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) 1999-2004 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;
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)
192 cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
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 * cln::cl_float(1, cln::float_format(Digits));
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 * cln::cl_float(1, cln::float_format(Digits));
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 * cln::cl_float(1, cln::float_format(Digits));
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();
464 if (abs(conv) >= 1) {
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_F one = cln::cl_float(1, cln::float_format(Digits));
850 cln::cl_N xf = x * one;
851 //cln::cl_N xf = x * cln::cl_float(1, prec);
855 cln::cl_N factor = cln::expt(xf, p);
859 if (i-p >= ynlength) {
861 make_Yn_longer(ynlength*2, prec);
863 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? ...
864 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
865 factor = factor * xf;
867 } while (res != resbuf);
873 // helper function for S(n,p,x)
874 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
877 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
879 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
880 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
882 for (int s=0; s<n; s++) {
884 for (int r=0; r<p; r++) {
885 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
886 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
888 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
894 return S_do_sum(n, p, x, prec);
898 // helper function for S(n,p,x)
899 numeric S_num(int n, int p, const numeric& x)
903 // [Kol] (2.22) with (2.21)
904 return cln::zeta(p+1);
909 return cln::zeta(n+1);
914 for (int nu=0; nu<n; nu++) {
915 for (int rho=0; rho<=p; rho++) {
916 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
917 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
920 result = result * cln::expt(cln::cl_I(-1),n+p-1);
927 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
929 // throw std::runtime_error("don't know how to evaluate this function!");
932 // what is the desired float format?
933 // first guess: default format
934 cln::float_format_t prec = cln::default_float_format;
935 const cln::cl_N value = x.to_cl_N();
936 // second guess: the argument's format
937 if (!x.real().is_rational())
938 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
939 else if (!x.imag().is_rational())
940 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
944 if (cln::realpart(value) < -0.5) {
946 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
947 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
949 for (int s=0; s<n; s++) {
951 for (int r=0; r<p; r++) {
952 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
953 * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
955 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
962 if (cln::abs(value) > 1) {
966 for (int s=0; s<p; s++) {
967 for (int r=0; r<=s; r++) {
968 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
969 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
970 * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
973 result = result * cln::expt(cln::cl_I(-1),n);
976 for (int r=0; r<n; r++) {
977 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
979 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
981 result = result + cln::expt(cln::cl_I(-1),p) * res2;
986 return S_projection(n, p, value, prec);
991 } // end of anonymous namespace
994 //////////////////////////////////////////////////////////////////////
996 // Nielsen's generalized polylogarithm S(n,p,x)
1000 //////////////////////////////////////////////////////////////////////
1003 static ex S_evalf(const ex& n, const ex& p, const ex& x)
1005 if (n.info(info_flags::posint) && p.info(info_flags::posint) && is_a<numeric>(x)) {
1006 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1008 return S(n, p, x).hold();
1012 static ex S_eval(const ex& n, const ex& p, const ex& x)
1014 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1020 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
1028 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
1029 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1032 return S(n, p, x).hold();
1036 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
1039 seq.push_back(expair(S(n, p, x), 0));
1040 return pseries(rel, seq);
1044 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
1046 GINAC_ASSERT(deriv_param < 3);
1047 if (deriv_param < 2) {
1051 return S(n-1, p, x) / x;
1053 return S(n, p-1, x) / (1-x);
1058 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
1060 c.s << "\\mbox{S}_{";
1070 REGISTER_FUNCTION(S,
1071 evalf_func(S_evalf).
1073 series_func(S_series).
1074 derivative_func(S_deriv).
1075 print_func<print_latex>(S_print_latex).
1076 do_not_evalf_params());
1079 //////////////////////////////////////////////////////////////////////
1081 // Harmonic polylogarithm H(m,x)
1085 //////////////////////////////////////////////////////////////////////
1088 // anonymous namespace for helper functions
1092 // regulates the pole (used by 1/x-transformation)
1093 symbol H_polesign("IMSIGN");
1096 // convert parameters from H to Li representation
1097 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
1098 // returns true if some parameters are negative
1099 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
1101 // expand parameter list
1103 for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
1105 for (ex count=*it-1; count > 0; count--) {
1109 } else if (*it < -1) {
1110 for (ex count=*it+1; count < 0; count++) {
1121 bool has_negative_parameters = false;
1123 for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
1129 m.append((*it+acc-1) * signum);
1131 m.append((*it-acc+1) * signum);
1137 has_negative_parameters = true;
1140 if (has_negative_parameters) {
1141 for (int i=0; i<m.nops(); i++) {
1143 m.let_op(i) = -m.op(i);
1151 return has_negative_parameters;
1155 // recursivly transforms H to corresponding multiple polylogarithms
1156 struct map_trafo_H_convert_to_Li : public map_function
1158 ex operator()(const ex& e)
1160 if (is_a<add>(e) || is_a<mul>(e)) {
1161 return e.map(*this);
1163 if (is_a<function>(e)) {
1164 std::string name = ex_to<function>(e).get_name();
1167 if (is_a<lst>(e.op(0))) {
1168 parameter = ex_to<lst>(e.op(0));
1170 parameter = lst(e.op(0));
1177 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1178 s.let_op(0) = s.op(0) * arg;
1179 return pf * Li(m, s).hold();
1181 for (int i=0; i<m.nops(); i++) {
1184 s.let_op(0) = s.op(0) * arg;
1185 return Li(m, s).hold();
1194 // recursivly transforms H to corresponding zetas
1195 struct map_trafo_H_convert_to_zeta : public map_function
1197 ex operator()(const ex& e)
1199 if (is_a<add>(e) || is_a<mul>(e)) {
1200 return e.map(*this);
1202 if (is_a<function>(e)) {
1203 std::string name = ex_to<function>(e).get_name();
1206 if (is_a<lst>(e.op(0))) {
1207 parameter = ex_to<lst>(e.op(0));
1209 parameter = lst(e.op(0));
1215 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1216 return pf * zeta(m, s);
1227 // remove trailing zeros from H-parameters
1228 struct map_trafo_H_reduce_trailing_zeros : public map_function
1230 ex operator()(const ex& e)
1232 if (is_a<add>(e) || is_a<mul>(e)) {
1233 return e.map(*this);
1235 if (is_a<function>(e)) {
1236 std::string name = ex_to<function>(e).get_name();
1239 if (is_a<lst>(e.op(0))) {
1240 parameter = ex_to<lst>(e.op(0));
1242 parameter = lst(e.op(0));
1245 if (parameter.op(parameter.nops()-1) == 0) {
1248 if (parameter.nops() == 1) {
1253 lst::const_iterator it = parameter.begin();
1254 while ((it != parameter.end()) && (*it == 0)) {
1257 if (it == parameter.end()) {
1258 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
1262 parameter.remove_last();
1263 int lastentry = parameter.nops();
1264 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
1269 ex result = log(arg) * H(parameter,arg).hold();
1271 for (ex i=0; i<lastentry; i++) {
1272 if (parameter[i] > 0) {
1274 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
1277 } else if (parameter[i] < 0) {
1279 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
1287 if (lastentry < parameter.nops()) {
1288 result = result / (parameter.nops()-lastentry+1);
1289 return result.map(*this);
1301 // returns an expression with zeta functions corresponding to the parameter list for H
1302 ex convert_H_to_zeta(const lst& m)
1304 symbol xtemp("xtemp");
1305 map_trafo_H_reduce_trailing_zeros filter;
1306 map_trafo_H_convert_to_zeta filter2;
1307 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
1311 // convert signs form Li to H representation
1312 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
1315 lst::const_iterator itm = m.begin();
1316 lst::const_iterator itx = ++x.begin();
1321 while (itx != x.end()) {
1324 res.append((*itm) * signum);
1332 // multiplies an one-dimensional H with another H
1334 ex trafo_H_mult(const ex& h1, const ex& h2)
1339 ex h1nops = h1.op(0).nops();
1340 ex h2nops = h2.op(0).nops();
1342 hshort = h2.op(0).op(0);
1343 hlong = ex_to<lst>(h1.op(0));
1345 hshort = h1.op(0).op(0);
1347 hlong = ex_to<lst>(h2.op(0));
1349 hlong = h2.op(0).op(0);
1352 for (int i=0; i<=hlong.nops(); i++) {
1356 newparameter.append(hlong[j]);
1358 newparameter.append(hshort);
1359 for (; j<hlong.nops(); j++) {
1360 newparameter.append(hlong[j]);
1362 res += H(newparameter, h1.op(1)).hold();
1368 // applies trafo_H_mult recursively on expressions
1369 struct map_trafo_H_mult : public map_function
1371 ex operator()(const ex& e)
1374 return e.map(*this);
1382 for (int pos=0; pos<e.nops(); pos++) {
1383 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
1384 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
1386 for (ex i=0; i<e.op(pos).op(1); i++) {
1387 Hlst.append(e.op(pos).op(0));
1391 } else if (is_a<function>(e.op(pos))) {
1392 std::string name = ex_to<function>(e.op(pos)).get_name();
1394 if (e.op(pos).op(0).nops() > 1) {
1397 Hlst.append(e.op(pos));
1402 result *= e.op(pos);
1405 if (Hlst.nops() > 0) {
1406 firstH = Hlst[Hlst.nops()-1];
1413 if (Hlst.nops() > 0) {
1414 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
1416 for (int i=1; i<Hlst.nops(); i++) {
1417 result *= Hlst.op(i);
1419 result = result.expand();
1420 map_trafo_H_mult recursion;
1421 return recursion(result);
1432 // do integration [ReV] (55)
1433 // put parameter 0 in front of existing parameters
1434 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
1438 if (is_a<function>(e)) {
1439 name = ex_to<function>(e).get_name();
1444 for (int i=0; i<e.nops(); i++) {
1445 if (is_a<function>(e.op(i))) {
1446 std::string name = ex_to<function>(e.op(i)).get_name();
1454 lst newparameter = ex_to<lst>(h.op(0));
1455 newparameter.prepend(0);
1456 ex addzeta = convert_H_to_zeta(newparameter);
1457 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1459 return e * (-H(lst(0),1/arg).hold());
1464 // do integration [ReV] (55)
1465 // put parameter -1 in front of existing parameters
1466 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
1470 if (is_a<function>(e)) {
1471 name = ex_to<function>(e).get_name();
1476 for (int i=0; i<e.nops(); i++) {
1477 if (is_a<function>(e.op(i))) {
1478 std::string name = ex_to<function>(e.op(i)).get_name();
1486 lst newparameter = ex_to<lst>(h.op(0));
1487 newparameter.prepend(-1);
1488 ex addzeta = convert_H_to_zeta(newparameter);
1489 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1491 ex addzeta = convert_H_to_zeta(lst(-1));
1492 return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
1497 // do integration [ReV] (55)
1498 // put parameter -1 in front of existing parameters
1499 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
1503 if (is_a<function>(e)) {
1504 name = ex_to<function>(e).get_name();
1509 for (int i=0; i<e.nops(); i++) {
1510 if (is_a<function>(e.op(i))) {
1511 std::string name = ex_to<function>(e.op(i)).get_name();
1519 lst newparameter = ex_to<lst>(h.op(0));
1520 newparameter.prepend(-1);
1521 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1523 return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
1528 // do integration [ReV] (55)
1529 // put parameter 1 in front of existing parameters
1530 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
1534 if (is_a<function>(e)) {
1535 name = ex_to<function>(e).get_name();
1540 for (int i=0; i<e.nops(); i++) {
1541 if (is_a<function>(e.op(i))) {
1542 std::string name = ex_to<function>(e.op(i)).get_name();
1550 lst newparameter = ex_to<lst>(h.op(0));
1551 newparameter.prepend(1);
1552 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1554 return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
1559 // do x -> 1/x transformation
1560 struct map_trafo_H_1overx : public map_function
1562 ex operator()(const ex& e)
1564 if (is_a<add>(e) || is_a<mul>(e)) {
1565 return e.map(*this);
1568 if (is_a<function>(e)) {
1569 std::string name = ex_to<function>(e).get_name();
1572 lst parameter = ex_to<lst>(e.op(0));
1575 // special cases if all parameters are either 0, 1 or -1
1576 bool allthesame = true;
1577 if (parameter.op(0) == 0) {
1578 for (int i=1; i<parameter.nops(); i++) {
1579 if (parameter.op(i) != 0) {
1585 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
1587 } else if (parameter.op(0) == -1) {
1588 for (int i=1; i<parameter.nops(); i++) {
1589 if (parameter.op(i) != -1) {
1595 map_trafo_H_mult unify;
1596 return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
1597 / factorial(parameter.nops())).expand());
1600 for (int i=1; i<parameter.nops(); i++) {
1601 if (parameter.op(i) != 1) {
1607 map_trafo_H_mult unify;
1608 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() + H_polesign, parameter.nops())
1609 / factorial(parameter.nops())).expand());
1613 lst newparameter = parameter;
1614 newparameter.remove_first();
1616 if (parameter.op(0) == 0) {
1619 ex res = convert_H_to_zeta(parameter);
1620 map_trafo_H_1overx recursion;
1621 ex buffer = recursion(H(newparameter, arg).hold());
1622 if (is_a<add>(buffer)) {
1623 for (int i=0; i<buffer.nops(); i++) {
1624 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
1627 res += trafo_H_1tx_prepend_zero(buffer, arg);
1631 } else if (parameter.op(0) == -1) {
1633 // leading negative one
1634 ex res = convert_H_to_zeta(parameter);
1635 map_trafo_H_1overx recursion;
1636 ex buffer = recursion(H(newparameter, arg).hold());
1637 if (is_a<add>(buffer)) {
1638 for (int i=0; i<buffer.nops(); i++) {
1639 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
1642 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
1649 map_trafo_H_1overx recursion;
1650 map_trafo_H_mult unify;
1651 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1653 while (parameter.op(firstzero) == 1) {
1656 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1660 newparameter.append(parameter[j+1]);
1662 newparameter.append(1);
1663 for (; j<parameter.nops()-1; j++) {
1664 newparameter.append(parameter[j+1]);
1666 res -= H(newparameter, arg).hold();
1668 res = recursion(res).expand() / firstzero;
1680 // do x -> (1-x)/(1+x) transformation
1681 struct map_trafo_H_1mxt1px : public map_function
1683 ex operator()(const ex& e)
1685 if (is_a<add>(e) || is_a<mul>(e)) {
1686 return e.map(*this);
1689 if (is_a<function>(e)) {
1690 std::string name = ex_to<function>(e).get_name();
1693 lst parameter = ex_to<lst>(e.op(0));
1696 // special cases if all parameters are either 0, 1 or -1
1697 bool allthesame = true;
1698 if (parameter.op(0) == 0) {
1699 for (int i=1; i<parameter.nops(); i++) {
1700 if (parameter.op(i) != 0) {
1706 map_trafo_H_mult unify;
1707 return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1708 / factorial(parameter.nops())).expand());
1710 } else if (parameter.op(0) == -1) {
1711 for (int i=1; i<parameter.nops(); i++) {
1712 if (parameter.op(i) != -1) {
1718 map_trafo_H_mult unify;
1719 return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1720 / factorial(parameter.nops())).expand());
1723 for (int i=1; i<parameter.nops(); i++) {
1724 if (parameter.op(i) != 1) {
1730 map_trafo_H_mult unify;
1731 return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1732 / factorial(parameter.nops())).expand());
1736 lst newparameter = parameter;
1737 newparameter.remove_first();
1739 if (parameter.op(0) == 0) {
1742 ex res = convert_H_to_zeta(parameter);
1743 map_trafo_H_1mxt1px recursion;
1744 ex buffer = recursion(H(newparameter, arg).hold());
1745 if (is_a<add>(buffer)) {
1746 for (int i=0; i<buffer.nops(); i++) {
1747 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1750 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1754 } else if (parameter.op(0) == -1) {
1756 // leading negative one
1757 ex res = convert_H_to_zeta(parameter);
1758 map_trafo_H_1mxt1px recursion;
1759 ex buffer = recursion(H(newparameter, arg).hold());
1760 if (is_a<add>(buffer)) {
1761 for (int i=0; i<buffer.nops(); i++) {
1762 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1765 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1772 map_trafo_H_1mxt1px recursion;
1773 map_trafo_H_mult unify;
1774 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1776 while (parameter.op(firstzero) == 1) {
1779 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1783 newparameter.append(parameter[j+1]);
1785 newparameter.append(1);
1786 for (; j<parameter.nops()-1; j++) {
1787 newparameter.append(parameter[j+1]);
1789 res -= H(newparameter, arg).hold();
1791 res = recursion(res).expand() / firstzero;
1803 // do the actual summation.
1804 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
1806 const int j = m.size();
1808 std::vector<cln::cl_N> t(j);
1810 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1811 cln::cl_N factor = cln::expt(x, j) * one;
1817 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
1818 for (int k=j-2; k>=1; k--) {
1819 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
1821 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
1822 factor = factor * x;
1823 } while (t[0] != t0buf);
1829 } // end of anonymous namespace
1832 //////////////////////////////////////////////////////////////////////
1834 // Harmonic polylogarithm H(m,x)
1838 //////////////////////////////////////////////////////////////////////
1841 static ex H_evalf(const ex& x1, const ex& x2)
1843 if (is_a<lst>(x1) && is_a<numeric>(x2)) {
1844 for (int i=0; i<x1.nops(); i++) {
1845 if (!x1.op(i).info(info_flags::integer)) {
1846 return H(x1,x2).hold();
1849 if (x1.nops() < 1) {
1850 return H(x1,x2).hold();
1853 cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
1855 const lst& morg = ex_to<lst>(x1);
1856 // remove trailing zeros ...
1857 if (*(--morg.end()) == 0) {
1858 symbol xtemp("xtemp");
1859 map_trafo_H_reduce_trailing_zeros filter;
1860 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
1862 // ... and expand parameter notation
1864 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
1866 for (ex count=*it-1; count > 0; count--) {
1870 } else if (*it < -1) {
1871 for (ex count=*it+1; count < 0; count++) {
1880 // since the transformations produce a lot of terms, they are only efficient for
1881 // argument near one.
1882 // no transformation needed -> do summation
1883 if (cln::abs(x) < 0.95) {
1887 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
1888 // negative parameters -> s_lst is filled
1889 std::vector<int> m_int;
1890 std::vector<cln::cl_N> x_cln;
1891 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
1892 it_int != m_lst.end(); it_int++, it_cln++) {
1893 m_int.push_back(ex_to<numeric>(*it_int).to_int());
1894 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
1896 x_cln.front() = x_cln.front() * x;
1897 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
1899 // only positive parameters
1901 if (m_lst.nops() == 1) {
1902 return Li(m_lst.op(0), x2).evalf();
1904 std::vector<int> m_int;
1905 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
1906 m_int.push_back(ex_to<numeric>(*it).to_int());
1908 return numeric(H_do_sum(m_int, x));
1914 // ensure that the realpart of the argument is positive
1915 if (cln::realpart(x) < 0) {
1917 for (int i=0; i<m.nops(); i++) {
1919 m.let_op(i) = -m.op(i);
1925 // choose transformations
1926 symbol xtemp("xtemp");
1927 if (cln::abs(x-1) < 1.4142) {
1929 map_trafo_H_1mxt1px trafo;
1930 res *= trafo(H(m, xtemp));
1933 map_trafo_H_1overx trafo;
1934 res *= trafo(H(m, xtemp));
1935 if (cln::imagpart(x) <= 0) {
1936 res = res.subs(H_polesign == -I*Pi);
1938 res = res.subs(H_polesign == I*Pi);
1944 // map_trafo_H_convert converter;
1945 // res = converter(res).expand();
1947 // res.find(H(wild(1),wild(2)), ll);
1948 // res.find(zeta(wild(1)), ll);
1949 // res.find(zeta(wild(1),wild(2)), ll);
1950 // res = res.collect(ll);
1952 return res.subs(xtemp == numeric(x)).evalf();
1955 return H(x1,x2).hold();
1959 static ex H_eval(const ex& m_, const ex& x)
1962 if (is_a<lst>(m_)) {
1967 if (m.nops() == 0) {
1975 if (*m.begin() > _ex1) {
1981 } else if (*m.begin() < _ex_1) {
1987 } else if (*m.begin() == _ex0) {
1994 for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
1995 if ((*it).info(info_flags::integer)) {
2006 } else if (*it < _ex_1) {
2026 } else if (step == 1) {
2038 // if some m_i is not an integer
2039 return H(m_, x).hold();
2042 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
2043 return convert_H_to_zeta(m);
2049 return H(m_, x).hold();
2051 return pow(log(x), m.nops()) / factorial(m.nops());
2054 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
2056 } else if ((step == 1) && (pos1 == _ex0)){
2061 return pow(-1, p) * S(n, p, -x);
2067 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2068 return H(m_, x).evalf();
2070 return H(m_, x).hold();
2074 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
2077 seq.push_back(expair(H(m, x), 0));
2078 return pseries(rel, seq);
2082 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
2084 GINAC_ASSERT(deriv_param < 2);
2085 if (deriv_param == 0) {
2089 if (is_a<lst>(m_)) {
2105 return 1/(1-x) * H(m, x);
2106 } else if (mb == _ex_1) {
2107 return 1/(1+x) * H(m, x);
2114 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
2117 if (is_a<lst>(m_)) {
2122 c.s << "\\mbox{H}_{";
2123 lst::const_iterator itm = m.begin();
2126 for (; itm != m.end(); itm++) {
2136 REGISTER_FUNCTION(H,
2137 evalf_func(H_evalf).
2139 series_func(H_series).
2140 derivative_func(H_deriv).
2141 print_func<print_latex>(H_print_latex).
2142 do_not_evalf_params());
2145 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
2146 ex convert_H_to_Li(const ex& m, const ex& x)
2148 map_trafo_H_reduce_trailing_zeros filter;
2149 map_trafo_H_convert_to_Li filter2;
2151 return filter2(filter(H(m, x).hold()));
2153 return filter2(filter(H(lst(m), x).hold()));
2158 //////////////////////////////////////////////////////////////////////
2160 // Multiple zeta values zeta(x) and zeta(x,s)
2164 //////////////////////////////////////////////////////////////////////
2167 // anonymous namespace for helper functions
2171 // parameters and data for [Cra] algorithm
2172 const cln::cl_N lambda = cln::cl_N("319/320");
2175 std::vector<std::vector<cln::cl_N> > f_kj;
2176 std::vector<cln::cl_N> crB;
2177 std::vector<std::vector<cln::cl_N> > crG;
2178 std::vector<cln::cl_N> crX;
2181 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
2183 const int size = a.size();
2184 for (int n=0; n<size; n++) {
2186 for (int m=0; m<=n; m++) {
2187 c[n] = c[n] + a[m]*b[n-m];
2194 void initcX(const std::vector<int>& s)
2196 const int k = s.size();
2202 for (int i=0; i<=L2; i++) {
2203 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
2208 for (int m=0; m<k-1; m++) {
2209 std::vector<cln::cl_N> crGbuf;
2212 for (int i=0; i<=L2; i++) {
2213 crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
2215 crG.push_back(crGbuf);
2220 for (int m=0; m<k-1; m++) {
2221 std::vector<cln::cl_N> Xbuf;
2222 for (int i=0; i<=L2; i++) {
2223 Xbuf.push_back(crX[i] * crG[m][i]);
2225 halfcyclic_convolute(Xbuf, crB, crX);
2231 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
2233 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2234 cln::cl_N factor = cln::expt(lambda, Sqk);
2235 cln::cl_N res = factor / Sqk * crX[0] * one;
2240 factor = factor * lambda;
2242 res = res + crX[N] * factor / (N+Sqk);
2243 } while ((res != resbuf) || cln::zerop(crX[N]));
2249 void calc_f(int maxr)
2254 cln::cl_N t0, t1, t2, t3, t4;
2256 std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
2257 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2259 t0 = cln::exp(-lambda);
2261 for (k=1; k<=L1; k++) {
2264 for (j=1; j<=maxr; j++) {
2267 for (i=2; i<=j; i++) {
2271 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
2279 cln::cl_N crandall_Z(const std::vector<int>& s)
2281 const int j = s.size();
2290 t0 = t0 + f_kj[q+j-2][s[0]-1];
2291 } while (t0 != t0buf);
2293 return t0 / cln::factorial(s[0]-1);
2296 std::vector<cln::cl_N> t(j);
2303 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
2304 for (int k=j-2; k>=1; k--) {
2305 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
2307 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
2308 } while (t[0] != t0buf);
2310 return t[0] / cln::factorial(s[0]-1);
2315 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
2317 std::vector<int> r = s;
2318 const int j = r.size();
2320 // decide on maximal size of f_kj for crandall_Z
2324 L1 = Digits * 3 + j*2;
2327 // decide on maximal size of crX for crandall_Y
2330 } else if (Digits < 86) {
2332 } else if (Digits < 192) {
2334 } else if (Digits < 394) {
2336 } else if (Digits < 808) {
2346 for (int i=0; i<j; i++) {
2355 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
2357 std::vector<int> rz;
2360 for (int k=r.size()-1; k>0; k--) {
2362 rz.insert(rz.begin(), r.back());
2363 skp1buf = rz.front();
2369 for (int q=0; q<skp1buf; q++) {
2371 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
2372 cln::cl_N pp2 = crandall_Z(rz);
2377 res = res - pp1 * pp2 / cln::factorial(q);
2379 res = res + pp1 * pp2 / cln::factorial(q);
2382 rz.front() = skp1buf;
2384 rz.insert(rz.begin(), r.back());
2388 res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
2394 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
2396 const int j = r.size();
2398 // buffer for subsums
2399 std::vector<cln::cl_N> t(j);
2400 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2407 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
2408 for (int k=j-2; k>=0; k--) {
2409 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
2411 } while (t[0] != t0buf);
2417 // does Hoelder convolution. see [BBB] (7.0)
2418 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
2420 // prepare parameters
2421 // holds Li arguments in [BBB] notation
2422 std::vector<int> s = s_;
2423 std::vector<int> m_p = m_;
2424 std::vector<int> m_q;
2425 // holds Li arguments in nested sums notation
2426 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
2427 s_p[0] = s_p[0] * cln::cl_N("1/2");
2428 // convert notations
2430 for (int i=0; i<s_.size(); i++) {
2435 s[i] = sig * std::abs(s[i]);
2437 std::vector<cln::cl_N> s_q;
2438 cln::cl_N signum = 1;
2441 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
2446 // change parameters
2447 if (s.front() > 0) {
2448 if (m_p.front() == 1) {
2449 m_p.erase(m_p.begin());
2450 s_p.erase(s_p.begin());
2451 if (s_p.size() > 0) {
2452 s_p.front() = s_p.front() * cln::cl_N("1/2");
2458 m_q.insert(m_q.begin(), 1);
2459 if (s_q.size() > 0) {
2460 s_q.front() = s_q.front() * 2;
2462 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2465 if (m_p.front() == 1) {
2466 m_p.erase(m_p.begin());
2467 cln::cl_N spbuf = s_p.front();
2468 s_p.erase(s_p.begin());
2469 if (s_p.size() > 0) {
2470 s_p.front() = s_p.front() * spbuf;
2473 m_q.insert(m_q.begin(), 1);
2474 if (s_q.size() > 0) {
2475 s_q.front() = s_q.front() * 4;
2477 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
2481 m_q.insert(m_q.begin(), 1);
2482 if (s_q.size() > 0) {
2483 s_q.front() = s_q.front() * 2;
2485 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2490 if (m_p.size() == 0) break;
2492 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
2497 res = res + signum * multipleLi_do_sum(m_q, s_q);
2503 } // end of anonymous namespace
2506 //////////////////////////////////////////////////////////////////////
2508 // Multiple zeta values zeta(x)
2512 //////////////////////////////////////////////////////////////////////
2515 static ex zeta1_evalf(const ex& x)
2517 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
2519 // multiple zeta value
2520 const int count = x.nops();
2521 const lst& xlst = ex_to<lst>(x);
2522 std::vector<int> r(count);
2524 // check parameters and convert them
2525 lst::const_iterator it1 = xlst.begin();
2526 std::vector<int>::iterator it2 = r.begin();
2528 if (!(*it1).info(info_flags::posint)) {
2529 return zeta(x).hold();
2531 *it2 = ex_to<numeric>(*it1).to_int();
2534 } while (it2 != r.end());
2536 // check for divergence
2538 return zeta(x).hold();
2541 // decide on summation algorithm
2542 // this is still a bit clumsy
2543 int limit = (Digits>17) ? 10 : 6;
2544 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
2545 return numeric(zeta_do_sum_Crandall(r));
2547 return numeric(zeta_do_sum_simple(r));
2551 // single zeta value
2552 if (is_exactly_a<numeric>(x) && (x != 1)) {
2554 return zeta(ex_to<numeric>(x));
2555 } catch (const dunno &e) { }
2558 return zeta(x).hold();
2562 static ex zeta1_eval(const ex& m)
2564 if (is_exactly_a<lst>(m)) {
2565 if (m.nops() == 1) {
2566 return zeta(m.op(0));
2568 return zeta(m).hold();
2571 if (m.info(info_flags::numeric)) {
2572 const numeric& y = ex_to<numeric>(m);
2573 // trap integer arguments:
2574 if (y.is_integer()) {
2578 if (y.is_equal(_num1)) {
2579 return zeta(m).hold();
2581 if (y.info(info_flags::posint)) {
2582 if (y.info(info_flags::odd)) {
2583 return zeta(m).hold();
2585 return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
2588 if (y.info(info_flags::odd)) {
2589 return -bernoulli(_num1-y) / (_num1-y);
2596 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
2597 return zeta1_evalf(m);
2600 return zeta(m).hold();
2604 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
2606 GINAC_ASSERT(deriv_param==0);
2608 if (is_exactly_a<lst>(m)) {
2611 return zetaderiv(_ex1, m);
2616 static void zeta1_print_latex(const ex& m_, const print_context& c)
2619 if (is_a<lst>(m_)) {
2620 const lst& m = ex_to<lst>(m_);
2621 lst::const_iterator it = m.begin();
2624 for (; it != m.end(); it++) {
2635 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta").
2636 evalf_func(zeta1_evalf).
2637 eval_func(zeta1_eval).
2638 derivative_func(zeta1_deriv).
2639 print_func<print_latex>(zeta1_print_latex).
2640 do_not_evalf_params().
2644 //////////////////////////////////////////////////////////////////////
2646 // Alternating Euler sum zeta(x,s)
2650 //////////////////////////////////////////////////////////////////////
2653 static ex zeta2_evalf(const ex& x, const ex& s)
2655 if (is_exactly_a<lst>(x)) {
2657 // alternating Euler sum
2658 const int count = x.nops();
2659 const lst& xlst = ex_to<lst>(x);
2660 const lst& slst = ex_to<lst>(s);
2661 std::vector<int> xi(count);
2662 std::vector<int> si(count);
2664 // check parameters and convert them
2665 lst::const_iterator it_xread = xlst.begin();
2666 lst::const_iterator it_sread = slst.begin();
2667 std::vector<int>::iterator it_xwrite = xi.begin();
2668 std::vector<int>::iterator it_swrite = si.begin();
2670 if (!(*it_xread).info(info_flags::posint)) {
2671 return zeta(x, s).hold();
2673 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
2674 if (*it_sread > 0) {
2683 } while (it_xwrite != xi.end());
2685 // check for divergence
2686 if ((xi[0] == 1) && (si[0] == 1)) {
2687 return zeta(x, s).hold();
2690 // use Hoelder convolution
2691 return numeric(zeta_do_Hoelder_convolution(xi, si));
2694 return zeta(x, s).hold();
2698 static ex zeta2_eval(const ex& m, const ex& s_)
2700 if (is_exactly_a<lst>(s_)) {
2701 const lst& s = ex_to<lst>(s_);
2702 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
2703 if ((*it).info(info_flags::positive)) {
2706 return zeta(m, s_).hold();
2709 } else if (s_.info(info_flags::positive)) {
2713 return zeta(m, s_).hold();
2717 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
2719 GINAC_ASSERT(deriv_param==0);
2721 if (is_exactly_a<lst>(m)) {
2724 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
2725 return zetaderiv(_ex1, m);
2732 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
2735 if (is_a<lst>(m_)) {
2741 if (is_a<lst>(s_)) {
2747 lst::const_iterator itm = m.begin();
2748 lst::const_iterator its = s.begin();
2750 c.s << "\\overline{";
2758 for (; itm != m.end(); itm++, its++) {
2761 c.s << "\\overline{";
2772 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta").
2773 evalf_func(zeta2_evalf).
2774 eval_func(zeta2_eval).
2775 derivative_func(zeta2_deriv).
2776 print_func<print_latex>(zeta2_print_latex).
2777 do_not_evalf_params().
2781 } // namespace GiNaC