1 /** @file inifcns_nstdsums.cpp
3 * Implementation of some special functions that have a representation as nested sums.
6 * classical polylogarithm Li(n,x)
7 * multiple polylogarithm Li(lst(m_1,...,m_k),lst(x_1,...,x_k))
8 * nielsen's generalized polylogarithm S(n,p,x)
9 * harmonic polylogarithm H(m,x) or H(lst(m_1,...,m_k),x)
10 * multiple zeta value zeta(m) or zeta(lst(m_1,...,m_k))
11 * alternating Euler sum zeta(m,s) or zeta(lst(m_1,...,m_k),lst(s_1,...,s_k))
15 * - All formulae used can be looked up in the following publications:
16 * [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
17 * [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
18 * [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
19 * [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 * - The order of parameters and arguments of Li and zeta is defined according to the nested sums
22 * representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
23 * 0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single
24 * number --- notation.
26 * - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in
27 * the whole complex plane. Multiple polylogarithms evaluate only if for each argument x_i the product
28 * x_1 * x_2 * ... * x_i is smaller than one. The parameters for Li, zeta and S must be positive integers.
29 * If you want to have an alternating Euler sum, you have to give the signs of the parameters as a
30 * second argument s to zeta(m,s) containing 1 and -1.
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.
36 * - The functions have no series expansion into nested sums. To do this, you have to convert these functions
37 * into the appropriate objects from the nestedsums library, do the expansion and convert the
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.
49 * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
51 * This program is free software; you can redistribute it and/or modify
52 * it under the terms of the GNU General Public License as published by
53 * the Free Software Foundation; either version 2 of the License, or
54 * (at your option) any later version.
56 * This program is distributed in the hope that it will be useful,
57 * but WITHOUT ANY WARRANTY; without even the implied warranty of
58 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
59 * GNU General Public License for more details.
61 * You should have received a copy of the GNU General Public License
62 * along with this program; if not, write to the Free Software
63 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
77 #include "operators.h"
80 #include "relational.h"
89 //////////////////////////////////////////////////////////////////////
91 // Classical polylogarithm Li(n,x)
95 //////////////////////////////////////////////////////////////////////
98 // anonymous namespace for helper functions
102 // lookup table for factors built from Bernoulli numbers
104 std::vector<std::vector<cln::cl_N> > Xn;
105 // initial size of Xn that should suffice for 32bit machines (must be even)
106 const int xninitsizestep = 26;
107 int xninitsize = xninitsizestep;
111 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
112 // With these numbers the polylogs can be calculated as follows:
113 // Li_p (x) = \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with u = -log(1-x)
114 // X_0(n) = B_n (Bernoulli numbers)
115 // X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
116 // The calculation of Xn depends on X0 and X{n-1}.
117 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
118 // This results in a slightly more complicated algorithm for the X_n.
119 // The first index in Xn corresponds to the index of the polylog minus 2.
120 // The second index in Xn corresponds to the index from the actual sum.
124 // calculate X_2 and higher (corresponding to Li_4 and higher)
125 std::vector<cln::cl_N> buf(xninitsize);
126 std::vector<cln::cl_N>::iterator it = buf.begin();
128 *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
130 for (int i=2; i<=xninitsize; i++) {
132 result = 0; // k == 0
134 result = Xn[0][i/2-1]; // k == 0
136 for (int k=1; k<i-1; k++) {
137 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
138 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
141 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
142 result = result + Xn[n-1][i-1] / (i+1); // k == i
149 // special case to handle the X_0 correct
150 std::vector<cln::cl_N> buf(xninitsize);
151 std::vector<cln::cl_N>::iterator it = buf.begin();
153 *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
155 *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
157 for (int i=3; i<=xninitsize; i++) {
159 result = -Xn[0][(i-3)/2]/2;
160 *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
163 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
164 for (int k=1; k<i/2; k++) {
165 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
174 std::vector<cln::cl_N> buf(xninitsize/2);
175 std::vector<cln::cl_N>::iterator it = buf.begin();
176 for (int i=1; i<=xninitsize/2; i++) {
177 *it = bernoulli(i*2).to_cl_N();
187 // doubles the number of entries in each Xn[]
190 const int pos0 = xninitsize / 2;
192 for (int i=1; i<=xninitsizestep/2; ++i) {
193 Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N());
196 int xend = xninitsize + xninitsizestep;
199 for (int i=xninitsize+1; i<=xend; ++i) {
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);
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);
208 Xn[1].push_back(result);
212 for (int n=2; n<Xn.size(); ++n) {
213 for (int i=xninitsize+1; i<=xend; ++i) {
215 result = 0; // k == 0
217 result = Xn[0][i/2-1]; // k == 0
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);
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);
230 xninitsize += xninitsizestep;
234 // calculates Li(2,x) without Xn
235 cln::cl_N Li2_do_sum(const cln::cl_N& x)
239 cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
240 cln::cl_I den = 1; // n^2 = 1
245 den = den + i; // n^2 = 4, 9, 16, ...
247 res = res + num / den;
248 } while (res != resbuf);
253 // calculates Li(2,x) with Xn
254 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
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 res = u - u*u/4;
265 factor = factor * u*u / (2*i * (2*i+1));
266 res = res + (*it) * factor;
270 it = Xn[0].begin() + (i-1);
273 } while (res != resbuf);
278 // calculates Li(n,x), n>2 without Xn
279 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
281 cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits));
288 res = res + factor / cln::expt(cln::cl_I(i),n);
290 } while (res != resbuf);
295 // calculates Li(n,x), n>2 with Xn
296 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
298 std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
299 std::vector<cln::cl_N>::const_iterator xend = Xn[n-2].end();
300 cln::cl_N u = -cln::log(1-x);
301 cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
307 factor = factor * u / i;
308 res = res + (*it) * factor;
312 it = Xn[n-2].begin() + (i-2);
313 xend = Xn[n-2].end();
315 } while (res != resbuf);
320 // forward declaration needed by function Li_projection and C below
321 numeric S_num(int n, int p, const numeric& x);
324 // helper function for classical polylog Li
325 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
327 // treat n=2 as special case
329 // check if precalculated X0 exists
334 if (cln::realpart(x) < 0.5) {
335 // choose the faster algorithm
336 // the switching point was empirically determined. the optimal point
337 // depends on hardware, Digits, ... so an approx value is okay.
338 // it solves also the problem with precision due to the u=-log(1-x) transformation
339 if (cln::abs(cln::realpart(x)) < 0.25) {
341 return Li2_do_sum(x);
343 return Li2_do_sum_Xn(x);
346 // choose the faster algorithm
347 if (cln::abs(cln::realpart(x)) > 0.75) {
348 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
350 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
354 // check if precalculated Xn exist
356 for (int i=xnsize; i<n-1; i++) {
361 if (cln::realpart(x) < 0.5) {
362 // choose the faster algorithm
363 // with n>=12 the "normal" summation always wins against the method with Xn
364 if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
365 return Lin_do_sum(n, x);
367 return Lin_do_sum_Xn(n, x);
370 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
371 for (int j=0; j<n-1; j++) {
372 result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x).to_cl_N())
373 * cln::expt(cln::log(x), j) / cln::factorial(j);
381 // helper function for classical polylog Li
382 numeric Li_num(int n, const numeric& x)
386 return -cln::log(1-x.to_cl_N());
397 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
400 // what is the desired float format?
401 // first guess: default format
402 cln::float_format_t prec = cln::default_float_format;
403 const cln::cl_N value = x.to_cl_N();
404 // second guess: the argument's format
405 if (!x.real().is_rational())
406 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
407 else if (!x.imag().is_rational())
408 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
411 if (cln::abs(value) > 1) {
412 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
413 // check if argument is complex. if it is real, the new polylog has to be conjugated.
414 if (cln::zerop(cln::imagpart(value))) {
416 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
419 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
424 result = result + Li_projection(n, cln::recip(value), prec);
427 result = result - Li_projection(n, cln::recip(value), prec);
431 for (int j=0; j<n-1; j++) {
432 add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
433 * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
435 result = result - add;
439 return Li_projection(n, value, prec);
444 } // end of anonymous namespace
447 //////////////////////////////////////////////////////////////////////
449 // Multiple polylogarithm Li(n,x)
453 //////////////////////////////////////////////////////////////////////
456 // anonymous namespace for helper function
460 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
462 const int j = s.size();
464 std::vector<cln::cl_N> t(j);
465 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
473 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
474 for (int k=j-2; k>=0; k--) {
475 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]);
477 // ... and do it again (to avoid premature drop out due to special arguments)
479 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
480 for (int k=j-2; k>=0; k--) {
481 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]);
483 } while (t[0] != t0buf);
488 // forward declaration for Li_eval()
489 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
492 } // end of anonymous namespace
495 //////////////////////////////////////////////////////////////////////
497 // Classical polylogarithm and multiple polylogarithm Li(n,x)
501 //////////////////////////////////////////////////////////////////////
504 static ex Li_evalf(const ex& x1, const ex& x2)
506 // classical polylogs
507 if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
508 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
511 else if (is_a<lst>(x1) && is_a<lst>(x2)) {
513 for (int i=0; i<x1.nops(); i++) {
514 if (!x1.op(i).info(info_flags::posint)) {
515 return Li(x1, x2).hold();
517 if (!is_a<numeric>(x2.op(i))) {
518 return Li(x1, x2).hold();
521 if (abs(conv) >= 1) {
522 return Li(x1, x2).hold();
527 std::vector<cln::cl_N> x;
528 for (int i=0; i<ex_to<numeric>(x1.nops()).to_int(); i++) {
529 m.push_back(ex_to<numeric>(x1.op(i)).to_int());
530 x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
533 return numeric(multipleLi_do_sum(m, x));
536 return Li(x1,x2).hold();
540 static ex Li_eval(const ex& m_, const ex& x_)
562 return (pow(2,1-m)-1) * zeta(m);
567 if (m.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
568 return Li_num(ex_to<numeric>(m).to_int(), ex_to<numeric>(x));
574 bool doevalf = false;
575 bool doevalfveto = true;
576 const lst& m = ex_to<lst>(m_);
577 const lst& x = ex_to<lst>(x_);
578 lst::const_iterator itm = m.begin();
579 lst::const_iterator itx = x.begin();
580 for (; itm != m.end(); itm++, itx++) {
581 if (!(*itm).info(info_flags::posint)) {
582 return Li(m_, x_).hold();
584 if ((*itx != _ex1) && (*itx != _ex_1)) {
585 if (itx != x.begin()) {
593 if (!(*itx).info(info_flags::numeric)) {
596 if (!(*itx).info(info_flags::crational)) {
608 lst newm = convert_parameter_Li_to_H(m, x, pf);
609 return pf * H(newm, x[0]);
611 if (doevalfveto && doevalf) {
612 return Li(m_, x_).evalf();
615 return Li(m_, x_).hold();
619 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
622 seq.push_back(expair(Li(m, x), 0));
623 return pseries(rel, seq);
627 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
629 GINAC_ASSERT(deriv_param < 2);
630 if (deriv_param == 0) {
634 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
649 return Li(m-1, x) / x;
656 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
670 c.s << "\\mbox{Li}_{";
671 lst::const_iterator itm = m.begin();
674 for (; itm != m.end(); itm++) {
679 lst::const_iterator itx = x.begin();
682 for (; itx != x.end(); itx++) {
690 REGISTER_FUNCTION(Li,
691 evalf_func(Li_evalf).
693 series_func(Li_series).
694 derivative_func(Li_deriv).
695 print_func<print_latex>(Li_print_latex).
696 do_not_evalf_params());
699 //////////////////////////////////////////////////////////////////////
701 // Nielsen's generalized polylogarithm S(n,p,x)
705 //////////////////////////////////////////////////////////////////////
708 // anonymous namespace for helper functions
712 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
714 std::vector<std::vector<cln::cl_N> > Yn;
715 int ynsize = 0; // number of Yn[]
716 int ynlength = 100; // initial length of all Yn[i]
719 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
720 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
721 // representing S_{n,p}(x).
722 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
724 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
725 // representing S_{n,p}(x).
726 // The calculation of Y_n uses the values from Y_{n-1}.
727 void fill_Yn(int n, const cln::float_format_t& prec)
729 const int initsize = ynlength;
730 //const int initsize = initsize_Yn;
731 cln::cl_N one = cln::cl_float(1, prec);
734 std::vector<cln::cl_N> buf(initsize);
735 std::vector<cln::cl_N>::iterator it = buf.begin();
736 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
737 *it = (*itprev) / cln::cl_N(n+1) * one;
740 // sums with an index smaller than the depth are zero and need not to be calculated.
741 // calculation starts with depth, which is n+2)
742 for (int i=n+2; i<=initsize+n; i++) {
743 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
749 std::vector<cln::cl_N> buf(initsize);
750 std::vector<cln::cl_N>::iterator it = buf.begin();
753 for (int i=2; i<=initsize; i++) {
754 *it = *(it-1) + 1 / cln::cl_N(i) * one;
763 // make Yn longer ...
764 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
767 cln::cl_N one = cln::cl_float(1, prec);
769 Yn[0].resize(newsize);
770 std::vector<cln::cl_N>::iterator it = Yn[0].begin();
772 for (int i=ynlength+1; i<=newsize; i++) {
773 *it = *(it-1) + 1 / cln::cl_N(i) * one;
777 for (int n=1; n<ynsize; n++) {
778 Yn[n].resize(newsize);
779 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
780 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
783 for (int i=ynlength+n+1; i<=newsize+n; i++) {
784 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
794 // helper function for S(n,p,x)
796 cln::cl_N C(int n, int p)
800 for (int k=0; k<p; k++) {
801 for (int j=0; j<=(n+k-1)/2; j++) {
805 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
808 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
815 result = result + cln::factorial(n+k-1)
816 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
817 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
820 result = result - cln::factorial(n+k-1)
821 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
822 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
827 result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
828 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
831 result = result + cln::factorial(n+k-1)
832 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
833 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
841 if (((np)/2+n) & 1) {
842 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
845 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
853 // helper function for S(n,p,x)
854 // [Kol] remark to (9.1)
864 for (int m=2; m<=k; m++) {
865 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
872 // helper function for S(n,p,x)
873 // [Kol] remark to (9.1)
883 for (int m=2; m<=k; m++) {
884 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
891 // helper function for S(n,p,x)
892 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
895 return Li_projection(n+1, x, prec);
898 // check if precalculated values are sufficient
900 for (int i=ynsize; i<p-1; i++) {
905 // should be done otherwise
906 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
907 cln::cl_N xf = x * one;
908 //cln::cl_N xf = x * cln::cl_float(1, prec);
912 cln::cl_N factor = cln::expt(xf, p);
916 if (i-p >= ynlength) {
918 make_Yn_longer(ynlength*2, prec);
920 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? ...
921 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
922 factor = factor * xf;
924 } while (res != resbuf);
930 // helper function for S(n,p,x)
931 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
934 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
936 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
937 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
939 for (int s=0; s<n; s++) {
941 for (int r=0; r<p; r++) {
942 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
943 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
945 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
951 return S_do_sum(n, p, x, prec);
955 // helper function for S(n,p,x)
956 numeric S_num(int n, int p, const numeric& x)
960 // [Kol] (2.22) with (2.21)
961 return cln::zeta(p+1);
966 return cln::zeta(n+1);
971 for (int nu=0; nu<n; nu++) {
972 for (int rho=0; rho<=p; rho++) {
973 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
974 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
977 result = result * cln::expt(cln::cl_I(-1),n+p-1);
984 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
986 // throw std::runtime_error("don't know how to evaluate this function!");
989 // what is the desired float format?
990 // first guess: default format
991 cln::float_format_t prec = cln::default_float_format;
992 const cln::cl_N value = x.to_cl_N();
993 // second guess: the argument's format
994 if (!x.real().is_rational())
995 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
996 else if (!x.imag().is_rational())
997 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1000 if ((cln::realpart(value) < -0.5) || (n == 0)) {
1002 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
1003 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
1005 for (int s=0; s<n; s++) {
1007 for (int r=0; r<p; r++) {
1008 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
1009 * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
1011 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
1018 if (cln::abs(value) > 1) {
1022 for (int s=0; s<p; s++) {
1023 for (int r=0; r<=s; r++) {
1024 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
1025 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
1026 * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
1029 result = result * cln::expt(cln::cl_I(-1),n);
1032 for (int r=0; r<n; r++) {
1033 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
1035 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
1037 result = result + cln::expt(cln::cl_I(-1),p) * res2;
1042 return S_projection(n, p, value, prec);
1047 } // end of anonymous namespace
1050 //////////////////////////////////////////////////////////////////////
1052 // Nielsen's generalized polylogarithm S(n,p,x)
1056 //////////////////////////////////////////////////////////////////////
1059 static ex S_evalf(const ex& n, const ex& p, const ex& x)
1061 if (n.info(info_flags::posint) && p.info(info_flags::posint) && is_a<numeric>(x)) {
1062 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1064 return S(n, p, x).hold();
1068 static ex S_eval(const ex& n, const ex& p, const ex& x)
1070 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1076 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
1084 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
1085 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1090 return pow(-log(1-x), p) / factorial(p);
1092 return S(n, p, x).hold();
1096 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
1099 seq.push_back(expair(S(n, p, x), 0));
1100 return pseries(rel, seq);
1104 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
1106 GINAC_ASSERT(deriv_param < 3);
1107 if (deriv_param < 2) {
1111 return S(n-1, p, x) / x;
1113 return S(n, p-1, x) / (1-x);
1118 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
1120 c.s << "\\mbox{S}_{";
1130 REGISTER_FUNCTION(S,
1131 evalf_func(S_evalf).
1133 series_func(S_series).
1134 derivative_func(S_deriv).
1135 print_func<print_latex>(S_print_latex).
1136 do_not_evalf_params());
1139 //////////////////////////////////////////////////////////////////////
1141 // Harmonic polylogarithm H(m,x)
1145 //////////////////////////////////////////////////////////////////////
1148 // anonymous namespace for helper functions
1152 // regulates the pole (used by 1/x-transformation)
1153 symbol H_polesign("IMSIGN");
1156 // convert parameters from H to Li representation
1157 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
1158 // returns true if some parameters are negative
1159 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
1161 // expand parameter list
1163 for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
1165 for (ex count=*it-1; count > 0; count--) {
1169 } else if (*it < -1) {
1170 for (ex count=*it+1; count < 0; count++) {
1181 bool has_negative_parameters = false;
1183 for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
1189 m.append((*it+acc-1) * signum);
1191 m.append((*it-acc+1) * signum);
1197 has_negative_parameters = true;
1200 if (has_negative_parameters) {
1201 for (int i=0; i<m.nops(); i++) {
1203 m.let_op(i) = -m.op(i);
1211 return has_negative_parameters;
1215 // recursivly transforms H to corresponding multiple polylogarithms
1216 struct map_trafo_H_convert_to_Li : public map_function
1218 ex operator()(const ex& e)
1220 if (is_a<add>(e) || is_a<mul>(e)) {
1221 return e.map(*this);
1223 if (is_a<function>(e)) {
1224 std::string name = ex_to<function>(e).get_name();
1227 if (is_a<lst>(e.op(0))) {
1228 parameter = ex_to<lst>(e.op(0));
1230 parameter = lst(e.op(0));
1237 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1238 s.let_op(0) = s.op(0) * arg;
1239 return pf * Li(m, s).hold();
1241 for (int i=0; i<m.nops(); i++) {
1244 s.let_op(0) = s.op(0) * arg;
1245 return Li(m, s).hold();
1254 // recursivly transforms H to corresponding zetas
1255 struct map_trafo_H_convert_to_zeta : public map_function
1257 ex operator()(const ex& e)
1259 if (is_a<add>(e) || is_a<mul>(e)) {
1260 return e.map(*this);
1262 if (is_a<function>(e)) {
1263 std::string name = ex_to<function>(e).get_name();
1266 if (is_a<lst>(e.op(0))) {
1267 parameter = ex_to<lst>(e.op(0));
1269 parameter = lst(e.op(0));
1275 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1276 return pf * zeta(m, s);
1287 // remove trailing zeros from H-parameters
1288 struct map_trafo_H_reduce_trailing_zeros : public map_function
1290 ex operator()(const ex& e)
1292 if (is_a<add>(e) || is_a<mul>(e)) {
1293 return e.map(*this);
1295 if (is_a<function>(e)) {
1296 std::string name = ex_to<function>(e).get_name();
1299 if (is_a<lst>(e.op(0))) {
1300 parameter = ex_to<lst>(e.op(0));
1302 parameter = lst(e.op(0));
1305 if (parameter.op(parameter.nops()-1) == 0) {
1308 if (parameter.nops() == 1) {
1313 lst::const_iterator it = parameter.begin();
1314 while ((it != parameter.end()) && (*it == 0)) {
1317 if (it == parameter.end()) {
1318 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
1322 parameter.remove_last();
1323 int lastentry = parameter.nops();
1324 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
1329 ex result = log(arg) * H(parameter,arg).hold();
1331 for (ex i=0; i<lastentry; i++) {
1332 if (parameter[i] > 0) {
1334 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
1337 } else if (parameter[i] < 0) {
1339 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
1347 if (lastentry < parameter.nops()) {
1348 result = result / (parameter.nops()-lastentry+1);
1349 return result.map(*this);
1361 // returns an expression with zeta functions corresponding to the parameter list for H
1362 ex convert_H_to_zeta(const lst& m)
1364 symbol xtemp("xtemp");
1365 map_trafo_H_reduce_trailing_zeros filter;
1366 map_trafo_H_convert_to_zeta filter2;
1367 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
1371 // convert signs form Li to H representation
1372 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
1375 lst::const_iterator itm = m.begin();
1376 lst::const_iterator itx = ++x.begin();
1381 while (itx != x.end()) {
1384 res.append((*itm) * signum);
1392 // multiplies an one-dimensional H with another H
1394 ex trafo_H_mult(const ex& h1, const ex& h2)
1399 ex h1nops = h1.op(0).nops();
1400 ex h2nops = h2.op(0).nops();
1402 hshort = h2.op(0).op(0);
1403 hlong = ex_to<lst>(h1.op(0));
1405 hshort = h1.op(0).op(0);
1407 hlong = ex_to<lst>(h2.op(0));
1409 hlong = h2.op(0).op(0);
1412 for (int i=0; i<=hlong.nops(); i++) {
1416 newparameter.append(hlong[j]);
1418 newparameter.append(hshort);
1419 for (; j<hlong.nops(); j++) {
1420 newparameter.append(hlong[j]);
1422 res += H(newparameter, h1.op(1)).hold();
1428 // applies trafo_H_mult recursively on expressions
1429 struct map_trafo_H_mult : public map_function
1431 ex operator()(const ex& e)
1434 return e.map(*this);
1442 for (int pos=0; pos<e.nops(); pos++) {
1443 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
1444 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
1446 for (ex i=0; i<e.op(pos).op(1); i++) {
1447 Hlst.append(e.op(pos).op(0));
1451 } else if (is_a<function>(e.op(pos))) {
1452 std::string name = ex_to<function>(e.op(pos)).get_name();
1454 if (e.op(pos).op(0).nops() > 1) {
1457 Hlst.append(e.op(pos));
1462 result *= e.op(pos);
1465 if (Hlst.nops() > 0) {
1466 firstH = Hlst[Hlst.nops()-1];
1473 if (Hlst.nops() > 0) {
1474 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
1476 for (int i=1; i<Hlst.nops(); i++) {
1477 result *= Hlst.op(i);
1479 result = result.expand();
1480 map_trafo_H_mult recursion;
1481 return recursion(result);
1492 // do integration [ReV] (55)
1493 // put parameter 0 in front of existing parameters
1494 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
1498 if (is_a<function>(e)) {
1499 name = ex_to<function>(e).get_name();
1504 for (int i=0; i<e.nops(); i++) {
1505 if (is_a<function>(e.op(i))) {
1506 std::string name = ex_to<function>(e.op(i)).get_name();
1514 lst newparameter = ex_to<lst>(h.op(0));
1515 newparameter.prepend(0);
1516 ex addzeta = convert_H_to_zeta(newparameter);
1517 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1519 return e * (-H(lst(0),1/arg).hold());
1524 // do integration [ReV] (55)
1525 // put parameter -1 in front of existing parameters
1526 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
1530 if (is_a<function>(e)) {
1531 name = ex_to<function>(e).get_name();
1536 for (int i=0; i<e.nops(); i++) {
1537 if (is_a<function>(e.op(i))) {
1538 std::string name = ex_to<function>(e.op(i)).get_name();
1546 lst newparameter = ex_to<lst>(h.op(0));
1547 newparameter.prepend(-1);
1548 ex addzeta = convert_H_to_zeta(newparameter);
1549 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1551 ex addzeta = convert_H_to_zeta(lst(-1));
1552 return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
1557 // do integration [ReV] (55)
1558 // put parameter -1 in front of existing parameters
1559 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
1563 if (is_a<function>(e)) {
1564 name = ex_to<function>(e).get_name();
1569 for (int i=0; i<e.nops(); i++) {
1570 if (is_a<function>(e.op(i))) {
1571 std::string name = ex_to<function>(e.op(i)).get_name();
1579 lst newparameter = ex_to<lst>(h.op(0));
1580 newparameter.prepend(-1);
1581 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1583 return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
1588 // do integration [ReV] (55)
1589 // put parameter 1 in front of existing parameters
1590 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
1594 if (is_a<function>(e)) {
1595 name = ex_to<function>(e).get_name();
1600 for (int i=0; i<e.nops(); i++) {
1601 if (is_a<function>(e.op(i))) {
1602 std::string name = ex_to<function>(e.op(i)).get_name();
1610 lst newparameter = ex_to<lst>(h.op(0));
1611 newparameter.prepend(1);
1612 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1614 return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
1619 // do x -> 1/x transformation
1620 struct map_trafo_H_1overx : public map_function
1622 ex operator()(const ex& e)
1624 if (is_a<add>(e) || is_a<mul>(e)) {
1625 return e.map(*this);
1628 if (is_a<function>(e)) {
1629 std::string name = ex_to<function>(e).get_name();
1632 lst parameter = ex_to<lst>(e.op(0));
1635 // special cases if all parameters are either 0, 1 or -1
1636 bool allthesame = true;
1637 if (parameter.op(0) == 0) {
1638 for (int i=1; i<parameter.nops(); i++) {
1639 if (parameter.op(i) != 0) {
1645 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
1647 } else if (parameter.op(0) == -1) {
1648 for (int i=1; i<parameter.nops(); i++) {
1649 if (parameter.op(i) != -1) {
1655 map_trafo_H_mult unify;
1656 return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
1657 / factorial(parameter.nops())).expand());
1660 for (int i=1; i<parameter.nops(); i++) {
1661 if (parameter.op(i) != 1) {
1667 map_trafo_H_mult unify;
1668 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() + H_polesign, parameter.nops())
1669 / factorial(parameter.nops())).expand());
1673 lst newparameter = parameter;
1674 newparameter.remove_first();
1676 if (parameter.op(0) == 0) {
1679 ex res = convert_H_to_zeta(parameter);
1680 map_trafo_H_1overx recursion;
1681 ex buffer = recursion(H(newparameter, arg).hold());
1682 if (is_a<add>(buffer)) {
1683 for (int i=0; i<buffer.nops(); i++) {
1684 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
1687 res += trafo_H_1tx_prepend_zero(buffer, arg);
1691 } else if (parameter.op(0) == -1) {
1693 // leading negative one
1694 ex res = convert_H_to_zeta(parameter);
1695 map_trafo_H_1overx recursion;
1696 ex buffer = recursion(H(newparameter, arg).hold());
1697 if (is_a<add>(buffer)) {
1698 for (int i=0; i<buffer.nops(); i++) {
1699 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
1702 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
1709 map_trafo_H_1overx recursion;
1710 map_trafo_H_mult unify;
1711 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1713 while (parameter.op(firstzero) == 1) {
1716 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1720 newparameter.append(parameter[j+1]);
1722 newparameter.append(1);
1723 for (; j<parameter.nops()-1; j++) {
1724 newparameter.append(parameter[j+1]);
1726 res -= H(newparameter, arg).hold();
1728 res = recursion(res).expand() / firstzero;
1740 // do x -> (1-x)/(1+x) transformation
1741 struct map_trafo_H_1mxt1px : public map_function
1743 ex operator()(const ex& e)
1745 if (is_a<add>(e) || is_a<mul>(e)) {
1746 return e.map(*this);
1749 if (is_a<function>(e)) {
1750 std::string name = ex_to<function>(e).get_name();
1753 lst parameter = ex_to<lst>(e.op(0));
1756 // special cases if all parameters are either 0, 1 or -1
1757 bool allthesame = true;
1758 if (parameter.op(0) == 0) {
1759 for (int i=1; i<parameter.nops(); i++) {
1760 if (parameter.op(i) != 0) {
1766 map_trafo_H_mult unify;
1767 return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1768 / factorial(parameter.nops())).expand());
1770 } else if (parameter.op(0) == -1) {
1771 for (int i=1; i<parameter.nops(); i++) {
1772 if (parameter.op(i) != -1) {
1778 map_trafo_H_mult unify;
1779 return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1780 / factorial(parameter.nops())).expand());
1783 for (int i=1; i<parameter.nops(); i++) {
1784 if (parameter.op(i) != 1) {
1790 map_trafo_H_mult unify;
1791 return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1792 / factorial(parameter.nops())).expand());
1796 lst newparameter = parameter;
1797 newparameter.remove_first();
1799 if (parameter.op(0) == 0) {
1802 ex res = convert_H_to_zeta(parameter);
1803 map_trafo_H_1mxt1px recursion;
1804 ex buffer = recursion(H(newparameter, arg).hold());
1805 if (is_a<add>(buffer)) {
1806 for (int i=0; i<buffer.nops(); i++) {
1807 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1810 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1814 } else if (parameter.op(0) == -1) {
1816 // leading negative one
1817 ex res = convert_H_to_zeta(parameter);
1818 map_trafo_H_1mxt1px recursion;
1819 ex buffer = recursion(H(newparameter, arg).hold());
1820 if (is_a<add>(buffer)) {
1821 for (int i=0; i<buffer.nops(); i++) {
1822 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1825 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1832 map_trafo_H_1mxt1px recursion;
1833 map_trafo_H_mult unify;
1834 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1836 while (parameter.op(firstzero) == 1) {
1839 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1843 newparameter.append(parameter[j+1]);
1845 newparameter.append(1);
1846 for (; j<parameter.nops()-1; j++) {
1847 newparameter.append(parameter[j+1]);
1849 res -= H(newparameter, arg).hold();
1851 res = recursion(res).expand() / firstzero;
1863 // do the actual summation.
1864 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
1866 const int j = m.size();
1868 std::vector<cln::cl_N> t(j);
1870 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1871 cln::cl_N factor = cln::expt(x, j) * one;
1877 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
1878 for (int k=j-2; k>=1; k--) {
1879 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
1881 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
1882 factor = factor * x;
1883 } while (t[0] != t0buf);
1889 } // end of anonymous namespace
1892 //////////////////////////////////////////////////////////////////////
1894 // Harmonic polylogarithm H(m,x)
1898 //////////////////////////////////////////////////////////////////////
1901 static ex H_evalf(const ex& x1, const ex& x2)
1903 if (is_a<lst>(x1) && is_a<numeric>(x2)) {
1904 for (int i=0; i<x1.nops(); i++) {
1905 if (!x1.op(i).info(info_flags::integer)) {
1906 return H(x1,x2).hold();
1909 if (x1.nops() < 1) {
1910 return H(x1,x2).hold();
1913 cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
1915 const lst& morg = ex_to<lst>(x1);
1916 // remove trailing zeros ...
1917 if (*(--morg.end()) == 0) {
1918 symbol xtemp("xtemp");
1919 map_trafo_H_reduce_trailing_zeros filter;
1920 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
1922 // ... and expand parameter notation
1924 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
1926 for (ex count=*it-1; count > 0; count--) {
1930 } else if (*it < -1) {
1931 for (ex count=*it+1; count < 0; count++) {
1940 // since the transformations produce a lot of terms, they are only efficient for
1941 // argument near one.
1942 // no transformation needed -> do summation
1943 if (cln::abs(x) < 0.95) {
1947 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
1948 // negative parameters -> s_lst is filled
1949 std::vector<int> m_int;
1950 std::vector<cln::cl_N> x_cln;
1951 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
1952 it_int != m_lst.end(); it_int++, it_cln++) {
1953 m_int.push_back(ex_to<numeric>(*it_int).to_int());
1954 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
1956 x_cln.front() = x_cln.front() * x;
1957 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
1959 // only positive parameters
1961 if (m_lst.nops() == 1) {
1962 return Li(m_lst.op(0), x2).evalf();
1964 std::vector<int> m_int;
1965 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
1966 m_int.push_back(ex_to<numeric>(*it).to_int());
1968 return numeric(H_do_sum(m_int, x));
1974 // ensure that the realpart of the argument is positive
1975 if (cln::realpart(x) < 0) {
1977 for (int i=0; i<m.nops(); i++) {
1979 m.let_op(i) = -m.op(i);
1985 // choose transformations
1986 symbol xtemp("xtemp");
1987 if (cln::abs(x-1) < 1.4142) {
1989 map_trafo_H_1mxt1px trafo;
1990 res *= trafo(H(m, xtemp));
1993 map_trafo_H_1overx trafo;
1994 res *= trafo(H(m, xtemp));
1995 if (cln::imagpart(x) <= 0) {
1996 res = res.subs(H_polesign == -I*Pi);
1998 res = res.subs(H_polesign == I*Pi);
2004 // map_trafo_H_convert converter;
2005 // res = converter(res).expand();
2007 // res.find(H(wild(1),wild(2)), ll);
2008 // res.find(zeta(wild(1)), ll);
2009 // res.find(zeta(wild(1),wild(2)), ll);
2010 // res = res.collect(ll);
2012 return res.subs(xtemp == numeric(x)).evalf();
2015 return H(x1,x2).hold();
2019 static ex H_eval(const ex& m_, const ex& x)
2022 if (is_a<lst>(m_)) {
2027 if (m.nops() == 0) {
2035 if (*m.begin() > _ex1) {
2041 } else if (*m.begin() < _ex_1) {
2047 } else if (*m.begin() == _ex0) {
2054 for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
2055 if ((*it).info(info_flags::integer)) {
2066 } else if (*it < _ex_1) {
2086 } else if (step == 1) {
2098 // if some m_i is not an integer
2099 return H(m_, x).hold();
2102 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
2103 return convert_H_to_zeta(m);
2109 return H(m_, x).hold();
2111 return pow(log(x), m.nops()) / factorial(m.nops());
2114 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
2116 } else if ((step == 1) && (pos1 == _ex0)){
2121 return pow(-1, p) * S(n, p, -x);
2127 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2128 return H(m_, x).evalf();
2130 return H(m_, x).hold();
2134 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
2137 seq.push_back(expair(H(m, x), 0));
2138 return pseries(rel, seq);
2142 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
2144 GINAC_ASSERT(deriv_param < 2);
2145 if (deriv_param == 0) {
2149 if (is_a<lst>(m_)) {
2165 return 1/(1-x) * H(m, x);
2166 } else if (mb == _ex_1) {
2167 return 1/(1+x) * H(m, x);
2174 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
2177 if (is_a<lst>(m_)) {
2182 c.s << "\\mbox{H}_{";
2183 lst::const_iterator itm = m.begin();
2186 for (; itm != m.end(); itm++) {
2196 REGISTER_FUNCTION(H,
2197 evalf_func(H_evalf).
2199 series_func(H_series).
2200 derivative_func(H_deriv).
2201 print_func<print_latex>(H_print_latex).
2202 do_not_evalf_params());
2205 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
2206 ex convert_H_to_Li(const ex& m, const ex& x)
2208 map_trafo_H_reduce_trailing_zeros filter;
2209 map_trafo_H_convert_to_Li filter2;
2211 return filter2(filter(H(m, x).hold()));
2213 return filter2(filter(H(lst(m), x).hold()));
2218 //////////////////////////////////////////////////////////////////////
2220 // Multiple zeta values zeta(x) and zeta(x,s)
2224 //////////////////////////////////////////////////////////////////////
2227 // anonymous namespace for helper functions
2231 // parameters and data for [Cra] algorithm
2232 const cln::cl_N lambda = cln::cl_N("319/320");
2235 std::vector<std::vector<cln::cl_N> > f_kj;
2236 std::vector<cln::cl_N> crB;
2237 std::vector<std::vector<cln::cl_N> > crG;
2238 std::vector<cln::cl_N> crX;
2241 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
2243 const int size = a.size();
2244 for (int n=0; n<size; n++) {
2246 for (int m=0; m<=n; m++) {
2247 c[n] = c[n] + a[m]*b[n-m];
2254 void initcX(const std::vector<int>& s)
2256 const int k = s.size();
2262 for (int i=0; i<=L2; i++) {
2263 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
2268 for (int m=0; m<k-1; m++) {
2269 std::vector<cln::cl_N> crGbuf;
2272 for (int i=0; i<=L2; i++) {
2273 crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
2275 crG.push_back(crGbuf);
2280 for (int m=0; m<k-1; m++) {
2281 std::vector<cln::cl_N> Xbuf;
2282 for (int i=0; i<=L2; i++) {
2283 Xbuf.push_back(crX[i] * crG[m][i]);
2285 halfcyclic_convolute(Xbuf, crB, crX);
2291 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
2293 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2294 cln::cl_N factor = cln::expt(lambda, Sqk);
2295 cln::cl_N res = factor / Sqk * crX[0] * one;
2300 factor = factor * lambda;
2302 res = res + crX[N] * factor / (N+Sqk);
2303 } while ((res != resbuf) || cln::zerop(crX[N]));
2309 void calc_f(int maxr)
2314 cln::cl_N t0, t1, t2, t3, t4;
2316 std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
2317 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2319 t0 = cln::exp(-lambda);
2321 for (k=1; k<=L1; k++) {
2324 for (j=1; j<=maxr; j++) {
2327 for (i=2; i<=j; i++) {
2331 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
2339 cln::cl_N crandall_Z(const std::vector<int>& s)
2341 const int j = s.size();
2350 t0 = t0 + f_kj[q+j-2][s[0]-1];
2351 } while (t0 != t0buf);
2353 return t0 / cln::factorial(s[0]-1);
2356 std::vector<cln::cl_N> t(j);
2363 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
2364 for (int k=j-2; k>=1; k--) {
2365 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
2367 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
2368 } while (t[0] != t0buf);
2370 return t[0] / cln::factorial(s[0]-1);
2375 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
2377 std::vector<int> r = s;
2378 const int j = r.size();
2380 // decide on maximal size of f_kj for crandall_Z
2384 L1 = Digits * 3 + j*2;
2387 // decide on maximal size of crX for crandall_Y
2390 } else if (Digits < 86) {
2392 } else if (Digits < 192) {
2394 } else if (Digits < 394) {
2396 } else if (Digits < 808) {
2406 for (int i=0; i<j; i++) {
2415 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
2417 std::vector<int> rz;
2420 for (int k=r.size()-1; k>0; k--) {
2422 rz.insert(rz.begin(), r.back());
2423 skp1buf = rz.front();
2429 for (int q=0; q<skp1buf; q++) {
2431 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
2432 cln::cl_N pp2 = crandall_Z(rz);
2437 res = res - pp1 * pp2 / cln::factorial(q);
2439 res = res + pp1 * pp2 / cln::factorial(q);
2442 rz.front() = skp1buf;
2444 rz.insert(rz.begin(), r.back());
2448 res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
2454 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
2456 const int j = r.size();
2458 // buffer for subsums
2459 std::vector<cln::cl_N> t(j);
2460 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2467 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
2468 for (int k=j-2; k>=0; k--) {
2469 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
2471 } while (t[0] != t0buf);
2477 // does Hoelder convolution. see [BBB] (7.0)
2478 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
2480 // prepare parameters
2481 // holds Li arguments in [BBB] notation
2482 std::vector<int> s = s_;
2483 std::vector<int> m_p = m_;
2484 std::vector<int> m_q;
2485 // holds Li arguments in nested sums notation
2486 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
2487 s_p[0] = s_p[0] * cln::cl_N("1/2");
2488 // convert notations
2490 for (int i=0; i<s_.size(); i++) {
2495 s[i] = sig * std::abs(s[i]);
2497 std::vector<cln::cl_N> s_q;
2498 cln::cl_N signum = 1;
2501 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
2506 // change parameters
2507 if (s.front() > 0) {
2508 if (m_p.front() == 1) {
2509 m_p.erase(m_p.begin());
2510 s_p.erase(s_p.begin());
2511 if (s_p.size() > 0) {
2512 s_p.front() = s_p.front() * cln::cl_N("1/2");
2518 m_q.insert(m_q.begin(), 1);
2519 if (s_q.size() > 0) {
2520 s_q.front() = s_q.front() * 2;
2522 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2525 if (m_p.front() == 1) {
2526 m_p.erase(m_p.begin());
2527 cln::cl_N spbuf = s_p.front();
2528 s_p.erase(s_p.begin());
2529 if (s_p.size() > 0) {
2530 s_p.front() = s_p.front() * spbuf;
2533 m_q.insert(m_q.begin(), 1);
2534 if (s_q.size() > 0) {
2535 s_q.front() = s_q.front() * 4;
2537 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
2541 m_q.insert(m_q.begin(), 1);
2542 if (s_q.size() > 0) {
2543 s_q.front() = s_q.front() * 2;
2545 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2550 if (m_p.size() == 0) break;
2552 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
2557 res = res + signum * multipleLi_do_sum(m_q, s_q);
2563 } // end of anonymous namespace
2566 //////////////////////////////////////////////////////////////////////
2568 // Multiple zeta values zeta(x)
2572 //////////////////////////////////////////////////////////////////////
2575 static ex zeta1_evalf(const ex& x)
2577 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
2579 // multiple zeta value
2580 const int count = x.nops();
2581 const lst& xlst = ex_to<lst>(x);
2582 std::vector<int> r(count);
2584 // check parameters and convert them
2585 lst::const_iterator it1 = xlst.begin();
2586 std::vector<int>::iterator it2 = r.begin();
2588 if (!(*it1).info(info_flags::posint)) {
2589 return zeta(x).hold();
2591 *it2 = ex_to<numeric>(*it1).to_int();
2594 } while (it2 != r.end());
2596 // check for divergence
2598 return zeta(x).hold();
2601 // decide on summation algorithm
2602 // this is still a bit clumsy
2603 int limit = (Digits>17) ? 10 : 6;
2604 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
2605 return numeric(zeta_do_sum_Crandall(r));
2607 return numeric(zeta_do_sum_simple(r));
2611 // single zeta value
2612 if (is_exactly_a<numeric>(x) && (x != 1)) {
2614 return zeta(ex_to<numeric>(x));
2615 } catch (const dunno &e) { }
2618 return zeta(x).hold();
2622 static ex zeta1_eval(const ex& m)
2624 if (is_exactly_a<lst>(m)) {
2625 if (m.nops() == 1) {
2626 return zeta(m.op(0));
2628 return zeta(m).hold();
2631 if (m.info(info_flags::numeric)) {
2632 const numeric& y = ex_to<numeric>(m);
2633 // trap integer arguments:
2634 if (y.is_integer()) {
2638 if (y.is_equal(_num1)) {
2639 return zeta(m).hold();
2641 if (y.info(info_flags::posint)) {
2642 if (y.info(info_flags::odd)) {
2643 return zeta(m).hold();
2645 return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
2648 if (y.info(info_flags::odd)) {
2649 return -bernoulli(_num1-y) / (_num1-y);
2656 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
2657 return zeta1_evalf(m);
2660 return zeta(m).hold();
2664 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
2666 GINAC_ASSERT(deriv_param==0);
2668 if (is_exactly_a<lst>(m)) {
2671 return zetaderiv(_ex1, m);
2676 static void zeta1_print_latex(const ex& m_, const print_context& c)
2679 if (is_a<lst>(m_)) {
2680 const lst& m = ex_to<lst>(m_);
2681 lst::const_iterator it = m.begin();
2684 for (; it != m.end(); it++) {
2695 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
2696 evalf_func(zeta1_evalf).
2697 eval_func(zeta1_eval).
2698 derivative_func(zeta1_deriv).
2699 print_func<print_latex>(zeta1_print_latex).
2700 do_not_evalf_params().
2704 //////////////////////////////////////////////////////////////////////
2706 // Alternating Euler sum zeta(x,s)
2710 //////////////////////////////////////////////////////////////////////
2713 static ex zeta2_evalf(const ex& x, const ex& s)
2715 if (is_exactly_a<lst>(x)) {
2717 // alternating Euler sum
2718 const int count = x.nops();
2719 const lst& xlst = ex_to<lst>(x);
2720 const lst& slst = ex_to<lst>(s);
2721 std::vector<int> xi(count);
2722 std::vector<int> si(count);
2724 // check parameters and convert them
2725 lst::const_iterator it_xread = xlst.begin();
2726 lst::const_iterator it_sread = slst.begin();
2727 std::vector<int>::iterator it_xwrite = xi.begin();
2728 std::vector<int>::iterator it_swrite = si.begin();
2730 if (!(*it_xread).info(info_flags::posint)) {
2731 return zeta(x, s).hold();
2733 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
2734 if (*it_sread > 0) {
2743 } while (it_xwrite != xi.end());
2745 // check for divergence
2746 if ((xi[0] == 1) && (si[0] == 1)) {
2747 return zeta(x, s).hold();
2750 // use Hoelder convolution
2751 return numeric(zeta_do_Hoelder_convolution(xi, si));
2754 return zeta(x, s).hold();
2758 static ex zeta2_eval(const ex& m, const ex& s_)
2760 if (is_exactly_a<lst>(s_)) {
2761 const lst& s = ex_to<lst>(s_);
2762 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
2763 if ((*it).info(info_flags::positive)) {
2766 return zeta(m, s_).hold();
2769 } else if (s_.info(info_flags::positive)) {
2773 return zeta(m, s_).hold();
2777 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
2779 GINAC_ASSERT(deriv_param==0);
2781 if (is_exactly_a<lst>(m)) {
2784 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
2785 return zetaderiv(_ex1, m);
2792 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
2795 if (is_a<lst>(m_)) {
2801 if (is_a<lst>(s_)) {
2807 lst::const_iterator itm = m.begin();
2808 lst::const_iterator its = s.begin();
2810 c.s << "\\overline{";
2818 for (; itm != m.end(); itm++, its++) {
2821 c.s << "\\overline{";
2832 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
2833 evalf_func(zeta2_evalf).
2834 eval_func(zeta2_eval).
2835 derivative_func(zeta2_deriv).
2836 print_func<print_latex>(zeta2_print_latex).
2837 do_not_evalf_params().
2841 } // namespace GiNaC