specified representation label. */
 519 static bool is_color_tinfo(unsigned ti, unsigned char rl)
 520 {
 521         return ti == (TINFO_color + rl);
 522 }
 523
 524 ex color_trace(const ex & e, unsigned char rl)
 525 {
 526         if (is_a<color>(e)) {
 527
 528                 if (ex_to<color>(e).get_representation_label() == rl
 529                  && is_a<su3one>(e.op(0)))
 530                         return _ex3;
 531                 else
 532                         return _ex0;
 533
 534         } else if (is_exactly_a<mul>(e)) {
 535
 536                 // Trace of product: pull out non-color factors
 537                 ex prod = _ex1;
 538                 for (size_t i=0; i<e.nops(); i++) {
 539                         const ex &o = e.op(i);
 540                         if (is_color_tinfo(o.return_type_tinfo(), rl))
 541                                 prod *= color_trace(o, rl);
 542                         else
 543                                 prod *= o;
 544                 }
 545                 return prod;
 546
 547         } else if (is_exactly_a<ncmul>(e)) {
 548
 549                 if (!is_color_tinfo(e.return_type_tinfo(), rl))
 550                         return _ex0;
 551
 552                 // Expand product, if necessary
 553                 ex e_expanded = e.expand();
 554                 if (!is_a<ncmul>(e_expanded))
 555                         return color_trace(e_expanded, rl);
 556
 557                 size_t num = e.nops();
 558
 559                 if (num == 2) {
 560
 561                         // Tr T_a T_b = 1/2 delta_a_b
 562                         return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
 563
 564                 } else if (num == 3) {
 565
 566                         // Tr T_a T_b T_c = 1/4 h_a_b_c
 567                         return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
 568
 569                 } else {
 570
 571                         // Traces of 4 or more generators are computed recursively:
 572                         // Tr T_a1 .. T_an =
 573                         //     1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
 574                         //   + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
 575                         const ex &last_index = e.op(num - 1).op(1);
 576                         const ex &next_to_last_index = e.op(num - 2).op(1);
 577                         idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
 578
 579                         exvector v1;
 580                         v1.reserve(num - 2);
 581                         for (size_t i=0; i<num-2; i++)
 582                                 v1.push_back(e.op(i));
 583
 584                         exvector v2 = v1;
 585                         v2.push_back(color_T(summation_index, rl));
 586
 587                         return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
 588                                + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
 589                 }
 590
 591         } else if (e.nops() > 0) {
 592
 593                 // Trace maps to all other container classes (this includes sums)
 594                 pointer_to_map_function_1arg<unsigned char> fcn(color_trace, rl);
 595                 return e.map(fcn);
 596
 597         } else
 598                 return _ex0;
 599 }
 600
 601 } // namespace GiNaC