ginac/inifcns_nstdsums.cpp

   1 /** @file inifcns_nstdsums.cpp
   2  *
   3  *  Implementation of some special functions that have a representation as nested sums.
   4  *
   5  *  The functions are:
   6  *    classical polylogarithm              Li(n,x)
   7  *    multiple polylogarithm               Li(lst(m_1,...,m_k),lst(x_1,...,x_k))
   8  *                                         G(lst(a_1,...,a_k),y) or G(lst(a_1,...,a_k),lst(s_1,...,s_k),y)
   9  *    Nielsen's generalized polylogarithm  S(n,p,x)
  10  *    harmonic polylogarithm               H(m,x) or H(lst(m_1,...,m_k),x)
  11  *    multiple zeta value                  zeta(m) or zeta(lst(m_1,...,m_k))
  12  *    alternating Euler sum                zeta(m,s) or zeta(lst(m_1,...,m_k),lst(s_1,...,s_k))
  13  *
  14  *  Some remarks:
  15  *
  16  *    - All formulae used can be looked up in the following publications:
  17  *      [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
  18  *      [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
  19  *      [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
  20  *      [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
  21  *      [VSW] Numerical evaluation of multiple polylogarithms, J.Vollinga, S.Weinzierl, hep-ph/0410259
  22  *
  23  *    - The order of parameters and arguments of Li and zeta is defined according to the nested sums
  24  *      representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
  25  *      0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single
  26  *      number --- notation.
  27  *
  28  *    - All functions can be nummerically evaluated with arguments in the whole complex plane. The parameters
  29  *      for Li, zeta and S must be positive integers. If you want to have an alternating Euler sum, you have
  30  *      to give the signs of the parameters as a second argument s to zeta(m,s) containing 1 and -1.
  31  *
  32  *    - The calculation of classical polylogarithms is speeded up by using Bernoulli numbers and
  33  *      look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in
  34  *      [Cra] and [BBB] for speed up. Multiple polylogarithms use Hoelder convolution [BBB].
  35  *
  36  *    - The functions have no means to do a series expansion into nested sums. To do this, you have to convert
  37  *      these functions into the appropriate objects from the nestedsums library, do the expansion and convert
  38  *      the result back.
  39  *
  40  *    - Numerical testing of this implementation has been performed by doing a comparison of results
  41  *      between this software and the commercial M.......... 4.1. Multiple zeta values have been checked
  42  *      by means of evaluations into simple zeta values. Harmonic polylogarithms have been checked by
  43  *      comparison to S(n,p,x) for corresponding parameter combinations and by continuity checks
  44  *      around |x|=1 along with comparisons to corresponding zeta functions. Multiple polylogarithms were
  45  *      checked against H and zeta and by means of shuffle and quasi-shuffle relations.
  46  *
  47  */
  48
  49 /*
  50  *  GiNaC Copyright (C) 1999-2011 Johannes Gutenberg University Mainz, Germany
  51  *
  52  *  This program is free software; you can redistribute it and/or modify
  53  *  it under the terms of the GNU General Public License as published by
  54  *  the Free Software Foundation; either version 2 of the License, or
  55  *  (at your option) any later version.
  56  *
  57  *  This program is distributed in the hope that it will be useful,
  58  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  59  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  60  *  GNU General Public License for more details.
  61  *
  62  *  You should have received a copy of the GNU General Public License
  63  *  along with this program; if not, write to the Free Software
  64  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  65  */
  66
  67 #include "inifcns.h"
  68
  69 #include "add.h"
  70 #include "constant.h"
  71 #include "lst.h"
  72 #include "mul.h"
  73 #include "numeric.h"
  74 #include "operators.h"
  75 #include "power.h"
  76 #include "pseries.h"
  77 #include "relational.h"
  78 #include "symbol.h"
  79 #include "utils.h"
  80 #include "wildcard.h"
  81
  82 #include <cln/cln.h>
  83 #include <sstream>
  84 #include <stdexcept>
  85 #include <vector>
  86
  87 namespace GiNaC {
  88
  89
  90 //////////////////////////////////////////////////////////////////////
  91 //
  92 // Classical polylogarithm  Li(n,x)
  93 //
  94 // helper functions
  95 //
  96 //////////////////////////////////////////////////////////////////////
  97
  98
  99 // anonymous namespace for helper functions
 100 namespace {
 101
 102
 103 // lookup table for factors built from Bernoulli numbers
 104 // see fill_Xn()
 105 std::vector<std::vector<cln::cl_N> > Xn;
 106 // initial size of Xn that should suffice for 32bit machines (must be even)
 107 const int xninitsizestep = 26;
 108 int xninitsize = xninitsizestep;
 109 int xnsize = 0;
 110
 111
 112 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
 113 // With these numbers the polylogs can be calculated as follows:
 114 //   Li_p (x)  =  \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with  u = -log(1-x)
 115 //   X_0(n) = B_n (Bernoulli numbers)
 116 //   X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
 117 // The calculation of Xn depends on X0 and X{n-1}.
 118 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
 119 // This results in a slightly more complicated algorithm for the X_n.
 120 // The first index in Xn corresponds to the index of the polylog minus 2.
 121 // The second index in Xn corresponds to the index from the actual sum.
 122 void fill_Xn(int n)
 123 {
 124         if (n>1) {
 125                 // calculate X_2 and higher (corresponding to Li_4 and higher)
 126                 std::vector<cln::cl_N> buf(xninitsize);
 127                 std::vector<cln::cl_N>::iterator it = buf.begin();
 128                 cln::cl_N result;
 129                 *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
 130                 it++;
 131                 for (int i=2; i<=xninitsize; i++) {
 132                         if (i&1) {
 133                                 result = 0; // k == 0
 134                         } else {
 135                                 result = Xn[0][i/2-1]; // k == 0
 136                         }
 137                         for (int k=1; k<i-1; k++) {
 138                                 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
 139                                         result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
 140                                 }
 141                         }
 142                         result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
 143                         result = result + Xn[n-1][i-1] / (i+1); // k == i
 144
 145                         *it = result;
 146                         it++;
 147                 }
 148                 Xn.push_back(buf);
 149         } else if (n==1) {
 150                 // special case to handle the X_0 correct
 151                 std::vector<cln::cl_N> buf(xninitsize);
 152                 std::vector<cln::cl_N>::iterator it = buf.begin();
 153                 cln::cl_N result;
 154                 *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
 155                 it++;
 156                 *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
 157                 it++;
 158                 for (int i=3; i<=xninitsize; i++) {
 159                         if (i & 1) {
 160                                 result = -Xn[0][(i-3)/2]/2;
 161                                 *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
 162                                 it++;
 163                         } else {
 164                                 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
 165                                 for (int k=1; k<i/2; k++) {
 166                                         result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
 167                                 }
 168                                 *it = result;
 169                                 it++;
 170                         }
 171                 }
 172                 Xn.push_back(buf);
 173         } else {
 174                 // calculate X_0
 175                 std::vector<cln::cl_N> buf(xninitsize/2);
 176                 std::vector<cln::cl_N>::iterator it = buf.begin();
 177                 for (int i=1; i<=xninitsize/2; i++) {
 178                         *it = bernoulli(i*2).to_cl_N();
 179                         it++;
 180                 }
 181                 Xn.push_back(buf);
 182         }
 183
 184         xnsize++;
 185 }
 186
 187 // doubles the number of entries in each Xn[]
 188 void double_Xn()
 189 {
 190         const int pos0 = xninitsize / 2;
 191         // X_0
 192         for (int i=1; i<=xninitsizestep/2; ++i) {
 193                 Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N());
 194         }
 195         if (Xn.size() > 1) {
 196                 int xend = xninitsize + xninitsizestep;
 197                 cln::cl_N result;
 198                 // X_1
 199                 for (int i=xninitsize+1; i<=xend; ++i) {
 200                         if (i & 1) {
 201                                 result = -Xn[0][(i-3)/2]/2;
 202                                 Xn[1].push_back((cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result);
 203                         } else {
 204                                 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
 205                                 for (int k=1; k<i/2; k++) {
 206                                         result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
 207                                 }
 208                                 Xn[1].push_back(result);
 209                         }
 210                 }
 211                 // X_n
 212                 for (size_t n=2; n<Xn.size(); ++n) {
 213                         for (int i=xninitsize+1; i<=xend; ++i) {
 214                                 if (i & 1) {
 215                                         result = 0; // k == 0
 216                                 } else {
 217                                         result = Xn[0][i/2-1]; // k == 0
 218                                 }
 219                                 for (int k=1; k<i-1; ++k) {
 220                                         if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
 221                                                 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
 222                                         }
 223                                 }
 224                                 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
 225                                 result = result + Xn[n-1][i-1] / (i+1); // k == i
 226                                 Xn[n].push_back(result);
 227                         }
 228                 }
 229         }
 230         xninitsize += xninitsizestep;
 231 }
 232
 233
 234 // calculates Li(2,x) without Xn
 235 cln::cl_N Li2_do_sum(const cln::cl_N& x)
 236 {
 237         cln::cl_N res = x;
 238         cln::cl_N resbuf;
 239         cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
 240         cln::cl_I den = 1; // n^2 = 1
 241         unsigned i = 3;
 242         do {
 243                 resbuf = res;
 244                 num = num * x;
 245                 den = den + i;  // n^2 = 4, 9, 16, ...
 246                 i += 2;
 247                 res = res + num / den;
 248         } while (res != resbuf);
 249         return res;
 250 }
 251
 252
 253 // calculates Li(2,x) with Xn
 254 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
 255 {
 256         std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
 257         std::vector<cln::cl_N>::const_iterator xend = Xn[0].end();
 258         cln::cl_N u = -cln::log(1-x);
 259         cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
 260         cln::cl_N uu = cln::square(u);
 261         cln::cl_N res = u - uu/4;
 262         cln::cl_N resbuf;
 263         unsigned i = 1;
 264         do {
 265                 resbuf = res;
 266                 factor = factor * uu / (2*i * (2*i+1));
 267                 res = res + (*it) * factor;
 268                 i++;
 269                 if (++it == xend) {
 270                         double_Xn();
 271                         it = Xn[0].begin() + (i-1);
 272                         xend = Xn[0].end();
 273                 }
 274         } while (res != resbuf);
 275         return res;
 276 }
 277
 278
 279 // calculates Li(n,x), n>2 without Xn
 280 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
 281 {
 282         cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits));
 283         cln::cl_N res = x;
 284         cln::cl_N resbuf;
 285         int i=2;
 286         do {
 287                 resbuf = res;
 288                 factor = factor * x;
 289                 res = res + factor / cln::expt(cln::cl_I(i),n);
 290                 i++;
 291         } while (res != resbuf);
 292         return res;
 293 }
 294
 295
 296 // calculates Li(n,x), n>2 with Xn
 297 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
 298 {
 299         std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
 300         std::vector<cln::cl_N>::const_iterator xend = Xn[n-2].end();
 301         cln::cl_N u = -cln::log(1-x);
 302         cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
 303         cln::cl_N res = u;
 304         cln::cl_N resbuf;
 305         unsigned i=2;
 306         do {
 307                 resbuf = res;
 308                 factor = factor * u / i;
 309                 res = res + (*it) * factor;
 310                 i++;
 311                 if (++it == xend) {
 312                         double_Xn();
 313                         it = Xn[n-2].begin() + (i-2);
 314                         xend = Xn[n-2].end();
 315                 }
 316         } while (res != resbuf);
 317         return res;
 318 }
 319
 320
 321 // forward declaration needed by function Li_projection and C below
 322 const cln::cl_N S_num(int n, int p, const cln::cl_N& x);
 323
 324
 325 // helper function for classical polylog Li
 326 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
 327 {
 328         // treat n=2 as special case
 329         if (n == 2) {
 330                 // check if precalculated X0 exists
 331                 if (xnsize == 0) {
 332                         fill_Xn(0);
 333                 }
 334
 335                 if (cln::realpart(x) < 0.5) {
 336                         // choose the faster algorithm
 337                         // the switching point was empirically determined. the optimal point
 338                         // depends on hardware, Digits, ... so an approx value is okay.
 339                         // it solves also the problem with precision due to the u=-log(1-x) transformation
 340                         if (cln::abs(cln::realpart(x)) < 0.25) {
 341
 342                                 return Li2_do_sum(x);
 343                         } else {
 344                                 return Li2_do_sum_Xn(x);
 345                         }
 346                 } else {
 347                         // choose the faster algorithm
 348                         if (cln::abs(cln::realpart(x)) > 0.75) {
 349                                 if ( x == 1 ) {
 350                                         return cln::zeta(2);
 351                                 } else {
 352                                         return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
 353                                 }
 354                         } else {
 355                                 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
 356                         }
 357                 }
 358         } else {
 359                 // check if precalculated Xn exist
 360                 if (n > xnsize+1) {
 361                         for (int i=xnsize; i<n-1; i++) {
 362                                 fill_Xn(i);
 363                         }
 364                 }
 365
 366                 if (cln::realpart(x) < 0.5) {
 367                         // choose the faster algorithm
 368                         // with n>=12 the "normal" summation always wins against the method with Xn
 369                         if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
 370                                 return Lin_do_sum(n, x);
 371                         } else {
 372                                 return Lin_do_sum_Xn(n, x);
 373                         }
 374                 } else {
 375                         cln::cl_N result = 0;
 376                         if ( x != 1 ) result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
 377                         for (int j=0; j<n-1; j++) {
 378                                 result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
 379                                                   * cln::expt(cln::log(x), j) / cln::factorial(j);
 380                         }
 381                         return result;
 382                 }
 383         }
 384 }
 385
 386 // helper function for classical polylog Li
 387 const cln::cl_N Lin_numeric(const int n, const cln::cl_N& x)
 388 {
 389         if (n == 1) {
 390                 // just a log
 391                 return -cln::log(1-x);
 392         }
 393         if (zerop(x)) {
 394                 return 0;
 395         }
 396         if (x == 1) {
 397                 // [Kol] (2.22)
 398                 return cln::zeta(n);
 399         }
 400         else if (x == -1) {
 401                 // [Kol] (2.22)
 402                 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
 403         }
 404         if (cln::abs(realpart(x)) < 0.4 && cln::abs(cln::abs(x)-1) < 0.01) {
 405                 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
 406                 for (int j=0; j<n-1; j++) {
 407                         result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
 408                                 * cln::expt(cln::log(x), j) / cln::factorial(j);
 409                 }
 410                 return result;
 411         }
 412
 413         // what is the desired float format?
 414         // first guess: default format
 415         cln::float_format_t prec = cln::default_float_format;
 416         const cln::cl_N value = x;
 417         // second guess: the argument's format
 418         if (!instanceof(realpart(x), cln::cl_RA_ring))
 419                 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
 420         else if (!instanceof(imagpart(x), cln::cl_RA_ring))
 421                 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
 422
 423         // [Kol] (5.15)
 424         if (cln::abs(value) > 1) {
 425                 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
 426                 // check if argument is complex. if it is real, the new polylog has to be conjugated.
 427                 if (cln::zerop(cln::imagpart(value))) {
 428                         if (n & 1) {
 429                                 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
 430                         }
 431                         else {
 432                                 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
 433                         }
 434                 }
 435                 else {
 436                         if (n & 1) {
 437                                 result = result + Li_projection(n, cln::recip(value), prec);
 438                         }
 439                         else {
 440                                 result = result - Li_projection(n, cln::recip(value), prec);
 441                         }
 442                 }
 443                 cln::cl_N add;
 444                 for (int j=0; j<n-1; j++) {
 445                         add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
 446                                     * Lin_numeric(n-j,1) * cln::expt(cln::log(-value),j) / cln::factorial(j);
 447                 }
 448                 result = result - add;
 449                 return result;
 450         }
 451         else {
 452                 return Li_projection(n, value, prec);
 453         }
 454 }
 455
 456
 457 } // end of anonymous namespace
 458
 459
 460 //////////////////////////////////////////////////////////////////////
 461 //
 462 // Multiple polylogarithm  Li(n,x)
 463 //
 464 // helper function
 465 //
 466 //////////////////////////////////////////////////////////////////////
 467
 468
 469 // anonymous namespace for helper function
 470 namespace {
 471
 472
 473 // performs the actual series summation for multiple polylogarithms
 474 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
 475 {
 476         // ensure all x <> 0.
 477         for (std::vector<cln::cl_N>::const_iterator it = x.begin(); it != x.end(); ++it) {
 478                 if ( *it == 0 ) return cln::cl_float(0, cln::float_format(Digits));
 479         }
 480
 481         const int j = s.size();
 482         bool flag_accidental_zero = false;
 483
 484         std::vector<cln::cl_N> t(j);
 485         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
 486
 487         cln::cl_N t0buf;
 488         int q = 0;
 489         do {
 490                 t0buf = t[0];
 491                 q++;
 492                 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
 493                 for (int k=j-2; k>=0; k--) {
 494                         t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
 495                 }
 496                 q++;
 497                 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
 498                 for (int k=j-2; k>=0; k--) {
 499                         flag_accidental_zero = cln::zerop(t[k+1]);
 500                         t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
 501                 }
 502         } while ( (t[0] != t0buf) || cln::zerop(t[0]) || flag_accidental_zero );
 503
 504         return t[0];
 505 }
 506
 507
 508 // forward declaration for Li_eval()
 509 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
 510
 511
 512 // type used by the transformation functions for G
 513 typedef std::vector<int> Gparameter;
 514
 515
 516 // G_eval1-function for G transformations
 517 ex G_eval1(int a, int scale, const exvector& gsyms)
 518 {
 519         if (a != 0) {
 520                 const ex& scs = gsyms[std::abs(scale)];
 521                 const ex& as = gsyms[std::abs(a)];
 522                 if (as != scs) {
 523                         return -log(1 - scs/as);
 524                 } else {
 525                         return -zeta(1);
 526                 }
 527         } else {
 528                 return log(gsyms[std::abs(scale)]);
 529         }
 530 }
 531
 532
 533 // G_eval-function for G transformations
 534 ex G_eval(const Gparameter& a, int scale, const exvector& gsyms)
 535 {
 536         // check for properties of G
 537         ex sc = gsyms[std::abs(scale)];
 538         lst newa;
 539         bool all_zero = true;
 540         bool all_ones = true;
 541         int count_ones = 0;
 542         for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
 543                 if (*it != 0) {
 544                         const ex sym = gsyms[std::abs(*it)];
 545                         newa.append(sym);
 546                         all_zero = false;
 547                         if (sym != sc) {
 548                                 all_ones = false;
 549                         }
 550                         if (all_ones) {
 551                                 ++count_ones;
 552                         }
 553                 } else {
 554                         all_ones = false;
 555                 }
 556         }
 557
 558         // care about divergent G: shuffle to separate divergencies that will be canceled
 559         // later on in the transformation
 560         if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) {
 561                 // do shuffle
 562                 Gparameter short_a;
 563                 Gparameter::const_iterator it = a.begin();
 564                 ++it;
 565                 for (; it != a.end(); ++it) {
 566                         short_a.push_back(*it);
 567                 }
 568                 ex result = G_eval1(a.front(), scale, gsyms) * G_eval(short_a, scale, gsyms);
 569                 it = short_a.begin();
 570                 for (int i=1; i<count_ones; ++i) {
 571                         ++it;
 572                 }
 573                 for (; it != short_a.end(); ++it) {
 574
 575                         Gparameter newa;
 576                         Gparameter::const_iterator it2 = short_a.begin();
 577                         for (; it2 != it; ++it2) {
 578                                 newa.push_back(*it2);
 579                         }
 580                         newa.push_back(*it);
 581                         newa.push_back(a[0]);
 582                         it2 = it;
 583                         ++it2;
 584                         for (; it2 != short_a.end(); ++it2) {
 585                                 newa.push_back(*it2);
 586                         }
 587                         result -= G_eval(newa, scale, gsyms);
 588                 }
 589                 return result / count_ones;
 590         }
 591
 592         // G({1,...,1};y) -> G({1};y)^k / k!
 593         if (all_ones && a.size() > 1) {
 594                 return pow(G_eval1(a.front(),scale, gsyms), count_ones) / factorial(count_ones);
 595         }
 596
 597         // G({0,...,0};y) -> log(y)^k / k!
 598         if (all_zero) {
 599                 return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size());
 600         }
 601
 602         // no special cases anymore -> convert it into Li
 603         lst m;
 604         lst x;
 605         ex argbuf = gsyms[std::abs(scale)];
 606         ex mval = _ex1;
 607         for (Gparameter::const_iterator it=a.begin(); it!=a.end(); ++it) {
 608                 if (*it != 0) {
 609                         const ex& sym = gsyms[std::abs(*it)];
 610                         x.append(argbuf / sym);
 611                         m.append(mval);
 612                         mval = _ex1;
 613                         argbuf = sym;
 614                 } else {
 615                         ++mval;
 616                 }
 617         }
 618         return pow(-1, x.nops()) * Li(m, x);
 619 }
 620
 621
 622 // converts data for G: pending_integrals -> a
 623 Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals)
 624 {
 625         GINAC_ASSERT(pending_integrals.size() != 1);
 626
 627         if (pending_integrals.size() > 0) {
 628                 // get rid of the first element, which would stand for the new upper limit
 629                 Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end());
 630                 return new_a;
 631         } else {
 632                 // just return empty parameter list
 633                 Gparameter new_a;
 634                 return new_a;
 635         }
 636 }
 637
 638
 639 // check the parameters a and scale for G and return information about convergence, depth, etc.
 640 // convergent     : true if G(a,scale) is convergent
 641 // depth          : depth of G(a,scale)
 642 // trailing_zeros : number of trailing zeros of a
 643 // min_it         : iterator of a pointing on the smallest element in a
 644 Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale,
 645                 bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it)
 646 {
 647         convergent = true;
 648         depth = 0;
 649         trailing_zeros = 0;
 650         min_it = a.end();
 651         Gparameter::const_iterator lastnonzero = a.end();
 652         for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
 653                 if (std::abs(*it) > 0) {
 654                         ++depth;
 655                         trailing_zeros = 0;
 656                         lastnonzero = it;
 657                         if (std::abs(*it) < scale) {
 658                                 convergent = false;
 659                                 if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) {
 660                                         min_it = it;
 661                                 }
 662                         }
 663                 } else {
 664                         ++trailing_zeros;
 665                 }
 666         }
 667         if (lastnonzero == a.end())
 668                 return a.end();
 669         return ++lastnonzero;
 670 }
 671
 672
 673 // add scale to pending_integrals if pending_integrals is empty
 674 Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale)
 675 {
 676         GINAC_ASSERT(pending_integrals.size() != 1);
 677
 678         if (pending_integrals.size() > 0) {
 679                 return pending_integrals;
 680         } else {
 681                 Gparameter new_pending_integrals;
 682                 new_pending_integrals.push_back(scale);
 683                 return new_pending_integrals;
 684         }
 685 }
 686
 687
 688 // handles trailing zeroes for an otherwise convergent integral
 689 ex trailing_zeros_G(const Gparameter& a, int scale, const exvector& gsyms)
 690 {
 691         bool convergent;
 692         int depth, trailing_zeros;
 693         Gparameter::const_iterator last, dummyit;
 694         last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit);
 695
 696         GINAC_ASSERT(convergent);
 697
 698         if ((trailing_zeros > 0) && (depth > 0)) {
 699                 ex result;
 700                 Gparameter new_a(a.begin(), a.end()-1);
 701                 result += G_eval1(0, scale, gsyms) * trailing_zeros_G(new_a, scale, gsyms);
 702                 for (Gparameter::const_iterator it = a.begin(); it != last; ++it) {
 703                         Gparameter new_a(a.begin(), it);
 704                         new_a.push_back(0);
 705                         new_a.insert(new_a.end(), it, a.end()-1);
 706                         result -= trailing_zeros_G(new_a, scale, gsyms);
 707                 }
 708
 709                 return result / trailing_zeros;
 710         } else {
 711                 return G_eval(a, scale, gsyms);
 712         }
 713 }
 714
 715
 716 // G transformation [VSW] (57),(58)
 717 ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale, const exvector& gsyms)
 718 {
 719         // pendint = ( y1, b1, ..., br )
 720         //       a = ( 0, ..., 0, amin )
 721         //   scale = y2
 722         //
 723         // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2)
 724         // where sr replaces amin
 725
 726         GINAC_ASSERT(a.back() != 0);
 727         GINAC_ASSERT(a.size() > 0);
 728
 729         ex result;
 730         Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back()));
 731         const int psize = pending_integrals.size();
 732
 733         // length == 1
 734         // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+})
 735
 736         if (a.size() == 1) {
 737
 738           // ln(-y2_{-+})
 739           result += log(gsyms[ex_to<numeric>(scale).to_int()]);
 740                 if (a.back() > 0) {
 741                         new_pending_integrals.push_back(-scale);
 742                         result += I*Pi;
 743                 } else {
 744                         new_pending_integrals.push_back(scale);
 745                         result -= I*Pi;
 746                 }
 747                 if (psize) {
 748                         result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
 749                                                    pending_integrals.front(),
 750                                                    gsyms);
 751                 }
 752
 753                 // G(y2_{-+}; sr)
 754                 result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
 755                                            new_pending_integrals.front(),
 756                                            gsyms);
 757
 758                 // G(0; sr)
 759                 new_pending_integrals.back() = 0;
 760                 result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
 761                                            new_pending_integrals.front(),
 762                                            gsyms);
 763
 764                 return result;
 765         }
 766
 767         // length > 1
 768         // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
 769         //                            - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
 770
 771         //term zeta_m
 772         result -= zeta(a.size());
 773         if (psize) {
 774                 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
 775                                            pending_integrals.front(),
 776                                            gsyms);
 777         }
 778
 779         // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
 780         //    = int_0^sr dt/t G_{m-1}( t_{+-}; y2 )
 781         Gparameter new_a(a.begin()+1, a.end());
 782         new_pending_integrals.push_back(0);
 783         result -= depth_one_trafo_G(new_pending_integrals, new_a, scale, gsyms);
 784
 785         // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
 786         //    = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 )
 787         Gparameter new_pending_integrals_2;
 788         new_pending_integrals_2.push_back(scale);
 789         new_pending_integrals_2.push_back(0);
 790         if (psize) {
 791                 result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
 792                                            pending_integrals.front(),
 793                                            gsyms)
 794                           * depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
 795         } else {
 796                 result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
 797         }
 798
 799         return result;
 800 }
 801
 802
 803 // forward declaration
 804 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
 805              const Gparameter& pendint, const Gparameter& a_old, int scale,
 806              const exvector& gsyms);
 807
 808
 809 // G transformation [VSW]
 810 ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale,
 811                const exvector& gsyms)
 812 {
 813         // main recursion routine
 814         //
 815         // pendint = ( y1, b1, ..., br )
 816         //       a = ( a1, ..., amin, ..., aw )
 817         //   scale = y2
 818         //
 819         // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
 820         // where sr replaces amin
 821
 822         // find smallest alpha, determine depth and trailing zeros, and check for convergence
 823         bool convergent;
 824         int depth, trailing_zeros;
 825         Gparameter::const_iterator min_it;
 826         Gparameter::const_iterator firstzero =
 827                 check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it);
 828         int min_it_pos = min_it - a.begin();
 829
 830         // special case: all a's are zero
 831         if (depth == 0) {
 832                 ex result;
 833
 834                 if (a.size() == 0) {
 835                   result = 1;
 836                 } else {
 837                   result = G_eval(a, scale, gsyms);
 838                 }
 839                 if (pendint.size() > 0) {
 840                   result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
 841                                              pendint.front(),
 842                                              gsyms);
 843                 }
 844                 return result;
 845         }
 846
 847         // handle trailing zeros
 848         if (trailing_zeros > 0) {
 849                 ex result;
 850                 Gparameter new_a(a.begin(), a.end()-1);
 851                 result += G_eval1(0, scale, gsyms) * G_transform(pendint, new_a, scale, gsyms);
 852                 for (Gparameter::const_iterator it = a.begin(); it != firstzero; ++it) {
 853                         Gparameter new_a(a.begin(), it);
 854                         new_a.push_back(0);
 855                         new_a.insert(new_a.end(), it, a.end()-1);
 856                         result -= G_transform(pendint, new_a, scale, gsyms);
 857                 }
 858                 return result / trailing_zeros;
 859         }
 860
 861         // convergence case
 862         if (convergent) {
 863                 if (pendint.size() > 0) {
 864                         return G_eval(convert_pending_integrals_G(pendint),
 865                                       pendint.front(), gsyms)*
 866                                 G_eval(a, scale, gsyms);
 867                 } else {
 868                         return G_eval(a, scale, gsyms);
 869                 }
 870         }
 871
 872         // call basic transformation for depth equal one
 873         if (depth == 1) {
 874                 return depth_one_trafo_G(pendint, a, scale, gsyms);
 875         }
 876
 877         // do recursion
 878         // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
 879         //  =  int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2)
 880         //   + int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) int_0^{sr} ds_{r+1} d/ds_{r+1} G(a1,...,s_{r+1},...,aw,y2)
 881
 882         // smallest element in last place
 883         if (min_it + 1 == a.end()) {
 884                 do { --min_it; } while (*min_it == 0);
 885                 Gparameter empty;
 886                 Gparameter a1(a.begin(),min_it+1);
 887                 Gparameter a2(min_it+1,a.end());
 888
 889                 ex result = G_transform(pendint, a2, scale, gsyms)*
 890                         G_transform(empty, a1, scale, gsyms);
 891
 892                 result -= shuffle_G(empty, a1, a2, pendint, a, scale, gsyms);
 893                 return result;
 894         }
 895
 896         Gparameter empty;
 897         Gparameter::iterator changeit;
 898
 899         // first term G(a_1,..,0,...,a_w;a_0)
 900         Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]);
 901         Gparameter new_a = a;
 902         new_a[min_it_pos] = 0;
 903         ex result = G_transform(empty, new_a, scale, gsyms);
 904         if (pendint.size() > 0) {
 905                 result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
 906                                            pendint.front(), gsyms);
 907         }
 908
 909         // other terms
 910         changeit = new_a.begin() + min_it_pos;
 911         changeit = new_a.erase(changeit);
 912         if (changeit != new_a.begin()) {
 913                 // smallest in the middle
 914                 new_pendint.push_back(*changeit);
 915                 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
 916                                            new_pendint.front(), gsyms)*
 917                         G_transform(empty, new_a, scale, gsyms);
 918                 int buffer = *changeit;
 919                 *changeit = *min_it;
 920                 result += G_transform(new_pendint, new_a, scale, gsyms);
 921                 *changeit = buffer;
 922                 new_pendint.pop_back();
 923                 --changeit;
 924                 new_pendint.push_back(*changeit);
 925                 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
 926                                            new_pendint.front(), gsyms)*
 927                         G_transform(empty, new_a, scale, gsyms);
 928                 *changeit = *min_it;
 929                 result -= G_transform(new_pendint, new_a, scale, gsyms);
 930         } else {
 931                 // smallest at the front
 932                 new_pendint.push_back(scale);
 933                 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
 934                                            new_pendint.front(), gsyms)*
 935                         G_transform(empty, new_a, scale, gsyms);
 936                 new_pendint.back() =  *changeit;
 937                 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
 938                                            new_pendint.front(), gsyms)*
 939                         G_transform(empty, new_a, scale, gsyms);
 940                 *changeit = *min_it;
 941                 result += G_transform(new_pendint, new_a, scale, gsyms);
 942         }
 943         return result;
 944 }
 945
 946
 947 // shuffles the two parameter list a1 and a2 and calls G_transform for every term except
 948 // for the one that is equal to a_old
 949 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
 950              const Gparameter& pendint, const Gparameter& a_old, int scale,
 951              const exvector& gsyms)
 952 {
 953         if (a1.size()==0 && a2.size()==0) {
 954                 // veto the one configuration we don't want
 955                 if ( a0 == a_old ) return 0;
 956
 957                 return G_transform(pendint, a0, scale, gsyms);
 958         }
 959
 960         if (a2.size()==0) {
 961                 Gparameter empty;
 962                 Gparameter aa0 = a0;
 963                 aa0.insert(aa0.end(),a1.begin(),a1.end());
 964                 return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms);
 965         }
 966
 967         if (a1.size()==0) {
 968                 Gparameter empty;
 969                 Gparameter aa0 = a0;
 970                 aa0.insert(aa0.end(),a2.begin(),a2.end());
 971                 return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms);
 972         }
 973
 974         Gparameter a1_removed(a1.begin()+1,a1.end());
 975         Gparameter a2_removed(a2.begin()+1,a2.end());
 976
 977         Gparameter a01 = a0;
 978         Gparameter a02 = a0;
 979
 980         a01.push_back( a1[0] );
 981         a02.push_back( a2[0] );
 982
 983         return shuffle_G(a01, a1_removed, a2, pendint, a_old, scale, gsyms)
 984              + shuffle_G(a02, a1, a2_removed, pendint, a_old, scale, gsyms);
 985 }
 986
 987 // handles the transformations and the numerical evaluation of G
 988 // the parameter x, s and y must only contain numerics
 989 static cln::cl_N
 990 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
 991           const cln::cl_N& y);
 992
 993 // do acceleration transformation (hoelder convolution [BBB])
 994 // the parameter x, s and y must only contain numerics
 995 static cln::cl_N
 996 G_do_hoelder(std::vector<cln::cl_N> x, /* yes, it's passed by value */
 997              const std::vector<int>& s, const cln::cl_N& y)
 998 {
 999         cln::cl_N result;
1000         const std::size_t size = x.size();
1001         for (std::size_t i = 0; i < size; ++i)
1002                 x[i] = x[i]/y;
1003
1004         for (std::size_t r = 0; r <= size; ++r) {
1005                 cln::cl_N buffer(1 & r ? -1 : 1);
1006                 cln::cl_RA p(2);
1007                 bool adjustp;
1008                 do {
1009                         adjustp = false;
1010                         for (std::size_t i = 0; i < size; ++i) {
1011                                 if (x[i] == cln::cl_RA(1)/p) {
1012                                         p = p/2 + cln::cl_RA(3)/2;
1013                                         adjustp = true;
1014                                         continue;
1015                                 }
1016                         }
1017                 } while (adjustp);
1018                 cln::cl_RA q = p/(p-1);
1019                 std::vector<cln::cl_N> qlstx;
1020                 std::vector<int> qlsts;
1021                 for (std::size_t j = r; j >= 1; --j) {
1022                         qlstx.push_back(cln::cl_N(1) - x[j-1]);
1023                         if (instanceof(x[j-1], cln::cl_R_ring) &&
1024                             realpart(x[j-1]) > 1 && realpart(x[j-1]) <= 2) {
1025                                 qlsts.push_back(s[j-1]);
1026                         } else {
1027                                 qlsts.push_back(-s[j-1]);
1028                         }
1029                 }
1030                 if (qlstx.size() > 0) {
1031                         buffer = buffer*G_numeric(qlstx, qlsts, 1/q);
1032                 }
1033                 std::vector<cln::cl_N> plstx;
1034                 std::vector<int> plsts;
1035                 for (std::size_t j = r+1; j <= size; ++j) {
1036                         plstx.push_back(x[j-1]);
1037                         plsts.push_back(s[j-1]);
1038                 }
1039                 if (plstx.size() > 0) {
1040                         buffer = buffer*G_numeric(plstx, plsts, 1/p);
1041                 }
1042                 result = result + buffer;
1043         }
1044         return result;
1045 }
1046
1047 // convergence transformation, used for numerical evaluation of G function.
1048 // the parameter x, s and y must only contain numerics
1049 static cln::cl_N
1050 G_do_trafo(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1051            const cln::cl_N& y)
1052 {
1053         // sort (|x|<->position) to determine indices
1054         typedef std::multimap<cln::cl_R, std::size_t> sortmap_t;
1055         sortmap_t sortmap;
1056         std::size_t size = 0;
1057         for (std::size_t i = 0; i < x.size(); ++i) {
1058                 if (!zerop(x[i])) {
1059                         sortmap.insert(std::make_pair(abs(x[i]), i));
1060                         ++size;
1061                 }
1062         }
1063         // include upper limit (scale)
1064         sortmap.insert(std::make_pair(abs(y), x.size()));
1065
1066         // generate missing dummy-symbols
1067         int i = 1;
1068         // holding dummy-symbols for the G/Li transformations
1069         exvector gsyms;
1070         gsyms.push_back(symbol("GSYMS_ERROR"));
1071         cln::cl_N lastentry(0);
1072         for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1073                 if (it != sortmap.begin()) {
1074                         if (it->second < x.size()) {
1075                                 if (x[it->second] == lastentry) {
1076                                         gsyms.push_back(gsyms.back());
1077                                         continue;
1078                                 }
1079                         } else {
1080                                 if (y == lastentry) {
1081                                         gsyms.push_back(gsyms.back());
1082                                         continue;
1083                                 }
1084                         }
1085                 }
1086                 std::ostringstream os;
1087                 os << "a" << i;
1088                 gsyms.push_back(symbol(os.str()));
1089                 ++i;
1090                 if (it->second < x.size()) {
1091                         lastentry = x[it->second];
1092                 } else {
1093                         lastentry = y;
1094                 }
1095         }
1096
1097         // fill position data according to sorted indices and prepare substitution list
1098         Gparameter a(x.size());
1099         exmap subslst;
1100         std::size_t pos = 1;
1101         int scale = pos;
1102         for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1103                 if (it->second < x.size()) {
1104                         if (s[it->second] > 0) {
1105                                 a[it->second] = pos;
1106                         } else {
1107                                 a[it->second] = -int(pos);
1108                         }
1109                         subslst[gsyms[pos]] = numeric(x[it->second]);
1110                 } else {
1111                         scale = pos;
1112                         subslst[gsyms[pos]] = numeric(y);
1113                 }
1114                 ++pos;
1115         }
1116
1117         // do transformation
1118         Gparameter pendint;
1119         ex result = G_transform(pendint, a, scale, gsyms);
1120         // replace dummy symbols with their values
1121         result = result.eval().expand();
1122         result = result.subs(subslst).evalf();
1123         if (!is_a<numeric>(result))
1124                 throw std::logic_error("G_do_trafo: G_transform returned non-numeric result");
1125
1126         cln::cl_N ret = ex_to<numeric>(result).to_cl_N();
1127         return ret;
1128 }
1129
1130 // handles the transformations and the numerical evaluation of G
1131 // the parameter x, s and y must only contain numerics
1132 static cln::cl_N
1133 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1134           const cln::cl_N& y)
1135 {
1136         // check for convergence and necessary accelerations
1137         bool need_trafo = false;
1138         bool need_hoelder = false;
1139         std::size_t depth = 0;
1140         for (std::size_t i = 0; i < x.size(); ++i) {
1141                 if (!zerop(x[i])) {
1142                         ++depth;
1143                         const cln::cl_N x_y = abs(x[i]) - y;
1144                         if (instanceof(x_y, cln::cl_R_ring) &&
1145                             realpart(x_y) < cln::least_negative_float(cln::float_format(Digits - 2)))
1146                                 need_trafo = true;
1147
1148                         if (abs(abs(x[i]/y) - 1) < 0.01)
1149                                 need_hoelder = true;
1150                 }
1151         }
1152         if (zerop(x[x.size() - 1]))
1153                 need_trafo = true;
1154
1155         if (depth == 1 && x.size() == 2 && !need_trafo)
1156                 return - Li_projection(2, y/x[1], cln::float_format(Digits));
1157
1158         // do acceleration transformation (hoelder convolution [BBB])
1159         if (need_hoelder)
1160                 return G_do_hoelder(x, s, y);
1161
1162         // convergence transformation
1163         if (need_trafo)
1164                 return G_do_trafo(x, s, y);
1165
1166         // do summation
1167         std::vector<cln::cl_N> newx;
1168         newx.reserve(x.size());
1169         std::vector<int> m;
1170         m.reserve(x.size());
1171         int mcount = 1;
1172         int sign = 1;
1173         cln::cl_N factor = y;
1174         for (std::size_t i = 0; i < x.size(); ++i) {
1175                 if (zerop(x[i])) {
1176                         ++mcount;
1177                 } else {
1178                         newx.push_back(factor/x[i]);
1179                         factor = x[i];
1180                         m.push_back(mcount);
1181                         mcount = 1;
1182                         sign = -sign;
1183                 }
1184         }
1185
1186         return sign*multipleLi_do_sum(m, newx);
1187 }
1188
1189
1190 ex mLi_numeric(const lst& m, const lst& x)
1191 {
1192         // let G_numeric do the transformation
1193         std::vector<cln::cl_N> newx;
1194         newx.reserve(x.nops());
1195         std::vector<int> s;
1196         s.reserve(x.nops());
1197         cln::cl_N factor(1);
1198         for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1199                 for (int i = 1; i < *itm; ++i) {
1200                         newx.push_back(cln::cl_N(0));
1201                         s.push_back(1);
1202                 }
1203                 const cln::cl_N xi = ex_to<numeric>(*itx).to_cl_N();
1204                 newx.push_back(factor/xi);
1205                 factor = factor/xi;
1206                 s.push_back(1);
1207         }
1208         return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1)));
1209 }
1210
1211
1212 } // end of anonymous namespace
1213
1214
1215 //////////////////////////////////////////////////////////////////////
1216 //
1217 // Generalized multiple polylogarithm  G(x, y) and G(x, s, y)
1218 //
1219 // GiNaC function
1220 //
1221 //////////////////////////////////////////////////////////////////////
1222
1223
1224 static ex G2_evalf(const ex& x_, const ex& y)
1225 {
1226         if (!y.info(info_flags::positive)) {
1227                 return G(x_, y).hold();
1228         }
1229         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1230         if (x.nops() == 0) {
1231                 return _ex1;
1232         }
1233         if (x.op(0) == y) {
1234                 return G(x_, y).hold();
1235         }
1236         std::vector<int> s;
1237         s.reserve(x.nops());
1238         bool all_zero = true;
1239         for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1240                 if (!(*it).info(info_flags::numeric)) {
1241                         return G(x_, y).hold();
1242                 }
1243                 if (*it != _ex0) {
1244                         all_zero = false;
1245                 }
1246                 if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
1247                         s.push_back(-1);
1248                 }
1249                 else {
1250                         s.push_back(1);
1251                 }
1252         }
1253         if (all_zero) {
1254                 return pow(log(y), x.nops()) / factorial(x.nops());
1255         }
1256         std::vector<cln::cl_N> xv;
1257         xv.reserve(x.nops());
1258         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1259                 xv.push_back(ex_to<numeric>(*it).to_cl_N());
1260         cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1261         return numeric(result);
1262 }
1263
1264
1265 static ex G2_eval(const ex& x_, const ex& y)
1266 {
1267         //TODO eval to MZV or H or S or Lin
1268
1269         if (!y.info(info_flags::positive)) {
1270                 return G(x_, y).hold();
1271         }
1272         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1273         if (x.nops() == 0) {
1274                 return _ex1;
1275         }
1276         if (x.op(0) == y) {
1277                 return G(x_, y).hold();
1278         }
1279         std::vector<int> s;
1280         s.reserve(x.nops());
1281         bool all_zero = true;
1282         bool crational = true;
1283         for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1284                 if (!(*it).info(info_flags::numeric)) {
1285                         return G(x_, y).hold();
1286                 }
1287                 if (!(*it).info(info_flags::crational)) {
1288                         crational = false;
1289                 }
1290                 if (*it != _ex0) {
1291                         all_zero = false;
1292                 }
1293                 if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
1294                         s.push_back(-1);
1295                 }
1296                 else {
1297                         s.push_back(+1);
1298                 }
1299         }
1300         if (all_zero) {
1301                 return pow(log(y), x.nops()) / factorial(x.nops());
1302         }
1303         if (!y.info(info_flags::crational)) {
1304                 crational = false;
1305         }
1306         if (crational) {
1307                 return G(x_, y).hold();
1308         }
1309         std::vector<cln::cl_N> xv;
1310         xv.reserve(x.nops());
1311         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1312                 xv.push_back(ex_to<numeric>(*it).to_cl_N());
1313         cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1314         return numeric(result);
1315 }
1316
1317
1318 unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
1319                                 evalf_func(G2_evalf).
1320                                 eval_func(G2_eval).
1321                                 do_not_evalf_params().
1322                                 overloaded(2));
1323 //TODO
1324 //                                derivative_func(G2_deriv).
1325 //                                print_func<print_latex>(G2_print_latex).
1326
1327
1328 static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
1329 {
1330         if (!y.info(info_flags::positive)) {
1331                 return G(x_, s_, y).hold();
1332         }
1333         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1334         lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1335         if (x.nops() != s.nops()) {
1336                 return G(x_, s_, y).hold();
1337         }
1338         if (x.nops() == 0) {
1339                 return _ex1;
1340         }
1341         if (x.op(0) == y) {
1342                 return G(x_, s_, y).hold();
1343         }
1344         std::vector<int> sn;
1345         sn.reserve(s.nops());
1346         bool all_zero = true;
1347         for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1348                 if (!(*itx).info(info_flags::numeric)) {
1349                         return G(x_, y).hold();
1350                 }
1351                 if (!(*its).info(info_flags::real)) {
1352                         return G(x_, y).hold();
1353                 }
1354                 if (*itx != _ex0) {
1355                         all_zero = false;
1356                 }
1357                 if ( ex_to<numeric>(*itx).is_real() ) {
1358                         if ( *its >= 0 ) {
1359                                 sn.push_back(1);
1360                         }
1361                         else {
1362                                 sn.push_back(-1);
1363                         }
1364                 }
1365                 else {
1366                         if ( ex_to<numeric>(*itx).imag() > 0 ) {
1367                                 sn.push_back(1);
1368                         }
1369                         else {
1370                                 sn.push_back(-1);
1371                         }
1372                 }
1373         }
1374         if (all_zero) {
1375                 return pow(log(y), x.nops()) / factorial(x.nops());
1376         }
1377         std::vector<cln::cl_N> xn;
1378         xn.reserve(x.nops());
1379         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1380                 xn.push_back(ex_to<numeric>(*it).to_cl_N());
1381         cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1382         return numeric(result);
1383 }
1384
1385
1386 static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
1387 {
1388         //TODO eval to MZV or H or S or Lin
1389
1390         if (!y.info(info_flags::positive)) {
1391                 return G(x_, s_, y).hold();
1392         }
1393         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1394         lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1395         if (x.nops() != s.nops()) {
1396                 return G(x_, s_, y).hold();
1397         }
1398         if (x.nops() == 0) {
1399                 return _ex1;
1400         }
1401         if (x.op(0) == y) {
1402                 return G(x_, s_, y).hold();
1403         }
1404         std::vector<int> sn;
1405         sn.reserve(s.nops());
1406         bool all_zero = true;
1407         bool crational = true;
1408         for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1409                 if (!(*itx).info(info_flags::numeric)) {
1410                         return G(x_, s_, y).hold();
1411                 }
1412                 if (!(*its).info(info_flags::real)) {
1413                         return G(x_, s_, y).hold();
1414                 }
1415                 if (!(*itx).info(info_flags::crational)) {
1416                         crational = false;
1417                 }
1418                 if (*itx != _ex0) {
1419                         all_zero = false;
1420                 }
1421                 if ( ex_to<numeric>(*itx).is_real() ) {
1422                         if ( *its >= 0 ) {
1423                                 sn.push_back(1);
1424                         }
1425                         else {
1426                                 sn.push_back(-1);
1427                         }
1428                 }
1429                 else {
1430                         if ( ex_to<numeric>(*itx).imag() > 0 ) {
1431                                 sn.push_back(1);
1432                         }
1433                         else {
1434                                 sn.push_back(-1);
1435                         }
1436                 }
1437         }
1438         if (all_zero) {
1439                 return pow(log(y), x.nops()) / factorial(x.nops());
1440         }
1441         if (!y.info(info_flags::crational)) {
1442                 crational = false;
1443         }
1444         if (crational) {
1445                 return G(x_, s_, y).hold();
1446         }
1447         std::vector<cln::cl_N> xn;
1448         xn.reserve(x.nops());
1449         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1450                 xn.push_back(ex_to<numeric>(*it).to_cl_N());
1451         cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1452         return numeric(result);
1453 }
1454
1455
1456 unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
1457                                 evalf_func(G3_evalf).
1458                                 eval_func(G3_eval).
1459                                 do_not_evalf_params().
1460                                 overloaded(2));
1461 //TODO
1462 //                                derivative_func(G3_deriv).
1463 //                                print_func<print_latex>(G3_print_latex).
1464
1465
1466 //////////////////////////////////////////////////////////////////////
1467 //
1468 // Classical polylogarithm and multiple polylogarithm  Li(m,x)
1469 //
1470 // GiNaC function
1471 //
1472 //////////////////////////////////////////////////////////////////////
1473
1474
1475 static ex Li_evalf(const ex& m_, const ex& x_)
1476 {
1477         // classical polylogs
1478         if (m_.info(info_flags::posint)) {
1479                 if (x_.info(info_flags::numeric)) {
1480                         int m__ = ex_to<numeric>(m_).to_int();
1481                         const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1482                         const cln::cl_N result = Lin_numeric(m__, x__);
1483                         return numeric(result);
1484                 } else {
1485                         // try to numerically evaluate second argument
1486                         ex x_val = x_.evalf();
1487                         if (x_val.info(info_flags::numeric)) {
1488                                 int m__ = ex_to<numeric>(m_).to_int();
1489                                 const cln::cl_N x__ = ex_to<numeric>(x_val).to_cl_N();
1490                                 const cln::cl_N result = Lin_numeric(m__, x__);
1491                                 return numeric(result);
1492                         }
1493                 }
1494         }
1495         // multiple polylogs
1496         if (is_a<lst>(m_) && is_a<lst>(x_)) {
1497
1498                 const lst& m = ex_to<lst>(m_);
1499                 const lst& x = ex_to<lst>(x_);
1500                 if (m.nops() != x.nops()) {
1501                         return Li(m_,x_).hold();
1502                 }
1503                 if (x.nops() == 0) {
1504                         return _ex1;
1505                 }
1506                 if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
1507                         return Li(m_,x_).hold();
1508                 }
1509
1510                 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1511                         if (!(*itm).info(info_flags::posint)) {
1512                                 return Li(m_, x_).hold();
1513                         }
1514                         if (!(*itx).info(info_flags::numeric)) {
1515                                 return Li(m_, x_).hold();
1516                         }
1517                         if (*itx == _ex0) {
1518                                 return _ex0;
1519                         }
1520                 }
1521
1522                 return mLi_numeric(m, x);
1523         }
1524
1525         return Li(m_,x_).hold();
1526 }
1527
1528
1529 static ex Li_eval(const ex& m_, const ex& x_)
1530 {
1531         if (is_a<lst>(m_)) {
1532                 if (is_a<lst>(x_)) {
1533                         // multiple polylogs
1534                         const lst& m = ex_to<lst>(m_);
1535                         const lst& x = ex_to<lst>(x_);
1536                         if (m.nops() != x.nops()) {
1537                                 return Li(m_,x_).hold();
1538                         }
1539                         if (x.nops() == 0) {
1540                                 return _ex1;
1541                         }
1542                         bool is_H = true;
1543                         bool is_zeta = true;
1544                         bool do_evalf = true;
1545                         bool crational = true;
1546                         for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1547                                 if (!(*itm).info(info_flags::posint)) {
1548                                         return Li(m_,x_).hold();
1549                                 }
1550                                 if ((*itx != _ex1) && (*itx != _ex_1)) {
1551                                         if (itx != x.begin()) {
1552                                                 is_H = false;
1553                                         }
1554                                         is_zeta = false;
1555                                 }
1556                                 if (*itx == _ex0) {
1557                                         return _ex0;
1558                                 }
1559                                 if (!(*itx).info(info_flags::numeric)) {
1560                                         do_evalf = false;
1561                                 }
1562                                 if (!(*itx).info(info_flags::crational)) {
1563                                         crational = false;
1564                                 }
1565                         }
1566                         if (is_zeta) {
1567                                 return zeta(m_,x_);
1568                         }
1569                         if (is_H) {
1570                                 ex prefactor;
1571                                 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1572                                 return prefactor * H(newm, x[0]);
1573                         }
1574                         if (do_evalf && !crational) {
1575                                 return mLi_numeric(m,x);
1576                         }
1577                 }
1578                 return Li(m_, x_).hold();
1579         } else if (is_a<lst>(x_)) {
1580                 return Li(m_, x_).hold();
1581         }
1582
1583         // classical polylogs
1584         if (x_ == _ex0) {
1585                 return _ex0;
1586         }
1587         if (x_ == _ex1) {
1588                 return zeta(m_);
1589         }
1590         if (x_ == _ex_1) {
1591                 return (pow(2,1-m_)-1) * zeta(m_);
1592         }
1593         if (m_ == _ex1) {
1594                 return -log(1-x_);
1595         }
1596         if (m_ == _ex2) {
1597                 if (x_.is_equal(I)) {
1598                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
1599                 }
1600                 if (x_.is_equal(-I)) {
1601                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
1602                 }
1603         }
1604         if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1605                 int m__ = ex_to<numeric>(m_).to_int();
1606                 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1607                 const cln::cl_N result = Lin_numeric(m__, x__);
1608                 return numeric(result);
1609         }
1610
1611         return Li(m_, x_).hold();
1612 }
1613
1614
1615 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1616 {
1617         if (is_a<lst>(m) || is_a<lst>(x)) {
1618                 // multiple polylog
1619                 epvector seq;
1620                 seq.push_back(expair(Li(m, x), 0));
1621                 return pseries(rel, seq);
1622         }
1623
1624         // classical polylog
1625         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1626         if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1627                 // First special case: x==0 (derivatives have poles)
1628                 if (x_pt.is_zero()) {
1629                         const symbol s;
1630                         ex ser;
1631                         // manually construct the primitive expansion
1632                         for (int i=1; i<order; ++i)
1633                                 ser += pow(s,i) / pow(numeric(i), m);
1634                         // substitute the argument's series expansion
1635                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1636                         // maybe that was terminating, so add a proper order term
1637                         epvector nseq;
1638                         nseq.push_back(expair(Order(_ex1), order));
1639                         ser += pseries(rel, nseq);
1640                         // reexpanding it will collapse the series again
1641                         return ser.series(rel, order);
1642                 }
1643                 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1644                 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1645         }
1646         // all other cases should be safe, by now:
1647         throw do_taylor();  // caught by function::series()
1648 }
1649
1650
1651 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1652 {
1653         GINAC_ASSERT(deriv_param < 2);
1654         if (deriv_param == 0) {
1655                 return _ex0;
1656         }
1657         if (m_.nops() > 1) {
1658                 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1659         }
1660         ex m;
1661         if (is_a<lst>(m_)) {
1662                 m = m_.op(0);
1663         } else {
1664                 m = m_;
1665         }
1666         ex x;
1667         if (is_a<lst>(x_)) {
1668                 x = x_.op(0);
1669         } else {
1670                 x = x_;
1671         }
1672         if (m > 0) {
1673                 return Li(m-1, x) / x;
1674         } else {
1675                 return 1/(1-x);
1676         }
1677 }
1678
1679
1680 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1681 {
1682         lst m;
1683         if (is_a<lst>(m_)) {
1684                 m = ex_to<lst>(m_);
1685         } else {
1686                 m = lst(m_);
1687         }
1688         lst x;
1689         if (is_a<lst>(x_)) {
1690                 x = ex_to<lst>(x_);
1691         } else {
1692                 x = lst(x_);
1693         }
1694         c.s << "\\mathrm{Li}_{";
1695         lst::const_iterator itm = m.begin();
1696         (*itm).print(c);
1697         itm++;
1698         for (; itm != m.end(); itm++) {
1699                 c.s << ",";
1700                 (*itm).print(c);
1701         }
1702         c.s << "}(";
1703         lst::const_iterator itx = x.begin();
1704         (*itx).print(c);
1705         itx++;
1706         for (; itx != x.end(); itx++) {
1707                 c.s << ",";
1708                 (*itx).print(c);
1709         }
1710         c.s << ")";
1711 }
1712
1713
1714 REGISTER_FUNCTION(Li,
1715                   evalf_func(Li_evalf).
1716                   eval_func(Li_eval).
1717                   series_func(Li_series).
1718                   derivative_func(Li_deriv).
1719                   print_func<print_latex>(Li_print_latex).
1720                   do_not_evalf_params());
1721
1722
1723 //////////////////////////////////////////////////////////////////////
1724 //
1725 // Nielsen's generalized polylogarithm  S(n,p,x)
1726 //
1727 // helper functions
1728 //
1729 //////////////////////////////////////////////////////////////////////
1730
1731
1732 // anonymous namespace for helper functions
1733 namespace {
1734
1735
1736 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1737 // see fill_Yn()
1738 std::vector<std::vector<cln::cl_N> > Yn;
1739 int ynsize = 0; // number of Yn[]
1740 int ynlength = 100; // initial length of all Yn[i]
1741
1742
1743 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1744 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1745 // representing S_{n,p}(x).
1746 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1747 // equivalent Z-sum.
1748 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1749 // representing S_{n,p}(x).
1750 // The calculation of Y_n uses the values from Y_{n-1}.
1751 void fill_Yn(int n, const cln::float_format_t& prec)
1752 {
1753         const int initsize = ynlength;
1754         //const int initsize = initsize_Yn;
1755         cln::cl_N one = cln::cl_float(1, prec);
1756
1757         if (n) {
1758                 std::vector<cln::cl_N> buf(initsize);
1759                 std::vector<cln::cl_N>::iterator it = buf.begin();
1760                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1761                 *it = (*itprev) / cln::cl_N(n+1) * one;
1762                 it++;
1763                 itprev++;
1764                 // sums with an index smaller than the depth are zero and need not to be calculated.
1765                 // calculation starts with depth, which is n+2)
1766                 for (int i=n+2; i<=initsize+n; i++) {
1767                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1768                         it++;
1769                         itprev++;
1770                 }
1771                 Yn.push_back(buf);
1772         } else {
1773                 std::vector<cln::cl_N> buf(initsize);
1774                 std::vector<cln::cl_N>::iterator it = buf.begin();
1775                 *it = 1 * one;
1776                 it++;
1777                 for (int i=2; i<=initsize; i++) {
1778                         *it = *(it-1) + 1 / cln::cl_N(i) * one;
1779                         it++;
1780                 }
1781                 Yn.push_back(buf);
1782         }
1783         ynsize++;
1784 }
1785
1786
1787 // make Yn longer ...
1788 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1789 {
1790
1791         cln::cl_N one = cln::cl_float(1, prec);
1792
1793         Yn[0].resize(newsize);
1794         std::vector<cln::cl_N>::iterator it = Yn[0].begin();
1795         it += ynlength;
1796         for (int i=ynlength+1; i<=newsize; i++) {
1797                 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1798                 it++;
1799         }
1800
1801         for (int n=1; n<ynsize; n++) {
1802                 Yn[n].resize(newsize);
1803                 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
1804                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1805                 it += ynlength;
1806                 itprev += ynlength;
1807                 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1808                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1809                         it++;
1810                         itprev++;
1811                 }
1812         }
1813
1814         ynlength = newsize;
1815 }
1816
1817
1818 // helper function for S(n,p,x)
1819 // [Kol] (7.2)
1820 cln::cl_N C(int n, int p)
1821 {
1822         cln::cl_N result;
1823
1824         for (int k=0; k<p; k++) {
1825                 for (int j=0; j<=(n+k-1)/2; j++) {
1826                         if (k == 0) {
1827                                 if (n & 1) {
1828                                         if (j & 1) {
1829                                                 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1830                                         }
1831                                         else {
1832                                                 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1833                                         }
1834                                 }
1835                         }
1836                         else {
1837                                 if (k & 1) {
1838                                         if (j & 1) {
1839                                                 result = result + cln::factorial(n+k-1)
1840                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1841                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1842                                         }
1843                                         else {
1844                                                 result = result - cln::factorial(n+k-1)
1845                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1846                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1847                                         }
1848                                 }
1849                                 else {
1850                                         if (j & 1) {
1851                                                 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1852                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1853                                         }
1854                                         else {
1855                                                 result = result + cln::factorial(n+k-1)
1856                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1857                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1858                                         }
1859                                 }
1860                         }
1861                 }
1862         }
1863         int np = n+p;
1864         if ((np-1) & 1) {
1865                 if (((np)/2+n) & 1) {
1866                         result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1867                 }
1868                 else {
1869                         result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1870                 }
1871         }
1872
1873         return result;
1874 }
1875
1876
1877 // helper function for S(n,p,x)
1878 // [Kol] remark to (9.1)
1879 cln::cl_N a_k(int k)
1880 {
1881         cln::cl_N result;
1882
1883         if (k == 0) {
1884                 return 1;
1885         }
1886
1887         result = result;
1888         for (int m=2; m<=k; m++) {
1889                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1890         }
1891
1892         return -result / k;
1893 }
1894
1895
1896 // helper function for S(n,p,x)
1897 // [Kol] remark to (9.1)
1898 cln::cl_N b_k(int k)
1899 {
1900         cln::cl_N result;
1901
1902         if (k == 0) {
1903                 return 1;
1904         }
1905
1906         result = result;
1907         for (int m=2; m<=k; m++) {
1908                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1909         }
1910
1911         return result / k;
1912 }
1913
1914
1915 // helper function for S(n,p,x)
1916 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1917 {
1918         static cln::float_format_t oldprec = cln::default_float_format;
1919
1920         if (p==1) {
1921                 return Li_projection(n+1, x, prec);
1922         }
1923
1924         // precision has changed, we need to clear lookup table Yn
1925         if ( oldprec != prec ) {
1926                 Yn.clear();
1927                 ynsize = 0;
1928                 ynlength = 100;
1929                 oldprec = prec;
1930         }
1931
1932         // check if precalculated values are sufficient
1933         if (p > ynsize+1) {
1934                 for (int i=ynsize; i<p-1; i++) {
1935                         fill_Yn(i, prec);
1936                 }
1937         }
1938
1939         // should be done otherwise
1940         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1941         cln::cl_N xf = x * one;
1942         //cln::cl_N xf = x * cln::cl_float(1, prec);
1943
1944         cln::cl_N res;
1945         cln::cl_N resbuf;
1946         cln::cl_N factor = cln::expt(xf, p);
1947         int i = p;
1948         do {
1949                 resbuf = res;
1950                 if (i-p >= ynlength) {
1951                         // make Yn longer
1952                         make_Yn_longer(ynlength*2, prec);
1953                 }
1954                 res = res + factor / cln::expt(cln::cl_I(i),n+1) * Yn[p-2][i-p]; // should we check it? or rely on magic number? ...
1955                 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
1956                 factor = factor * xf;
1957                 i++;
1958         } while (res != resbuf);
1959
1960         return res;
1961 }
1962
1963
1964 // helper function for S(n,p,x)
1965 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1966 {
1967         // [Kol] (5.3)
1968         if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
1969
1970                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
1971                                    * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
1972
1973                 for (int s=0; s<n; s++) {
1974                         cln::cl_N res2;
1975                         for (int r=0; r<p; r++) {
1976                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
1977                                               * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
1978                         }
1979                         result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
1980                 }
1981
1982                 return result;
1983         }
1984
1985         return S_do_sum(n, p, x, prec);
1986 }
1987
1988
1989 // helper function for S(n,p,x)
1990 const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
1991 {
1992         if (x == 1) {
1993                 if (n == 1) {
1994                     // [Kol] (2.22) with (2.21)
1995                         return cln::zeta(p+1);
1996                 }
1997
1998                 if (p == 1) {
1999                     // [Kol] (2.22)
2000                         return cln::zeta(n+1);
2001                 }
2002
2003                 // [Kol] (9.1)
2004                 cln::cl_N result;
2005                 for (int nu=0; nu<n; nu++) {
2006                         for (int rho=0; rho<=p; rho++) {
2007                                 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
2008                                                   * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
2009                         }
2010                 }
2011                 result = result * cln::expt(cln::cl_I(-1),n+p-1);
2012
2013                 return result;
2014         }
2015         else if (x == -1) {
2016                 // [Kol] (2.22)
2017                 if (p == 1) {
2018                         return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
2019                 }
2020 //              throw std::runtime_error("don't know how to evaluate this function!");
2021         }
2022
2023         // what is the desired float format?
2024         // first guess: default format
2025         cln::float_format_t prec = cln::default_float_format;
2026         const cln::cl_N value = x;
2027         // second guess: the argument's format
2028         if (!instanceof(realpart(value), cln::cl_RA_ring))
2029                 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
2030         else if (!instanceof(imagpart(value), cln::cl_RA_ring))
2031                 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
2032
2033         // [Kol] (5.3)
2034         if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95))) {
2035
2036                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
2037                                    * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
2038
2039                 for (int s=0; s<n; s++) {
2040                         cln::cl_N res2;
2041                         for (int r=0; r<p; r++) {
2042                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
2043                                               * S_num(p-r,n-s,1-value) / cln::factorial(r);
2044                         }
2045                         result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2046                 }
2047
2048                 return result;
2049
2050         }
2051         // [Kol] (5.12)
2052         if (cln::abs(value) > 1) {
2053
2054                 cln::cl_N result;
2055
2056                 for (int s=0; s<p; s++) {
2057                         for (int r=0; r<=s; r++) {
2058                                 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
2059                                                   / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
2060                                                   * S_num(n+s-r,p-s,cln::recip(value));
2061                         }
2062                 }
2063                 result = result * cln::expt(cln::cl_I(-1),n);
2064
2065                 cln::cl_N res2;
2066                 for (int r=0; r<n; r++) {
2067                         res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
2068                 }
2069                 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
2070
2071                 result = result + cln::expt(cln::cl_I(-1),p) * res2;
2072
2073                 return result;
2074         }
2075         else {
2076                 return S_projection(n, p, value, prec);
2077         }
2078 }
2079
2080
2081 } // end of anonymous namespace
2082
2083
2084 //////////////////////////////////////////////////////////////////////
2085 //
2086 // Nielsen's generalized polylogarithm  S(n,p,x)
2087 //
2088 // GiNaC function
2089 //
2090 //////////////////////////////////////////////////////////////////////
2091
2092
2093 static ex S_evalf(const ex& n, const ex& p, const ex& x)
2094 {
2095         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2096                 const int n_ = ex_to<numeric>(n).to_int();
2097                 const int p_ = ex_to<numeric>(p).to_int();
2098                 if (is_a<numeric>(x)) {
2099                         const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2100                         const cln::cl_N result = S_num(n_, p_, x_);
2101                         return numeric(result);
2102                 } else {
2103                         ex x_val = x.evalf();
2104                         if (is_a<numeric>(x_val)) {
2105                                 const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
2106                                 const cln::cl_N result = S_num(n_, p_, x_val_);
2107                                 return numeric(result);
2108                         }
2109                 }
2110         }
2111         return S(n, p, x).hold();
2112 }
2113
2114
2115 static ex S_eval(const ex& n, const ex& p, const ex& x)
2116 {
2117         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2118                 if (x == 0) {
2119                         return _ex0;
2120                 }
2121                 if (x == 1) {
2122                         lst m(n+1);
2123                         for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2124                                 m.append(1);
2125                         }
2126                         return zeta(m);
2127                 }
2128                 if (p == 1) {
2129                         return Li(n+1, x);
2130                 }
2131                 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2132                         int n_ = ex_to<numeric>(n).to_int();
2133                         int p_ = ex_to<numeric>(p).to_int();
2134                         const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2135                         const cln::cl_N result = S_num(n_, p_, x_);
2136                         return numeric(result);
2137                 }
2138         }
2139         if (n.is_zero()) {
2140                 // [Kol] (5.3)
2141                 return pow(-log(1-x), p) / factorial(p);
2142         }
2143         return S(n, p, x).hold();
2144 }
2145
2146
2147 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2148 {
2149         if (p == _ex1) {
2150                 return Li(n+1, x).series(rel, order, options);
2151         }
2152
2153         const ex x_pt = x.subs(rel, subs_options::no_pattern);
2154         if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2155                 // First special case: x==0 (derivatives have poles)
2156                 if (x_pt.is_zero()) {
2157                         const symbol s;
2158                         ex ser;
2159                         // manually construct the primitive expansion
2160                         // subsum = Euler-Zagier-Sum is needed
2161                         // dirty hack (slow ...) calculation of subsum:
2162                         std::vector<ex> presubsum, subsum;
2163                         subsum.push_back(0);
2164                         for (int i=1; i<order-1; ++i) {
2165                                 subsum.push_back(subsum[i-1] + numeric(1, i));
2166                         }
2167                         for (int depth=2; depth<p; ++depth) {
2168                                 presubsum = subsum;
2169                                 for (int i=1; i<order-1; ++i) {
2170                                         subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2171                                 }
2172                         }
2173
2174                         for (int i=1; i<order; ++i) {
2175                                 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2176                         }
2177                         // substitute the argument's series expansion
2178                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2179                         // maybe that was terminating, so add a proper order term
2180                         epvector nseq;
2181                         nseq.push_back(expair(Order(_ex1), order));
2182                         ser += pseries(rel, nseq);
2183                         // reexpanding it will collapse the series again
2184                         return ser.series(rel, order);
2185                 }
2186                 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2187                 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2188         }
2189         // all other cases should be safe, by now:
2190         throw do_taylor();  // caught by function::series()
2191 }
2192
2193
2194 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2195 {
2196         GINAC_ASSERT(deriv_param < 3);
2197         if (deriv_param < 2) {
2198                 return _ex0;
2199         }
2200         if (n > 0) {
2201                 return S(n-1, p, x) / x;
2202         } else {
2203                 return S(n, p-1, x) / (1-x);
2204         }
2205 }
2206
2207
2208 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2209 {
2210         c.s << "\\mathrm{S}_{";
2211         n.print(c);
2212         c.s << ",";
2213         p.print(c);
2214         c.s << "}(";
2215         x.print(c);
2216         c.s << ")";
2217 }
2218
2219
2220 REGISTER_FUNCTION(S,
2221                   evalf_func(S_evalf).
2222                   eval_func(S_eval).
2223                   series_func(S_series).
2224                   derivative_func(S_deriv).
2225                   print_func<print_latex>(S_print_latex).
2226                   do_not_evalf_params());
2227
2228
2229 //////////////////////////////////////////////////////////////////////
2230 //
2231 // Harmonic polylogarithm  H(m,x)
2232 //
2233 // helper functions
2234 //
2235 //////////////////////////////////////////////////////////////////////
2236
2237
2238 // anonymous namespace for helper functions
2239 namespace {
2240
2241
2242 // regulates the pole (used by 1/x-transformation)
2243 symbol H_polesign("IMSIGN");
2244
2245
2246 // convert parameters from H to Li representation
2247 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2248 // returns true if some parameters are negative
2249 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2250 {
2251         // expand parameter list
2252         lst mexp;
2253         for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
2254                 if (*it > 1) {
2255                         for (ex count=*it-1; count > 0; count--) {
2256                                 mexp.append(0);
2257                         }
2258                         mexp.append(1);
2259                 } else if (*it < -1) {
2260                         for (ex count=*it+1; count < 0; count++) {
2261                                 mexp.append(0);
2262                         }
2263                         mexp.append(-1);
2264                 } else {
2265                         mexp.append(*it);
2266                 }
2267         }
2268
2269         ex signum = 1;
2270         pf = 1;
2271         bool has_negative_parameters = false;
2272         ex acc = 1;
2273         for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
2274                 if (*it == 0) {
2275                         acc++;
2276                         continue;
2277                 }
2278                 if (*it > 0) {
2279                         m.append((*it+acc-1) * signum);
2280                 } else {
2281                         m.append((*it-acc+1) * signum);
2282                 }
2283                 acc = 1;
2284                 signum = *it;
2285                 pf *= *it;
2286                 if (pf < 0) {
2287                         has_negative_parameters = true;
2288                 }
2289         }
2290         if (has_negative_parameters) {
2291                 for (std::size_t i=0; i<m.nops(); i++) {
2292                         if (m.op(i) < 0) {
2293                                 m.let_op(i) = -m.op(i);
2294                                 s.append(-1);
2295                         } else {
2296                                 s.append(1);
2297                         }
2298                 }
2299         }
2300
2301         return has_negative_parameters;
2302 }
2303
2304
2305 // recursivly transforms H to corresponding multiple polylogarithms
2306 struct map_trafo_H_convert_to_Li : public map_function
2307 {
2308         ex operator()(const ex& e)
2309         {
2310                 if (is_a<add>(e) || is_a<mul>(e)) {
2311                         return e.map(*this);
2312                 }
2313                 if (is_a<function>(e)) {
2314                         std::string name = ex_to<function>(e).get_name();
2315                         if (name == "H") {
2316                                 lst parameter;
2317                                 if (is_a<lst>(e.op(0))) {
2318                                                 parameter = ex_to<lst>(e.op(0));
2319                                 } else {
2320                                         parameter = lst(e.op(0));
2321                                 }
2322                                 ex arg = e.op(1);
2323
2324                                 lst m;
2325                                 lst s;
2326                                 ex pf;
2327                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2328                                         s.let_op(0) = s.op(0) * arg;
2329                                         return pf * Li(m, s).hold();
2330                                 } else {
2331                                         for (std::size_t i=0; i<m.nops(); i++) {
2332                                                 s.append(1);
2333                                         }
2334                                         s.let_op(0) = s.op(0) * arg;
2335                                         return Li(m, s).hold();
2336                                 }
2337                         }
2338                 }
2339                 return e;
2340         }
2341 };
2342
2343
2344 // recursivly transforms H to corresponding zetas
2345 struct map_trafo_H_convert_to_zeta : public map_function
2346 {
2347         ex operator()(const ex& e)
2348         {
2349                 if (is_a<add>(e) || is_a<mul>(e)) {
2350                         return e.map(*this);
2351                 }
2352                 if (is_a<function>(e)) {
2353                         std::string name = ex_to<function>(e).get_name();
2354                         if (name == "H") {
2355                                 lst parameter;
2356                                 if (is_a<lst>(e.op(0))) {
2357                                                 parameter = ex_to<lst>(e.op(0));
2358                                 } else {
2359                                         parameter = lst(e.op(0));
2360                                 }
2361
2362                                 lst m;
2363                                 lst s;
2364                                 ex pf;
2365                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2366                                         return pf * zeta(m, s);
2367                                 } else {
2368                                         return zeta(m);
2369                                 }
2370                         }
2371                 }
2372                 return e;
2373         }
2374 };
2375
2376
2377 // remove trailing zeros from H-parameters
2378 struct map_trafo_H_reduce_trailing_zeros : public map_function
2379 {
2380         ex operator()(const ex& e)
2381         {
2382                 if (is_a<add>(e) || is_a<mul>(e)) {
2383                         return e.map(*this);
2384                 }
2385                 if (is_a<function>(e)) {
2386                         std::string name = ex_to<function>(e).get_name();
2387                         if (name == "H") {
2388                                 lst parameter;
2389                                 if (is_a<lst>(e.op(0))) {
2390                                         parameter = ex_to<lst>(e.op(0));
2391                                 } else {
2392                                         parameter = lst(e.op(0));
2393                                 }
2394                                 ex arg = e.op(1);
2395                                 if (parameter.op(parameter.nops()-1) == 0) {
2396
2397                                         //
2398                                         if (parameter.nops() == 1) {
2399                                                 return log(arg);
2400                                         }
2401
2402                                         //
2403                                         lst::const_iterator it = parameter.begin();
2404                                         while ((it != parameter.end()) && (*it == 0)) {
2405                                                 it++;
2406                                         }
2407                                         if (it == parameter.end()) {
2408                                                 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2409                                         }
2410
2411                                         //
2412                                         parameter.remove_last();
2413                                         std::size_t lastentry = parameter.nops();
2414                                         while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2415                                                 lastentry--;
2416                                         }
2417
2418                                         //
2419                                         ex result = log(arg) * H(parameter,arg).hold();
2420                                         ex acc = 0;
2421                                         for (ex i=0; i<lastentry; i++) {
2422                                                 if (parameter[i] > 0) {
2423                                                         parameter[i]++;
2424                                                         result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2425                                                         parameter[i]--;
2426                                                         acc = 0;
2427                                                 } else if (parameter[i] < 0) {
2428                                                         parameter[i]--;
2429                                                         result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2430                                                         parameter[i]++;
2431                                                         acc = 0;
2432                                                 } else {
2433                                                         acc++;
2434                                                 }
2435                                         }
2436
2437                                         if (lastentry < parameter.nops()) {
2438                                                 result = result / (parameter.nops()-lastentry+1);
2439                                                 return result.map(*this);
2440                                         } else {
2441                                                 return result;
2442                                         }
2443                                 }
2444                         }
2445                 }
2446                 return e;
2447         }
2448 };
2449
2450
2451 // returns an expression with zeta functions corresponding to the parameter list for H
2452 ex convert_H_to_zeta(const lst& m)
2453 {
2454         symbol xtemp("xtemp");
2455         map_trafo_H_reduce_trailing_zeros filter;
2456         map_trafo_H_convert_to_zeta filter2;
2457         return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2458 }
2459
2460
2461 // convert signs form Li to H representation
2462 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2463 {
2464         lst res;
2465         lst::const_iterator itm = m.begin();
2466         lst::const_iterator itx = ++x.begin();
2467         int signum = 1;
2468         pf = _ex1;
2469         res.append(*itm);
2470         itm++;
2471         while (itx != x.end()) {
2472                 signum *= (*itx > 0) ? 1 : -1;
2473                 pf *= signum;
2474                 res.append((*itm) * signum);
2475                 itm++;
2476                 itx++;
2477         }
2478         return res;
2479 }
2480
2481
2482 // multiplies an one-dimensional H with another H
2483 // [ReV] (18)
2484 ex trafo_H_mult(const ex& h1, const ex& h2)
2485 {
2486         ex res;
2487         ex hshort;
2488         lst hlong;
2489         ex h1nops = h1.op(0).nops();
2490         ex h2nops = h2.op(0).nops();
2491         if (h1nops > 1) {
2492                 hshort = h2.op(0).op(0);
2493                 hlong = ex_to<lst>(h1.op(0));
2494         } else {
2495                 hshort = h1.op(0).op(0);
2496                 if (h2nops > 1) {
2497                         hlong = ex_to<lst>(h2.op(0));
2498                 } else {
2499                         hlong = h2.op(0).op(0);
2500                 }
2501         }
2502         for (std::size_t i=0; i<=hlong.nops(); i++) {
2503                 lst newparameter;
2504                 std::size_t j=0;
2505                 for (; j<i; j++) {
2506                         newparameter.append(hlong[j]);
2507                 }
2508                 newparameter.append(hshort);
2509                 for (; j<hlong.nops(); j++) {
2510                         newparameter.append(hlong[j]);
2511                 }
2512                 res += H(newparameter, h1.op(1)).hold();
2513         }
2514         return res;
2515 }
2516
2517
2518 // applies trafo_H_mult recursively on expressions
2519 struct map_trafo_H_mult : public map_function
2520 {
2521         ex operator()(const ex& e)
2522         {
2523                 if (is_a<add>(e)) {
2524                         return e.map(*this);
2525                 }
2526
2527                 if (is_a<mul>(e)) {
2528
2529                         ex result = 1;
2530                         ex firstH;
2531                         lst Hlst;
2532                         for (std::size_t pos=0; pos<e.nops(); pos++) {
2533                                 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2534                                         std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2535                                         if (name == "H") {
2536                                                 for (ex i=0; i<e.op(pos).op(1); i++) {
2537                                                         Hlst.append(e.op(pos).op(0));
2538                                                 }
2539                                                 continue;
2540                                         }
2541                                 } else if (is_a<function>(e.op(pos))) {
2542                                         std::string name = ex_to<function>(e.op(pos)).get_name();
2543                                         if (name == "H") {
2544                                                 if (e.op(pos).op(0).nops() > 1) {
2545                                                         firstH = e.op(pos);
2546                                                 } else {
2547                                                         Hlst.append(e.op(pos));
2548                                                 }
2549                                                 continue;
2550                                         }
2551                                 }
2552                                 result *= e.op(pos);
2553                         }
2554                         if (firstH == 0) {
2555                                 if (Hlst.nops() > 0) {
2556                                         firstH = Hlst[Hlst.nops()-1];
2557                                         Hlst.remove_last();
2558                                 } else {
2559                                         return e;
2560                                 }
2561                         }
2562
2563                         if (Hlst.nops() > 0) {
2564                                 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2565                                 result *= buffer;
2566                                 for (std::size_t i=1; i<Hlst.nops(); i++) {
2567                                         result *= Hlst.op(i);
2568                                 }
2569                                 result = result.expand();
2570                                 map_trafo_H_mult recursion;
2571                                 return recursion(result);
2572                         } else {
2573                                 return e;
2574                         }
2575
2576                 }
2577                 return e;
2578         }
2579 };
2580
2581
2582 // do integration [ReV] (55)
2583 // put parameter 0 in front of existing parameters
2584 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
2585 {
2586         ex h;
2587         std::string name;
2588         if (is_a<function>(e)) {
2589                 name = ex_to<function>(e).get_name();
2590         }
2591         if (name == "H") {
2592                 h = e;
2593         } else {
2594                 for (std::size_t i=0; i<e.nops(); i++) {
2595                         if (is_a<function>(e.op(i))) {
2596                                 std::string name = ex_to<function>(e.op(i)).get_name();
2597                                 if (name == "H") {
2598                                         h = e.op(i);
2599                                 }
2600                         }
2601                 }
2602         }
2603         if (h != 0) {
2604                 lst newparameter = ex_to<lst>(h.op(0));
2605                 newparameter.prepend(0);
2606                 ex addzeta = convert_H_to_zeta(newparameter);
2607                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2608         } else {
2609                 return e * (-H(lst(ex(0)),1/arg).hold());
2610         }
2611 }
2612
2613
2614 // do integration [ReV] (49)
2615 // put parameter 1 in front of existing parameters
2616 ex trafo_H_prepend_one(const ex& e, const ex& arg)
2617 {
2618         ex h;
2619         std::string name;
2620         if (is_a<function>(e)) {
2621                 name = ex_to<function>(e).get_name();
2622         }
2623         if (name == "H") {
2624                 h = e;
2625         } else {
2626                 for (std::size_t i=0; i<e.nops(); i++) {
2627                         if (is_a<function>(e.op(i))) {
2628                                 std::string name = ex_to<function>(e.op(i)).get_name();
2629                                 if (name == "H") {
2630                                         h = e.op(i);
2631                                 }
2632                         }
2633                 }
2634         }
2635         if (h != 0) {
2636                 lst newparameter = ex_to<lst>(h.op(0));
2637                 newparameter.prepend(1);
2638                 return e.subs(h == H(newparameter, h.op(1)).hold());
2639         } else {
2640                 return e * H(lst(ex(1)),1-arg).hold();
2641         }
2642 }
2643
2644
2645 // do integration [ReV] (55)
2646 // put parameter -1 in front of existing parameters
2647 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2648 {
2649         ex h;
2650         std::string name;
2651         if (is_a<function>(e)) {
2652                 name = ex_to<function>(e).get_name();
2653         }
2654         if (name == "H") {
2655                 h = e;
2656         } else {
2657                 for (std::size_t i=0; i<e.nops(); i++) {
2658                         if (is_a<function>(e.op(i))) {
2659                                 std::string name = ex_to<function>(e.op(i)).get_name();
2660                                 if (name == "H") {
2661                                         h = e.op(i);
2662                                 }
2663                         }
2664                 }
2665         }
2666         if (h != 0) {
2667                 lst newparameter = ex_to<lst>(h.op(0));
2668                 newparameter.prepend(-1);
2669                 ex addzeta = convert_H_to_zeta(newparameter);
2670                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2671         } else {
2672                 ex addzeta = convert_H_to_zeta(lst(ex(-1)));
2673                 return (e * (addzeta - H(lst(ex(-1)),1/arg).hold())).expand();
2674         }
2675 }
2676
2677
2678 // do integration [ReV] (55)
2679 // put parameter -1 in front of existing parameters
2680 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
2681 {
2682         ex h;
2683         std::string name;
2684         if (is_a<function>(e)) {
2685                 name = ex_to<function>(e).get_name();
2686         }
2687         if (name == "H") {
2688                 h = e;
2689         } else {
2690                 for (std::size_t i = 0; i < e.nops(); i++) {
2691                         if (is_a<function>(e.op(i))) {
2692                                 std::string name = ex_to<function>(e.op(i)).get_name();
2693                                 if (name == "H") {
2694                                         h = e.op(i);
2695                                 }
2696                         }
2697                 }
2698         }
2699         if (h != 0) {
2700                 lst newparameter = ex_to<lst>(h.op(0));
2701                 newparameter.prepend(-1);
2702                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2703         } else {
2704                 return (e * H(lst(ex(-1)),(1-arg)/(1+arg)).hold()).expand();
2705         }
2706 }
2707
2708
2709 // do integration [ReV] (55)
2710 // put parameter 1 in front of existing parameters
2711 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2712 {
2713         ex h;
2714         std::string name;
2715         if (is_a<function>(e)) {
2716                 name = ex_to<function>(e).get_name();
2717         }
2718         if (name == "H") {
2719                 h = e;
2720         } else {
2721                 for (std::size_t i = 0; i < e.nops(); i++) {
2722                         if (is_a<function>(e.op(i))) {
2723                                 std::string name = ex_to<function>(e.op(i)).get_name();
2724                                 if (name == "H") {
2725                                         h = e.op(i);
2726                                 }
2727                         }
2728                 }
2729         }
2730         if (h != 0) {
2731                 lst newparameter = ex_to<lst>(h.op(0));
2732                 newparameter.prepend(1);
2733                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2734         } else {
2735                 return (e * H(lst(ex(1)),(1-arg)/(1+arg)).hold()).expand();
2736         }
2737 }
2738
2739
2740 // do x -> 1-x transformation
2741 struct map_trafo_H_1mx : public map_function
2742 {
2743         ex operator()(const ex& e)
2744         {
2745                 if (is_a<add>(e) || is_a<mul>(e)) {
2746                         return e.map(*this);
2747                 }
2748
2749                 if (is_a<function>(e)) {
2750                         std::string name = ex_to<function>(e).get_name();
2751                         if (name == "H") {
2752
2753                                 lst parameter = ex_to<lst>(e.op(0));
2754                                 ex arg = e.op(1);
2755
2756                                 // special cases if all parameters are either 0, 1 or -1
2757                                 bool allthesame = true;
2758                                 if (parameter.op(0) == 0) {
2759                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2760                                                 if (parameter.op(i) != 0) {
2761                                                         allthesame = false;
2762                                                         break;
2763                                                 }
2764                                         }
2765                                         if (allthesame) {
2766                                                 lst newparameter;
2767                                                 for (int i=parameter.nops(); i>0; i--) {
2768                                                         newparameter.append(1);
2769                                                 }
2770                                                 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2771                                         }
2772                                 } else if (parameter.op(0) == -1) {
2773                                         throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2774                                 } else {
2775                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2776                                                 if (parameter.op(i) != 1) {
2777                                                         allthesame = false;
2778                                                         break;
2779                                                 }
2780                                         }
2781                                         if (allthesame) {
2782                                                 lst newparameter;
2783                                                 for (int i=parameter.nops(); i>0; i--) {
2784                                                         newparameter.append(0);
2785                                                 }
2786                                                 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2787                                         }
2788                                 }
2789
2790                                 lst newparameter = parameter;
2791                                 newparameter.remove_first();
2792
2793                                 if (parameter.op(0) == 0) {
2794
2795                                         // leading zero
2796                                         ex res = convert_H_to_zeta(parameter);
2797                                         //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2798                                         map_trafo_H_1mx recursion;
2799                                         ex buffer = recursion(H(newparameter, arg).hold());
2800                                         if (is_a<add>(buffer)) {
2801                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2802                                                         res -= trafo_H_prepend_one(buffer.op(i), arg);
2803                                                 }
2804                                         } else {
2805                                                 res -= trafo_H_prepend_one(buffer, arg);
2806                                         }
2807                                         return res;
2808
2809                                 } else {
2810
2811                                         // leading one
2812                                         map_trafo_H_1mx recursion;
2813                                         map_trafo_H_mult unify;
2814                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2815                                         std::size_t firstzero = 0;
2816                                         while (parameter.op(firstzero) == 1) {
2817                                                 firstzero++;
2818                                         }
2819                                         for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) {
2820                                                 lst newparameter;
2821                                                 std::size_t j=0;
2822                                                 for (; j<=i; j++) {
2823                                                         newparameter.append(parameter[j+1]);
2824                                                 }
2825                                                 newparameter.append(1);
2826                                                 for (; j<parameter.nops()-1; j++) {
2827                                                         newparameter.append(parameter[j+1]);
2828                                                 }
2829                                                 res -= H(newparameter, arg).hold();
2830                                         }
2831                                         res = recursion(res).expand() / firstzero;
2832                                         return unify(res);
2833                                 }
2834                         }
2835                 }
2836                 return e;
2837         }
2838 };
2839
2840
2841 // do x -> 1/x transformation
2842 struct map_trafo_H_1overx : public map_function
2843 {
2844         ex operator()(const ex& e)
2845         {
2846                 if (is_a<add>(e) || is_a<mul>(e)) {
2847                         return e.map(*this);
2848                 }
2849
2850                 if (is_a<function>(e)) {
2851                         std::string name = ex_to<function>(e).get_name();
2852                         if (name == "H") {
2853
2854                                 lst parameter = ex_to<lst>(e.op(0));
2855                                 ex arg = e.op(1);
2856
2857                                 // special cases if all parameters are either 0, 1 or -1
2858                                 bool allthesame = true;
2859                                 if (parameter.op(0) == 0) {
2860                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2861                                                 if (parameter.op(i) != 0) {
2862                                                         allthesame = false;
2863                                                         break;
2864                                                 }
2865                                         }
2866                                         if (allthesame) {
2867                                                 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2868                                         }
2869                                 } else if (parameter.op(0) == -1) {
2870                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2871                                                 if (parameter.op(i) != -1) {
2872                                                         allthesame = false;
2873                                                         break;
2874                                                 }
2875                                         }
2876                                         if (allthesame) {
2877                                                 map_trafo_H_mult unify;
2878                                                 return unify((pow(H(lst(ex(-1)),1/arg).hold() - H(lst(ex(0)),1/arg).hold(), parameter.nops())
2879                                                        / factorial(parameter.nops())).expand());
2880                                         }
2881                                 } else {
2882                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2883                                                 if (parameter.op(i) != 1) {
2884                                                         allthesame = false;
2885                                                         break;
2886                                                 }
2887                                         }
2888                                         if (allthesame) {
2889                                                 map_trafo_H_mult unify;
2890                                                 return unify((pow(H(lst(ex(1)),1/arg).hold() + H(lst(ex(0)),1/arg).hold() + H_polesign, parameter.nops())
2891                                                        / factorial(parameter.nops())).expand());
2892                                         }
2893                                 }
2894
2895                                 lst newparameter = parameter;
2896                                 newparameter.remove_first();
2897
2898                                 if (parameter.op(0) == 0) {
2899
2900                                         // leading zero
2901                                         ex res = convert_H_to_zeta(parameter);
2902                                         map_trafo_H_1overx recursion;
2903                                         ex buffer = recursion(H(newparameter, arg).hold());
2904                                         if (is_a<add>(buffer)) {
2905                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2906                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2907                                                 }
2908                                         } else {
2909                                                 res += trafo_H_1tx_prepend_zero(buffer, arg);
2910                                         }
2911                                         return res;
2912
2913                                 } else if (parameter.op(0) == -1) {
2914
2915                                         // leading negative one
2916                                         ex res = convert_H_to_zeta(parameter);
2917                                         map_trafo_H_1overx recursion;
2918                                         ex buffer = recursion(H(newparameter, arg).hold());
2919                                         if (is_a<add>(buffer)) {
2920                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2921                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
2922                                                 }
2923                                         } else {
2924                                                 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
2925                                         }
2926                                         return res;
2927
2928                                 } else {
2929
2930                                         // leading one
2931                                         map_trafo_H_1overx recursion;
2932                                         map_trafo_H_mult unify;
2933                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2934                                         std::size_t firstzero = 0;
2935                                         while (parameter.op(firstzero) == 1) {
2936                                                 firstzero++;
2937                                         }
2938                                         for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) {
2939                                                 lst newparameter;
2940                                                 std::size_t j = 0;
2941                                                 for (; j<=i; j++) {
2942                                                         newparameter.append(parameter[j+1]);
2943                                                 }
2944                                                 newparameter.append(1);
2945                                                 for (; j<parameter.nops()-1; j++) {
2946                                                         newparameter.append(parameter[j+1]);
2947                                                 }
2948                                                 res -= H(newparameter, arg).hold();
2949                                         }
2950                                         res = recursion(res).expand() / firstzero;
2951                                         return unify(res);
2952
2953                                 }
2954
2955                         }
2956                 }
2957                 return e;
2958         }
2959 };
2960
2961
2962 // do x -> (1-x)/(1+x) transformation
2963 struct map_trafo_H_1mxt1px : public map_function
2964 {
2965         ex operator()(const ex& e)
2966         {
2967                 if (is_a<add>(e) || is_a<mul>(e)) {
2968                         return e.map(*this);
2969                 }
2970
2971                 if (is_a<function>(e)) {
2972                         std::string name = ex_to<function>(e).get_name();
2973                         if (name == "H") {
2974
2975                                 lst parameter = ex_to<lst>(e.op(0));
2976                                 ex arg = e.op(1);
2977
2978                                 // special cases if all parameters are either 0, 1 or -1
2979                                 bool allthesame = true;
2980                                 if (parameter.op(0) == 0) {
2981                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2982                                                 if (parameter.op(i) != 0) {
2983                                                         allthesame = false;
2984                                                         break;
2985                                                 }
2986                                         }
2987                                         if (allthesame) {
2988                                                 map_trafo_H_mult unify;
2989                                                 return unify((pow(-H(lst(ex(1)),(1-arg)/(1+arg)).hold() - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
2990                                                        / factorial(parameter.nops())).expand());
2991                                         }
2992                                 } else if (parameter.op(0) == -1) {
2993                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2994                                                 if (parameter.op(i) != -1) {
2995                                                         allthesame = false;
2996                                                         break;
2997                                                 }
2998                                         }
2999                                         if (allthesame) {
3000                                                 map_trafo_H_mult unify;
3001                                                 return unify((pow(log(2) - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3002                                                        / factorial(parameter.nops())).expand());
3003                                         }
3004                                 } else {
3005                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
3006                                                 if (parameter.op(i) != 1) {
3007                                                         allthesame = false;
3008                                                         break;
3009                                                 }
3010                                         }
3011                                         if (allthesame) {
3012                                                 map_trafo_H_mult unify;
3013                                                 return unify((pow(-log(2) - H(lst(ex(0)),(1-arg)/(1+arg)).hold() + H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3014                                                        / factorial(parameter.nops())).expand());
3015                                         }
3016                                 }
3017
3018                                 lst newparameter = parameter;
3019                                 newparameter.remove_first();
3020
3021                                 if (parameter.op(0) == 0) {
3022
3023                                         // leading zero
3024                                         ex res = convert_H_to_zeta(parameter);
3025                                         map_trafo_H_1mxt1px recursion;
3026                                         ex buffer = recursion(H(newparameter, arg).hold());
3027                                         if (is_a<add>(buffer)) {
3028                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
3029                                                         res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3030                                                 }
3031                                         } else {
3032                                                 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3033                                         }
3034                                         return res;
3035
3036                                 } else if (parameter.op(0) == -1) {
3037
3038                                         // leading negative one
3039                                         ex res = convert_H_to_zeta(parameter);
3040                                         map_trafo_H_1mxt1px recursion;
3041                                         ex buffer = recursion(H(newparameter, arg).hold());
3042                                         if (is_a<add>(buffer)) {
3043                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
3044                                                         res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3045                                                 }
3046                                         } else {
3047                                                 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3048                                         }
3049                                         return res;
3050
3051                                 } else {
3052
3053                                         // leading one
3054                                         map_trafo_H_1mxt1px recursion;
3055                                         map_trafo_H_mult unify;
3056                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
3057                                         std::size_t firstzero = 0;
3058                                         while (parameter.op(firstzero) == 1) {
3059                                                 firstzero++;
3060                                         }
3061                                         for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) {
3062                                                 lst newparameter;
3063                                                 std::size_t j=0;
3064                                                 for (; j<=i; j++) {
3065                                                         newparameter.append(parameter[j+1]);
3066                                                 }
3067                                                 newparameter.append(1);
3068                                                 for (; j<parameter.nops()-1; j++) {
3069                                                         newparameter.append(parameter[j+1]);
3070                                                 }
3071                                                 res -= H(newparameter, arg).hold();
3072                                         }
3073                                         res = recursion(res).expand() / firstzero;
3074                                         return unify(res);
3075
3076                                 }
3077
3078                         }
3079                 }
3080                 return e;
3081         }
3082 };
3083
3084
3085 // do the actual summation.
3086 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
3087 {
3088         const int j = m.size();
3089
3090         std::vector<cln::cl_N> t(j);
3091
3092         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3093         cln::cl_N factor = cln::expt(x, j) * one;
3094         cln::cl_N t0buf;
3095         int q = 0;
3096         do {
3097                 t0buf = t[0];
3098                 q++;
3099                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
3100                 for (int k=j-2; k>=1; k--) {
3101                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
3102                 }
3103                 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
3104                 factor = factor * x;
3105         } while (t[0] != t0buf);
3106
3107         return t[0];
3108 }
3109
3110
3111 } // end of anonymous namespace
3112
3113
3114 //////////////////////////////////////////////////////////////////////
3115 //
3116 // Harmonic polylogarithm  H(m,x)
3117 //
3118 // GiNaC function
3119 //
3120 //////////////////////////////////////////////////////////////////////
3121
3122
3123 static ex H_evalf(const ex& x1, const ex& x2)
3124 {
3125         if (is_a<lst>(x1)) {
3126
3127                 cln::cl_N x;
3128                 if (is_a<numeric>(x2)) {
3129                         x = ex_to<numeric>(x2).to_cl_N();
3130                 } else {
3131                         ex x2_val = x2.evalf();
3132                         if (is_a<numeric>(x2_val)) {
3133                                 x = ex_to<numeric>(x2_val).to_cl_N();
3134                         }
3135                 }
3136
3137                 for (std::size_t i = 0; i < x1.nops(); i++) {
3138                         if (!x1.op(i).info(info_flags::integer)) {
3139                                 return H(x1, x2).hold();
3140                         }
3141                 }
3142                 if (x1.nops() < 1) {
3143                         return H(x1, x2).hold();
3144                 }
3145
3146                 const lst& morg = ex_to<lst>(x1);
3147                 // remove trailing zeros ...
3148                 if (*(--morg.end()) == 0) {
3149                         symbol xtemp("xtemp");
3150                         map_trafo_H_reduce_trailing_zeros filter;
3151                         return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3152                 }
3153                 // ... and expand parameter notation
3154                 bool has_minus_one = false;
3155                 lst m;
3156                 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
3157                         if (*it > 1) {
3158                                 for (ex count=*it-1; count > 0; count--) {
3159                                         m.append(0);
3160                                 }
3161                                 m.append(1);
3162                         } else if (*it <= -1) {
3163                                 for (ex count=*it+1; count < 0; count++) {
3164                                         m.append(0);
3165                                 }
3166                                 m.append(-1);
3167                                 has_minus_one = true;
3168                         } else {
3169                                 m.append(*it);
3170                         }
3171                 }
3172
3173                 // do summation
3174                 if (cln::abs(x) < 0.95) {
3175                         lst m_lst;
3176                         lst s_lst;
3177                         ex pf;
3178                         if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3179                                 // negative parameters -> s_lst is filled
3180                                 std::vector<int> m_int;
3181                                 std::vector<cln::cl_N> x_cln;
3182                                 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
3183                                      it_int != m_lst.end(); it_int++, it_cln++) {
3184                                         m_int.push_back(ex_to<numeric>(*it_int).to_int());
3185                                         x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3186                                 }
3187                                 x_cln.front() = x_cln.front() * x;
3188                                 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3189                         } else {
3190                                 // only positive parameters
3191                                 //TODO
3192                                 if (m_lst.nops() == 1) {
3193                                         return Li(m_lst.op(0), x2).evalf();
3194                                 }
3195                                 std::vector<int> m_int;
3196                                 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
3197                                         m_int.push_back(ex_to<numeric>(*it).to_int());
3198                                 }
3199                                 return numeric(H_do_sum(m_int, x));
3200                         }
3201                 }
3202
3203                 symbol xtemp("xtemp");
3204                 ex res = 1;
3205
3206                 // ensure that the realpart of the argument is positive
3207                 if (cln::realpart(x) < 0) {
3208                         x = -x;
3209                         for (std::size_t i = 0; i < m.nops(); i++) {
3210                                 if (m.op(i) != 0) {
3211                                         m.let_op(i) = -m.op(i);
3212                                         res *= -1;
3213                                 }
3214                         }
3215                 }
3216
3217                 // x -> 1/x
3218                 if (cln::abs(x) >= 2.0) {
3219                         map_trafo_H_1overx trafo;
3220                         res *= trafo(H(m, xtemp));
3221                         if (cln::imagpart(x) <= 0) {
3222                                 res = res.subs(H_polesign == -I*Pi);
3223                         } else {
3224                                 res = res.subs(H_polesign == I*Pi);
3225                         }
3226                         return res.subs(xtemp == numeric(x)).evalf();
3227                 }
3228
3229                 // check transformations for 0.95 <= |x| < 2.0
3230
3231                 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3232                 if (cln::abs(x-9.53) <= 9.47) {
3233                         // x -> (1-x)/(1+x)
3234                         map_trafo_H_1mxt1px trafo;
3235                         res *= trafo(H(m, xtemp));
3236                 } else {
3237                         // x -> 1-x
3238                         if (has_minus_one) {
3239                                 map_trafo_H_convert_to_Li filter;
3240                                 return filter(H(m, numeric(x)).hold()).evalf();
3241                         }
3242                         map_trafo_H_1mx trafo;
3243                         res *= trafo(H(m, xtemp));
3244                 }
3245
3246                 return res.subs(xtemp == numeric(x)).evalf();
3247         }
3248
3249         return H(x1,x2).hold();
3250 }
3251
3252
3253 static ex H_eval(const ex& m_, const ex& x)
3254 {
3255         lst m;
3256         if (is_a<lst>(m_)) {
3257                 m = ex_to<lst>(m_);
3258         } else {
3259                 m = lst(m_);
3260         }
3261         if (m.nops() == 0) {
3262                 return _ex1;
3263         }
3264         ex pos1;
3265         ex pos2;
3266         ex n;
3267         ex p;
3268         int step = 0;
3269         if (*m.begin() > _ex1) {
3270                 step++;
3271                 pos1 = _ex0;
3272                 pos2 = _ex1;
3273                 n = *m.begin()-1;
3274                 p = _ex1;
3275         } else if (*m.begin() < _ex_1) {
3276                 step++;
3277                 pos1 = _ex0;
3278                 pos2 = _ex_1;
3279                 n = -*m.begin()-1;
3280                 p = _ex1;
3281         } else if (*m.begin() == _ex0) {
3282                 pos1 = _ex0;
3283                 n = _ex1;
3284         } else {
3285                 pos1 = *m.begin();
3286                 p = _ex1;
3287         }
3288         for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
3289                 if ((*it).info(info_flags::integer)) {
3290                         if (step == 0) {
3291                                 if (*it > _ex1) {
3292                                         if (pos1 == _ex0) {
3293                                                 step = 1;
3294                                                 pos2 = _ex1;
3295                                                 n += *it-1;
3296                                                 p = _ex1;
3297                                         } else {
3298                                                 step = 2;
3299                                         }
3300                                 } else if (*it < _ex_1) {
3301                                         if (pos1 == _ex0) {
3302                                                 step = 1;
3303                                                 pos2 = _ex_1;
3304                                                 n += -*it-1;
3305                                                 p = _ex1;
3306                                         } else {
3307                                                 step = 2;
3308                                         }
3309                                 } else {
3310                                         if (*it != pos1) {
3311                                                 step = 1;
3312                                                 pos2 = *it;
3313                                         }
3314                                         if (*it == _ex0) {
3315                                                 n++;
3316                                         } else {
3317                                                 p++;
3318                                         }
3319                                 }
3320                         } else if (step == 1) {
3321                                 if (*it != pos2) {
3322                                         step = 2;
3323                                 } else {
3324                                         if (*it == _ex0) {
3325                                                 n++;
3326                                         } else {
3327                                                 p++;
3328                                         }
3329                                 }
3330                         }
3331                 } else {
3332                         // if some m_i is not an integer
3333                         return H(m_, x).hold();
3334                 }
3335         }
3336         if ((x == _ex1) && (*(--m.end()) != _ex0)) {
3337                 return convert_H_to_zeta(m);
3338         }
3339         if (step == 0) {
3340                 if (pos1 == _ex0) {
3341                         // all zero
3342                         if (x =