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) 1999-2001 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 <algorithm>
  24 #include <stdexcept>
  25
  26 #include "color.h"
  27 #include "idx.h"
  28 #include "ncmul.h"
  29 #include "symmetry.h"
  30 #include "numeric.h"
  31 #include "power.h" // for sqrt()
  32 #include "symbol.h"
  33 #include "print.h"
  34 #include "archive.h"
  35 #include "debugmsg.h"
  36 #include "utils.h"
  37
  38 namespace GiNaC {
  39
  40 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(su3one, tensor)
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(su3t, tensor)
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(su3f, tensor)
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(su3d, tensor)
  45
  46 //////////
  47 // default constructor, destructor, copy constructor assignment operator and helpers
  48 //////////
  49
  50 color::color() : representation_label(0)
  51 {
  52         debugmsg("color default constructor", LOGLEVEL_CONSTRUCT);
  53         tinfo_key = TINFO_color;
  54 }
  55
  56 void color::copy(const color & other)
  57 {
  58         inherited::copy(other);
  59         representation_label = other.representation_label;
  60 }
  61
  62 DEFAULT_DESTROY(color)
  63 DEFAULT_CTORS(su3one)
  64 DEFAULT_CTORS(su3t)
  65 DEFAULT_CTORS(su3f)
  66 DEFAULT_CTORS(su3d)
  67
  68 //////////
  69 // other constructors
  70 //////////
  71
  72 /** Construct object without any color index. This constructor is for
  73  *  internal use only. Use the color_ONE() function instead.
  74  *  @see color_ONE */
  75 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  76 {
  77         debugmsg("color constructor from ex,unsigned char", LOGLEVEL_CONSTRUCT);
  78         tinfo_key = TINFO_color;
  79 }
  80
  81 /** Construct object with one color index. This constructor is for internal
  82  *  use only. Use the color_T() function instead.
  83  *  @see color_T */
  84 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
  85 {
  86         debugmsg("color constructor from ex,ex,unsigned char", LOGLEVEL_CONSTRUCT);
  87         tinfo_key = TINFO_color;
  88 }
  89
  90 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
  91 {
  92         debugmsg("color constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
  93         tinfo_key = TINFO_color;
  94 }
  95
  96 color::color(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
  97 {
  98         debugmsg("color constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
  99         tinfo_key = TINFO_color;
 100 }
 101
 102 //////////
 103 // archiving
 104 //////////
 105
 106 color::color(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 107 {
 108         debugmsg("color constructor from archive_node", LOGLEVEL_CONSTRUCT);
 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 bases classes
 128 //////////
 129
 130 int color::compare_same_type(const basic & other) const
 131 {
 132         GINAC_ASSERT(other.tinfo() == TINFO_color);
 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 DEFAULT_COMPARE(su3one)
 144 DEFAULT_COMPARE(su3t)
 145 DEFAULT_COMPARE(su3f)
 146 DEFAULT_COMPARE(su3d)
 147
 148 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
 149 DEFAULT_PRINT(su3t, "T")
 150 DEFAULT_PRINT(su3f, "f")
 151 DEFAULT_PRINT(su3d, "d")
 152
 153 /** Perform automatic simplification on noncommutative product of color
 154  *  objects. This removes superfluous ONEs. */
 155 ex color::simplify_ncmul(const exvector & v) const
 156 {
 157         exvector s;
 158         s.reserve(v.size());
 159
 160         // Remove superfluous ONEs
 161         exvector::const_iterator it = v.begin(), itend = v.end();
 162         while (it != itend) {
 163                 if (!is_ex_of_type(it->op(0), su3one))
 164                         s.push_back(*it);
 165                 it++;
 166         }
 167
 168         if (s.size() == 0)
 169                 return color(su3one(), representation_label);
 170         else
 171                 return simplified_ncmul(s);
 172 }
 173
 174 ex color::thisexprseq(const exvector & v) const
 175 {
 176         return color(representation_label, v);
 177 }
 178
 179 ex color::thisexprseq(exvector * vp) const
 180 {
 181         return color(representation_label, vp);
 182 }
 183
 184 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
 185  *  is a subset of iv3, return the (free) index that is in iv3 but not in
 186  *  iv2 and the sign introduced by permuting that index to the front.
 187  *
 188  *  @param iv3 Vector of 3 indices
 189  *  @param iv2 Vector of 2 indices, must be a subset of iv3
 190  *  @param sig Returs sign introduced by index permutation
 191  *  @return the free index (the one that is in iv3 but not in iv2) */
 192 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
 193 {
 194         GINAC_ASSERT(iv3.size() == 3);
 195         GINAC_ASSERT(iv2.size() == 2);
 196
 197         sig = 1;
 198
 199 #define TEST_PERMUTATION(A,B,C,P) \
 200         if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
 201                 sig = P; \
 202                 return iv3[A]; \
 203         }
 204
 205         TEST_PERMUTATION(0,1,2,  1);
 206         TEST_PERMUTATION(0,2,1, -1);
 207         TEST_PERMUTATION(1,0,2, -1);
 208         TEST_PERMUTATION(1,2,0,  1);
 209         TEST_PERMUTATION(2,0,1,  1);
 210         TEST_PERMUTATION(2,1,0, -1);
 211
 212         throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
 213 }
 214
 215 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
 216 ex su3d::eval_indexed(const basic & i) const
 217 {
 218         GINAC_ASSERT(is_of_type(i, indexed));
 219         GINAC_ASSERT(i.nops() == 4);
 220         GINAC_ASSERT(is_ex_of_type(i.op(0), su3d));
 221
 222         // Convolutions are zero
 223         if (static_cast<const indexed &>(i).get_dummy_indices().size() != 0)
 224                 return _ex0();
 225
 226         // Numeric evaluation
 227         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 228
 229                 // Sort indices
 230                 int v[3];
 231                 for (unsigned j=0; j<3; j++)
 232                         v[j] = ex_to_numeric(ex_to_idx(i.op(j + 1)).get_value()).to_int();
 233                 if (v[0] > v[1]) std::swap(v[0], v[1]);
 234                 if (v[0] > v[2]) std::swap(v[0], v[2]);
 235                 if (v[1] > v[2]) std::swap(v[1], v[2]);
 236
 237 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
 238
 239                 // Check for non-zero elements
 240                 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
 241                  || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
 242                         return _ex1_2();
 243                 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
 244                         return _ex_1_2();
 245                 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
 246                         return sqrt(_ex3())/3;
 247                 else if (CMPINDICES(8,8,8))
 248                         return -sqrt(_ex3())/3;
 249                 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
 250                       || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
 251                         return -sqrt(_ex3())/6;
 252                 else
 253                         return _ex0();
 254         }
 255
 256         // No further simplifications
 257         return i.hold();
 258 }
 259
 260 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
 261 ex su3f::eval_indexed(const basic & i) const
 262 {
 263         GINAC_ASSERT(is_of_type(i, indexed));
 264         GINAC_ASSERT(i.nops() == 4);
 265         GINAC_ASSERT(is_ex_of_type(i.op(0), su3f));
 266
 267         // Numeric evaluation
 268         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 269
 270                 // Sort indices, remember permutation sign
 271                 int v[3];
 272                 for (unsigned j=0; j<3; j++)
 273                         v[j] = ex_to_numeric(ex_to_idx(i.op(j + 1)).get_value()).to_int();
 274                 int sign = 1;
 275                 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
 276                 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
 277                 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
 278
 279                 // Check for non-zero elements
 280                 if (CMPINDICES(1,2,3))
 281                         return sign;
 282                 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
 283                       || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
 284                         return _ex1_2() * sign;
 285                 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
 286                         return _ex_1_2() * sign;
 287                 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
 288                         return sqrt(_ex3())/2 * sign;
 289                 else
 290                         return _ex0();
 291         }
 292
 293         // No further simplifications
 294         return i.hold();
 295 }
 296
 297
 298 /** Contraction of generator with something else. */
 299 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 300 {
 301         GINAC_ASSERT(is_ex_of_type(*self, indexed));
 302         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 303         GINAC_ASSERT(self->nops() == 2);
 304         GINAC_ASSERT(is_ex_of_type(self->op(0), su3t));
 305         unsigned char rl = ex_to_color(*self).get_representation_label();
 306
 307         if (is_ex_exactly_of_type(other->op(0), su3t)) {
 308
 309                 // T.a T.a = 4/3 ONE
 310                 if (other - self == 1) {
 311                         *self = numeric(4, 3);
 312                         *other = color_ONE(rl);
 313                         return true;
 314
 315                 // T.a T.b T.a = -1/6 T.b
 316                 } else if (other - self == 2
 317                         && is_ex_of_type(self[1], color)) {
 318                         *self = numeric(-1, 6);
 319                         *other = _ex1();
 320                         return true;
 321
 322                 // T.a S T.a = 1/2 Tr(S) - 1/6 S
 323                 } else {
 324                         exvector::iterator it = self + 1;
 325                         while (it != other) {
 326                                 if (!is_ex_of_type(*it, color)) {
 327                                         return false;
 328                                 }
 329                                 it++;
 330                         }
 331
 332                         it = self + 1;
 333                         ex S = _ex1();
 334                         while (it != other) {
 335                                 S *= *it;
 336                                 *it++ = _ex1();
 337                         }
 338
 339                         *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
 340                         *other = _ex1();
 341                         return true;
 342                 }
 343         }
 344
 345         return false;
 346 }
 347
 348 /** Contraction of an indexed symmetric structure constant with something else. */
 349 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 350 {
 351         GINAC_ASSERT(is_ex_of_type(*self, indexed));
 352         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 353         GINAC_ASSERT(self->nops() == 4);
 354         GINAC_ASSERT(is_ex_of_type(self->op(0), su3d));
 355
 356         if (is_ex_exactly_of_type(other->op(0), su3d)) {
 357
 358                 // Find the dummy indices of the contraction
 359                 exvector self_indices = ex_to_indexed(*self).get_indices();
 360                 exvector other_indices = ex_to_indexed(*other).get_indices();
 361                 exvector all_indices = self_indices;
 362                 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
 363                 exvector free_indices, dummy_indices;
 364                 find_free_and_dummy(all_indices, free_indices, dummy_indices);
 365
 366                 // d.abc d.abc = 40/3
 367                 if (dummy_indices.size() == 3) {
 368                         *self = numeric(40, 3);
 369                         *other = _ex1();
 370                         return true;
 371
 372                 // d.akl d.bkl = 5/3 delta.ab
 373                 } else if (dummy_indices.size() == 2) {
 374                         exvector a;
 375                         std::back_insert_iterator<exvector> ita(a);
 376                         ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 377                         ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 378                         GINAC_ASSERT(a.size() == 2);
 379                         *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
 380                         *other = _ex1();
 381                         return true;
 382                 }
 383
 384         } else if (is_ex_exactly_of_type(other->op(0), su3t)) {
 385
 386                 // d.abc T.b T.c = 5/6 T.a
 387                 if (other+1 != v.end()
 388                  && is_ex_exactly_of_type(other[1].op(0), su3t)
 389                  && ex_to_indexed(*self).has_dummy_index_for(other[1].op(1))) {
 390
 391                         exvector self_indices = ex_to_indexed(*self).get_indices();
 392                         exvector dummy_indices;
 393                         dummy_indices.push_back(other[0].op(1));
 394                         dummy_indices.push_back(other[1].op(1));
 395                         int sig;
 396                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 397                         *self = numeric(5, 6);
 398                         other[0] = color_T(a, ex_to_color(other[0]).get_representation_label());
 399                         other[1] = _ex1();
 400                         return true;
 401                 }
 402         }
 403
 404         return false;
 405 }
 406
 407 /** Contraction of an indexed antisymmetric structure constant with something else. */
 408 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 409 {
 410         GINAC_ASSERT(is_ex_of_type(*self, indexed));
 411         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 412         GINAC_ASSERT(self->nops() == 4);
 413         GINAC_ASSERT(is_ex_of_type(self->op(0), su3f));
 414
 415         if (is_ex_exactly_of_type(other->op(0), su3f)) { // f*d is handled by su3d class
 416
 417                 // Find the dummy indices of the contraction
 418                 exvector dummy_indices;
 419                 dummy_indices = ex_to_indexed(*self).get_dummy_indices(ex_to_indexed(*other));
 420
 421                 // f.abc f.abc = 24
 422                 if (dummy_indices.size() == 3) {
 423                         *self = 24;
 424                         *other = _ex1();
 425                         return true;
 426
 427                 // f.akl f.bkl = 3 delta.ab
 428                 } else if (dummy_indices.size() == 2) {
 429                         int sign1, sign2;
 430                         ex a = permute_free_index_to_front(ex_to_indexed(*self).get_indices(), dummy_indices, sign1);
 431                         ex b = permute_free_index_to_front(ex_to_indexed(*other).get_indices(), dummy_indices, sign2);
 432                         *self = sign1 * sign2 * 3 * delta_tensor(a, b);
 433                         *other = _ex1();
 434                         return true;
 435                 }
 436
 437         } else if (is_ex_exactly_of_type(other->op(0), su3t)) {
 438
 439                 // f.abc T.b T.c = 3/2 I T.a
 440                 if (other+1 != v.end()
 441                  && is_ex_exactly_of_type(other[1].op(0), su3t)
 442                  && ex_to_indexed(*self).has_dummy_index_for(other[1].op(1))) {
 443
 444                         exvector self_indices = ex_to_indexed(*self).get_indices();
 445                         exvector dummy_indices;
 446                         dummy_indices.push_back(other[0].op(1));
 447                         dummy_indices.push_back(other[1].op(1));
 448                         int sig;
 449                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 450                         *self = numeric(3, 2) * sig * I;
 451                         other[0] = color_T(a, ex_to_color(other[0]).get_representation_label());
 452                         other[1] = _ex1();
 453                         return true;
 454                 }
 455         }
 456
 457         return false;
 458 }
 459
 460 //////////
 461 // global functions
 462 //////////
 463
 464 ex color_ONE(unsigned char rl)
 465 {
 466         return color(su3one(), rl);
 467 }
 468
 469 ex color_T(const ex & a, unsigned char rl)
 470 {
 471         if (!is_ex_of_type(a, idx))
 472                 throw(std::invalid_argument("indices of color_T must be of type idx"));
 473         if (!ex_to_idx(a).get_dim().is_equal(8))
 474                 throw(std::invalid_argument("index dimension for color_T must be 8"));
 475
 476         return color(su3t(), a, rl);
 477 }
 478
 479 ex color_f(const ex & a, const ex & b, const ex & c)
 480 {
 481         if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
 482                 throw(std::invalid_argument("indices of color_f must be of type idx"));
 483         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))
 484                 throw(std::invalid_argument("index dimension for color_f must be 8"));
 485
 486         return indexed(su3f(), sy_anti(), a, b, c);
 487 }
 488
 489 ex color_d(const ex & a, const ex & b, const ex & c)
 490 {
 491         if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
 492                 throw(std::invalid_argument("indices of color_d must be of type idx"));
 493         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))
 494                 throw(std::invalid_argument("index dimension for color_d must be 8"));
 495
 496         return indexed(su3d(), sy_symm(), a, b, c);
 497 }
 498
 499 ex color_h(const ex & a, const ex & b, const ex & c)
 500 {
 501         return color_d(a, b, c) + I * color_f(a, b, c);
 502 }
 503
 504 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 505  *  is that of a color object with the specified representation label. */
 506 static bool is_color_tinfo(unsigned ti, unsigned char rl)
 507 {
 508         return ti == (TINFO_color + rl);
 509 }
 510
 511 ex color_trace(const ex & e, unsigned char rl)
 512 {
 513         if (is_ex_of_type(e, color)) {
 514
 515                 if (ex_to_color(e).get_representation_label() == rl
 516                  && is_ex_of_type(e.op(0), su3one))
 517                         return _ex3();
 518                 else
 519                         return _ex0();
 520
 521         } else if (is_ex_exactly_of_type(e, mul)) {
 522
 523                 // Trace of product: pull out non-color factors
 524                 ex prod = _ex1();
 525                 for (unsigned i=0; i<e.nops(); i++) {
 526                         const ex &o = e.op(i);
 527                         if (is_color_tinfo(o.return_type_tinfo(), rl))
 528                                 prod *= color_trace(o, rl);
 529                         else
 530                                 prod *= o;
 531                 }
 532                 return prod;
 533
 534         } else if (is_ex_exactly_of_type(e, ncmul)) {
 535
 536                 if (!is_color_tinfo(e.return_type_tinfo(), rl))
 537                         return _ex0();
 538
 539                 // Expand product, if necessary
 540                 ex e_expanded = e.expand();
 541                 if (!is_ex_of_type(e_expanded, ncmul))
 542                         return color_trace(e_expanded, rl);
 543
 544                 unsigned num = e.nops();
 545
 546                 if (num == 2) {
 547
 548                         // Tr T_a T_b = 1/2 delta_a_b
 549                         return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
 550
 551                 } else if (num == 3) {
 552
 553                         // Tr T_a T_b T_c = 1/4 h_a_b_c
 554                         return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
 555
 556                 } else {
 557
 558                         // Traces of 4 or more generators are computed recursively:
 559                         // Tr T_a1 .. T_an =
 560                         //     1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
 561                         //   + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
 562                         const ex &last_index = e.op(num - 1).op(1);
 563                         const ex &next_to_last_index = e.op(num - 2).op(1);
 564                         idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
 565
 566                         exvector v1;
 567                         v1.reserve(num - 2);
 568                         for (int i=0; i<num-2; i++)
 569                                 v1.push_back(e.op(i));
 570
 571                         exvector v2 = v1;
 572                         v2.push_back(color_T(summation_index, rl));
 573
 574                         return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
 575                                + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
 576                 }
 577
 578         } else if (e.nops() > 0) {
 579
 580                 // Trace maps to all other container classes (this includes sums)
 581                 pointer_to_map_function_1arg<unsigned char> fcn(color_trace, rl);
 582                 return e.map(fcn);
 583
 584         } else
 585                 return _ex0();
 586 }
 587
 588 } // namespace GiNaC