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