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) && realpart(x[j-1]) > 1) {
1024                                 qlsts.push_back(1);
1025                         } else {
1026                                 qlsts.push_back(-s[j-1]);
1027                         }
1028                 }
1029                 if (qlstx.size() > 0) {
1030                         buffer = buffer*G_numeric(qlstx, qlsts, 1/q);
1031                 }
1032                 std::vector<cln::cl_N> plstx;
1033                 std::vector<int> plsts;
1034                 for (std::size_t j = r+1; j <= size; ++j) {
1035                         plstx.push_back(x[j-1]);
1036                         plsts.push_back(s[j-1]);
1037                 }
1038                 if (plstx.size() > 0) {
1039                         buffer = buffer*G_numeric(plstx, plsts, 1/p);
1040                 }
1041                 result = result + buffer;
1042         }
1043         return result;
1044 }
1045
1046 // convergence transformation, used for numerical evaluation of G function.
1047 // the parameter x, s and y must only contain numerics
1048 static cln::cl_N
1049 G_do_trafo(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1050            const cln::cl_N& y)
1051 {
1052         // sort (|x|<->position) to determine indices
1053         typedef std::multimap<cln::cl_R, std::size_t> sortmap_t;
1054         sortmap_t sortmap;
1055         std::size_t size = 0;
1056         for (std::size_t i = 0; i < x.size(); ++i) {
1057                 if (!zerop(x[i])) {
1058                         sortmap.insert(std::make_pair(abs(x[i]), i));
1059                         ++size;
1060                 }
1061         }
1062         // include upper limit (scale)
1063         sortmap.insert(std::make_pair(abs(y), x.size()));
1064
1065         // generate missing dummy-symbols
1066         int i = 1;
1067         // holding dummy-symbols for the G/Li transformations
1068         exvector gsyms;
1069         gsyms.push_back(symbol("GSYMS_ERROR"));
1070         cln::cl_N lastentry(0);
1071         for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1072                 if (it != sortmap.begin()) {
1073                         if (it->second < x.size()) {
1074                                 if (x[it->second] == lastentry) {
1075                                         gsyms.push_back(gsyms.back());
1076                                         continue;
1077                                 }
1078                         } else {
1079                                 if (y == lastentry) {
1080                                         gsyms.push_back(gsyms.back());
1081                                         continue;
1082                                 }
1083                         }
1084                 }
1085                 std::ostringstream os;
1086                 os << "a" << i;
1087                 gsyms.push_back(symbol(os.str()));
1088                 ++i;
1089                 if (it->second < x.size()) {
1090                         lastentry = x[it->second];
1091                 } else {
1092                         lastentry = y;
1093                 }
1094         }
1095
1096         // fill position data according to sorted indices and prepare substitution list
1097         Gparameter a(x.size());
1098         exmap subslst;
1099         std::size_t pos = 1;
1100         int scale = pos;
1101         for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1102                 if (it->second < x.size()) {
1103                         if (s[it->second] > 0) {
1104                                 a[it->second] = pos;
1105                         } else {
1106                                 a[it->second] = -int(pos);
1107                         }
1108                         subslst[gsyms[pos]] = numeric(x[it->second]);
1109                 } else {
1110                         scale = pos;
1111                         subslst[gsyms[pos]] = numeric(y);
1112                 }
1113                 ++pos;
1114         }
1115
1116         // do transformation
1117         Gparameter pendint;
1118         ex result = G_transform(pendint, a, scale, gsyms);
1119         // replace dummy symbols with their values
1120         result = result.eval().expand();
1121         result = result.subs(subslst).evalf();
1122         if (!is_a<numeric>(result))
1123                 throw std::logic_error("G_do_trafo: G_transform returned non-numeric result");
1124
1125         cln::cl_N ret = ex_to<numeric>(result).to_cl_N();
1126         return ret;
1127 }
1128
1129 // handles the transformations and the numerical evaluation of G
1130 // the parameter x, s and y must only contain numerics
1131 static cln::cl_N
1132 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1133           const cln::cl_N& y)
1134 {
1135         // check for convergence and necessary accelerations
1136         bool need_trafo = false;
1137         bool need_hoelder = false;
1138         bool have_trailing_zero = 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         have_trailing_zero = zerop(x.back());
1153         if (have_trailing_zero) {
1154                 need_trafo = true;
1155                 if (y != 1) {
1156                         need_hoelder = false;
1157                 }
1158         }
1159
1160         if (depth == 1 && x.size() == 2 && !need_trafo)
1161                 return - Li_projection(2, y/x[1], cln::float_format(Digits));
1162
1163         // do acceleration transformation (hoelder convolution [BBB])
1164         if (need_hoelder)
1165                 return G_do_hoelder(x, s, y);
1166
1167         // convergence transformation
1168         if (need_trafo)
1169                 return G_do_trafo(x, s, y);
1170
1171         // do summation
1172         std::vector<cln::cl_N> newx;
1173         newx.reserve(x.size());
1174         std::vector<int> m;
1175         m.reserve(x.size());
1176         int mcount = 1;
1177         int sign = 1;
1178         cln::cl_N factor = y;
1179         for (std::size_t i = 0; i < x.size(); ++i) {
1180                 if (zerop(x[i])) {
1181                         ++mcount;
1182                 } else {
1183                         newx.push_back(factor/x[i]);
1184                         factor = x[i];
1185                         m.push_back(mcount);
1186                         mcount = 1;
1187                         sign = -sign;
1188                 }
1189         }
1190
1191         return sign*multipleLi_do_sum(m, newx);
1192 }
1193
1194
1195 ex mLi_numeric(const lst& m, const lst& x)
1196 {
1197         // let G_numeric do the transformation
1198         std::vector<cln::cl_N> newx;
1199         newx.reserve(x.nops());
1200         std::vector<int> s;
1201         s.reserve(x.nops());
1202         cln::cl_N factor(1);
1203         for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1204                 for (int i = 1; i < *itm; ++i) {
1205                         newx.push_back(cln::cl_N(0));
1206                         s.push_back(1);
1207                 }
1208                 const cln::cl_N xi = ex_to<numeric>(*itx).to_cl_N();
1209                 factor = factor/xi;
1210                 newx.push_back(factor);
1211                 if ( !instanceof(factor, cln::cl_R_ring) && imagpart(factor) < 0 ) {
1212                         s.push_back(-1);
1213                 }
1214                 else {
1215                         s.push_back(1);
1216                 }
1217         }
1218         return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1)));
1219 }
1220
1221
1222 } // end of anonymous namespace
1223
1224
1225 //////////////////////////////////////////////////////////////////////
1226 //
1227 // Generalized multiple polylogarithm  G(x, y) and G(x, s, y)
1228 //
1229 // GiNaC function
1230 //
1231 //////////////////////////////////////////////////////////////////////
1232
1233
1234 static ex G2_evalf(const ex& x_, const ex& y)
1235 {
1236         if (!y.info(info_flags::positive)) {
1237                 return G(x_, y).hold();
1238         }
1239         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1240         if (x.nops() == 0) {
1241                 return _ex1;
1242         }
1243         if (x.op(0) == y) {
1244                 return G(x_, y).hold();
1245         }
1246         std::vector<int> s;
1247         s.reserve(x.nops());
1248         bool all_zero = true;
1249         for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1250                 if (!(*it).info(info_flags::numeric)) {
1251                         return G(x_, y).hold();
1252                 }
1253                 if (*it != _ex0) {
1254                         all_zero = false;
1255                 }
1256                 if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
1257                         s.push_back(-1);
1258                 }
1259                 else {
1260                         s.push_back(1);
1261                 }
1262         }
1263         if (all_zero) {
1264                 return pow(log(y), x.nops()) / factorial(x.nops());
1265         }
1266         std::vector<cln::cl_N> xv;
1267         xv.reserve(x.nops());
1268         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1269                 xv.push_back(ex_to<numeric>(*it).to_cl_N());
1270         cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1271         return numeric(result);
1272 }
1273
1274
1275 static ex G2_eval(const ex& x_, const ex& y)
1276 {
1277         //TODO eval to MZV or H or S or Lin
1278
1279         if (!y.info(info_flags::positive)) {
1280                 return G(x_, y).hold();
1281         }
1282         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1283         if (x.nops() == 0) {
1284                 return _ex1;
1285         }
1286         if (x.op(0) == y) {
1287                 return G(x_, y).hold();
1288         }
1289         std::vector<int> s;
1290         s.reserve(x.nops());
1291         bool all_zero = true;
1292         bool crational = true;
1293         for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1294                 if (!(*it).info(info_flags::numeric)) {
1295                         return G(x_, y).hold();
1296                 }
1297                 if (!(*it).info(info_flags::crational)) {
1298                         crational = false;
1299                 }
1300                 if (*it != _ex0) {
1301                         all_zero = false;
1302                 }
1303                 if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
1304                         s.push_back(-1);
1305                 }
1306                 else {
1307                         s.push_back(+1);
1308                 }
1309         }
1310         if (all_zero) {
1311                 return pow(log(y), x.nops()) / factorial(x.nops());
1312         }
1313         if (!y.info(info_flags::crational)) {
1314                 crational = false;
1315         }
1316         if (crational) {
1317                 return G(x_, y).hold();
1318         }
1319         std::vector<cln::cl_N> xv;
1320         xv.reserve(x.nops());
1321         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1322                 xv.push_back(ex_to<numeric>(*it).to_cl_N());
1323         cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1324         return numeric(result);
1325 }
1326
1327
1328 unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
1329                                 evalf_func(G2_evalf).
1330                                 eval_func(G2_eval).
1331                                 do_not_evalf_params().
1332                                 overloaded(2));
1333 //TODO
1334 //                                derivative_func(G2_deriv).
1335 //                                print_func<print_latex>(G2_print_latex).
1336
1337
1338 static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
1339 {
1340         if (!y.info(info_flags::positive)) {
1341                 return G(x_, s_, y).hold();
1342         }
1343         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1344         lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1345         if (x.nops() != s.nops()) {
1346                 return G(x_, s_, y).hold();
1347         }
1348         if (x.nops() == 0) {
1349                 return _ex1;
1350         }
1351         if (x.op(0) == y) {
1352                 return G(x_, s_, y).hold();
1353         }
1354         std::vector<int> sn;
1355         sn.reserve(s.nops());
1356         bool all_zero = true;
1357         for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1358                 if (!(*itx).info(info_flags::numeric)) {
1359                         return G(x_, y).hold();
1360                 }
1361                 if (!(*its).info(info_flags::real)) {
1362                         return G(x_, y).hold();
1363                 }
1364                 if (*itx != _ex0) {
1365                         all_zero = false;
1366                 }
1367                 if ( ex_to<numeric>(*itx).is_real() ) {
1368                         if ( ex_to<numeric>(*itx).is_positive() ) {
1369                                 if ( *its >= 0 ) {
1370                                         sn.push_back(1);
1371                                 }
1372                                 else {
1373                                         sn.push_back(-1);
1374                                 }
1375                         } else {
1376                                 sn.push_back(1);
1377                         }
1378                 }
1379                 else {
1380                         if ( ex_to<numeric>(*itx).imag() > 0 ) {
1381                                 sn.push_back(1);
1382                         }
1383                         else {
1384                                 sn.push_back(-1);
1385                         }
1386                 }
1387         }
1388         if (all_zero) {
1389                 return pow(log(y), x.nops()) / factorial(x.nops());
1390         }
1391         std::vector<cln::cl_N> xn;
1392         xn.reserve(x.nops());
1393         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1394                 xn.push_back(ex_to<numeric>(*it).to_cl_N());
1395         cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1396         return numeric(result);
1397 }
1398
1399
1400 static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
1401 {
1402         //TODO eval to MZV or H or S or Lin
1403
1404         if (!y.info(info_flags::positive)) {
1405                 return G(x_, s_, y).hold();
1406         }
1407         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1408         lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1409         if (x.nops() != s.nops()) {
1410                 return G(x_, s_, y).hold();
1411         }
1412         if (x.nops() == 0) {
1413                 return _ex1;
1414         }
1415         if (x.op(0) == y) {
1416                 return G(x_, s_, y).hold();
1417         }
1418         std::vector<int> sn;
1419         sn.reserve(s.nops());
1420         bool all_zero = true;
1421         bool crational = true;
1422         for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1423                 if (!(*itx).info(info_flags::numeric)) {
1424                         return G(x_, s_, y).hold();
1425                 }
1426                 if (!(*its).info(info_flags::real)) {
1427                         return G(x_, s_, y).hold();
1428                 }
1429                 if (!(*itx).info(info_flags::crational)) {
1430                         crational = false;
1431                 }
1432                 if (*itx != _ex0) {
1433                         all_zero = false;
1434                 }
1435                 if ( ex_to<numeric>(*itx).is_real() ) {
1436                         if ( ex_to<numeric>(*itx).is_positive() ) {
1437                                 if ( *its >= 0 ) {
1438                                         sn.push_back(1);
1439                                 }
1440                                 else {
1441                                         sn.push_back(-1);
1442                                 }
1443                         } else {
1444                                 sn.push_back(1);
1445                         }
1446                 }
1447                 else {
1448                         if ( ex_to<numeric>(*itx).imag() > 0 ) {
1449                                 sn.push_back(1);
1450                         }
1451                         else {
1452                                 sn.push_back(-1);
1453                         }
1454                 }
1455         }
1456         if (all_zero) {
1457                 return pow(log(y), x.nops()) / factorial(x.nops());
1458         }
1459         if (!y.info(info_flags::crational)) {
1460                 crational = false;
1461         }
1462         if (crational) {
1463                 return G(x_, s_, y).hold();
1464         }
1465         std::vector<cln::cl_N> xn;
1466         xn.reserve(x.nops());
1467         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1468                 xn.push_back(ex_to<numeric>(*it).to_cl_N());
1469         cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1470         return numeric(result);
1471 }
1472
1473
1474 unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
1475                                 evalf_func(G3_evalf).
1476                                 eval_func(G3_eval).
1477                                 do_not_evalf_params().
1478                                 overloaded(2));
1479 //TODO
1480 //                                derivative_func(G3_deriv).
1481 //                                print_func<print_latex>(G3_print_latex).
1482
1483
1484 //////////////////////////////////////////////////////////////////////
1485 //
1486 // Classical polylogarithm and multiple polylogarithm  Li(m,x)
1487 //
1488 // GiNaC function
1489 //
1490 //////////////////////////////////////////////////////////////////////
1491
1492
1493 static ex Li_evalf(const ex& m_, const ex& x_)
1494 {
1495         // classical polylogs
1496         if (m_.info(info_flags::posint)) {
1497                 if (x_.info(info_flags::numeric)) {
1498                         int m__ = ex_to<numeric>(m_).to_int();
1499                         const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1500                         const cln::cl_N result = Lin_numeric(m__, x__);
1501                         return numeric(result);
1502                 } else {
1503                         // try to numerically evaluate second argument
1504                         ex x_val = x_.evalf();
1505                         if (x_val.info(info_flags::numeric)) {
1506                                 int m__ = ex_to<numeric>(m_).to_int();
1507                                 const cln::cl_N x__ = ex_to<numeric>(x_val).to_cl_N();
1508                                 const cln::cl_N result = Lin_numeric(m__, x__);
1509                                 return numeric(result);
1510                         }
1511                 }
1512         }
1513         // multiple polylogs
1514         if (is_a<lst>(m_) && is_a<lst>(x_)) {
1515
1516                 const lst& m = ex_to<lst>(m_);
1517                 const lst& x = ex_to<lst>(x_);
1518                 if (m.nops() != x.nops()) {
1519                         return Li(m_,x_).hold();
1520                 }
1521                 if (x.nops() == 0) {
1522                         return _ex1;
1523                 }
1524                 if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
1525                         return Li(m_,x_).hold();
1526                 }
1527
1528                 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1529                         if (!(*itm).info(info_flags::posint)) {
1530                                 return Li(m_, x_).hold();
1531                         }
1532                         if (!(*itx).info(info_flags::numeric)) {
1533                                 return Li(m_, x_).hold();
1534                         }
1535                         if (*itx == _ex0) {
1536                                 return _ex0;
1537                         }
1538                 }
1539
1540                 return mLi_numeric(m, x);
1541         }
1542
1543         return Li(m_,x_).hold();
1544 }
1545
1546
1547 static ex Li_eval(const ex& m_, const ex& x_)
1548 {
1549         if (is_a<lst>(m_)) {
1550                 if (is_a<lst>(x_)) {
1551                         // multiple polylogs
1552                         const lst& m = ex_to<lst>(m_);
1553                         const lst& x = ex_to<lst>(x_);
1554                         if (m.nops() != x.nops()) {
1555                                 return Li(m_,x_).hold();
1556                         }
1557                         if (x.nops() == 0) {
1558                                 return _ex1;
1559                         }
1560                         bool is_H = true;
1561                         bool is_zeta = true;
1562                         bool do_evalf = true;
1563                         bool crational = true;
1564                         for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1565                                 if (!(*itm).info(info_flags::posint)) {
1566                                         return Li(m_,x_).hold();
1567                                 }
1568                                 if ((*itx != _ex1) && (*itx != _ex_1)) {
1569                                         if (itx != x.begin()) {
1570                                                 is_H = false;
1571                                         }
1572                                         is_zeta = false;
1573                                 }
1574                                 if (*itx == _ex0) {
1575                                         return _ex0;
1576                                 }
1577                                 if (!(*itx).info(info_flags::numeric)) {
1578                                         do_evalf = false;
1579                                 }
1580                                 if (!(*itx).info(info_flags::crational)) {
1581                                         crational = false;
1582                                 }
1583                         }
1584                         if (is_zeta) {
1585                                 lst newx;
1586                                 for (lst::const_iterator itx = x.begin(); itx != x.end(); ++itx) {
1587                                         GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
1588                                         // XXX: 1 + 0.0*I is considered equal to 1. However
1589                                         // the former is a not automatically converted
1590                                         // to a real number. Do the conversion explicitly
1591                                         // to avoid the "numeric::operator>(): complex inequality"
1592                                         // exception (and similar problems).
1593                                         newx.append(*itx != _ex_1 ? _ex1 : _ex_1);
1594                                 }
1595                                 return zeta(m_, newx);
1596                         }
1597                         if (is_H) {
1598                                 ex prefactor;
1599                                 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1600                                 return prefactor * H(newm, x[0]);
1601                         }
1602                         if (do_evalf && !crational) {
1603                                 return mLi_numeric(m,x);
1604                         }
1605                 }
1606                 return Li(m_, x_).hold();
1607         } else if (is_a<lst>(x_)) {
1608                 return Li(m_, x_).hold();
1609         }
1610
1611         // classical polylogs
1612         if (x_ == _ex0) {
1613                 return _ex0;
1614         }
1615         if (x_ == _ex1) {
1616                 return zeta(m_);
1617         }
1618         if (x_ == _ex_1) {
1619                 return (pow(2,1-m_)-1) * zeta(m_);
1620         }
1621         if (m_ == _ex1) {
1622                 return -log(1-x_);
1623         }
1624         if (m_ == _ex2) {
1625                 if (x_.is_equal(I)) {
1626                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
1627                 }
1628                 if (x_.is_equal(-I)) {
1629                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
1630                 }
1631         }
1632         if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1633                 int m__ = ex_to<numeric>(m_).to_int();
1634                 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1635                 const cln::cl_N result = Lin_numeric(m__, x__);
1636                 return numeric(result);
1637         }
1638
1639         return Li(m_, x_).hold();
1640 }
1641
1642
1643 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1644 {
1645         if (is_a<lst>(m) || is_a<lst>(x)) {
1646                 // multiple polylog
1647                 epvector seq;
1648                 seq.push_back(expair(Li(m, x), 0));
1649                 return pseries(rel, seq);
1650         }
1651
1652         // classical polylog
1653         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1654         if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1655                 // First special case: x==0 (derivatives have poles)
1656                 if (x_pt.is_zero()) {
1657                         const symbol s;
1658                         ex ser;
1659                         // manually construct the primitive expansion
1660                         for (int i=1; i<order; ++i)
1661                                 ser += pow(s,i) / pow(numeric(i), m);
1662                         // substitute the argument's series expansion
1663                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1664                         // maybe that was terminating, so add a proper order term
1665                         epvector nseq;
1666                         nseq.push_back(expair(Order(_ex1), order));
1667                         ser += pseries(rel, nseq);
1668                         // reexpanding it will collapse the series again
1669                         return ser.series(rel, order);
1670                 }
1671                 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1672                 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1673         }
1674         // all other cases should be safe, by now:
1675         throw do_taylor();  // caught by function::series()
1676 }
1677
1678
1679 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1680 {
1681         GINAC_ASSERT(deriv_param < 2);
1682         if (deriv_param == 0) {
1683                 return _ex0;
1684         }
1685         if (m_.nops() > 1) {
1686                 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1687         }
1688         ex m;
1689         if (is_a<lst>(m_)) {
1690                 m = m_.op(0);
1691         } else {
1692                 m = m_;
1693         }
1694         ex x;
1695         if (is_a<lst>(x_)) {
1696                 x = x_.op(0);
1697         } else {
1698                 x = x_;
1699         }
1700         if (m > 0) {
1701                 return Li(m-1, x) / x;
1702         } else {
1703                 return 1/(1-x);
1704         }
1705 }
1706
1707
1708 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1709 {
1710         lst m;
1711         if (is_a<lst>(m_)) {
1712                 m = ex_to<lst>(m_);
1713         } else {
1714                 m = lst(m_);
1715         }
1716         lst x;
1717         if (is_a<lst>(x_)) {
1718                 x = ex_to<lst>(x_);
1719         } else {
1720                 x = lst(x_);
1721         }
1722         c.s << "\\mathrm{Li}_{";
1723         lst::const_iterator itm = m.begin();
1724         (*itm).print(c);
1725         itm++;
1726         for (; itm != m.end(); itm++) {
1727                 c.s << ",";
1728                 (*itm).print(c);
1729         }
1730         c.s << "}(";
1731         lst::const_iterator itx = x.begin();
1732         (*itx).print(c);
1733         itx++;
1734         for (; itx != x.end(); itx++) {
1735                 c.s << ",";
1736                 (*itx).print(c);
1737         }
1738         c.s << ")";
1739 }
1740
1741
1742 REGISTER_FUNCTION(Li,
1743                   evalf_func(Li_evalf).
1744                   eval_func(Li_eval).
1745                   series_func(Li_series).
1746                   derivative_func(Li_deriv).
1747                   print_func<print_latex>(Li_print_latex).
1748                   do_not_evalf_params());
1749
1750
1751 //////////////////////////////////////////////////////////////////////
1752 //
1753 // Nielsen's generalized polylogarithm  S(n,p,x)
1754 //
1755 // helper functions
1756 //
1757 //////////////////////////////////////////////////////////////////////
1758
1759
1760 // anonymous namespace for helper functions
1761 namespace {
1762
1763
1764 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1765 // see fill_Yn()
1766 std::vector<std::vector<cln::cl_N> > Yn;
1767 int ynsize = 0; // number of Yn[]
1768 int ynlength = 100; // initial length of all Yn[i]
1769
1770
1771 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1772 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1773 // representing S_{n,p}(x).
1774 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1775 // equivalent Z-sum.
1776 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1777 // representing S_{n,p}(x).
1778 // The calculation of Y_n uses the values from Y_{n-1}.
1779 void fill_Yn(int n, const cln::float_format_t& prec)
1780 {
1781         const int initsize = ynlength;
1782         //const int initsize = initsize_Yn;
1783         cln::cl_N one = cln::cl_float(1, prec);
1784
1785         if (n) {
1786                 std::vector<cln::cl_N> buf(initsize);
1787                 std::vector<cln::cl_N>::iterator it = buf.begin();
1788                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1789                 *it = (*itprev) / cln::cl_N(n+1) * one;
1790                 it++;
1791                 itprev++;
1792                 // sums with an index smaller than the depth are zero and need not to be calculated.
1793                 // calculation starts with depth, which is n+2)
1794                 for (int i=n+2; i<=initsize+n; i++) {
1795                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1796                         it++;
1797                         itprev++;
1798                 }
1799                 Yn.push_back(buf);
1800         } else {
1801                 std::vector<cln::cl_N> buf(initsize);
1802                 std::vector<cln::cl_N>::iterator it = buf.begin();
1803                 *it = 1 * one;
1804                 it++;
1805                 for (int i=2; i<=initsize; i++) {
1806                         *it = *(it-1) + 1 / cln::cl_N(i) * one;
1807                         it++;
1808                 }
1809                 Yn.push_back(buf);
1810         }
1811         ynsize++;
1812 }
1813
1814
1815 // make Yn longer ...
1816 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1817 {
1818
1819         cln::cl_N one = cln::cl_float(1, prec);
1820
1821         Yn[0].resize(newsize);
1822         std::vector<cln::cl_N>::iterator it = Yn[0].begin();
1823         it += ynlength;
1824         for (int i=ynlength+1; i<=newsize; i++) {
1825                 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1826                 it++;
1827         }
1828
1829         for (int n=1; n<ynsize; n++) {
1830                 Yn[n].resize(newsize);
1831                 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
1832                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1833                 it += ynlength;
1834                 itprev += ynlength;
1835                 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1836                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1837                         it++;
1838                         itprev++;
1839                 }
1840         }
1841
1842         ynlength = newsize;
1843 }
1844
1845
1846 // helper function for S(n,p,x)
1847 // [Kol] (7.2)
1848 cln::cl_N C(int n, int p)
1849 {
1850         cln::cl_N result;
1851
1852         for (int k=0; k<p; k++) {
1853                 for (int j=0; j<=(n+k-1)/2; j++) {
1854                         if (k == 0) {
1855                                 if (n & 1) {
1856                                         if (j & 1) {
1857                                                 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1858                                         }
1859                                         else {
1860                                                 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1861                                         }
1862                                 }
1863                         }
1864                         else {
1865                                 if (k & 1) {
1866                                         if (j & 1) {
1867                                                 result = result + cln::factorial(n+k-1)
1868                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1869                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1870                                         }
1871                                         else {
1872                                                 result = result - cln::factorial(n+k-1)
1873                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1874                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1875                                         }
1876                                 }
1877                                 else {
1878                                         if (j & 1) {
1879                                                 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1880                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1881                                         }
1882                                         else {
1883                                                 result = result + cln::factorial(n+k-1)
1884                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1885                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1886                                         }
1887                                 }
1888                         }
1889                 }
1890         }
1891         int np = n+p;
1892         if ((np-1) & 1) {
1893                 if (((np)/2+n) & 1) {
1894                         result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1895                 }
1896                 else {
1897                         result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1898                 }
1899         }
1900
1901         return result;
1902 }
1903
1904
1905 // helper function for S(n,p,x)
1906 // [Kol] remark to (9.1)
1907 cln::cl_N a_k(int k)
1908 {
1909         cln::cl_N result;
1910
1911         if (k == 0) {
1912                 return 1;
1913         }
1914
1915         result = result;
1916         for (int m=2; m<=k; m++) {
1917                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1918         }
1919
1920         return -result / k;
1921 }
1922
1923
1924 // helper function for S(n,p,x)
1925 // [Kol] remark to (9.1)
1926 cln::cl_N b_k(int k)
1927 {
1928         cln::cl_N result;
1929
1930         if (k == 0) {
1931                 return 1;
1932         }
1933
1934         result = result;
1935         for (int m=2; m<=k; m++) {
1936                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1937         }
1938
1939         return result / k;
1940 }
1941
1942
1943 // helper function for S(n,p,x)
1944 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1945 {
1946         static cln::float_format_t oldprec = cln::default_float_format;
1947
1948         if (p==1) {
1949                 return Li_projection(n+1, x, prec);
1950         }
1951
1952         // precision has changed, we need to clear lookup table Yn
1953         if ( oldprec != prec ) {
1954                 Yn.clear();
1955                 ynsize = 0;
1956                 ynlength = 100;
1957                 oldprec = prec;
1958         }
1959
1960         // check if precalculated values are sufficient
1961         if (p > ynsize+1) {
1962                 for (int i=ynsize; i<p-1; i++) {
1963                         fill_Yn(i, prec);
1964                 }
1965         }
1966
1967         // should be done otherwise
1968         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1969         cln::cl_N xf = x * one;
1970         //cln::cl_N xf = x * cln::cl_float(1, prec);
1971
1972         cln::cl_N res;
1973         cln::cl_N resbuf;
1974         cln::cl_N factor = cln::expt(xf, p);
1975         int i = p;
1976         do {
1977                 resbuf = res;
1978                 if (i-p >= ynlength) {
1979                         // make Yn longer
1980                         make_Yn_longer(ynlength*2, prec);
1981                 }
1982                 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? ...
1983                 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
1984                 factor = factor * xf;
1985                 i++;
1986         } while (res != resbuf);
1987
1988         return res;
1989 }
1990
1991
1992 // helper function for S(n,p,x)
1993 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1994 {
1995         // [Kol] (5.3)
1996         if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
1997
1998                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
1999                                    * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
2000
2001                 for (int s=0; s<n; s++) {
2002                         cln::cl_N res2;
2003                         for (int r=0; r<p; r++) {
2004                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
2005                                               * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
2006                         }
2007                         result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2008                 }
2009
2010                 return result;
2011         }
2012
2013         return S_do_sum(n, p, x, prec);
2014 }
2015
2016
2017 // helper function for S(n,p,x)
2018 const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
2019 {
2020         if (x == 1) {
2021                 if (n == 1) {
2022                     // [Kol] (2.22) with (2.21)
2023                         return cln::zeta(p+1);
2024                 }
2025
2026                 if (p == 1) {
2027                     // [Kol] (2.22)
2028                         return cln::zeta(n+1);
2029                 }
2030
2031                 // [Kol] (9.1)
2032                 cln::cl_N result;
2033                 for (int nu=0; nu<n; nu++) {
2034                         for (int rho=0; rho<=p; rho++) {
2035                                 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
2036                                                   * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
2037                         }
2038                 }
2039                 result = result * cln::expt(cln::cl_I(-1),n+p-1);
2040
2041                 return result;
2042         }
2043         else if (x == -1) {
2044                 // [Kol] (2.22)
2045                 if (p == 1) {
2046                         return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
2047                 }
2048 //              throw std::runtime_error("don't know how to evaluate this function!");
2049         }
2050
2051         // what is the desired float format?
2052         // first guess: default format
2053         cln::float_format_t prec = cln::default_float_format;
2054         const cln::cl_N value = x;
2055         // second guess: the argument's format
2056         if (!instanceof(realpart(value), cln::cl_RA_ring))
2057                 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
2058         else if (!instanceof(imagpart(value), cln::cl_RA_ring))
2059                 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
2060
2061         // [Kol] (5.3)
2062         // the condition abs(1-value)>1 avoids an infinite recursion in the region abs(value)<=1 && abs(value)>0.95 && abs(1-value)<=1 && abs(1-value)>0.95
2063         // we don't care here about abs(value)<1 && real(value)>0.5, this will be taken care of in S_projection
2064         if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95) && (cln::abs(1-value) > 1) )) {
2065
2066                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
2067                                    * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
2068
2069                 for (int s=0; s<n; s++) {
2070                         cln::cl_N res2;
2071                         for (int r=0; r<p; r++) {
2072                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
2073                                               * S_num(p-r,n-s,1-value) / cln::factorial(r);
2074                         }
2075                         result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2076                 }
2077
2078                 return result;
2079
2080         }
2081         // [Kol] (5.12)
2082         if (cln::abs(value) > 1) {
2083
2084                 cln::cl_N result;
2085
2086                 for (int s=0; s<p; s++) {
2087                         for (int r=0; r<=s; r++) {
2088                                 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
2089                                                   / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
2090                                                   * S_num(n+s-r,p-s,cln::recip(value));
2091                         }
2092                 }
2093                 result = result * cln::expt(cln::cl_I(-1),n);
2094
2095                 cln::cl_N res2;
2096                 for (int r=0; r<n; r++) {
2097                         res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
2098                 }
2099                 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
2100
2101                 result = result + cln::expt(cln::cl_I(-1),p) * res2;
2102
2103                 return result;
2104         }
2105
2106         if ((cln::abs(value) > 0.95) && (cln::abs(value-9.53) < 9.47)) {
2107                 lst m;
2108                 m.append(n+1);
2109                 for (int s=0; s<p-1; s++)
2110                         m.append(1);
2111
2112                 ex res = H(m,numeric(value)).evalf();
2113                 return ex_to<numeric>(res).to_cl_N();
2114         }
2115         else {
2116                 return S_projection(n, p, value, prec);
2117         }
2118 }
2119
2120
2121 } // end of anonymous namespace
2122
2123
2124 //////////////////////////////////////////////////////////////////////
2125 //
2126 // Nielsen's generalized polylogarithm  S(n,p,x)
2127 //
2128 // GiNaC function
2129 //
2130 //////////////////////////////////////////////////////////////////////
2131
2132
2133 static ex S_evalf(const ex& n, const ex& p, const ex& x)
2134 {
2135         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2136                 const int n_ = ex_to<numeric>(n).to_int();
2137                 const int p_ = ex_to<numeric>(p).to_int();
2138                 if (is_a<numeric>(x)) {
2139                         const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2140                         const cln::cl_N result = S_num(n_, p_, x_);
2141                         return numeric(result);
2142                 } else {
2143                         ex x_val = x.evalf();
2144                         if (is_a<numeric>(x_val)) {
2145                                 const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
2146                                 const cln::cl_N result = S_num(n_, p_, x_val_);
2147                                 return numeric(result);
2148                         }
2149                 }
2150         }
2151         return S(n, p, x).hold();
2152 }
2153
2154
2155 static ex S_eval(const ex& n, const ex& p, const ex& x)
2156 {
2157         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2158                 if (x == 0) {
2159                         return _ex0;
2160                 }
2161                 if (x == 1) {
2162                         lst m(n+1);
2163                         for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2164                                 m.append(1);
2165                         }
2166                         return zeta(m);
2167                 }
2168                 if (p == 1) {
2169                         return Li(n+1, x);
2170                 }
2171                 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2172                         int n_ = ex_to<numeric>(n).to_int();
2173                         int p_ = ex_to<numeric>(p).to_int();
2174                         const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2175                         const cln::cl_N result = S_num(n_, p_, x_);
2176                         return numeric(result);
2177                 }
2178         }
2179         if (n.is_zero()) {
2180                 // [Kol] (5.3)
2181                 return pow(-log(1-x), p) / factorial(p);
2182         }
2183         return S(n, p, x).hold();
2184 }
2185
2186
2187 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2188 {
2189         if (p == _ex1) {
2190                 return Li(n+1, x).series(rel, order, options);
2191         }
2192
2193         const ex x_pt = x.subs(rel, subs_options::no_pattern);
2194         if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2195                 // First special case: x==0 (derivatives have poles)
2196                 if (x_pt.is_zero()) {
2197                         const symbol s;
2198                         ex ser;
2199                         // manually construct the primitive expansion
2200                         // subsum = Euler-Zagier-Sum is needed
2201                         // dirty hack (slow ...) calculation of subsum:
2202                         std::vector<ex> presubsum, subsum;
2203                         subsum.push_back(0);
2204                         for (int i=1; i<order-1; ++i) {
2205                                 subsum.push_back(subsum[i-1] + numeric(1, i));
2206                         }
2207                         for (int depth=2; depth<p; ++depth) {
2208                                 presubsum = subsum;
2209                                 for (int i=1; i<order-1; ++i) {
2210                                         subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2211                                 }
2212                         }
2213
2214                         for (int i=1; i<order; ++i) {
2215                                 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2216                         }
2217                         // substitute the argument's series expansion
2218                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2219                         // maybe that was terminating, so add a proper order term
2220                         epvector nseq;
2221                         nseq.push_back(expair(Order(_ex1), order));
2222                         ser += pseries(rel, nseq);
2223                         // reexpanding it will collapse the series again
2224                         return ser.series(rel, order);
2225                 }
2226                 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2227                 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2228         }
2229         // all other cases should be safe, by now:
2230         throw do_taylor();  // caught by function::series()
2231 }
2232
2233
2234 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2235 {
2236         GINAC_ASSERT(deriv_param < 3);
2237         if (deriv_param < 2) {
2238                 return _ex0;
2239         }
2240         if (n > 0) {
2241                 return S(n-1, p, x) / x;
2242         } else {
2243                 return S(n, p-1, x) / (1-x);
2244         }
2245 }
2246
2247
2248 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2249 {
2250         c.s << "\\mathrm{S}_{";
2251         n.print(c);
2252         c.s << ",";
2253         p.print(c);
2254         c.s << "}(";
2255         x.print(c);
2256         c.s << ")";
2257 }
2258
2259
2260 REGISTER_FUNCTION(S,
2261                   evalf_func(S_evalf).
2262                   eval_func(S_eval).
2263                   series_func(S_series).
2264                   derivative_func(S_deriv).
2265                   print_func<print_latex>(S_print_latex).
2266                   do_not_evalf_params());
2267
2268
2269 //////////////////////////////////////////////////////////////////////
2270 //
2271 // Harmonic polylogarithm  H(m,x)
2272 //
2273 // helper functions
2274 //
2275 //////////////////////////////////////////////////////////////////////
2276
2277
2278 // anonymous namespace for helper functions
2279 namespace {
2280
2281
2282 // regulates the pole (used by 1/x-transformation)
2283 symbol H_polesign("IMSIGN");
2284
2285
2286 // convert parameters from H to Li representation
2287 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2288 // returns true if some parameters are negative
2289 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2290 {
2291         // expand parameter list
2292         lst mexp;
2293         for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
2294                 if (*it > 1) {
2295                         for (ex count=*it-1; count > 0; count--) {
2296                                 mexp.append(0);
2297                         }
2298                         mexp.append(1);
2299                 } else if (*it < -1) {
2300                         for (ex count=*it+1; count < 0; count++) {
2301                                 mexp.append(0);
2302                         }
2303                         mexp.append(-1);
2304                 } else {
2305                         mexp.append(*it);
2306                 }
2307         }
2308
2309         ex signum = 1;
2310         pf = 1;
2311         bool has_negative_parameters = false;
2312         ex acc = 1;
2313         for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
2314                 if (*it == 0) {
2315                         acc++;
2316                         continue;
2317                 }
2318                 if (*it > 0) {
2319                         m.append((*it+acc-1) * signum);
2320                 } else {
2321                         m.append((*it-acc+1) * signum);
2322                 }
2323                 acc = 1;
2324                 signum = *it;
2325                 pf *= *it;
2326                 if (pf < 0) {
2327                         has_negative_parameters = true;
2328                 }
2329         }
2330         if (has_negative_parameters) {
2331                 for (std::size_t i=0; i<m.nops(); i++) {
2332                         if (m.op(i) < 0) {
2333                                 m.let_op(i) = -m.op(i);
2334                                 s.append(-1);
2335                         } else {
2336                                 s.append(1);
2337                         }
2338                 }
2339         }
2340
2341         return has_negative_parameters;
2342 }
2343
2344
2345 // recursivly transforms H to corresponding multiple polylogarithms
2346 struct map_trafo_H_convert_to_Li : public map_function
2347 {
2348         ex operator()(const ex& e)
2349         {
2350                 if (is_a<add>(e) || is_a<mul>(e)) {
2351                         return e.map(*this);
2352                 }
2353                 if (is_a<function>(e)) {
2354                         std::string name = ex_to<function>(e).get_name();
2355                         if (name == "H") {
2356                                 lst parameter;
2357                                 if (is_a<lst>(e.op(0))) {
2358                                                 parameter = ex_to<lst>(e.op(0));
2359                                 } else {
2360                                         parameter = lst(e.op(0));
2361                                 }
2362                                 ex arg = e.op(1);
2363
2364                                 lst m;
2365                                 lst s;
2366                                 ex pf;
2367                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2368                                         s.let_op(0) = s.op(0) * arg;
2369                                         return pf * Li(m, s).hold();
2370                                 } else {
2371                                         for (std::size_t i=0; i<m.nops(); i++) {
2372                                                 s.append(1);
2373                                         }
2374                                         s.let_op(0) = s.op(0) * arg;
2375                                         return Li(m, s).hold();
2376                                 }
2377                         }
2378                 }
2379                 return e;
2380         }
2381 };
2382
2383
2384 // recursivly transforms H to corresponding zetas
2385 struct map_trafo_H_convert_to_zeta : public map_function
2386 {
2387         ex operator()(const ex& e)
2388         {
2389                 if (is_a<add>(e) || is_a<mul>(e)) {
2390                         return e.map(*this);
2391                 }
2392                 if (is_a<function>(e)) {
2393                         std::string name = ex_to<function>(e).get_name();
2394                         if (name == "H") {
2395                                 lst parameter;
2396                                 if (is_a<lst>(e.op(0))) {
2397                                                 parameter = ex_to<lst>(e.op(0));
2398                                 } else {
2399                                         parameter = lst(e.op(0));
2400                                 }
2401
2402                                 lst m;
2403                                 lst s;
2404                                 ex pf;
2405                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2406                                         return pf * zeta(m, s);
2407                                 } else {
2408                                         return zeta(m);
2409                                 }
2410                         }
2411                 }
2412                 return e;
2413         }
2414 };
2415
2416
2417 // remove trailing zeros from H-parameters
2418 struct map_trafo_H_reduce_trailing_zeros : public map_function
2419 {
2420         ex operator()(const ex& e)
2421         {
2422                 if (is_a<add>(e) || is_a<mul>(e)) {
2423                         return e.map(*this);
2424                 }
2425                 if (is_a<function>(e)) {
2426                         std::string name = ex_to<function>(e).get_name();
2427                         if (name == "H") {
2428                                 lst parameter;
2429                                 if (is_a<lst>(e.op(0))) {
2430                                         parameter = ex_to<lst>(e.op(0));
2431                                 } else {
2432                                         parameter = lst(e.op(0));
2433                                 }
2434                                 ex arg = e.op(1);
2435                                 if (parameter.op(parameter.nops()-1) == 0) {
2436
2437                                         //
2438                                         if (parameter.nops() == 1) {
2439                                                 return log(arg);
2440                                         }
2441
2442                                         //
2443                                         lst::const_iterator it = parameter.begin();
2444                                         while ((it != parameter.end()) && (*it == 0)) {
2445                                                 it++;
2446                                         }
2447                                         if (it == parameter.end()) {
2448                                                 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2449                                         }
2450
2451                                         //
2452                                         parameter.remove_last();
2453                                         std::size_t lastentry = parameter.nops();
2454                                         while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2455                                                 lastentry--;
2456                                         }
2457
2458                                         //
2459                                         ex result = log(arg) * H(parameter,arg).hold();
2460                                         ex acc = 0;
2461                                         for (ex i=0; i<lastentry; i++) {
2462                                                 if (parameter[i] > 0) {
2463                                                         parameter[i]++;
2464                                                         result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2465                                                         parameter[i]--;
2466                                                         acc = 0;
2467                                                 } else if (parameter[i] < 0) {
2468                                                         parameter[i]--;
2469                                                         result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2470                                                         parameter[i]++;
2471                                                         acc = 0;
2472                                                 } else {
2473                                                         acc++;
2474                                                 }
2475                                         }
2476
2477                                         if (lastentry < parameter.nops()) {
2478                                                 result = result / (parameter.nops()-lastentry+1);
2479                                                 return result.map(*this);
2480                                         } else {
2481                                                 return result;
2482                                         }
2483                                 }
2484                         }
2485                 }
2486                 return e;
2487         }
2488 };
2489
2490
2491 // returns an expression with zeta functions corresponding to the parameter list for H
2492 ex convert_H_to_zeta(const lst& m)
2493 {
2494         symbol xtemp("xtemp");
2495         map_trafo_H_reduce_trailing_zeros filter;
2496         map_trafo_H_convert_to_zeta filter2;
2497         return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2498 }
2499
2500
2501 // convert signs form Li to H representation
2502 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2503 {
2504         lst res;
2505         lst::const_iterator itm = m.begin();
2506         lst::const_iterator itx = ++x.begin();
2507         int signum = 1;
2508         pf = _ex1;
2509         res.append(*itm);
2510         itm++;
2511         while (itx != x.end()) {
2512                 GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
2513                 // XXX: 1 + 0.0*I is considered equal to 1. However the former
2514                 // is not automatically converted to a real number.
2515                 // Do the conversion explicitly to avoid the
2516                 // "numeric::operator>(): complex inequality" exception.
2517                 signum *= (*itx != _ex_1) ? 1 : -1;
2518                 pf *= signum;
2519                 res.append((*itm) * signum);
2520                 itm++;
2521                 itx++;
2522         }
2523         return res;
2524 }
2525
2526
2527 // multiplies an one-dimensional H with another H
2528 // [ReV] (18)
2529 ex trafo_H_mult(const ex& h1, const ex& h2)
2530 {
2531         ex res;
2532         ex hshort;
2533         lst hlong;
2534         ex h1nops = h1.op(0).nops();
2535         ex h2nops = h2.op(0).nops();
2536         if (h1nops > 1) {
2537                 hshort = h2.op(0).op(0);
2538                 hlong = ex_to<lst>(h1.op(0));
2539         } else {
2540                 hshort = h1.op(0).op(0);
2541                 if (h2nops > 1) {
2542                         hlong = ex_to<lst>(h2.op(0));
2543                 } else {
2544                         hlong = h2.op(0).op(0);
2545                 }
2546         }
2547         for (std::size_t i=0; i<=hlong.nops(); i++) {
2548                 lst newparameter;
2549                 std::size_t j=0;
2550                 for (; j<i; j++) {
2551                         newparameter.append(hlong[j]);
2552                 }
2553                 newparameter.append(hshort);
2554                 for (; j<hlong.nops(); j++) {
2555                         newparameter.append(hlong[j]);
2556                 }
2557                 res += H(newparameter, h1.op(1)).hold();
2558         }
2559         return res;
2560 }
2561
2562
2563 // applies trafo_H_mult recursively on expressions
2564 struct map_trafo_H_mult : public map_function
2565 {
2566         ex operator()(const ex& e)
2567         {
2568                 if (is_a<add>(e)) {
2569                         return e.map(*this);
2570                 }
2571
2572                 if (is_a<mul>(e)) {
2573
2574                         ex result = 1;
2575                         ex firstH;
2576                         lst Hlst;
2577                         for (std::size_t pos=0; pos<e.nops(); pos++) {
2578                                 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2579                                         std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2580                                         if (name == "H") {
2581                                                 for (ex i=0; i<e.op(pos).op(1); i++) {
2582                                                         Hlst.append(e.op(pos).op(0));
2583                                                 }
2584                                                 continue;
2585                                         }
2586                                 } else if (is_a<function>(e.op(pos))) {
2587                                         std::string name = ex_to<function>(e.op(pos)).get_name();
2588                                         if (name == "H") {
2589                                                 if (e.op(pos).op(0).nops() > 1) {
2590                                                         firstH = e.op(pos);
2591                                                 } else {
2592                                                         Hlst.append(e.op(pos));
2593                                                 }
2594                                                 continue;
2595                                         }
2596                                 }
2597                                 result *= e.op(pos);
2598                         }
2599                         if (firstH == 0) {
2600                                 if (Hlst.nops() > 0) {
2601                                         firstH = Hlst[Hlst.nops()-1];
2602                                         Hlst.remove_last();
2603                                 } else {
2604                                         return e;
2605                                 }
2606                         }
2607
2608                         if (Hlst.nops() > 0) {
2609                                 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2610                                 result *= buffer;
2611                                 for (std::size_t i=1; i<Hlst.nops(); i++) {
2612                                         result *= Hlst.op(i);
2613                                 }
2614                                 result = result.expand();
2615                                 map_trafo_H_mult recursion;
2616                                 return recursion(result);
2617                         } else {
2618                                 return e;
2619                         }
2620
2621                 }
2622                 return e;
2623         }
2624 };
2625
2626
2627 // do integration [ReV] (55)
2628 // put parameter 0 in front of existing parameters
2629 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
2630 {
2631         ex h;
2632         std::string name;
2633         if (is_a<function>(e)) {
2634                 name = ex_to<function>(e).get_name();
2635         }
2636         if (name == "H") {
2637                 h = e;
2638         } else {
2639                 for (std::size_t i=0; i<e.nops(); i++) {
2640                         if (is_a<function>(e.op(i))) {
2641                                 std::string name = ex_to<function>(e.op(i)).get_name();
2642                                 if (name == "H") {
2643                                         h = e.op(i);
2644                                 }
2645                         }
2646                 }
2647         }
2648         if (h != 0) {
2649                 lst newparameter = ex_to<lst>(h.op(0));
2650                 newparameter.prepend(0);
2651                 ex addzeta = convert_H_to_zeta(newparameter);
2652                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2653         } else {
2654                 return e * (-H(lst(ex(0)),1/arg).hold());
2655         }
2656 }
2657
2658
2659 // do integration [ReV] (49)
2660 // put parameter 1 in front of existing parameters
2661 ex trafo_H_prepend_one(const ex& e, const ex& arg)
2662 {
2663         ex h;
2664         std::string name;
2665         if (is_a<function>(e)) {
2666                 name = ex_to<function>(e).get_name();
2667         }
2668         if (name == "H") {
2669                 h = e;
2670         } else {
2671                 for (std::size_t i=0; i<e.nops(); i++) {
2672                         if (is_a<function>(e.op(i))) {
2673                                 std::string name = ex_to<function>(e.op(i)).get_name();
2674                                 if (name == "H") {
2675                                         h = e.op(i);
2676                                 }
2677                         }
2678                 }
2679         }
2680         if (h != 0) {
2681                 lst newparameter = ex_to<lst>(h.op(0));
2682                 newparameter.prepend(1);
2683                 return e.subs(h == H(newparameter, h.op(1)).hold());
2684         } else {
2685                 return e * H(lst(ex(1)),1-arg).hold();
2686         }
2687 }
2688
2689
2690 // do integration [ReV] (55)
2691 // put parameter -1 in front of existing parameters
2692 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2693 {
2694         ex h;
2695         std::string name;
2696         if (is_a<function>(e)) {
2697                 name = ex_to<function>(e).get_name();
2698         }
2699         if (name == "H") {
2700                 h = e;
2701         } else {
2702                 for (std::size_t i=0; i<e.nops(); i++) {
2703                         if (is_a<function>(e.op(i))) {
2704                                 std::string name = ex_to<function>(e.op(i)).get_name();
2705                                 if (name == "H") {
2706                                         h = e.op(i);
2707                                 }
2708                         }
2709                 }
2710         }
2711         if (h != 0) {
2712                 lst newparameter = ex_to<lst>(h.op(0));
2713                 newparameter.prepend(-1);
2714                 ex addzeta = convert_H_to_zeta(newparameter);
2715                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2716         } else {
2717                 ex addzeta = convert_H_to_zeta(lst(ex(-1)));
2718                 return (e * (addzeta - H(lst(ex(-1)),1/arg).hold())).expand();
2719         }
2720 }
2721
2722
2723 // do integration [ReV] (55)
2724 // put parameter -1 in front of existing parameters
2725 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
2726 {
2727         ex h;
2728         std::string name;
2729         if (is_a<function>(e)) {
2730                 name = ex_to<function>(e).get_name();
2731         }
2732         if (name == "H") {
2733                 h = e;
2734         } else {
2735                 for (std::size_t i = 0; i < e.nops(); i++) {
2736                         if (is_a<function>(e.op(i))) {
2737                                 std::string name = ex_to<function>(e.op(i)).get_name();
2738                                 if (name == "H") {
2739                                         h = e.op(i);
2740                                 }
2741                         }
2742                 }
2743         }
2744         if (h != 0) {
2745                 lst newparameter = ex_to<lst>(h.op(0));
2746                 newparameter.prepend(-1);
2747                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2748         } else {
2749                 return (e * H(lst(ex(-1)),(1-arg)/(1+arg)).hold()).expand();
2750         }
2751 }
2752
2753
2754 // do integration [ReV] (55)
2755 // put parameter 1 in front of existing parameters
2756 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2757 {
2758         ex h;
2759         std::string name;
2760         if (is_a<function>(e)) {
2761                 name = ex_to<function>(e).get_name();
2762         }
2763         if (name == "H") {
2764                 h = e;
2765         } else {
2766                 for (std::size_t i = 0; i < e.nops(); i++) {
2767                         if (is_a<function>(e.op(i))) {
2768                                 std::string name = ex_to<function>(e.op(i)).get_name();
2769                                 if (name == "H") {
2770                                         h = e.op(i);
2771                                 }
2772                         }
2773                 }
2774         }
2775         if (h != 0) {
2776                 lst newparameter = ex_to<lst>(h.op(0));
2777                 newparameter.prepend(1);
2778                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2779         } else {
2780                 return (e * H(lst(ex(1)),(1-arg)/(1+arg)).hold()).expand();
2781         }
2782 }
2783
2784
2785 // do x -> 1-x transformation
2786 struct map_trafo_H_1mx : public map_function
2787 {
2788         ex operator()(const ex& e)
2789         {
2790                 if (is_a<add>(e) || is_a<mul>(e)) {
2791                         return e.map(*this);
2792                 }
2793
2794                 if (is_a<function>(e)) {
2795                         std::string name = ex_to<function>(e).get_name();
2796                         if (name == "H") {
2797
2798                                 lst parameter = ex_to<lst>(e.op(0));
2799                                 ex arg = e.op(1);
2800
2801                                 // special cases if all parameters are either 0, 1 or -1
2802                                 bool allthesame = true;
2803                                 if (parameter.op(0) == 0) {
2804                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2805                                                 if (parameter.op(i) != 0) {
2806                                                         allthesame = false;
2807                                                         break;
2808                                                 }
2809                                         }
2810                                         if (allthesame) {
2811                                                 lst newparameter;
2812                                                 for (int i=parameter.nops(); i>0; i--) {
2813                                                         newparameter.append(1);
2814                                                 }
2815                                                 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2816                                         }
2817                                 } else if (parameter.op(0) == -1) {
2818                                         throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2819                                 } else {
2820                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2821                                                 if (parameter.op(i) != 1) {
2822                                                         allthesame = false;
2823                                                         break;
2824                                                 }
2825                                         }
2826                                         if (allthesame) {
2827                                                 lst newparameter;
2828                                                 for (int i=parameter.nops(); i>0; i--) {
2829                                                         newparameter.append(0);
2830                                                 }
2831                                                 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2832                                         }
2833                                 }
2834
2835                                 lst newparameter = parameter;
2836                                 newparameter.remove_first();
2837
2838                                 if (parameter.op(0) == 0) {
2839
2840                                         // leading zero
2841                                         ex res = convert_H_to_zeta(parameter);
2842                                         //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2843                                         map_trafo_H_1mx recursion;
2844                                         ex buffer = recursion(H(newparameter, arg).hold());
2845                                         if (is_a<add>(buffer)) {
2846                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2847                                                         res -= trafo_H_prepend_one(buffer.op(i), arg);
2848                                                 }
2849                                         } else {
2850                                                 res -= trafo_H_prepend_one(buffer, arg);
2851                                         }
2852                                         return res;
2853
2854                                 } else {
2855
2856                                         // leading one
2857                                         map_trafo_H_1mx recursion;
2858                                         map_trafo_H_mult unify;
2859                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2860                                         std::size_t firstzero = 0;
2861                                         while (parameter.op(firstzero) == 1) {
2862                                                 firstzero++;
2863                                         }
2864                                         for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) {
2865                                                 lst newparameter;
2866                                                 std::size_t j=0;
2867                                                 for (; j<=i; j++) {
2868                                                         newparameter.append(parameter[j+1]);
2869                                                 }
2870                                                 newparameter.append(1);
2871                                                 for (; j<parameter.nops()-1; j++) {
2872                                                         newparameter.append(parameter[j+1]);
2873                                                 }
2874                                                 res -= H(newparameter, arg).hold();
2875                                         }
2876                                         res = recursion(res).expand() / firstzero;
2877                                         return unify(res);
2878                                 }
2879                         }
2880                 }
2881                 return e;
2882         }
2883 };
2884
2885
2886 // do x -> 1/x transformation
2887 struct map_trafo_H_1overx : public map_function
2888 {
2889         ex operator()(const ex& e)
2890         {
2891                 if (is_a<add>(e) || is_a<mul>(e)) {
2892                         return e.map(*this);
2893                 }
2894
2895                 if (is_a<function>(e)) {
2896                         std::string name = ex_to<function>(e).get_name();
2897                         if (name == "H") {
2898
2899                                 lst parameter = ex_to<lst>(e.op(0));
2900                                 ex arg = e.op(1);
2901
2902                                 // special cases if all parameters are either 0, 1 or -1
2903                                 bool allthesame = true;
2904                                 if (parameter.op(0) == 0) {
2905                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2906                                                 if (parameter.op(i) != 0) {
2907                                                         allthesame = false;
2908                                                         break;
2909                                                 }
2910                                         }
2911                                         if (allthesame) {
2912                                                 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2913                                         }
2914                                 } else if (parameter.op(0) == -1) {
2915                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2916                                                 if (parameter.op(i) != -1) {
2917                                                         allthesame = false;
2918                                                         break;
2919                                                 }
2920                                         }
2921                                         if (allthesame) {
2922                                                 map_trafo_H_mult unify;
2923                                                 return unify((pow(H(lst(ex(-1)),1/arg).hold() - H(lst(ex(0)),1/arg).hold(), parameter.nops())
2924                                                        / factorial(parameter.nops())).expand());
2925                                         }
2926                                 } else {
2927                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2928                                                 if (parameter.op(i) != 1) {
2929                                                         allthesame = false;
2930                                                         break;
2931                                                 }
2932                                         }
2933                                         if (allthesame) {
2934                                                 map_trafo_H_mult unify;
2935                                                 return unify((pow(H(lst(ex(1)),1/arg).hold() + H(lst(ex(0)),1/arg).hold() + H_polesign, parameter.nops())
2936                                                        / factorial(parameter.nops())).expand());
2937                                         }
2938                                 }
2939
2940                                 lst newparameter = parameter;
2941                                 newparameter.remove_first();
2942
2943                                 if (parameter.op(0) == 0) {
2944
2945                                         // leading zero
2946                                         ex res = convert_H_to_zeta(parameter);
2947                                         map_trafo_H_1overx recursion;
2948                                         ex buffer = recursion(H(newparameter, arg).hold());
2949                                         if (is_a<add>(buffer)) {
2950                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2951                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2952                                                 }
2953                                         } else {
2954                                                 res += trafo_H_1tx_prepend_zero(buffer, arg);
2955                                         }
2956                                         return res;
2957
2958                                 } else if (parameter.op(0) == -1) {
2959
2960                                         // leading negative one
2961                                         ex res = convert_H_to_zeta(parameter);
2962                                         map_trafo_H_1overx recursion;
2963                                         ex buffer = recursion(H(newparameter, arg).hold());
2964                                         if (is_a<add>(buffer)) {
2965                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2966                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
2967                                                 }
2968                                         } else {
2969                                                 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
2970                                         }
2971                                         return res;
2972
2973                                 } else {
2974
2975                                         // leading one
2976                                         map_trafo_H_1overx recursion;
2977                                         map_trafo_H_mult unify;
2978                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2979                                         std::size_t firstzero = 0;
2980                                         while (parameter.op(firstzero) == 1) {
2981                                                 firstzero++;
2982                                         }
2983                                         for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) {
2984                                                 lst newparameter;
2985                                                 std::size_t j = 0;
2986                                                 for (; j<=i; j++) {
2987                                                         newparameter.append(parameter[j+1]);
2988                                                 }
2989                                                 newparameter.append(1);
2990                                                 for (; j<parameter.nops()-1; j++) {
2991                                                         newparameter.append(parameter[j+1]);
2992                                                 }
2993                                                 res -= H(newparameter, arg).hold();
2994                                         }
2995                                         res = recursion(res).expand() / firstzero;
2996                                         return unify(res);
2997
2998                                 }
2999
3000                         }
3001                 }
3002                 return e;
3003         }
3004 };
3005
3006
3007 // do x -> (1-x)/(1+x) transformation
3008 struct map_trafo_H_1mxt1px : public map_function
3009 {
3010         ex operator()(const ex& e)
3011         {
3012                 if (is_a<add>(e) || is_a<mul>(e)) {
3013                         return e.map(*this);
3014                 }
3015
3016                 if (is_a<function>(e)) {
3017                         std::string name = ex_to<function>(e).get_name();
3018                         if (name == "H") {
3019
3020                                 lst parameter = ex_to<lst>(e.op(0));
3021                                 ex arg = e.op(1);
3022
3023                                 // special cases if all parameters are either 0, 1 or -1
3024                                 bool allthesame = true;
3025                                 if (parameter.op(0) == 0) {
3026                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
3027                                                 if (parameter.op(i) != 0) {
3028                                                         allthesame = false;
3029                                                         break;
3030                                                 }
3031                                         }
3032                                         if (allthesame) {
3033                                                 map_trafo_H_mult unify;
3034                                                 return unify((pow(-H(lst(ex(1)),(1-arg)/(1+arg)).hold() - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3035                                                        / factorial(parameter.nops())).expand());
3036                                         }
3037                                 } else if (parameter.op(0) == -1) {
3038                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
3039                                                 if (parameter.op(i) != -1) {
3040                                                         allthesame = false;
3041                                                         break;
3042                                                 }
3043                                         }
3044                                         if (allthesame) {
3045                                                 map_trafo_H_mult unify;
3046                                                 return unify((pow(log(2) - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3047                                                        / factorial(parameter.nops())).expand());
3048                                         }
3049                                 } else {
3050                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
3051                                                 if (parameter.op(i) != 1) {
3052                                                         allthesame = false;
3053                                                         break;
3054                                                 }
3055                                         }
3056                                         if (allthesame) {
3057                                                 map_trafo_H_mult unify;
3058                                                 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())
3059                                                        / factorial(parameter.nops())).expand());
3060                                         }
3061                                 }
3062
3063                                 lst newparameter = parameter;
3064                                 newparameter.remove_first();
3065
3066                                 if (parameter.op(0) == 0) {
3067
3068                                         // leading zero
3069                                         ex res = convert_H_to_zeta(parameter);
3070                                         map_trafo_H_1mxt1px recursion;
3071                                         ex buffer = recursion(H(newparameter, arg).hold());
3072                                         if (is_a<add>(buffer)) {
3073                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
3074                                                         res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3075                                                 }
3076                                         } else {
3077                                                 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3078                                         }
3079                                         return res;
3080
3081                                 } else if (parameter.op(0) == -1) {
3082
3083                                         // leading negative one
3084                                         ex res = convert_H_to_zeta(parameter);
3085                                         map_trafo_H_1mxt1px recursion;
3086                                         ex buffer = recursion(H(newparameter, arg).hold());
3087                                         if (is_a<add>(buffer)) {
3088                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
3089                                                         res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3090                                                 }
3091                                         } else {
3092                                                 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3093                                         }
3094                                         return res;
3095
3096                                 } else {
3097
3098                                         // leading one
3099                                         map_trafo_H_1mxt1px recursion;
3100                                         map_trafo_H_mult unify;
3101                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
3102                                         std::size_t firstzero = 0;
3103                                         while (parameter.op(firstzero) == 1) {
3104                                                 firstzero++;
3105                                         }
3106                                         for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) {
3107                                                 lst newparameter;
3108                                                 std::size_t j=0;
3109                                                 for (; j<=i; j++) {
3110                                                         newparameter.append(parameter[j+1]);
3111                                                 }
3112                                                 newparameter.append(1);
3113                                                 for (; j<parameter.nops()-1; j++) {
3114                                                         newparameter.append(parameter[j+1]);
3115                                                 }
3116                                                 res -= H(newparameter, arg).hold();
3117                                         }
3118                                         res = recursion(res).expand() / firstzero;
3119                                         return unify(res);
3120
3121                                 }
3122
3123                         }
3124                 }
3125                 return e;
3126         }
3127 };
3128
3129
3130 // do the actual summation.
3131 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
3132 {
3133         const int j = m.size();
3134
3135         std::vector<cln::cl_N> t(j);
3136
3137         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3138         cln::cl_N factor = cln::expt(x, j) * one;
3139         cln::cl_N t0buf;
3140         int q = 0;
3141         do {
3142                 t0buf = t[0];
3143                 q++;
3144                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
3145                 for (int k=j-2; k>=1; k--) {
3146                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
3147                 }
3148                 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
3149                 factor = factor * x;
3150         } while (t[0] != t0buf);
3151
3152         return t[0];
3153 }
3154
3155
3156 } // end of anonymous namespace
3157
3158
3159 //////////////////////////////////////////////////////////////////////
3160 //
3161 // Harmonic polylogarithm  H(m,x)
3162 //
3163 // GiNaC function
3164 //
3165 //////////////////////////////////////////////////////////////////////
3166
3167
3168 static ex H_evalf(const ex& x1, const ex& x2)
3169 {
3170         if (is_a<lst>(x1)) {
3171
3172                 cln::cl_N x;
3173                 if (is_a<numeric>(x2)) {
3174                         x = ex_to<numeric>(x2).to_cl_N();
3175                 } else {
3176                         ex x2_val = x2.evalf();
3177                         if (is_a<numeric>(x2_val)) {
3178                                 x = ex_to<numeric>(x2_val).to_cl_N();
3179                         }
3180                 }
3181
3182                 for (std::size_t i = 0; i < x1.nops(); i++) {
3183                         if (!x1.op(i).info(info_flags::integer)) {
3184                                 return H(x1, x2).hold();
3185                         }
3186                 }
3187                 if (x1.nops() < 1) {
3188                         return H(x1, x2).hold();
3189                 }
3190
3191                 const lst& morg = ex_to<lst>(x1);
3192                 // remove trailing zeros ...
3193                 if (*(--morg.end()) == 0) {
3194                         symbol xtemp("xtemp");
3195                         map_trafo_H_reduce_trailing_zeros filter;
3196                         return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3197                 }
3198                 // ... and expand parameter notation
3199                 bool has_minus_one = false;
3200                 lst m;
3201                 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
3202                         if (*it > 1) {
3203                                 for (ex count=*it-1; count > 0; count--) {
3204                                         m.append(0);
3205                                 }
3206                                 m.append(1);
3207                         } else if (*it <= -1) {
3208                                 for (ex count=*it+1; count < 0; count++) {
3209                                         m.append(0);
3210                                 }
3211                                 m.append(-1);
3212                                 has_minus_one = true;
3213                         } else {
3214                                 m.append(*it);
3215                         }
3216                 }
3217
3218                 // do summation
3219                 if (cln::abs(x) < 0.95) {
3220                         lst m_lst;
3221                         lst s_lst;
3222                         ex pf;
3223                         if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3224                                 // negative parameters -> s_lst is filled
3225                                 std::vector<int> m_int;
3226                                 std::vector<cln::cl_N> x_cln;
3227                                 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
3228                                      it_int != m_lst.end(); it_int++, it_cln++) {
3229                                         m_int.push_back(ex_to<numeric>(*it_int).to_int());
3230                                         x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3231                                 }
3232                                 x_cln.front() = x_cln.front() * x;
3233                                 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3234                         } else {
3235                                 // only positive parameters
3236                                 //TODO
3237                                 if (m_lst.nops() == 1) {
3238                                         return Li(m_lst.op(0), x2).evalf();
3239                                 }
3240                                 std::vector<int> m_int;
3241                                 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
3242                                         m_int.push_back(ex_to<numeric>(*it).to_int());
3243                                 }
3244                                 return numeric(H_do_sum(m_int, x));
3245                         }
3246                 }
3247
3248                 symbol xtemp("xtemp");
3249                 ex res = 1;
3250
3251                 // ensure that the realpart of the argument is positive
3252                 if (cln::realpart(x) < 0) {
3253                         x = -x;
3254                         for (std::size_t i = 0; i < m.nops(); i++) {
3255                                 if (m.op(i) != 0) {
3256                                         m.let_op(i) = -m.op(i);
3257                                         res *= -1;
3258                                 }
3259                         }
3260                 }
3261
3262                 // x -> 1/x
3263                 if (cln::abs(x) >= 2.0) {
3264                         map_trafo_H_1overx trafo;
3265                         res *= trafo(H(m, xtemp).hold());
3266                         if (cln::imagpart(x) <= 0) {
3267                                 res = res.subs(H_polesign == -I*Pi);
3268                         } else {
3269                                 res = res.subs(H_polesign == I*Pi);
3270                         }
3271                         return res.subs(xtemp == numeric(x)).evalf();
3272                 }
3273
3274                 // check transformations for 0.95 <= |x| < 2.0
3275
3276                 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3277                 if (cln::abs(x-9.53) <= 9.47) {
3278                         // x -> (1-x)/(1+x)
3279                         map_trafo_H_1mxt1px trafo;
3280                         res *= trafo(H(m, xtemp).hold());
3281                 } else {
3282                         // x -> 1-x
3283                         if (has_minus_one) {
3284                                 map_trafo_H_convert_to_Li filter;
3285                                 return filter(H(m, numeric(x)).hold()).evalf();
3286                         }
3287                         map_trafo_H_1mx trafo;
3288                         res *= trafo(H(m, xtemp).hold());
3289                 }
3290
3291                 return res.subs(xtemp == numeric(x)).evalf();
3292         }
3293
3294         return H(x1,x2).hold();
3295 }
3296
3297
3298 static ex H_eval(const ex& m_, const ex& x)
3299 {
3300         lst m;
3301         if (is_a<lst>(m_)) {
3302                 m = ex_to<lst>(m_);
3303         } else {
3304                 m = lst(m_);
3305         }
3306         if (m.nops() == 0) {
3307                 return _ex1;
3308         }
3309         ex pos1;
3310         ex pos2;
3311         ex n;
3312         ex p;
3313         int step = 0;
3314         if (*m.begin() > _ex1) {
3315                 step++;
3316                 pos1 = _ex0;
3317                 pos2 = _ex1;
3318                 n = *m.begin()-1;
3319                 p = _ex1;
3320         } else if (*m.begin() < _ex_1) {
3321                 step++;
3322                 pos1 = _ex0;
3323                 pos2 = _ex_1;
3324                 n = -*m.begin()-1;
3325                 p = _ex1;
3326         } else if (*m.begin() == _ex0) {
3327                 pos1 = _ex0;
3328                 n = _ex1;
3329         } else {
3330                 pos1 = *m.begin();
3331                 p = _ex1;
3332         }
3333         for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
3334                 if ((*it).info(info_flags::integer)) {
3335                         if (step == 0) {
3336                                 if (*it > _ex1) {
3337                                         if (pos1 == _ex0) {
3338                                                 step = 1;
3339                                                 pos2 = _ex1;
3340                                                 n += *it-1;
3341                                                 p = _ex1;
3342                                         } else {
3343                                                 step = 2;
3344                                         }
3345                                 } else if (*it < _ex_1) {
3346                                         if (pos1 == _ex0) {
3347                                                 step = 1;
3348                                                 pos2 = _ex_1;
3349                                                 n += -*it-1;
3350                                                 p = _ex1;
3351                                         } else {
3352                                                 step = 2;
3353                                         }
3354                                 } else {
3355                                         if (*it != pos1) {
3356                                                 step = 1;
3357                                                 pos2 = *it;
3358                                         }
3359                                         if (*it == _ex0) {
3360                                                 n++;
3361                                         } else {
3362                                                 p++;
3363                                         }
3364                                 }
3365                         } else if (step == 1) {
3366                                 if (*it != pos2) {
3367                                         step = 2;
3368                                 } else {
3369                                         if (*it == _ex0) {
3370                                                 n++;
3371                                         } else {
3372                                                 p++;
3373                                         }
3374                                 }
3375                         }
3376                 } else {
3377                         // if some m_i is not an integer
3378                         return H(m_, x).hold();
3379                 }
3380         }
3381         if ((x == _ex1) && (*(--m.end()) != _ex0)) {
3382                 return convert_H_to_zeta(m);
3383         }
3384         if (step == 0) {
3385                 if (pos1 == _ex0) {
3386                         // all zero
3387                         if (x == _ex0) {
3388                                 return H(m_, x).hold();
3389                         }
3390                         return pow(log(x), m.nops()) / factorial(m.nops());
3391                 } else {
3392                         // all (minus) one
3393                         return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
3394                 }
3395         } else if ((step == 1) && (pos1 == _ex0)){
3396                 // convertible to S
3397                 if (pos2 == _ex1) {
3398                         return S(n, p, x);
3399                 } else {
3400                         return pow(-1, p) * S(n, p, -x);
3401                 }
3402         }
3403         if (x == _ex0) {
3404                 return _ex0;
3405         }
3406         if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
3407                 return H(m_, x).evalf();
3408         }
3409         return H(m_, x).hold();
3410 }
3411
3412
3413 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
3414 {
3415         epvector seq;
3416         seq.push_back(expair(H(m, x), 0));
3417         return pseries(rel, seq);
3418 }
3419
3420
3421 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
3422 {
3423         GINAC_ASSERT(deriv_param < 2);
3424         if (deriv_param == 0) {
3425                 return _ex0;
3426         }
3427         lst m;
3428         if (is_a<lst>(m_)) {
3429                 m = ex_to<lst>(m_);
3430         } else {
3431                 m = lst(m_);
3432         }
3433         ex mb = *m.begin();
3434         if (mb > _ex1) {
3435                 m[0]--;
3436                 return H(m, x) / x;
3437         }
3438         if (mb < _ex_1) {
3439                 m[0]++;
3440                 return H(m, x) / x;
3441         }
3442         m.remove_first();
3443         if (mb == _ex1) {
3444                 return 1/(1-x) * H(m, x);
3445         } else if (mb == _ex_1) {
3446                 return 1/(1+x) * H(m, x);
3447         } else {
3448                 return H(m, x) / x;
3449         }
3450 }
3451
3452
3453 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
3454 {
3455         lst m;
3456         if (is_a<lst>(m_)) {
3457                 m = ex_to<lst>(m_);
3458         } else {
3459                 m = lst(m_);
3460         }
3461         c.s << "\\mathrm{H}_{";
3462         lst::const_iterator itm = m.begin();
3463         (*itm).print(c);
3464         itm++;
3465         for (; itm != m.end(); itm++) {
3466                 c.s << ",";
3467                 (*itm).print(c);
3468         }
3469         c.s << "}(";
3470         x.print(c);
3471         c.s << ")";
3472 }
3473
3474
3475 REGISTER_FUNCTION(H,
3476                   evalf_func(H_evalf).
3477                   eval_func(H_eval).
3478                   series_func(H_series).
3479                   derivative_func(H_deriv).
3480                   print_func<print_latex>(H_print_latex).
3481                   do_not_evalf_params());
3482
3483
3484 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
3485 ex convert_H_to_Li(const ex& m, const ex& x)
3486 {
3487         map_trafo_H_reduce_trailing_zeros filter;
3488         map_trafo_H_convert_to_Li filter2;
3489         if (is_a<lst>(m)) {
3490                 return filter2(filter(H(m, x).hold()));
3491         } else {
3492                 return filter2(filter(H(lst(m), x).hold()));
3493         }
3494 }
3495
3496
3497 //////////////////////////////////////////////////////////////////////
3498 //
3499 // Multiple zeta values  zeta(x) and zeta(x,s)
3500 //
3501 // helper functions
3502 //
3503 //////////////////////////////////////////////////////////////////////
3504
3505
3506 // anonymous namespace for helper functions
3507 namespace {
3508
3509
3510 // parameters and data for [Cra] algorithm
3511 const cln::cl_N lambda = cln::cl_N("319/320");
3512
3513 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
3514 {
3515         const int size = a.size();
3516         for (int n=0; n<size; n++) {
3517                 c[n] = 0;
3518                 for (int m=0; m<=n; m++) {
3519                         c[n] = c[n] + a[m]*b[n-m];
3520                 }
3521         }
3522 }
3523
3524
3525 // [Cra] section 4
3526 static void initcX(std::vector<cln::cl_N>& crX,
3527                    const std::vector<int>& s,
3528                    const int L2)
3529 {
3530         std::vector<cln::cl_N> crB(L2 + 1);
3531         for (int i=0; i<=L2; i++)
3532                 crB[i] = bernoulli(i).to_cl_N() / cln::factorial(i);
3533
3534         int Sm = 0;
3535         int Smp1 = 0;
3536         std::vector<std::vector<cln::cl_N> > crG(s.size() - 1, std::vector<cln::cl_N>(L2 + 1));
3537         for (int m=0; m < (int)s.size() - 1; m++) {
3538                 Sm += s[m];
3539                 Smp1 = Sm + s[m+1];
3540                 for (int i = 0; i <= L2; i++)
3541                         crG[m][i] = cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2);
3542         }
3543
3544         crX = crB;
3545
3546         for (std::size_t m = 0; m < s.size() - 1; m++) {
3547                 std::vector<cln::cl_N> Xbuf(L2 + 1);
3548                 for (int i = 0; i <= L2; i++)
3549                         Xbuf[i] = crX[i] * crG[m][i];
3550
3551                 halfcyclic_convolute(Xbuf, crB, crX);
3552         }
3553 }
3554
3555
3556 // [Cra] section 4
3557 static cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk,
3558                                  const std::vector<cln::cl_N>& crX)
3559 {
3560         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3561         cln::cl_N factor = cln::expt(lambda, Sqk);
3562         cln::cl_N res = factor / Sqk * crX[0] * one;
3563         cln::cl_N resbuf;
3564         int N = 0;
3565         do {
3566                 resbuf = res;
3567                 factor = factor * lambda;
3568                 N++;
3569                 res = res + crX[N] * factor / (N+Sqk);
3570         } while ((res != resbuf) || cln::zerop(crX[N]));
3571         return res;
3572 }
3573
3574
3575 // [Cra] section 4
3576 static void calc_f(std::vector<std::vector<cln::cl_N> >& f_kj,
3577                    const int maxr, const int L1)
3578 {
3579         cln::cl_N t0, t1, t2, t3, t4;
3580         int i, j, k;
3581         std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
3582         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3583
3584         t0 = cln::exp(-lambda);
3585         t2 = 1;
3586         for (k=1; k<=L1; k++) {
3587                 t1 = k * lambda;
3588                 t2 = t0 * t2;
3589                 for (j=1; j<=maxr; j++) {
3590                         t3 = 1;
3591                         t4 = 1;
3592                         for (i=2; i<=j; i++) {
3593                                 t4 = t4 * (j-i+1);
3594                                 t3 = t1 * t3 + t4;
3595                         }
3596                         (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
3597                 }
3598                 it++;
3599         }
3600 }
3601
3602
3603 // [Cra] (3.1)
3604 static cln::cl_N crandall_Z(const std::vector<int>& s,
3605                             const std::vector<std::vector<cln::cl_N> >& f_kj)
3606 {
3607         const int j = s.size();
3608
3609         if (j == 1) {
3610                 cln::cl_N t0;
3611                 cln::cl_N t0buf;
3612                 int q = 0;
3613                 do {
3614                         t0buf = t0;
3615                         q++;
3616                         t0 = t0 + f_kj[q+j-2][s[0]-1];
3617                 } while (t0 != t0buf);
3618
3619                 return t0 / cln::factorial(s[0]-1);
3620         }
3621
3622         std::vector<cln::cl_N> t(j);
3623
3624         cln::cl_N t0buf;
3625         int q = 0;
3626         do {
3627                 t0buf = t[0];
3628                 q++;
3629                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
3630                 for (int k=j-2; k>=1; k--) {
3631                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
3632                 }
3633                 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
3634         } while (t[0] != t0buf);
3635
3636         return t[0] / cln::factorial(s[0]-1);
3637 }
3638
3639
3640 // [Cra] (2.4)
3641 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
3642 {
3643         std::vector<int> r = s;
3644         const int j = r.size();
3645
3646         std::size_t L1;
3647
3648         // decide on maximal size of f_kj for crandall_Z
3649         if (Digits < 50) {
3650                 L1 = 150;
3651         } else {
3652                 L1 = Digits * 3 + j*2;
3653         }
3654
3655         std::size_t L2;
3656         // decide on maximal size of crX for crandall_Y
3657         if (Digits < 38) {
3658                 L2 = 63;
3659         } else if (Digits < 86) {
3660                 L2 = 127;
3661         } else if (Digits < 192) {
3662                 L2 = 255;
3663         } else if (Digits < 394) {
3664                 L2 = 511;
3665         } else if (Digits < 808) {
3666                 L2 = 1023;
3667         } else {
3668                 L2 = 2047;
3669         }
3670
3671         cln::cl_N res;
3672
3673         int maxr = 0;
3674         int S = 0;
3675         for (int i=0; i<j; i++) {
3676                 S += r[i];
3677                 if (r[i] > maxr) {
3678                         maxr = r[i];
3679                 }
3680         }
3681
3682         std::vector<std::vector<cln::cl_N> > f_kj(L1);
3683         calc_f(f_kj, maxr, L1);
3684
3685         const cln::cl_N r0factorial = cln::factorial(r[0]-1);
3686
3687         std::vector<int> rz;
3688         int skp1buf;
3689         int Srun = S;
3690         for (int k=r.size()-1; k>0; k--) {
3691
3692                 rz.insert(rz.begin(), r.back());
3693                 skp1buf = rz.front();
3694                 Srun -= skp1buf;
3695                 r.pop_back();
3696
3697                 std::vector<cln::cl_N> crX;
3698                 initcX(crX, r, L2);
3699
3700                 for (int q=0; q<skp1buf; q++) {
3701
3702                         cln::cl_N pp1 = crandall_Y_loop(Srun+q-k, crX);
3703                         cln::cl_N pp2 = crandall_Z(rz, f_kj);
3704
3705                         rz.front()--;
3706
3707                         if (q & 1) {
3708                                 res = res - pp1 * pp2 / cln::factorial(q);
3709                         } else {
3710                                 res = res + pp1 * pp2 / cln::factorial(q);
3711                         }
3712                 }
3713                 rz.front() = skp1buf;
3714         }
3715         rz.insert(rz.begin(), r.back());
3716
3717         std::vector<cln::cl_N> crX;
3718         initcX(crX, rz, L2);
3719
3720         res = (res + crandall_Y_loop(S-j, crX)) / r0factorial
3721                 + crandall_Z(rz, f_kj);
3722
3723         return res;
3724 }
3725
3726
3727 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
3728 {
3729         const int j = r.size();
3730
3731         // buffer for subsums
3732         std::vector<cln::cl_N> t(j);
3733         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3734
3735         cln::cl_N t0buf;
3736         int q = 0;
3737         do {
3738                 t0buf = t[0];
3739                 q++;
3740                 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
3741                 for (int k=j-2; k>=0; k--) {
3742                         t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
3743                 }
3744         } while (t[0] != t0buf);
3745
3746         return t[0];
3747 }
3748
3749
3750 // does Hoelder convolution. see [BBB] (7.0)
3751 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
3752 {
3753         // prepare parameters
3754         // holds Li arguments in [BBB] notation
3755         std::vector<int> s = s_;
3756         std::vector<int> m_p = m_;
3757         std::vector<int> m_q;
3758         // holds Li arguments in nested sums notation
3759         std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
3760         s_p[0] = s_p[0] * cln::cl_N("1/2");
3761         // convert notations
3762         int sig = 1;
3763         for (std::size_t i = 0; i < s_.size(); i++) {
3764                 if (s_[i] < 0) {
3765                         sig = -sig;
3766                         s_p[i] = -s_p[i];
3767                 }
3768                 s[i] = sig * std::abs(s[i]);
3769         }
3770         std::vector<cln::cl_N> s_q;
3771         cln::cl_N signum = 1;
3772
3773         // first term
3774         cln::cl_N res = multipleLi_do_sum(m_p, s_p);
3775
3776         // middle terms
3777         do {
3778
3779                 // change parameters
3780                 if (s.front() > 0) {
3781                         if (m_p.front() == 1) {
3782                                 m_p.erase(m_p.begin());
3783                                 s_p.erase(s_p.begin());
3784                                 if (s_p.size() > 0) {
3785                                         s_p.front() = s_p.front() * cln::cl_N("1/2");
3786                                 }
3787                                 s.erase(s.begin());
3788                                 m_q.front()++;
3789                         } else {
3790                                 m_p.front()--;
3791                                 m_q.insert(m_q.begin(), 1);
3792                                 if (s_q.size() > 0) {
3793                                         s_q.front() = s_q.front() * 2;
3794                                 }
3795                                 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3796                         }
3797                 } else {
3798                         if (m_p.front() == 1) {
3799                                 m_p.erase(m_p.begin());
3800                                 cln::cl_N spbuf = s_p.front();
3801                                 s_p.erase(s_p.begin());
3802                                 if (s_p.size() > 0) {
3803                                         s_p.front() = s_p.front() * spbuf;
3804                                 }
3805                                 s.erase(s.begin());
3806                                 m_q.insert(m_q.begin(), 1);
3807                                 if (s_q.size() > 0) {
3808                                         s_q.front() = s_q.front() * 4;
3809                                 }
3810                                 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
3811                                 signum = -signum;
3812                         } else {
3813                                 m_p.front()--;
3814                                 m_q.insert(m_q.begin(), 1);
3815                                 if (s_q.size() > 0) {
3816                                         s_q.front() = s_q.front() * 2;
3817                                 }
3818                                 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3819                         }
3820                 }
3821
3822                 // exiting the loop
3823                 if (m_p.size() == 0) break;
3824
3825                 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
3826
3827         } while (true);
3828
3829         // last term
3830         res = res + signum * multipleLi_do_sum(m_q, s_q);
3831
3832         return res;
3833 }
3834
3835
3836 } // end of anonymous namespace
3837
3838
3839 //////////////////////////////////////////////////////////////////////
3840 //
3841 // Multiple zeta values  zeta(x)
3842 //
3843 // GiNaC function
3844 //
3845 //////////////////////////////////////////////////////////////////////
3846
3847
3848 static ex zeta1_evalf(const ex& x)
3849 {
3850         if (is_exactly_a<lst>(x) && (x.nops()>1)) {
3851
3852                 // multiple zeta value
3853                 const int count = x.nops();
3854                 const lst& xlst = ex_to<lst>(x);
3855                 std::vector<int> r(count);
3856
3857                 // check parameters and convert them
3858                 lst::const_iterator it1 = xlst.begin();
3859                 std::vector<int>::iterator it2 = r.begin();
3860                 do {
3861                         if (!(*it1).info(info_flags::posint)) {
3862                                 return zeta(x).hold();
3863                         }
3864                         *it2 = ex_to<numeric>(*it1).to_int();
3865                         it1++;
3866                         it2++;
3867                 } while (it2 != r.end());
3868
3869                 // check for divergence
3870                 if (r[0] == 1) {
3871                         return zeta(x).hold();
3872                 }
3873
3874                 // decide on summation algorithm
3875                 // this is still a bit clumsy
3876                 int limit = (Digits>17) ? 10 : 6;
3877                 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
3878                         return numeric(zeta_do_sum_Crandall(r));
3879                 } else {
3880                         return numeric(zeta_do_sum_simple(r));
3881                 }
3882         }
3883
3884         // single zeta value
3885         if (is_exactly_a<numeric>(x) && (x != 1)) {
3886                 try {
3887                         return zeta(ex_to<numeric>(x));
3888                 } catch (const dunno &e) { }
3889         }
3890
3891         return zeta(x).hold();
3892 }
3893
3894
3895 static ex zeta1_eval(const ex& m)
3896 {
3897         if (is_exactly_a<lst>(m)) {
3898                 if (m.nops() == 1) {
3899                         return zeta(m.op(0));
3900                 }
3901                 return zeta(m).hold();
3902         }
3903
3904         if (m.info(info_flags::numeric)) {
3905                 const numeric& y = ex_to<numeric>(m);
3906                 // trap integer arguments:
3907                 if (y.is_integer()) {
3908                         if (y.is_zero()) {
3909                                 return _ex_1_2;
3910                         }
3911                         if (y.is_equal(*_num1_p)) {
3912                                 return zeta(m).hold();
3913                         }
3914                         if (y.info(info_flags::posint)) {
3915                                 if (y.info(info_flags::odd)) {
3916                                         return zeta(m).hold();
3917                                 } else {
3918                                         return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
3919                                 }
3920                         } else {
3921                                 if (y.info(info_flags::odd)) {
3922                                         return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
3923                                 } else {
3924                                         return _ex0;
3925                                 }
3926                         }
3927                 }
3928                 // zeta(float)
3929                 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
3930                         return zeta1_evalf(m);
3931                 }
3932         }
3933         return zeta(m).hold();
3934 }
3935
3936
3937 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
3938 {
3939         GINAC_ASSERT(deriv_param==0);
3940
3941         if (is_exactly_a<lst>(m)) {
3942                 return _ex0;
3943         } else {
3944                 return zetaderiv(_ex1, m);
3945         }
3946 }
3947
3948
3949 static void zeta1_print_latex(const ex& m_, const print_context& c)
3950 {
3951         c.s << "\\zeta(";
3952         if (is_a<lst>(m_)) {
3953                 const lst& m = ex_to<lst>(m_);
3954                 lst::const_iterator it = m.begin();
3955                 (*it).print(c);
3956                 it++;
3957                 for (; it != m.end(); it++) {
3958                         c.s << ",";
3959                         (*it).print(c);
3960                 }
3961         } else {
3962                 m_.print(c);
3963         }
3964         c.s << ")";
3965 }
3966
3967
3968 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
3969                                 evalf_func(zeta1_evalf).
3970                                 eval_func(zeta1_eval).
3971                                 derivative_func(zeta1_deriv).
3972                                 print_func<print_latex>(zeta1_print_latex).
3973                                 do_not_evalf_params().
3974                                 overloaded(2));
3975
3976
3977 //////////////////////////////////////////////////////////////////////
3978 //
3979 // Alternating Euler sum  zeta(x,s)
3980 //
3981 // GiNaC function
3982 //
3983 //////////////////////////////////////////////////////////////////////
3984
3985
3986 static ex zeta2_evalf(const ex& x, const ex& s)
3987 {
3988         if (is_exactly_a<lst>(x)) {
3989
3990                 // alternating Euler sum
3991                 const int count = x.nops();
3992                 const lst& xlst = ex_to<lst>(x);
3993                 const lst& slst = ex_to<lst>(s);
3994                 std::vector<int> xi(count);
3995                 std::vector<int> si(count);
3996
3997                 // check parameters and convert them
3998                 lst::const_iterator it_xread = xlst.begin();
3999                 lst::const_iterator it_sread = slst.begin();
4000                 std::vector<int>::iterator it_xwrite = xi.begin();
4001                 std::vector<int>::iterator it_swrite = si.begin();
4002                 do {
4003                         if (!(*it_xread).info(info_flags::posint)) {
4004                                 return zeta(x, s).hold();
4005                         }
4006                         *it_xwrite = ex_to<numeric>(*it_xread).to_int();
4007                         if (*it_sread > 0) {
4008                                 *it_swrite = 1;
4009                         } else {
4010                                 *it_swrite = -1;
4011                         }
4012                         it_xread++;
4013                         it_sread++;
4014                         it_xwrite++;
4015                         it_swrite++;
4016                 } while (it_xwrite != xi.end());
4017
4018                 // check for divergence
4019                 if ((xi[0] == 1) && (si[0] == 1)) {
4020                         return zeta(x, s).hold();
4021                 }
4022
4023                 // use Hoelder convolution
4024                 return numeric(zeta_do_Hoelder_convolution(xi, si));
4025         }
4026
4027         return zeta(x, s).hold();
4028 }
4029
4030
4031 static ex zeta2_eval(const ex& m, const ex& s_)
4032 {
4033         if (is_exactly_a<lst>(s_)) {
4034                 const lst& s = ex_to<lst>(s_);
4035                 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
4036                         if ((*it).info(info_flags::positive)) {
4037                                 continue;
4038                         }
4039                         return zeta(m, s_).hold();
4040                 }
4041                 return zeta(m);
4042         } else if (s_.info(info_flags::positive)) {
4043                 return zeta(m);
4044         }
4045
4046         return zeta(m, s_).hold();
4047 }
4048
4049
4050 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
4051 {
4052         GINAC_ASSERT(deriv_param==0);
4053
4054         if (is_exactly_a<lst>(m)) {
4055                 return _ex0;
4056         } else {
4057                 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
4058                         return zetaderiv(_ex1, m);
4059                 }
4060                 return _ex0;
4061         }
4062 }
4063
4064
4065 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
4066 {
4067         lst m;
4068         if (is_a<lst>(m_)) {
4069                 m = ex_to<lst>(m_);
4070         } else {
4071                 m = lst(m_);
4072         }
4073         lst s;
4074         if (is_a<lst>(s_)) {
4075                 s = ex_to<lst>(s_);
4076         } else {
4077                 s = lst(s_);
4078         }
4079         c.s << "\\zeta(";
4080         lst::const_iterator itm = m.begin();
4081         lst::const_iterator its = s.begin();
4082         if (*its < 0) {
4083                 c.s << "\\overline{";
4084                 (*itm).print(c);
4085                 c.s << "}";
4086         } else {
4087                 (*itm).print(c);
4088         }
4089         its++;
4090         itm++;
4091         for (; itm != m.end(); itm++, its++) {
4092                 c.s << ",";
4093                 if (*its < 0) {
4094                         c.s << "\\overline{";
4095                         (*itm).print(c);
4096                         c.s << "}";
4097                 } else {
4098                         (*itm).print(c);
4099                 }
4100         }
4101         c.s << ")";
4102 }
4103
4104
4105 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
4106                                 evalf_func(zeta2_evalf).
4107                                 eval_func(zeta2_eval).
4108                                 derivative_func(zeta2_deriv).
4109                                 print_func<print_latex>(zeta2_print_latex).
4110                                 do_not_evalf_params().
4111                                 overloaded(2));
4112
4113
4114 } // namespace GiNaC
4115