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 speed 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) 1999-2003 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;
108 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
109 // With these numbers the polylogs can be calculated as follows:
110 // Li_p (x) = \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with u = -log(1-x)
111 // X_0(n) = B_n (Bernoulli numbers)
112 // X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
113 // The calculation of Xn depends on X0 and X{n-1}.
114 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
115 // This results in a slightly more complicated algorithm for the X_n.
116 // The first index in Xn corresponds to the index of the polylog minus 2.
117 // The second index in Xn corresponds to the index from the actual sum.
120 // rule of thumb. needs to be improved. TODO
121 const int initsize = Digits * 3 / 2;
124 // calculate X_2 and higher (corresponding to Li_4 and higher)
125 std::vector<cln::cl_N> buf(initsize);
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<=initsize; 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(initsize);
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<=initsize; 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(initsize/2);
175 std::vector<cln::cl_N>::iterator it = buf.begin();
176 for (int i=1; i<=initsize/2; i++) {
177 *it = bernoulli(i*2).to_cl_N();
187 // calculates Li(2,x) without Xn
188 cln::cl_N Li2_do_sum(const cln::cl_N& x)
193 cln::cl_I den = 1; // n^2 = 1
198 den = den + i; // n^2 = 4, 9, 16, ...
200 res = res + num / den;
201 } while (res != resbuf);
206 // calculates Li(2,x) with Xn
207 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
209 std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
210 cln::cl_N u = -cln::log(1-x);
211 cln::cl_N factor = u;
212 cln::cl_N res = u - u*u/4;
217 factor = factor * u*u / (2*i * (2*i+1));
218 res = res + (*it) * factor;
219 it++; // should we check it? or rely on initsize? ...
221 } while (res != resbuf);
226 // calculates Li(n,x), n>2 without Xn
227 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
229 cln::cl_N factor = x;
236 res = res + factor / cln::expt(cln::cl_I(i),n);
238 } while (res != resbuf);
243 // calculates Li(n,x), n>2 with Xn
244 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
246 std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
247 cln::cl_N u = -cln::log(1-x);
248 cln::cl_N factor = u;
254 factor = factor * u / i;
255 res = res + (*it) * factor;
256 it++; // should we check it? or rely on initsize? ...
258 } while (res != resbuf);
263 // forward declaration needed by function Li_projection and C below
264 numeric S_num(int n, int p, const numeric& x);
267 // helper function for classical polylog Li
268 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
270 // treat n=2 as special case
272 // check if precalculated X0 exists
277 if (cln::realpart(x) < 0.5) {
278 // choose the faster algorithm
279 // the switching point was empirically determined. the optimal point
280 // depends on hardware, Digits, ... so an approx value is okay.
281 // it solves also the problem with precision due to the u=-log(1-x) transformation
282 if (cln::abs(cln::realpart(x)) < 0.25) {
284 return Li2_do_sum(x);
286 return Li2_do_sum_Xn(x);
289 // choose the faster algorithm
290 if (cln::abs(cln::realpart(x)) > 0.75) {
291 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
293 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
297 // check if precalculated Xn exist
299 for (int i=xnsize; i<n-1; i++) {
304 if (cln::realpart(x) < 0.5) {
305 // choose the faster algorithm
306 // with n>=12 the "normal" summation always wins against the method with Xn
307 if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
308 return Lin_do_sum(n, x);
310 return Lin_do_sum_Xn(n, x);
313 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
314 for (int j=0; j<n-1; j++) {
315 result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x).to_cl_N())
316 * cln::expt(cln::log(x), j) / cln::factorial(j);
324 // helper function for classical polylog Li
325 numeric Li_num(int n, const numeric& x)
329 return -cln::log(1-x.to_cl_N());
340 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
343 // what is the desired float format?
344 // first guess: default format
345 cln::float_format_t prec = cln::default_float_format;
346 const cln::cl_N value = x.to_cl_N();
347 // second guess: the argument's format
348 if (!x.real().is_rational())
349 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
350 else if (!x.imag().is_rational())
351 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
354 if (cln::abs(value) > 1) {
355 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
356 // check if argument is complex. if it is real, the new polylog has to be conjugated.
357 if (cln::zerop(cln::imagpart(value))) {
359 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
362 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
367 result = result + Li_projection(n, cln::recip(value), prec);
370 result = result - Li_projection(n, cln::recip(value), prec);
374 for (int j=0; j<n-1; j++) {
375 add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
376 * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
378 result = result - add;
382 return Li_projection(n, value, prec);
387 } // end of anonymous namespace
390 //////////////////////////////////////////////////////////////////////
392 // Multiple polylogarithm Li(n,x)
396 //////////////////////////////////////////////////////////////////////
399 // anonymous namespace for helper function
403 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
405 const int j = s.size();
407 std::vector<cln::cl_N> t(j);
408 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
416 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
417 for (int k=j-2; k>=0; k--) {
418 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]);
420 // ... and do it again (to avoid premature drop out due to special arguments)
422 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
423 for (int k=j-2; k>=0; k--) {
424 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]);
426 } while (t[0] != t0buf);
431 // forward declaration for Li_eval()
432 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
435 } // end of anonymous namespace
438 //////////////////////////////////////////////////////////////////////
440 // Classical polylogarithm and multiple polylogarithm Li(n,x)
444 //////////////////////////////////////////////////////////////////////
447 static ex Li_evalf(const ex& x1, const ex& x2)
449 // classical polylogs
450 if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
451 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
454 else if (is_a<lst>(x1) && is_a<lst>(x2)) {
456 for (int i=0; i<x1.nops(); i++) {
457 if (!x1.op(i).info(info_flags::posint)) {
458 return Li(x1, x2).hold();
460 if (!is_a<numeric>(x2.op(i))) {
461 return Li(x1, x2).hold();
465 return Li(x1, x2).hold();
470 std::vector<cln::cl_N> x;
471 for (int i=0; i<ex_to<numeric>(x1.nops()).to_int(); i++) {
472 m.push_back(ex_to<numeric>(x1.op(i)).to_int());
473 x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
476 return numeric(multipleLi_do_sum(m, x));
479 return Li(x1,x2).hold();
483 static ex Li_eval(const ex& m_, const ex& x_)
505 return (pow(2,1-m)-1) * zeta(m);
510 if (m.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
511 return Li_num(ex_to<numeric>(m).to_int(), ex_to<numeric>(x));
517 bool doevalf = false;
518 bool doevalfveto = true;
519 const lst& m = ex_to<lst>(m_);
520 const lst& x = ex_to<lst>(x_);
521 lst::const_iterator itm = m.begin();
522 lst::const_iterator itx = x.begin();
523 for (; itm != m.end(); itm++, itx++) {
524 if (!(*itm).info(info_flags::posint)) {
525 return Li(m_, x_).hold();
527 if ((*itx != _ex1) && (*itx != _ex_1)) {
528 if (itx != x.begin()) {
536 if (!(*itx).info(info_flags::numeric)) {
539 if (!(*itx).info(info_flags::crational)) {
551 lst newm = convert_parameter_Li_to_H(m, x, pf);
552 return pf * H(newm, x[0]);
554 if (doevalfveto && doevalf) {
555 return Li(m_, x_).evalf();
558 return Li(m_, x_).hold();
562 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
565 seq.push_back(expair(Li(m, x), 0));
566 return pseries(rel, seq);
570 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
572 GINAC_ASSERT(deriv_param < 2);
573 if (deriv_param == 0) {
577 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
592 return Li(m-1, x) / x;
599 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
613 c.s << "\\mbox{Li}_{";
614 lst::const_iterator itm = m.begin();
617 for (; itm != m.end(); itm++) {
622 lst::const_iterator itx = x.begin();
625 for (; itx != x.end(); itx++) {
633 REGISTER_FUNCTION(Li,
634 evalf_func(Li_evalf).
636 series_func(Li_series).
637 derivative_func(Li_deriv).
638 print_func<print_latex>(Li_print_latex).
639 do_not_evalf_params());
642 //////////////////////////////////////////////////////////////////////
644 // Nielsen's generalized polylogarithm S(n,p,x)
648 //////////////////////////////////////////////////////////////////////
651 // anonymous namespace for helper functions
655 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
657 std::vector<std::vector<cln::cl_N> > Yn;
658 int ynsize = 0; // number of Yn[]
659 int ynlength = 100; // initial length of all Yn[i]
662 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
663 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
664 // representing S_{n,p}(x).
665 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
667 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
668 // representing S_{n,p}(x).
669 // The calculation of Y_n uses the values from Y_{n-1}.
670 void fill_Yn(int n, const cln::float_format_t& prec)
672 const int initsize = ynlength;
673 //const int initsize = initsize_Yn;
674 cln::cl_N one = cln::cl_float(1, prec);
677 std::vector<cln::cl_N> buf(initsize);
678 std::vector<cln::cl_N>::iterator it = buf.begin();
679 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
680 *it = (*itprev) / cln::cl_N(n+1) * one;
683 // sums with an index smaller than the depth are zero and need not to be calculated.
684 // calculation starts with depth, which is n+2)
685 for (int i=n+2; i<=initsize+n; i++) {
686 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
692 std::vector<cln::cl_N> buf(initsize);
693 std::vector<cln::cl_N>::iterator it = buf.begin();
696 for (int i=2; i<=initsize; i++) {
697 *it = *(it-1) + 1 / cln::cl_N(i) * one;
706 // make Yn longer ...
707 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
710 cln::cl_N one = cln::cl_float(1, prec);
712 Yn[0].resize(newsize);
713 std::vector<cln::cl_N>::iterator it = Yn[0].begin();
715 for (int i=ynlength+1; i<=newsize; i++) {
716 *it = *(it-1) + 1 / cln::cl_N(i) * one;
720 for (int n=1; n<ynsize; n++) {
721 Yn[n].resize(newsize);
722 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
723 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
726 for (int i=ynlength+n+1; i<=newsize+n; i++) {
727 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
737 // helper function for S(n,p,x)
739 cln::cl_N C(int n, int p)
743 for (int k=0; k<p; k++) {
744 for (int j=0; j<=(n+k-1)/2; j++) {
748 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
751 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
758 result = result + cln::factorial(n+k-1)
759 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
760 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
763 result = result - cln::factorial(n+k-1)
764 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
765 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
770 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()
771 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
774 result = result + cln::factorial(n+k-1)
775 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
776 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
784 if (((np)/2+n) & 1) {
785 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
788 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
796 // helper function for S(n,p,x)
797 // [Kol] remark to (9.1)
807 for (int m=2; m<=k; m++) {
808 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
815 // helper function for S(n,p,x)
816 // [Kol] remark to (9.1)
826 for (int m=2; m<=k; m++) {
827 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
834 // helper function for S(n,p,x)
835 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
838 return Li_projection(n+1, x, prec);
841 // check if precalculated values are sufficient
843 for (int i=ynsize; i<p-1; i++) {
848 // should be done otherwise
849 cln::cl_N xf = x * cln::cl_float(1, prec);
853 cln::cl_N factor = cln::expt(xf, p);
857 if (i-p >= ynlength) {
859 make_Yn_longer(ynlength*2, prec);
861 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? ...
862 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
863 factor = factor * xf;
865 } while (res != resbuf);
871 // helper function for S(n,p,x)
872 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
875 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
877 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
878 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
880 for (int s=0; s<n; s++) {
882 for (int r=0; r<p; r++) {
883 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
884 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
886 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
892 return S_do_sum(n, p, x, prec);
896 // helper function for S(n,p,x)
897 numeric S_num(int n, int p, const numeric& x)
901 // [Kol] (2.22) with (2.21)
902 return cln::zeta(p+1);
907 return cln::zeta(n+1);
912 for (int nu=0; nu<n; nu++) {
913 for (int rho=0; rho<=p; rho++) {
914 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
915 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
918 result = result * cln::expt(cln::cl_I(-1),n+p-1);
925 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
927 // throw std::runtime_error("don't know how to evaluate this function!");
930 // what is the desired float format?
931 // first guess: default format
932 cln::float_format_t prec = cln::default_float_format;
933 const cln::cl_N value = x.to_cl_N();
934 // second guess: the argument's format
935 if (!x.real().is_rational())
936 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
937 else if (!x.imag().is_rational())
938 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
942 if (cln::realpart(value) < -0.5) {
944 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
945 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
947 for (int s=0; s<n; s++) {
949 for (int r=0; r<p; r++) {
950 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
951 * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
953 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
960 if (cln::abs(value) > 1) {
964 for (int s=0; s<p; s++) {
965 for (int r=0; r<=s; r++) {
966 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
967 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
968 * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
971 result = result * cln::expt(cln::cl_I(-1),n);
974 for (int r=0; r<n; r++) {
975 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
977 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
979 result = result + cln::expt(cln::cl_I(-1),p) * res2;
984 return S_projection(n, p, value, prec);
989 } // end of anonymous namespace
992 //////////////////////////////////////////////////////////////////////
994 // Nielsen's generalized polylogarithm S(n,p,x)
998 //////////////////////////////////////////////////////////////////////
1001 static ex S_evalf(const ex& n, const ex& p, const ex& x)
1003 if (n.info(info_flags::posint) && p.info(info_flags::posint) && is_a<numeric>(x)) {
1004 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1006 return S(n, p, x).hold();
1010 static ex S_eval(const ex& n, const ex& p, const ex& x)
1012 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1018 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
1026 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
1027 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1030 return S(n, p, x).hold();
1034 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
1037 seq.push_back(expair(S(n, p, x), 0));
1038 return pseries(rel, seq);
1042 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
1044 GINAC_ASSERT(deriv_param < 3);
1045 if (deriv_param < 2) {
1049 return S(n-1, p, x) / x;
1051 return S(n, p-1, x) / (1-x);
1056 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
1058 c.s << "\\mbox{S}_{";
1068 REGISTER_FUNCTION(S,
1069 evalf_func(S_evalf).
1071 series_func(S_series).
1072 derivative_func(S_deriv).
1073 print_func<print_latex>(S_print_latex).
1074 do_not_evalf_params());
1077 //////////////////////////////////////////////////////////////////////
1079 // Harmonic polylogarithm H(m,x)
1083 //////////////////////////////////////////////////////////////////////
1086 // anonymous namespace for helper functions
1090 // convert parameters from H to Li representation
1091 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
1092 // returns true if some parameters are negative
1093 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
1095 // expand parameter list
1097 for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
1099 for (ex count=*it-1; count > 0; count--) {
1103 } else if (*it < -1) {
1104 for (ex count=*it+1; count < 0; count++) {
1115 bool has_negative_parameters = false;
1117 for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
1123 m.append((*it+acc-1) * signum);
1125 m.append((*it-acc+1) * signum);
1131 has_negative_parameters = true;
1134 if (has_negative_parameters) {
1135 for (int i=0; i<m.nops(); i++) {
1137 m.let_op(i) = -m.op(i);
1145 return has_negative_parameters;
1149 // recursivly transforms H to corresponding multiple polylogarithms
1150 struct map_trafo_H_convert_to_Li : public map_function
1152 ex operator()(const ex& e)
1154 if (is_a<add>(e) || is_a<mul>(e)) {
1155 return e.map(*this);
1157 if (is_a<function>(e)) {
1158 std::string name = ex_to<function>(e).get_name();
1161 if (is_a<lst>(e.op(0))) {
1162 parameter = ex_to<lst>(e.op(0));
1164 parameter = lst(e.op(0));
1171 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1172 s.let_op(0) = s.op(0) * arg;
1173 return pf * Li(m, s).hold();
1175 for (int i=0; i<m.nops(); i++) {
1178 s.let_op(0) = s.op(0) * arg;
1179 return Li(m, s).hold();
1188 // recursivly transforms H to corresponding zetas
1189 struct map_trafo_H_convert_to_zeta : public map_function
1191 ex operator()(const ex& e)
1193 if (is_a<add>(e) || is_a<mul>(e)) {
1194 return e.map(*this);
1196 if (is_a<function>(e)) {
1197 std::string name = ex_to<function>(e).get_name();
1200 if (is_a<lst>(e.op(0))) {
1201 parameter = ex_to<lst>(e.op(0));
1203 parameter = lst(e.op(0));
1209 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1210 return pf * zeta(m, s);
1221 // remove trailing zeros from H-parameters
1222 struct map_trafo_H_reduce_trailing_zeros : public map_function
1224 ex operator()(const ex& e)
1226 if (is_a<add>(e) || is_a<mul>(e)) {
1227 return e.map(*this);
1229 if (is_a<function>(e)) {
1230 std::string name = ex_to<function>(e).get_name();
1233 if (is_a<lst>(e.op(0))) {
1234 parameter = ex_to<lst>(e.op(0));
1236 parameter = lst(e.op(0));
1239 if (parameter.op(parameter.nops()-1) == 0) {
1242 if (parameter.nops() == 1) {
1247 lst::const_iterator it = parameter.begin();
1248 while ((it != parameter.end()) && (*it == 0)) {
1251 if (it == parameter.end()) {
1252 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
1256 parameter.remove_last();
1257 int lastentry = parameter.nops();
1258 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
1263 ex result = log(arg) * H(parameter,arg).hold();
1265 for (ex i=0; i<lastentry; i++) {
1266 if (parameter[i] > 0) {
1268 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
1271 } else if (parameter[i] < 0) {
1273 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
1281 if (lastentry < parameter.nops()) {
1282 result = result / (parameter.nops()-lastentry+1);
1283 return result.map(*this);
1295 // returns an expression with zeta functions corresponding to the parameter list for H
1296 ex convert_H_to_zeta(const lst& m)
1298 symbol xtemp("xtemp");
1299 map_trafo_H_reduce_trailing_zeros filter;
1300 map_trafo_H_convert_to_zeta filter2;
1301 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
1305 // convert signs form Li to H representation
1306 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
1309 lst::const_iterator itm = m.begin();
1310 lst::const_iterator itx = ++x.begin();
1315 while (itx != x.end()) {
1318 res.append((*itm) * signum);
1326 // multiplies an one-dimensional H with another H
1328 ex trafo_H_mult(const ex& h1, const ex& h2)
1333 ex h1nops = h1.op(0).nops();
1334 ex h2nops = h2.op(0).nops();
1336 hshort = h2.op(0).op(0);
1337 hlong = ex_to<lst>(h1.op(0));
1339 hshort = h1.op(0).op(0);
1341 hlong = ex_to<lst>(h2.op(0));
1343 hlong = h2.op(0).op(0);
1346 for (int i=0; i<=hlong.nops(); i++) {
1350 newparameter.append(hlong[j]);
1352 newparameter.append(hshort);
1353 for (; j<hlong.nops(); j++) {
1354 newparameter.append(hlong[j]);
1356 res += H(newparameter, h1.op(1)).hold();
1362 // applies trafo_H_mult recursively on expressions
1363 struct map_trafo_H_mult : public map_function
1365 ex operator()(const ex& e)
1368 return e.map(*this);
1376 for (int pos=0; pos<e.nops(); pos++) {
1377 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
1378 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
1380 for (ex i=0; i<e.op(pos).op(1); i++) {
1381 Hlst.append(e.op(pos).op(0));
1385 } else if (is_a<function>(e.op(pos))) {
1386 std::string name = ex_to<function>(e.op(pos)).get_name();
1388 if (e.op(pos).op(0).nops() > 1) {
1391 Hlst.append(e.op(pos));
1396 result *= e.op(pos);
1399 if (Hlst.nops() > 0) {
1400 firstH = Hlst[Hlst.nops()-1];
1407 if (Hlst.nops() > 0) {
1408 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
1410 for (int i=1; i<Hlst.nops(); i++) {
1411 result *= Hlst.op(i);
1413 result = result.expand();
1414 map_trafo_H_mult recursion;
1415 return recursion(result);
1426 // do integration [ReV] (55)
1427 // put parameter 0 in front of existing parameters
1428 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
1432 if (is_a<function>(e)) {
1433 name = ex_to<function>(e).get_name();
1438 for (int i=0; i<e.nops(); i++) {
1439 if (is_a<function>(e.op(i))) {
1440 std::string name = ex_to<function>(e.op(i)).get_name();
1448 lst newparameter = ex_to<lst>(h.op(0));
1449 newparameter.prepend(0);
1450 ex addzeta = convert_H_to_zeta(newparameter);
1451 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1453 return e * (-H(lst(0),1/arg).hold());
1458 // do integration [ReV] (55)
1459 // put parameter -1 in front of existing parameters
1460 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
1464 if (is_a<function>(e)) {
1465 name = ex_to<function>(e).get_name();
1470 for (int i=0; i<e.nops(); i++) {
1471 if (is_a<function>(e.op(i))) {
1472 std::string name = ex_to<function>(e.op(i)).get_name();
1480 lst newparameter = ex_to<lst>(h.op(0));
1481 newparameter.prepend(-1);
1482 ex addzeta = convert_H_to_zeta(newparameter);
1483 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1485 ex addzeta = convert_H_to_zeta(lst(-1));
1486 return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
1491 // do integration [ReV] (55)
1492 // put parameter -1 in front of existing parameters
1493 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
1497 if (is_a<function>(e)) {
1498 name = ex_to<function>(e).get_name();
1503 for (int i=0; i<e.nops(); i++) {
1504 if (is_a<function>(e.op(i))) {
1505 std::string name = ex_to<function>(e.op(i)).get_name();
1513 lst newparameter = ex_to<lst>(h.op(0));
1514 newparameter.prepend(-1);
1515 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1517 return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
1522 // do integration [ReV] (55)
1523 // put parameter 1 in front of existing parameters
1524 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
1528 if (is_a<function>(e)) {
1529 name = ex_to<function>(e).get_name();
1534 for (int i=0; i<e.nops(); i++) {
1535 if (is_a<function>(e.op(i))) {
1536 std::string name = ex_to<function>(e.op(i)).get_name();
1544 lst newparameter = ex_to<lst>(h.op(0));
1545 newparameter.prepend(1);
1546 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1548 return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
1553 // do x -> 1/x transformation
1554 struct map_trafo_H_1overx : public map_function
1556 ex operator()(const ex& e)
1558 if (is_a<add>(e) || is_a<mul>(e)) {
1559 return e.map(*this);
1562 if (is_a<function>(e)) {
1563 std::string name = ex_to<function>(e).get_name();
1566 lst parameter = ex_to<lst>(e.op(0));
1569 // special cases if all parameters are either 0, 1 or -1
1570 bool allthesame = true;
1571 if (parameter.op(0) == 0) {
1572 for (int i=1; i<parameter.nops(); i++) {
1573 if (parameter.op(i) != 0) {
1579 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
1581 } else if (parameter.op(0) == -1) {
1582 for (int i=1; i<parameter.nops(); i++) {
1583 if (parameter.op(i) != -1) {
1589 map_trafo_H_mult unify;
1590 return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
1591 / factorial(parameter.nops())).expand());
1594 for (int i=1; i<parameter.nops(); i++) {
1595 if (parameter.op(i) != 1) {
1601 map_trafo_H_mult unify;
1602 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() - I*Pi, parameter.nops())
1603 / factorial(parameter.nops())).expand());
1607 lst newparameter = parameter;
1608 newparameter.remove_first();
1610 if (parameter.op(0) == 0) {
1613 ex res = convert_H_to_zeta(parameter);
1614 map_trafo_H_1overx recursion;
1615 ex buffer = recursion(H(newparameter, arg).hold());
1616 if (is_a<add>(buffer)) {
1617 for (int i=0; i<buffer.nops(); i++) {
1618 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
1621 res += trafo_H_1tx_prepend_zero(buffer, arg);
1625 } else if (parameter.op(0) == -1) {
1627 // leading negative one
1628 ex res = convert_H_to_zeta(parameter);
1629 map_trafo_H_1overx recursion;
1630 ex buffer = recursion(H(newparameter, arg).hold());
1631 if (is_a<add>(buffer)) {
1632 for (int i=0; i<buffer.nops(); i++) {
1633 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
1636 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
1643 map_trafo_H_1overx recursion;
1644 map_trafo_H_mult unify;
1645 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1647 while (parameter.op(firstzero) == 1) {
1650 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1654 newparameter.append(parameter[j+1]);
1656 newparameter.append(1);
1657 for (; j<parameter.nops()-1; j++) {
1658 newparameter.append(parameter[j+1]);
1660 res -= H(newparameter, arg).hold();
1662 res = recursion(res).expand() / firstzero;
1674 // do x -> (1-x)/(1+x) transformation
1675 struct map_trafo_H_1mxt1px : public map_function
1677 ex operator()(const ex& e)
1679 if (is_a<add>(e) || is_a<mul>(e)) {
1680 return e.map(*this);
1683 if (is_a<function>(e)) {
1684 std::string name = ex_to<function>(e).get_name();
1687 lst parameter = ex_to<lst>(e.op(0));
1690 // special cases if all parameters are either 0, 1 or -1
1691 bool allthesame = true;
1692 if (parameter.op(0) == 0) {
1693 for (int i=1; i<parameter.nops(); i++) {
1694 if (parameter.op(i) != 0) {
1700 map_trafo_H_mult unify;
1701 return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1702 / factorial(parameter.nops())).expand());
1704 } else if (parameter.op(0) == -1) {
1705 for (int i=1; i<parameter.nops(); i++) {
1706 if (parameter.op(i) != -1) {
1712 map_trafo_H_mult unify;
1713 return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1714 / factorial(parameter.nops())).expand());
1717 for (int i=1; i<parameter.nops(); i++) {
1718 if (parameter.op(i) != 1) {
1724 map_trafo_H_mult unify;
1725 return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1726 / factorial(parameter.nops())).expand());
1730 lst newparameter = parameter;
1731 newparameter.remove_first();
1733 if (parameter.op(0) == 0) {
1736 ex res = convert_H_to_zeta(parameter);
1737 map_trafo_H_1mxt1px recursion;
1738 ex buffer = recursion(H(newparameter, arg).hold());
1739 if (is_a<add>(buffer)) {
1740 for (int i=0; i<buffer.nops(); i++) {
1741 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1744 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1748 } else if (parameter.op(0) == -1) {
1750 // leading negative one
1751 ex res = convert_H_to_zeta(parameter);
1752 map_trafo_H_1mxt1px recursion;
1753 ex buffer = recursion(H(newparameter, arg).hold());
1754 if (is_a<add>(buffer)) {
1755 for (int i=0; i<buffer.nops(); i++) {
1756 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1759 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1766 map_trafo_H_1mxt1px recursion;
1767 map_trafo_H_mult unify;
1768 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1770 while (parameter.op(firstzero) == 1) {
1773 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1777 newparameter.append(parameter[j+1]);
1779 newparameter.append(1);
1780 for (; j<parameter.nops()-1; j++) {
1781 newparameter.append(parameter[j+1]);
1783 res -= H(newparameter, arg).hold();
1785 res = recursion(res).expand() / firstzero;
1797 // do the actual summation.
1798 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
1800 const int j = m.size();
1802 std::vector<cln::cl_N> t(j);
1804 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1805 cln::cl_N factor = cln::expt(x, j) * one;
1811 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
1812 for (int k=j-2; k>=1; k--) {
1813 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
1815 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
1816 factor = factor * x;
1817 } while (t[0] != t0buf);
1823 } // end of anonymous namespace
1826 //////////////////////////////////////////////////////////////////////
1828 // Harmonic polylogarithm H(m,x)
1832 //////////////////////////////////////////////////////////////////////
1835 static ex H_evalf(const ex& x1, const ex& x2)
1837 if (is_a<lst>(x1) && is_a<numeric>(x2)) {
1838 for (int i=0; i<x1.nops(); i++) {
1839 if (!x1.op(i).info(info_flags::integer)) {
1840 return H(x1,x2).hold();
1843 if (x1.nops() < 1) {
1844 return H(x1,x2).hold();
1847 cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
1849 const lst& morg = ex_to<lst>(x1);
1850 // remove trailing zeros ...
1851 if (*(--morg.end()) == 0) {
1852 symbol xtemp("xtemp");
1853 map_trafo_H_reduce_trailing_zeros filter;
1854 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
1856 // ... and expand parameter notation
1858 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
1860 for (ex count=*it-1; count > 0; count--) {
1864 } else if (*it < -1) {
1865 for (ex count=*it+1; count < 0; count++) {
1874 // since the transformations produce a lot of terms, they are only efficient for
1875 // argument near one.
1876 // no transformation needed -> do summation
1877 if (cln::abs(x) < 0.95) {
1881 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
1882 // negative parameters -> s_lst is filled
1883 std::vector<int> m_int;
1884 std::vector<cln::cl_N> x_cln;
1885 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
1886 it_int != m_lst.end(); it_int++, it_cln++) {
1887 m_int.push_back(ex_to<numeric>(*it_int).to_int());
1888 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
1890 x_cln.front() = x_cln.front() * x;
1891 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
1893 // only positive parameters
1895 if (m_lst.nops() == 1) {
1896 return Li(m_lst.op(0), x2).evalf();
1898 std::vector<int> m_int;
1899 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
1900 m_int.push_back(ex_to<numeric>(*it).to_int());
1902 return numeric(H_do_sum(m_int, x));
1908 // ensure that the realpart of the argument is positive
1909 if (cln::realpart(x) < 0) {
1911 for (int i=0; i<m.nops(); i++) {
1913 m.let_op(i) = -m.op(i);
1919 // choose transformations
1920 symbol xtemp("xtemp");
1921 if (cln::abs(x-1) < 1.4142) {
1923 map_trafo_H_1mxt1px trafo;
1924 res *= trafo(H(m, xtemp));
1927 map_trafo_H_1overx trafo;
1928 res *= trafo(H(m, xtemp));
1933 // map_trafo_H_convert converter;
1934 // res = converter(res).expand();
1936 // res.find(H(wild(1),wild(2)), ll);
1937 // res.find(zeta(wild(1)), ll);
1938 // res.find(zeta(wild(1),wild(2)), ll);
1939 // res = res.collect(ll);
1941 return res.subs(xtemp == numeric(x)).evalf();
1944 return H(x1,x2).hold();
1948 static ex H_eval(const ex& m_, const ex& x)
1951 if (is_a<lst>(m_)) {
1956 if (m.nops() == 0) {
1964 if (*m.begin() > _ex1) {
1970 } else if (*m.begin() < _ex_1) {
1976 } else if (*m.begin() == _ex0) {
1983 for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
1984 if ((*it).info(info_flags::integer)) {
1995 } else if (*it < _ex_1) {
2015 } else if (step == 1) {
2027 // if some m_i is not an integer
2028 return H(m_, x).hold();
2031 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
2032 return convert_H_to_zeta(m);
2038 return H(m_, x).hold();
2040 return pow(log(x), m.nops()) / factorial(m.nops());
2043 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
2045 } else if ((step == 1) && (pos1 == _ex0)){
2050 return pow(-1, p) * S(n, p, -x);
2056 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2057 return H(m_, x).evalf();
2059 return H(m_, x).hold();
2063 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
2066 seq.push_back(expair(H(m, x), 0));
2067 return pseries(rel, seq);
2071 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
2073 GINAC_ASSERT(deriv_param < 2);
2074 if (deriv_param == 0) {
2078 if (is_a<lst>(m_)) {
2094 return 1/(1-x) * H(m, x);
2095 } else if (mb == _ex_1) {
2096 return 1/(1+x) * H(m, x);
2103 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
2106 if (is_a<lst>(m_)) {
2111 c.s << "\\mbox{H}_{";
2112 lst::const_iterator itm = m.begin();
2115 for (; itm != m.end(); itm++) {
2125 REGISTER_FUNCTION(H,
2126 evalf_func(H_evalf).
2128 series_func(H_series).
2129 derivative_func(H_deriv).
2130 print_func<print_latex>(H_print_latex).
2131 do_not_evalf_params());
2134 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
2135 ex convert_H_to_Li(const ex& m, const ex& x)
2137 map_trafo_H_reduce_trailing_zeros filter;
2138 map_trafo_H_convert_to_Li filter2;
2140 return filter2(filter(H(m, x).hold()));
2142 return filter2(filter(H(lst(m), x).hold()));
2147 //////////////////////////////////////////////////////////////////////
2149 // Multiple zeta values zeta(x) and zeta(x,s)
2153 //////////////////////////////////////////////////////////////////////
2156 // anonymous namespace for helper functions
2160 // parameters and data for [Cra] algorithm
2161 const cln::cl_N lambda = cln::cl_N("319/320");
2164 std::vector<std::vector<cln::cl_N> > f_kj;
2165 std::vector<cln::cl_N> crB;
2166 std::vector<std::vector<cln::cl_N> > crG;
2167 std::vector<cln::cl_N> crX;
2170 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
2172 const int size = a.size();
2173 for (int n=0; n<size; n++) {
2175 for (int m=0; m<=n; m++) {
2176 c[n] = c[n] + a[m]*b[n-m];
2183 void initcX(const std::vector<int>& s)
2185 const int k = s.size();
2191 for (int i=0; i<=L2; i++) {
2192 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
2197 for (int m=0; m<k-1; m++) {
2198 std::vector<cln::cl_N> crGbuf;
2201 for (int i=0; i<=L2; i++) {
2202 crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
2204 crG.push_back(crGbuf);
2209 for (int m=0; m<k-1; m++) {
2210 std::vector<cln::cl_N> Xbuf;
2211 for (int i=0; i<=L2; i++) {
2212 Xbuf.push_back(crX[i] * crG[m][i]);
2214 halfcyclic_convolute(Xbuf, crB, crX);
2220 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
2222 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2223 cln::cl_N factor = cln::expt(lambda, Sqk);
2224 cln::cl_N res = factor / Sqk * crX[0] * one;
2229 factor = factor * lambda;
2231 res = res + crX[N] * factor / (N+Sqk);
2232 } while ((res != resbuf) || cln::zerop(crX[N]));
2238 void calc_f(int maxr)
2243 cln::cl_N t0, t1, t2, t3, t4;
2245 std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
2246 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2248 t0 = cln::exp(-lambda);
2250 for (k=1; k<=L1; k++) {
2253 for (j=1; j<=maxr; j++) {
2256 for (i=2; i<=j; i++) {
2260 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
2268 cln::cl_N crandall_Z(const std::vector<int>& s)
2270 const int j = s.size();
2279 t0 = t0 + f_kj[q+j-2][s[0]-1];
2280 } while (t0 != t0buf);
2282 return t0 / cln::factorial(s[0]-1);
2285 std::vector<cln::cl_N> t(j);
2292 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
2293 for (int k=j-2; k>=1; k--) {
2294 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
2296 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
2297 } while (t[0] != t0buf);
2299 return t[0] / cln::factorial(s[0]-1);
2304 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
2306 std::vector<int> r = s;
2307 const int j = r.size();
2309 // decide on maximal size of f_kj for crandall_Z
2313 L1 = Digits * 3 + j*2;
2316 // decide on maximal size of crX for crandall_Y
2319 } else if (Digits < 86) {
2321 } else if (Digits < 192) {
2323 } else if (Digits < 394) {
2325 } else if (Digits < 808) {
2335 for (int i=0; i<j; i++) {
2344 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
2346 std::vector<int> rz;
2349 for (int k=r.size()-1; k>0; k--) {
2351 rz.insert(rz.begin(), r.back());
2352 skp1buf = rz.front();
2358 for (int q=0; q<skp1buf; q++) {
2360 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
2361 cln::cl_N pp2 = crandall_Z(rz);
2366 res = res - pp1 * pp2 / cln::factorial(q);
2368 res = res + pp1 * pp2 / cln::factorial(q);
2371 rz.front() = skp1buf;
2373 rz.insert(rz.begin(), r.back());
2377 res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
2383 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
2385 const int j = r.size();
2387 // buffer for subsums
2388 std::vector<cln::cl_N> t(j);
2389 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2396 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
2397 for (int k=j-2; k>=0; k--) {
2398 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
2400 } while (t[0] != t0buf);
2406 // does Hoelder convolution. see [BBB] (7.0)
2407 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
2409 // prepare parameters
2410 // holds Li arguments in [BBB] notation
2411 std::vector<int> s = s_;
2412 std::vector<int> m_p = m_;
2413 std::vector<int> m_q;
2414 // holds Li arguments in nested sums notation
2415 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
2416 s_p[0] = s_p[0] * cln::cl_N("1/2");
2417 // convert notations
2419 for (int i=0; i<s_.size(); i++) {
2424 s[i] = sig * std::abs(s[i]);
2426 std::vector<cln::cl_N> s_q;
2427 cln::cl_N signum = 1;
2430 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
2435 // change parameters
2436 if (s.front() > 0) {
2437 if (m_p.front() == 1) {
2438 m_p.erase(m_p.begin());
2439 s_p.erase(s_p.begin());
2440 if (s_p.size() > 0) {
2441 s_p.front() = s_p.front() * cln::cl_N("1/2");
2447 m_q.insert(m_q.begin(), 1);
2448 if (s_q.size() > 0) {
2449 s_q.front() = s_q.front() * 2;
2451 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2454 if (m_p.front() == 1) {
2455 m_p.erase(m_p.begin());
2456 cln::cl_N spbuf = s_p.front();
2457 s_p.erase(s_p.begin());
2458 if (s_p.size() > 0) {
2459 s_p.front() = s_p.front() * spbuf;
2462 m_q.insert(m_q.begin(), 1);
2463 if (s_q.size() > 0) {
2464 s_q.front() = s_q.front() * 4;
2466 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
2470 m_q.insert(m_q.begin(), 1);
2471 if (s_q.size() > 0) {
2472 s_q.front() = s_q.front() * 2;
2474 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2479 if (m_p.size() == 0) break;
2481 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
2486 res = res + signum * multipleLi_do_sum(m_q, s_q);
2492 } // end of anonymous namespace
2495 //////////////////////////////////////////////////////////////////////
2497 // Multiple zeta values zeta(x)
2501 //////////////////////////////////////////////////////////////////////
2504 static ex zeta1_evalf(const ex& x)
2506 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
2508 // multiple zeta value
2509 const int count = x.nops();
2510 const lst& xlst = ex_to<lst>(x);
2511 std::vector<int> r(count);
2513 // check parameters and convert them
2514 lst::const_iterator it1 = xlst.begin();
2515 std::vector<int>::iterator it2 = r.begin();
2517 if (!(*it1).info(info_flags::posint)) {
2518 return zeta(x).hold();
2520 *it2 = ex_to<numeric>(*it1).to_int();
2523 } while (it2 != r.end());
2525 // check for divergence
2527 return zeta(x).hold();
2530 // decide on summation algorithm
2531 // this is still a bit clumsy
2532 int limit = (Digits>17) ? 10 : 6;
2533 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
2534 return numeric(zeta_do_sum_Crandall(r));
2536 return numeric(zeta_do_sum_simple(r));
2540 // single zeta value
2541 if (is_exactly_a<numeric>(x) && (x != 1)) {
2543 return zeta(ex_to<numeric>(x));
2544 } catch (const dunno &e) { }
2547 return zeta(x).hold();
2551 static ex zeta1_eval(const ex& m)
2553 if (is_exactly_a<lst>(m)) {
2554 if (m.nops() == 1) {
2555 return zeta(m.op(0));
2557 return zeta(m).hold();
2560 if (m.info(info_flags::numeric)) {
2561 const numeric& y = ex_to<numeric>(m);
2562 // trap integer arguments:
2563 if (y.is_integer()) {
2567 if (y.is_equal(_num1)) {
2568 return zeta(m).hold();
2570 if (y.info(info_flags::posint)) {
2571 if (y.info(info_flags::odd)) {
2572 return zeta(m).hold();
2574 return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
2577 if (y.info(info_flags::odd)) {
2578 return -bernoulli(_num1-y) / (_num1-y);
2585 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
2586 return zeta1_evalf(m);
2589 return zeta(m).hold();
2593 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
2595 GINAC_ASSERT(deriv_param==0);
2597 if (is_exactly_a<lst>(m)) {
2600 return zetaderiv(_ex1, m);
2605 static void zeta1_print_latex(const ex& m_, const print_context& c)
2608 if (is_a<lst>(m_)) {
2609 const lst& m = ex_to<lst>(m_);
2610 lst::const_iterator it = m.begin();
2613 for (; it != m.end(); it++) {
2624 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta").
2625 evalf_func(zeta1_evalf).
2626 eval_func(zeta1_eval).
2627 derivative_func(zeta1_deriv).
2628 print_func<print_latex>(zeta1_print_latex).
2629 do_not_evalf_params().
2633 //////////////////////////////////////////////////////////////////////
2635 // Alternating Euler sum zeta(x,s)
2639 //////////////////////////////////////////////////////////////////////
2642 static ex zeta2_evalf(const ex& x, const ex& s)
2644 if (is_exactly_a<lst>(x)) {
2646 // alternating Euler sum
2647 const int count = x.nops();
2648 const lst& xlst = ex_to<lst>(x);
2649 const lst& slst = ex_to<lst>(s);
2650 std::vector<int> xi(count);
2651 std::vector<int> si(count);
2653 // check parameters and convert them
2654 lst::const_iterator it_xread = xlst.begin();
2655 lst::const_iterator it_sread = slst.begin();
2656 std::vector<int>::iterator it_xwrite = xi.begin();
2657 std::vector<int>::iterator it_swrite = si.begin();
2659 if (!(*it_xread).info(info_flags::posint)) {
2660 return zeta(x, s).hold();
2662 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
2663 if (*it_sread > 0) {
2672 } while (it_xwrite != xi.end());
2674 // check for divergence
2675 if ((xi[0] == 1) && (si[0] == 1)) {
2676 return zeta(x, s).hold();
2679 // use Hoelder convolution
2680 return numeric(zeta_do_Hoelder_convolution(xi, si));
2683 return zeta(x, s).hold();
2687 static ex zeta2_eval(const ex& m, const ex& s_)
2689 if (is_exactly_a<lst>(s_)) {
2690 const lst& s = ex_to<lst>(s_);
2691 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
2692 if ((*it).info(info_flags::positive)) {
2695 return zeta(m, s_).hold();
2698 } else if (s_.info(info_flags::positive)) {
2702 return zeta(m, s_).hold();
2706 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
2708 GINAC_ASSERT(deriv_param==0);
2710 if (is_exactly_a<lst>(m)) {
2713 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
2714 return zetaderiv(_ex1, m);
2721 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
2724 if (is_a<lst>(m_)) {
2730 if (is_a<lst>(s_)) {
2736 lst::const_iterator itm = m.begin();
2737 lst::const_iterator its = s.begin();
2739 c.s << "\\overline{";
2747 for (; itm != m.end(); itm++, its++) {
2750 c.s << "\\overline{";
2761 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta").
2762 evalf_func(zeta2_evalf).
2763 eval_func(zeta2_eval).
2764 derivative_func(zeta2_deriv).
2765 print_func<print_latex>(zeta2_print_latex).
2766 do_not_evalf_params().
2770 } // namespace GiNaC