9876075c6526fdd4fd6983cc5a37798ca3179eb2
[ginac.git] / 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                                 return zeta(m_,x_);
1581                         }
1582                         if (is_H) {
1583                                 ex prefactor;
1584                                 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1585                                 return prefactor * H(newm, x[0]);
1586                         }
1587                         if (do_evalf && !crational) {
1588                                 return mLi_numeric(m,x);
1589                         }
1590                 }
1591                 return Li(m_, x_).hold();
1592         } else if (is_a<lst>(x_)) {
1593                 return Li(m_, x_).hold();
1594         }
1595
1596         // classical polylogs
1597         if (x_ == _ex0) {
1598                 return _ex0;
1599         }
1600         if (x_ == _ex1) {
1601                 return zeta(m_);
1602         }
1603         if (x_ == _ex_1) {
1604                 return (pow(2,1-m_)-1) * zeta(m_);
1605         }
1606         if (m_ == _ex1) {
1607                 return -log(1-x_);
1608         }
1609         if (m_ == _ex2) {
1610                 if (x_.is_equal(I)) {
1611                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
1612                 }
1613                 if (x_.is_equal(-I)) {
1614                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
1615                 }
1616         }
1617         if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1618                 int m__ = ex_to<numeric>(m_).to_int();
1619                 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1620                 const cln::cl_N result = Lin_numeric(m__, x__);
1621                 return numeric(result);
1622         }
1623
1624         return Li(m_, x_).hold();
1625 }
1626
1627
1628 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1629 {
1630         if (is_a<lst>(m) || is_a<lst>(x)) {
1631                 // multiple polylog
1632                 epvector seq;
1633                 seq.push_back(expair(Li(m, x), 0));
1634                 return pseries(rel, seq);
1635         }
1636         
1637         // classical polylog
1638         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1639         if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1640                 // First special case: x==0 (derivatives have poles)
1641                 if (x_pt.is_zero()) {
1642                         const symbol s;
1643                         ex ser;
1644                         // manually construct the primitive expansion
1645                         for (int i=1; i<order; ++i)
1646                                 ser += pow(s,i) / pow(numeric(i), m);
1647                         // substitute the argument's series expansion
1648                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1649                         // maybe that was terminating, so add a proper order term
1650                         epvector nseq;
1651                         nseq.push_back(expair(Order(_ex1), order));
1652                         ser += pseries(rel, nseq);
1653                         // reexpanding it will collapse the series again
1654                         return ser.series(rel, order);
1655                 }
1656                 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1657                 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1658         }
1659         // all other cases should be safe, by now:
1660         throw do_taylor();  // caught by function::series()
1661 }
1662
1663
1664 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1665 {
1666         GINAC_ASSERT(deriv_param < 2);
1667         if (deriv_param == 0) {
1668                 return _ex0;
1669         }
1670         if (m_.nops() > 1) {
1671                 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1672         }
1673         ex m;
1674         if (is_a<lst>(m_)) {
1675                 m = m_.op(0);
1676         } else {
1677                 m = m_;
1678         }
1679         ex x;
1680         if (is_a<lst>(x_)) {
1681                 x = x_.op(0);
1682         } else {
1683                 x = x_;
1684         }
1685         if (m > 0) {
1686                 return Li(m-1, x) / x;
1687         } else {
1688                 return 1/(1-x);
1689         }
1690 }
1691
1692
1693 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1694 {
1695         lst m;
1696         if (is_a<lst>(m_)) {
1697                 m = ex_to<lst>(m_);
1698         } else {
1699                 m = lst(m_);
1700         }
1701         lst x;
1702         if (is_a<lst>(x_)) {
1703                 x = ex_to<lst>(x_);
1704         } else {
1705                 x = lst(x_);
1706         }
1707         c.s << "\\mathrm{Li}_{";
1708         lst::const_iterator itm = m.begin();
1709         (*itm).print(c);
1710         itm++;
1711         for (; itm != m.end(); itm++) {
1712                 c.s << ",";
1713                 (*itm).print(c);
1714         }
1715         c.s << "}(";
1716         lst::const_iterator itx = x.begin();
1717         (*itx).print(c);
1718         itx++;
1719         for (; itx != x.end(); itx++) {
1720                 c.s << ",";
1721                 (*itx).print(c);
1722         }
1723         c.s << ")";
1724 }
1725
1726
1727 REGISTER_FUNCTION(Li,
1728                   evalf_func(Li_evalf).
1729                   eval_func(Li_eval).
1730                   series_func(Li_series).
1731                   derivative_func(Li_deriv).
1732                   print_func<print_latex>(Li_print_latex).
1733                   do_not_evalf_params());
1734
1735
1736 //////////////////////////////////////////////////////////////////////
1737 //
1738 // Nielsen's generalized polylogarithm  S(n,p,x)
1739 //
1740 // helper functions
1741 //
1742 //////////////////////////////////////////////////////////////////////
1743
1744
1745 // anonymous namespace for helper functions
1746 namespace {
1747
1748
1749 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1750 // see fill_Yn()
1751 std::vector<std::vector<cln::cl_N> > Yn;
1752 int ynsize = 0; // number of Yn[]
1753 int ynlength = 100; // initial length of all Yn[i]
1754
1755
1756 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1757 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1758 // representing S_{n,p}(x).
1759 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1760 // equivalent Z-sum.
1761 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1762 // representing S_{n,p}(x).
1763 // The calculation of Y_n uses the values from Y_{n-1}.
1764 void fill_Yn(int n, const cln::float_format_t& prec)
1765 {
1766         const int initsize = ynlength;
1767         //const int initsize = initsize_Yn;
1768         cln::cl_N one = cln::cl_float(1, prec);
1769
1770         if (n) {
1771                 std::vector<cln::cl_N> buf(initsize);
1772                 std::vector<cln::cl_N>::iterator it = buf.begin();
1773                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1774                 *it = (*itprev) / cln::cl_N(n+1) * one;
1775                 it++;
1776                 itprev++;
1777                 // sums with an index smaller than the depth are zero and need not to be calculated.
1778                 // calculation starts with depth, which is n+2)
1779                 for (int i=n+2; i<=initsize+n; i++) {
1780                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1781                         it++;
1782                         itprev++;
1783                 }
1784                 Yn.push_back(buf);
1785         } else {
1786                 std::vector<cln::cl_N> buf(initsize);
1787                 std::vector<cln::cl_N>::iterator it = buf.begin();
1788                 *it = 1 * one;
1789                 it++;
1790                 for (int i=2; i<=initsize; i++) {
1791                         *it = *(it-1) + 1 / cln::cl_N(i) * one;
1792                         it++;
1793                 }
1794                 Yn.push_back(buf);
1795         }
1796         ynsize++;
1797 }
1798
1799
1800 // make Yn longer ... 
1801 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1802 {
1803
1804         cln::cl_N one = cln::cl_float(1, prec);
1805
1806         Yn[0].resize(newsize);
1807         std::vector<cln::cl_N>::iterator it = Yn[0].begin();
1808         it += ynlength;
1809         for (int i=ynlength+1; i<=newsize; i++) {
1810                 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1811                 it++;
1812         }
1813
1814         for (int n=1; n<ynsize; n++) {
1815                 Yn[n].resize(newsize);
1816                 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
1817                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1818                 it += ynlength;
1819                 itprev += ynlength;
1820                 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1821                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1822                         it++;
1823                         itprev++;
1824                 }
1825         }
1826         
1827         ynlength = newsize;
1828 }
1829
1830
1831 // helper function for S(n,p,x)
1832 // [Kol] (7.2)
1833 cln::cl_N C(int n, int p)
1834 {
1835         cln::cl_N result;
1836
1837         for (int k=0; k<p; k++) {
1838                 for (int j=0; j<=(n+k-1)/2; j++) {
1839                         if (k == 0) {
1840                                 if (n & 1) {
1841                                         if (j & 1) {
1842                                                 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1843                                         }
1844                                         else {
1845                                                 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1846                                         }
1847                                 }
1848                         }
1849                         else {
1850                                 if (k & 1) {
1851                                         if (j & 1) {
1852                                                 result = result + cln::factorial(n+k-1)
1853                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1854                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1855                                         }
1856                                         else {
1857                                                 result = result - cln::factorial(n+k-1)
1858                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1859                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1860                                         }
1861                                 }
1862                                 else {
1863                                         if (j & 1) {
1864                                                 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1865                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1866                                         }
1867                                         else {
1868                                                 result = result + cln::factorial(n+k-1)
1869                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1870                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1871                                         }
1872                                 }
1873                         }
1874                 }
1875         }
1876         int np = n+p;
1877         if ((np-1) & 1) {
1878                 if (((np)/2+n) & 1) {
1879                         result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1880                 }
1881                 else {
1882                         result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1883                 }
1884         }
1885
1886         return result;
1887 }
1888
1889
1890 // helper function for S(n,p,x)
1891 // [Kol] remark to (9.1)
1892 cln::cl_N a_k(int k)
1893 {
1894         cln::cl_N result;
1895
1896         if (k == 0) {
1897                 return 1;
1898         }
1899
1900         result = result;
1901         for (int m=2; m<=k; m++) {
1902                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1903         }
1904
1905         return -result / k;
1906 }
1907
1908
1909 // helper function for S(n,p,x)
1910 // [Kol] remark to (9.1)
1911 cln::cl_N b_k(int k)
1912 {
1913         cln::cl_N result;
1914
1915         if (k == 0) {
1916                 return 1;
1917         }
1918
1919         result = result;
1920         for (int m=2; m<=k; m++) {
1921                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1922         }
1923
1924         return result / k;
1925 }
1926
1927
1928 // helper function for S(n,p,x)
1929 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1930 {
1931         static cln::float_format_t oldprec = cln::default_float_format;
1932
1933         if (p==1) {
1934                 return Li_projection(n+1, x, prec);
1935         }
1936
1937         // precision has changed, we need to clear lookup table Yn
1938         if ( oldprec != prec ) {
1939                 Yn.clear();
1940                 ynsize = 0;
1941                 ynlength = 100;
1942                 oldprec = prec;
1943         }
1944                 
1945         // check if precalculated values are sufficient
1946         if (p > ynsize+1) {
1947                 for (int i=ynsize; i<p-1; i++) {
1948                         fill_Yn(i, prec);
1949                 }
1950         }
1951
1952         // should be done otherwise
1953         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1954         cln::cl_N xf = x * one;
1955         //cln::cl_N xf = x * cln::cl_float(1, prec);
1956
1957         cln::cl_N res;
1958         cln::cl_N resbuf;
1959         cln::cl_N factor = cln::expt(xf, p);
1960         int i = p;
1961         do {
1962                 resbuf = res;
1963                 if (i-p >= ynlength) {
1964                         // make Yn longer
1965                         make_Yn_longer(ynlength*2, prec);
1966                 }
1967                 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? ...
1968                 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
1969                 factor = factor * xf;
1970                 i++;
1971         } while (res != resbuf);
1972         
1973         return res;
1974 }
1975
1976
1977 // helper function for S(n,p,x)
1978 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1979 {
1980         // [Kol] (5.3)
1981         if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
1982
1983                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
1984                                    * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
1985
1986                 for (int s=0; s<n; s++) {
1987                         cln::cl_N res2;
1988                         for (int r=0; r<p; r++) {
1989                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
1990                                               * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
1991                         }
1992                         result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
1993                 }
1994
1995                 return result;
1996         }
1997         
1998         return S_do_sum(n, p, x, prec);
1999 }
2000
2001
2002 // helper function for S(n,p,x)
2003 const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
2004 {
2005         if (x == 1) {
2006                 if (n == 1) {
2007                     // [Kol] (2.22) with (2.21)
2008                         return cln::zeta(p+1);
2009                 }
2010
2011                 if (p == 1) {
2012                     // [Kol] (2.22)
2013                         return cln::zeta(n+1);
2014                 }
2015
2016                 // [Kol] (9.1)
2017                 cln::cl_N result;
2018                 for (int nu=0; nu<n; nu++) {
2019                         for (int rho=0; rho<=p; rho++) {
2020                                 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
2021                                                   * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
2022                         }
2023                 }
2024                 result = result * cln::expt(cln::cl_I(-1),n+p-1);
2025
2026                 return result;
2027         }
2028         else if (x == -1) {
2029                 // [Kol] (2.22)
2030                 if (p == 1) {
2031                         return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
2032                 }
2033 //              throw std::runtime_error("don't know how to evaluate this function!");
2034         }
2035
2036         // what is the desired float format?
2037         // first guess: default format
2038         cln::float_format_t prec = cln::default_float_format;
2039         const cln::cl_N value = x;
2040         // second guess: the argument's format
2041         if (!instanceof(realpart(value), cln::cl_RA_ring))
2042                 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
2043         else if (!instanceof(imagpart(value), cln::cl_RA_ring))
2044                 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
2045
2046         // [Kol] (5.3)
2047         // 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
2048         // we don't care here about abs(value)<1 && real(value)>0.5, this will be taken care of in S_projection
2049         if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95) && (cln::abs(1-value) > 1) )) {
2050
2051                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
2052                                    * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
2053
2054                 for (int s=0; s<n; s++) {
2055                         cln::cl_N res2;
2056                         for (int r=0; r<p; r++) {
2057                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
2058                                               * S_num(p-r,n-s,1-value) / cln::factorial(r);
2059                         }
2060                         result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2061                 }
2062
2063                 return result;
2064                 
2065         }
2066         // [Kol] (5.12)
2067         if (cln::abs(value) > 1) {
2068                 
2069                 cln::cl_N result;
2070
2071                 for (int s=0; s<p; s++) {
2072                         for (int r=0; r<=s; r++) {
2073                                 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
2074                                                   / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
2075                                                   * S_num(n+s-r,p-s,cln::recip(value));
2076                         }
2077                 }
2078                 result = result * cln::expt(cln::cl_I(-1),n);
2079
2080                 cln::cl_N res2;
2081                 for (int r=0; r<n; r++) {
2082                         res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
2083                 }
2084                 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
2085
2086                 result = result + cln::expt(cln::cl_I(-1),p) * res2;
2087
2088                 return result;
2089         }
2090
2091         if ((cln::abs(value) > 0.95) && (cln::abs(value-9.53) < 9.47)) {
2092                 lst m;
2093                 m.append(n+1);
2094                 for (int s=0; s<p-1; s++)
2095                         m.append(1);
2096
2097                 ex res = H(m,numeric(value)).evalf();
2098                 return ex_to<numeric>(res).to_cl_N();
2099         }
2100         else {
2101                 return S_projection(n, p, value, prec);
2102         }
2103 }
2104
2105
2106 } // end of anonymous namespace
2107
2108
2109 //////////////////////////////////////////////////////////////////////
2110 //
2111 // Nielsen's generalized polylogarithm  S(n,p,x)
2112 //
2113 // GiNaC function
2114 //
2115 //////////////////////////////////////////////////////////////////////
2116
2117
2118 static ex S_evalf(const ex& n, const ex& p, const ex& x)
2119 {
2120         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2121                 const int n_ = ex_to<numeric>(n).to_int();
2122                 const int p_ = ex_to<numeric>(p).to_int();
2123                 if (is_a<numeric>(x)) {
2124                         const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2125                         const cln::cl_N result = S_num(n_, p_, x_);
2126                         return numeric(result);
2127                 } else {
2128                         ex x_val = x.evalf();
2129                         if (is_a<numeric>(x_val)) {
2130                                 const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
2131                                 const cln::cl_N result = S_num(n_, p_, x_val_);
2132                                 return numeric(result);
2133                         }
2134                 }
2135         }
2136         return S(n, p, x).hold();
2137 }
2138
2139
2140 static ex S_eval(const ex& n, const ex& p, const ex& x)
2141 {
2142         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2143                 if (x == 0) {
2144                         return _ex0;
2145                 }
2146                 if (x == 1) {
2147                         lst m(n+1);
2148                         for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2149                                 m.append(1);
2150                         }
2151                         return zeta(m);
2152                 }
2153                 if (p == 1) {
2154                         return Li(n+1, x);
2155                 }
2156                 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2157                         int n_ = ex_to<numeric>(n).to_int();
2158                         int p_ = ex_to<numeric>(p).to_int();
2159                         const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2160                         const cln::cl_N result = S_num(n_, p_, x_);
2161                         return numeric(result);
2162                 }
2163         }
2164         if (n.is_zero()) {
2165                 // [Kol] (5.3)
2166                 return pow(-log(1-x), p) / factorial(p);
2167         }
2168         return S(n, p, x).hold();
2169 }
2170
2171
2172 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2173 {
2174         if (p == _ex1) {
2175                 return Li(n+1, x).series(rel, order, options);
2176         }
2177
2178         const ex x_pt = x.subs(rel, subs_options::no_pattern);
2179         if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2180                 // First special case: x==0 (derivatives have poles)
2181                 if (x_pt.is_zero()) {
2182                         const symbol s;
2183                         ex ser;
2184                         // manually construct the primitive expansion
2185                         // subsum = Euler-Zagier-Sum is needed
2186                         // dirty hack (slow ...) calculation of subsum:
2187                         std::vector<ex> presubsum, subsum;
2188                         subsum.push_back(0);
2189                         for (int i=1; i<order-1; ++i) {
2190                                 subsum.push_back(subsum[i-1] + numeric(1, i));
2191                         }
2192                         for (int depth=2; depth<p; ++depth) {
2193                                 presubsum = subsum;
2194                                 for (int i=1; i<order-1; ++i) {
2195                                         subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2196                                 }
2197                         }
2198                                 
2199                         for (int i=1; i<order; ++i) {
2200                                 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2201                         }
2202                         // substitute the argument's series expansion
2203                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2204                         // maybe that was terminating, so add a proper order term
2205                         epvector nseq;
2206                         nseq.push_back(expair(Order(_ex1), order));
2207                         ser += pseries(rel, nseq);
2208                         // reexpanding it will collapse the series again
2209                         return ser.series(rel, order);
2210                 }
2211                 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2212                 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2213         }
2214         // all other cases should be safe, by now:
2215         throw do_taylor();  // caught by function::series()
2216 }
2217
2218
2219 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2220 {
2221         GINAC_ASSERT(deriv_param < 3);
2222         if (deriv_param < 2) {
2223                 return _ex0;
2224         }
2225         if (n > 0) {
2226                 return S(n-1, p, x) / x;
2227         } else {
2228                 return S(n, p-1, x) / (1-x);
2229         }
2230 }
2231
2232
2233 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2234 {
2235         c.s << "\\mathrm{S}_{";
2236         n.print(c);
2237         c.s << ",";
2238         p.print(c);
2239         c.s << "}(";
2240         x.print(c);
2241         c.s << ")";
2242 }
2243
2244
2245 REGISTER_FUNCTION(S,
2246                   evalf_func(S_evalf).
2247                   eval_func(S_eval).
2248                   series_func(S_series).
2249                   derivative_func(S_deriv).
2250                   print_func<print_latex>(S_print_latex).
2251                   do_not_evalf_params());
2252
2253
2254 //////////////////////////////////////////////////////////////////////
2255 //
2256 // Harmonic polylogarithm  H(m,x)
2257 //
2258 // helper functions
2259 //
2260 //////////////////////////////////////////////////////////////////////
2261
2262
2263 // anonymous namespace for helper functions
2264 namespace {
2265
2266         
2267 // regulates the pole (used by 1/x-transformation)
2268 symbol H_polesign("IMSIGN");
2269
2270
2271 // convert parameters from H to Li representation
2272 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2273 // returns true if some parameters are negative
2274 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2275 {
2276         // expand parameter list
2277         lst mexp;
2278         for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
2279                 if (*it > 1) {
2280                         for (ex count=*it-1; count > 0; count--) {
2281                                 mexp.append(0);
2282                         }
2283                         mexp.append(1);
2284                 } else if (*it < -1) {
2285                         for (ex count=*it+1; count < 0; count++) {
2286                                 mexp.append(0);
2287                         }
2288                         mexp.append(-1);
2289                 } else {
2290                         mexp.append(*it);
2291                 }
2292         }
2293         
2294         ex signum = 1;
2295         pf = 1;
2296         bool has_negative_parameters = false;
2297         ex acc = 1;
2298         for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
2299                 if (*it == 0) {
2300                         acc++;
2301                         continue;
2302                 }
2303                 if (*it > 0) {
2304                         m.append((*it+acc-1) * signum);
2305                 } else {
2306                         m.append((*it-acc+1) * signum);
2307                 }
2308                 acc = 1;
2309                 signum = *it;
2310                 pf *= *it;
2311                 if (pf < 0) {
2312                         has_negative_parameters = true;
2313                 }
2314         }
2315         if (has_negative_parameters) {
2316                 for (std::size_t i=0; i<m.nops(); i++) {
2317                         if (m.op(i) < 0) {
2318                                 m.let_op(i) = -m.op(i);
2319                                 s.append(-1);
2320                         } else {
2321                                 s.append(1);
2322                         }
2323                 }
2324         }
2325         
2326         return has_negative_parameters;
2327 }
2328
2329
2330 // recursivly transforms H to corresponding multiple polylogarithms
2331 struct map_trafo_H_convert_to_Li : public map_function
2332 {
2333         ex operator()(const ex& e)
2334         {
2335                 if (is_a<add>(e) || is_a<mul>(e)) {
2336                         return e.map(*this);
2337                 }
2338                 if (is_a<function>(e)) {
2339                         std::string name = ex_to<function>(e).get_name();
2340                         if (name == "H") {
2341                                 lst parameter;
2342                                 if (is_a<lst>(e.op(0))) {
2343                                                 parameter = ex_to<lst>(e.op(0));
2344                                 } else {
2345                                         parameter = lst(e.op(0));
2346                                 }
2347                                 ex arg = e.op(1);
2348
2349                                 lst m;
2350                                 lst s;
2351                                 ex pf;
2352                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2353                                         s.let_op(0) = s.op(0) * arg;
2354                                         return pf * Li(m, s).hold();
2355                                 } else {
2356                                         for (std::size_t i=0; i<m.nops(); i++) {
2357                                                 s.append(1);
2358                                         }
2359                                         s.let_op(0) = s.op(0) * arg;
2360                                         return Li(m, s).hold();
2361                                 }
2362                         }
2363                 }
2364                 return e;
2365         }
2366 };
2367
2368
2369 // recursivly transforms H to corresponding zetas
2370 struct map_trafo_H_convert_to_zeta : public map_function
2371 {
2372         ex operator()(const ex& e)
2373         {
2374                 if (is_a<add>(e) || is_a<mul>(e)) {
2375                         return e.map(*this);
2376                 }
2377                 if (is_a<function>(e)) {
2378                         std::string name = ex_to<function>(e).get_name();
2379                         if (name == "H") {
2380                                 lst parameter;
2381                                 if (is_a<lst>(e.op(0))) {
2382                                                 parameter = ex_to<lst>(e.op(0));
2383                                 } else {
2384                                         parameter = lst(e.op(0));
2385                                 }
2386
2387                                 lst m;
2388                                 lst s;
2389                                 ex pf;
2390                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2391                                         return pf * zeta(m, s);
2392                                 } else {
2393                                         return zeta(m);
2394                                 }
2395                         }
2396                 }
2397                 return e;
2398         }
2399 };
2400
2401
2402 // remove trailing zeros from H-parameters
2403 struct map_trafo_H_reduce_trailing_zeros : public map_function
2404 {
2405         ex operator()(const ex& e)
2406         {
2407                 if (is_a<add>(e) || is_a<mul>(e)) {
2408                         return e.map(*this);
2409                 }
2410                 if (is_a<function>(e)) {
2411                         std::string name = ex_to<function>(e).get_name();
2412                         if (name == "H") {
2413                                 lst parameter;
2414                                 if (is_a<lst>(e.op(0))) {
2415                                         parameter = ex_to<lst>(e.op(0));
2416                                 } else {
2417                                         parameter = lst(e.op(0));
2418                                 }
2419                                 ex arg = e.op(1);
2420                                 if (parameter.op(parameter.nops()-1) == 0) {
2421                                         
2422                                         //
2423                                         if (parameter.nops() == 1) {
2424                                                 return log(arg);
2425                                         }
2426                                         
2427                                         //
2428                                         lst::const_iterator it = parameter.begin();
2429                                         while ((it != parameter.end()) && (*it == 0)) {
2430                                                 it++;
2431                                         }
2432                                         if (it == parameter.end()) {
2433                                                 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2434                                         }
2435                                         
2436                                         //
2437                                         parameter.remove_last();
2438                                         std::size_t lastentry = parameter.nops();
2439                                         while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2440                                                 lastentry--;
2441                                         }
2442                                         
2443                                         //
2444                                         ex result = log(arg) * H(parameter,arg).hold();
2445                                         ex acc = 0;
2446                                         for (ex i=0; i<lastentry; i++) {
2447                                                 if (parameter[i] > 0) {
2448                                                         parameter[i]++;
2449                                                         result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2450                                                         parameter[i]--;
2451                                                         acc = 0;
2452                                                 } else if (parameter[i] < 0) {
2453                                                         parameter[i]--;
2454                                                         result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2455                                                         parameter[i]++;
2456                                                         acc = 0;
2457                                                 } else {
2458                                                         acc++;
2459                                                 }
2460                                         }
2461                                         
2462                                         if (lastentry < parameter.nops()) {
2463                                                 result = result / (parameter.nops()-lastentry+1);
2464                                                 return result.map(*this);
2465                                         } else {
2466                                                 return result;
2467                                         }
2468                                 }
2469                         }
2470                 }
2471                 return e;
2472         }
2473 };
2474
2475
2476 // returns an expression with zeta functions corresponding to the parameter list for H
2477 ex convert_H_to_zeta(const lst& m)
2478 {
2479         symbol xtemp("xtemp");
2480         map_trafo_H_reduce_trailing_zeros filter;
2481         map_trafo_H_convert_to_zeta filter2;
2482         return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2483 }
2484
2485
2486 // convert signs form Li to H representation
2487 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2488 {
2489         lst res;
2490         lst::const_iterator itm = m.begin();
2491         lst::const_iterator itx = ++x.begin();
2492         int signum = 1;
2493         pf = _ex1;
2494         res.append(*itm);
2495         itm++;
2496         while (itx != x.end()) {
2497                 signum *= (*itx > 0) ? 1 : -1;
2498                 pf *= signum;
2499                 res.append((*itm) * signum);
2500                 itm++;
2501                 itx++;
2502         }
2503         return res;
2504 }
2505
2506
2507 // multiplies an one-dimensional H with another H
2508 // [ReV] (18)
2509 ex trafo_H_mult(const ex& h1, const ex& h2)
2510 {
2511         ex res;
2512         ex hshort;
2513         lst hlong;
2514         ex h1nops = h1.op(0).nops();
2515         ex h2nops = h2.op(0).nops();
2516         if (h1nops > 1) {
2517                 hshort = h2.op(0).op(0);
2518                 hlong = ex_to<lst>(h1.op(0));
2519         } else {
2520                 hshort = h1.op(0).op(0);
2521                 if (h2nops > 1) {
2522                         hlong = ex_to<lst>(h2.op(0));
2523                 } else {
2524                         hlong = h2.op(0).op(0);
2525                 }
2526         }
2527         for (std::size_t i=0; i<=hlong.nops(); i++) {
2528                 lst newparameter;
2529                 std::size_t j=0;
2530                 for (; j<i; j++) {
2531                         newparameter.append(hlong[j]);
2532                 }
2533                 newparameter.append(hshort);
2534                 for (; j<hlong.nops(); j++) {
2535                         newparameter.append(hlong[j]);
2536                 }
2537                 res += H(newparameter, h1.op(1)).hold();
2538         }
2539         return res;
2540 }
2541
2542
2543 // applies trafo_H_mult recursively on expressions
2544 struct map_trafo_H_mult : public map_function
2545 {
2546         ex operator()(const ex& e)
2547         {
2548                 if (is_a<add>(e)) {
2549                         return e.map(*this);
2550                 }
2551
2552                 if (is_a<mul>(e)) {
2553
2554                         ex result = 1;
2555                         ex firstH;
2556                         lst Hlst;
2557                         for (std::size_t pos=0; pos<e.nops(); pos++) {
2558                                 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2559                                         std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2560                                         if (name == "H") {
2561                                                 for (ex i=0; i<e.op(pos).op(1); i++) {
2562                                                         Hlst.append(e.op(pos).op(0));
2563                                                 }
2564                                                 continue;
2565                                         }
2566                                 } else if (is_a<function>(e.op(pos))) {
2567                                         std::string name = ex_to<function>(e.op(pos)).get_name();
2568                                         if (name == "H") {
2569                                                 if (e.op(pos).op(0).nops() > 1) {
2570                                                         firstH = e.op(pos);
2571                                                 } else {
2572                                                         Hlst.append(e.op(pos));
2573                                                 }
2574                                                 continue;
2575                                         }
2576                                 }
2577                                 result *= e.op(pos);
2578                         }
2579                         if (firstH == 0) {
2580                                 if (Hlst.nops() > 0) {
2581                                         firstH = Hlst[Hlst.nops()-1];
2582                                         Hlst.remove_last();
2583                                 } else {
2584                                         return e;
2585                                 }
2586                         }
2587
2588                         if (Hlst.nops() > 0) {
2589                                 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2590                                 result *= buffer;
2591                                 for (std::size_t i=1; i<Hlst.nops(); i++) {
2592                                         result *= Hlst.op(i);
2593                                 }
2594                                 result = result.expand();
2595                                 map_trafo_H_mult recursion;
2596                                 return recursion(result);
2597                         } else {
2598                                 return e;
2599                         }
2600
2601                 }
2602                 return e;
2603         }
2604 };
2605
2606
2607 // do integration [ReV] (55)
2608 // put parameter 0 in front of existing parameters
2609 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
2610 {
2611         ex h;
2612         std::string name;
2613         if (is_a<function>(e)) {
2614                 name = ex_to<function>(e).get_name();
2615         }
2616         if (name == "H") {
2617                 h = e;
2618         } else {
2619                 for (std::size_t i=0; i<e.nops(); i++) {
2620                         if (is_a<function>(e.op(i))) {
2621                                 std::string name = ex_to<function>(e.op(i)).get_name();
2622                                 if (name == "H") {
2623                                         h = e.op(i);
2624                                 }
2625                         }
2626                 }
2627         }
2628         if (h != 0) {
2629                 lst newparameter = ex_to<lst>(h.op(0));
2630                 newparameter.prepend(0);
2631                 ex addzeta = convert_H_to_zeta(newparameter);
2632                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2633         } else {
2634                 return e * (-H(lst(ex(0)),1/arg).hold());
2635         }
2636 }
2637
2638
2639 // do integration [ReV] (49)
2640 // put parameter 1 in front of existing parameters
2641 ex trafo_H_prepend_one(const ex& e, const ex& arg)
2642 {
2643         ex h;
2644         std::string name;
2645         if (is_a<function>(e)) {
2646                 name = ex_to<function>(e).get_name();
2647         }
2648         if (name == "H") {
2649                 h = e;
2650         } else {
2651                 for (std::size_t i=0; i<e.nops(); i++) {
2652                         if (is_a<function>(e.op(i))) {
2653                                 std::string name = ex_to<function>(e.op(i)).get_name();
2654                                 if (name == "H") {
2655                                         h = e.op(i);
2656                                 }
2657                         }
2658                 }
2659         }
2660         if (h != 0) {
2661                 lst newparameter = ex_to<lst>(h.op(0));
2662                 newparameter.prepend(1);
2663                 return e.subs(h == H(newparameter, h.op(1)).hold());
2664         } else {
2665                 return e * H(lst(ex(1)),1-arg).hold();
2666         }
2667 }
2668
2669
2670 // do integration [ReV] (55)
2671 // put parameter -1 in front of existing parameters
2672 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2673 {
2674         ex h;
2675         std::string name;
2676         if (is_a<function>(e)) {
2677                 name = ex_to<function>(e).get_name();
2678         }
2679         if (name == "H") {
2680                 h = e;
2681         } else {
2682                 for (std::size_t i=0; i<e.nops(); i++) {
2683                         if (is_a<function>(e.op(i))) {
2684                                 std::string name = ex_to<function>(e.op(i)).get_name();
2685                                 if (name == "H") {
2686                                         h = e.op(i);
2687                                 }
2688                         }
2689                 }
2690         }
2691         if (h != 0) {
2692                 lst newparameter = ex_to<lst>(h.op(0));
2693                 newparameter.prepend(-1);
2694                 ex addzeta = convert_H_to_zeta(newparameter);
2695                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2696         } else {
2697                 ex addzeta = convert_H_to_zeta(lst(ex(-1)));
2698                 return (e * (addzeta - H(lst(ex(-1)),1/arg).hold())).expand();
2699         }
2700 }
2701
2702
2703 // do integration [ReV] (55)
2704 // put parameter -1 in front of existing parameters
2705 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
2706 {
2707         ex h;
2708         std::string name;
2709         if (is_a<function>(e)) {
2710                 name = ex_to<function>(e).get_name();
2711         }
2712         if (name == "H") {
2713                 h = e;
2714         } else {
2715                 for (std::size_t i = 0; i < e.nops(); i++) {
2716                         if (is_a<function>(e.op(i))) {
2717                                 std::string name = ex_to<function>(e.op(i)).get_name();
2718                                 if (name == "H") {
2719                                         h = e.op(i);
2720                                 }
2721                         }
2722                 }
2723         }
2724         if (h != 0) {
2725                 lst newparameter = ex_to<lst>(h.op(0));
2726                 newparameter.prepend(-1);
2727                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2728         } else {
2729                 return (e * H(lst(ex(-1)),(1-arg)/(1+arg)).hold()).expand();
2730         }
2731 }
2732
2733
2734 // do integration [ReV] (55)
2735 // put parameter 1 in front of existing parameters
2736 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2737 {
2738         ex h;
2739         std::string name;
2740         if (is_a<function>(e)) {
2741                 name = ex_to<function>(e).get_name();
2742         }
2743         if (name == "H") {
2744                 h = e;
2745         } else {
2746                 for (std::size_t i = 0; i < e.nops(); i++) {
2747                         if (is_a<function>(e.op(i))) {
2748                                 std::string name = ex_to<function>(e.op(i)).get_name();
2749                                 if (name == "H") {
2750                                         h = e.op(i);
2751                                 }
2752                         }
2753                 }
2754         }
2755         if (h != 0) {
2756                 lst newparameter = ex_to<lst>(h.op(0));
2757                 newparameter.prepend(1);
2758                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2759         } else {
2760                 return (e * H(lst(ex(1)),(1-arg)/(1+arg)).hold()).expand();
2761         }
2762 }
2763
2764
2765 // do x -> 1-x transformation
2766 struct map_trafo_H_1mx : public map_function
2767 {
2768         ex operator()(const ex& e)
2769         {
2770                 if (is_a<add>(e) || is_a<mul>(e)) {
2771                         return e.map(*this);
2772                 }
2773                 
2774                 if (is_a<function>(e)) {
2775                         std::string name = ex_to<function>(e).get_name();
2776                         if (name == "H") {
2777
2778                                 lst parameter = ex_to<lst>(e.op(0));
2779                                 ex arg = e.op(1);
2780
2781                                 // special cases if all parameters are either 0, 1 or -1
2782                                 bool allthesame = true;
2783                                 if (parameter.op(0) == 0) {
2784                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2785                                                 if (parameter.op(i) != 0) {
2786                                                         allthesame = false;
2787                                                         break;
2788                                                 }
2789                                         }
2790                                         if (allthesame) {
2791                                                 lst newparameter;
2792                                                 for (int i=parameter.nops(); i>0; i--) {
2793                                                         newparameter.append(1);
2794                                                 }
2795                                                 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2796                                         }
2797                                 } else if (parameter.op(0) == -1) {
2798                                         throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2799                                 } else {
2800                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2801                                                 if (parameter.op(i) != 1) {
2802                                                         allthesame = false;
2803                                                         break;
2804                                                 }
2805                                         }
2806                                         if (allthesame) {
2807                                                 lst newparameter;
2808                                                 for (int i=parameter.nops(); i>0; i--) {
2809                                                         newparameter.append(0);
2810                                                 }
2811                                                 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2812                                         }
2813                                 }
2814
2815                                 lst newparameter = parameter;
2816                                 newparameter.remove_first();
2817
2818                                 if (parameter.op(0) == 0) {
2819
2820                                         // leading zero
2821                                         ex res = convert_H_to_zeta(parameter);
2822                                         //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2823                                         map_trafo_H_1mx recursion;
2824                                         ex buffer = recursion(H(newparameter, arg).hold());
2825                                         if (is_a<add>(buffer)) {
2826                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2827                                                         res -= trafo_H_prepend_one(buffer.op(i), arg);
2828                                                 }
2829                                         } else {
2830                                                 res -= trafo_H_prepend_one(buffer, arg);
2831                                         }
2832                                         return res;
2833
2834                                 } else {
2835
2836                                         // leading one
2837                                         map_trafo_H_1mx recursion;
2838                                         map_trafo_H_mult unify;
2839                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2840                                         std::size_t firstzero = 0;
2841                                         while (parameter.op(firstzero) == 1) {
2842                                                 firstzero++;
2843                                         }
2844                                         for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) {
2845                                                 lst newparameter;
2846                                                 std::size_t j=0;
2847                                                 for (; j<=i; j++) {
2848                                                         newparameter.append(parameter[j+1]);
2849                                                 }
2850                                                 newparameter.append(1);
2851                                                 for (; j<parameter.nops()-1; j++) {
2852                                                         newparameter.append(parameter[j+1]);
2853                                                 }
2854                                                 res -= H(newparameter, arg).hold();
2855                                         }
2856                                         res = recursion(res).expand() / firstzero;
2857                                         return unify(res);
2858                                 }
2859                         }
2860                 }
2861                 return e;
2862         }
2863 };
2864
2865
2866 // do x -> 1/x transformation
2867 struct map_trafo_H_1overx : public map_function
2868 {
2869         ex operator()(const ex& e)
2870         {
2871                 if (is_a<add>(e) || is_a<mul>(e)) {
2872                         return e.map(*this);
2873                 }
2874
2875                 if (is_a<function>(e)) {
2876                         std::string name = ex_to<function>(e).get_name();
2877                         if (name == "H") {
2878
2879                                 lst parameter = ex_to<lst>(e.op(0));
2880                                 ex arg = e.op(1);
2881
2882                                 // special cases if all parameters are either 0, 1 or -1
2883                                 bool allthesame = true;
2884                                 if (parameter.op(0) == 0) {
2885                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2886                                                 if (parameter.op(i) != 0) {
2887                                                         allthesame = false;
2888                                                         break;
2889                                                 }
2890                                         }
2891                                         if (allthesame) {
2892                                                 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2893                                         }
2894                                 } else if (parameter.op(0) == -1) {
2895                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2896                                                 if (parameter.op(i) != -1) {
2897                                                         allthesame = false;
2898                                                         break;
2899                                                 }
2900                                         }
2901                                         if (allthesame) {
2902                                                 map_trafo_H_mult unify;
2903                                                 return unify((pow(H(lst(ex(-1)),1/arg).hold() - H(lst(ex(0)),1/arg).hold(), parameter.nops())
2904                                                        / factorial(parameter.nops())).expand());
2905                                         }
2906                                 } else {
2907                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2908                                                 if (parameter.op(i) != 1) {
2909                                                         allthesame = false;
2910                                                         break;
2911                                                 }
2912                                         }
2913                                         if (allthesame) {
2914                                                 map_trafo_H_mult unify;
2915                                                 return unify((pow(H(lst(ex(1)),1/arg).hold() + H(lst(ex(0)),1/arg).hold() + H_polesign, parameter.nops())
2916                                                        / factorial(parameter.nops())).expand());
2917                                         }
2918                                 }
2919
2920                                 lst newparameter = parameter;
2921                                 newparameter.remove_first();
2922
2923                                 if (parameter.op(0) == 0) {
2924                                         
2925                                         // leading zero
2926                                         ex res = convert_H_to_zeta(parameter);
2927                                         map_trafo_H_1overx recursion;
2928                                         ex buffer = recursion(H(newparameter, arg).hold());
2929                                         if (is_a<add>(buffer)) {
2930                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2931                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2932                                                 }
2933                                         } else {
2934                                                 res += trafo_H_1tx_prepend_zero(buffer, arg);
2935                                         }
2936                                         return res;
2937
2938                                 } else if (parameter.op(0) == -1) {
2939
2940                                         // leading negative one
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) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
2947                                                 }
2948                                         } else {
2949                                                 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
2950                                         }
2951                                         return res;
2952
2953                                 } else {
2954
2955                                         // leading one
2956                                         map_trafo_H_1overx recursion;
2957                                         map_trafo_H_mult unify;
2958                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2959                                         std::size_t firstzero = 0;
2960                                         while (parameter.op(firstzero) == 1) {
2961                                                 firstzero++;
2962                                         }
2963                                         for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) {
2964                                                 lst newparameter;
2965                                                 std::size_t j = 0;
2966                                                 for (; j<=i; j++) {
2967                                                         newparameter.append(parameter[j+1]);
2968                                                 }
2969                                                 newparameter.append(1);
2970                                                 for (; j<parameter.nops()-1; j++) {
2971                                                         newparameter.append(parameter[j+1]);
2972                                                 }
2973                                                 res -= H(newparameter, arg).hold();
2974                                         }
2975                                         res = recursion(res).expand() / firstzero;
2976                                         return unify(res);
2977
2978                                 }
2979
2980                         }
2981                 }
2982                 return e;
2983         }
2984 };
2985
2986
2987 // do x -> (1-x)/(1+x) transformation
2988 struct map_trafo_H_1mxt1px : public map_function
2989 {
2990         ex operator()(const ex& e)
2991         {
2992                 if (is_a<add>(e) || is_a<mul>(e)) {
2993                         return e.map(*this);
2994                 }
2995
2996                 if (is_a<function>(e)) {
2997                         std::string name = ex_to<function>(e).get_name();
2998                         if (name == "H") {
2999
3000                                 lst parameter = ex_to<lst>(e.op(0));
3001                                 ex arg = e.op(1);
3002
3003                                 // special cases if all parameters are either 0, 1 or -1
3004                                 bool allthesame = true;
3005                                 if (parameter.op(0) == 0) {
3006                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
3007                                                 if (parameter.op(i) != 0) {
3008                                                         allthesame = false;
3009                                                         break;
3010                                                 }
3011                                         }
3012                                         if (allthesame) {
3013                                                 map_trafo_H_mult unify;
3014                                                 return unify((pow(-H(lst(ex(1)),(1-arg)/(1+arg)).hold() - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3015                                                        / factorial(parameter.nops())).expand());
3016                                         }
3017                                 } else if (parameter.op(0) == -1) {
3018                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
3019                                                 if (parameter.op(i) != -1) {
3020                                                         allthesame = false;
3021                                                         break;
3022                                                 }
3023                                         }
3024                                         if (allthesame) {
3025                                                 map_trafo_H_mult unify;
3026                                                 return unify((pow(log(2) - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3027                                                        / factorial(parameter.nops())).expand());
3028                                         }
3029                                 } else {
3030                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
3031                                                 if (parameter.op(i) != 1) {
3032                                                         allthesame = false;
3033                                                         break;
3034                                                 }
3035                                         }
3036                                         if (allthesame) {
3037                                                 map_trafo_H_mult unify;
3038                                                 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())
3039                                                        / factorial(parameter.nops())).expand());
3040                                         }
3041                                 }
3042
3043                                 lst newparameter = parameter;
3044                                 newparameter.remove_first();
3045
3046                                 if (parameter.op(0) == 0) {
3047
3048                                         // leading zero
3049                                         ex res = convert_H_to_zeta(parameter);
3050                                         map_trafo_H_1mxt1px recursion;
3051                                         ex buffer = recursion(H(newparameter, arg).hold());
3052                                         if (is_a<add>(buffer)) {
3053                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
3054                                                         res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3055                                                 }
3056                                         } else {
3057                                                 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3058                                         }
3059                                         return res;
3060
3061                                 } else if (parameter.op(0) == -1) {
3062
3063                                         // leading negative one
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_minusone(buffer.op(i), arg);
3070                                                 }
3071                                         } else {
3072                                                 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3073                                         }
3074                                         return res;
3075
3076                                 } else {
3077
3078                                         // leading one
3079                                         map_trafo_H_1mxt1px recursion;
3080                                         map_trafo_H_mult unify;
3081                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
3082                                         std::size_t firstzero = 0;
3083                                         while (parameter.op(firstzero) == 1) {
3084                                                 firstzero++;
3085                                         }
3086                                         for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) {
3087                                                 lst newparameter;
3088                                                 std::size_t j=0;
3089                                                 for (; j<=i; j++) {
3090                                                         newparameter.append(parameter[j+1]);
3091                                                 }
3092                                                 newparameter.append(1);
3093                                                 for (; j<parameter.nops()-1; j++) {
3094                                                         newparameter.append(parameter[j+1]);
3095                                                 }
3096                                                 res -= H(newparameter, arg).hold();
3097                                         }
3098                                         res = recursion(res).expand() / firstzero;
3099                                         return unify(res);
3100
3101                                 }
3102
3103                         }
3104                 }
3105                 return e;
3106         }
3107 };
3108
3109
3110 // do the actual summation.
3111 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
3112 {
3113         const int j = m.size();
3114
3115         std::vector<cln::cl_N> t(j);
3116
3117         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3118         cln::cl_N factor = cln::expt(x, j) * one;
3119         cln::cl_N t0buf;
3120         int q = 0;
3121         do {
3122                 t0buf = t[0];
3123                 q++;
3124                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
3125                 for (int k=j-2; k>=1; k--) {
3126                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
3127                 }
3128                 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
3129                 factor = factor * x;
3130         } while (t[0] != t0buf);
3131
3132         return t[0];
3133 }
3134
3135
3136 } // end of anonymous namespace
3137
3138
3139 //////////////////////////////////////////////////////////////////////
3140 //
3141 // Harmonic polylogarithm  H(m,x)
3142 //
3143 // GiNaC function
3144 //
3145 //////////////////////////////////////////////////////////////////////
3146
3147
3148 static ex H_evalf(const ex& x1, const ex& x2)
3149 {
3150         if (is_a<lst>(x1)) {
3151                 
3152                 cln::cl_N x;
3153                 if (is_a<numeric>(x2)) {
3154                         x = ex_to<numeric>(x2).to_cl_N();
3155                 } else {
3156                         ex x2_val = x2.evalf();
3157                         if (is_a<numeric>(x2_val)) {
3158                                 x = ex_to<numeric>(x2_val).to_cl_N();
3159                         }
3160                 }
3161
3162                 for (std::size_t i = 0; i < x1.nops(); i++) {
3163                         if (!x1.op(i).info(info_flags::integer)) {
3164                                 return H(x1, x2).hold();
3165                         }
3166                 }
3167                 if (x1.nops() < 1) {
3168                         return H(x1, x2).hold();
3169                 }
3170
3171                 const lst& morg = ex_to<lst>(x1);
3172                 // remove trailing zeros ...
3173                 if (*(--morg.end()) == 0) {
3174                         symbol xtemp("xtemp");
3175                         map_trafo_H_reduce_trailing_zeros filter;
3176                         return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3177                 }
3178                 // ... and expand parameter notation
3179                 bool has_minus_one = false;
3180                 lst m;
3181                 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
3182                         if (*it > 1) {
3183                                 for (ex count=*it-1; count > 0; count--) {
3184                                         m.append(0);
3185                                 }
3186                                 m.append(1);
3187                         } else if (*it <= -1) {
3188                                 for (ex count=*it+1; count < 0; count++) {
3189                                         m.append(0);
3190                                 }
3191                                 m.append(-1);
3192                                 has_minus_one = true;
3193                         } else {
3194                                 m.append(*it);
3195                         }
3196                 }
3197
3198                 // do summation
3199                 if (cln::abs(x) < 0.95) {
3200                         lst m_lst;
3201                         lst s_lst;
3202                         ex pf;
3203                         if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3204                                 // negative parameters -> s_lst is filled
3205                                 std::vector<int> m_int;
3206                                 std::vector<cln::cl_N> x_cln;
3207                                 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin(); 
3208                                      it_int != m_lst.end(); it_int++, it_cln++) {
3209                                         m_int.push_back(ex_to<numeric>(*it_int).to_int());
3210                                         x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3211                                 }
3212                                 x_cln.front() = x_cln.front() * x;
3213                                 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3214                         } else {
3215                                 // only positive parameters
3216                                 //TODO
3217                                 if (m_lst.nops() == 1) {
3218                                         return Li(m_lst.op(0), x2).evalf();
3219                                 }
3220                                 std::vector<int> m_int;
3221                                 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
3222                                         m_int.push_back(ex_to<numeric>(*it).to_int());
3223                                 }
3224                                 return numeric(H_do_sum(m_int, x));
3225                         }
3226                 }
3227
3228                 symbol xtemp("xtemp");
3229                 ex res = 1;     
3230                 
3231                 // ensure that the realpart of the argument is positive
3232                 if (cln::realpart(x) < 0) {
3233                         x = -x;
3234                         for (std::size_t i = 0; i < m.nops(); i++) {
3235                                 if (m.op(i) != 0) {
3236                                         m.let_op(i) = -m.op(i);
3237                                         res *= -1;
3238                                 }
3239                         }
3240                 }
3241
3242                 // x -> 1/x
3243                 if (cln::abs(x) >= 2.0) {
3244                         map_trafo_H_1overx trafo;
3245                         res *= trafo(H(m, xtemp).hold());
3246                         if (cln::imagpart(x) <= 0) {
3247                                 res = res.subs(H_polesign == -I*Pi);
3248                         } else {
3249                                 res = res.subs(H_polesign == I*Pi);
3250                         }
3251                         return res.subs(xtemp == numeric(x)).evalf();
3252                 }
3253                 
3254                 // check transformations for 0.95 <= |x| < 2.0
3255                 
3256                 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3257                 if (cln::abs(x-9.53) <= 9.47) {
3258                         // x -> (1-x)/(1+x)
3259                         map_trafo_H_1mxt1px trafo;
3260                         res *= trafo(H(m, xtemp).hold());
3261                 } else {
3262                         // x -> 1-x
3263                         if (has_minus_one) {
3264                                 map_trafo_H_convert_to_Li filter;
3265                                 return filter(H(m, numeric(x)).hold()).evalf();
3266                         }
3267                         map_trafo_H_1mx trafo;
3268                         res *= trafo(H(m, xtemp).hold());
3269                 }
3270
3271                 return res.subs(xtemp == numeric(x)).evalf();
3272         }
3273
3274         return H(x1,x2).hold();
3275 }
3276
3277
3278 static ex H_eval(const ex& m_, const ex& x)
3279 {
3280         lst m;
3281         if (is_a<lst>(m_)) {
3282                 m = ex_to<lst>(m_);
3283         } else {
3284                 m = lst(m_);
3285         }
3286         if (m.nops() == 0) {
3287                 return _ex1;
3288         }
3289         ex pos1;
3290         ex pos2;
3291         ex n;
3292         ex p;
3293         int step = 0;
3294         if (*m.begin() > _ex1) {
3295                 step++;
3296                 pos1 = _ex0;
3297                 pos2 = _ex1;
3298                 n = *m.begin()-1;
3299                 p = _ex1;
3300         } else if (*m.begin() < _ex_1) {
3301                 step++;
3302                 pos1 = _ex0;
3303                 pos2 = _ex_1;
3304                 n = -*m.begin()-1;
3305                 p = _ex1;
3306         } else if (*m.begin() == _ex0) {
3307                 pos1 = _ex0;
3308                 n = _ex1;
3309         } else {
3310                 pos1 = *m.begin();
3311                 p = _ex1;
3312         }
3313         for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
3314                 if ((*it).info(info_flags::integer)) {
3315                         if (step == 0) {
3316                                 if (*it > _ex1) {
3317                                         if (pos1 == _ex0) {
3318                                                 step = 1;
3319                                                 pos2 = _ex1;
3320                                                 n += *it-1;
3321                                                 p = _ex1;
3322                                         } else {
3323                                                 step = 2;
3324                                         }
3325                                 } else if (*it < _ex_1) {
3326                                         if (pos1 == _ex0) {
3327                                                 step = 1;
3328                                                 pos2 = _ex_1;
3329                                                 n += -*it-1;
3330                                                 p = _ex1;
3331                                         } else {
3332                                                 step = 2;
3333                                         }
3334                                 } else {
3335                                         if (*it != pos1) {
3336                                                 step = 1;
3337                                                 pos2 = *it;
3338                                         }
3339                                         if (*it == _ex0) {
3340                                                 n++;
3341                                         } else {
3342                                               &n