ginac/inifcns_nstdsums.cpp

   1 /** @file inifcns_nstdsums.cpp
   2  *
   3  *  Implementation of some special functions that have a representation as nested sums.
   4  *
   5  *  The functions are:
   6  *    classical polylogarithm              Li(n,x)
   7  *    multiple polylogarithm               Li(lst(m_1,...,m_k),lst(x_1,...,x_k))
   8  *                                         G(lst(a_1,...,a_k),y) or G(lst(a_1,...,a_k),lst(s_1,...,s_k),y)
   9  *    Nielsen's generalized polylogarithm  S(n,p,x)
  10  *    harmonic polylogarithm               H(m,x) or H(lst(m_1,...,m_k),x)
  11  *    multiple zeta value                  zeta(m) or zeta(lst(m_1,...,m_k))
  12  *    alternating Euler sum                zeta(m,s) or zeta(lst(m_1,...,m_k),lst(s_1,...,s_k))
  13  *
  14  *  Some remarks:
  15  *
  16  *    - All formulae used can be looked up in the following publications:
  17  *      [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
  18  *      [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
  19  *      [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
  20  *      [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
  21  *      [VSW] Numerical evaluation of multiple polylogarithms, J.Vollinga, S.Weinzierl, hep-ph/0410259
  22  *
  23  *    - The order of parameters and arguments of Li and zeta is defined according to the nested sums
  24  *      representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
  25  *      0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single
  26  *      number --- notation.
  27  *
  28  *    - All functions can be nummerically evaluated with arguments in the whole complex plane. The parameters
  29  *      for Li, zeta and S must be positive integers. If you want to have an alternating Euler sum, you have
  30  *      to give the signs of the parameters as a second argument s to zeta(m,s) containing 1 and -1.
  31  *
  32  *    - The calculation of classical polylogarithms is speeded up by using Bernoulli numbers and
  33  *      look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in
  34  *      [Cra] and [BBB] for speed up. Multiple polylogarithms use Hoelder convolution [BBB].
  35  *
  36  *    - The functions have no means to do a series expansion into nested sums. To do this, you have to convert
  37  *      these functions into the appropriate objects from the nestedsums library, do the expansion and convert
  38  *      the result back.
  39  *
  40  *    - Numerical testing of this implementation has been performed by doing a comparison of results
  41  *      between this software and the commercial M.......... 4.1. Multiple zeta values have been checked
  42  *      by means of evaluations into simple zeta values. Harmonic polylogarithms have been checked by
  43  *      comparison to S(n,p,x) for corresponding parameter combinations and by continuity checks
  44  *      around |x|=1 along with comparisons to corresponding zeta functions. Multiple polylogarithms were
  45  *      checked against H and zeta and by means of shuffle and quasi-shuffle relations.
  46  *
  47  */
  48
  49 /*
  50  *  GiNaC Copyright (C) 1999-2011 Johannes Gutenberg University Mainz, Germany
  51  *
  52  *  This program is free software; you can redistribute it and/or modify
  53  *  it under the terms of the GNU General Public License as published by
  54  *  the Free Software Foundation; either version 2 of the License, or
  55  *  (at your option) any later version.
  56  *
  57  *  This program is distributed in the hope that it will be useful,
  58  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  59  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  60  *  GNU General Public License for more details.
  61  *
  62  *  You should have received a copy of the GNU General Public License
  63  *  along with this program; if not, write to the Free Software
  64  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  65  */
  66
  67 #include "inifcns.h"
  68
  69 #include "add.h"
  70 #include "constant.h"
  71 #include "lst.h"
  72 #include "mul.h"
  73 #include "numeric.h"
  74 #include "operators.h"
  75 #include "power.h"
  76 #include "pseries.h"
  77 #include "relational.h"
  78 #include "symbol.h"
  79 #include "utils.h"
  80 #include "wildcard.h"
  81
  82 #include <cln/cln.h>
  83 #include <sstream>
  84 #include <stdexcept>
  85 #include <vector>
  86
  87 namespace GiNaC {
  88
  89
  90 //////////////////////////////////////////////////////////////////////
  91 //
  92 // Classical polylogarithm  Li(n,x)
  93 //
  94 // helper functions
  95 //
  96 //////////////////////////////////////////////////////////////////////
  97
  98
  99 // anonymous namespace for helper functions
 100 namespace {
 101
 102
 103 // lookup table for factors built from Bernoulli numbers
 104 // see fill_Xn()
 105 std::vector<std::vector<cln::cl_N> > Xn;
 106 // initial size of Xn that should suffice for 32bit machines (must be even)
 107 const int xninitsizestep = 26;
 108 int xninitsize = xninitsizestep;
 109 int xnsize = 0;
 110
 111
 112 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
 113 // With these numbers the polylogs can be calculated as follows:
 114 //   Li_p (x)  =  \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with  u = -log(1-x)
 115 //   X_0(n) = B_n (Bernoulli numbers)
 116 //   X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
 117 // The calculation of Xn depends on X0 and X{n-1}.
 118 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
 119 // This results in a slightly more complicated algorithm for the X_n.
 120 // The first index in Xn corresponds to the index of the polylog minus 2.
 121 // The second index in Xn corresponds to the index from the actual sum.
 122 void fill_Xn(int n)
 123 {
 124         if (n>1) {
 125                 // calculate X_2 and higher (corresponding to Li_4 and higher)
 126                 std::vector<cln::cl_N> buf(xninitsize);
 127                 std::vector<cln::cl_N>::iterator it = buf.begin();
 128                 cln::cl_N result;
 129                 *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
 130                 it++;
 131                 for (int i=2; i<=xninitsize; i++) {
 132                         if (i&1) {
 133                                 result = 0; // k == 0
 134                         } else {
 135                                 result = Xn[0][i/2-1]; // k == 0
 136                         }
 137                         for (int k=1; k<i-1; k++) {
 138                                 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
 139                                         result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
 140                                 }
 141                         }
 142                         result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
 143                         result = result + Xn[n-1][i-1] / (i+1); // k == i
 144
 145                         *it = result;
 146                         it++;
 147                 }
 148                 Xn.push_back(buf);
 149         } else if (n==1) {
 150                 // special case to handle the X_0 correct
 151                 std::vector<cln::cl_N> buf(xninitsize);
 152                 std::vector<cln::cl_N>::iterator it = buf.begin();
 153                 cln::cl_N result;
 154                 *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
 155                 it++;
 156                 *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
 157                 it++;
 158                 for (int i=3; i<=xninitsize; i++) {
 159                         if (i & 1) {
 160                                 result = -Xn[0][(i-3)/2]/2;
 161                                 *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
 162                                 it++;
 163                         } else {
 164                                 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
 165                                 for (int k=1; k<i/2; k++) {
 166                                         result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
 167                                 }
 168                                 *it = result;
 169                                 it++;
 170                         }
 171                 }
 172                 Xn.push_back(buf);
 173         } else {
 174                 // calculate X_0
 175                 std::vector<cln::cl_N> buf(xninitsize/2);
 176                 std::vector<cln::cl_N>::iterator it = buf.begin();
 177                 for (int i=1; i<=xninitsize/2; i++) {
 178                         *it = bernoulli(i*2).to_cl_N();
 179                         it++;
 180                 }
 181                 Xn.push_back(buf);
 182         }
 183
 184         xnsize++;
 185 }
 186
 187 // doubles the number of entries in each Xn[]
 188 void double_Xn()
 189 {
 190         const int pos0 = xninitsize / 2;
 191         // X_0
 192         for (int i=1; i<=xninitsizestep/2; ++i) {
 193                 Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N());
 194         }
 195         if (Xn.size() > 1) {
 196                 int xend = xninitsize + xninitsizestep;
 197                 cln::cl_N result;
 198                 // X_1
 199                 for (int i=xninitsize+1; i<=xend; ++i) {
 200                         if (i & 1) {
 201                                 result = -Xn[0][(i-3)/2]/2;
 202                                 Xn[1].push_back((cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result);
 203                         } else {
 204                                 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
 205                                 for (int k=1; k<i/2; k++) {
 206                                         result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
 207                                 }
 208                                 Xn[1].push_back(result);
 209                         }
 210                 }
 211                 // X_n
 212                 for (size_t n=2; n<Xn.size(); ++n) {
 213                         for (int i=xninitsize+1; i<=xend; ++i) {
 214                                 if (i & 1) {
 215                                         result = 0; // k == 0
 216                                 } else {
 217                                         result = Xn[0][i/2-1]; // k == 0
 218                                 }
 219                                 for (int k=1; k<i-1; ++k) {
 220                                         if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
 221                                                 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
 222                                         }
 223                                 }
 224                                 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
 225                                 result = result + Xn[n-1][i-1] / (i+1); // k == i
 226                                 Xn[n].push_back(result);
 227                         }
 228                 }
 229         }
 230         xninitsize += xninitsizestep;
 231 }
 232
 233
 234 // calculates Li(2,x) without Xn
 235 cln::cl_N Li2_do_sum(const cln::cl_N& x)
 236 {
 237         cln::cl_N res = x;
 238         cln::cl_N resbuf;
 239         cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
 240         cln::cl_I den = 1; // n^2 = 1
 241         unsigned i = 3;
 242         do {
 243                 resbuf = res;
 244                 num = num * x;
 245                 den = den + i;  // n^2 = 4, 9, 16, ...
 246                 i += 2;
 247                 res = res + num / den;
 248         } while (res != resbuf);
 249         return res;
 250 }
 251
 252
 253 // calculates Li(2,x) with Xn
 254 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
 255 {
 256         std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
 257         std::vector<cln::cl_N>::const_iterator xend = Xn[0].end();
 258         cln::cl_N u = -cln::log(1-x);
 259         cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
 260         cln::cl_N uu = cln::square(u);
 261         cln::cl_N res = u - uu/4;
 262         cln::cl_N resbuf;
 263         unsigned i = 1;
 264         do {
 265                 resbuf = res;
 266                 factor = factor * uu / (2*i * (2*i+1));
 267                 res = res + (*it) * factor;
 268                 i++;
 269                 if (++it == xend) {
 270                         double_Xn();
 271                         it = Xn[0].begin() + (i-1);
 272                         xend = Xn[0].end();
 273                 }
 274         } while (res != resbuf);
 275         return res;
 276 }
 277
 278
 279 // calculates Li(n,x), n>2 without Xn
 280 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
 281 {
 282         cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits));
 283         cln::cl_N res = x;
 284         cln::cl_N resbuf;
 285         int i=2;
 286         do {
 287                 resbuf = res;
 288                 factor = factor * x;
 289                 res = res + factor / cln::expt(cln::cl_I(i),n);
 290                 i++;
 291         } while (res != resbuf);
 292         return res;
 293 }
 294
 295
 296 // calculates Li(n,x), n>2 with Xn
 297 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
 298 {
 299         std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
 300         std::vector<cln::cl_N>::const_iterator xend = Xn[n-2].end();
 301         cln::cl_N u = -cln::log(1-x);
 302         cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
 303         cln::cl_N res = u;
 304         cln::cl_N resbuf;
 305         unsigned i=2;
 306         do {
 307                 resbuf = res;
 308                 factor = factor * u / i;
 309                 res = res + (*it) * factor;
 310                 i++;
 311                 if (++it == xend) {
 312                         double_Xn();
 313                         it = Xn[n-2].begin() + (i-2);
 314                         xend = Xn[n-2].end();
 315                 }
 316         } while (res != resbuf);
 317         return res;
 318 }
 319
 320
 321 // forward declaration needed by function Li_projection and C below
 322 const cln::cl_N S_num(int n, int p, const cln::cl_N& x);
 323
 324
 325 // helper function for classical polylog Li
 326 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
 327 {
 328         // treat n=2 as special case
 329         if (n == 2) {
 330                 // check if precalculated X0 exists
 331                 if (xnsize == 0) {
 332                         fill_Xn(0);
 333                 }
 334
 335                 if (cln::realpart(x) < 0.5) {
 336                         // choose the faster algorithm
 337                         // the switching point was empirically determined. the optimal point
 338                         // depends on hardware, Digits, ... so an approx value is okay.
 339                         // it solves also the problem with precision due to the u=-log(1-x) transformation
 340                         if (cln::abs(cln::realpart(x)) < 0.25) {
 341
 342                                 return Li2_do_sum(x);
 343                         } else {
 344                                 return Li2_do_sum_Xn(x);
 345                         }
 346                 } else {
 347                         // choose the faster algorithm
 348                         if (cln::abs(cln::realpart(x)) > 0.75) {
 349                                 if ( x == 1 ) {
 350                                         return cln::zeta(2);
 351                                 } else {
 352                                         return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
 353                                 }
 354                         } else {
 355                                 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
 356                         }
 357                 }
 358         } else {
 359                 // check if precalculated Xn exist
 360                 if (n > xnsize+1) {
 361                         for (int i=xnsize; i<n-1; i++) {
 362                                 fill_Xn(i);
 363                         }
 364                 }
 365
 366                 if (cln::realpart(x) < 0.5) {
 367                         // choose the faster algorithm
 368                         // with n>=12 the "normal" summation always wins against the method with Xn
 369                         if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
 370                                 return Lin_do_sum(n, x);
 371                         } else {
 372                                 return Lin_do_sum_Xn(n, x);
 373                         }
 374                 } else {
 375                         cln::cl_N result = 0;
 376                         if ( x != 1 ) result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
 377                         for (int j=0; j<n-1; j++) {
 378                                 result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
 379                                                   * cln::expt(cln::log(x), j) / cln::factorial(j);
 380                         }
 381                         return result;
 382                 }
 383         }
 384 }
 385
 386 // helper function for classical polylog Li
 387 const cln::cl_N Lin_numeric(const int n, const cln::cl_N& x)
 388 {
 389         if (n == 1) {
 390                 // just a log
 391                 return -cln::log(1-x);
 392         }
 393         if (zerop(x)) {
 394                 return 0;
 395         }
 396         if (x == 1) {
 397                 // [Kol] (2.22)
 398                 return cln::zeta(n);
 399         }
 400         else if (x == -1) {
 401                 // [Kol] (2.22)
 402                 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
 403         }
 404         if (cln::abs(realpart(x)) < 0.4 && cln::abs(cln::abs(x)-1) < 0.01) {
 405                 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
 406                 for (int j=0; j<n-1; j++) {
 407                         result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
 408                                 * cln::expt(cln::log(x), j) / cln::factorial(j);
 409                 }
 410                 return result;
 411         }
 412
 413         // what is the desired float format?
 414         // first guess: default format
 415         cln::float_format_t prec = cln::default_float_format;
 416         const cln::cl_N value = x;
 417         // second guess: the argument's format
 418         if (!instanceof(realpart(x), cln::cl_RA_ring))
 419                 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
 420         else if (!instanceof(imagpart(x), cln::cl_RA_ring))
 421                 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
 422
 423         // [Kol] (5.15)
 424         if (cln::abs(value) > 1) {
 425                 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
 426                 // check if argument is complex. if it is real, the new polylog has to be conjugated.
 427                 if (cln::zerop(cln::imagpart(value))) {
 428                         if (n & 1) {
 429                                 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
 430                         }
 431                         else {
 432                                 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
 433                         }
 434                 }
 435                 else {
 436                         if (n & 1) {
 437                                 result = result + Li_projection(n, cln::recip(value), prec);
 438                         }
 439                         else {
 440                                 result = result - Li_projection(n, cln::recip(value), prec);
 441                         }
 442                 }
 443                 cln::cl_N add;
 444                 for (int j=0; j<n-1; j++) {
 445                         add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
 446                                     * Lin_numeric(n-j,1) * cln::expt(cln::log(-value),j) / cln::factorial(j);
 447                 }
 448                 result = result - add;
 449                 return result;
 450         }
 451         else {
 452                 return Li_projection(n, value, prec);
 453         }
 454 }
 455
 456
 457 } // end of anonymous namespace
 458
 459
 460 //////////////////////////////////////////////////////////////////////
 461 //
 462 // Multiple polylogarithm  Li(n,x)
 463 //
 464 // helper function
 465 //
 466 //////////////////////////////////////////////////////////////////////
 467
 468
 469 // anonymous namespace for helper function
 470 namespace {
 471
 472
 473 // performs the actual series summation for multiple polylogarithms
 474 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
 475 {
 476         // ensure all x <> 0.
 477         for (std::vector<cln::cl_N>::const_iterator it = x.begin(); it != x.end(); ++it) {
 478                 if ( *it == 0 ) return cln::cl_float(0, cln::float_format(Digits));
 479         }
 480
 481         const int j = s.size();
 482         bool flag_accidental_zero = false;
 483
 484         std::vector<cln::cl_N> t(j);
 485         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
 486
 487         cln::cl_N t0buf;
 488         int q = 0;
 489         do {
 490                 t0buf = t[0];
 491                 q++;
 492                 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
 493                 for (int k=j-2; k>=0; k--) {
 494                         t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
 495                 }
 496                 q++;
 497                 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
 498                 for (int k=j-2; k>=0; k--) {
 499                         flag_accidental_zero = cln::zerop(t[k+1]);
 500                         t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
 501                 }
 502         } while ( (t[0] != t0buf) || cln::zerop(t[0]) || flag_accidental_zero );
 503
 504         return t[0];
 505 }
 506
 507
 508 // forward declaration for Li_eval()
 509 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
 510
 511
 512 // type used by the transformation functions for G
 513 typedef std::vector<int> Gparameter;
 514
 515
 516 // G_eval1-function for G transformations
 517 ex G_eval1(int a, int scale, const exvector& gsyms)
 518 {
 519         if (a != 0) {
 520                 const ex& scs = gsyms[std::abs(scale)];
 521                 const ex& as = gsyms[std::abs(a)];
 522                 if (as != scs) {
 523                         return -log(1 - scs/as);
 524                 } else {
 525                         return -zeta(1);
 526                 }
 527         } else {
 528                 return log(gsyms[std::abs(scale)]);
 529         }
 530 }
 531
 532
 533 // G_eval-function for G transformations
 534 ex G_eval(const Gparameter& a, int scale, const exvector& gsyms)
 535 {
 536         // check for properties of G
 537         ex sc = gsyms[std::abs(scale)];
 538         lst newa;
 539         bool all_zero = true;
 540         bool all_ones = true;
 541         int count_ones = 0;
 542         for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
 543                 if (*it != 0) {
 544                         const ex sym = gsyms[std::abs(*it)];
 545                         newa.append(sym);
 546                         all_zero = false;
 547                         if (sym != sc) {
 548                                 all_ones = false;
 549                         }
 550                         if (all_ones) {
 551                                 ++count_ones;
 552                         }
 553                 } else {
 554                         all_ones = false;
 555                 }
 556         }
 557
 558         // care about divergent G: shuffle to separate divergencies that will be canceled
 559         // later on in the transformation
 560         if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) {
 561                 // do shuffle
 562                 Gparameter short_a;
 563                 Gparameter::const_iterator it = a.begin();
 564                 ++it;
 565                 for (; it != a.end(); ++it) {
 566                         short_a.push_back(*it);
 567                 }
 568                 ex result = G_eval1(a.front(), scale, gsyms) * G_eval(short_a, scale, gsyms);
 569                 it = short_a.begin();
 570                 for (int i=1; i<count_ones; ++i) {
 571                         ++it;
 572                 }
 573                 for (; it != short_a.end(); ++it) {
 574
 575                         Gparameter newa;
 576                         Gparameter::const_iterator it2 = short_a.begin();
 577                         for (; it2 != it; ++it2) {
 578                                 newa.push_back(*it2);
 579                         }
 580                         newa.push_back(*it);
 581                         newa.push_back(a[0]);
 582                         it2 = it;
 583                         ++it2;
 584                         for (; it2 != short_a.end(); ++it2) {
 585                                 newa.push_back(*it2);
 586                         }
 587                         result -= G_eval(newa, scale, gsyms);
 588                 }
 589                 return result / count_ones;
 590         }
 591
 592         // G({1,...,1};y) -> G({1};y)^k / k!
 593         if (all_ones && a.size() > 1) {
 594                 return pow(G_eval1(a.front(),scale, gsyms), count_ones) / factorial(count_ones);
 595         }
 596
 597         // G({0,...,0};y) -> log(y)^k / k!
 598         if (all_zero) {
 599                 return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size());
 600         }
 601
 602         // no special cases anymore -> convert it into Li
 603         lst m;
 604         lst x;
 605         ex argbuf = gsyms[std::abs(scale)];
 606         ex mval = _ex1;
 607         for (Gparameter::const_iterator it=a.begin(); it!=a.end(); ++it) {
 608                 if (*it != 0) {
 609                         const ex& sym = gsyms[std::abs(*it)];
 610                         x.append(argbuf / sym);
 611                         m.append(mval);
 612                         mval = _ex1;
 613                         argbuf = sym;
 614                 } else {
 615                         ++mval;
 616                 }
 617         }
 618         return pow(-1, x.nops()) * Li(m, x);
 619 }
 620
 621
 622 // converts data for G: pending_integrals -> a
 623 Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals)
 624 {
 625         GINAC_ASSERT(pending_integrals.size() != 1);
 626
 627         if (pending_integrals.size() > 0) {
 628                 // get rid of the first element, which would stand for the new upper limit
 629                 Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end());
 630                 return new_a;
 631         } else {
 632                 // just return empty parameter list
 633                 Gparameter new_a;
 634                 return new_a;
 635         }
 636 }
 637
 638
 639 // check the parameters a and scale for G and return information about convergence, depth, etc.
 640 // convergent     : true if G(a,scale) is convergent
 641 // depth          : depth of G(a,scale)
 642 // trailing_zeros : number of trailing zeros of a
 643 // min_it         : iterator of a pointing on the smallest element in a
 644 Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale,
 645                 bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it)
 646 {
 647         convergent = true;
 648         depth = 0;
 649         trailing_zeros = 0;
 650         min_it = a.end();
 651         Gparameter::const_iterator lastnonzero = a.end();
 652         for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
 653                 if (std::abs(*it) > 0) {
 654                         ++depth;
 655                         trailing_zeros = 0;
 656                         lastnonzero = it;
 657                         if (std::abs(*it) < scale) {
 658                                 convergent = false;
 659                                 if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) {
 660                                         min_it = it;
 661                                 }
 662                         }
 663                 } else {
 664                         ++trailing_zeros;
 665                 }
 666         }
 667         if (lastnonzero == a.end())
 668                 return a.end();
 669         return ++lastnonzero;
 670 }
 671
 672
 673 // add scale to pending_integrals if pending_integrals is empty
 674 Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale)
 675 {
 676         GINAC_ASSERT(pending_integrals.size() != 1);
 677
 678         if (pending_integrals.size() > 0) {
 679                 return pending_integrals;
 680         } else {
 681                 Gparameter new_pending_integrals;
 682                 new_pending_integrals.push_back(scale);
 683                 return new_pending_integrals;
 684         }
 685 }
 686
 687
 688 // handles trailing zeroes for an otherwise convergent integral
 689 ex trailing_zeros_G(const Gparameter& a, int scale, const exvector& gsyms)
 690 {
 691         bool convergent;
 692         int depth, trailing_zeros;
 693         Gparameter::const_iterator last, dummyit;
 694         last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit);
 695
 696         GINAC_ASSERT(convergent);
 697
 698         if ((trailing_zeros > 0) && (depth > 0)) {
 699                 ex result;
 700                 Gparameter new_a(a.begin(), a.end()-1);
 701                 result += G_eval1(0, scale, gsyms) * trailing_zeros_G(new_a, scale, gsyms);
 702                 for (Gparameter::const_iterator it = a.begin(); it != last; ++it) {
 703                         Gparameter new_a(a.begin(), it);
 704                         new_a.push_back(0);
 705                         new_a.insert(new_a.end(), it, a.end()-1);
 706                         result -= trailing_zeros_G(new_a, scale, gsyms);
 707                 }
 708
 709                 return result / trailing_zeros;
 710         } else {
 711                 return G_eval(a, scale, gsyms);
 712         }
 713 }
 714
 715
 716 // G transformation [VSW] (57),(58)
 717 ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale, const exvector& gsyms)
 718 {
 719         // pendint = ( y1, b1, ..., br )
 720         //       a = ( 0, ..., 0, amin )
 721         //   scale = y2
 722         //
 723         // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2)
 724         // where sr replaces amin
 725
 726         GINAC_ASSERT(a.back() != 0);
 727         GINAC_ASSERT(a.size() > 0);
 728
 729         ex result;
 730         Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back()));
 731         const int psize = pending_integrals.size();
 732
 733         // length == 1
 734         // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+})
 735
 736         if (a.size() == 1) {
 737
 738           // ln(-y2_{-+})
 739           result += log(gsyms[ex_to<numeric>(scale).to_int()]);
 740                 if (a.back() > 0) {
 741                         new_pending_integrals.push_back(-scale);
 742                         result += I*Pi;
 743                 } else {
 744                         new_pending_integrals.push_back(scale);
 745                         result -= I*Pi;
 746                 }
 747                 if (psize) {
 748                         result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
 749                                                    pending_integrals.front(),
 750                                                    gsyms);
 751                 }
 752
 753                 // G(y2_{-+}; sr)
 754                 result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
 755                                            new_pending_integrals.front(),
 756                                            gsyms);
 757
 758                 // G(0; sr)
 759                 new_pending_integrals.back() = 0;
 760                 result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
 761                                            new_pending_integrals.front(),
 762                                            gsyms);
 763
 764                 return result;
 765         }
 766
 767         // length > 1
 768         // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
 769         //                            - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
 770
 771         //term zeta_m
 772         result -= zeta(a.size());
 773         if (psize) {
 774                 result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
 775                                            pending_integrals.front(),
 776                                            gsyms);
 777         }
 778
 779         // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
 780         //    = int_0^sr dt/t G_{m-1}( t_{+-}; y2 )
 781         Gparameter new_a(a.begin()+1, a.end());
 782         new_pending_integrals.push_back(0);
 783         result -= depth_one_trafo_G(new_pending_integrals, new_a, scale, gsyms);
 784
 785         // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
 786         //    = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 )
 787         Gparameter new_pending_integrals_2;
 788         new_pending_integrals_2.push_back(scale);
 789         new_pending_integrals_2.push_back(0);
 790         if (psize) {
 791                 result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
 792                                            pending_integrals.front(),
 793                                            gsyms)
 794                           * depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
 795         } else {
 796                 result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
 797         }
 798
 799         return result;
 800 }
 801
 802
 803 // forward declaration
 804 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
 805              const Gparameter& pendint, const Gparameter& a_old, int scale,
 806              const exvector& gsyms);
 807
 808
 809 // G transformation [VSW]
 810 ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale,
 811                const exvector& gsyms)
 812 {
 813         // main recursion routine
 814         //
 815         // pendint = ( y1, b1, ..., br )
 816         //       a = ( a1, ..., amin, ..., aw )
 817         //   scale = y2
 818         //
 819         // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
 820         // where sr replaces amin
 821
 822         // find smallest alpha, determine depth and trailing zeros, and check for convergence
 823         bool convergent;
 824         int depth, trailing_zeros;
 825         Gparameter::const_iterator min_it;
 826         Gparameter::const_iterator firstzero =
 827                 check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it);
 828         int min_it_pos = min_it - a.begin();
 829
 830         // special case: all a's are zero
 831         if (depth == 0) {
 832                 ex result;
 833
 834                 if (a.size() == 0) {
 835                   result = 1;
 836                 } else {
 837                   result = G_eval(a, scale, gsyms);
 838                 }
 839                 if (pendint.size() > 0) {
 840                   result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
 841                                              pendint.front(),
 842                                              gsyms);
 843                 }
 844                 return result;
 845         }
 846
 847         // handle trailing zeros
 848         if (trailing_zeros > 0) {
 849                 ex result;
 850                 Gparameter new_a(a.begin(), a.end()-1);
 851                 result += G_eval1(0, scale, gsyms) * G_transform(pendint, new_a, scale, gsyms);
 852                 for (Gparameter::const_iterator it = a.begin(); it != firstzero; ++it) {
 853                         Gparameter new_a(a.begin(), it);
 854                         new_a.push_back(0);
 855                         new_a.insert(new_a.end(), it, a.end()-1);
 856                         result -= G_transform(pendint, new_a, scale, gsyms);
 857                 }
 858                 return result / trailing_zeros;
 859         }
 860
 861         // convergence case
 862         if (convergent) {
 863                 if (pendint.size() > 0) {
 864                         return G_eval(convert_pending_integrals_G(pendint),
 865                                       pendint.front(), gsyms)*
 866                                 G_eval(a, scale, gsyms);
 867                 } else {
 868                         return G_eval(a, scale, gsyms);
 869                 }
 870         }
 871
 872         // call basic transformation for depth equal one
 873         if (depth == 1) {
 874                 return depth_one_trafo_G(pendint, a, scale, gsyms);
 875         }
 876
 877         // do recursion
 878         // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
 879         //  =  int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2)
 880         //   + int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) int_0^{sr} ds_{r+1} d/ds_{r+1} G(a1,...,s_{r+1},...,aw,y2)
 881
 882         // smallest element in last place
 883         if (min_it + 1 == a.end()) {
 884                 do { --min_it; } while (*min_it == 0);
 885                 Gparameter empty;
 886                 Gparameter a1(a.begin(),min_it+1);
 887                 Gparameter a2(min_it+1,a.end());
 888
 889                 ex result = G_transform(pendint, a2, scale, gsyms)*
 890                         G_transform(empty, a1, scale, gsyms);
 891
 892                 result -= shuffle_G(empty, a1, a2, pendint, a, scale, gsyms);
 893                 return result;
 894         }
 895
 896         Gparameter empty;
 897         Gparameter::iterator changeit;
 898
 899         // first term G(a_1,..,0,...,a_w;a_0)
 900         Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]);
 901         Gparameter new_a = a;
 902         new_a[min_it_pos] = 0;
 903         ex result = G_transform(empty, new_a, scale, gsyms);
 904         if (pendint.size() > 0) {
 905                 result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
 906                                            pendint.front(), gsyms);
 907         }
 908
 909         // other terms
 910         changeit = new_a.begin() + min_it_pos;
 911         changeit = new_a.erase(changeit);
 912         if (changeit != new_a.begin()) {
 913                 // smallest in the middle
 914                 new_pendint.push_back(*changeit);
 915                 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
 916                                            new_pendint.front(), gsyms)*
 917                         G_transform(empty, new_a, scale, gsyms);
 918                 int buffer = *changeit;
 919                 *changeit = *min_it;
 920                 result += G_transform(new_pendint, new_a, scale, gsyms);
 921                 *changeit = buffer;
 922                 new_pendint.pop_back();
 923                 --changeit;
 924                 new_pendint.push_back(*changeit);
 925                 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
 926                                            new_pendint.front(), gsyms)*
 927                         G_transform(empty, new_a, scale, gsyms);
 928                 *changeit = *min_it;
 929                 result -= G_transform(new_pendint, new_a, scale, gsyms);
 930         } else {
 931                 // smallest at the front
 932                 new_pendint.push_back(scale);
 933                 result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
 934                                            new_pendint.front(), gsyms)*
 935                         G_transform(empty, new_a, scale, gsyms);
 936                 new_pendint.back() =  *changeit;
 937                 result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
 938                                            new_pendint.front(), gsyms)*
 939                         G_transform(empty, new_a, scale, gsyms);
 940                 *changeit = *min_it;
 941                 result += G_transform(new_pendint, new_a, scale, gsyms);
 942         }
 943         return result;
 944 }
 945
 946
 947 // shuffles the two parameter list a1 and a2 and calls G_transform for every term except
 948 // for the one that is equal to a_old
 949 ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
 950              const Gparameter& pendint, const Gparameter& a_old, int scale,
 951              const exvector& gsyms)
 952 {
 953         if (a1.size()==0 && a2.size()==0) {
 954                 // veto the one configuration we don't want
 955                 if ( a0 == a_old ) return 0;
 956
 957                 return G_transform(pendint, a0, scale, gsyms);
 958         }
 959
 960         if (a2.size()==0) {
 961                 Gparameter empty;
 962                 Gparameter aa0 = a0;
 963                 aa0.insert(aa0.end(),a1.begin(),a1.end());
 964                 return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms);
 965         }
 966
 967         if (a1.size()==0) {
 968                 Gparameter empty;
 969                 Gparameter aa0 = a0;
 970                 aa0.insert(aa0.end(),a2.begin(),a2.end());
 971                 return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms);
 972         }
 973
 974         Gparameter a1_removed(a1.begin()+1,a1.end());
 975         Gparameter a2_removed(a2.begin()+1,a2.end());
 976
 977         Gparameter a01 = a0;
 978         Gparameter a02 = a0;
 979
 980         a01.push_back( a1[0] );
 981         a02.push_back( a2[0] );
 982
 983         return shuffle_G(a01, a1_removed, a2, pendint, a_old, scale, gsyms)
 984              + shuffle_G(a02, a1, a2_removed, pendint, a_old, scale, gsyms);
 985 }
 986
 987 // handles the transformations and the numerical evaluation of G
 988 // the parameter x, s and y must only contain numerics
 989 static cln::cl_N
 990 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
 991           const cln::cl_N& y);
 992
 993 // do acceleration transformation (hoelder convolution [BBB])
 994 // the parameter x, s and y must only contain numerics
 995 static cln::cl_N
 996 G_do_hoelder(std::vector<cln::cl_N> x, /* yes, it's passed by value */
 997              const std::vector<int>& s, const cln::cl_N& y)
 998 {
 999         cln::cl_N result;
1000         const std::size_t size = x.size();
1001         for (std::size_t i = 0; i < size; ++i)
1002                 x[i] = x[i]/y;
1003
1004         for (std::size_t r = 0; r <= size; ++r) {
1005                 cln::cl_N buffer(1 & r ? -1 : 1);
1006                 cln::cl_RA p(2);
1007                 bool adjustp;
1008                 do {
1009                         adjustp = false;
1010                         for (std::size_t i = 0; i < size; ++i) {
1011                                 if (x[i] == cln::cl_RA(1)/p) {
1012                                         p = p/2 + cln::cl_RA(3)/2;
1013                                         adjustp = true;
1014                                         continue;
1015                                 }
1016                         }
1017                 } while (adjustp);
1018                 cln::cl_RA q = p/(p-1);
1019                 std::vector<cln::cl_N> qlstx;
1020                 std::vector<int> qlsts;
1021                 for (std::size_t j = r; j >= 1; --j) {
1022                         qlstx.push_back(cln::cl_N(1) - x[j-1]);
1023                         if (instanceof(x[j-1], cln::cl_R_ring) &&
1024                             realpart(x[j-1]) > 1 && realpart(x[j-1]) <= 2) {
1025                                 qlsts.push_back(s[j-1]);
1026                         } else {
1027                                 qlsts.push_back(-s[j-1]);
1028                         }
1029                 }
1030                 if (qlstx.size() > 0) {
1031                         buffer = buffer*G_numeric(qlstx, qlsts, 1/q);
1032                 }
1033                 std::vector<cln::cl_N> plstx;
1034                 std::vector<int> plsts;
1035                 for (std::size_t j = r+1; j <= size; ++j) {
1036                         plstx.push_back(x[j-1]);
1037                         plsts.push_back(s[j-1]);
1038                 }
1039                 if (plstx.size() > 0) {
1040                         buffer = buffer*G_numeric(plstx, plsts, 1/p);
1041                 }
1042                 result = result + buffer;
1043         }
1044         return result;
1045 }
1046
1047 // convergence transformation, used for numerical evaluation of G function.
1048 // the parameter x, s and y must only contain numerics
1049 static cln::cl_N
1050 G_do_trafo(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1051            const cln::cl_N& y)
1052 {
1053         // sort (|x|<->position) to determine indices
1054         typedef std::multimap<cln::cl_R, std::size_t> sortmap_t;
1055         sortmap_t sortmap;
1056         std::size_t size = 0;
1057         for (std::size_t i = 0; i < x.size(); ++i) {
1058                 if (!zerop(x[i])) {
1059                         sortmap.insert(std::make_pair(abs(x[i]), i));
1060                         ++size;
1061                 }
1062         }
1063         // include upper limit (scale)
1064         sortmap.insert(std::make_pair(abs(y), x.size()));
1065
1066         // generate missing dummy-symbols
1067         int i = 1;
1068         // holding dummy-symbols for the G/Li transformations
1069         exvector gsyms;
1070         gsyms.push_back(symbol("GSYMS_ERROR"));
1071         cln::cl_N lastentry(0);
1072         for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1073                 if (it != sortmap.begin()) {
1074                         if (it->second < x.size()) {
1075                                 if (x[it->second] == lastentry) {
1076                                         gsyms.push_back(gsyms.back());
1077                                         continue;
1078                                 }
1079                         } else {
1080                                 if (y == lastentry) {
1081                                         gsyms.push_back(gsyms.back());
1082                                         continue;
1083                                 }
1084                         }
1085                 }
1086                 std::ostringstream os;
1087                 os << "a" << i;
1088                 gsyms.push_back(symbol(os.str()));
1089                 ++i;
1090                 if (it->second < x.size()) {
1091                         lastentry = x[it->second];
1092                 } else {
1093                         lastentry = y;
1094                 }
1095         }
1096
1097         // fill position data according to sorted indices and prepare substitution list
1098         Gparameter a(x.size());
1099         exmap subslst;
1100         std::size_t pos = 1;
1101         int scale = pos;
1102         for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1103                 if (it->second < x.size()) {
1104                         if (s[it->second] > 0) {
1105                                 a[it->second] = pos;
1106                         } else {
1107                                 a[it->second] = -int(pos);
1108                         }
1109                         subslst[gsyms[pos]] = numeric(x[it->second]);
1110                 } else {
1111                         scale = pos;
1112                         subslst[gsyms[pos]] = numeric(y);
1113                 }
1114                 ++pos;
1115         }
1116
1117         // do transformation
1118         Gparameter pendint;
1119         ex result = G_transform(pendint, a, scale, gsyms);
1120         // replace dummy symbols with their values
1121         result = result.eval().expand();
1122         result = result.subs(subslst).evalf();
1123         if (!is_a<numeric>(result))
1124                 throw std::logic_error("G_do_trafo: G_transform returned non-numeric result");
1125
1126         cln::cl_N ret = ex_to<numeric>(result).to_cl_N();
1127         return ret;
1128 }
1129
1130 // handles the transformations and the numerical evaluation of G
1131 // the parameter x, s and y must only contain numerics
1132 static cln::cl_N
1133 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1134           const cln::cl_N& y)
1135 {
1136         // check for convergence and necessary accelerations
1137         bool need_trafo = false;
1138         bool need_hoelder = false;
1139         std::size_t depth = 0;
1140         for (std::size_t i = 0; i < x.size(); ++i) {
1141                 if (!zerop(x[i])) {
1142                         ++depth;
1143                         const cln::cl_N x_y = abs(x[i]) - y;
1144                         if (instanceof(x_y, cln::cl_R_ring) &&
1145                             realpart(x_y) < cln::least_negative_float(cln::float_format(Digits - 2)))
1146                                 need_trafo = true;
1147
1148                         if (abs(abs(x[i]/y) - 1) < 0.01)
1149                                 need_hoelder = true;
1150                 }
1151         }
1152         if (zerop(x[x.size() - 1]))
1153                 need_trafo = true;
1154
1155         if (depth == 1 && x.size() == 2 && !need_trafo)
1156                 return - Li_projection(2, y/x[1], cln::float_format(Digits));
1157
1158         // do acceleration transformation (hoelder convolution [BBB])
1159         if (need_hoelder)
1160                 return G_do_hoelder(x, s, y);
1161
1162         // convergence transformation
1163         if (need_trafo)
1164                 return G_do_trafo(x, s, y);
1165
1166         // do summation
1167         std::vector<cln::cl_N> newx;
1168         newx.reserve(x.size());
1169         std::vector<int> m;
1170         m.reserve(x.size());
1171         int mcount = 1;
1172         int sign = 1;
1173         cln::cl_N factor = y;
1174         for (std::size_t i = 0; i < x.size(); ++i) {
1175                 if (zerop(x[i])) {
1176                         ++mcount;
1177                 } else {
1178                         newx.push_back(factor/x[i]);
1179                         factor = x[i];
1180                         m.push_back(mcount);
1181                         mcount = 1;
1182                         sign = -sign;
1183                 }
1184         }
1185
1186         return sign*multipleLi_do_sum(m, newx);
1187 }
1188
1189
1190 ex mLi_numeric(const lst& m, const lst& x)
1191 {
1192         // let G_numeric do the transformation
1193         std::vector<cln::cl_N> newx;
1194         newx.reserve(x.nops());
1195         std::vector<int> s;
1196         s.reserve(x.nops());
1197         cln::cl_N factor(1);
1198         for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1199                 for (int i = 1; i < *itm; ++i) {
1200                         newx.push_back(cln::cl_N(0));
1201                         s.push_back(1);
1202                 }
1203                 const cln::cl_N xi = ex_to<numeric>(*itx).to_cl_N();
1204                 factor = factor/xi;
1205                 newx.push_back(factor);
1206                 if ( !instanceof(factor, cln::cl_R_ring) && imagpart(factor) < 0 ) {
1207                         s.push_back(-1);
1208                 }
1209                 else {
1210                         s.push_back(1);
1211                 }
1212         }
1213         return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1)));
1214 }
1215
1216
1217 } // end of anonymous namespace
1218
1219
1220 //////////////////////////////////////////////////////////////////////
1221 //
1222 // Generalized multiple polylogarithm  G(x, y) and G(x, s, y)
1223 //
1224 // GiNaC function
1225 //
1226 //////////////////////////////////////////////////////////////////////
1227
1228
1229 static ex G2_evalf(const ex& x_, const ex& y)
1230 {
1231         if (!y.info(info_flags::positive)) {
1232                 return G(x_, y).hold();
1233         }
1234         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1235         if (x.nops() == 0) {
1236                 return _ex1;
1237         }
1238         if (x.op(0) == y) {
1239                 return G(x_, y).hold();
1240         }
1241         std::vector<int> s;
1242         s.reserve(x.nops());
1243         bool all_zero = true;
1244         for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1245                 if (!(*it).info(info_flags::numeric)) {
1246                         return G(x_, y).hold();
1247                 }
1248                 if (*it != _ex0) {
1249                         all_zero = false;
1250                 }
1251                 if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
1252                         s.push_back(-1);
1253                 }
1254                 else {
1255                         s.push_back(1);
1256                 }
1257         }
1258         if (all_zero) {
1259                 return pow(log(y), x.nops()) / factorial(x.nops());
1260         }
1261         std::vector<cln::cl_N> xv;
1262         xv.reserve(x.nops());
1263         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1264                 xv.push_back(ex_to<numeric>(*it).to_cl_N());
1265         cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1266         return numeric(result);
1267 }
1268
1269
1270 static ex G2_eval(const ex& x_, const ex& y)
1271 {
1272         //TODO eval to MZV or H or S or Lin
1273
1274         if (!y.info(info_flags::positive)) {
1275                 return G(x_, y).hold();
1276         }
1277         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1278         if (x.nops() == 0) {
1279                 return _ex1;
1280         }
1281         if (x.op(0) == y) {
1282                 return G(x_, y).hold();
1283         }
1284         std::vector<int> s;
1285         s.reserve(x.nops());
1286         bool all_zero = true;
1287         bool crational = true;
1288         for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1289                 if (!(*it).info(info_flags::numeric)) {
1290                         return G(x_, y).hold();
1291                 }
1292                 if (!(*it).info(info_flags::crational)) {
1293                         crational = false;
1294                 }
1295                 if (*it != _ex0) {
1296                         all_zero = false;
1297                 }
1298                 if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
1299                         s.push_back(-1);
1300                 }
1301                 else {
1302                         s.push_back(+1);
1303                 }
1304         }
1305         if (all_zero) {
1306                 return pow(log(y), x.nops()) / factorial(x.nops());
1307         }
1308         if (!y.info(info_flags::crational)) {
1309                 crational = false;
1310         }
1311         if (crational) {
1312                 return G(x_, y).hold();
1313         }
1314         std::vector<cln::cl_N> xv;
1315         xv.reserve(x.nops());
1316         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1317                 xv.push_back(ex_to<numeric>(*it).to_cl_N());
1318         cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1319         return numeric(result);
1320 }
1321
1322
1323 unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
1324                                 evalf_func(G2_evalf).
1325                                 eval_func(G2_eval).
1326                                 do_not_evalf_params().
1327                                 overloaded(2));
1328 //TODO
1329 //                                derivative_func(G2_deriv).
1330 //                                print_func<print_latex>(G2_print_latex).
1331
1332
1333 static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
1334 {
1335         if (!y.info(info_flags::positive)) {
1336                 return G(x_, s_, y).hold();
1337         }
1338         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1339         lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1340         if (x.nops() != s.nops()) {
1341                 return G(x_, s_, y).hold();
1342         }
1343         if (x.nops() == 0) {
1344                 return _ex1;
1345         }
1346         if (x.op(0) == y) {
1347                 return G(x_, s_, y).hold();
1348         }
1349         std::vector<int> sn;
1350         sn.reserve(s.nops());
1351         bool all_zero = true;
1352         for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1353                 if (!(*itx).info(info_flags::numeric)) {
1354                         return G(x_, y).hold();
1355                 }
1356                 if (!(*its).info(info_flags::real)) {
1357                         return G(x_, y).hold();
1358                 }
1359                 if (*itx != _ex0) {
1360                         all_zero = false;
1361                 }
1362                 if ( ex_to<numeric>(*itx).is_real() ) {
1363                         if ( ex_to<numeric>(*itx).is_positive() ) {
1364                                 if ( *its >= 0 ) {
1365                                         sn.push_back(1);
1366                                 }
1367                                 else {
1368                                         sn.push_back(-1);
1369                                 }
1370                         } else {
1371                                 sn.push_back(1);
1372                         }
1373                 }
1374                 else {
1375                         if ( ex_to<numeric>(*itx).imag() > 0 ) {
1376                                 sn.push_back(1);
1377                         }
1378                         else {
1379                                 sn.push_back(-1);
1380                         }
1381                 }
1382         }
1383         if (all_zero) {
1384                 return pow(log(y), x.nops()) / factorial(x.nops());
1385         }
1386         std::vector<cln::cl_N> xn;
1387         xn.reserve(x.nops());
1388         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1389                 xn.push_back(ex_to<numeric>(*it).to_cl_N());
1390         cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1391         return numeric(result);
1392 }
1393
1394
1395 static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
1396 {
1397         //TODO eval to MZV or H or S or Lin
1398
1399         if (!y.info(info_flags::positive)) {
1400                 return G(x_, s_, y).hold();
1401         }
1402         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1403         lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1404         if (x.nops() != s.nops()) {
1405                 return G(x_, s_, y).hold();
1406         }
1407         if (x.nops() == 0) {
1408                 return _ex1;
1409         }
1410         if (x.op(0) == y) {
1411                 return G(x_, s_, y).hold();
1412         }
1413         std::vector<int> sn;
1414         sn.reserve(s.nops());
1415         bool all_zero = true;
1416         bool crational = true;
1417         for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1418                 if (!(*itx).info(info_flags::numeric)) {
1419                         return G(x_, s_, y).hold();
1420                 }
1421                 if (!(*its).info(info_flags::real)) {
1422                         return G(x_, s_, y).hold();
1423                 }
1424                 if (!(*itx).info(info_flags::crational)) {
1425                         crational = false;
1426                 }
1427                 if (*itx != _ex0) {
1428                         all_zero = false;
1429                 }
1430                 if ( ex_to<numeric>(*itx).is_real() ) {
1431                         if ( ex_to<numeric>(*itx).is_positive() ) {
1432                                 if ( *its >= 0 ) {
1433                                         sn.push_back(1);
1434                                 }
1435                                 else {
1436                                         sn.push_back(-1);
1437                                 }
1438                         } else {
1439                                 sn.push_back(1);
1440                         }
1441                 }
1442                 else {
1443                         if ( ex_to<numeric>(*itx).imag() > 0 ) {
1444                                 sn.push_back(1);
1445                         }
1446                         else {
1447                                 sn.push_back(-1);
1448                         }
1449                 }
1450         }
1451         if (all_zero) {
1452                 return pow(log(y), x.nops()) / factorial(x.nops());
1453         }
1454         if (!y.info(info_flags::crational)) {
1455                 crational = false;
1456         }
1457         if (crational) {
1458                 return G(x_, s_, y).hold();
1459         }
1460         std::vector<cln::cl_N> xn;
1461         xn.reserve(x.nops());
1462         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1463                 xn.push_back(ex_to<numeric>(*it).to_cl_N());
1464         cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1465         return numeric(result);
1466 }
1467
1468
1469 unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
1470                                 evalf_func(G3_evalf).
1471                                 eval_func(G3_eval).
1472                                 do_not_evalf_params().
1473                                 overloaded(2));
1474 //TODO
1475 //                                derivative_func(G3_deriv).
1476 //                                print_func<print_latex>(G3_print_latex).
1477
1478
1479 //////////////////////////////////////////////////////////////////////
1480 //
1481 // Classical polylogarithm and multiple polylogarithm  Li(m,x)
1482 //
1483 // GiNaC function
1484 //
1485 //////////////////////////////////////////////////////////////////////
1486
1487
1488 static ex Li_evalf(const ex& m_, const ex& x_)
1489 {
1490         // classical polylogs
1491         if (m_.info(info_flags::posint)) {
1492                 if (x_.info(info_flags::numeric)) {
1493                         int m__ = ex_to<numeric>(m_).to_int();
1494                         const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1495                         const cln::cl_N result = Lin_numeric(m__, x__);
1496                         return numeric(result);
1497                 } else {
1498                         // try to numerically evaluate second argument
1499                         ex x_val = x_.evalf();
1500                         if (x_val.info(info_flags::numeric)) {
1501                                 int m__ = ex_to<numeric>(m_).to_int();
1502                                 const cln::cl_N x__ = ex_to<numeric>(x_val).to_cl_N();
1503                                 const cln::cl_N result = Lin_numeric(m__, x__);
1504                                 return numeric(result);
1505                         }
1506                 }
1507         }
1508         // multiple polylogs
1509         if (is_a<lst>(m_) && is_a<lst>(x_)) {
1510
1511                 const lst& m = ex_to<lst>(m_);
1512                 const lst& x = ex_to<lst>(x_);
1513                 if (m.nops() != x.nops()) {
1514                         return Li(m_,x_).hold();
1515                 }
1516                 if (x.nops() == 0) {
1517                         return _ex1;
1518                 }
1519                 if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
1520                         return Li(m_,x_).hold();
1521                 }
1522
1523                 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1524                         if (!(*itm).info(info_flags::posint)) {
1525                                 return Li(m_, x_).hold();
1526                         }
1527                         if (!(*itx).info(info_flags::numeric)) {
1528                                 return Li(m_, x_).hold();
1529                         }
1530                         if (*itx == _ex0) {
1531                                 return _ex0;
1532                         }
1533                 }
1534
1535                 return mLi_numeric(m, x);
1536         }
1537
1538         return Li(m_,x_).hold();
1539 }
1540
1541
1542 static ex Li_eval(const ex& m_, const ex& x_)
1543 {
1544         if (is_a<lst>(m_)) {
1545                 if (is_a<lst>(x_)) {
1546                         // multiple polylogs
1547                         const lst& m = ex_to<lst>(m_);
1548                         const lst& x = ex_to<lst>(x_);
1549                         if (m.nops() != x.nops()) {
1550                                 return Li(m_,x_).hold();
1551                         }
1552                         if (x.nops() == 0) {
1553                                 return _ex1;
1554                         }
1555                         bool is_H = true;
1556                         bool is_zeta = true;
1557                         bool do_evalf = true;
1558                         bool crational = true;
1559                         for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1560                                 if (!(*itm).info(info_flags::posint)) {
1561                                         return Li(m_,x_).hold();
1562                                 }
1563                                 if ((*itx != _ex1) && (*itx != _ex_1)) {
1564                                         if (itx != x.begin()) {
1565                                                 is_H = false;
1566                                         }
1567                                         is_zeta = false;
1568                                 }
1569                                 if (*itx == _ex0) {
1570                                         return _ex0;
1571                                 }
1572                                 if (!(*itx).info(info_flags::numeric)) {
1573                                         do_evalf = false;
1574                                 }
1575                                 if (!(*itx).info(info_flags::crational)) {
1576                                         crational = false;
1577                                 }
1578                         }
1579                         if (is_zeta) {
1580                                 lst newx;
1581                                 for (lst::const_iterator itx = x.begin(); itx != x.end(); ++itx) {
1582                                         GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
1583                                         // XXX: 1 + 0.0*I is considered equal to 1. However
1584                                         // the former is a not automatically converted
1585                                         // to a real number. Do the conversion explicitly
1586                                         // to avoid the "numeric::operator>(): complex inequality"
1587                                         // exception (and similar problems).
1588                                         newx.append(*itx != _ex_1 ? _ex1 : _ex_1);
1589                                 }
1590                                 return zeta(m_, newx);
1591                         }
1592                         if (is_H) {
1593                                 ex prefactor;
1594                                 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1595                                 return prefactor * H(newm, x[0]);
1596                         }
1597                         if (do_evalf && !crational) {
1598                                 return mLi_numeric(m,x);
1599                         }
1600                 }
1601                 return Li(m_, x_).hold();
1602         } else if (is_a<lst>(x_)) {
1603                 return Li(m_, x_).hold();
1604         }
1605
1606         // classical polylogs
1607         if (x_ == _ex0) {
1608                 return _ex0;
1609         }
1610         if (x_ == _ex1) {
1611                 return zeta(m_);
1612         }
1613         if (x_ == _ex_1) {
1614                 return (pow(2,1-m_)-1) * zeta(m_);
1615         }
1616         if (m_ == _ex1) {
1617                 return -log(1-x_);
1618         }
1619         if (m_ == _ex2) {
1620                 if (x_.is_equal(I)) {
1621                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
1622                 }
1623                 if (x_.is_equal(-I)) {
1624                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
1625                 }
1626         }
1627         if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1628                 int m__ = ex_to<numeric>(m_).to_int();
1629                 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1630                 const cln::cl_N result = Lin_numeric(m__, x__);
1631                 return numeric(result);
1632         }
1633
1634         return Li(m_, x_).hold();
1635 }
1636
1637
1638 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1639 {
1640         if (is_a<lst>(m) || is_a<lst>(x)) {
1641                 // multiple polylog
1642                 epvector seq;
1643                 seq.push_back(expair(Li(m, x), 0));
1644                 return pseries(rel, seq);
1645         }
1646
1647         // classical polylog
1648         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1649         if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1650                 // First special case: x==0 (derivatives have poles)
1651                 if (x_pt.is_zero()) {
1652                         const symbol s;
1653                         ex ser;
1654                         // manually construct the primitive expansion
1655                         for (int i=1; i<order; ++i)
1656                                 ser += pow(s,i) / pow(numeric(i), m);
1657                         // substitute the argument's series expansion
1658                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1659                         // maybe that was terminating, so add a proper order term
1660                         epvector nseq;
1661                         nseq.push_back(expair(Order(_ex1), order));
1662                         ser += pseries(rel, nseq);
1663                         // reexpanding it will collapse the series again
1664                         return ser.series(rel, order);
1665                 }
1666                 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1667                 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1668         }
1669         // all other cases should be safe, by now:
1670         throw do_taylor();  // caught by function::series()
1671 }
1672
1673
1674 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1675 {
1676         GINAC_ASSERT(deriv_param < 2);
1677         if (deriv_param == 0) {
1678                 return _ex0;
1679         }
1680         if (m_.nops() > 1) {
1681                 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1682         }
1683         ex m;
1684         if (is_a<lst>(m_)) {
1685                 m = m_.op(0);
1686         } else {
1687                 m = m_;
1688         }
1689         ex x;
1690         if (is_a<lst>(x_)) {
1691                 x = x_.op(0);
1692         } else {
1693                 x = x_;
1694         }
1695         if (m > 0) {
1696                 return Li(m-1, x) / x;
1697         } else {
1698                 return 1/(1-x);
1699         }
1700 }
1701
1702
1703 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1704 {
1705         lst m;
1706         if (is_a<lst>(m_)) {
1707                 m = ex_to<lst>(m_);
1708         } else {
1709                 m = lst(m_);
1710         }
1711         lst x;
1712         if (is_a<lst>(x_)) {
1713                 x = ex_to<lst>(x_);
1714         } else {
1715                 x = lst(x_);
1716         }
1717         c.s << "\\mathrm{Li}_{";
1718         lst::const_iterator itm = m.begin();
1719         (*itm).print(c);
1720         itm++;
1721         for (; itm != m.end(); itm++) {
1722                 c.s << ",";
1723                 (*itm).print(c);
1724         }
1725         c.s << "}(";
1726         lst::const_iterator itx = x.begin();
1727         (*itx).print(c);
1728         itx++;
1729         for (; itx != x.end(); itx++) {
1730                 c.s << ",";
1731                 (*itx).print(c);
1732         }
1733         c.s << ")";
1734 }
1735
1736
1737 REGISTER_FUNCTION(Li,
1738                   evalf_func(Li_evalf).
1739                   eval_func(Li_eval).
1740                   series_func(Li_series).
1741                   derivative_func(Li_deriv).
1742                   print_func<print_latex>(Li_print_latex).
1743                   do_not_evalf_params());
1744
1745
1746 //////////////////////////////////////////////////////////////////////
1747 //
1748 // Nielsen's generalized polylogarithm  S(n,p,x)
1749 //
1750 // helper functions
1751 //
1752 //////////////////////////////////////////////////////////////////////
1753
1754
1755 // anonymous namespace for helper functions
1756 namespace {
1757
1758
1759 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1760 // see fill_Yn()
1761 std::vector<std::vector<cln::cl_N> > Yn;
1762 int ynsize = 0; // number of Yn[]
1763 int ynlength = 100; // initial length of all Yn[i]
1764
1765
1766 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1767 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1768 // representing S_{n,p}(x).
1769 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1770 // equivalent Z-sum.
1771 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1772 // representing S_{n,p}(x).
1773 // The calculation of Y_n uses the values from Y_{n-1}.
1774 void fill_Yn(int n, const cln::float_format_t& prec)
1775 {
1776         const int initsize = ynlength;
1777         //const int initsize = initsize_Yn;
1778         cln::cl_N one = cln::cl_float(1, prec);
1779
1780         if (n) {
1781                 std::vector<cln::cl_N> buf(initsize);
1782                 std::vector<cln::cl_N>::iterator it = buf.begin();
1783                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1784                 *it = (*itprev) / cln::cl_N(n+1) * one;
1785                 it++;
1786                 itprev++;
1787                 // sums with an index smaller than the depth are zero and need not to be calculated.
1788                 // calculation starts with depth, which is n+2)
1789                 for (int i=n+2; i<=initsize+n; i++) {
1790                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1791                         it++;
1792                         itprev++;
1793                 }
1794                 Yn.push_back(buf);
1795         } else {
1796                 std::vector<cln::cl_N> buf(initsize);
1797                 std::vector<cln::cl_N>::iterator it = buf.begin();
1798                 *it = 1 * one;
1799                 it++;
1800                 for (int i=2; i<=initsize; i++) {
1801                         *it = *(it-1) + 1 / cln::cl_N(i) * one;
1802                         it++;
1803                 }
1804                 Yn.push_back(buf);
1805         }
1806         ynsize++;
1807 }
1808
1809
1810 // make Yn longer ...
1811 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1812 {
1813
1814         cln::cl_N one = cln::cl_float(1, prec);
1815
1816         Yn[0].resize(newsize);
1817         std::vector<cln::cl_N>::iterator it = Yn[0].begin();
1818         it += ynlength;
1819         for (int i=ynlength+1; i<=newsize; i++) {
1820                 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1821                 it++;
1822         }
1823
1824         for (int n=1; n<ynsize; n++) {
1825                 Yn[n].resize(newsize);
1826                 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
1827                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1828                 it += ynlength;
1829                 itprev += ynlength;
1830                 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1831                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1832                         it++;
1833                         itprev++;
1834                 }
1835         }
1836
1837         ynlength = newsize;
1838 }
1839
1840
1841 // helper function for S(n,p,x)
1842 // [Kol] (7.2)
1843 cln::cl_N C(int n, int p)
1844 {
1845         cln::cl_N result;
1846
1847         for (int k=0; k<p; k++) {
1848                 for (int j=0; j<=(n+k-1)/2; j++) {
1849                         if (k == 0) {
1850                                 if (n & 1) {
1851                                         if (j & 1) {
1852                                                 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1853                                         }
1854                                         else {
1855                                                 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1856                                         }
1857                                 }
1858                         }
1859                         else {
1860                                 if (k & 1) {
1861                                         if (j & 1) {
1862                                                 result = result + cln::factorial(n+k-1)
1863                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1864                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1865                                         }
1866                                         else {
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                                 }
1872                                 else {
1873                                         if (j & 1) {
1874                                                 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1875                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1876                                         }
1877                                         else {
1878                                                 result = result + cln::factorial(n+k-1)
1879                                                                   * 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                                 }
1883                         }
1884                 }
1885         }
1886         int np = n+p;
1887         if ((np-1) & 1) {
1888                 if (((np)/2+n) & 1) {
1889                         result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1890                 }
1891                 else {
1892                         result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1893                 }
1894         }
1895
1896         return result;
1897 }
1898
1899
1900 // helper function for S(n,p,x)
1901 // [Kol] remark to (9.1)
1902 cln::cl_N a_k(int k)
1903 {
1904         cln::cl_N result;
1905
1906         if (k == 0) {
1907                 return 1;
1908         }
1909
1910         result = result;
1911         for (int m=2; m<=k; m++) {
1912                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1913         }
1914
1915         return -result / k;
1916 }
1917
1918
1919 // helper function for S(n,p,x)
1920 // [Kol] remark to (9.1)
1921 cln::cl_N b_k(int k)
1922 {
1923         cln::cl_N result;
1924
1925         if (k == 0) {
1926                 return 1;
1927         }
1928
1929         result = result;
1930         for (int m=2; m<=k; m++) {
1931                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1932         }
1933
1934         return result / k;
1935 }
1936
1937
1938 // helper function for S(n,p,x)
1939 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1940 {
1941         static cln::float_format_t oldprec = cln::default_float_format;
1942
1943         if (p==1) {
1944                 return Li_projection(n+1, x, prec);
1945         }
1946
1947         // precision has changed, we need to clear lookup table Yn
1948         if ( oldprec != prec ) {
1949                 Yn.clear();
1950                 ynsize = 0;
1951                 ynlength = 100;
1952                 oldprec = prec;
1953         }
1954
1955         // check if precalculated values are sufficient
1956         if (p > ynsize+1) {
1957                 for (int i=ynsize; i<p-1; i++) {
1958                         fill_Yn(i, prec);
1959                 }
1960         }
1961
1962         // should be done otherwise
1963         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1964         cln::cl_N xf = x * one;
1965         //cln::cl_N xf = x * cln::cl_float(1, prec);
1966
1967         cln::cl_N res;
1968         cln::cl_N resbuf;
1969         cln::cl_N factor = cln::expt(xf, p);
1970         int i = p;
1971         do {
1972                 resbuf = res;
1973                 if (i-p >= ynlength) {
1974                         // make Yn longer
1975                         make_Yn_longer(ynlength*2, prec);
1976                 }
1977                 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? ...
1978                 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
1979                 factor = factor * xf;
1980                 i++;
1981         } while (res != resbuf);
1982
1983         return res;
1984 }
1985
1986
1987 // helper function for S(n,p,x)
1988 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1989 {
1990         // [Kol] (5.3)
1991         if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
1992
1993                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
1994                                    * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
1995
1996                 for (int s=0; s<n; s++) {
1997                         cln::cl_N res2;
1998                         for (int r=0; r<p; r++) {
1999                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
2000                                               * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
2001                         }
2002                         result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2003                 }
2004
2005                 return result;
2006         }
2007
2008         return S_do_sum(n, p, x, prec);
2009 }
2010
2011
2012 // helper function for S(n,p,x)
2013 const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
2014 {
2015         if (x == 1) {
2016                 if (n == 1) {
2017                     // [Kol] (2.22) with (2.21)
2018                         return cln::zeta(p+1);
2019                 }
2020
2021                 if (p == 1) {
2022                     // [Kol] (2.22)
2023                         return cln::zeta(n+1);
2024                 }
2025
2026                 // [Kol] (9.1)
2027                 cln::cl_N result;
2028                 for (int nu=0; nu<n; nu++) {
2029                         for (int rho=0; rho<=p; rho++) {
2030                                 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
2031                                                   * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
2032                         }
2033                 }
2034                 result = result * cln::expt(cln::cl_I(-1),n+p-1);
2035
2036                 return result;
2037         }
2038         else if (x == -1) {
2039                 // [Kol] (2.22)
2040                 if (p == 1) {
2041                         return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
2042                 }
2043 //              throw std::runtime_error("don't know how to evaluate this function!");
2044         }
2045
2046         // what is the desired float format?
2047         // first guess: default format
2048         cln::float_format_t prec = cln::default_float_format;
2049         const cln::cl_N value = x;
2050         // second guess: the argument's format
2051         if (!instanceof(realpart(value), cln::cl_RA_ring))
2052                 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
2053         else if (!instanceof(imagpart(value), cln::cl_RA_ring))
2054                 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
2055
2056         // [Kol] (5.3)
2057         // 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
2058         // we don't care here about abs(value)<1 && real(value)>0.5, this will be taken care of in S_projection
2059         if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95) && (cln::abs(1-value) > 1) )) {
2060
2061                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
2062                                    * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
2063
2064                 for (int s=0; s<n; s++) {
2065                         cln::cl_N res2;
2066                         for (int r=0; r<p; r++) {
2067                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
2068                                               * S_num(p-r,n-s,1-value) / cln::factorial(r);
2069                         }
2070                         result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2071                 }
2072
2073                 return result;
2074
2075         }
2076         // [Kol] (5.12)
2077         if (cln::abs(value) > 1) {
2078
2079                 cln::cl_N result;
2080
2081                 for (int s=0; s<p; s++) {
2082                         for (int r=0; r<=s; r++) {
2083                                 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
2084                                                   / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
2085                                                   * S_num(n+s-r,p-s,cln::recip(value));
2086                         }
2087                 }
2088                 result = result * cln::expt(cln::cl_I(-1),n);
2089
2090                 cln::cl_N res2;
2091                 for (int r=0; r<n; r++) {
2092                         res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
2093                 }
2094                 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
2095
2096                 result = result + cln::expt(cln::cl_I(-1),p) * res2;
2097
2098                 return result;
2099         }
2100
2101         if ((cln::abs(value) > 0.95) && (cln::abs(value-9.53) < 9.47)) {
2102                 lst m;
2103                 m.append(n+1);
2104                 for (int s=0; s<p-1; s++)
2105                         m.append(1);
2106
2107                 ex res = H(m,numeric(value)).evalf();
2108                 return ex_to<numeric>(res).to_cl_N();
2109         }
2110         else {
2111                 return S_projection(n, p, value, prec);
2112         }
2113 }
2114
2115
2116 } // end of anonymous namespace
2117
2118
2119 //////////////////////////////////////////////////////////////////////
2120 //
2121 // Nielsen's generalized polylogarithm  S(n,p,x)
2122 //
2123 // GiNaC function
2124 //
2125 //////////////////////////////////////////////////////////////////////
2126
2127
2128 static ex S_evalf(const ex& n, const ex& p, const ex& x)
2129 {
2130         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2131                 const int n_ = ex_to<numeric>(n).to_int();
2132                 const int p_ = ex_to<numeric>(p).to_int();
2133                 if (is_a<numeric>(x)) {
2134                         const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2135                         const cln::cl_N result = S_num(n_, p_, x_);
2136                         return numeric(result);
2137                 } else {
2138                         ex x_val = x.evalf();
2139                         if (is_a<numeric>(x_val)) {
2140                                 const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
2141                                 const cln::cl_N result = S_num(n_, p_, x_val_);
2142                                 return numeric(result);
2143                         }
2144                 }
2145         }
2146         return S(n, p, x).hold();
2147 }
2148
2149
2150 static ex S_eval(const ex& n, const ex& p, const ex& x)
2151 {
2152         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2153                 if (x == 0) {
2154                         return _ex0;
2155                 }
2156                 if (x == 1) {
2157                         lst m(n+1);
2158                         for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2159                                 m.append(1);
2160                         }
2161                         return zeta(m);
2162                 }
2163                 if (p == 1) {
2164                         return Li(n+1, x);
2165                 }
2166                 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2167                         int n_ = ex_to<numeric>(n).to_int();
2168                         int p_ = ex_to<numeric>(p).to_int();
2169                         const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2170                         const cln::cl_N result = S_num(n_, p_, x_);
2171                         return numeric(result);
2172                 }
2173         }
2174         if (n.is_zero()) {
2175                 // [Kol] (5.3)
2176                 return pow(-log(1-x), p) / factorial(p);
2177         }
2178         return S(n, p, x).hold();
2179 }
2180
2181
2182 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2183 {
2184         if (p == _ex1) {
2185                 return Li(n+1, x).series(rel, order, options);
2186         }
2187
2188         const ex x_pt = x.subs(rel, subs_options::no_pattern);
2189         if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2190                 // First special case: x==0 (derivatives have poles)
2191                 if (x_pt.is_zero()) {
2192                         const symbol s;
2193                         ex ser;
2194                         // manually construct the primitive expansion
2195                         // subsum = Euler-Zagier-Sum is needed
2196                         // dirty hack (slow ...) calculation of subsum:
2197                         std::vector<ex> presubsum, subsum;
2198                         subsum.push_back(0);
2199                         for (int i=1; i<order-1; ++i) {
2200                                 subsum.push_back(subsum[i-1] + numeric(1, i));
2201                         }
2202                         for (int depth=2; depth<p; ++depth) {
2203                                 presubsum = subsum;
2204                                 for (int i=1; i<order-1; ++i) {
2205                                         subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2206                                 }
2207                         }
2208
2209                         for (int i=1; i<order; ++i) {
2210                                 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2211                         }
2212                         // substitute the argument's series expansion
2213                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2214                         // maybe that was terminating, so add a proper order term
2215                         epvector nseq;
2216                         nseq.push_back(expair(Order(_ex1), order));
2217                         ser += pseries(rel, nseq);
2218                         // reexpanding it will collapse the series again
2219                         return ser.series(rel, order);
2220                 }
2221                 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2222                 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2223         }
2224         // all other cases should be safe, by now:
2225         throw do_taylor();  // caught by function::series()
2226 }
2227
2228
2229 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2230 {
2231         GINAC_ASSERT(deriv_param < 3);
2232         if (deriv_param < 2) {
2233                 return _ex0;
2234         }
2235         if (n > 0) {
2236                 return S(n-1, p, x) / x;
2237         } else {
2238                 return S(n, p-1, x) / (1-x);
2239         }
2240 }
2241
2242
2243 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2244 {
2245         c.s << "\\mathrm{S}_{";
2246         n.print(c);
2247         c.s << ",";
2248         p.print(c);
2249         c.s << "}(";
2250         x.print(c);
2251         c.s << ")";
2252 }
2253
2254
2255 REGISTER_FUNCTION(S,
2256                   evalf_func(S_evalf).
2257                   eval_func(S_eval).
2258                   series_func(S_series).
2259                   derivative_func(S_deriv).
2260                   print_func<print_latex>(S_print_latex).
2261                   do_not_evalf_params());
2262
2263
2264 //////////////////////////////////////////////////////////////////////
2265 //
2266 // Harmonic polylogarithm  H(m,x)
2267 //
2268 // helper functions
2269 //
2270 //////////////////////////////////////////////////////////////////////
2271
2272
2273 // anonymous namespace for helper functions
2274 namespace {
2275
2276
2277 // regulates the pole (used by 1/x-transformation)
2278 symbol H_polesign("IMSIGN");
2279
2280
2281 // convert parameters from H to Li representation
2282 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2283 // returns true if some parameters are negative
2284 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2285 {
2286         // expand parameter list
2287         lst mexp;
2288         for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
2289                 if (*it > 1) {
2290                         for (ex count=*it-1; count > 0; count--) {
2291                                 mexp.append(0);
2292                         }
2293                         mexp.append(1);
2294                 } else if (*it < -1) {
2295                         for (ex count=*it+1; count < 0; count++) {
2296                                 mexp.append(0);
2297                         }
2298                         mexp.append(-1);
2299                 } else {
2300                         mexp.append(*it);
2301                 }
2302         }
2303
2304         ex signum = 1;
2305         pf = 1;
2306         bool has_negative_parameters = false;
2307         ex acc = 1;
2308         for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
2309                 if (*it == 0) {
2310                         acc++;
2311                         continue;
2312                 }
2313                 if (*it > 0) {
2314                         m.append((*it+acc-1) * signum);
2315                 } else {
2316                         m.append((*it-acc+1) * signum);
2317                 }
2318                 acc = 1;
2319                 signum = *it;
2320                 pf *= *it;
2321                 if (pf < 0) {
2322                         has_negative_parameters = true;
2323                 }
2324         }
2325         if (has_negative_parameters) {
2326                 for (std::size_t i=0; i<m.nops(); i++) {
2327                         if (m.op(i) < 0) {
2328                                 m.let_op(i) = -m.op(i);
2329                                 s.append(-1);
2330                         } else {
2331                                 s.append(1);
2332                         }
2333                 }
2334         }
2335
2336         return has_negative_parameters;
2337 }
2338
2339
2340 // recursivly transforms H to corresponding multiple polylogarithms
2341 struct map_trafo_H_convert_to_Li : public map_function
2342 {
2343         ex operator()(const ex& e)
2344         {
2345                 if (is_a<add>(e) || is_a<mul>(e)) {
2346                         return e.map(*this);
2347                 }
2348                 if (is_a<function>(e)) {
2349                         std::string name = ex_to<function>(e).get_name();
2350                         if (name == "H") {
2351                                 lst parameter;
2352                                 if (is_a<lst>(e.op(0))) {
2353                                                 parameter = ex_to<lst>(e.op(0));
2354                                 } else {
2355                                         parameter = lst(e.op(0));
2356                                 }
2357                                 ex arg = e.op(1);
2358
2359                                 lst m;
2360                                 lst s;
2361                                 ex pf;
2362                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2363                                         s.let_op(0) = s.op(0) * arg;
2364                                         return pf * Li(m, s).hold();
2365                                 } else {
2366                                         for (std::size_t i=0; i<m.nops(); i++) {
2367                                                 s.append(1);
2368                                         }
2369                                         s.let_op(0) = s.op(0) * arg;
2370                                         return Li(m, s).hold();
2371                                 }
2372                         }
2373                 }
2374                 return e;
2375         }
2376 };
2377
2378
2379 // recursivly transforms H to corresponding zetas
2380 struct map_trafo_H_convert_to_zeta : public map_function
2381 {
2382         ex operator()(const ex& e)
2383         {
2384                 if (is_a<add>(e) || is_a<mul>(e)) {
2385                         return e.map(*this);
2386                 }
2387                 if (is_a<function>(e)) {
2388                         std::string name = ex_to<function>(e).get_name();
2389                         if (name == "H") {
2390                                 lst parameter;
2391                                 if (is_a<lst>(e.op(0))) {
2392                                                 parameter = ex_to<lst>(e.op(0));
2393                                 } else {
2394                                         parameter = lst(e.op(0));
2395                                 }
2396
2397                                 lst m;
2398                                 lst s;
2399                                 ex pf;
2400                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2401                                         return pf * zeta(m, s);
2402                                 } else {
2403                                         return zeta(m);
2404                                 }
2405                         }
2406                 }
2407                 return e;
2408         }
2409 };
2410
2411
2412 // remove trailing zeros from H-parameters
2413 struct map_trafo_H_reduce_trailing_zeros : public map_function
2414 {
2415         ex operator()(const ex& e)
2416         {
2417                 if (is_a<add>(e) || is_a<mul>(e)) {
2418                         return e.map(*this);
2419                 }
2420                 if (is_a<function>(e)) {
2421                         std::string name = ex_to<function>(e).get_name();
2422                         if (name == "H") {
2423                                 lst parameter;
2424                                 if (is_a<lst>(e.op(0))) {
2425                                         parameter = ex_to<lst>(e.op(0));
2426                                 } else {
2427                                         parameter = lst(e.op(0));
2428                                 }
2429                                 ex arg = e.op(1);
2430                                 if (parameter.op(parameter.nops()-1) == 0) {
2431
2432                                         //
2433                                         if (parameter.nops() == 1) {
2434                                                 return log(arg);
2435                                         }
2436
2437                                         //
2438                                         lst::const_iterator it = parameter.begin();
2439                                         while ((it != parameter.end()) && (*it == 0)) {
2440                                                 it++;
2441                                         }
2442                                         if (it == parameter.end()) {
2443                                                 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2444                                         }
2445
2446                                         //
2447                                         parameter.remove_last();
2448                                         std::size_t lastentry = parameter.nops();
2449                                         while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2450                                                 lastentry--;
2451                                         }
2452
2453                                         //
2454                                         ex result = log(arg) * H(parameter,arg).hold();
2455                                         ex acc = 0;
2456                                         for (ex i=0; i<lastentry; i++) {
2457                                                 if (parameter[i] > 0) {
2458                                                         parameter[i]++;
2459                                                         result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2460                                                         parameter[i]--;
2461                                                         acc = 0;
2462                                                 } else if (parameter[i] < 0) {
2463                                                         parameter[i]--;
2464                                                         result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2465                                                         parameter[i]++;
2466                                                         acc = 0;
2467                                                 } else {
2468                                                         acc++;
2469                                                 }
2470                                         }
2471
2472                                         if (lastentry < parameter.nops()) {
2473                                                 result = result / (parameter.nops()-lastentry+1);
2474                                                 return result.map(*this);
2475                                         } else {
2476                                                 return result;
2477                                         }
2478                                 }
2479                         }
2480                 }
2481                 return e;
2482         }
2483 };
2484
2485
2486 // returns an expression with zeta functions corresponding to the parameter list for H
2487 ex convert_H_to_zeta(const lst& m)
2488 {
2489         symbol xtemp("xtemp");
2490         map_trafo_H_reduce_trailing_zeros filter;
2491         map_trafo_H_convert_to_zeta filter2;
2492         return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2493 }
2494
2495
2496 // convert signs form Li to H representation
2497 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2498 {
2499         lst res;
2500         lst::const_iterator itm = m.begin();
2501         lst::const_iterator itx = ++x.begin();
2502         int signum = 1;
2503         pf = _ex1;
2504         res.append(*itm);
2505         itm++;
2506         while (itx != x.end()) {
2507                 GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
2508                 // XXX: 1 + 0.0*I is considered equal to 1. However the former
2509                 // is not automatically converted to a real number.
2510                 // Do the conversion explicitly to avoid the
2511                 // "numeric::operator>(): complex inequality" exception.
2512                 signum *= (*itx != _ex_1) ? 1 : -1;
2513                 pf *= signum;
2514                 res.append((*itm) * signum);
2515                 itm++;
2516                 itx++;
2517         }
2518         return res;
2519 }
2520
2521
2522 // multiplies an one-dimensional H with another H
2523 // [ReV] (18)
2524 ex trafo_H_mult(const ex& h1, const ex& h2)
2525 {
2526         ex res;
2527         ex hshort;
2528         lst hlong;
2529         ex h1nops = h1.op(0).nops();
2530         ex h2nops = h2.op(0).nops();
2531         if (h1nops > 1) {
2532                 hshort = h2.op(0).op(0);
2533                 hlong = ex_to<lst>(h1.op(0));
2534         } else {
2535                 hshort = h1.op(0).op(0);
2536                 if (h2nops > 1) {
2537                         hlong = ex_to<lst>(h2.op(0));
2538                 } else {
2539                         hlong = h2.op(0).op(0);
2540                 }
2541         }
2542         for (std::size_t i=0; i<=hlong.nops(); i++) {
2543                 lst newparameter;
2544                 std::size_t j=0;
2545                 for (; j<i; j++) {
2546                         newparameter.append(hlong[j]);
2547                 }
2548                 newparameter.append(hshort);
2549                 for (; j<hlong.nops(); j++) {
2550                         newparameter.append(hlong[j]);
2551                 }
2552                 res += H(newparameter, h1.op(1)).hold();
2553         }
2554         return res;
2555 }
2556
2557
2558 // applies trafo_H_mult recursively on expressions
2559 struct map_trafo_H_mult : public map_function
2560 {
2561         ex operator()(const ex& e)
2562         {
2563                 if (is_a<add>(e)) {
2564                         return e.map(*this);
2565                 }
2566
2567                 if (is_a<mul>(e)) {
2568
2569                         ex result = 1;
2570                         ex firstH;
2571                         lst Hlst;
2572                         for (std::size_t pos=0; pos<e.nops(); pos++) {
2573                                 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2574                                         std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2575                                         if (name == "H") {
2576                                                 for (ex i=0; i<e.op(pos).op(1); i++) {
2577                                                         Hlst.append(e.op(pos).op(0));
2578                                                 }
2579                                                 continue;
2580                                         }
2581                                 } else if (is_a<function>(e.op(pos))) {
2582                                         std::string name = ex_to<function>(e.op(pos)).get_name();
2583                                         if (name == "H") {
2584                                                 if (e.op(pos).op(0).nops() > 1) {
2585                                                         firstH = e.op(pos);
2586                                                 } else {
2587                                                         Hlst.append(e.op(pos));
2588                                                 }
2589                                                 continue;
2590                                         }
2591                                 }
2592                                 result *= e.op(pos);
2593                         }
2594                         if (firstH == 0) {
2595                                 if (Hlst.nops() > 0) {
2596                                         firstH = Hlst[Hlst.nops()-1];
2597                                         Hlst.remove_last();
2598                                 } else {
2599                                         return e;
2600                                 }
2601                         }
2602
2603                         if (Hlst.nops() > 0) {
2604                                 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2605                                 result *= buffer;
2606                                 for (std::size_t i=1; i<Hlst.nops(); i++) {
2607                                         result *= Hlst.op(i);
2608                                 }
2609                                 result = result.expand();
2610                                 map_trafo_H_mult recursion;
2611                                 return recursion(result);
2612                         } else {
2613                                 return e;
2614                         }
2615
2616                 }
2617                 return e;
2618         }
2619 };
2620
2621
2622 // do integration [ReV] (55)
2623 // put parameter 0 in front of existing parameters
2624 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
2625 {
2626         ex h;
2627         std::string name;
2628         if (is_a<function>(e)) {
2629                 name = ex_to<function>(e).get_name();
2630         }
2631         if (name == "H") {
2632                 h = e;
2633         } else {
2634                 for (std::size_t i=0; i<e.nops(); i++) {
2635                         if (is_a<function>(e.op(i))) {
2636                                 std::string name = ex_to<function>(e.op(i)).get_name();
2637                                 if (name == "H") {
2638                                         h = e.op(i);
2639                                 }
2640                         }
2641                 }
2642         }
2643         if (h != 0) {
2644                 lst newparameter = ex_to<lst>(h.op(0));
2645                 newparameter.prepend(0);
2646                 ex addzeta = convert_H_to_zeta(newparameter);
2647                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2648         } else {
2649                 return e * (-H(lst(ex(0)),1/arg).hold());
2650         }
2651 }
2652
2653
2654 // do integration [ReV] (49)
2655 // put parameter 1 in front of existing parameters
2656 ex trafo_H_prepend_one(const ex& e, const ex& arg)
2657 {
2658         ex h;
2659         std::string name;
2660         if (is_a<function>(e)) {
2661                 name = ex_to<function>(e).get_name();
2662         }
2663         if (name == "H") {
2664                 h = e;
2665         } else {
2666                 for (std::size_t i=0; i<e.nops(); i++) {
2667                         if (is_a<function>(e.op(i))) {
2668                                 std::string name = ex_to<function>(e.op(i)).get_name();
2669                                 if (name == "H") {
2670                                         h = e.op(i);
2671                                 }
2672                         }
2673                 }
2674         }
2675         if (h != 0) {
2676                 lst newparameter = ex_to<lst>(h.op(0));
2677                 newparameter.prepend(1);
2678                 return e.subs(h == H(newparameter, h.op(1)).hold());
2679         } else {
2680                 return e * H(lst(ex(1)),1-arg).hold();
2681         }
2682 }
2683
2684
2685 // do integration [ReV] (55)
2686 // put parameter -1 in front of existing parameters
2687 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2688 {
2689         ex h;
2690         std::string name;
2691         if (is_a<function>(e)) {
2692                 name = ex_to<function>(e).get_name();
2693         }
2694         if (name == "H") {
2695                 h = e;
2696         } else {
2697                 for (std::size_t i=0; i<e.nops(); i++) {
2698                         if (is_a<function>(e.op(i))) {
2699                                 std::string name = ex_to<function>(e.op(i)).get_name();
2700                                 if (name == "H") {
2701                                         h = e.op(i);
2702                                 }
2703                         }
2704                 }
2705         }
2706         if (h != 0) {
2707                 lst newparameter = ex_to<lst>(h.op(0));
2708                 newparameter.prepend(-1);
2709                 ex addzeta = convert_H_to_zeta(newparameter);
2710                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2711         } else {
2712                 ex addzeta = convert_H_to_zeta(lst(ex(-1)));
2713                 return (e * (addzeta - H(lst(ex(-1)),1/arg).hold())).expand();
2714         }
2715 }
2716
2717
2718 // do integration [ReV] (55)
2719 // put parameter -1 in front of existing parameters
2720 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
2721 {
2722         ex h;
2723         std::string name;
2724         if (is_a<function>(e)) {
2725                 name = ex_to<function>(e).get_name();
2726         }
2727         if (name == "H") {
2728                 h = e;
2729         } else {
2730                 for (std::size_t i = 0; i < e.nops(); i++) {
2731                         if (is_a<function>(e.op(i))) {
2732                                 std::string name = ex_to<function>(e.op(i)).get_name();
2733                                 if (name == "H") {
2734                                         h = e.op(i);
2735                                 }
2736                         }
2737                 }
2738         }
2739         if (h != 0) {
2740                 lst newparameter = ex_to<lst>(h.op(0));
2741                 newparameter.prepend(-1);
2742                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2743         } else {
2744                 return (e * H(lst(ex(-1)),(1-arg)/(1+arg)).hold()).expand();
2745         }
2746 }
2747
2748
2749 // do integration [ReV] (55)
2750 // put parameter 1 in front of existing parameters
2751 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2752 {
2753         ex h;
2754         std::string name;
2755         if (is_a<function>(e)) {
2756                 name = ex_to<function>(e).get_name();
2757         }
2758         if (name == "H") {
2759                 h = e;
2760         } else {
2761                 for (std::size_t i = 0; i < e.nops(); i++) {
2762                         if (is_a<function>(e.op(i))) {
2763                                 std::string name = ex_to<function>(e.op(i)).get_name();
2764                                 if (name == "H") {
2765                                         h = e.op(i);
2766                                 }
2767                         }
2768                 }
2769         }
2770         if (h != 0) {
2771                 lst newparameter = ex_to<lst>(h.op(0));
2772                 newparameter.prepend(1);
2773                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2774         } else {
2775                 return (e * H(lst(ex(1)),(1-arg)/(1+arg)).hold()).expand();
2776         }
2777 }
2778
2779
2780 // do x -> 1-x transformation
2781 struct map_trafo_H_1mx : public map_function
2782 {
2783         ex operator()(const ex& e)
2784         {
2785                 if (is_a<add>(e) || is_a<mul>(e)) {
2786                         return e.map(*this);
2787                 }
2788
2789                 if (is_a<function>(e)) {
2790                         std::string name = ex_to<function>(e).get_name();
2791                         if (name == "H") {
2792
2793                                 lst parameter = ex_to<lst>(e.op(0));
2794                                 ex arg = e.op(1);
2795
2796                                 // special cases if all parameters are either 0, 1 or -1
2797                                 bool allthesame = true;
2798                                 if (parameter.op(0) == 0) {
2799                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2800                                                 if (parameter.op(i) != 0) {
2801                                                         allthesame = false;
2802                                                         break;
2803                                                 }
2804                                         }
2805                                         if (allthesame) {
2806                                                 lst newparameter;
2807                                                 for (int i=parameter.nops(); i>0; i--) {
2808                                                         newparameter.append(1);
2809                                                 }
2810                                                 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2811                                         }
2812                                 } else if (parameter.op(0) == -1) {
2813                                         throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2814                                 } else {
2815                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2816                                                 if (parameter.op(i) != 1) {
2817                                                         allthesame = false;
2818                                                         break;
2819                                                 }
2820                                         }
2821                                         if (allthesame) {
2822                                                 lst newparameter;
2823                                                 for (int i=parameter.nops(); i>0; i--) {
2824                                                         newparameter.append(0);
2825                                                 }
2826                                                 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2827                                         }
2828                                 }
2829
2830                                 lst newparameter = parameter;
2831                                 newparameter.remove_first();
2832
2833                                 if (parameter.op(0) == 0) {
2834
2835                                         // leading zero
2836                                         ex res = convert_H_to_zeta(parameter);
2837                                         //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2838                                         map_trafo_H_1mx recursion;
2839                                         ex buffer = recursion(H(newparameter, arg).hold());
2840                                         if (is_a<add>(buffer)) {
2841                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2842                                                         res -= trafo_H_prepend_one(buffer.op(i), arg);
2843                                                 }
2844                                         } else {
2845                                                 res -= trafo_H_prepend_one(buffer, arg);
2846                                         }
2847                                         return res;
2848
2849                                 } else {
2850
2851                                         // leading one
2852                                         map_trafo_H_1mx recursion;
2853                                         map_trafo_H_mult unify;
2854                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2855                                         std::size_t firstzero = 0;
2856                                         while (parameter.op(firstzero) == 1) {
2857                                                 firstzero++;
2858                                         }
2859                                         for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) {
2860                                                 lst newparameter;
2861                                                 std::size_t j=0;
2862                                                 for (; j<=i; j++) {
2863                                                         newparameter.append(parameter[j+1]);
2864                                                 }
2865                                                 newparameter.append(1);
2866                                                 for (; j<parameter.nops()-1; j++) {
2867                                                         newparameter.append(parameter[j+1]);
2868                                                 }
2869                                                 res -= H(newparameter, arg).hold();
2870                                         }
2871                                         res = recursion(res).expand() / firstzero;
2872                                         return unify(res);
2873                                 }
2874                         }
2875                 }
2876                 return e;
2877         }
2878 };
2879
2880
2881 // do x -> 1/x transformation
2882 struct map_trafo_H_1overx : public map_function
2883 {
2884         ex operator()(const ex& e)
2885         {
2886                 if (is_a<add>(e) || is_a<mul>(e)) {
2887                         return e.map(*this);
2888                 }
2889
2890                 if (is_a<function>(e)) {
2891                         std::string name = ex_to<function>(e).get_name();
2892                         if (name == "H") {
2893
2894                                 lst parameter = ex_to<lst>(e.op(0));
2895                                 ex arg = e.op(1);
2896
2897                                 // special cases if all parameters are either 0, 1 or -1
2898                                 bool allthesame = true;
2899                                 if (parameter.op(0) == 0) {
2900                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2901                                                 if (parameter.op(i) != 0) {
2902                                                         allthesame = false;
2903                                                         break;
2904                                                 }
2905                                         }
2906                                         if (allthesame) {
2907                                                 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2908                                         }
2909                                 } else if (parameter.op(0) == -1) {
2910                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2911                                                 if (parameter.op(i) != -1) {
2912                                                         allthesame = false;
2913                                                         break;
2914                                                 }
2915                                         }
2916                                         if (allthesame) {
2917                                                 map_trafo_H_mult unify;
2918                                                 return unify((pow(H(lst(ex(-1)),1/arg).hold() - H(lst(ex(0)),1/arg).hold(), parameter.nops())
2919                                                        / factorial(parameter.nops())).expand());
2920                                         }
2921                                 } else {
2922                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2923                                                 if (parameter.op(i) != 1) {
2924                                                         allthesame = false;
2925                                                         break;
2926                                                 }
2927                                         }
2928                                         if (allthesame) {
2929                                                 map_trafo_H_mult unify;
2930                                                 return unify((pow(H(lst(ex(1)),1/arg).hold() + H(lst(ex(0)),1/arg).hold() + H_polesign, parameter.nops())
2931                                                        / factorial(parameter.nops())).expand());
2932                                         }
2933                                 }
2934
2935                                 lst newparameter = parameter;
2936                                 newparameter.remove_first();
2937
2938                                 if (parameter.op(0) == 0) {
2939
2940                                         // leading zero
2941                                         ex res = convert_H_to_zeta(parameter);
2942                                         map_trafo_H_1overx recursion;
2943                                         ex buffer = recursion(H(newparameter, arg).hold());
2944                                         if (is_a<add>(buffer)) {
2945                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2946                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2947                                                 }
2948                                         } else {
2949                                                 res += trafo_H_1tx_prepend_zero(buffer, arg);
2950                                         }
2951                                         return res;
2952
2953                                 } else if (parameter.op(0) == -1) {
2954
2955                                         // leading negative one
2956                                         ex res = convert_H_to_zeta(parameter);
2957                                         map_trafo_H_1overx recursion;
2958                                         ex buffer = recursion(H(newparameter, arg).hold());
2959                                         if (is_a<add>(buffer)) {
2960                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2961                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
2962                                                 }
2963                                         } else {
2964                                                 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
2965                                         }
2966                                         return res;
2967
2968                                 } else {
2969
2970                                         // leading one
2971                                         map_trafo_H_1overx recursion;
2972                                         map_trafo_H_mult unify;
2973                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2974                                         std::size_t firstzero = 0;
2975                                         while (parameter.op(firstzero) == 1) {
2976                                                 firstzero++;
2977                                         }
2978                                         for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) {
2979                                                 lst newparameter;
2980                                                 std::size_t j = 0;
2981                                                 for (; j<=i; j++) {
2982                                                         newparameter.append(parameter[j+1]);
2983                                                 }
2984                                                 newparameter.append(1);
2985                                                 for (; j<parameter.nops()-1; j++) {
2986                                                         newparameter.append(parameter[j+1]);
2987                                                 }
2988                                                 res -= H(newparameter, arg).hold();
2989                                         }
2990                                         res = recursion(res).expand() / firstzero;
2991                                         return unify(res);
2992
2993                                 }
2994
2995                         }
2996                 }
2997                 return e;
2998         }
2999 };
3000
3001
3002 // do x -> (1-x)/(1+x) transformation
3003 struct map_trafo_H_1mxt1px : public map_function
3004 {
3005         ex operator()(const ex& e)
3006         {
3007                 if (is_a<add>(e) || is_a<mul>(e)) {
3008                         return e.map(*this);
3009                 }
3010
3011                 if (is_a<function>(e)) {
3012                         std::string name = ex_to<function>(e).get_name();
3013                         if (name == "H") {
3014
3015                                 lst parameter = ex_to<lst>(e.op(0));
3016                                 ex arg = e.op(1);
3017
3018                                 // special cases if all parameters are either 0, 1 or -1
3019                                 bool allthesame = true;
3020                                 if (parameter.op(0) == 0) {
3021                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
3022                                                 if (parameter.op(i) != 0) {
3023                                                         allthesame = false;
3024                                                         break;
3025                                                 }
3026                                         }
3027                                         if (allthesame) {
3028                                                 map_trafo_H_mult unify;
3029                                                 return unify((pow(-H(lst(ex(1)),(1-arg)/(1+arg)).hold() - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3030                                                        / factorial(parameter.nops())).expand());
3031                                         }
3032                                 } else if (parameter.op(0) == -1) {
3033                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
3034                                                 if (parameter.op(i) != -1) {
3035                                                         allthesame = false;
3036                                                         break;
3037                                                 }
3038                                         }
3039                                         if (allthesame) {
3040                                                 map_trafo_H_mult unify;
3041                                                 return unify((pow(log(2) - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3042                                                        / factorial(parameter.nops())).expand());
3043                                         }
3044                                 } else {
3045                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
3046                                                 if (parameter.op(i) != 1) {
3047                                                         allthesame = false;
3048                                                         break;
3049                                                 }
3050                                         }
3051                                         if (allthesame) {
3052                                                 map_trafo_H_mult unify;
3053                                                 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())
3054                                                        / factorial(parameter.nops())).expand());
3055                                         }
3056                                 }
3057
3058                                 lst newparameter = parameter;
3059                                 newparameter.remove_first();
3060
3061                                 if (parameter.op(0) == 0) {
3062
3063                                         // leading zero
3064                                         ex res = convert_H_to_zeta(parameter);
3065                                         map_trafo_H_1mxt1px recursion;
3066                                         ex buffer = recursion(H(newparameter, arg).hold());
3067                                         if (is_a<add>(buffer)) {
3068                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
3069                                                         res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3070                                                 }
3071                                         } else {
3072                                                 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3073                                         }
3074                                         return res;
3075
3076                                 } else if (parameter.op(0) == -1) {
3077
3078                                         // leading negative one
3079                                         ex res = convert_H_to_zeta(parameter);
3080                                         map_trafo_H_1mxt1px recursion;
3081                                         ex buffer = recursion(H(newparameter, arg).hold());
3082                                         if (is_a<add>(buffer)) {
3083                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
3084                                                         res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3085                                                 }
3086                                         } else {
3087                                                 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3088                                         }
3089                                         return res;
3090
3091                                 } else {
3092
3093                                         // leading one
3094                                         map_trafo_H_1mxt1px recursion;
3095                                         map_trafo_H_mult unify;
3096                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
3097                                         std::size_t firstzero = 0;
3098                                         while (parameter.op(firstzero) == 1) {
3099                                                 firstzero++;
3100                                         }
3101                                         for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) {
3102                                                 lst newparameter;
3103                                                 std::size_t j=0;
3104                                                 for (; j<=i; j++) {
3105                                                         newparameter.append(parameter[j+1]);
3106                                                 }
3107                                                 newparameter.append(1);
3108                                                 for (; j<parameter.nops()-1; j++) {
3109                                                         newparameter.append(parameter[j+1]);
3110                                                 }
3111                                                 res -= H(newparameter, arg).hold();
3112                                         }
3113                                         res = recursion(res).expand() / firstzero;
3114                                         return unify(res);
3115
3116                                 }
3117
3118                         }
3119                 }
3120                 return e;
3121         }
3122 };
3123
3124
3125 // do the actual summation.
3126 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
3127 {
3128         const int j = m.size();
3129
3130         std::vector<cln::cl_N> t(j);
3131
3132         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3133         cln::cl_N factor = cln::expt(x, j) * one;
3134         cln::cl_N t0buf;
3135         int q = 0;
3136         do {
3137                 t0buf = t[0];
3138                 q++;
3139                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
3140                 for (int k=j-2; k>=1; k--) {
3141                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
3142                 }
3143                 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
3144                 factor = factor * x;
3145         } while (t[0] != t0buf);
3146
3147         return t[0];
3148 }
3149
3150
3151 } // end of anonymous namespace
3152
3153
3154 //////////////////////////////////////////////////////////////////////
3155 //
3156 // Harmonic polylogarithm  H(m,x)
3157 //
3158 // GiNaC function
3159 //
3160 //////////////////////////////////////////////////////////////////////
3161
3162
3163 static ex H_evalf(const ex& x1, const ex& x2)
3164 {
3165         if (is_a<lst>(x1)) {
3166
3167                 cln::cl_N x;
3168                 if (is_a<numeric>(x2)) {
3169                         x = ex_to<numeric>(x2).to_cl_N();
3170                 } else {
3171                         ex x2_val = x2.evalf();
3172                         if (is_a<numeric>(x2_val)) {
3173                                 x = ex_to<numeric>(x2_val).to_cl_N();
3174                         }
3175                 }
3176
3177                 for (std::size_t i = 0; i < x1.nops(); i++) {
3178                         if (!x1.op(i).info(info_flags::integer)) {
3179                                 return H(x1, x2).hold();
3180                         }
3181                 }
3182                 if (x1.nops() < 1) {
3183                         return H(x1, x2).hold();
3184                 }
3185
3186                 const lst& morg = ex_to<lst>(x1);
3187                 // remove trailing zeros ...
3188                 if (*(--morg.end()) == 0) {
3189                         symbol xtemp("xtemp");
3190                         map_trafo_H_reduce_trailing_zeros filter;
3191                         return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3192                 }
3193                 // ... and expand parameter notation
3194                 bool has_minus_one = false;
3195                 lst m;
3196                 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
3197                         if (*it > 1) {
3198                                 for (ex count=*it-1; count > 0; count--) {
3199                                         m.append(0);
3200                                 }
3201                                 m.append(1);
3202                         } else if (*it <= -1) {
3203                                 for (ex count=*it+1; count < 0; count++) {
3204                                         m.append(0);
3205                                 }
3206                                 m.append(-1);
3207                                 has_minus_one = true;
3208                         } else {
3209                                 m.append(*it);
3210                         }
3211                 }
3212
3213                 // do summation
3214                 if (cln::abs(x) < 0.95) {
3215                         lst m_lst;
3216                         lst s_lst;
3217                         ex pf;
3218                         if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3219                                 // negative parameters -> s_lst is filled
3220                                 std::vector<int> m_int;
3221                                 std::vector<cln::cl_N> x_cln;
3222                                 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
3223                                      it_int != m_lst.end(); it_int++, it_cln++) {
3224                                         m_int.push_back(ex_to<numeric>(*it_int).to_int());
3225                                         x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3226                                 }
3227                                 x_cln.front() = x_cln.front() * x;
3228                                 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3229                         } else {
3230                                 // only positive parameters
3231                                 //TODO
3232                                 if (m_lst.nops() == 1) {
3233                                         return Li(m_lst.op(0), x2).evalf();
3234                                 }
3235                                 std::vector<int> m_int;
3236                                 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
3237                                         m_int.push_back(ex_to<numeric>(*it).to_int());
3238                                 }
3239                                 return numeric(H_do_sum(m_int, x));
3240                         }
3241                 }
3242
3243                 symbol xtemp("xtemp");
3244                 ex res = 1;
3245
3246                 // ensure that the realpart of the argument is positive
3247                 if (cln::realpart(x) < 0) {
3248                         x = -x;
3249                         for (std::size_t i = 0; i < m.nops(); i++) {
3250                                 if (m.op(i) != 0) {
3251                                         m.let_op(i) = -m.op(i);
3252                                         res *= -1;
3253                                 }
3254                         }
3255                 }
3256
3257                 // x -> 1/x
3258                 if (cln::abs(x) >= 2.0) {
3259                         map_trafo_H_1overx trafo;
3260                         res *= trafo(H(m, xtemp).hold());
3261                         if (cln::imagpart(x) <= 0) {
3262                                 res = res.subs(H_polesign == -I*Pi);
3263                         } else {
3264                                 res = res.subs(H_polesign == I*Pi);
3265                         }
3266                         return res.subs(xtemp == numeric(x)).evalf();
3267                 }
3268
3269                 // check transformations for 0.95 <= |x| < 2.0
3270
3271                 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3272                 if (cln::abs(x-9.53) <= 9.47) {
3273                         // x -> (1-x)/(1+x)
3274                         map_trafo_H_1mxt1px trafo;
3275                         res *= trafo(H(m, xtemp).hold());
3276                 } else {
3277                         // x -> 1-x
3278                         if (has_minus_one) {
3279                                 map_trafo_H_convert_to_Li filter;
3280                                 return filter(H(m, numeric(x)).hold()).evalf();
3281                         }
3282                         map_trafo_H_1mx trafo;
3283                         res *= trafo(H(m, xtemp).hold());
3284                 }
3285
3286                 return res.subs(xtemp == numeric(x)).evalf();
3287         }
3288
3289         return H(x1,x2).hold();
3290 }
3291
3292
3293 static ex H_eval(const ex& m_, const ex& x)
3294 {
3295         lst m;
3296         if (is_a<lst>(m_)) {
3297                 m = ex_to<lst>(m_);
3298         } else {
3299                 m = lst(m_);
3300         }
3301         if (m.nops() == 0) {
3302                 return _ex1;
3303         }
3304         ex pos1;
3305         ex pos2;
3306         ex n;
3307         ex p;
3308         int step = 0;
3309         if (*m.begin() > _ex1) {
3310                 step++;
3311                 pos1 = _ex0;
3312                 pos2 = _ex1;
3313                 n = *m.begin()-1;
3314                 p = _ex1;
3315         } else if (*m.begin() < _ex_1) {
3316                 step++;
3317                 pos1 = _ex0;
3318                 pos2 = _ex_1;
3319                 n = -*m.begin()-1;
3320                 p = _ex1;
3321         } else if (*m.begin() == _ex0) {
3322                 pos1 = _ex0;
3323                 n = _ex1;
3324         } else {
3325                 pos1 = *m.begin();
3326                 p = _ex1;
3327         }
3328         for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
3329                 if ((*it).info(info_flags::integer)) {
3330                         if (step == 0) {
3331                                 if (*it > _ex1) {
3332                                         if (pos1 == _ex0) {
3333                                                 step = 1;
3334                                                 pos2 = _ex1;
3335                                                 n += *it-1;
3336                                                 p = _ex1;
3337                                         } else {
3338                                                 step = 2;
3339                                         }
3340                                 } else if (*it < _ex_1) {
3341                                         if (pos1 == _ex0) {
3342                                                 step = 1;
3343                                                 pos2 = _ex_1;
3344                                                 n += -*it-1;
3345                                                 p = _ex1;
3346                                         } else {
3347                                                 step = 2;
3348                                         }
3349                                 } else {
3350                                         if (*it != pos1) {
3351                                                 step = 1;
3352                                                 pos2 = *it;
3353                                         }
3354                                         if (*it == _ex0) {
3355                                                 n++;
3356                                         } else {
3357                                                 p++;
3358                                         }
3359                                 }
3360                         } else if (step == 1) {
3361                                 if (*it != pos2) {
3362                                         step = 2;
3363                                 } else {
3364                                         if (*it == _ex0) {
3365                                                 n++;
3366                                         } else {
3367                                                 p++;
3368                                         }
3369                                 }
3370                         }
3371                 } else {
3372                         // if some m_i is not an integer
3373                         return H(m_, x).hold();
3374                 }
3375         }
3376         if ((x == _ex1) && (*(--m.end()) != _ex0)) {
3377                 return convert_H_to_zeta(m);
3378         }
3379         if (step == 0) {
3380                 if (pos1 == _ex0) {
3381                         // all zero
3382                         if (x == _ex0) {
3383                                 return H(m_, x).hold();
3384                         }
3385                         return pow(log(x), m.nops()) / factorial(m.nops());
3386                 } else {
3387                         // all (minus) one
3388                         return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
3389                 }
3390         } else if ((step == 1) && (pos1 == _ex0)){
3391                 // convertible to S
3392                 if (pos2 == _ex1) {
3393                         return S(n, p, x);
3394                 } else {
3395                         return pow(-1, p) * S(n, p, -x);
3396                 }
3397         }
3398         if (x == _ex0) {
3399                 return _ex0;
3400         }
3401         if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
3402                 return H(m_, x).evalf();
3403         }
3404         return H(m_, x).hold();
3405 }
3406
3407
3408 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
3409 {
3410         epvector seq;
3411         seq.push_back(expair(H(m, x), 0));
3412         return pseries(rel, seq);
3413 }
3414
3415
3416 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
3417 {
3418         GINAC_ASSERT(deriv_param < 2);
3419         if (deriv_param == 0) {
3420                 return _ex0;
3421         }
3422         lst m;
3423         if (is_a<lst>(m_)) {
3424                 m = ex_to<lst>(m_);
3425         } else {
3426                 m = lst(m_);
3427         }
3428         ex mb = *m.begin();
3429         if (mb > _ex1) {
3430                 m[0]--;
3431                 return H(m, x) / x;
3432         }
3433         if (mb < _ex_1) {
3434                 m[0]++;
3435                 return H(m, x) / x;
3436         }
3437         m.remove_first();
3438         if (mb == _ex1) {
3439                 return 1/(1-x) * H(m, x);
3440         } else if (mb == _ex_1) {
3441                 return 1/(1+x) * H(m, x);
3442         } else {
3443                 return H(m, x) / x;
3444         }
3445 }
3446
3447
3448 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
3449 {
3450         lst m;
3451         if (is_a<lst>(m_)) {
3452                 m = ex_to<lst>(m_);
3453         } else {
3454                 m = lst(m_);
3455         }
3456         c.s << "\\mathrm{H}_{";
3457         lst::const_iterator itm = m.begin();
3458         (*itm).print(c);
3459         itm++;
3460         for (; itm != m.end(); itm++) {
3461                 c.s << ",";
3462                 (*itm).print(c);
3463         }
3464         c.s << "}(";
3465         x.print(c);
3466         c.s << ")";
3467 }
3468
3469
3470 REGISTER_FUNCTION(H,
3471                   evalf_func(H_evalf).
3472                   eval_func(H_eval).
3473                   series_func(H_series).
3474                   derivative_func(H_deriv).
3475                   print_func<print_latex>(H_print_latex).
3476                   do_not_evalf_params());
3477
3478
3479 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
3480 ex convert_H_to_Li(const ex& m, const ex& x)
3481 {
3482         map_trafo_H_reduce_trailing_zeros filter;
3483         map_trafo_H_convert_to_Li filter2;
3484         if (is_a<lst>(m)) {
3485                 return filter2(filter(H(m, x).hold()));
3486         } else {
3487                 return filter2(filter(H(lst(m), x).hold()));
3488         }
3489 }
3490
3491
3492 //////////////////////////////////////////////////////////////////////
3493 //
3494 // Multiple zeta values  zeta(x) and zeta(x,s)
3495 //
3496 // helper functions
3497 //
3498 //////////////////////////////////////////////////////////////////////
3499
3500
3501 // anonymous namespace for helper functions
3502 namespace {
3503
3504
3505 // parameters and data for [Cra] algorithm
3506 const cln::cl_N lambda = cln::cl_N("319/320");
3507
3508 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
3509 {
3510         const int size = a.size();
3511         for (int n=0; n<size; n++) {
3512                 c[n] = 0;
3513                 for (int m=0; m<=n; m++) {
3514                         c[n] = c[n] + a[m]*b[n-m];
3515                 }
3516         }
3517 }
3518
3519
3520 // [Cra] section 4
3521 static void initcX(std::vector<cln::cl_N>& crX,
3522                    const std::vector<int>& s,
3523                    const int L2)
3524 {
3525         std::vector<cln::cl_N> crB(L2 + 1);
3526         for (int i=0; i<=L2; i++)
3527                 crB[i] = bernoulli(i).to_cl_N() / cln::factorial(i);
3528
3529         int Sm = 0;
3530         int Smp1 = 0;
3531         std::vector<std::vector<cln::cl_N> > crG(s.size() - 1, std::vector<cln::cl_N>(L2 + 1));
3532         for (int m=0; m < (int)s.size() - 1; m++) {
3533                 Sm += s[m];
3534                 Smp1 = Sm + s[m+1];
3535                 for (int i = 0; i <= L2; i++)
3536                         crG[m][i] = cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2);
3537         }
3538
3539         crX = crB;
3540
3541         for (std::size_t m = 0; m < s.size() - 1; m++) {
3542                 std::vector<cln::cl_N> Xbuf(L2 + 1);
3543                 for (int i = 0; i <= L2; i++)
3544                         Xbuf[i] = crX[i] * crG[m][i];
3545
3546                 halfcyclic_convolute(Xbuf, crB, crX);
3547         }
3548 }
3549
3550
3551 // [Cra] section 4
3552 static cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk,
3553                                  const std::vector<cln::cl_N>& crX)
3554 {
3555         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3556         cln::cl_N factor = cln::expt(lambda, Sqk);
3557         cln::cl_N res = factor / Sqk * crX[0] * one;
3558         cln::cl_N resbuf;
3559         int N = 0;
3560         do {
3561                 resbuf = res;
3562                 factor = factor * lambda;
3563                 N++;
3564                 res = res + crX[N] * factor / (N+Sqk);
3565         } while ((res != resbuf) || cln::zerop(crX[N]));
3566         return res;
3567 }
3568
3569
3570 // [Cra] section 4
3571 static void calc_f(std::vector<std::vector<cln::cl_N> >& f_kj,
3572                    const int maxr, const int L1)
3573 {
3574         cln::cl_N t0, t1, t2, t3, t4;
3575         int i, j, k;
3576         std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
3577         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3578
3579         t0 = cln::exp(-lambda);
3580         t2 = 1;
3581         for (k=1; k<=L1; k++) {
3582                 t1 = k * lambda;
3583                 t2 = t0 * t2;
3584                 for (j=1; j<=maxr; j++) {
3585                         t3 = 1;
3586                         t4 = 1;
3587                         for (i=2; i<=j; i++) {
3588                                 t4 = t4 * (j-i+1);
3589                                 t3 = t1 * t3 + t4;
3590                         }
3591                         (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
3592                 }
3593                 it++;
3594         }
3595 }
3596
3597
3598 // [Cra] (3.1)
3599 static cln::cl_N crandall_Z(const std::vector<int>& s,
3600                             const std::vector<std::vector<cln::cl_N> >& f_kj)
3601 {
3602         const int j = s.size();
3603
3604         if (j == 1) {
3605                 cln::cl_N t0;
3606                 cln::cl_N t0buf;
3607                 int q = 0;
3608                 do {
3609                         t0buf = t0;
3610                         q++;
3611                         t0 = t0 + f_kj[q+j-2][s[0]-1];
3612                 } while (t0 != t0buf);
3613
3614                 return t0 / cln::factorial(s[0]-1);
3615         }
3616
3617         std::vector<cln::cl_N> t(j);
3618
3619         cln::cl_N t0buf;
3620         int q = 0;
3621         do {
3622                 t0buf = t[0];
3623                 q++;
3624                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
3625                 for (int k=j-2; k>=1; k--) {
3626                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
3627                 }
3628                 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
3629         } while (t[0] != t0buf);
3630
3631         return t[0] / cln::factorial(s[0]-1);
3632 }
3633
3634
3635 // [Cra] (2.4)
3636 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
3637 {
3638         std::vector<int> r = s;
3639         const int j = r.size();
3640
3641         std::size_t L1;
3642
3643         // decide on maximal size of f_kj for crandall_Z
3644         if (Digits < 50) {
3645                 L1 = 150;
3646         } else {
3647                 L1 = Digits * 3 + j*2;
3648         }
3649
3650         std::size_t L2;
3651         // decide on maximal size of crX for crandall_Y
3652         if (Digits < 38) {
3653                 L2 = 63;
3654         } else if (Digits < 86) {
3655                 L2 = 127;
3656         } else if (Digits < 192) {
3657                 L2 = 255;
3658         } else if (Digits < 394) {
3659                 L2 = 511;
3660         } else if (Digits < 808) {
3661                 L2 = 1023;
3662         } else {
3663                 L2 = 2047;
3664         }
3665
3666         cln::cl_N res;
3667
3668         int maxr = 0;
3669         int S = 0;
3670         for (int i=0; i<j; i++) {
3671                 S += r[i];
3672                 if (r[i] > maxr) {
3673                         maxr = r[i];
3674                 }
3675         }
3676
3677         std::vector<std::vector<cln::cl_N> > f_kj(L1);
3678         calc_f(f_kj, maxr, L1);
3679
3680         const cln::cl_N r0factorial = cln::factorial(r[0]-1);
3681
3682         std::vector<int> rz;
3683         int skp1buf;
3684         int Srun = S;
3685         for (int k=r.size()-1; k>0; k--) {
3686
3687                 rz.insert(rz.begin(), r.back());
3688                 skp1buf = rz.front();
3689                 Srun -= skp1buf;
3690                 r.pop_back();
3691
3692                 std::vector<cln::cl_N> crX;
3693                 initcX(crX, r, L2);
3694
3695                 for (int q=0; q<skp1buf; q++) {
3696
3697                         cln::cl_N pp1 = crandall_Y_loop(Srun+q-k, crX);
3698                         cln::cl_N pp2 = crandall_Z(rz, f_kj);
3699
3700                         rz.front()--;
3701
3702                         if (q & 1) {
3703                                 res = res - pp1 * pp2 / cln::factorial(q);
3704                         } else {
3705                                 res = res + pp1 * pp2 / cln::factorial(q);
3706                         }
3707                 }
3708                 rz.front() = skp1buf;
3709         }
3710         rz.insert(rz.begin(), r.back());
3711
3712         std::vector<cln::cl_N> crX;
3713         initcX(crX, rz, L2);
3714
3715         res = (res + crandall_Y_loop(S-j, crX)) / r0factorial
3716                 + crandall_Z(rz, f_kj);
3717
3718         return res;
3719 }
3720
3721
3722 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
3723 {
3724         const int j = r.size();
3725
3726         // buffer for subsums
3727         std::vector<cln::cl_N> t(j);
3728         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3729
3730         cln::cl_N t0buf;
3731         int q = 0;
3732         do {
3733                 t0buf = t[0];
3734                 q++;
3735                 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
3736                 for (int k=j-2; k>=0; k--) {
3737                         t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
3738                 }
3739         } while (t[0] != t0buf);
3740
3741         return t[0];
3742 }
3743
3744
3745 // does Hoelder convolution. see [BBB] (7.0)
3746 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
3747 {
3748         // prepare parameters
3749         // holds Li arguments in [BBB] notation
3750         std::vector<int> s = s_;
3751         std::vector<int> m_p = m_;
3752         std::vector<int> m_q;
3753         // holds Li arguments in nested sums notation
3754         std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
3755         s_p[0] = s_p[0] * cln::cl_N("1/2");
3756         // convert notations
3757         int sig = 1;
3758         for (std::size_t i = 0; i < s_.size(); i++) {
3759                 if (s_[i] < 0) {
3760                         sig = -sig;
3761                         s_p[i] = -s_p[i];
3762                 }
3763                 s[i] = sig * std::abs(s[i]);
3764         }
3765         std::vector<cln::cl_N> s_q;
3766         cln::cl_N signum = 1;
3767
3768         // first term
3769         cln::cl_N res = multipleLi_do_sum(m_p, s_p);
3770
3771         // middle terms
3772         do {
3773
3774                 // change parameters
3775                 if (s.front() > 0) {
3776                         if (m_p.front() == 1) {
3777                                 m_p.erase(m_p.begin());
3778                                 s_p.erase(s_p.begin());
3779                                 if (s_p.size() > 0) {
3780                                         s_p.front() = s_p.front() * cln::cl_N("1/2");
3781                                 }
3782                                 s.erase(s.begin());
3783                                 m_q.front()++;
3784                         } else {
3785                                 m_p.front()--;
3786                                 m_q.insert(m_q.begin(), 1);
3787                                 if (s_q.size() > 0) {
3788                                         s_q.front() = s_q.front() * 2;
3789                                 }
3790                                 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3791                         }
3792                 } else {
3793                         if (m_p.front() == 1) {
3794                                 m_p.erase(m_p.begin());
3795                                 cln::cl_N spbuf = s_p.front();
3796                                 s_p.erase(s_p.begin());
3797                                 if (s_p.size() > 0) {
3798                                         s_p.front() = s_p.front() * spbuf;
3799                                 }
3800                                 s.erase(s.begin());
3801                                 m_q.insert(m_q.begin(), 1);
3802                                 if (s_q.size() > 0) {
3803                                         s_q.front() = s_q.front() * 4;
3804                                 }
3805                                 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
3806                                 signum = -signum;
3807                         } else {
3808                                 m_p.front()--;
3809                                 m_q.insert(m_q.begin(), 1);
3810                                 if (s_q.size() > 0) {
3811                                         s_q.front() = s_q.front() * 2;
3812                                 }
3813                                 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
3814                         }
3815                 }
3816
3817                 // exiting the loop
3818                 if (m_p.size() == 0) break;
3819
3820                 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
3821
3822         } while (true);
3823
3824         // last term
3825         res = res + signum * multipleLi_do_sum(m_q, s_q);
3826
3827         return res;
3828 }
3829
3830
3831 } // end of anonymous namespace
3832
3833
3834 //////////////////////////////////////////////////////////////////////
3835 //
3836 // Multiple zeta values  zeta(x)
3837 //
3838 // GiNaC function
3839 //
3840 //////////////////////////////////////////////////////////////////////
3841
3842
3843 static ex zeta1_evalf(const ex& x)
3844 {
3845         if (is_exactly_a<lst>(x) && (x.nops()>1)) {
3846
3847                 // multiple zeta value
3848                 const int count = x.nops();
3849                 const lst& xlst = ex_to<lst>(x);
3850                 std::vector<int> r(count);
3851
3852                 // check parameters and convert them
3853                 lst::const_iterator it1 = xlst.begin();
3854                 std::vector<int>::iterator it2 = r.begin();
3855                 do {
3856                         if (!(*it1).info(info_flags::posint)) {
3857                                 return zeta(x).hold();
3858                         }
3859                         *it2 = ex_to<numeric>(*it1).to_int();
3860                         it1++;
3861                         it2++;
3862                 } while (it2 != r.end());
3863
3864                 // check for divergence
3865                 if (r[0] == 1) {
3866                         return zeta(x).hold();
3867                 }
3868
3869                 // decide on summation algorithm
3870                 // this is still a bit clumsy
3871                 int limit = (Digits>17) ? 10 : 6;
3872                 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
3873                         return numeric(zeta_do_sum_Crandall(r));
3874                 } else {
3875                         return numeric(zeta_do_sum_simple(r));
3876                 }
3877         }
3878
3879         // single zeta value
3880         if (is_exactly_a<numeric>(x) && (x != 1)) {
3881                 try {
3882                         return zeta(ex_to<numeric>(x));
3883                 } catch (const dunno &e) { }
3884         }
3885
3886         return zeta(x).hold();
3887 }
3888
3889
3890 static ex zeta1_eval(const ex& m)
3891 {
3892         if (is_exactly_a<lst>(m)) {
3893                 if (m.nops() == 1) {
3894                         return zeta(m.op(0));
3895                 }
3896                 return zeta(m).hold();
3897         }
3898
3899         if (m.info(info_flags::numeric)) {
3900                 const numeric& y = ex_to<numeric>(m);
3901                 // trap integer arguments:
3902                 if (y.is_integer()) {
3903                         if (y.is_zero()) {
3904                                 return _ex_1_2;
3905                         }
3906                         if (y.is_equal(*_num1_p)) {
3907                                 return zeta(m).hold();
3908                         }
3909                         if (y.info(info_flags::posint)) {
3910                                 if (y.info(info_flags::odd)) {
3911                                         return zeta(m).hold();
3912                                 } else {
3913                                         return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
3914                                 }
3915                         } else {
3916                                 if (y.info(info_flags::odd)) {
3917                                         return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
3918                                 } else {
3919                                         return _ex0;
3920                                 }
3921                         }
3922                 }
3923                 // zeta(float)
3924                 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
3925                         return zeta1_evalf(m);
3926                 }
3927         }
3928         return zeta(m).hold();
3929 }
3930
3931
3932 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
3933 {
3934         GINAC_ASSERT(deriv_param==0);
3935
3936         if (is_exactly_a<lst>(m)) {
3937                 return _ex0;
3938         } else {
3939                 return zetaderiv(_ex1, m);
3940         }
3941 }
3942
3943
3944 static void zeta1_print_latex(const ex& m_, const print_context& c)
3945 {
3946         c.s << "\\zeta(";
3947         if (is_a<lst>(m_)) {
3948                 const lst& m = ex_to<lst>(m_);
3949                 lst::const_iterator it = m.begin();
3950                 (*it).print(c);
3951                 it++;
3952                 for (; it != m.end(); it++) {
3953                         c.s << ",";
3954                         (*it).print(c);
3955                 }
3956         } else {
3957                 m_.print(c);
3958         }
3959         c.s << ")";
3960 }
3961
3962
3963 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
3964                                 evalf_func(zeta1_evalf).
3965                                 eval_func(zeta1_eval).
3966                                 derivative_func(zeta1_deriv).
3967                                 print_func<print_latex>(zeta1_print_latex).
3968                                 do_not_evalf_params().
3969                                 overloaded(2));
3970
3971
3972 //////////////////////////////////////////////////////////////////////
3973 //
3974 // Alternating Euler sum  zeta(x,s)
3975 //
3976 // GiNaC function
3977 //
3978 //////////////////////////////////////////////////////////////////////
3979
3980
3981 static ex zeta2_evalf(const ex& x, const ex& s)
3982 {
3983         if (is_exactly_a<lst>(x)) {
3984
3985                 // alternating Euler sum
3986                 const int count = x.nops();
3987                 const lst& xlst = ex_to<lst>(x);
3988                 const lst& slst = ex_to<lst>(s);
3989                 std::vector<int> xi(count);
3990                 std::vector<int> si(count);
3991
3992                 // check parameters and convert them
3993                 lst::const_iterator it_xread = xlst.begin();
3994                 lst::const_iterator it_sread = slst.begin();
3995                 std::vector<int>::iterator it_xwrite = xi.begin();
3996                 std::vector<int>::iterator it_swrite = si.begin();
3997                 do {
3998                         if (!(*it_xread).info(info_flags::posint)) {
3999                                 return zeta(x, s).hold();
4000                         }
4001                         *it_xwrite = ex_to<numeric>(*it_xread).to_int();
4002                         if (*it_sread > 0) {
4003                                 *it_swrite = 1;
4004                         } else {
4005                                 *it_swrite = -1;
4006                         }
4007                         it_xread++;
4008                         it_sread++;
4009                         it_xwrite++;
4010                         it_swrite++;
4011                 } while (it_xwrite != xi.end());
4012
4013                 // check for divergence
4014                 if ((xi[0] == 1) && (si[0] == 1)) {
4015                         return zeta(x, s).hold();
4016                 }
4017
4018                 // use Hoelder convolution
4019                 return numeric(zeta_do_Hoelder_convolution(xi, si));
4020         }
4021
4022         return zeta(x, s).hold();
4023 }
4024
4025
4026 static ex zeta2_eval(const ex& m, const ex& s_)
4027 {
4028         if (is_exactly_a<lst>(s_)) {
4029                 const lst& s = ex_to<lst>(s_);
4030                 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
4031                         if ((*it).info(info_flags::positive)) {
4032                                 continue;
4033                         }
4034                         return zeta(m, s_).hold();
4035                 }
4036                 return zeta(m);
4037         } else if (s_.info(info_flags::positive)) {
4038                 return zeta(m);
4039         }
4040
4041         return zeta(m, s_).hold();
4042 }
4043
4044
4045 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
4046 {
4047         GINAC_ASSERT(deriv_param==0);
4048
4049         if (is_exactly_a<lst>(m)) {
4050                 return _ex0;
4051         } else {
4052                 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
4053                         return zetaderiv(_ex1, m);
4054                 }
4055                 return _ex0;
4056         }
4057 }
4058
4059
4060 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
4061 {
4062         lst m;
4063         if (is_a<lst>(m_)) {
4064                 m = ex_to<lst>(m_);
4065         } else {
4066                 m = lst(m_);
4067         }
4068         lst s;
4069         if (is_a<lst>(s_)) {
4070                 s = ex_to<lst>(s_);
4071         } else {
4072                 s = lst(s_);
4073         }
4074         c.s << "\\zeta(";
4075         lst::const_iterator itm = m.begin();
4076         lst::const_iterator its = s.begin();
4077         if (*its < 0) {
4078                 c.s << "\\overline{";
4079                 (*itm).print(c);
4080                 c.s << "}";
4081         } else {
4082                 (*itm).print(c);
4083         }
4084         its++;
4085         itm++;
4086         for (; itm != m.end(); itm++, its++) {
4087                 c.s << ",";
4088                 if (*its < 0) {
4089                         c.s << "\\overline{";
4090                         (*itm).print(c);
4091                         c.s << "}";
4092                 } else {
4093                         (*itm).print(c);
4094                 }
4095         }
4096         c.s << ")";
4097 }
4098
4099
4100 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
4101                                 evalf_func(zeta2_evalf).
4102                                 eval_func(zeta2_eval).
4103                                 derivative_func(zeta2_deriv).
4104                                 print_func<print_latex>(zeta2_print_latex).
4105                                 do_not_evalf_params().
4106                                 overloaded(2));
4107
4108
4109 } // namespace GiNaC
4110