1 /** @file inifcns_nstdsums.cpp
3 * Implementation of some special functions that have a representation as nested sums.
6 * classical polylogarithm Li(n,x)
7 * multiple polylogarithm Li(lst(m_1,...,m_k),lst(x_1,...,x_k))
8 * nielsen's generalized polylogarithm S(n,p,x)
9 * harmonic polylogarithm H(m,x) or H(lst(m_1,...,m_k),x)
10 * multiple zeta value zeta(m) or zeta(lst(m_1,...,m_k))
11 * alternating Euler sum zeta(m,s) or zeta(lst(m_1,...,m_k),lst(s_1,...,s_k))
15 * - All formulae used can be looked up in the following publications:
16 * [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
17 * [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
18 * [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
19 * [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
21 * - The order of parameters and arguments of Li and zeta is defined according to the nested sums
22 * representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
23 * 0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single
24 * number --- notation.
26 * - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in
27 * the whole complex plane. Multiple polylogarithms evaluate only if for each argument x_i the product
28 * x_1 * x_2 * ... * x_i is smaller than one. The parameters for Li, zeta and S must be positive integers.
29 * If you want to have an alternating Euler sum, you have to give the signs of the parameters as a
30 * second argument s to zeta(m,s) containing 1 and -1.
32 * - The calculation of classical polylogarithms is speeded up by using Bernoulli numbers and
33 * look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in
34 * [Cra] and [BBB] for speed up.
36 * - The functions have no series expansion into nested sums. To do this, you have to convert these functions
37 * into the appropriate objects from the nestedsums library, do the expansion and convert the
40 * - Numerical testing of this implementation has been performed by doing a comparison of results
41 * between this software and the commercial M.......... 4.1. Multiple zeta values have been checked
42 * by means of evaluations into simple zeta values. Harmonic polylogarithms have been checked by
43 * comparison to S(n,p,x) for corresponding parameter combinations and by continuity checks
44 * around |x|=1 along with comparisons to corresponding zeta functions.
49 * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
51 * This program is free software; you can redistribute it and/or modify
52 * it under the terms of the GNU General Public License as published by
53 * the Free Software Foundation; either version 2 of the License, or
54 * (at your option) any later version.
56 * This program is distributed in the hope that it will be useful,
57 * but WITHOUT ANY WARRANTY; without even the implied warranty of
58 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
59 * GNU General Public License for more details.
61 * You should have received a copy of the GNU General Public License
62 * along with this program; if not, write to the Free Software
63 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
77 #include "operators.h"
80 #include "relational.h"
89 //////////////////////////////////////////////////////////////////////
91 // Classical polylogarithm Li(n,x)
95 //////////////////////////////////////////////////////////////////////
98 // anonymous namespace for helper functions
102 // lookup table for factors built from Bernoulli numbers
104 std::vector<std::vector<cln::cl_N> > Xn;
105 // initial size of Xn that should suffice for 32bit machines (must be even)
106 const int xninitsizestep = 26;
107 int xninitsize = xninitsizestep;
111 // This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
112 // With these numbers the polylogs can be calculated as follows:
113 // Li_p (x) = \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with u = -log(1-x)
114 // X_0(n) = B_n (Bernoulli numbers)
115 // X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
116 // The calculation of Xn depends on X0 and X{n-1}.
117 // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
118 // This results in a slightly more complicated algorithm for the X_n.
119 // The first index in Xn corresponds to the index of the polylog minus 2.
120 // The second index in Xn corresponds to the index from the actual sum.
124 // calculate X_2 and higher (corresponding to Li_4 and higher)
125 std::vector<cln::cl_N> buf(xninitsize);
126 std::vector<cln::cl_N>::iterator it = buf.begin();
128 *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
130 for (int i=2; i<=xninitsize; i++) {
132 result = 0; // k == 0
134 result = Xn[0][i/2-1]; // k == 0
136 for (int k=1; k<i-1; k++) {
137 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
138 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
141 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
142 result = result + Xn[n-1][i-1] / (i+1); // k == i
149 // special case to handle the X_0 correct
150 std::vector<cln::cl_N> buf(xninitsize);
151 std::vector<cln::cl_N>::iterator it = buf.begin();
153 *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
155 *it = cln::cl_I(17)/cln::cl_I(36); // i == 2
157 for (int i=3; i<=xninitsize; i++) {
159 result = -Xn[0][(i-3)/2]/2;
160 *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
163 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
164 for (int k=1; k<i/2; k++) {
165 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
174 std::vector<cln::cl_N> buf(xninitsize/2);
175 std::vector<cln::cl_N>::iterator it = buf.begin();
176 for (int i=1; i<=xninitsize/2; i++) {
177 *it = bernoulli(i*2).to_cl_N();
187 // doubles the number of entries in each Xn[]
190 const int pos0 = xninitsize / 2;
192 for (int i=1; i<=xninitsizestep/2; ++i) {
193 Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N());
196 int xend = xninitsize + xninitsizestep;
199 for (int i=xninitsize+1; i<=xend; ++i) {
201 result = -Xn[0][(i-3)/2]/2;
202 Xn[1].push_back((cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result);
204 result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
205 for (int k=1; k<i/2; k++) {
206 result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
208 Xn[1].push_back(result);
212 for (int n=2; n<Xn.size(); ++n) {
213 for (int i=xninitsize+1; i<=xend; ++i) {
215 result = 0; // k == 0
217 result = Xn[0][i/2-1]; // k == 0
219 for (int k=1; k<i-1; ++k) {
220 if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
221 result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
224 result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
225 result = result + Xn[n-1][i-1] / (i+1); // k == i
226 Xn[n].push_back(result);
230 xninitsize += xninitsizestep;
234 // calculates Li(2,x) without Xn
235 cln::cl_N Li2_do_sum(const cln::cl_N& x)
239 cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
240 cln::cl_I den = 1; // n^2 = 1
245 den = den + i; // n^2 = 4, 9, 16, ...
247 res = res + num / den;
248 } while (res != resbuf);
253 // calculates Li(2,x) with Xn
254 cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
256 std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
257 std::vector<cln::cl_N>::const_iterator xend = Xn[0].end();
258 cln::cl_N u = -cln::log(1-x);
259 cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
260 cln::cl_N res = u - u*u/4;
265 factor = factor * u*u / (2*i * (2*i+1));
266 res = res + (*it) * factor;
270 it = Xn[0].begin() + (i-1);
273 } while (res != resbuf);
278 // calculates Li(n,x), n>2 without Xn
279 cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
281 cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits));
288 res = res + factor / cln::expt(cln::cl_I(i),n);
290 } while (res != resbuf);
295 // calculates Li(n,x), n>2 with Xn
296 cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
298 std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
299 std::vector<cln::cl_N>::const_iterator xend = Xn[n-2].end();
300 cln::cl_N u = -cln::log(1-x);
301 cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
307 factor = factor * u / i;
308 res = res + (*it) * factor;
312 it = Xn[n-2].begin() + (i-2);
313 xend = Xn[n-2].end();
315 } while (res != resbuf);
320 // forward declaration needed by function Li_projection and C below
321 numeric S_num(int n, int p, const numeric& x);
324 // helper function for classical polylog Li
325 cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
327 // treat n=2 as special case
329 // check if precalculated X0 exists
334 if (cln::realpart(x) < 0.5) {
335 // choose the faster algorithm
336 // the switching point was empirically determined. the optimal point
337 // depends on hardware, Digits, ... so an approx value is okay.
338 // it solves also the problem with precision due to the u=-log(1-x) transformation
339 if (cln::abs(cln::realpart(x)) < 0.25) {
341 return Li2_do_sum(x);
343 return Li2_do_sum_Xn(x);
346 // choose the faster algorithm
347 if (cln::abs(cln::realpart(x)) > 0.75) {
348 return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
350 return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
354 // check if precalculated Xn exist
356 for (int i=xnsize; i<n-1; i++) {
361 if (cln::realpart(x) < 0.5) {
362 // choose the faster algorithm
363 // with n>=12 the "normal" summation always wins against the method with Xn
364 if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
365 return Lin_do_sum(n, x);
367 return Lin_do_sum_Xn(n, x);
370 cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
371 for (int j=0; j<n-1; j++) {
372 result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x).to_cl_N())
373 * cln::expt(cln::log(x), j) / cln::factorial(j);
381 // helper function for classical polylog Li
382 numeric Li_num(int n, const numeric& x)
386 return -cln::log(1-x.to_cl_N());
397 return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
400 // what is the desired float format?
401 // first guess: default format
402 cln::float_format_t prec = cln::default_float_format;
403 const cln::cl_N value = x.to_cl_N();
404 // second guess: the argument's format
405 if (!x.real().is_rational())
406 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
407 else if (!x.imag().is_rational())
408 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
411 if (cln::abs(value) > 1) {
412 cln::cl_N result = -cln::expt(cln::log(-value),n) / cln::factorial(n);
413 // check if argument is complex. if it is real, the new polylog has to be conjugated.
414 if (cln::zerop(cln::imagpart(value))) {
416 result = result + conjugate(Li_projection(n, cln::recip(value), prec));
419 result = result - conjugate(Li_projection(n, cln::recip(value), prec));
424 result = result + Li_projection(n, cln::recip(value), prec);
427 result = result - Li_projection(n, cln::recip(value), prec);
431 for (int j=0; j<n-1; j++) {
432 add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
433 * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
435 result = result - add;
439 return Li_projection(n, value, prec);
444 } // end of anonymous namespace
447 //////////////////////////////////////////////////////////////////////
449 // Multiple polylogarithm Li(n,x)
453 //////////////////////////////////////////////////////////////////////
456 // anonymous namespace for helper function
460 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
462 const int j = s.size();
464 std::vector<cln::cl_N> t(j);
465 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
473 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
474 for (int k=j-2; k>=0; k--) {
475 t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
477 // ... and do it again (to avoid premature drop out due to special arguments)
479 t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
480 for (int k=j-2; k>=0; k--) {
481 t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]);
483 } while (t[0] != t0buf);
488 // forward declaration for Li_eval()
489 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
492 } // end of anonymous namespace
495 //////////////////////////////////////////////////////////////////////
497 // Classical polylogarithm and multiple polylogarithm Li(n,x)
501 //////////////////////////////////////////////////////////////////////
504 static ex Li_evalf(const ex& x1, const ex& x2)
506 // classical polylogs
507 if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
508 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
510 if (is_a<numeric>(x1) && !is_a<lst>(x2)) {
511 // try to numerically evaluate second argument
512 ex x2_val = x2.evalf();
513 if (is_a<numeric>(x2_val)) {
514 return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2_val));
516 return Li(x1, x2).hold();
520 else if (is_a<lst>(x1) && is_a<lst>(x2)) {
522 for (int i=0; i<x1.nops(); i++) {
523 if (!x1.op(i).info(info_flags::posint)) {
524 return Li(x1, x2).hold();
526 if (!is_a<numeric>(x2.op(i))) {
527 return Li(x1, x2).hold();
530 if (abs(conv) >= 1) {
531 return Li(x1, x2).hold();
536 std::vector<cln::cl_N> x;
537 for (int i=0; i<ex_to<numeric>(x1.nops()).to_int(); i++) {
538 m.push_back(ex_to<numeric>(x1.op(i)).to_int());
539 x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
542 return numeric(multipleLi_do_sum(m, x));
545 return Li(x1,x2).hold();
549 static ex Li_eval(const ex& m_, const ex& x_)
571 return (pow(2,1-m)-1) * zeta(m);
576 if (m.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
577 return Li_num(ex_to<numeric>(m).to_int(), ex_to<numeric>(x));
583 bool doevalf = false;
584 bool doevalfveto = true;
585 const lst& m = ex_to<lst>(m_);
586 const lst& x = ex_to<lst>(x_);
587 lst::const_iterator itm = m.begin();
588 lst::const_iterator itx = x.begin();
589 for (; itm != m.end(); itm++, itx++) {
590 if (!(*itm).info(info_flags::posint)) {
591 return Li(m_, x_).hold();
593 if ((*itx != _ex1) && (*itx != _ex_1)) {
594 if (itx != x.begin()) {
602 if (!(*itx).info(info_flags::numeric)) {
605 if (!(*itx).info(info_flags::crational)) {
617 lst newm = convert_parameter_Li_to_H(m, x, pf);
618 return pf * H(newm, x[0]);
620 if (doevalfveto && doevalf) {
621 return Li(m_, x_).evalf();
624 return Li(m_, x_).hold();
628 static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
631 seq.push_back(expair(Li(m, x), 0));
632 return pseries(rel, seq);
636 static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
638 GINAC_ASSERT(deriv_param < 2);
639 if (deriv_param == 0) {
643 throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
658 return Li(m-1, x) / x;
665 static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
679 c.s << "\\mbox{Li}_{";
680 lst::const_iterator itm = m.begin();
683 for (; itm != m.end(); itm++) {
688 lst::const_iterator itx = x.begin();
691 for (; itx != x.end(); itx++) {
699 REGISTER_FUNCTION(Li,
700 evalf_func(Li_evalf).
702 series_func(Li_series).
703 derivative_func(Li_deriv).
704 print_func<print_latex>(Li_print_latex).
705 do_not_evalf_params());
708 //////////////////////////////////////////////////////////////////////
710 // Nielsen's generalized polylogarithm S(n,p,x)
714 //////////////////////////////////////////////////////////////////////
717 // anonymous namespace for helper functions
721 // lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
723 std::vector<std::vector<cln::cl_N> > Yn;
724 int ynsize = 0; // number of Yn[]
725 int ynlength = 100; // initial length of all Yn[i]
728 // This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
729 // The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
730 // representing S_{n,p}(x).
731 // The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
733 // The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
734 // representing S_{n,p}(x).
735 // The calculation of Y_n uses the values from Y_{n-1}.
736 void fill_Yn(int n, const cln::float_format_t& prec)
738 const int initsize = ynlength;
739 //const int initsize = initsize_Yn;
740 cln::cl_N one = cln::cl_float(1, prec);
743 std::vector<cln::cl_N> buf(initsize);
744 std::vector<cln::cl_N>::iterator it = buf.begin();
745 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
746 *it = (*itprev) / cln::cl_N(n+1) * one;
749 // sums with an index smaller than the depth are zero and need not to be calculated.
750 // calculation starts with depth, which is n+2)
751 for (int i=n+2; i<=initsize+n; i++) {
752 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
758 std::vector<cln::cl_N> buf(initsize);
759 std::vector<cln::cl_N>::iterator it = buf.begin();
762 for (int i=2; i<=initsize; i++) {
763 *it = *(it-1) + 1 / cln::cl_N(i) * one;
772 // make Yn longer ...
773 void make_Yn_longer(int newsize, const cln::float_format_t& prec)
776 cln::cl_N one = cln::cl_float(1, prec);
778 Yn[0].resize(newsize);
779 std::vector<cln::cl_N>::iterator it = Yn[0].begin();
781 for (int i=ynlength+1; i<=newsize; i++) {
782 *it = *(it-1) + 1 / cln::cl_N(i) * one;
786 for (int n=1; n<ynsize; n++) {
787 Yn[n].resize(newsize);
788 std::vector<cln::cl_N>::iterator it = Yn[n].begin();
789 std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
792 for (int i=ynlength+n+1; i<=newsize+n; i++) {
793 *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
803 // helper function for S(n,p,x)
805 cln::cl_N C(int n, int p)
809 for (int k=0; k<p; k++) {
810 for (int j=0; j<=(n+k-1)/2; j++) {
814 result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
817 result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
824 result = result + cln::factorial(n+k-1)
825 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
826 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
829 result = result - cln::factorial(n+k-1)
830 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
831 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
836 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()
837 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
840 result = result + cln::factorial(n+k-1)
841 * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N()
842 / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
850 if (((np)/2+n) & 1) {
851 result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
854 result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
862 // helper function for S(n,p,x)
863 // [Kol] remark to (9.1)
873 for (int m=2; m<=k; m++) {
874 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
881 // helper function for S(n,p,x)
882 // [Kol] remark to (9.1)
892 for (int m=2; m<=k; m++) {
893 result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
900 // helper function for S(n,p,x)
901 cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
904 return Li_projection(n+1, x, prec);
907 // check if precalculated values are sufficient
909 for (int i=ynsize; i<p-1; i++) {
914 // should be done otherwise
915 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
916 cln::cl_N xf = x * one;
917 //cln::cl_N xf = x * cln::cl_float(1, prec);
921 cln::cl_N factor = cln::expt(xf, p);
925 if (i-p >= ynlength) {
927 make_Yn_longer(ynlength*2, prec);
929 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? ...
930 //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
931 factor = factor * xf;
933 } while (res != resbuf);
939 // helper function for S(n,p,x)
940 cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
943 if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
945 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
946 * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
948 for (int s=0; s<n; s++) {
950 for (int r=0; r<p; r++) {
951 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
952 * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
954 result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
960 return S_do_sum(n, p, x, prec);
964 // helper function for S(n,p,x)
965 numeric S_num(int n, int p, const numeric& x)
969 // [Kol] (2.22) with (2.21)
970 return cln::zeta(p+1);
975 return cln::zeta(n+1);
980 for (int nu=0; nu<n; nu++) {
981 for (int rho=0; rho<=p; rho++) {
982 result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
983 * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
986 result = result * cln::expt(cln::cl_I(-1),n+p-1);
993 return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
995 // throw std::runtime_error("don't know how to evaluate this function!");
998 // what is the desired float format?
999 // first guess: default format
1000 cln::float_format_t prec = cln::default_float_format;
1001 const cln::cl_N value = x.to_cl_N();
1002 // second guess: the argument's format
1003 if (!x.real().is_rational())
1004 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1005 else if (!x.imag().is_rational())
1006 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1009 if ((cln::realpart(value) < -0.5) || (n == 0)) {
1011 cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
1012 * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
1014 for (int s=0; s<n; s++) {
1016 for (int r=0; r<p; r++) {
1017 res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
1018 * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
1020 result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
1027 if (cln::abs(value) > 1) {
1031 for (int s=0; s<p; s++) {
1032 for (int r=0; r<=s; r++) {
1033 result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
1034 / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
1035 * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
1038 result = result * cln::expt(cln::cl_I(-1),n);
1041 for (int r=0; r<n; r++) {
1042 res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
1044 res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
1046 result = result + cln::expt(cln::cl_I(-1),p) * res2;
1051 return S_projection(n, p, value, prec);
1056 } // end of anonymous namespace
1059 //////////////////////////////////////////////////////////////////////
1061 // Nielsen's generalized polylogarithm S(n,p,x)
1065 //////////////////////////////////////////////////////////////////////
1068 static ex S_evalf(const ex& n, const ex& p, const ex& x)
1070 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1071 if (is_a<numeric>(x)) {
1072 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1074 ex x_val = x.evalf();
1075 if (is_a<numeric>(x_val)) {
1076 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x_val));
1080 return S(n, p, x).hold();
1084 static ex S_eval(const ex& n, const ex& p, const ex& x)
1086 if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
1092 for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
1100 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
1101 return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
1106 return pow(-log(1-x), p) / factorial(p);
1108 return S(n, p, x).hold();
1112 static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
1115 seq.push_back(expair(S(n, p, x), 0));
1116 return pseries(rel, seq);
1120 static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
1122 GINAC_ASSERT(deriv_param < 3);
1123 if (deriv_param < 2) {
1127 return S(n-1, p, x) / x;
1129 return S(n, p-1, x) / (1-x);
1134 static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
1136 c.s << "\\mbox{S}_{";
1146 REGISTER_FUNCTION(S,
1147 evalf_func(S_evalf).
1149 series_func(S_series).
1150 derivative_func(S_deriv).
1151 print_func<print_latex>(S_print_latex).
1152 do_not_evalf_params());
1155 //////////////////////////////////////////////////////////////////////
1157 // Harmonic polylogarithm H(m,x)
1161 //////////////////////////////////////////////////////////////////////
1164 // anonymous namespace for helper functions
1168 // regulates the pole (used by 1/x-transformation)
1169 symbol H_polesign("IMSIGN");
1172 // convert parameters from H to Li representation
1173 // parameters are expected to be in expanded form, i.e. only 0, 1 and -1
1174 // returns true if some parameters are negative
1175 bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
1177 // expand parameter list
1179 for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
1181 for (ex count=*it-1; count > 0; count--) {
1185 } else if (*it < -1) {
1186 for (ex count=*it+1; count < 0; count++) {
1197 bool has_negative_parameters = false;
1199 for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
1205 m.append((*it+acc-1) * signum);
1207 m.append((*it-acc+1) * signum);
1213 has_negative_parameters = true;
1216 if (has_negative_parameters) {
1217 for (int i=0; i<m.nops(); i++) {
1219 m.let_op(i) = -m.op(i);
1227 return has_negative_parameters;
1231 // recursivly transforms H to corresponding multiple polylogarithms
1232 struct map_trafo_H_convert_to_Li : public map_function
1234 ex operator()(const ex& e)
1236 if (is_a<add>(e) || is_a<mul>(e)) {
1237 return e.map(*this);
1239 if (is_a<function>(e)) {
1240 std::string name = ex_to<function>(e).get_name();
1243 if (is_a<lst>(e.op(0))) {
1244 parameter = ex_to<lst>(e.op(0));
1246 parameter = lst(e.op(0));
1253 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1254 s.let_op(0) = s.op(0) * arg;
1255 return pf * Li(m, s).hold();
1257 for (int i=0; i<m.nops(); i++) {
1260 s.let_op(0) = s.op(0) * arg;
1261 return Li(m, s).hold();
1270 // recursivly transforms H to corresponding zetas
1271 struct map_trafo_H_convert_to_zeta : public map_function
1273 ex operator()(const ex& e)
1275 if (is_a<add>(e) || is_a<mul>(e)) {
1276 return e.map(*this);
1278 if (is_a<function>(e)) {
1279 std::string name = ex_to<function>(e).get_name();
1282 if (is_a<lst>(e.op(0))) {
1283 parameter = ex_to<lst>(e.op(0));
1285 parameter = lst(e.op(0));
1291 if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
1292 return pf * zeta(m, s);
1303 // remove trailing zeros from H-parameters
1304 struct map_trafo_H_reduce_trailing_zeros : public map_function
1306 ex operator()(const ex& e)
1308 if (is_a<add>(e) || is_a<mul>(e)) {
1309 return e.map(*this);
1311 if (is_a<function>(e)) {
1312 std::string name = ex_to<function>(e).get_name();
1315 if (is_a<lst>(e.op(0))) {
1316 parameter = ex_to<lst>(e.op(0));
1318 parameter = lst(e.op(0));
1321 if (parameter.op(parameter.nops()-1) == 0) {
1324 if (parameter.nops() == 1) {
1329 lst::const_iterator it = parameter.begin();
1330 while ((it != parameter.end()) && (*it == 0)) {
1333 if (it == parameter.end()) {
1334 return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
1338 parameter.remove_last();
1339 int lastentry = parameter.nops();
1340 while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
1345 ex result = log(arg) * H(parameter,arg).hold();
1347 for (ex i=0; i<lastentry; i++) {
1348 if (parameter[i] > 0) {
1350 result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
1353 } else if (parameter[i] < 0) {
1355 result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
1363 if (lastentry < parameter.nops()) {
1364 result = result / (parameter.nops()-lastentry+1);
1365 return result.map(*this);
1377 // returns an expression with zeta functions corresponding to the parameter list for H
1378 ex convert_H_to_zeta(const lst& m)
1380 symbol xtemp("xtemp");
1381 map_trafo_H_reduce_trailing_zeros filter;
1382 map_trafo_H_convert_to_zeta filter2;
1383 return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
1387 // convert signs form Li to H representation
1388 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
1391 lst::const_iterator itm = m.begin();
1392 lst::const_iterator itx = ++x.begin();
1397 while (itx != x.end()) {
1400 res.append((*itm) * signum);
1408 // multiplies an one-dimensional H with another H
1410 ex trafo_H_mult(const ex& h1, const ex& h2)
1415 ex h1nops = h1.op(0).nops();
1416 ex h2nops = h2.op(0).nops();
1418 hshort = h2.op(0).op(0);
1419 hlong = ex_to<lst>(h1.op(0));
1421 hshort = h1.op(0).op(0);
1423 hlong = ex_to<lst>(h2.op(0));
1425 hlong = h2.op(0).op(0);
1428 for (int i=0; i<=hlong.nops(); i++) {
1432 newparameter.append(hlong[j]);
1434 newparameter.append(hshort);
1435 for (; j<hlong.nops(); j++) {
1436 newparameter.append(hlong[j]);
1438 res += H(newparameter, h1.op(1)).hold();
1444 // applies trafo_H_mult recursively on expressions
1445 struct map_trafo_H_mult : public map_function
1447 ex operator()(const ex& e)
1450 return e.map(*this);
1458 for (int pos=0; pos<e.nops(); pos++) {
1459 if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
1460 std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
1462 for (ex i=0; i<e.op(pos).op(1); i++) {
1463 Hlst.append(e.op(pos).op(0));
1467 } else if (is_a<function>(e.op(pos))) {
1468 std::string name = ex_to<function>(e.op(pos)).get_name();
1470 if (e.op(pos).op(0).nops() > 1) {
1473 Hlst.append(e.op(pos));
1478 result *= e.op(pos);
1481 if (Hlst.nops() > 0) {
1482 firstH = Hlst[Hlst.nops()-1];
1489 if (Hlst.nops() > 0) {
1490 ex buffer = trafo_H_mult(firstH, Hlst.op(0));
1492 for (int i=1; i<Hlst.nops(); i++) {
1493 result *= Hlst.op(i);
1495 result = result.expand();
1496 map_trafo_H_mult recursion;
1497 return recursion(result);
1508 // do integration [ReV] (55)
1509 // put parameter 0 in front of existing parameters
1510 ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
1514 if (is_a<function>(e)) {
1515 name = ex_to<function>(e).get_name();
1520 for (int i=0; i<e.nops(); i++) {
1521 if (is_a<function>(e.op(i))) {
1522 std::string name = ex_to<function>(e.op(i)).get_name();
1530 lst newparameter = ex_to<lst>(h.op(0));
1531 newparameter.prepend(0);
1532 ex addzeta = convert_H_to_zeta(newparameter);
1533 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1535 return e * (-H(lst(0),1/arg).hold());
1540 // do integration [ReV] (55)
1541 // put parameter -1 in front of existing parameters
1542 ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
1546 if (is_a<function>(e)) {
1547 name = ex_to<function>(e).get_name();
1552 for (int i=0; i<e.nops(); i++) {
1553 if (is_a<function>(e.op(i))) {
1554 std::string name = ex_to<function>(e.op(i)).get_name();
1562 lst newparameter = ex_to<lst>(h.op(0));
1563 newparameter.prepend(-1);
1564 ex addzeta = convert_H_to_zeta(newparameter);
1565 return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
1567 ex addzeta = convert_H_to_zeta(lst(-1));
1568 return (e * (addzeta - H(lst(-1),1/arg).hold())).expand();
1573 // do integration [ReV] (55)
1574 // put parameter -1 in front of existing parameters
1575 ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
1579 if (is_a<function>(e)) {
1580 name = ex_to<function>(e).get_name();
1585 for (int i=0; i<e.nops(); i++) {
1586 if (is_a<function>(e.op(i))) {
1587 std::string name = ex_to<function>(e.op(i)).get_name();
1595 lst newparameter = ex_to<lst>(h.op(0));
1596 newparameter.prepend(-1);
1597 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1599 return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand();
1604 // do integration [ReV] (55)
1605 // put parameter 1 in front of existing parameters
1606 ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
1610 if (is_a<function>(e)) {
1611 name = ex_to<function>(e).get_name();
1616 for (int i=0; i<e.nops(); i++) {
1617 if (is_a<function>(e.op(i))) {
1618 std::string name = ex_to<function>(e.op(i)).get_name();
1626 lst newparameter = ex_to<lst>(h.op(0));
1627 newparameter.prepend(1);
1628 return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
1630 return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand();
1635 // do x -> 1/x transformation
1636 struct map_trafo_H_1overx : public map_function
1638 ex operator()(const ex& e)
1640 if (is_a<add>(e) || is_a<mul>(e)) {
1641 return e.map(*this);
1644 if (is_a<function>(e)) {
1645 std::string name = ex_to<function>(e).get_name();
1648 lst parameter = ex_to<lst>(e.op(0));
1651 // special cases if all parameters are either 0, 1 or -1
1652 bool allthesame = true;
1653 if (parameter.op(0) == 0) {
1654 for (int i=1; i<parameter.nops(); i++) {
1655 if (parameter.op(i) != 0) {
1661 return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
1663 } else if (parameter.op(0) == -1) {
1664 for (int i=1; i<parameter.nops(); i++) {
1665 if (parameter.op(i) != -1) {
1671 map_trafo_H_mult unify;
1672 return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
1673 / factorial(parameter.nops())).expand());
1676 for (int i=1; i<parameter.nops(); i++) {
1677 if (parameter.op(i) != 1) {
1683 map_trafo_H_mult unify;
1684 return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() + H_polesign, parameter.nops())
1685 / factorial(parameter.nops())).expand());
1689 lst newparameter = parameter;
1690 newparameter.remove_first();
1692 if (parameter.op(0) == 0) {
1695 ex res = convert_H_to_zeta(parameter);
1696 map_trafo_H_1overx recursion;
1697 ex buffer = recursion(H(newparameter, arg).hold());
1698 if (is_a<add>(buffer)) {
1699 for (int i=0; i<buffer.nops(); i++) {
1700 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
1703 res += trafo_H_1tx_prepend_zero(buffer, arg);
1707 } else if (parameter.op(0) == -1) {
1709 // leading negative one
1710 ex res = convert_H_to_zeta(parameter);
1711 map_trafo_H_1overx recursion;
1712 ex buffer = recursion(H(newparameter, arg).hold());
1713 if (is_a<add>(buffer)) {
1714 for (int i=0; i<buffer.nops(); i++) {
1715 res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
1718 res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
1725 map_trafo_H_1overx recursion;
1726 map_trafo_H_mult unify;
1727 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1729 while (parameter.op(firstzero) == 1) {
1732 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1736 newparameter.append(parameter[j+1]);
1738 newparameter.append(1);
1739 for (; j<parameter.nops()-1; j++) {
1740 newparameter.append(parameter[j+1]);
1742 res -= H(newparameter, arg).hold();
1744 res = recursion(res).expand() / firstzero;
1756 // do x -> (1-x)/(1+x) transformation
1757 struct map_trafo_H_1mxt1px : public map_function
1759 ex operator()(const ex& e)
1761 if (is_a<add>(e) || is_a<mul>(e)) {
1762 return e.map(*this);
1765 if (is_a<function>(e)) {
1766 std::string name = ex_to<function>(e).get_name();
1769 lst parameter = ex_to<lst>(e.op(0));
1772 // special cases if all parameters are either 0, 1 or -1
1773 bool allthesame = true;
1774 if (parameter.op(0) == 0) {
1775 for (int i=1; i<parameter.nops(); i++) {
1776 if (parameter.op(i) != 0) {
1782 map_trafo_H_mult unify;
1783 return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1784 / factorial(parameter.nops())).expand());
1786 } else if (parameter.op(0) == -1) {
1787 for (int i=1; i<parameter.nops(); i++) {
1788 if (parameter.op(i) != -1) {
1794 map_trafo_H_mult unify;
1795 return unify((pow(log(2) - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1796 / factorial(parameter.nops())).expand());
1799 for (int i=1; i<parameter.nops(); i++) {
1800 if (parameter.op(i) != 1) {
1806 map_trafo_H_mult unify;
1807 return unify((pow(-log(2) - H(lst(0),(1-arg)/(1+arg)).hold() + H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
1808 / factorial(parameter.nops())).expand());
1812 lst newparameter = parameter;
1813 newparameter.remove_first();
1815 if (parameter.op(0) == 0) {
1818 ex res = convert_H_to_zeta(parameter);
1819 map_trafo_H_1mxt1px recursion;
1820 ex buffer = recursion(H(newparameter, arg).hold());
1821 if (is_a<add>(buffer)) {
1822 for (int i=0; i<buffer.nops(); i++) {
1823 res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1826 res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1830 } else if (parameter.op(0) == -1) {
1832 // leading negative one
1833 ex res = convert_H_to_zeta(parameter);
1834 map_trafo_H_1mxt1px recursion;
1835 ex buffer = recursion(H(newparameter, arg).hold());
1836 if (is_a<add>(buffer)) {
1837 for (int i=0; i<buffer.nops(); i++) {
1838 res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
1841 res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
1848 map_trafo_H_1mxt1px recursion;
1849 map_trafo_H_mult unify;
1850 ex res = H(lst(1), arg).hold() * H(newparameter, arg).hold();
1852 while (parameter.op(firstzero) == 1) {
1855 for (int i=firstzero-1; i<parameter.nops()-1; i++) {
1859 newparameter.append(parameter[j+1]);
1861 newparameter.append(1);
1862 for (; j<parameter.nops()-1; j++) {
1863 newparameter.append(parameter[j+1]);
1865 res -= H(newparameter, arg).hold();
1867 res = recursion(res).expand() / firstzero;
1879 // do the actual summation.
1880 cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
1882 const int j = m.size();
1884 std::vector<cln::cl_N> t(j);
1886 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
1887 cln::cl_N factor = cln::expt(x, j) * one;
1893 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
1894 for (int k=j-2; k>=1; k--) {
1895 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
1897 t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
1898 factor = factor * x;
1899 } while (t[0] != t0buf);
1905 } // end of anonymous namespace
1908 //////////////////////////////////////////////////////////////////////
1910 // Harmonic polylogarithm H(m,x)
1914 //////////////////////////////////////////////////////////////////////
1917 static ex H_evalf(const ex& x1, const ex& x2)
1919 if (is_a<lst>(x1)) {
1922 if (is_a<numeric>(x2)) {
1923 x = ex_to<numeric>(x2).to_cl_N();
1925 ex x2_val = x2.evalf();
1926 if (is_a<numeric>(x2_val)) {
1927 x = ex_to<numeric>(x2_val).to_cl_N();
1931 for (int i=0; i<x1.nops(); i++) {
1932 if (!x1.op(i).info(info_flags::integer)) {
1933 return H(x1, x2).hold();
1936 if (x1.nops() < 1) {
1937 return H(x1, x2).hold();
1940 const lst& morg = ex_to<lst>(x1);
1941 // remove trailing zeros ...
1942 if (*(--morg.end()) == 0) {
1943 symbol xtemp("xtemp");
1944 map_trafo_H_reduce_trailing_zeros filter;
1945 return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
1947 // ... and expand parameter notation
1949 for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
1951 for (ex count=*it-1; count > 0; count--) {
1955 } else if (*it < -1) {
1956 for (ex count=*it+1; count < 0; count++) {
1965 // since the transformations produce a lot of terms, they are only efficient for
1966 // argument near one.
1967 // no transformation needed -> do summation
1968 if (cln::abs(x) < 0.95) {
1972 if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
1973 // negative parameters -> s_lst is filled
1974 std::vector<int> m_int;
1975 std::vector<cln::cl_N> x_cln;
1976 for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
1977 it_int != m_lst.end(); it_int++, it_cln++) {
1978 m_int.push_back(ex_to<numeric>(*it_int).to_int());
1979 x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
1981 x_cln.front() = x_cln.front() * x;
1982 return pf * numeric(multipleLi_do_sum(m_int, x_cln));
1984 // only positive parameters
1986 if (m_lst.nops() == 1) {
1987 return Li(m_lst.op(0), x2).evalf();
1989 std::vector<int> m_int;
1990 for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
1991 m_int.push_back(ex_to<numeric>(*it).to_int());
1993 return numeric(H_do_sum(m_int, x));
1999 // ensure that the realpart of the argument is positive
2000 if (cln::realpart(x) < 0) {
2002 for (int i=0; i<m.nops(); i++) {
2004 m.let_op(i) = -m.op(i);
2010 // choose transformations
2011 symbol xtemp("xtemp");
2012 if (cln::abs(x-1) < 1.4142) {
2014 map_trafo_H_1mxt1px trafo;
2015 res *= trafo(H(m, xtemp));
2018 map_trafo_H_1overx trafo;
2019 res *= trafo(H(m, xtemp));
2020 if (cln::imagpart(x) <= 0) {
2021 res = res.subs(H_polesign == -I*Pi);
2023 res = res.subs(H_polesign == I*Pi);
2029 // map_trafo_H_convert converter;
2030 // res = converter(res).expand();
2032 // res.find(H(wild(1),wild(2)), ll);
2033 // res.find(zeta(wild(1)), ll);
2034 // res.find(zeta(wild(1),wild(2)), ll);
2035 // res = res.collect(ll);
2037 return res.subs(xtemp == numeric(x)).evalf();
2040 return H(x1,x2).hold();
2044 static ex H_eval(const ex& m_, const ex& x)
2047 if (is_a<lst>(m_)) {
2052 if (m.nops() == 0) {
2060 if (*m.begin() > _ex1) {
2066 } else if (*m.begin() < _ex_1) {
2072 } else if (*m.begin() == _ex0) {
2079 for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
2080 if ((*it).info(info_flags::integer)) {
2091 } else if (*it < _ex_1) {
2111 } else if (step == 1) {
2123 // if some m_i is not an integer
2124 return H(m_, x).hold();
2127 if ((x == _ex1) && (*(--m.end()) != _ex0)) {
2128 return convert_H_to_zeta(m);
2134 return H(m_, x).hold();
2136 return pow(log(x), m.nops()) / factorial(m.nops());
2139 return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
2141 } else if ((step == 1) && (pos1 == _ex0)){
2146 return pow(-1, p) * S(n, p, -x);
2152 if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
2153 return H(m_, x).evalf();
2155 return H(m_, x).hold();
2159 static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
2162 seq.push_back(expair(H(m, x), 0));
2163 return pseries(rel, seq);
2167 static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
2169 GINAC_ASSERT(deriv_param < 2);
2170 if (deriv_param == 0) {
2174 if (is_a<lst>(m_)) {
2190 return 1/(1-x) * H(m, x);
2191 } else if (mb == _ex_1) {
2192 return 1/(1+x) * H(m, x);
2199 static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
2202 if (is_a<lst>(m_)) {
2207 c.s << "\\mbox{H}_{";
2208 lst::const_iterator itm = m.begin();
2211 for (; itm != m.end(); itm++) {
2221 REGISTER_FUNCTION(H,
2222 evalf_func(H_evalf).
2224 series_func(H_series).
2225 derivative_func(H_deriv).
2226 print_func<print_latex>(H_print_latex).
2227 do_not_evalf_params());
2230 // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
2231 ex convert_H_to_Li(const ex& m, const ex& x)
2233 map_trafo_H_reduce_trailing_zeros filter;
2234 map_trafo_H_convert_to_Li filter2;
2236 return filter2(filter(H(m, x).hold()));
2238 return filter2(filter(H(lst(m), x).hold()));
2243 //////////////////////////////////////////////////////////////////////
2245 // Multiple zeta values zeta(x) and zeta(x,s)
2249 //////////////////////////////////////////////////////////////////////
2252 // anonymous namespace for helper functions
2256 // parameters and data for [Cra] algorithm
2257 const cln::cl_N lambda = cln::cl_N("319/320");
2260 std::vector<std::vector<cln::cl_N> > f_kj;
2261 std::vector<cln::cl_N> crB;
2262 std::vector<std::vector<cln::cl_N> > crG;
2263 std::vector<cln::cl_N> crX;
2266 void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
2268 const int size = a.size();
2269 for (int n=0; n<size; n++) {
2271 for (int m=0; m<=n; m++) {
2272 c[n] = c[n] + a[m]*b[n-m];
2279 void initcX(const std::vector<int>& s)
2281 const int k = s.size();
2287 for (int i=0; i<=L2; i++) {
2288 crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
2293 for (int m=0; m<k-1; m++) {
2294 std::vector<cln::cl_N> crGbuf;
2297 for (int i=0; i<=L2; i++) {
2298 crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
2300 crG.push_back(crGbuf);
2305 for (int m=0; m<k-1; m++) {
2306 std::vector<cln::cl_N> Xbuf;
2307 for (int i=0; i<=L2; i++) {
2308 Xbuf.push_back(crX[i] * crG[m][i]);
2310 halfcyclic_convolute(Xbuf, crB, crX);
2316 cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
2318 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2319 cln::cl_N factor = cln::expt(lambda, Sqk);
2320 cln::cl_N res = factor / Sqk * crX[0] * one;
2325 factor = factor * lambda;
2327 res = res + crX[N] * factor / (N+Sqk);
2328 } while ((res != resbuf) || cln::zerop(crX[N]));
2334 void calc_f(int maxr)
2339 cln::cl_N t0, t1, t2, t3, t4;
2341 std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
2342 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2344 t0 = cln::exp(-lambda);
2346 for (k=1; k<=L1; k++) {
2349 for (j=1; j<=maxr; j++) {
2352 for (i=2; i<=j; i++) {
2356 (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
2364 cln::cl_N crandall_Z(const std::vector<int>& s)
2366 const int j = s.size();
2375 t0 = t0 + f_kj[q+j-2][s[0]-1];
2376 } while (t0 != t0buf);
2378 return t0 / cln::factorial(s[0]-1);
2381 std::vector<cln::cl_N> t(j);
2388 t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
2389 for (int k=j-2; k>=1; k--) {
2390 t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
2392 t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
2393 } while (t[0] != t0buf);
2395 return t[0] / cln::factorial(s[0]-1);
2400 cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
2402 std::vector<int> r = s;
2403 const int j = r.size();
2405 // decide on maximal size of f_kj for crandall_Z
2409 L1 = Digits * 3 + j*2;
2412 // decide on maximal size of crX for crandall_Y
2415 } else if (Digits < 86) {
2417 } else if (Digits < 192) {
2419 } else if (Digits < 394) {
2421 } else if (Digits < 808) {
2431 for (int i=0; i<j; i++) {
2440 const cln::cl_N r0factorial = cln::factorial(r[0]-1);
2442 std::vector<int> rz;
2445 for (int k=r.size()-1; k>0; k--) {
2447 rz.insert(rz.begin(), r.back());
2448 skp1buf = rz.front();
2454 for (int q=0; q<skp1buf; q++) {
2456 cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
2457 cln::cl_N pp2 = crandall_Z(rz);
2462 res = res - pp1 * pp2 / cln::factorial(q);
2464 res = res + pp1 * pp2 / cln::factorial(q);
2467 rz.front() = skp1buf;
2469 rz.insert(rz.begin(), r.back());
2473 res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
2479 cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
2481 const int j = r.size();
2483 // buffer for subsums
2484 std::vector<cln::cl_N> t(j);
2485 cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
2492 t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
2493 for (int k=j-2; k>=0; k--) {
2494 t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
2496 } while (t[0] != t0buf);
2502 // does Hoelder convolution. see [BBB] (7.0)
2503 cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
2505 // prepare parameters
2506 // holds Li arguments in [BBB] notation
2507 std::vector<int> s = s_;
2508 std::vector<int> m_p = m_;
2509 std::vector<int> m_q;
2510 // holds Li arguments in nested sums notation
2511 std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
2512 s_p[0] = s_p[0] * cln::cl_N("1/2");
2513 // convert notations
2515 for (int i=0; i<s_.size(); i++) {
2520 s[i] = sig * std::abs(s[i]);
2522 std::vector<cln::cl_N> s_q;
2523 cln::cl_N signum = 1;
2526 cln::cl_N res = multipleLi_do_sum(m_p, s_p);
2531 // change parameters
2532 if (s.front() > 0) {
2533 if (m_p.front() == 1) {
2534 m_p.erase(m_p.begin());
2535 s_p.erase(s_p.begin());
2536 if (s_p.size() > 0) {
2537 s_p.front() = s_p.front() * cln::cl_N("1/2");
2543 m_q.insert(m_q.begin(), 1);
2544 if (s_q.size() > 0) {
2545 s_q.front() = s_q.front() * 2;
2547 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2550 if (m_p.front() == 1) {
2551 m_p.erase(m_p.begin());
2552 cln::cl_N spbuf = s_p.front();
2553 s_p.erase(s_p.begin());
2554 if (s_p.size() > 0) {
2555 s_p.front() = s_p.front() * spbuf;
2558 m_q.insert(m_q.begin(), 1);
2559 if (s_q.size() > 0) {
2560 s_q.front() = s_q.front() * 4;
2562 s_q.insert(s_q.begin(), cln::cl_N("1/4"));
2566 m_q.insert(m_q.begin(), 1);
2567 if (s_q.size() > 0) {
2568 s_q.front() = s_q.front() * 2;
2570 s_q.insert(s_q.begin(), cln::cl_N("1/2"));
2575 if (m_p.size() == 0) break;
2577 res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
2582 res = res + signum * multipleLi_do_sum(m_q, s_q);
2588 } // end of anonymous namespace
2591 //////////////////////////////////////////////////////////////////////
2593 // Multiple zeta values zeta(x)
2597 //////////////////////////////////////////////////////////////////////
2600 static ex zeta1_evalf(const ex& x)
2602 if (is_exactly_a<lst>(x) && (x.nops()>1)) {
2604 // multiple zeta value
2605 const int count = x.nops();
2606 const lst& xlst = ex_to<lst>(x);
2607 std::vector<int> r(count);
2609 // check parameters and convert them
2610 lst::const_iterator it1 = xlst.begin();
2611 std::vector<int>::iterator it2 = r.begin();
2613 if (!(*it1).info(info_flags::posint)) {
2614 return zeta(x).hold();
2616 *it2 = ex_to<numeric>(*it1).to_int();
2619 } while (it2 != r.end());
2621 // check for divergence
2623 return zeta(x).hold();
2626 // decide on summation algorithm
2627 // this is still a bit clumsy
2628 int limit = (Digits>17) ? 10 : 6;
2629 if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
2630 return numeric(zeta_do_sum_Crandall(r));
2632 return numeric(zeta_do_sum_simple(r));
2636 // single zeta value
2637 if (is_exactly_a<numeric>(x) && (x != 1)) {
2639 return zeta(ex_to<numeric>(x));
2640 } catch (const dunno &e) { }
2643 return zeta(x).hold();
2647 static ex zeta1_eval(const ex& m)
2649 if (is_exactly_a<lst>(m)) {
2650 if (m.nops() == 1) {
2651 return zeta(m.op(0));
2653 return zeta(m).hold();
2656 if (m.info(info_flags::numeric)) {
2657 const numeric& y = ex_to<numeric>(m);
2658 // trap integer arguments:
2659 if (y.is_integer()) {
2663 if (y.is_equal(_num1)) {
2664 return zeta(m).hold();
2666 if (y.info(info_flags::posint)) {
2667 if (y.info(info_flags::odd)) {
2668 return zeta(m).hold();
2670 return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y);
2673 if (y.info(info_flags::odd)) {
2674 return -bernoulli(_num1-y) / (_num1-y);
2681 if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
2682 return zeta1_evalf(m);
2685 return zeta(m).hold();
2689 static ex zeta1_deriv(const ex& m, unsigned deriv_param)
2691 GINAC_ASSERT(deriv_param==0);
2693 if (is_exactly_a<lst>(m)) {
2696 return zetaderiv(_ex1, m);
2701 static void zeta1_print_latex(const ex& m_, const print_context& c)
2704 if (is_a<lst>(m_)) {
2705 const lst& m = ex_to<lst>(m_);
2706 lst::const_iterator it = m.begin();
2709 for (; it != m.end(); it++) {
2720 unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
2721 evalf_func(zeta1_evalf).
2722 eval_func(zeta1_eval).
2723 derivative_func(zeta1_deriv).
2724 print_func<print_latex>(zeta1_print_latex).
2725 do_not_evalf_params().
2729 //////////////////////////////////////////////////////////////////////
2731 // Alternating Euler sum zeta(x,s)
2735 //////////////////////////////////////////////////////////////////////
2738 static ex zeta2_evalf(const ex& x, const ex& s)
2740 if (is_exactly_a<lst>(x)) {
2742 // alternating Euler sum
2743 const int count = x.nops();
2744 const lst& xlst = ex_to<lst>(x);
2745 const lst& slst = ex_to<lst>(s);
2746 std::vector<int> xi(count);
2747 std::vector<int> si(count);
2749 // check parameters and convert them
2750 lst::const_iterator it_xread = xlst.begin();
2751 lst::const_iterator it_sread = slst.begin();
2752 std::vector<int>::iterator it_xwrite = xi.begin();
2753 std::vector<int>::iterator it_swrite = si.begin();
2755 if (!(*it_xread).info(info_flags::posint)) {
2756 return zeta(x, s).hold();
2758 *it_xwrite = ex_to<numeric>(*it_xread).to_int();
2759 if (*it_sread > 0) {
2768 } while (it_xwrite != xi.end());
2770 // check for divergence
2771 if ((xi[0] == 1) && (si[0] == 1)) {
2772 return zeta(x, s).hold();
2775 // use Hoelder convolution
2776 return numeric(zeta_do_Hoelder_convolution(xi, si));
2779 return zeta(x, s).hold();
2783 static ex zeta2_eval(const ex& m, const ex& s_)
2785 if (is_exactly_a<lst>(s_)) {
2786 const lst& s = ex_to<lst>(s_);
2787 for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
2788 if ((*it).info(info_flags::positive)) {
2791 return zeta(m, s_).hold();
2794 } else if (s_.info(info_flags::positive)) {
2798 return zeta(m, s_).hold();
2802 static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
2804 GINAC_ASSERT(deriv_param==0);
2806 if (is_exactly_a<lst>(m)) {
2809 if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
2810 return zetaderiv(_ex1, m);
2817 static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
2820 if (is_a<lst>(m_)) {
2826 if (is_a<lst>(s_)) {
2832 lst::const_iterator itm = m.begin();
2833 lst::const_iterator its = s.begin();
2835 c.s << "\\overline{";
2843 for (; itm != m.end(); itm++, its++) {
2846 c.s << "\\overline{";
2857 unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
2858 evalf_func(zeta2_evalf).
2859 eval_func(zeta2_eval).
2860 derivative_func(zeta2_deriv).
2861 print_func<print_latex>(zeta2_print_latex).
2862 do_not_evalf_params().
2866 } // namespace GiNaC