7fbc774a5ef903fb4c79bc51e1336a8995d13ee5
[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 numerically 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) 1999-2015 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, bool flag_trailing_zeros_only);
807
808
809 // G transformation [VSW]
810 ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale,
811                const exvector& gsyms, bool flag_trailing_zeros_only)
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, flag_trailing_zeros_only);
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, flag_trailing_zeros_only);
857                 }
858                 return result / trailing_zeros;
859         }
860
861         // convergence case or flag_trailing_zeros_only
862         if (convergent || flag_trailing_zeros_only) {
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, flag_trailing_zeros_only)*
890                             G_transform(empty, a1, scale, gsyms, flag_trailing_zeros_only);
891
892                 result -= shuffle_G(empty, a1, a2, pendint, a, scale, gsyms, flag_trailing_zeros_only);
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, flag_trailing_zeros_only);
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, flag_trailing_zeros_only);
918                 int buffer = *changeit;
919                 *changeit = *min_it;
920                 result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
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, flag_trailing_zeros_only);
928                 *changeit = *min_it;
929                 result -= G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
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, flag_trailing_zeros_only);
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, flag_trailing_zeros_only);
940                 *changeit = *min_it;
941                 result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
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, bool flag_trailing_zeros_only) 
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, flag_trailing_zeros_only);
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, flag_trailing_zeros_only);
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, flag_trailing_zeros_only);
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, flag_trailing_zeros_only)
984              + shuffle_G(a02, a1, a2_removed, pendint, a_old, scale, gsyms, flag_trailing_zeros_only);
985 }
986
987 // handles the transformations and the numerical evaluation of G
988 // the parameter x, s and y must only contain numerics
989 static cln::cl_N
990 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
991           const cln::cl_N& y);
992
993 // do acceleration transformation (hoelder convolution [BBB])
994 // the parameter x, s and y must only contain numerics
995 static cln::cl_N
996 G_do_hoelder(std::vector<cln::cl_N> x, /* yes, it's passed by value */
997              const std::vector<int>& s, const cln::cl_N& y)
998 {
999         cln::cl_N result;
1000         const std::size_t size = x.size();
1001         for (std::size_t i = 0; i < size; ++i)
1002                 x[i] = x[i]/y;
1003
1004         for (std::size_t r = 0; r <= size; ++r) {
1005                 cln::cl_N buffer(1 & r ? -1 : 1);
1006                 cln::cl_RA p(2);
1007                 bool adjustp;
1008                 do {
1009                         adjustp = false;
1010                         for (std::size_t i = 0; i < size; ++i) {
1011                                 if (x[i] == cln::cl_RA(1)/p) {
1012                                         p = p/2 + cln::cl_RA(3)/2;
1013                                         adjustp = true;
1014                                         continue;
1015                                 }
1016                         }
1017                 } while (adjustp);
1018                 cln::cl_RA q = p/(p-1);
1019                 std::vector<cln::cl_N> qlstx;
1020                 std::vector<int> qlsts;
1021                 for (std::size_t j = r; j >= 1; --j) {
1022                         qlstx.push_back(cln::cl_N(1) - x[j-1]);
1023                         if (instanceof(x[j-1], cln::cl_R_ring) && realpart(x[j-1]) > 1) {
1024                                 qlsts.push_back(1);
1025                         } else {
1026                                 qlsts.push_back(-s[j-1]);
1027                         }
1028                 }
1029                 if (qlstx.size() > 0) {
1030                         buffer = buffer*G_numeric(qlstx, qlsts, 1/q);
1031                 }
1032                 std::vector<cln::cl_N> plstx;
1033                 std::vector<int> plsts;
1034                 for (std::size_t j = r+1; j <= size; ++j) {
1035                         plstx.push_back(x[j-1]);
1036                         plsts.push_back(s[j-1]);
1037                 }
1038                 if (plstx.size() > 0) {
1039                         buffer = buffer*G_numeric(plstx, plsts, 1/p);
1040                 }
1041                 result = result + buffer;
1042         }
1043         return result;
1044 }
1045
1046 class less_object_for_cl_N
1047 {
1048 public:
1049         bool operator() (const cln::cl_N & a, const cln::cl_N & b) const
1050         {
1051                 // absolute value?
1052                 if (abs(a) != abs(b))
1053                         return (abs(a) < abs(b)) ? true : false;
1054
1055                 // complex phase?
1056                 if (phase(a) != phase(b))
1057                         return (phase(a) < phase(b)) ? true : false;
1058
1059                 // equal, therefore "less" is not true
1060                 return false;
1061         }
1062 };
1063
1064
1065 // convergence transformation, used for numerical evaluation of G function.
1066 // the parameter x, s and y must only contain numerics
1067 static cln::cl_N
1068 G_do_trafo(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1069            const cln::cl_N& y, bool flag_trailing_zeros_only)
1070 {
1071         // sort (|x|<->position) to determine indices
1072         typedef std::multimap<cln::cl_N, std::size_t, less_object_for_cl_N> sortmap_t;
1073         sortmap_t sortmap;
1074         std::size_t size = 0;
1075         for (std::size_t i = 0; i < x.size(); ++i) {
1076                 if (!zerop(x[i])) {
1077                         sortmap.insert(std::make_pair(x[i], i));
1078                         ++size;
1079                 }
1080         }
1081         // include upper limit (scale)
1082         sortmap.insert(std::make_pair(y, x.size()));
1083
1084         // generate missing dummy-symbols
1085         int i = 1;
1086         // holding dummy-symbols for the G/Li transformations
1087         exvector gsyms;
1088         gsyms.push_back(symbol("GSYMS_ERROR"));
1089         cln::cl_N lastentry(0);
1090         for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1091                 if (it != sortmap.begin()) {
1092                         if (it->second < x.size()) {
1093                                 if (x[it->second] == lastentry) {
1094                                         gsyms.push_back(gsyms.back());
1095                                         continue;
1096                                 }
1097                         } else {
1098                                 if (y == lastentry) {
1099                                         gsyms.push_back(gsyms.back());
1100                                         continue;
1101                                 }
1102                         }
1103                 }
1104                 std::ostringstream os;
1105                 os << "a" << i;
1106                 gsyms.push_back(symbol(os.str()));
1107                 ++i;
1108                 if (it->second < x.size()) {
1109                         lastentry = x[it->second];
1110                 } else {
1111                         lastentry = y;
1112                 }
1113         }
1114
1115         // fill position data according to sorted indices and prepare substitution list
1116         Gparameter a(x.size());
1117         exmap subslst;
1118         std::size_t pos = 1;
1119         int scale = pos;
1120         for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
1121                 if (it->second < x.size()) {
1122                         if (s[it->second] > 0) {
1123                                 a[it->second] = pos;
1124                         } else {
1125                                 a[it->second] = -int(pos);
1126                         }
1127                         subslst[gsyms[pos]] = numeric(x[it->second]);
1128                 } else {
1129                         scale = pos;
1130                         subslst[gsyms[pos]] = numeric(y);
1131                 }
1132                 ++pos;
1133         }
1134
1135         // do transformation
1136         Gparameter pendint;
1137         ex result = G_transform(pendint, a, scale, gsyms, flag_trailing_zeros_only);
1138         // replace dummy symbols with their values
1139         result = result.eval().expand();
1140         result = result.subs(subslst).evalf();
1141         if (!is_a<numeric>(result))
1142                 throw std::logic_error("G_do_trafo: G_transform returned non-numeric result");
1143         
1144         cln::cl_N ret = ex_to<numeric>(result).to_cl_N();
1145         return ret;
1146 }
1147
1148 // handles the transformations and the numerical evaluation of G
1149 // the parameter x, s and y must only contain numerics
1150 static cln::cl_N
1151 G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
1152           const cln::cl_N& y)
1153 {
1154         // check for convergence and necessary accelerations
1155         bool need_trafo = false;
1156         bool need_hoelder = false;
1157         bool have_trailing_zero = false;
1158         std::size_t depth = 0;
1159         for (std::size_t i = 0; i < x.size(); ++i) {
1160                 if (!zerop(x[i])) {
1161                         ++depth;
1162                         const cln::cl_N x_y = abs(x[i]) - y;
1163                         if (instanceof(x_y, cln::cl_R_ring) &&
1164                             realpart(x_y) < cln::least_negative_float(cln::float_format(Digits - 2)))
1165                                 need_trafo = true;
1166
1167                         if (abs(abs(x[i]/y) - 1) < 0.01)
1168                                 need_hoelder = true;
1169                 }
1170         }
1171         if (zerop(x.back())) {
1172                 have_trailing_zero = true;
1173                 need_trafo = true;
1174         }
1175
1176         if (depth == 1 && x.size() == 2 && !need_trafo)
1177                 return - Li_projection(2, y/x[1], cln::float_format(Digits));
1178         
1179         // do acceleration transformation (hoelder convolution [BBB])
1180         if (need_hoelder && !have_trailing_zero)
1181                 return G_do_hoelder(x, s, y);
1182         
1183         // convergence transformation
1184         if (need_trafo)
1185                 return G_do_trafo(x, s, y, have_trailing_zero);
1186
1187         // do summation
1188         std::vector<cln::cl_N> newx;
1189         newx.reserve(x.size());
1190         std::vector<int> m;
1191         m.reserve(x.size());
1192         int mcount = 1;
1193         int sign = 1;
1194         cln::cl_N factor = y;
1195         for (std::size_t i = 0; i < x.size(); ++i) {
1196                 if (zerop(x[i])) {
1197                         ++mcount;
1198                 } else {
1199                         newx.push_back(factor/x[i]);
1200                         factor = x[i];
1201                         m.push_back(mcount);
1202                         mcount = 1;
1203                         sign = -sign;
1204                 }
1205         }
1206
1207         return sign*multipleLi_do_sum(m, newx);
1208 }
1209
1210
1211 ex mLi_numeric(const lst& m, const lst& x)
1212 {
1213         // let G_numeric do the transformation
1214         std::vector<cln::cl_N> newx;
1215         newx.reserve(x.nops());
1216         std::vector<int> s;
1217         s.reserve(x.nops());
1218         cln::cl_N factor(1);
1219         for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1220                 for (int i = 1; i < *itm; ++i) {
1221                         newx.push_back(cln::cl_N(0));
1222                         s.push_back(1);
1223                 }
1224                 const cln::cl_N xi = ex_to<numeric>(*itx).to_cl_N();
1225                 factor = factor/xi;
1226                 newx.push_back(factor);
1227                 if ( !instanceof(factor, cln::cl_R_ring) && imagpart(factor) < 0 ) {
1228                         s.push_back(-1);
1229                 }
1230                 else {
1231                         s.push_back(1);
1232                 }
1233         }
1234         return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1)));
1235 }
1236
1237
1238 } // end of anonymous namespace
1239
1240
1241 //////////////////////////////////////////////////////////////////////
1242 //
1243 // Generalized multiple polylogarithm  G(x, y) and G(x, s, y)
1244 //
1245 // GiNaC function
1246 //
1247 //////////////////////////////////////////////////////////////////////
1248
1249
1250 static ex G2_evalf(const ex& x_, const ex& y)
1251 {
1252         if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1253                 return G(x_, y).hold();
1254         }
1255         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1256         if (x.nops() == 0) {
1257                 return _ex1;
1258         }
1259         if (x.op(0) == y) {
1260                 return G(x_, y).hold();
1261         }
1262         std::vector<int> s;
1263         s.reserve(x.nops());
1264         bool all_zero = true;
1265         for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1266                 if (!(*it).info(info_flags::numeric)) {
1267                         return G(x_, y).hold();
1268                 }
1269                 if (*it != _ex0) {
1270                         all_zero = false;
1271                 }
1272                 if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
1273                         s.push_back(-1);
1274                 }
1275                 else {
1276                         s.push_back(1);
1277                 }
1278         }
1279         if (all_zero) {
1280                 return pow(log(y), x.nops()) / factorial(x.nops());
1281         }
1282         std::vector<cln::cl_N> xv;
1283         xv.reserve(x.nops());
1284         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1285                 xv.push_back(ex_to<numeric>(*it).to_cl_N());
1286         cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1287         return numeric(result);
1288 }
1289
1290
1291 static ex G2_eval(const ex& x_, const ex& y)
1292 {
1293         //TODO eval to MZV or H or S or Lin
1294
1295         if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1296                 return G(x_, y).hold();
1297         }
1298         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1299         if (x.nops() == 0) {
1300                 return _ex1;
1301         }
1302         if (x.op(0) == y) {
1303                 return G(x_, y).hold();
1304         }
1305         std::vector<int> s;
1306         s.reserve(x.nops());
1307         bool all_zero = true;
1308         bool crational = true;
1309         for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
1310                 if (!(*it).info(info_flags::numeric)) {
1311                         return G(x_, y).hold();
1312                 }
1313                 if (!(*it).info(info_flags::crational)) {
1314                         crational = false;
1315                 }
1316                 if (*it != _ex0) {
1317                         all_zero = false;
1318                 }
1319                 if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
1320                         s.push_back(-1);
1321                 }
1322                 else {
1323                         s.push_back(+1);
1324                 }
1325         }
1326         if (all_zero) {
1327                 return pow(log(y), x.nops()) / factorial(x.nops());
1328         }
1329         if (!y.info(info_flags::crational)) {
1330                 crational = false;
1331         }
1332         if (crational) {
1333                 return G(x_, y).hold();
1334         }
1335         std::vector<cln::cl_N> xv;
1336         xv.reserve(x.nops());
1337         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1338                 xv.push_back(ex_to<numeric>(*it).to_cl_N());
1339         cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
1340         return numeric(result);
1341 }
1342
1343
1344 // option do_not_evalf_params() removed.
1345 unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
1346                                 evalf_func(G2_evalf).
1347                                 eval_func(G2_eval).
1348                                 overloaded(2));
1349 //TODO
1350 //                                derivative_func(G2_deriv).
1351 //                                print_func<print_latex>(G2_print_latex).
1352
1353
1354 static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
1355 {
1356         if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1357                 return G(x_, s_, y).hold();
1358         }
1359         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1360         lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1361         if (x.nops() != s.nops()) {
1362                 return G(x_, s_, y).hold();
1363         }
1364         if (x.nops() == 0) {
1365                 return _ex1;
1366         }
1367         if (x.op(0) == y) {
1368                 return G(x_, s_, y).hold();
1369         }
1370         std::vector<int> sn;
1371         sn.reserve(s.nops());
1372         bool all_zero = true;
1373         for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1374                 if (!(*itx).info(info_flags::numeric)) {
1375                         return G(x_, y).hold();
1376                 }
1377                 if (!(*its).info(info_flags::real)) {
1378                         return G(x_, y).hold();
1379                 }
1380                 if (*itx != _ex0) {
1381                         all_zero = false;
1382                 }
1383                 if ( ex_to<numeric>(*itx).is_real() ) {
1384                         if ( ex_to<numeric>(*itx).is_positive() ) {
1385                                 if ( *its >= 0 ) {
1386                                         sn.push_back(1);
1387                                 }
1388                                 else {
1389                                         sn.push_back(-1);
1390                                 }
1391                         } else {
1392                                 sn.push_back(1);
1393                         }
1394                 }
1395                 else {
1396                         if ( ex_to<numeric>(*itx).imag() > 0 ) {
1397                                 sn.push_back(1);
1398                         }
1399                         else {
1400                                 sn.push_back(-1);
1401                         }
1402                 }
1403         }
1404         if (all_zero) {
1405                 return pow(log(y), x.nops()) / factorial(x.nops());
1406         }
1407         std::vector<cln::cl_N> xn;
1408         xn.reserve(x.nops());
1409         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1410                 xn.push_back(ex_to<numeric>(*it).to_cl_N());
1411         cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1412         return numeric(result);
1413 }
1414
1415
1416 static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
1417 {
1418         //TODO eval to MZV or H or S or Lin
1419
1420         if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
1421                 return G(x_, s_, y).hold();
1422         }
1423         lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
1424         lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
1425         if (x.nops() != s.nops()) {
1426                 return G(x_, s_, y).hold();
1427         }
1428         if (x.nops() == 0) {
1429                 return _ex1;
1430         }
1431         if (x.op(0) == y) {
1432                 return G(x_, s_, y).hold();
1433         }
1434         std::vector<int> sn;
1435         sn.reserve(s.nops());
1436         bool all_zero = true;
1437         bool crational = true;
1438         for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
1439                 if (!(*itx).info(info_flags::numeric)) {
1440                         return G(x_, s_, y).hold();
1441                 }
1442                 if (!(*its).info(info_flags::real)) {
1443                         return G(x_, s_, y).hold();
1444                 }
1445                 if (!(*itx).info(info_flags::crational)) {
1446                         crational = false;
1447                 }
1448                 if (*itx != _ex0) {
1449                         all_zero = false;
1450                 }
1451                 if ( ex_to<numeric>(*itx).is_real() ) {
1452                         if ( ex_to<numeric>(*itx).is_positive() ) {
1453                                 if ( *its >= 0 ) {
1454                                         sn.push_back(1);
1455                                 }
1456                                 else {
1457                                         sn.push_back(-1);
1458                                 }
1459                         } else {
1460                                 sn.push_back(1);
1461                         }
1462                 }
1463                 else {
1464                         if ( ex_to<numeric>(*itx).imag() > 0 ) {
1465                                 sn.push_back(1);
1466                         }
1467                         else {
1468                                 sn.push_back(-1);
1469                         }
1470                 }
1471         }
1472         if (all_zero) {
1473                 return pow(log(y), x.nops()) / factorial(x.nops());
1474         }
1475         if (!y.info(info_flags::crational)) {
1476                 crational = false;
1477         }
1478         if (crational) {
1479                 return G(x_, s_, y).hold();
1480         }
1481         std::vector<cln::cl_N> xn;
1482         xn.reserve(x.nops());
1483         for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
1484                 xn.push_back(ex_to<numeric>(*it).to_cl_N());
1485         cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
1486         return numeric(result);
1487 }
1488
1489
1490 // option do_not_evalf_params() removed.
1491 // This is safe: in the code above it only matters if s_ > 0 or s_ < 0,
1492 // s_ is allowed to be of floating type.
1493 unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
1494                                 evalf_func(G3_evalf).
1495                                 eval_func(G3_eval).
1496                                 overloaded(2));
1497 //TODO
1498 //                                derivative_func(G3_deriv).
1499 //                                print_func<print_latex>(G3_print_latex).
1500
1501
1502 //////////////////////////////////////////////////////////////////////
1503 //
1504 // Classical polylogarithm and multiple polylogarithm  Li(m,x)
1505 //
1506 // GiNaC function
1507 //
1508 //////////////////////////////////////////////////////////////////////
1509
1510
1511 static ex Li_evalf(const ex& m_, const ex& x_)
1512 {
1513         // classical polylogs
1514         if (m_.info(info_flags::posint)) {
1515                 if (x_.info(info_flags::numeric)) {
1516                         int m__ = ex_to<numeric>(m_).to_int();
1517                         const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1518                         const cln::cl_N result = Lin_numeric(m__, x__);
1519                         return numeric(result);
1520                 } else {
1521                         // try to numerically evaluate second argument
1522                         ex x_val = x_.evalf();
1523                         if (x_val.info(info_flags::numeric)) {
1524                                 int m__ = ex_to<numeric>(m_).to_int();
1525                                 const cln::cl_N x__ = ex_to<numeric>(x_val).to_cl_N();
1526                                 const cln::cl_N result = Lin_numeric(m__, x__);
1527                                 return numeric(result);
1528                         }
1529                 }
1530         }
1531         // multiple polylogs
1532         if (is_a<lst>(m_) && is_a<lst>(x_)) {
1533
1534                 const lst& m = ex_to<lst>(m_);
1535                 const lst& x = ex_to<lst>(x_);
1536                 if (m.nops() != x.nops()) {
1537                         return Li(m_,x_).hold();
1538                 }
1539                 if (x.nops() == 0) {
1540                         return _ex1;
1541                 }
1542                 if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
1543                         return Li(m_,x_).hold();
1544                 }
1545
1546                 for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1547                         if (!(*itm).info(info_flags::posint)) {
1548                                 return Li(m_, x_).hold();
1549                         }
1550                         if (!(*itx).info(info_flags::numeric)) {
1551                                 return Li(m_, x_).hold();
1552                         }
1553                         if (*itx == _ex0) {
1554                                 return _ex0;
1555                         }
1556                 }
1557
1558                 return mLi_numeric(m, x);
1559         }
1560
1561         return Li(m_,x_).hold();
1562 }
1563
1564
1565 static ex Li_eval(const ex& m_, const ex& x_)
1566 {
1567         if (is_a<lst>(m_)) {
1568                 if (is_a<lst>(x_)) {
1569                         // multiple polylogs
1570                         const lst& m = ex_to<lst>(m_);
1571                         const lst& x = ex_to<lst>(x_);
1572                         if (m.nops() != x.nops()) {
1573                                 return Li(m_,x_).hold();
1574                         }
1575                         if (x.nops() == 0) {
1576                                 return _ex1;
1577                         }
1578                         bool is_H = true;
1579                         bool is_zeta = true;
1580                         bool do_evalf = true;
1581                         bool crational = true;
1582                         for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
1583                                 if (!(*itm).info(info_flags::posint)) {
1584                                         return Li(m_,x_).hold();
1585                                 }
1586                                 if ((*itx != _ex1) && (*itx != _ex_1)) {
1587                                         if (itx != x.begin()) {
1588                                                 is_H = false;
1589                                         }
1590                                         is_zeta = false;
1591                                 }
1592                                 if (*itx == _ex0) {
1593                                         return _ex0;
1594                                 }
1595                                 if (!(*itx).info(info_flags::numeric)) {
1596                                         do_evalf = false;
1597                                 }
1598                                 if (!(*itx).info(info_flags::crational)) {
1599                                         crational = false;
1600                                 }
1601                         }
1602                         if (is_zeta) {
1603                                 lst newx;
1604                                 for (lst::const_iterator itx = x.begin(); itx != x.end(); ++itx) {
1605                                         GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
1606                                         // XXX: 1 + 0.0*I is considered equal to 1. However
1607                                         // the former is a not automatically converted
1608                                         // to a real number. Do the conversion explicitly
1609                                         // to avoid the "numeric::operator>(): complex inequality"
1610                                         // exception (and similar problems).
1611                                         newx.append(*itx != _ex_1 ? _ex1 : _ex_1);
1612                                 }
1613                                 return zeta(m_, newx);
1614                         }
1615                         if (is_H) {
1616                                 ex prefactor;
1617                                 lst newm = convert_parameter_Li_to_H(m, x, prefactor);
1618                                 return prefactor * H(newm, x[0]);
1619                         }
1620                         if (do_evalf && !crational) {
1621                                 return mLi_numeric(m,x);
1622                         }
1623                 }
1624                 return Li(m_, x_).hold();
1625         } else if (is_a<lst>(x_)) {
1626                 return Li(m_, x_).hold();
1627         }
1628
1629         // classical polylogs
1630         if (x_ == _ex0) {
1631                 return _ex0;
1632         }
1633         if (x_ == _ex1) {
1634                 return zeta(m_);
1635         }
1636         if (x_ == _ex_1) {
1637                 return (pow(2,1-m_)-1) * zeta(m_);
1638         }
1639         if (m_ == _ex1) {
1640                 return -log(1-x_);
1641         }
1642         if (m_ == _ex2) {
1643                 if (x_.is_equal(I)) {
1644                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
1645                 }
1646                 if (x_.is_equal(-I)) {
1647                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
1648                 }
1649         }
1650         if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
1651                 int m__ = ex_to<numeric>(m_).to_int();
1652                 const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
1653                 const cln::cl_N result = Lin_numeric(m__, x__);
1654                 return numeric(result);
1655         }
1656
1657         return Li(m_, x_).hold();
1658 }
1659
1660
1661 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
1662 {
1663         if (is_a<lst>(m) || is_a<lst>(x)) {
1664                 // multiple polylog
1665                 epvector seq { expair(Li(m, x), 0) };
1666                 return pseries(rel, std::move(seq));
1667         }
1668         
1669         // classical polylog
1670         const ex x_pt = x.subs(rel, subs_options::no_pattern);
1671         if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
1672                 // First special case: x==0 (derivatives have poles)
1673                 if (x_pt.is_zero()) {
1674                         const symbol s;
1675                         ex ser;
1676                         // manually construct the primitive expansion
1677                         for (int i=1; i<order; ++i)
1678                                 ser += pow(s,i) / pow(numeric(i), m);
1679                         // substitute the argument's series expansion
1680                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
1681                         // maybe that was terminating, so add a proper order term
1682                         epvector nseq { expair(Order(_ex1), order) };
1683                         ser += pseries(rel, std::move(nseq));
1684                         // reexpanding it will collapse the series again
1685                         return ser.series(rel, order);
1686                 }
1687                 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
1688                 throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
1689         }
1690         // all other cases should be safe, by now:
1691         throw do_taylor();  // caught by function::series()
1692 }
1693
1694
1695 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
1696 {
1697         GINAC_ASSERT(deriv_param < 2);
1698         if (deriv_param == 0) {
1699                 return _ex0;
1700         }
1701         if (m_.nops() > 1) {
1702                 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
1703         }
1704         ex m;
1705         if (is_a<lst>(m_)) {
1706                 m = m_.op(0);
1707         } else {
1708                 m = m_;
1709         }
1710         ex x;
1711         if (is_a<lst>(x_)) {
1712                 x = x_.op(0);
1713         } else {
1714                 x = x_;
1715         }
1716         if (m > 0) {
1717                 return Li(m-1, x) / x;
1718         } else {
1719                 return 1/(1-x);
1720         }
1721 }
1722
1723
1724 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
1725 {
1726         lst m;
1727         if (is_a<lst>(m_)) {
1728                 m = ex_to<lst>(m_);
1729         } else {
1730                 m = lst(m_);
1731         }
1732         lst x;
1733         if (is_a<lst>(x_)) {
1734                 x = ex_to<lst>(x_);
1735         } else {
1736                 x = lst(x_);
1737         }
1738         c.s << "\\mathrm{Li}_{";
1739         lst::const_iterator itm = m.begin();
1740         (*itm).print(c);
1741         itm++;
1742         for (; itm != m.end(); itm++) {
1743                 c.s << ",";
1744                 (*itm).print(c);
1745         }
1746         c.s << "}(";
1747         lst::const_iterator itx = x.begin();
1748         (*itx).print(c);
1749         itx++;
1750         for (; itx != x.end(); itx++) {
1751                 c.s << ",";
1752                 (*itx).print(c);
1753         }
1754         c.s << ")";
1755 }
1756
1757
1758 REGISTER_FUNCTION(Li,
1759                   evalf_func(Li_evalf).
1760                   eval_func(Li_eval).
1761                   series_func(Li_series).
1762                   derivative_func(Li_deriv).
1763                   print_func<print_latex>(Li_print_latex).
1764                   do_not_evalf_params());
1765
1766
1767 //////////////////////////////////////////////////////////////////////
1768 //
1769 // Nielsen's generalized polylogarithm  S(n,p,x)
1770 //
1771 // helper functions
1772 //
1773 //////////////////////////////////////////////////////////////////////
1774
1775
1776 // anonymous namespace for helper functions
1777 namespace {
1778
1779
1780 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
1781 // see fill_Yn()
1782 std::vector<std::vector<cln::cl_N>> Yn;
1783 int ynsize = 0; // number of Yn[]
1784 int ynlength = 100; // initial length of all Yn[i]
1785
1786
1787 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
1788 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
1789 // representing S_{n,p}(x).
1790 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
1791 // equivalent Z-sum.
1792 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
1793 // representing S_{n,p}(x).
1794 // The calculation of Y_n uses the values from Y_{n-1}.
1795 void fill_Yn(int n, const cln::float_format_t& prec)
1796 {
1797         const int initsize = ynlength;
1798         //const int initsize = initsize_Yn;
1799         cln::cl_N one = cln::cl_float(1, prec);
1800
1801         if (n) {
1802                 std::vector<cln::cl_N> buf(initsize);
1803                 std::vector<cln::cl_N>::iterator it = buf.begin();
1804                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1805                 *it = (*itprev) / cln::cl_N(n+1) * one;
1806                 it++;
1807                 itprev++;
1808                 // sums with an index smaller than the depth are zero and need not to be calculated.
1809                 // calculation starts with depth, which is n+2)
1810                 for (int i=n+2; i<=initsize+n; i++) {
1811                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1812                         it++;
1813                         itprev++;
1814                 }
1815                 Yn.push_back(buf);
1816         } else {
1817                 std::vector<cln::cl_N> buf(initsize);
1818                 std::vector<cln::cl_N>::iterator it = buf.begin();
1819                 *it = 1 * one;
1820                 it++;
1821                 for (int i=2; i<=initsize; i++) {
1822                         *it = *(it-1) + 1 / cln::cl_N(i) * one;
1823                         it++;
1824                 }
1825                 Yn.push_back(buf);
1826         }
1827         ynsize++;
1828 }
1829
1830
1831 // make Yn longer ... 
1832 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
1833 {
1834
1835         cln::cl_N one = cln::cl_float(1, prec);
1836
1837         Yn[0].resize(newsize);
1838         std::vector<cln::cl_N>::iterator it = Yn[0].begin();
1839         it += ynlength;
1840         for (int i=ynlength+1; i<=newsize; i++) {
1841                 *it = *(it-1) + 1 / cln::cl_N(i) * one;
1842                 it++;
1843         }
1844
1845         for (int n=1; n<ynsize; n++) {
1846                 Yn[n].resize(newsize);
1847                 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
1848                 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
1849                 it += ynlength;
1850                 itprev += ynlength;
1851                 for (int i=ynlength+n+1; i<=newsize+n; i++) {
1852                         *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
1853                         it++;
1854                         itprev++;
1855                 }
1856         }
1857         
1858         ynlength = newsize;
1859 }
1860
1861
1862 // helper function for S(n,p,x)
1863 // [Kol] (7.2)
1864 cln::cl_N C(int n, int p)
1865 {
1866         cln::cl_N result;
1867
1868         for (int k=0; k<p; k++) {
1869                 for (int j=0; j<=(n+k-1)/2; j++) {
1870                         if (k == 0) {
1871                                 if (n & 1) {
1872                                         if (j & 1) {
1873                                                 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1874                                         }
1875                                         else {
1876                                                 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
1877                                         }
1878                                 }
1879                         }
1880                         else {
1881                                 if (k & 1) {
1882                                         if (j & 1) {
1883                                                 result = result + cln::factorial(n+k-1)
1884                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1885                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1886                                         }
1887                                         else {
1888                                                 result = result - cln::factorial(n+k-1)
1889                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1890                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1891                                         }
1892                                 }
1893                                 else {
1894                                         if (j & 1) {
1895                                                 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1896                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1897                                         }
1898                                         else {
1899                                                 result = result + cln::factorial(n+k-1)
1900                                                                   * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
1901                                                                   / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
1902                                         }
1903                                 }
1904                         }
1905                 }
1906         }
1907         int np = n+p;
1908         if ((np-1) & 1) {
1909                 if (((np)/2+n) & 1) {
1910                         result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1911                 }
1912                 else {
1913                         result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
1914                 }
1915         }
1916
1917         return result;
1918 }
1919
1920
1921 // helper function for S(n,p,x)
1922 // [Kol] remark to (9.1)
1923 cln::cl_N a_k(int k)
1924 {
1925         cln::cl_N result;
1926
1927         if (k == 0) {
1928                 return 1;
1929         }
1930
1931         result = result;
1932         for (int m=2; m<=k; m++) {
1933                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
1934         }
1935
1936         return -result / k;
1937 }
1938
1939
1940 // helper function for S(n,p,x)
1941 // [Kol] remark to (9.1)
1942 cln::cl_N b_k(int k)
1943 {
1944         cln::cl_N result;
1945
1946         if (k == 0) {
1947                 return 1;
1948         }
1949
1950         result = result;
1951         for (int m=2; m<=k; m++) {
1952                 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
1953         }
1954
1955         return result / k;
1956 }
1957
1958
1959 // helper function for S(n,p,x)
1960 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
1961 {
1962         static cln::float_format_t oldprec = cln::default_float_format;
1963
1964         if (p==1) {
1965                 return Li_projection(n+1, x, prec);
1966         }
1967
1968         // precision has changed, we need to clear lookup table Yn
1969         if ( oldprec != prec ) {
1970                 Yn.clear();
1971                 ynsize = 0;
1972                 ynlength = 100;
1973                 oldprec = prec;
1974         }
1975                 
1976         // check if precalculated values are sufficient
1977         if (p > ynsize+1) {
1978                 for (int i=ynsize; i<p-1; i++) {
1979                         fill_Yn(i, prec);
1980                 }
1981         }
1982
1983         // should be done otherwise
1984         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1985         cln::cl_N xf = x * one;
1986         //cln::cl_N xf = x * cln::cl_float(1, prec);
1987
1988         cln::cl_N res;
1989         cln::cl_N resbuf;
1990         cln::cl_N factor = cln::expt(xf, p);
1991         int i = p;
1992         do {
1993                 resbuf = res;
1994                 if (i-p >= ynlength) {
1995                         // make Yn longer
1996                         make_Yn_longer(ynlength*2, prec);
1997                 }
1998                 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? ...
1999                 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
2000                 factor = factor * xf;
2001                 i++;
2002         } while (res != resbuf);
2003         
2004         return res;
2005 }
2006
2007
2008 // helper function for S(n,p,x)
2009 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
2010 {
2011         // [Kol] (5.3)
2012         if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
2013
2014                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
2015                                    * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
2016
2017                 for (int s=0; s<n; s++) {
2018                         cln::cl_N res2;
2019                         for (int r=0; r<p; r++) {
2020                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
2021                                               * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
2022                         }
2023                         result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2024                 }
2025
2026                 return result;
2027         }
2028         
2029         return S_do_sum(n, p, x, prec);
2030 }
2031
2032
2033 // helper function for S(n,p,x)
2034 const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
2035 {
2036         if (x == 1) {
2037                 if (n == 1) {
2038                     // [Kol] (2.22) with (2.21)
2039                         return cln::zeta(p+1);
2040                 }
2041
2042                 if (p == 1) {
2043                     // [Kol] (2.22)
2044                         return cln::zeta(n+1);
2045                 }
2046
2047                 // [Kol] (9.1)
2048                 cln::cl_N result;
2049                 for (int nu=0; nu<n; nu++) {
2050                         for (int rho=0; rho<=p; rho++) {
2051                                 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
2052                                                   * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
2053                         }
2054                 }
2055                 result = result * cln::expt(cln::cl_I(-1),n+p-1);
2056
2057                 return result;
2058         }
2059         else if (x == -1) {
2060                 // [Kol] (2.22)
2061                 if (p == 1) {
2062                         return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
2063                 }
2064 //              throw std::runtime_error("don't know how to evaluate this function!");
2065         }
2066
2067         // what is the desired float format?
2068         // first guess: default format
2069         cln::float_format_t prec = cln::default_float_format;
2070         const cln::cl_N value = x;
2071         // second guess: the argument's format
2072         if (!instanceof(realpart(value), cln::cl_RA_ring))
2073                 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
2074         else if (!instanceof(imagpart(value), cln::cl_RA_ring))
2075                 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
2076
2077         // [Kol] (5.3)
2078         // 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
2079         // we don't care here about abs(value)<1 && real(value)>0.5, this will be taken care of in S_projection
2080         if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95) && (cln::abs(1-value) > 1) )) {
2081
2082                 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
2083                                    * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
2084
2085                 for (int s=0; s<n; s++) {
2086                         cln::cl_N res2;
2087                         for (int r=0; r<p; r++) {
2088                                 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
2089                                               * S_num(p-r,n-s,1-value) / cln::factorial(r);
2090                         }
2091                         result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
2092                 }
2093
2094                 return result;
2095                 
2096         }
2097         // [Kol] (5.12)
2098         if (cln::abs(value) > 1) {
2099                 
2100                 cln::cl_N result;
2101
2102                 for (int s=0; s<p; s++) {
2103                         for (int r=0; r<=s; r++) {
2104                                 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
2105                                                   / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
2106                                                   * S_num(n+s-r,p-s,cln::recip(value));
2107                         }
2108                 }
2109                 result = result * cln::expt(cln::cl_I(-1),n);
2110
2111                 cln::cl_N res2;
2112                 for (int r=0; r<n; r++) {
2113                         res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
2114                 }
2115                 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
2116
2117                 result = result + cln::expt(cln::cl_I(-1),p) * res2;
2118
2119                 return result;
2120         }
2121
2122         if ((cln::abs(value) > 0.95) && (cln::abs(value-9.53) < 9.47)) {
2123                 lst m;
2124                 m.append(n+1);
2125                 for (int s=0; s<p-1; s++)
2126                         m.append(1);
2127
2128                 ex res = H(m,numeric(value)).evalf();
2129                 return ex_to<numeric>(res).to_cl_N();
2130         }
2131         else {
2132                 return S_projection(n, p, value, prec);
2133         }
2134 }
2135
2136
2137 } // end of anonymous namespace
2138
2139
2140 //////////////////////////////////////////////////////////////////////
2141 //
2142 // Nielsen's generalized polylogarithm  S(n,p,x)
2143 //
2144 // GiNaC function
2145 //
2146 //////////////////////////////////////////////////////////////////////
2147
2148
2149 static ex S_evalf(const ex& n, const ex& p, const ex& x)
2150 {
2151         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2152                 const int n_ = ex_to<numeric>(n).to_int();
2153                 const int p_ = ex_to<numeric>(p).to_int();
2154                 if (is_a<numeric>(x)) {
2155                         const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2156                         const cln::cl_N result = S_num(n_, p_, x_);
2157                         return numeric(result);
2158                 } else {
2159                         ex x_val = x.evalf();
2160                         if (is_a<numeric>(x_val)) {
2161                                 const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
2162                                 const cln::cl_N result = S_num(n_, p_, x_val_);
2163                                 return numeric(result);
2164                         }
2165                 }
2166         }
2167         return S(n, p, x).hold();
2168 }
2169
2170
2171 static ex S_eval(const ex& n, const ex& p, const ex& x)
2172 {
2173         if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
2174                 if (x == 0) {
2175                         return _ex0;
2176                 }
2177                 if (x == 1) {
2178                         lst m(n+1);
2179                         for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
2180                                 m.append(1);
2181                         }
2182                         return zeta(m);
2183                 }
2184                 if (p == 1) {
2185                         return Li(n+1, x);
2186                 }
2187                 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2188                         int n_ = ex_to<numeric>(n).to_int();
2189                         int p_ = ex_to<numeric>(p).to_int();
2190                         const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
2191                         const cln::cl_N result = S_num(n_, p_, x_);
2192                         return numeric(result);
2193                 }
2194         }
2195         if (n.is_zero()) {
2196                 // [Kol] (5.3)
2197                 return pow(-log(1-x), p) / factorial(p);
2198         }
2199         return S(n, p, x).hold();
2200 }
2201
2202
2203 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
2204 {
2205         if (p == _ex1) {
2206                 return Li(n+1, x).series(rel, order, options);
2207         }
2208
2209         const ex x_pt = x.subs(rel, subs_options::no_pattern);
2210         if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
2211                 // First special case: x==0 (derivatives have poles)
2212                 if (x_pt.is_zero()) {
2213                         const symbol s;
2214                         ex ser;
2215                         // manually construct the primitive expansion
2216                         // subsum = Euler-Zagier-Sum is needed
2217                         // dirty hack (slow ...) calculation of subsum:
2218                         std::vector<ex> presubsum, subsum;
2219                         subsum.push_back(0);
2220                         for (int i=1; i<order-1; ++i) {
2221                                 subsum.push_back(subsum[i-1] + numeric(1, i));
2222                         }
2223                         for (int depth=2; depth<p; ++depth) {
2224                                 presubsum = subsum;
2225                                 for (int i=1; i<order-1; ++i) {
2226                                         subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
2227                                 }
2228                         }
2229                                 
2230                         for (int i=1; i<order; ++i) {
2231                                 ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
2232                         }
2233                         // substitute the argument's series expansion
2234                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
2235                         // maybe that was terminating, so add a proper order term
2236                         epvector nseq { expair(Order(_ex1), order) };
2237                         ser += pseries(rel, std::move(nseq));
2238                         // reexpanding it will collapse the series again
2239                         return ser.series(rel, order);
2240                 }
2241                 // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
2242                 throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
2243         }
2244         // all other cases should be safe, by now:
2245         throw do_taylor();  // caught by function::series()
2246 }
2247
2248
2249 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
2250 {
2251         GINAC_ASSERT(deriv_param < 3);
2252         if (deriv_param < 2) {
2253                 return _ex0;
2254         }
2255         if (n > 0) {
2256                 return S(n-1, p, x) / x;
2257         } else {
2258                 return S(n, p-1, x) / (1-x);
2259         }
2260 }
2261
2262
2263 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
2264 {
2265         c.s << "\\mathrm{S}_{";
2266         n.print(c);
2267         c.s << ",";
2268         p.print(c);
2269         c.s << "}(";
2270         x.print(c);
2271         c.s << ")";
2272 }
2273
2274
2275 REGISTER_FUNCTION(S,
2276                   evalf_func(S_evalf).
2277                   eval_func(S_eval).
2278                   series_func(S_series).
2279                   derivative_func(S_deriv).
2280                   print_func<print_latex>(S_print_latex).
2281                   do_not_evalf_params());
2282
2283
2284 //////////////////////////////////////////////////////////////////////
2285 //
2286 // Harmonic polylogarithm  H(m,x)
2287 //
2288 // helper functions
2289 //
2290 //////////////////////////////////////////////////////////////////////
2291
2292
2293 // anonymous namespace for helper functions
2294 namespace {
2295
2296         
2297 // regulates the pole (used by 1/x-transformation)
2298 symbol H_polesign("IMSIGN");
2299
2300
2301 // convert parameters from H to Li representation
2302 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
2303 // returns true if some parameters are negative
2304 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
2305 {
2306         // expand parameter list
2307         lst mexp;
2308         for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
2309                 if (*it > 1) {
2310                         for (ex count=*it-1; count > 0; count--) {
2311                                 mexp.append(0);
2312                         }
2313                         mexp.append(1);
2314                 } else if (*it < -1) {
2315                         for (ex count=*it+1; count < 0; count++) {
2316                                 mexp.append(0);
2317                         }
2318                         mexp.append(-1);
2319                 } else {
2320                         mexp.append(*it);
2321                 }
2322         }
2323         
2324         ex signum = 1;
2325         pf = 1;
2326         bool has_negative_parameters = false;
2327         ex acc = 1;
2328         for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
2329                 if (*it == 0) {
2330                         acc++;
2331                         continue;
2332                 }
2333                 if (*it > 0) {
2334                         m.append((*it+acc-1) * signum);
2335                 } else {
2336                         m.append((*it-acc+1) * signum);
2337                 }
2338                 acc = 1;
2339                 signum = *it;
2340                 pf *= *it;
2341                 if (pf < 0) {
2342                         has_negative_parameters = true;
2343                 }
2344         }
2345         if (has_negative_parameters) {
2346                 for (std::size_t i=0; i<m.nops(); i++) {
2347                         if (m.op(i) < 0) {
2348                                 m.let_op(i) = -m.op(i);
2349                                 s.append(-1);
2350                         } else {
2351                                 s.append(1);
2352                         }
2353                 }
2354         }
2355         
2356         return has_negative_parameters;
2357 }
2358
2359
2360 // recursivly transforms H to corresponding multiple polylogarithms
2361 struct map_trafo_H_convert_to_Li : public map_function
2362 {
2363         ex operator()(const ex& e) override
2364         {
2365                 if (is_a<add>(e) || is_a<mul>(e)) {
2366                         return e.map(*this);
2367                 }
2368                 if (is_a<function>(e)) {
2369                         std::string name = ex_to<function>(e).get_name();
2370                         if (name == "H") {
2371                                 lst parameter;
2372                                 if (is_a<lst>(e.op(0))) {
2373                                                 parameter = ex_to<lst>(e.op(0));
2374                                 } else {
2375                                         parameter = lst(e.op(0));
2376                                 }
2377                                 ex arg = e.op(1);
2378
2379                                 lst m;
2380                                 lst s;
2381                                 ex pf;
2382                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2383                                         s.let_op(0) = s.op(0) * arg;
2384                                         return pf * Li(m, s).hold();
2385                                 } else {
2386                                         for (std::size_t i=0; i<m.nops(); i++) {
2387                                                 s.append(1);
2388                                         }
2389                                         s.let_op(0) = s.op(0) * arg;
2390                                         return Li(m, s).hold();
2391                                 }
2392                         }
2393                 }
2394                 return e;
2395         }
2396 };
2397
2398
2399 // recursivly transforms H to corresponding zetas
2400 struct map_trafo_H_convert_to_zeta : public map_function
2401 {
2402         ex operator()(const ex& e) override
2403         {
2404                 if (is_a<add>(e) || is_a<mul>(e)) {
2405                         return e.map(*this);
2406                 }
2407                 if (is_a<function>(e)) {
2408                         std::string name = ex_to<function>(e).get_name();
2409                         if (name == "H") {
2410                                 lst parameter;
2411                                 if (is_a<lst>(e.op(0))) {
2412                                                 parameter = ex_to<lst>(e.op(0));
2413                                 } else {
2414                                         parameter = lst(e.op(0));
2415                                 }
2416
2417                                 lst m;
2418                                 lst s;
2419                                 ex pf;
2420                                 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
2421                                         return pf * zeta(m, s);
2422                                 } else {
2423                                         return zeta(m);
2424                                 }
2425                         }
2426                 }
2427                 return e;
2428         }
2429 };
2430
2431
2432 // remove trailing zeros from H-parameters
2433 struct map_trafo_H_reduce_trailing_zeros : public map_function
2434 {
2435         ex operator()(const ex& e) override
2436         {
2437                 if (is_a<add>(e) || is_a<mul>(e)) {
2438                         return e.map(*this);
2439                 }
2440                 if (is_a<function>(e)) {
2441                         std::string name = ex_to<function>(e).get_name();
2442                         if (name == "H") {
2443                                 lst parameter;
2444                                 if (is_a<lst>(e.op(0))) {
2445                                         parameter = ex_to<lst>(e.op(0));
2446                                 } else {
2447                                         parameter = lst(e.op(0));
2448                                 }
2449                                 ex arg = e.op(1);
2450                                 if (parameter.op(parameter.nops()-1) == 0) {
2451                                         
2452                                         //
2453                                         if (parameter.nops() == 1) {
2454                                                 return log(arg);
2455                                         }
2456                                         
2457                                         //
2458                                         lst::const_iterator it = parameter.begin();
2459                                         while ((it != parameter.end()) && (*it == 0)) {
2460                                                 it++;
2461                                         }
2462                                         if (it == parameter.end()) {
2463                                                 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
2464                                         }
2465                                         
2466                                         //
2467                                         parameter.remove_last();
2468                                         std::size_t lastentry = parameter.nops();
2469                                         while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
2470                                                 lastentry--;
2471                                         }
2472                                         
2473                                         //
2474                                         ex result = log(arg) * H(parameter,arg).hold();
2475                                         ex acc = 0;
2476                                         for (ex i=0; i<lastentry; i++) {
2477                                                 if (parameter[i] > 0) {
2478                                                         parameter[i]++;
2479                                                         result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
2480                                                         parameter[i]--;
2481                                                         acc = 0;
2482                                                 } else if (parameter[i] < 0) {
2483                                                         parameter[i]--;
2484                                                         result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
2485                                                         parameter[i]++;
2486                                                         acc = 0;
2487                                                 } else {
2488                                                         acc++;
2489                                                 }
2490                                         }
2491                                         
2492                                         if (lastentry < parameter.nops()) {
2493                                                 result = result / (parameter.nops()-lastentry+1);
2494                                                 return result.map(*this);
2495                                         } else {
2496                                                 return result;
2497                                         }
2498                                 }
2499                         }
2500                 }
2501                 return e;
2502         }
2503 };
2504
2505
2506 // returns an expression with zeta functions corresponding to the parameter list for H
2507 ex convert_H_to_zeta(const lst& m)
2508 {
2509         symbol xtemp("xtemp");
2510         map_trafo_H_reduce_trailing_zeros filter;
2511         map_trafo_H_convert_to_zeta filter2;
2512         return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
2513 }
2514
2515
2516 // convert signs form Li to H representation
2517 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
2518 {
2519         lst res;
2520         lst::const_iterator itm = m.begin();
2521         lst::const_iterator itx = ++x.begin();
2522         int signum = 1;
2523         pf = _ex1;
2524         res.append(*itm);
2525         itm++;
2526         while (itx != x.end()) {
2527                 GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
2528                 // XXX: 1 + 0.0*I is considered equal to 1. However the former
2529                 // is not automatically converted to a real number.
2530                 // Do the conversion explicitly to avoid the
2531                 // "numeric::operator>(): complex inequality" exception.
2532                 signum *= (*itx != _ex_1) ? 1 : -1;
2533                 pf *= signum;
2534                 res.append((*itm) * signum);
2535                 itm++;
2536                 itx++;
2537         }
2538         return res;
2539 }
2540
2541
2542 // multiplies an one-dimensional H with another H
2543 // [ReV] (18)
2544 ex trafo_H_mult(const ex& h1, const ex& h2)
2545 {
2546         ex res;
2547         ex hshort;
2548         lst hlong;
2549         ex h1nops = h1.op(0).nops();
2550         ex h2nops = h2.op(0).nops();
2551         if (h1nops > 1) {
2552                 hshort = h2.op(0).op(0);
2553                 hlong = ex_to<lst>(h1.op(0));
2554         } else {
2555                 hshort = h1.op(0).op(0);
2556                 if (h2nops > 1) {
2557                         hlong = ex_to<lst>(h2.op(0));
2558                 } else {
2559                         hlong = h2.op(0).op(0);
2560                 }
2561         }
2562         for (std::size_t i=0; i<=hlong.nops(); i++) {
2563                 lst newparameter;
2564                 std::size_t j=0;
2565                 for (; j<i; j++) {
2566                         newparameter.append(hlong[j]);
2567                 }
2568                 newparameter.append(hshort);
2569                 for (; j<hlong.nops(); j++) {
2570                         newparameter.append(hlong[j]);
2571                 }
2572                 res += H(newparameter, h1.op(1)).hold();
2573         }
2574         return res;
2575 }
2576
2577
2578 // applies trafo_H_mult recursively on expressions
2579 struct map_trafo_H_mult : public map_function
2580 {
2581         ex operator()(const ex& e) override
2582         {
2583                 if (is_a<add>(e)) {
2584                         return e.map(*this);
2585                 }
2586
2587                 if (is_a<mul>(e)) {
2588
2589                         ex result = 1;
2590                         ex firstH;
2591                         lst Hlst;
2592                         for (std::size_t pos=0; pos<e.nops(); pos++) {
2593                                 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
2594                                         std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
2595                                         if (name == "H") {
2596                                                 for (ex i=0; i<e.op(pos).op(1); i++) {
2597                                                         Hlst.append(e.op(pos).op(0));
2598                                                 }
2599                                                 continue;
2600                                         }
2601                                 } else if (is_a<function>(e.op(pos))) {
2602                                         std::string name = ex_to<function>(e.op(pos)).get_name();
2603                                         if (name == "H") {
2604                                                 if (e.op(pos).op(0).nops() > 1) {
2605                                                         firstH = e.op(pos);
2606                                                 } else {
2607                                                         Hlst.append(e.op(pos));
2608                                                 }
2609                                                 continue;
2610                                         }
2611                                 }
2612                                 result *= e.op(pos);
2613                         }
2614                         if (firstH == 0) {
2615                                 if (Hlst.nops() > 0) {
2616                                         firstH = Hlst[Hlst.nops()-1];
2617                                         Hlst.remove_last();
2618                                 } else {
2619                                         return e;
2620                                 }
2621                         }
2622
2623                         if (Hlst.nops() > 0) {
2624                                 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
2625                                 result *= buffer;
2626                                 for (std::size_t i=1; i<Hlst.nops(); i++) {
2627                                         result *= Hlst.op(i);
2628                                 }
2629                                 result = result.expand();
2630                                 map_trafo_H_mult recursion;
2631                                 return recursion(result);
2632                         } else {
2633                                 return e;
2634                         }
2635
2636                 }
2637                 return e;
2638         }
2639 };
2640
2641
2642 // do integration [ReV] (55)
2643 // put parameter 0 in front of existing parameters
2644 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
2645 {
2646         ex h;
2647         std::string name;
2648         if (is_a<function>(e)) {
2649                 name = ex_to<function>(e).get_name();
2650         }
2651         if (name == "H") {
2652                 h = e;
2653         } else {
2654                 for (std::size_t i=0; i<e.nops(); i++) {
2655                         if (is_a<function>(e.op(i))) {
2656                                 std::string name = ex_to<function>(e.op(i)).get_name();
2657                                 if (name == "H") {
2658                                         h = e.op(i);
2659                                 }
2660                         }
2661                 }
2662         }
2663         if (h != 0) {
2664                 lst newparameter = ex_to<lst>(h.op(0));
2665                 newparameter.prepend(0);
2666                 ex addzeta = convert_H_to_zeta(newparameter);
2667                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2668         } else {
2669                 return e * (-H(lst(ex(0)),1/arg).hold());
2670         }
2671 }
2672
2673
2674 // do integration [ReV] (49)
2675 // put parameter 1 in front of existing parameters
2676 ex trafo_H_prepend_one(const ex& e, const ex& arg)
2677 {
2678         ex h;
2679         std::string name;
2680         if (is_a<function>(e)) {
2681                 name = ex_to<function>(e).get_name();
2682         }
2683         if (name == "H") {
2684                 h = e;
2685         } else {
2686                 for (std::size_t i=0; i<e.nops(); i++) {
2687                         if (is_a<function>(e.op(i))) {
2688                                 std::string name = ex_to<function>(e.op(i)).get_name();
2689                                 if (name == "H") {
2690                                         h = e.op(i);
2691                                 }
2692                         }
2693                 }
2694         }
2695         if (h != 0) {
2696                 lst newparameter = ex_to<lst>(h.op(0));
2697                 newparameter.prepend(1);
2698                 return e.subs(h == H(newparameter, h.op(1)).hold());
2699         } else {
2700                 return e * H(lst(ex(1)),1-arg).hold();
2701         }
2702 }
2703
2704
2705 // do integration [ReV] (55)
2706 // put parameter -1 in front of existing parameters
2707 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
2708 {
2709         ex h;
2710         std::string name;
2711         if (is_a<function>(e)) {
2712                 name = ex_to<function>(e).get_name();
2713         }
2714         if (name == "H") {
2715                 h = e;
2716         } else {
2717                 for (std::size_t i=0; i<e.nops(); i++) {
2718                         if (is_a<function>(e.op(i))) {
2719                                 std::string name = ex_to<function>(e.op(i)).get_name();
2720                                 if (name == "H") {
2721                                         h = e.op(i);
2722                                 }
2723                         }
2724                 }
2725         }
2726         if (h != 0) {
2727                 lst newparameter = ex_to<lst>(h.op(0));
2728                 newparameter.prepend(-1);
2729                 ex addzeta = convert_H_to_zeta(newparameter);
2730                 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
2731         } else {
2732                 ex addzeta = convert_H_to_zeta(lst(ex(-1)));
2733                 return (e * (addzeta - H(lst(ex(-1)),1/arg).hold())).expand();
2734         }
2735 }
2736
2737
2738 // do integration [ReV] (55)
2739 // put parameter -1 in front of existing parameters
2740 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
2741 {
2742         ex h;
2743         std::string name;
2744         if (is_a<function>(e)) {
2745                 name = ex_to<function>(e).get_name();
2746         }
2747         if (name == "H") {
2748                 h = e;
2749         } else {
2750                 for (std::size_t i = 0; i < e.nops(); i++) {
2751                         if (is_a<function>(e.op(i))) {
2752                                 std::string name = ex_to<function>(e.op(i)).get_name();
2753                                 if (name == "H") {
2754                                         h = e.op(i);
2755                                 }
2756                         }
2757                 }
2758         }
2759         if (h != 0) {
2760                 lst newparameter = ex_to<lst>(h.op(0));
2761                 newparameter.prepend(-1);
2762                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2763         } else {
2764                 return (e * H(lst(ex(-1)),(1-arg)/(1+arg)).hold()).expand();
2765         }
2766 }
2767
2768
2769 // do integration [ReV] (55)
2770 // put parameter 1 in front of existing parameters
2771 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
2772 {
2773         ex h;
2774         std::string name;
2775         if (is_a<function>(e)) {
2776                 name = ex_to<function>(e).get_name();
2777         }
2778         if (name == "H") {
2779                 h = e;
2780         } else {
2781                 for (std::size_t i = 0; i < e.nops(); i++) {
2782                         if (is_a<function>(e.op(i))) {
2783                                 std::string name = ex_to<function>(e.op(i)).get_name();
2784                                 if (name == "H") {
2785                                         h = e.op(i);
2786                                 }
2787                         }
2788                 }
2789         }
2790         if (h != 0) {
2791                 lst newparameter = ex_to<lst>(h.op(0));
2792                 newparameter.prepend(1);
2793                 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
2794         } else {
2795                 return (e * H(lst(ex(1)),(1-arg)/(1+arg)).hold()).expand();
2796         }
2797 }
2798
2799
2800 // do x -> 1-x transformation
2801 struct map_trafo_H_1mx : public map_function
2802 {
2803         ex operator()(const ex& e) override
2804         {
2805                 if (is_a<add>(e) || is_a<mul>(e)) {
2806                         return e.map(*this);
2807                 }
2808                 
2809                 if (is_a<function>(e)) {
2810                         std::string name = ex_to<function>(e).get_name();
2811                         if (name == "H") {
2812
2813                                 lst parameter = ex_to<lst>(e.op(0));
2814                                 ex arg = e.op(1);
2815
2816                                 // special cases if all parameters are either 0, 1 or -1
2817                                 bool allthesame = true;
2818                                 if (parameter.op(0) == 0) {
2819                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2820                                                 if (parameter.op(i) != 0) {
2821                                                         allthesame = false;
2822                                                         break;
2823                                                 }
2824                                         }
2825                                         if (allthesame) {
2826                                                 lst newparameter;
2827                                                 for (int i=parameter.nops(); i>0; i--) {
2828                                                         newparameter.append(1);
2829                                                 }
2830                                                 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2831                                         }
2832                                 } else if (parameter.op(0) == -1) {
2833                                         throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
2834                                 } else {
2835                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2836                                                 if (parameter.op(i) != 1) {
2837                                                         allthesame = false;
2838                                                         break;
2839                                                 }
2840                                         }
2841                                         if (allthesame) {
2842                                                 lst newparameter;
2843                                                 for (int i=parameter.nops(); i>0; i--) {
2844                                                         newparameter.append(0);
2845                                                 }
2846                                                 return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
2847                                         }
2848                                 }
2849
2850                                 lst newparameter = parameter;
2851                                 newparameter.remove_first();
2852
2853                                 if (parameter.op(0) == 0) {
2854
2855                                         // leading zero
2856                                         ex res = convert_H_to_zeta(parameter);
2857                                         //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
2858                                         map_trafo_H_1mx recursion;
2859                                         ex buffer = recursion(H(newparameter, arg).hold());
2860                                         if (is_a<add>(buffer)) {
2861                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2862                                                         res -= trafo_H_prepend_one(buffer.op(i), arg);
2863                                                 }
2864                                         } else {
2865                                                 res -= trafo_H_prepend_one(buffer, arg);
2866                                         }
2867                                         return res;
2868
2869                                 } else {
2870
2871                                         // leading one
2872                                         map_trafo_H_1mx recursion;
2873                                         map_trafo_H_mult unify;
2874                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2875                                         std::size_t firstzero = 0;
2876                                         while (parameter.op(firstzero) == 1) {
2877                                                 firstzero++;
2878                                         }
2879                                         for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) {
2880                                                 lst newparameter;
2881                                                 std::size_t j=0;
2882                                                 for (; j<=i; j++) {
2883                                                         newparameter.append(parameter[j+1]);
2884                                                 }
2885                                                 newparameter.append(1);
2886                                                 for (; j<parameter.nops()-1; j++) {
2887                                                         newparameter.append(parameter[j+1]);
2888                                                 }
2889                                                 res -= H(newparameter, arg).hold();
2890                                         }
2891                                         res = recursion(res).expand() / firstzero;
2892                                         return unify(res);
2893                                 }
2894                         }
2895                 }
2896                 return e;
2897         }
2898 };
2899
2900
2901 // do x -> 1/x transformation
2902 struct map_trafo_H_1overx : public map_function
2903 {
2904         ex operator()(const ex& e) override
2905         {
2906                 if (is_a<add>(e) || is_a<mul>(e)) {
2907                         return e.map(*this);
2908                 }
2909
2910                 if (is_a<function>(e)) {
2911                         std::string name = ex_to<function>(e).get_name();
2912                         if (name == "H") {
2913
2914                                 lst parameter = ex_to<lst>(e.op(0));
2915                                 ex arg = e.op(1);
2916
2917                                 // special cases if all parameters are either 0, 1 or -1
2918                                 bool allthesame = true;
2919                                 if (parameter.op(0) == 0) {
2920                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2921                                                 if (parameter.op(i) != 0) {
2922                                                         allthesame = false;
2923                                                         break;
2924                                                 }
2925                                         }
2926                                         if (allthesame) {
2927                                                 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
2928                                         }
2929                                 } else if (parameter.op(0) == -1) {
2930                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2931                                                 if (parameter.op(i) != -1) {
2932                                                         allthesame = false;
2933                                                         break;
2934                                                 }
2935                                         }
2936                                         if (allthesame) {
2937                                                 map_trafo_H_mult unify;
2938                                                 return unify((pow(H(lst(ex(-1)),1/arg).hold() - H(lst(ex(0)),1/arg).hold(), parameter.nops())
2939                                                        / factorial(parameter.nops())).expand());
2940                                         }
2941                                 } else {
2942                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
2943                                                 if (parameter.op(i) != 1) {
2944                                                         allthesame = false;
2945                                                         break;
2946                                                 }
2947                                         }
2948                                         if (allthesame) {
2949                                                 map_trafo_H_mult unify;
2950                                                 return unify((pow(H(lst(ex(1)),1/arg).hold() + H(lst(ex(0)),1/arg).hold() + H_polesign, parameter.nops())
2951                                                        / factorial(parameter.nops())).expand());
2952                                         }
2953                                 }
2954
2955                                 lst newparameter = parameter;
2956                                 newparameter.remove_first();
2957
2958                                 if (parameter.op(0) == 0) {
2959                                         
2960                                         // leading zero
2961                                         ex res = convert_H_to_zeta(parameter);
2962                                         map_trafo_H_1overx recursion;
2963                                         ex buffer = recursion(H(newparameter, arg).hold());
2964                                         if (is_a<add>(buffer)) {
2965                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2966                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
2967                                                 }
2968                                         } else {
2969                                                 res += trafo_H_1tx_prepend_zero(buffer, arg);
2970                                         }
2971                                         return res;
2972
2973                                 } else if (parameter.op(0) == -1) {
2974
2975                                         // leading negative one
2976                                         ex res = convert_H_to_zeta(parameter);
2977                                         map_trafo_H_1overx recursion;
2978                                         ex buffer = recursion(H(newparameter, arg).hold());
2979                                         if (is_a<add>(buffer)) {
2980                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
2981                                                         res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
2982                                                 }
2983                                         } else {
2984                                                 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
2985                                         }
2986                                         return res;
2987
2988                                 } else {
2989
2990                                         // leading one
2991                                         map_trafo_H_1overx recursion;
2992                                         map_trafo_H_mult unify;
2993                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
2994                                         std::size_t firstzero = 0;
2995                                         while (parameter.op(firstzero) == 1) {
2996                                                 firstzero++;
2997                                         }
2998                                         for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) {
2999                                                 lst newparameter;
3000                                                 std::size_t j = 0;
3001                                                 for (; j<=i; j++) {
3002                                                         newparameter.append(parameter[j+1]);
3003                                                 }
3004                                                 newparameter.append(1);
3005                                                 for (; j<parameter.nops()-1; j++) {
3006                                                         newparameter.append(parameter[j+1]);
3007                                                 }
3008                                                 res -= H(newparameter, arg).hold();
3009                                         }
3010                                         res = recursion(res).expand() / firstzero;
3011                                         return unify(res);
3012
3013                                 }
3014
3015                         }
3016                 }
3017                 return e;
3018         }
3019 };
3020
3021
3022 // do x -> (1-x)/(1+x) transformation
3023 struct map_trafo_H_1mxt1px : public map_function
3024 {
3025         ex operator()(const ex& e) override
3026         {
3027                 if (is_a<add>(e) || is_a<mul>(e)) {
3028                         return e.map(*this);
3029                 }
3030
3031                 if (is_a<function>(e)) {
3032                         std::string name = ex_to<function>(e).get_name();
3033                         if (name == "H") {
3034
3035                                 lst parameter = ex_to<lst>(e.op(0));
3036                                 ex arg = e.op(1);
3037
3038                                 // special cases if all parameters are either 0, 1 or -1
3039                                 bool allthesame = true;
3040                                 if (parameter.op(0) == 0) {
3041                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
3042                                                 if (parameter.op(i) != 0) {
3043                                                         allthesame = false;
3044                                                         break;
3045                                                 }
3046                                         }
3047                                         if (allthesame) {
3048                                                 map_trafo_H_mult unify;
3049                                                 return unify((pow(-H(lst(ex(1)),(1-arg)/(1+arg)).hold() - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3050                                                        / factorial(parameter.nops())).expand());
3051                                         }
3052                                 } else if (parameter.op(0) == -1) {
3053                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
3054                                                 if (parameter.op(i) != -1) {
3055                                                         allthesame = false;
3056                                                         break;
3057                                                 }
3058                                         }
3059                                         if (allthesame) {
3060                                                 map_trafo_H_mult unify;
3061                                                 return unify((pow(log(2) - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
3062                                                        / factorial(parameter.nops())).expand());
3063                                         }
3064                                 } else {
3065                                         for (std::size_t i = 1; i < parameter.nops(); i++) {
3066                                                 if (parameter.op(i) != 1) {
3067                                                         allthesame = false;
3068                                                         break;
3069                                                 }
3070                                         }
3071                                         if (allthesame) {
3072                                                 map_trafo_H_mult unify;
3073                                                 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())
3074                                                        / factorial(parameter.nops())).expand());
3075                                         }
3076                                 }
3077
3078                                 lst newparameter = parameter;
3079                                 newparameter.remove_first();
3080
3081                                 if (parameter.op(0) == 0) {
3082
3083                                         // leading zero
3084                                         ex res = convert_H_to_zeta(parameter);
3085                                         map_trafo_H_1mxt1px recursion;
3086                                         ex buffer = recursion(H(newparameter, arg).hold());
3087                                         if (is_a<add>(buffer)) {
3088                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
3089                                                         res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3090                                                 }
3091                                         } else {
3092                                                 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3093                                         }
3094                                         return res;
3095
3096                                 } else if (parameter.op(0) == -1) {
3097
3098                                         // leading negative one
3099                                         ex res = convert_H_to_zeta(parameter);
3100                                         map_trafo_H_1mxt1px recursion;
3101                                         ex buffer = recursion(H(newparameter, arg).hold());
3102                                         if (is_a<add>(buffer)) {
3103                                                 for (std::size_t i = 0; i < buffer.nops(); i++) {
3104                                                         res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
3105                                                 }
3106                                         } else {
3107                                                 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
3108                                         }
3109                                         return res;
3110
3111                                 } else {
3112
3113                                         // leading one
3114                                         map_trafo_H_1mxt1px recursion;
3115                                         map_trafo_H_mult unify;
3116                                         ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
3117                                         std::size_t firstzero = 0;
3118                                         while (parameter.op(firstzero) == 1) {
3119                                                 firstzero++;
3120                                         }
3121                                         for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) {
3122                                                 lst newparameter;
3123                                                 std::size_t j=0;
3124                                                 for (; j<=i; j++) {
3125                                                         newparameter.append(parameter[j+1]);
3126                                                 }
3127                                                 newparameter.append(1);
3128                                                 for (; j<parameter.nops()-1; j++) {
3129                                                         newparameter.append(parameter[j+1]);
3130                                                 }
3131                                                 res -= H(newparameter, arg).hold();
3132                                         }
3133                                         res = recursion(res).expand() / firstzero;
3134                                         return unify(res);
3135
3136                                 }
3137
3138                         }
3139                 }
3140                 return e;
3141         }
3142 };
3143
3144
3145 // do the actual summation.
3146 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
3147 {
3148         const int j = m.size();
3149
3150         std::vector<cln::cl_N> t(j);
3151
3152         cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
3153         cln::cl_N factor = cln::expt(x, j) * one;
3154         cln::cl_N t0buf;
3155         int q = 0;
3156         do {
3157                 t0buf = t[0];
3158                 q++;
3159                 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
3160                 for (int k=j-2; k>=1; k--) {
3161                         t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
3162                 }
3163                 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
3164                 factor = factor * x;
3165         } while (t[0] != t0buf);
3166
3167         return t[0];
3168 }
3169
3170
3171 } // end of anonymous namespace
3172
3173
3174 //////////////////////////////////////////////////////////////////////
3175 //
3176 // Harmonic polylogarithm  H(m,x)
3177 //
3178 // GiNaC function
3179 //
3180 //////////////////////////////////////////////////////////////////////
3181
3182
3183 static ex H_evalf(const ex& x1, const ex& x2)
3184 {
3185         if (is_a<lst>(x1)) {
3186                 
3187                 cln::cl_N x;
3188                 if (is_a<numeric>(x2)) {
3189                         x = ex_to<numeric>(x2).to_cl_N();
3190                 } else {
3191                         ex x2_val = x2.evalf();
3192                         if (is_a<numeric>(x2_val)) {
3193                                 x = ex_to<numeric>(x2_val).to_cl_N();
3194                         }
3195                 }
3196
3197                 for (std::size_t i = 0; i < x1.nops(); i++) {
3198                         if (!x1.op(i).info(info_flags::integer)) {
3199                                 return H(x1, x2).hold();
3200                         }
3201                 }
3202                 if (x1.nops() < 1) {
3203                         return H(x1, x2).hold();
3204                 }
3205
3206                 const lst& morg = ex_to<lst>(x1);
3207                 // remove trailing zeros ...
3208                 if (*(--morg.end()) == 0) {
3209                         symbol xtemp("xtemp");
3210                         map_trafo_H_reduce_trailing_zeros filter;
3211                         return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
3212                 }
3213                 // ... and expand parameter notation
3214                 bool has_minus_one = false;
3215                 lst m;
3216                 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
3217                         if (*it > 1) {
3218                                 for (ex count=*it-1; count > 0; count--) {
3219                                         m.append(0);
3220                                 }
3221                                 m.append(1);
3222                         } else if (*it <= -1) {
3223                                 for (ex count=*it+1; count < 0; count++) {
3224                                         m.append(0);
3225                                 }
3226                                 m.append(-1);
3227                                 has_minus_one = true;
3228                         } else {
3229                                 m.append(*it);
3230                         }
3231                 }
3232
3233                 // do summation
3234                 if (cln::abs(x) < 0.95) {
3235                         lst m_lst;
3236                         lst s_lst;
3237                         ex pf;
3238                         if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
3239                                 // negative parameters -> s_lst is filled
3240                                 std::vector<int> m_int;
3241                                 std::vector<cln::cl_N> x_cln;
3242                                 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin(); 
3243                                      it_int != m_lst.end(); it_int++, it_cln++) {
3244                                         m_int.push_back(ex_to<numeric>(*it_int).to_int());
3245                                         x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
3246                                 }
3247                                 x_cln.front() = x_cln.front() * x;
3248                                 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
3249                         } else {
3250                                 // only positive parameters
3251                                 //TODO
3252                                 if (m_lst.nops() == 1) {
3253                                         return Li(m_lst.op(0), x2).evalf();
3254                                 }
3255                                 std::vector<int> m_int;
3256                                 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
3257                                         m_int.push_back(ex_to<numeric>(*it).to_int());
3258                                 }
3259                                 return numeric(H_do_sum(m_int, x));
3260                         }
3261                 }
3262
3263                 symbol xtemp("xtemp");
3264                 ex res = 1;     
3265                 
3266                 // ensure that the realpart of the argument is positive
3267                 if (cln::realpart(x) < 0) {
3268                         x = -x;
3269                         for (std::size_t i = 0; i < m.nops(); i++) {
3270                                 if (m.op(i) != 0) {
3271                                         m.let_op(i) = -m.op(i);
3272                                         res *= -1;
3273                                 }
3274                         }
3275                 }
3276
3277                 // x -> 1/x
3278                 if (cln::abs(x) >= 2.0) {
3279                         map_trafo_H_1overx trafo;
3280                         res *= trafo(H(m, xtemp).hold());
3281                         if (cln::imagpart(x) <= 0) {
3282                                 res = res.subs(H_polesign == -I*Pi);
3283                         } else {
3284                                 res = res.subs(H_polesign == I*Pi);
3285                         }
3286                         return res.subs(xtemp == numeric(x)).evalf();
3287                 }
3288                 
3289                 // check transformations for 0.95 <= |x| < 2.0
3290                 
3291                 // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
3292                 if (cln::abs(x-9.53) <= 9.47) {
3293                         // x -> (1-x)/(1+x)
3294                         map_trafo_H_1mxt1px trafo;
3295                         res *= trafo(H(m, xtemp).hold());
3296                 } else {
3297                         // x -> 1-x
3298                         if (has_minus_one) {
3299                                 map_trafo_H_convert_to_Li filter;
3300                                 return filter(H(m, numeric(x)).hold()).evalf();
3301                         }
3302                         map_trafo_H_1mx trafo;
3303                         res *= trafo(H(m, xtemp).hold());
3304                 }
3305
3306                 return res.subs(xtemp == numeric(x)).evalf();
3307         }
3308
3309         return H(x1,x2).hold();
3310 }
3311
3312
3313 static ex H_eval(const ex& m_, const ex& x)
3314 {
3315         lst m;
3316         if (is_a<lst>(m_)) {
3317                 m = ex_to<lst>(m_);
3318         } else {
3319                 m = lst(m_);
3320         }
3321         if (m.nops() == 0) {
3322                 return _ex1;
3323         }
3324         ex pos1;
3325         ex pos2;
3326         ex n;
3327         ex p;
3328         int step = 0;
3329         if (*m.begin() > _ex1) {
3330                 step++;
3331                 pos1 = _ex0;
3332                 pos2 = _ex1;
3333                 n = *m.begin()-1;
3334                 p = _ex1;
3335         } else if (*m.begin() < _ex_1) {
3336                 step++;
3337                 pos1 = _ex0;
3338                 pos2 = _ex_1;
3339                 n = -*m.begin()-1;
3340                 p = _ex1;
3341         } else if (*m.begin() == _ex0) {
3342                 pos1 = _ex0;
3343                 n = _ex1;
3344         } else {
3345                 pos1 = *m.begin();
3346                 p = _ex1;
3347         }
3348         for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
3349                 if ((*it).info(info_flags::integer)) {