/** @file inifcns_nstdsums.cpp
*
* Implementation of some special functions that have a representation as nested sums.
- * The functions are:
+ *
+ * The functions are:
* classical polylogarithm Li(n,x)
- * multiple polylogarithm Li(lst(n_1,...,n_k),lst(x_1,...,x_k)
- * nielsen's generalized polylogarithm S(n,p,x)
- * harmonic polylogarithm H(lst(m_1,...,m_k),x)
- * multiple zeta value mZeta(lst(m_1,...,m_k))
+ * multiple polylogarithm Li(lst(m_1,...,m_k),lst(x_1,...,x_k))
+ * G(lst(a_1,...,a_k),y) or G(lst(a_1,...,a_k),lst(s_1,...,s_k),y)
+ * Nielsen's generalized polylogarithm S(n,p,x)
+ * harmonic polylogarithm H(m,x) or H(lst(m_1,...,m_k),x)
+ * multiple zeta value zeta(m) or zeta(lst(m_1,...,m_k))
+ * alternating Euler sum zeta(m,s) or zeta(lst(m_1,...,m_k),lst(s_1,...,s_k))
*
* Some remarks:
- * - All formulae used can be looked up in the following publication:
- * Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
- * This document will be referenced as [Kol] throughout this source code.
- * - Classical polylogarithms (Li) and nielsen's generalized polylogarithms (S) can be numerically
- * evaluated in the whole complex plane. And of course, there is still room for speed optimizations ;-).
- * - The calculation of classical polylogarithms is speed up by using Euler-MacLaurin summation (EuMac).
- * - The remaining functions can only be numerically evaluated with arguments lying in the unit sphere
- * at the moment. Sorry. The evaluation especially for mZeta is very slow ... better not use it
- * right now.
- * - The functions have no series expansion. To do it, you have to convert these functions
- * into the appropriate objects from the nestedsums library, do the expansion and convert the
- * result back.
+ *
+ * - All formulae used can be looked up in the following publications:
+ * [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258.
+ * [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
+ * [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
+ * [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
+ * [VSW] Numerical evaluation of multiple polylogarithms, J.Vollinga, S.Weinzierl, hep-ph/0410259
+ *
+ * - The order of parameters and arguments of Li and zeta is defined according to the nested sums
+ * representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
+ * 0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single
+ * number --- notation.
+ *
+ * - All functions can be nummerically evaluated with arguments in the whole complex plane. The parameters
+ * for Li, zeta and S must be positive integers. If you want to have an alternating Euler sum, you have
+ * to give the signs of the parameters as a second argument s to zeta(m,s) containing 1 and -1.
+ *
+ * - The calculation of classical polylogarithms is speeded up by using Bernoulli numbers and
+ * look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in
+ * [Cra] and [BBB] for speed up. Multiple polylogarithms use Hoelder convolution [BBB].
+ *
+ * - The functions have no means to do a series expansion into nested sums. To do this, you have to convert
+ * these functions into the appropriate objects from the nestedsums library, do the expansion and convert
+ * the result back.
+ *
* - Numerical testing of this implementation has been performed by doing a comparison of results
- * between this software and the commercial M.......... 4.1.
+ * between this software and the commercial M.......... 4.1. Multiple zeta values have been checked
+ * by means of evaluations into simple zeta values. Harmonic polylogarithms have been checked by
+ * comparison to S(n,p,x) for corresponding parameter combinations and by continuity checks
+ * around |x|=1 along with comparisons to corresponding zeta functions. Multiple polylogarithms were
+ * checked against H and zeta and by means of shuffle and quasi-shuffle relations.
*
*/
/*
- * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2015 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
- * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
-#include <stdexcept>
-#include <vector>
-#include <cln/cln.h>
-
#include "inifcns.h"
+
+#include "add.h"
+#include "constant.h"
#include "lst.h"
+#include "mul.h"
#include "numeric.h"
#include "operators.h"
-#include "relational.h"
+#include "power.h"
#include "pseries.h"
+#include "relational.h"
+#include "symbol.h"
+#include "utils.h"
+#include "wildcard.h"
+#include <cln/cln.h>
+#include <sstream>
+#include <stdexcept>
+#include <vector>
namespace GiNaC {
-
-// lookup table for Euler-MacLaurin optimization
-// see fill_Xn()
-std::vector<std::vector<cln::cl_N> > Xn;
-int xnsize = 0;
+//////////////////////////////////////////////////////////////////////
+//
+// Classical polylogarithm Li(n,x)
+//
+// helper functions
+//
+//////////////////////////////////////////////////////////////////////
-// lookup table for Euler-Zagier-Sums (used for S_n,p(x))
-// see fill_Yn()
-std::vector<std::vector<cln::cl_N> > Yn;
-int ynsize = 0; // number of Yn[]
-int ynlength = 100; // initial length of all Yn[i]
+
+// anonymous namespace for helper functions
+namespace {
-//////////////////////
-// helper functions //
-//////////////////////
+// lookup table for factors built from Bernoulli numbers
+// see fill_Xn()
+std::vector<std::vector<cln::cl_N> > Xn;
+// initial size of Xn that should suffice for 32bit machines (must be even)
+const int xninitsizestep = 26;
+int xninitsize = xninitsizestep;
+int xnsize = 0;
-// This function calculates the X_n. The X_n are needed for the Euler-MacLaurin summation (EMS) of
-// classical polylogarithms.
-// With EMS the polylogs can be calculated as follows:
+// This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms.
+// With these numbers the polylogs can be calculated as follows:
// Li_p (x) = \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with u = -log(1-x)
// X_0(n) = B_n (Bernoulli numbers)
// X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k)
// X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater.
// This results in a slightly more complicated algorithm for the X_n.
// The first index in Xn corresponds to the index of the polylog minus 2.
-// The second index in Xn corresponds to the index from the EMS.
-static void fill_Xn(int n)
+// The second index in Xn corresponds to the index from the actual sum.
+void fill_Xn(int n)
{
- // rule of thumb. needs to be improved. TODO
- const int initsize = Digits * 3 / 2;
-
if (n>1) {
// calculate X_2 and higher (corresponding to Li_4 and higher)
- std::vector<cln::cl_N> buf(initsize);
+ std::vector<cln::cl_N> buf(xninitsize);
std::vector<cln::cl_N>::iterator it = buf.begin();
cln::cl_N result;
*it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1
it++;
- for (int i=2; i<=initsize; i++) {
+ for (int i=2; i<=xninitsize; i++) {
if (i&1) {
result = 0; // k == 0
} else {
Xn.push_back(buf);
} else if (n==1) {
// special case to handle the X_0 correct
- std::vector<cln::cl_N> buf(initsize);
+ std::vector<cln::cl_N> buf(xninitsize);
std::vector<cln::cl_N>::iterator it = buf.begin();
cln::cl_N result;
*it = cln::cl_I(-3)/cln::cl_I(4); // i == 1
it++;
*it = cln::cl_I(17)/cln::cl_I(36); // i == 2
it++;
- for (int i=3; i<=initsize; i++) {
+ for (int i=3; i<=xninitsize; i++) {
if (i & 1) {
result = -Xn[0][(i-3)/2]/2;
*it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result;
Xn.push_back(buf);
} else {
// calculate X_0
- std::vector<cln::cl_N> buf(initsize/2);
+ std::vector<cln::cl_N> buf(xninitsize/2);
std::vector<cln::cl_N>::iterator it = buf.begin();
- for (int i=1; i<=initsize/2; i++) {
+ for (int i=1; i<=xninitsize/2; i++) {
*it = bernoulli(i*2).to_cl_N();
it++;
}
xnsize++;
}
-
-// This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
-// The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
-// representing S_{n,p}(x).
-// The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
-// equivalent Z-sum.
-// The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
-// representing S_{n,p}(x).
-// The calculation of Y_n uses the values from Y_{n-1}.
-static void fill_Yn(int n, const cln::float_format_t& prec)
-{
- // TODO -> get rid of the magic number
- const int initsize = ynlength;
- //const int initsize = initsize_Yn;
- cln::cl_N one = cln::cl_float(1, prec);
-
- if (n) {
- std::vector<cln::cl_N> buf(initsize);
- std::vector<cln::cl_N>::iterator it = buf.begin();
- std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
- *it = (*itprev) / cln::cl_N(n+1) * one;
- it++;
- itprev++;
- // sums with an index smaller than the depth are zero and need not to be calculated.
- // calculation starts with depth, which is n+2)
- for (int i=n+2; i<=initsize+n; i++) {
- *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
- it++;
- itprev++;
- }
- Yn.push_back(buf);
- } else {
- std::vector<cln::cl_N> buf(initsize);
- std::vector<cln::cl_N>::iterator it = buf.begin();
- *it = 1 * one;
- it++;
- for (int i=2; i<=initsize; i++) {
- *it = *(it-1) + 1 / cln::cl_N(i) * one;
- it++;
- }
- Yn.push_back(buf);
- }
- ynsize++;
-}
-
-
-// make Yn longer ...
-static void make_Yn_longer(int newsize, const cln::float_format_t& prec)
+// doubles the number of entries in each Xn[]
+void double_Xn()
{
-
- cln::cl_N one = cln::cl_float(1, prec);
-
- Yn[0].resize(newsize);
- std::vector<cln::cl_N>::iterator it = Yn[0].begin();
- it += ynlength;
- for (int i=ynlength+1; i<=newsize; i++) {
- *it = *(it-1) + 1 / cln::cl_N(i) * one;
- it++;
+ const int pos0 = xninitsize / 2;
+ // X_0
+ for (int i=1; i<=xninitsizestep/2; ++i) {
+ Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N());
}
-
- for (int n=1; n<ynsize; n++) {
- Yn[n].resize(newsize);
- std::vector<cln::cl_N>::iterator it = Yn[n].begin();
- std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
- it += ynlength;
- itprev += ynlength;
- for (int i=ynlength+n+1; i<=newsize+n; i++) {
- *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
- it++;
- itprev++;
+ if (Xn.size() > 1) {
+ int xend = xninitsize + xninitsizestep;
+ cln::cl_N result;
+ // X_1
+ for (int i=xninitsize+1; i<=xend; ++i) {
+ if (i & 1) {
+ result = -Xn[0][(i-3)/2]/2;
+ Xn[1].push_back((cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result);
+ } else {
+ result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1);
+ for (int k=1; k<i/2; k++) {
+ result = result + cln::binomial(i,k*2) * Xn[0][k-1] * Xn[0][i/2-k-1] / (k*2+1);
+ }
+ Xn[1].push_back(result);
+ }
+ }
+ // X_n
+ for (size_t n=2; n<Xn.size(); ++n) {
+ for (int i=xninitsize+1; i<=xend; ++i) {
+ if (i & 1) {
+ result = 0; // k == 0
+ } else {
+ result = Xn[0][i/2-1]; // k == 0
+ }
+ for (int k=1; k<i-1; ++k) {
+ if ( !(((i-k) & 1) && ((i-k) > 1)) ) {
+ result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1);
+ }
+ }
+ result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1
+ result = result + Xn[n-1][i-1] / (i+1); // k == i
+ Xn[n].push_back(result);
+ }
}
}
-
- ynlength = newsize;
+ xninitsize += xninitsizestep;
}
-// calculates Li(2,x) without EuMac
-static cln::cl_N Li2_series(const cln::cl_N& x)
+// calculates Li(2,x) without Xn
+cln::cl_N Li2_do_sum(const cln::cl_N& x)
{
cln::cl_N res = x;
cln::cl_N resbuf;
- cln::cl_N num = x;
+ cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits));
cln::cl_I den = 1; // n^2 = 1
unsigned i = 3;
do {
}
-// calculates Li(2,x) with EuMac
-static cln::cl_N Li2_series_EuMac(const cln::cl_N& x)
+// calculates Li(2,x) with Xn
+cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x)
{
std::vector<cln::cl_N>::const_iterator it = Xn[0].begin();
+ std::vector<cln::cl_N>::const_iterator xend = Xn[0].end();
cln::cl_N u = -cln::log(1-x);
- cln::cl_N factor = u;
- cln::cl_N res = u - u*u/4;
+ cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
+ cln::cl_N uu = cln::square(u);
+ cln::cl_N res = u - uu/4;
cln::cl_N resbuf;
unsigned i = 1;
do {
resbuf = res;
- factor = factor * u*u / (2*i * (2*i+1));
+ factor = factor * uu / (2*i * (2*i+1));
res = res + (*it) * factor;
- it++; // should we check it? or rely on initsize? ...
i++;
+ if (++it == xend) {
+ double_Xn();
+ it = Xn[0].begin() + (i-1);
+ xend = Xn[0].end();
+ }
} while (res != resbuf);
return res;
}
-// calculates Li(n,x), n>2 without EuMac
-static cln::cl_N Lin_series(int n, const cln::cl_N& x)
+// calculates Li(n,x), n>2 without Xn
+cln::cl_N Lin_do_sum(int n, const cln::cl_N& x)
{
- cln::cl_N factor = x;
+ cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits));
cln::cl_N res = x;
cln::cl_N resbuf;
int i=2;
}
-// calculates Li(n,x), n>2 with EuMac
-static cln::cl_N Lin_series_EuMac(int n, const cln::cl_N& x)
+// calculates Li(n,x), n>2 with Xn
+cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x)
{
std::vector<cln::cl_N>::const_iterator it = Xn[n-2].begin();
+ std::vector<cln::cl_N>::const_iterator xend = Xn[n-2].end();
cln::cl_N u = -cln::log(1-x);
- cln::cl_N factor = u;
+ cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
cln::cl_N res = u;
cln::cl_N resbuf;
unsigned i=2;
resbuf = res;
factor = factor * u / i;
res = res + (*it) * factor;
- it++; // should we check it? or rely on initsize? ...
i++;
+ if (++it == xend) {
+ double_Xn();
+ it = Xn[n-2].begin() + (i-2);
+ xend = Xn[n-2].end();
+ }
} while (res != resbuf);
return res;
}
// forward declaration needed by function Li_projection and C below
-static numeric S_num(int n, int p, const numeric& x);
+const cln::cl_N S_num(int n, int p, const cln::cl_N& x);
// helper function for classical polylog Li
-static cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
+cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec)
{
// treat n=2 as special case
if (n == 2) {
// it solves also the problem with precision due to the u=-log(1-x) transformation
if (cln::abs(cln::realpart(x)) < 0.25) {
- return Li2_series(x);
+ return Li2_do_sum(x);
} else {
- return Li2_series_EuMac(x);
+ return Li2_do_sum_Xn(x);
}
} else {
// choose the faster algorithm
if (cln::abs(cln::realpart(x)) > 0.75) {
- return -Li2_series(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
+ if ( x == 1 ) {
+ return cln::zeta(2);
+ } else {
+ return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
+ }
} else {
- return -Li2_series_EuMac(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
+ return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
}
}
} else {
if (cln::realpart(x) < 0.5) {
// choose the faster algorithm
- // with n>=12 the "normal" summation always wins against EuMac
+ // with n>=12 the "normal" summation always wins against the method with Xn
if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) {
- return Lin_series(n, x);
+ return Lin_do_sum(n, x);
} else {
- return Lin_series_EuMac(n, x);
+ return Lin_do_sum_Xn(n, x);
}
} else {
- cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
+ cln::cl_N result = 0;
+ if ( x != 1 ) result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
for (int j=0; j<n-1; j++) {
- result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x).to_cl_N())
- * cln::expt(cln::log(x), j) / cln::factorial(j);
+ result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
+ * cln::expt(cln::log(x), j) / cln::factorial(j);
}
return result;
}
}
}
-
// helper function for classical polylog Li
-static numeric Li_num(int n, const numeric& x)
+const cln::cl_N Lin_numeric(const int n, const cln::cl_N& x)
{
if (n == 1) {
// just a log
- return -cln::log(1-x.to_cl_N());
+ return -cln::log(1-x);
}
- if (x.is_zero()) {
+ if (zerop(x)) {
return 0;
}
if (x == 1) {
// [Kol] (2.22)
return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
}
-
+ if (cln::abs(realpart(x)) < 0.4 && cln::abs(cln::abs(x)-1) < 0.01) {
+ cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1);
+ for (int j=0; j<n-1; j++) {
+ result = result + (S_num(n-j-1, 1, 1) - S_num(1, n-j-1, 1-x))
+ * cln::expt(cln::log(x), j) / cln::factorial(j);
+ }
+ return result;
+ }
+
// what is the desired float format?
// first guess: default format
cln::float_format_t prec = cln::default_float_format;
- const cln::cl_N value = x.to_cl_N();
+ const cln::cl_N value = x;
// second guess: the argument's format
- if (!x.real().is_rational())
+ if (!instanceof(realpart(x), cln::cl_RA_ring))
prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
- else if (!x.imag().is_rational())
+ else if (!instanceof(imagpart(x), cln::cl_RA_ring))
prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
// [Kol] (5.15)
cln::cl_N add;
for (int j=0; j<n-1; j++) {
add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
- * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
+ * Lin_numeric(n-j,1) * cln::expt(cln::log(-value),j) / cln::factorial(j);
}
result = result - add;
return result;
}
-// helper function for S(n,p,x)
-static cln::cl_N numeric_nielsen(int n, int step)
+} // end of anonymous namespace
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Multiple polylogarithm Li(n,x)
+//
+// helper function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+// anonymous namespace for helper function
+namespace {
+
+
+// performs the actual series summation for multiple polylogarithms
+cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
{
- if (step) {
- cln::cl_N res;
- for (int i=1; i<n; i++) {
- res = res + numeric_nielsen(i, step-1) / cln::cl_I(i);
- }
- return res;
+ // ensure all x <> 0.
+ for (std::vector<cln::cl_N>::const_iterator it = x.begin(); it != x.end(); ++it) {
+ if ( *it == 0 ) return cln::cl_float(0, cln::float_format(Digits));
}
- else {
- return 1;
- }
-}
+ const int j = s.size();
+ bool flag_accidental_zero = false;
-// helper function for S(n,p,x)
-// [Kol] (7.2)
-static cln::cl_N C(int n, int p)
-{
- cln::cl_N result;
+ std::vector<cln::cl_N> t(j);
+ cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
- for (int k=0; k<p; k++) {
- for (int j=0; j<=(n+k-1)/2; j++) {
- if (k == 0) {
- if (n & 1) {
- if (j & 1) {
- result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
- }
- else {
- result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1).to_cl_N() / cln::factorial(2*j);
- }
- }
- }
- else {
- if (k & 1) {
- if (j & 1) {
- 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()
- / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
- }
- else {
- 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()
- / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
- }
- }
- else {
- if (j & 1) {
- 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()
- / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
- }
- else {
- 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()
- / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
- }
- }
- }
- }
- }
- int np = n+p;
- if ((np-1) & 1) {
- if (((np)/2+n) & 1) {
- result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
+ cln::cl_N t0buf;
+ int q = 0;
+ do {
+ t0buf = t[0];
+ q++;
+ t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
+ for (int k=j-2; k>=0; k--) {
+ 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]);
}
- else {
- result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
+ q++;
+ t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one;
+ for (int k=j-2; k>=0; k--) {
+ flag_accidental_zero = cln::zerop(t[k+1]);
+ 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]);
}
- }
+ } while ( (t[0] != t0buf) || cln::zerop(t[0]) || flag_accidental_zero );
- return result;
+ return t[0];
}
-// helper function for S(n,p,x)
-// [Kol] remark to (9.1)
-static cln::cl_N a_k(int k)
-{
- cln::cl_N result;
-
- if (k == 0) {
- return 1;
- }
+// forward declaration for Li_eval()
+lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
- result = result;
- for (int m=2; m<=k; m++) {
- result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
- }
- return -result / k;
-}
+// type used by the transformation functions for G
+typedef std::vector<int> Gparameter;
-// helper function for S(n,p,x)
-// [Kol] remark to (9.1)
-static cln::cl_N b_k(int k)
+// G_eval1-function for G transformations
+ex G_eval1(int a, int scale, const exvector& gsyms)
{
- cln::cl_N result;
-
- if (k == 0) {
- return 1;
- }
-
- result = result;
- for (int m=2; m<=k; m++) {
- result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
+ if (a != 0) {
+ const ex& scs = gsyms[std::abs(scale)];
+ const ex& as = gsyms[std::abs(a)];
+ if (as != scs) {
+ return -log(1 - scs/as);
+ } else {
+ return -zeta(1);
+ }
+ } else {
+ return log(gsyms[std::abs(scale)]);
}
-
- return result / k;
}
-// helper function for S(n,p,x)
-static cln::cl_N S_series(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
+// G_eval-function for G transformations
+ex G_eval(const Gparameter& a, int scale, const exvector& gsyms)
{
- if (p==1) {
- return Li_projection(n+1, x, prec);
+ // check for properties of G
+ ex sc = gsyms[std::abs(scale)];
+ lst newa;
+ bool all_zero = true;
+ bool all_ones = true;
+ int count_ones = 0;
+ for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
+ if (*it != 0) {
+ const ex sym = gsyms[std::abs(*it)];
+ newa.append(sym);
+ all_zero = false;
+ if (sym != sc) {
+ all_ones = false;
+ }
+ if (all_ones) {
+ ++count_ones;
+ }
+ } else {
+ all_ones = false;
+ }
}
-
- // TODO -> check for vector boundaries and do missing calculations
- // check if precalculated values are sufficient
- if (p > ynsize+1) {
- for (int i=ynsize; i<p-1; i++) {
- fill_Yn(i, prec);
+ // care about divergent G: shuffle to separate divergencies that will be canceled
+ // later on in the transformation
+ if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) {
+ // do shuffle
+ Gparameter short_a;
+ Gparameter::const_iterator it = a.begin();
+ ++it;
+ for (; it != a.end(); ++it) {
+ short_a.push_back(*it);
+ }
+ ex result = G_eval1(a.front(), scale, gsyms) * G_eval(short_a, scale, gsyms);
+ it = short_a.begin();
+ for (int i=1; i<count_ones; ++i) {
+ ++it;
+ }
+ for (; it != short_a.end(); ++it) {
+
+ Gparameter newa;
+ Gparameter::const_iterator it2 = short_a.begin();
+ for (; it2 != it; ++it2) {
+ newa.push_back(*it2);
+ }
+ newa.push_back(*it);
+ newa.push_back(a[0]);
+ it2 = it;
+ ++it2;
+ for (; it2 != short_a.end(); ++it2) {
+ newa.push_back(*it2);
+ }
+ result -= G_eval(newa, scale, gsyms);
}
+ return result / count_ones;
}
- // should be done otherwise
- cln::cl_N xf = x * cln::cl_float(1, prec);
+ // G({1,...,1};y) -> G({1};y)^k / k!
+ if (all_ones && a.size() > 1) {
+ return pow(G_eval1(a.front(),scale, gsyms), count_ones) / factorial(count_ones);
+ }
- cln::cl_N result;
- cln::cl_N resultbuffer;
- int i;
- for (i=p; true; i++) {
- resultbuffer = result;
- if (i-p >= ynlength) {
- // make Yn longer
- make_Yn_longer(ynlength*2, prec);
- }
- result = result + cln::expt(xf,i) / cln::expt(cln::cl_I(i),n+1) * Yn[p-2][i-p]; // should we check it? or rely on magic number? ...
- if (cln::zerop(result-resultbuffer)) {
- break;
+ // G({0,...,0};y) -> log(y)^k / k!
+ if (all_zero) {
+ return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size());
+ }
+
+ // no special cases anymore -> convert it into Li
+ lst m;
+ lst x;
+ ex argbuf = gsyms[std::abs(scale)];
+ ex mval = _ex1;
+ for (Gparameter::const_iterator it=a.begin(); it!=a.end(); ++it) {
+ if (*it != 0) {
+ const ex& sym = gsyms[std::abs(*it)];
+ x.append(argbuf / sym);
+ m.append(mval);
+ mval = _ex1;
+ argbuf = sym;
+ } else {
+ ++mval;
}
}
-
- return result;
+ return pow(-1, x.nops()) * Li(m, x);
}
-// helper function for S(n,p,x)
-static cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
+// converts data for G: pending_integrals -> a
+Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals)
{
- // [Kol] (5.3)
- if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
+ GINAC_ASSERT(pending_integrals.size() != 1);
- cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
- * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
+ if (pending_integrals.size() > 0) {
+ // get rid of the first element, which would stand for the new upper limit
+ Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end());
+ return new_a;
+ } else {
+ // just return empty parameter list
+ Gparameter new_a;
+ return new_a;
+ }
+}
- for (int s=0; s<n; s++) {
- cln::cl_N res2;
- for (int r=0; r<p; r++) {
- res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
- * S_series(p-r,n-s,1-x,prec) / cln::factorial(r);
+
+// check the parameters a and scale for G and return information about convergence, depth, etc.
+// convergent : true if G(a,scale) is convergent
+// depth : depth of G(a,scale)
+// trailing_zeros : number of trailing zeros of a
+// min_it : iterator of a pointing on the smallest element in a
+Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale,
+ bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it)
+{
+ convergent = true;
+ depth = 0;
+ trailing_zeros = 0;
+ min_it = a.end();
+ Gparameter::const_iterator lastnonzero = a.end();
+ for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
+ if (std::abs(*it) > 0) {
+ ++depth;
+ trailing_zeros = 0;
+ lastnonzero = it;
+ if (std::abs(*it) < scale) {
+ convergent = false;
+ if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) {
+ min_it = it;
+ }
}
- result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
+ } else {
+ ++trailing_zeros;
}
-
- return result;
}
-
- return S_series(n, p, x, prec);
+ if (lastnonzero == a.end())
+ return a.end();
+ return ++lastnonzero;
}
-// helper function for S(n,p,x)
-static numeric S_num(int n, int p, const numeric& x)
+// add scale to pending_integrals if pending_integrals is empty
+Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale)
{
- if (x == 1) {
- if (n == 1) {
- // [Kol] (2.22) with (2.21)
- return cln::zeta(p+1);
- }
+ GINAC_ASSERT(pending_integrals.size() != 1);
- if (p == 1) {
- // [Kol] (2.22)
- return cln::zeta(n+1);
+ if (pending_integrals.size() > 0) {
+ return pending_integrals;
+ } else {
+ Gparameter new_pending_integrals;
+ new_pending_integrals.push_back(scale);
+ return new_pending_integrals;
+ }
+}
+
+
+// handles trailing zeroes for an otherwise convergent integral
+ex trailing_zeros_G(const Gparameter& a, int scale, const exvector& gsyms)
+{
+ bool convergent;
+ int depth, trailing_zeros;
+ Gparameter::const_iterator last, dummyit;
+ last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit);
+
+ GINAC_ASSERT(convergent);
+
+ if ((trailing_zeros > 0) && (depth > 0)) {
+ ex result;
+ Gparameter new_a(a.begin(), a.end()-1);
+ result += G_eval1(0, scale, gsyms) * trailing_zeros_G(new_a, scale, gsyms);
+ for (Gparameter::const_iterator it = a.begin(); it != last; ++it) {
+ Gparameter new_a(a.begin(), it);
+ new_a.push_back(0);
+ new_a.insert(new_a.end(), it, a.end()-1);
+ result -= trailing_zeros_G(new_a, scale, gsyms);
}
- // [Kol] (9.1)
- cln::cl_N result;
- for (int nu=0; nu<n; nu++) {
- for (int rho=0; rho<=p; rho++) {
- result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
- * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
- }
+ return result / trailing_zeros;
+ } else {
+ return G_eval(a, scale, gsyms);
+ }
+}
+
+
+// G transformation [VSW] (57),(58)
+ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale, const exvector& gsyms)
+{
+ // pendint = ( y1, b1, ..., br )
+ // a = ( 0, ..., 0, amin )
+ // scale = y2
+ //
+ // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2)
+ // where sr replaces amin
+
+ GINAC_ASSERT(a.back() != 0);
+ GINAC_ASSERT(a.size() > 0);
+
+ ex result;
+ Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back()));
+ const int psize = pending_integrals.size();
+
+ // length == 1
+ // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+})
+
+ if (a.size() == 1) {
+
+ // ln(-y2_{-+})
+ result += log(gsyms[ex_to<numeric>(scale).to_int()]);
+ if (a.back() > 0) {
+ new_pending_integrals.push_back(-scale);
+ result += I*Pi;
+ } else {
+ new_pending_integrals.push_back(scale);
+ result -= I*Pi;
}
- result = result * cln::expt(cln::cl_I(-1),n+p-1);
+ if (psize) {
+ result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
+ pending_integrals.front(),
+ gsyms);
+ }
+
+ // G(y2_{-+}; sr)
+ result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
+ new_pending_integrals.front(),
+ gsyms);
+
+ // G(0; sr)
+ new_pending_integrals.back() = 0;
+ result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals),
+ new_pending_integrals.front(),
+ gsyms);
return result;
}
- else if (x == -1) {
- // [Kol] (2.22)
- if (p == 1) {
- return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
- }
-// throw std::runtime_error("don't know how to evaluate this function!");
+
+ // length > 1
+ // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
+ // - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
+
+ //term zeta_m
+ result -= zeta(a.size());
+ if (psize) {
+ result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
+ pending_integrals.front(),
+ gsyms);
+ }
+
+ // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
+ // = int_0^sr dt/t G_{m-1}( t_{+-}; y2 )
+ Gparameter new_a(a.begin()+1, a.end());
+ new_pending_integrals.push_back(0);
+ result -= depth_one_trafo_G(new_pending_integrals, new_a, scale, gsyms);
+
+ // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
+ // = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 )
+ Gparameter new_pending_integrals_2;
+ new_pending_integrals_2.push_back(scale);
+ new_pending_integrals_2.push_back(0);
+ if (psize) {
+ result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals),
+ pending_integrals.front(),
+ gsyms)
+ * depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
+ } else {
+ result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms);
}
- // what is the desired float format?
- // first guess: default format
- cln::float_format_t prec = cln::default_float_format;
- const cln::cl_N value = x.to_cl_N();
- // second guess: the argument's format
- if (!x.real().is_rational())
- prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
- else if (!x.imag().is_rational())
- prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
+ return result;
+}
- // [Kol] (5.3)
- if (cln::realpart(value) < -0.5) {
+// forward declaration
+ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
+ const Gparameter& pendint, const Gparameter& a_old, int scale,
+ const exvector& gsyms, bool flag_trailing_zeros_only);
- cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
- * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
- for (int s=0; s<n; s++) {
- cln::cl_N res2;
- for (int r=0; r<p; r++) {
- res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
- * S_num(p-r,n-s,1-value).to_cl_N() / cln::factorial(r);
- }
- result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1).to_cl_N() - res2) / cln::factorial(s);
+// G transformation [VSW]
+ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale,
+ const exvector& gsyms, bool flag_trailing_zeros_only)
+{
+ // main recursion routine
+ //
+ // pendint = ( y1, b1, ..., br )
+ // a = ( a1, ..., amin, ..., aw )
+ // scale = y2
+ //
+ // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
+ // where sr replaces amin
+
+ // find smallest alpha, determine depth and trailing zeros, and check for convergence
+ bool convergent;
+ int depth, trailing_zeros;
+ Gparameter::const_iterator min_it;
+ Gparameter::const_iterator firstzero =
+ check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it);
+ int min_it_pos = min_it - a.begin();
+
+ // special case: all a's are zero
+ if (depth == 0) {
+ ex result;
+
+ if (a.size() == 0) {
+ result = 1;
+ } else {
+ result = G_eval(a, scale, gsyms);
}
-
+ if (pendint.size() > 0) {
+ result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
+ pendint.front(),
+ gsyms);
+ }
return result;
-
}
- // [Kol] (5.12)
- if (cln::abs(value) > 1) {
-
- cln::cl_N result;
- for (int s=0; s<p; s++) {
- for (int r=0; r<=s; r++) {
- result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
- / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
- * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
- }
+ // handle trailing zeros
+ if (trailing_zeros > 0) {
+ ex result;
+ Gparameter new_a(a.begin(), a.end()-1);
+ result += G_eval1(0, scale, gsyms) * G_transform(pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
+ for (Gparameter::const_iterator it = a.begin(); it != firstzero; ++it) {
+ Gparameter new_a(a.begin(), it);
+ new_a.push_back(0);
+ new_a.insert(new_a.end(), it, a.end()-1);
+ result -= G_transform(pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
}
- result = result * cln::expt(cln::cl_I(-1),n);
+ return result / trailing_zeros;
+ }
- cln::cl_N res2;
- for (int r=0; r<n; r++) {
- res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
+ // convergence case or flag_trailing_zeros_only
+ if (convergent || flag_trailing_zeros_only) {
+ if (pendint.size() > 0) {
+ return G_eval(convert_pending_integrals_G(pendint),
+ pendint.front(), gsyms)*
+ G_eval(a, scale, gsyms);
+ } else {
+ return G_eval(a, scale, gsyms);
}
- res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
+ }
- result = result + cln::expt(cln::cl_I(-1),p) * res2;
+ // call basic transformation for depth equal one
+ if (depth == 1) {
+ return depth_one_trafo_G(pendint, a, scale, gsyms);
+ }
+ // do recursion
+ // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
+ // = int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2)
+ // + int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) int_0^{sr} ds_{r+1} d/ds_{r+1} G(a1,...,s_{r+1},...,aw,y2)
+
+ // smallest element in last place
+ if (min_it + 1 == a.end()) {
+ do { --min_it; } while (*min_it == 0);
+ Gparameter empty;
+ Gparameter a1(a.begin(),min_it+1);
+ Gparameter a2(min_it+1,a.end());
+
+ ex result = G_transform(pendint, a2, scale, gsyms, flag_trailing_zeros_only)*
+ G_transform(empty, a1, scale, gsyms, flag_trailing_zeros_only);
+
+ result -= shuffle_G(empty, a1, a2, pendint, a, scale, gsyms, flag_trailing_zeros_only);
return result;
}
- else {
- return S_projection(n, p, value, prec);
- }
-}
-
-// helper function for multiple polylogarithm
-static cln::cl_N numeric_zsum(int n, std::vector<cln::cl_N>& x, std::vector<cln::cl_N>& m)
-{
- cln::cl_N res;
- if (x.empty()) {
- return 1;
+ Gparameter empty;
+ Gparameter::iterator changeit;
+
+ // first term G(a_1,..,0,...,a_w;a_0)
+ Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]);
+ Gparameter new_a = a;
+ new_a[min_it_pos] = 0;
+ ex result = G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
+ if (pendint.size() > 0) {
+ result *= trailing_zeros_G(convert_pending_integrals_G(pendint),
+ pendint.front(), gsyms);
}
- for (int i=1; i<n; i++) {
- std::vector<cln::cl_N>::iterator be;
- std::vector<cln::cl_N>::iterator en;
- be = x.begin();
- be++;
- en = x.end();
- std::vector<cln::cl_N> xbuf(be, en);
- be = m.begin();
- be++;
- en = m.end();
- std::vector<cln::cl_N> mbuf(be, en);
- res = res + cln::expt(x[0],i) / cln::expt(i,m[0]) * numeric_zsum(i, xbuf, mbuf);
+
+ // other terms
+ changeit = new_a.begin() + min_it_pos;
+ changeit = new_a.erase(changeit);
+ if (changeit != new_a.begin()) {
+ // smallest in the middle
+ new_pendint.push_back(*changeit);
+ result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
+ new_pendint.front(), gsyms)*
+ G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
+ int buffer = *changeit;
+ *changeit = *min_it;
+ result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
+ *changeit = buffer;
+ new_pendint.pop_back();
+ --changeit;
+ new_pendint.push_back(*changeit);
+ result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
+ new_pendint.front(), gsyms)*
+ G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
+ *changeit = *min_it;
+ result -= G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
+ } else {
+ // smallest at the front
+ new_pendint.push_back(scale);
+ result += trailing_zeros_G(convert_pending_integrals_G(new_pendint),
+ new_pendint.front(), gsyms)*
+ G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
+ new_pendint.back() = *changeit;
+ result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint),
+ new_pendint.front(), gsyms)*
+ G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only);
+ *changeit = *min_it;
+ result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only);
}
- return res;
+ return result;
}
-// helper function for harmonic polylogarithm
-static cln::cl_N numeric_harmonic(int n, std::vector<cln::cl_N>& m)
+// shuffles the two parameter list a1 and a2 and calls G_transform for every term except
+// for the one that is equal to a_old
+ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
+ const Gparameter& pendint, const Gparameter& a_old, int scale,
+ const exvector& gsyms, bool flag_trailing_zeros_only)
{
- cln::cl_N res;
- if (m.empty()) {
- return 1;
+ if (a1.size()==0 && a2.size()==0) {
+ // veto the one configuration we don't want
+ if ( a0 == a_old ) return 0;
+
+ return G_transform(pendint, a0, scale, gsyms, flag_trailing_zeros_only);
}
- for (int i=1; i<n; i++) {
- std::vector<cln::cl_N>::iterator be;
- std::vector<cln::cl_N>::iterator en;
- be = m.begin();
- be++;
- en = m.end();
- std::vector<cln::cl_N> mbuf(be, en);
- res = res + cln::recip(cln::expt(i,m[0])) * numeric_harmonic(i, mbuf);
+
+ if (a2.size()==0) {
+ Gparameter empty;
+ Gparameter aa0 = a0;
+ aa0.insert(aa0.end(),a1.begin(),a1.end());
+ return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms, flag_trailing_zeros_only);
+ }
+
+ if (a1.size()==0) {
+ Gparameter empty;
+ Gparameter aa0 = a0;
+ aa0.insert(aa0.end(),a2.begin(),a2.end());
+ return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms, flag_trailing_zeros_only);
}
- return res;
-}
+ Gparameter a1_removed(a1.begin()+1,a1.end());
+ Gparameter a2_removed(a2.begin()+1,a2.end());
-/////////////////////////////
-// end of helper functions //
-/////////////////////////////
+ Gparameter a01 = a0;
+ Gparameter a02 = a0;
+ a01.push_back( a1[0] );
+ a02.push_back( a2[0] );
-// Polylogarithm and multiple polylogarithm
+ return shuffle_G(a01, a1_removed, a2, pendint, a_old, scale, gsyms, flag_trailing_zeros_only)
+ + shuffle_G(a02, a1, a2_removed, pendint, a_old, scale, gsyms, flag_trailing_zeros_only);
+}
-static ex Li_eval(const ex& x1, const ex& x2)
+// handles the transformations and the numerical evaluation of G
+// the parameter x, s and y must only contain numerics
+static cln::cl_N
+G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
+ const cln::cl_N& y);
+
+// do acceleration transformation (hoelder convolution [BBB])
+// the parameter x, s and y must only contain numerics
+static cln::cl_N
+G_do_hoelder(std::vector<cln::cl_N> x, /* yes, it's passed by value */
+ const std::vector<int>& s, const cln::cl_N& y)
{
- if (x2.is_zero()) {
- return 0;
- }
- else {
- if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational)))
- return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
- return Li(x1,x2).hold();
+ cln::cl_N result;
+ const std::size_t size = x.size();
+ for (std::size_t i = 0; i < size; ++i)
+ x[i] = x[i]/y;
+
+ for (std::size_t r = 0; r <= size; ++r) {
+ cln::cl_N buffer(1 & r ? -1 : 1);
+ cln::cl_RA p(2);
+ bool adjustp;
+ do {
+ adjustp = false;
+ for (std::size_t i = 0; i < size; ++i) {
+ if (x[i] == cln::cl_RA(1)/p) {
+ p = p/2 + cln::cl_RA(3)/2;
+ adjustp = true;
+ continue;
+ }
+ }
+ } while (adjustp);
+ cln::cl_RA q = p/(p-1);
+ std::vector<cln::cl_N> qlstx;
+ std::vector<int> qlsts;
+ for (std::size_t j = r; j >= 1; --j) {
+ qlstx.push_back(cln::cl_N(1) - x[j-1]);
+ if (instanceof(x[j-1], cln::cl_R_ring) && realpart(x[j-1]) > 1) {
+ qlsts.push_back(1);
+ } else {
+ qlsts.push_back(-s[j-1]);
+ }
+ }
+ if (qlstx.size() > 0) {
+ buffer = buffer*G_numeric(qlstx, qlsts, 1/q);
+ }
+ std::vector<cln::cl_N> plstx;
+ std::vector<int> plsts;
+ for (std::size_t j = r+1; j <= size; ++j) {
+ plstx.push_back(x[j-1]);
+ plsts.push_back(s[j-1]);
+ }
+ if (plstx.size() > 0) {
+ buffer = buffer*G_numeric(plstx, plsts, 1/p);
+ }
+ result = result + buffer;
}
+ return result;
}
-static ex Li_evalf(const ex& x1, const ex& x2)
+class less_object_for_cl_N
{
- // classical polylogs
- if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
- return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
+public:
+ bool operator() (const cln::cl_N & a, const cln::cl_N & b) const
+ {
+ // absolute value?
+ if (abs(a) != abs(b))
+ return (abs(a) < abs(b)) ? true : false;
+
+ // complex phase?
+ if (phase(a) != phase(b))
+ return (phase(a) < phase(b)) ? true : false;
+
+ // equal, therefore "less" is not true
+ return false;
}
- // multiple polylogs
- else if (is_a<lst>(x1) && is_a<lst>(x2)) {
- for (int i=0; i<x1.nops(); i++) {
- if (!is_a<numeric>(x1.op(i)))
- return Li(x1,x2).hold();
- if (!is_a<numeric>(x2.op(i)))
- return Li(x1,x2).hold();
- if (x2.op(i) >= 1)
- return Li(x1,x2).hold();
- }
+};
- cln::cl_N m_1 = ex_to<numeric>(x1.op(x1.nops()-1)).to_cl_N();
- cln::cl_N x_1 = ex_to<numeric>(x2.op(x2.nops()-1)).to_cl_N();
- std::vector<cln::cl_N> x;
- std::vector<cln::cl_N> m;
- const int nops = ex_to<numeric>(x1.nops()).to_int();
- for (int i=nops-2; i>=0; i--) {
- m.push_back(ex_to<numeric>(x1.op(i)).to_cl_N());
- x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
+
+// convergence transformation, used for numerical evaluation of G function.
+// the parameter x, s and y must only contain numerics
+static cln::cl_N
+G_do_trafo(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
+ const cln::cl_N& y, bool flag_trailing_zeros_only)
+{
+ // sort (|x|<->position) to determine indices
+ typedef std::multimap<cln::cl_N, std::size_t, less_object_for_cl_N> sortmap_t;
+ sortmap_t sortmap;
+ std::size_t size = 0;
+ for (std::size_t i = 0; i < x.size(); ++i) {
+ if (!zerop(x[i])) {
+ sortmap.insert(std::make_pair(x[i], i));
+ ++size;
+ }
+ }
+ // include upper limit (scale)
+ sortmap.insert(std::make_pair(y, x.size()));
+
+ // generate missing dummy-symbols
+ int i = 1;
+ // holding dummy-symbols for the G/Li transformations
+ exvector gsyms;
+ gsyms.push_back(symbol("GSYMS_ERROR"));
+ cln::cl_N lastentry(0);
+ for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
+ if (it != sortmap.begin()) {
+ if (it->second < x.size()) {
+ if (x[it->second] == lastentry) {
+ gsyms.push_back(gsyms.back());
+ continue;
+ }
+ } else {
+ if (y == lastentry) {
+ gsyms.push_back(gsyms.back());
+ continue;
+ }
+ }
}
+ std::ostringstream os;
+ os << "a" << i;
+ gsyms.push_back(symbol(os.str()));
+ ++i;
+ if (it->second < x.size()) {
+ lastentry = x[it->second];
+ } else {
+ lastentry = y;
+ }
+ }
- cln::cl_N res;
- cln::cl_N resbuf;
- for (int i=nops; true; i++) {
- resbuf = res;
- res = res + cln::expt(x_1,i) / cln::expt(i,m_1) * numeric_zsum(i, x, m);
- if (cln::zerop(res-resbuf))
- break;
+ // fill position data according to sorted indices and prepare substitution list
+ Gparameter a(x.size());
+ exmap subslst;
+ std::size_t pos = 1;
+ int scale = pos;
+ for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
+ if (it->second < x.size()) {
+ if (s[it->second] > 0) {
+ a[it->second] = pos;
+ } else {
+ a[it->second] = -int(pos);
+ }
+ subslst[gsyms[pos]] = numeric(x[it->second]);
+ } else {
+ scale = pos;
+ subslst[gsyms[pos]] = numeric(y);
}
+ ++pos;
+ }
+
+ // do transformation
+ Gparameter pendint;
+ ex result = G_transform(pendint, a, scale, gsyms, flag_trailing_zeros_only);
+ // replace dummy symbols with their values
+ result = result.eval().expand();
+ result = result.subs(subslst).evalf();
+ if (!is_a<numeric>(result))
+ throw std::logic_error("G_do_trafo: G_transform returned non-numeric result");
+
+ cln::cl_N ret = ex_to<numeric>(result).to_cl_N();
+ return ret;
+}
- return numeric(res);
+// handles the transformations and the numerical evaluation of G
+// the parameter x, s and y must only contain numerics
+static cln::cl_N
+G_numeric(const std::vector<cln::cl_N>& x, const std::vector<int>& s,
+ const cln::cl_N& y)
+{
+ // check for convergence and necessary accelerations
+ bool need_trafo = false;
+ bool need_hoelder = false;
+ bool have_trailing_zero = false;
+ std::size_t depth = 0;
+ for (std::size_t i = 0; i < x.size(); ++i) {
+ if (!zerop(x[i])) {
+ ++depth;
+ const cln::cl_N x_y = abs(x[i]) - y;
+ if (instanceof(x_y, cln::cl_R_ring) &&
+ realpart(x_y) < cln::least_negative_float(cln::float_format(Digits - 2)))
+ need_trafo = true;
+
+ if (abs(abs(x[i]/y) - 1) < 0.01)
+ need_hoelder = true;
+ }
+ }
+ if (zerop(x.back())) {
+ have_trailing_zero = true;
+ need_trafo = true;
+ }
+ if (depth == 1 && x.size() == 2 && !need_trafo)
+ return - Li_projection(2, y/x[1], cln::float_format(Digits));
+
+ // do acceleration transformation (hoelder convolution [BBB])
+ if (need_hoelder && !have_trailing_zero)
+ return G_do_hoelder(x, s, y);
+
+ // convergence transformation
+ if (need_trafo)
+ return G_do_trafo(x, s, y, have_trailing_zero);
+
+ // do summation
+ std::vector<cln::cl_N> newx;
+ newx.reserve(x.size());
+ std::vector<int> m;
+ m.reserve(x.size());
+ int mcount = 1;
+ int sign = 1;
+ cln::cl_N factor = y;
+ for (std::size_t i = 0; i < x.size(); ++i) {
+ if (zerop(x[i])) {
+ ++mcount;
+ } else {
+ newx.push_back(factor/x[i]);
+ factor = x[i];
+ m.push_back(mcount);
+ mcount = 1;
+ sign = -sign;
+ }
}
- return Li(x1,x2).hold();
+ return sign*multipleLi_do_sum(m, newx);
}
-static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
+
+ex mLi_numeric(const lst& m, const lst& x)
{
- epvector seq;
- seq.push_back(expair(Li(x1,x2), 0));
- return pseries(rel,seq);
+ // let G_numeric do the transformation
+ std::vector<cln::cl_N> newx;
+ newx.reserve(x.nops());
+ std::vector<int> s;
+ s.reserve(x.nops());
+ cln::cl_N factor(1);
+ for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
+ for (int i = 1; i < *itm; ++i) {
+ newx.push_back(cln::cl_N(0));
+ s.push_back(1);
+ }
+ const cln::cl_N xi = ex_to<numeric>(*itx).to_cl_N();
+ factor = factor/xi;
+ newx.push_back(factor);
+ if ( !instanceof(factor, cln::cl_R_ring) && imagpart(factor) < 0 ) {
+ s.push_back(-1);
+ }
+ else {
+ s.push_back(1);
+ }
+ }
+ return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1)));
}
-REGISTER_FUNCTION(Li, eval_func(Li_eval).evalf_func(Li_evalf).do_not_evalf_params().series_func(Li_series));
+
+} // end of anonymous namespace
-// Nielsen's generalized polylogarithm
+//////////////////////////////////////////////////////////////////////
+//
+// Generalized multiple polylogarithm G(x, y) and G(x, s, y)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
-static ex S_eval(const ex& x1, const ex& x2, const ex& x3)
+
+static ex G2_evalf(const ex& x_, const ex& y)
{
- if (x2 == 1) {
- return Li(x1+1,x3);
+ if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
+ return G(x_, y).hold();
}
- if (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational)) &&
- x1.info(info_flags::posint) && x2.info(info_flags::posint)) {
- return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
+ lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+ if (x.nops() == 0) {
+ return _ex1;
}
- return S(x1,x2,x3).hold();
-}
-
-static ex S_evalf(const ex& x1, const ex& x2, const ex& x3)
-{
- if (is_a<numeric>(x1) && is_a<numeric>(x2) && is_a<numeric>(x3)) {
- if ((x3 == -1) && (x2 != 1)) {
- // no formula to evaluate this ... sorry
-// return S(x1,x2,x3).hold();
+ if (x.op(0) == y) {
+ return G(x_, y).hold();
+ }
+ std::vector<int> s;
+ s.reserve(x.nops());
+ bool all_zero = true;
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+ if (!(*it).info(info_flags::numeric)) {
+ return G(x_, y).hold();
+ }
+ if (*it != _ex0) {
+ all_zero = false;
}
- return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
+ if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
+ s.push_back(-1);
+ }
+ else {
+ s.push_back(1);
+ }
+ }
+ if (all_zero) {
+ return pow(log(y), x.nops()) / factorial(x.nops());
}
- return S(x1,x2,x3).hold();
+ std::vector<cln::cl_N> xv;
+ xv.reserve(x.nops());
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
+ xv.push_back(ex_to<numeric>(*it).to_cl_N());
+ cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
+ return numeric(result);
}
-static ex S_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options)
+
+static ex G2_eval(const ex& x_, const ex& y)
{
- epvector seq;
- seq.push_back(expair(S(x1,x2,x3), 0));
- return pseries(rel,seq);
+ //TODO eval to MZV or H or S or Lin
+
+ if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
+ return G(x_, y).hold();
+ }
+ lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ if (x.op(0) == y) {
+ return G(x_, y).hold();
+ }
+ std::vector<int> s;
+ s.reserve(x.nops());
+ bool all_zero = true;
+ bool crational = true;
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+ if (!(*it).info(info_flags::numeric)) {
+ return G(x_, y).hold();
+ }
+ if (!(*it).info(info_flags::crational)) {
+ crational = false;
+ }
+ if (*it != _ex0) {
+ all_zero = false;
+ }
+ if ( !ex_to<numeric>(*it).is_real() && ex_to<numeric>(*it).imag() < 0 ) {
+ s.push_back(-1);
+ }
+ else {
+ s.push_back(+1);
+ }
+ }
+ if (all_zero) {
+ return pow(log(y), x.nops()) / factorial(x.nops());
+ }
+ if (!y.info(info_flags::crational)) {
+ crational = false;
+ }
+ if (crational) {
+ return G(x_, y).hold();
+ }
+ std::vector<cln::cl_N> xv;
+ xv.reserve(x.nops());
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
+ xv.push_back(ex_to<numeric>(*it).to_cl_N());
+ cln::cl_N result = G_numeric(xv, s, ex_to<numeric>(y).to_cl_N());
+ return numeric(result);
}
-REGISTER_FUNCTION(S, eval_func(S_eval).evalf_func(S_evalf).do_not_evalf_params().series_func(S_series));
+// option do_not_evalf_params() removed.
+unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
+ evalf_func(G2_evalf).
+ eval_func(G2_eval).
+ overloaded(2));
+//TODO
+// derivative_func(G2_deriv).
+// print_func<print_latex>(G2_print_latex).
-// Harmonic polylogarithm
-static ex H_eval(const ex& x1, const ex& x2)
+static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
{
- if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) {
- return H(x1,x2).evalf();
+ if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
+ return G(x_, s_, y).hold();
}
- return H(x1,x2).hold();
+ lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+ lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
+ if (x.nops() != s.nops()) {
+ return G(x_, s_, y).hold();
+ }
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ if (x.op(0) == y) {
+ return G(x_, s_, y).hold();
+ }
+ std::vector<int> sn;
+ sn.reserve(s.nops());
+ bool all_zero = true;
+ for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
+ if (!(*itx).info(info_flags::numeric)) {
+ return G(x_, y).hold();
+ }
+ if (!(*its).info(info_flags::real)) {
+ return G(x_, y).hold();
+ }
+ if (*itx != _ex0) {
+ all_zero = false;
+ }
+ if ( ex_to<numeric>(*itx).is_real() ) {
+ if ( ex_to<numeric>(*itx).is_positive() ) {
+ if ( *its >= 0 ) {
+ sn.push_back(1);
+ }
+ else {
+ sn.push_back(-1);
+ }
+ } else {
+ sn.push_back(1);
+ }
+ }
+ else {
+ if ( ex_to<numeric>(*itx).imag() > 0 ) {
+ sn.push_back(1);
+ }
+ else {
+ sn.push_back(-1);
+ }
+ }
+ }
+ if (all_zero) {
+ return pow(log(y), x.nops()) / factorial(x.nops());
+ }
+ std::vector<cln::cl_N> xn;
+ xn.reserve(x.nops());
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
+ xn.push_back(ex_to<numeric>(*it).to_cl_N());
+ cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
+ return numeric(result);
}
-static ex H_evalf(const ex& x1, const ex& x2)
+
+static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
{
- if (is_a<lst>(x1) && is_a<numeric>(x2)) {
- for (int i=0; i<x1.nops(); i++) {
- if (!is_a<numeric>(x1.op(i)))
- return H(x1,x2).hold();
+ //TODO eval to MZV or H or S or Lin
+
+ if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) {
+ return G(x_, s_, y).hold();
+ }
+ lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+ lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
+ if (x.nops() != s.nops()) {
+ return G(x_, s_, y).hold();
+ }
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ if (x.op(0) == y) {
+ return G(x_, s_, y).hold();
+ }
+ std::vector<int> sn;
+ sn.reserve(s.nops());
+ bool all_zero = true;
+ bool crational = true;
+ for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
+ if (!(*itx).info(info_flags::numeric)) {
+ return G(x_, s_, y).hold();
}
- if (x2 >= 1) {
- return H(x1,x2).hold();
+ if (!(*its).info(info_flags::real)) {
+ return G(x_, s_, y).hold();
}
-
- cln::cl_N m_1 = ex_to<numeric>(x1.op(x1.nops()-1)).to_cl_N();
- cln::cl_N x_1 = ex_to<numeric>(x2).to_cl_N();
- std::vector<cln::cl_N> m;
- const int nops = ex_to<numeric>(x1.nops()).to_int();
- for (int i=nops-2; i>=0; i--) {
- m.push_back(ex_to<numeric>(x1.op(i)).to_cl_N());
+ if (!(*itx).info(info_flags::crational)) {
+ crational = false;
}
-
- cln::cl_N res;
- cln::cl_N resbuf;
- for (int i=nops; true; i++) {
- resbuf = res;
- res = res + cln::expt(x_1,i) / cln::expt(i,m_1) * numeric_harmonic(i, m);
- if (cln::zerop(res-resbuf))
- break;
+ if (*itx != _ex0) {
+ all_zero = false;
+ }
+ if ( ex_to<numeric>(*itx).is_real() ) {
+ if ( ex_to<numeric>(*itx).is_positive() ) {
+ if ( *its >= 0 ) {
+ sn.push_back(1);
+ }
+ else {
+ sn.push_back(-1);
+ }
+ } else {
+ sn.push_back(1);
+ }
+ }
+ else {
+ if ( ex_to<numeric>(*itx).imag() > 0 ) {
+ sn.push_back(1);
+ }
+ else {
+ sn.push_back(-1);
+ }
}
-
- return numeric(res);
-
}
-
- return H(x1,x2).hold();
+ if (all_zero) {
+ return pow(log(y), x.nops()) / factorial(x.nops());
+ }
+ if (!y.info(info_flags::crational)) {
+ crational = false;
+ }
+ if (crational) {
+ return G(x_, s_, y).hold();
+ }
+ std::vector<cln::cl_N> xn;
+ xn.reserve(x.nops());
+ for (lst::const_iterator it = x.begin(); it != x.end(); ++it)
+ xn.push_back(ex_to<numeric>(*it).to_cl_N());
+ cln::cl_N result = G_numeric(xn, sn, ex_to<numeric>(y).to_cl_N());
+ return numeric(result);
}
-static ex H_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
-{
- epvector seq;
- seq.push_back(expair(H(x1,x2), 0));
- return pseries(rel,seq);
-}
-REGISTER_FUNCTION(H, eval_func(H_eval).evalf_func(H_evalf).do_not_evalf_params().series_func(H_series));
+// option do_not_evalf_params() removed.
+// This is safe: in the code above it only matters if s_ > 0 or s_ < 0,
+// s_ is allowed to be of floating type.
+unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
+ evalf_func(G3_evalf).
+ eval_func(G3_eval).
+ overloaded(2));
+//TODO
+// derivative_func(G3_deriv).
+// print_func<print_latex>(G3_print_latex).
-// Multiple zeta value
+//////////////////////////////////////////////////////////////////////
+//
+// Classical polylogarithm and multiple polylogarithm Li(m,x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
-static ex mZeta_eval(const ex& x1)
-{
- return mZeta(x1).hold();
-}
-static ex mZeta_evalf(const ex& x1)
+static ex Li_evalf(const ex& m_, const ex& x_)
{
- if (is_a<lst>(x1)) {
- for (int i=0; i<x1.nops(); i++) {
- if (!is_a<numeric>(x1.op(i)))
- return mZeta(x1).hold();
+ // classical polylogs
+ if (m_.info(info_flags::posint)) {
+ if (x_.info(info_flags::numeric)) {
+ int m__ = ex_to<numeric>(m_).to_int();
+ const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
+ const cln::cl_N result = Lin_numeric(m__, x__);
+ return numeric(result);
+ } else {
+ // try to numerically evaluate second argument
+ ex x_val = x_.evalf();
+ if (x_val.info(info_flags::numeric)) {
+ int m__ = ex_to<numeric>(m_).to_int();
+ const cln::cl_N x__ = ex_to<numeric>(x_val).to_cl_N();
+ const cln::cl_N result = Lin_numeric(m__, x__);
+ return numeric(result);
+ }
}
+ }
+ // multiple polylogs
+ if (is_a<lst>(m_) && is_a<lst>(x_)) {
- cln::cl_N m_1 = ex_to<numeric>(x1.op(x1.nops()-1)).to_cl_N();
-
- // check for divergence
- if (m_1 == 1) {
- return mZeta(x1).hold();
+ const lst& m = ex_to<lst>(m_);
+ const lst& x = ex_to<lst>(x_);
+ if (m.nops() != x.nops()) {
+ return Li(m_,x_).hold();
}
-
- std::vector<cln::cl_N> m;
- const int nops = ex_to<numeric>(x1.nops()).to_int();
- for (int i=nops-2; i>=0; i--) {
- m.push_back(ex_to<numeric>(x1.op(i)).to_cl_N());
+ if (x.nops() == 0) {
+ return _ex1;
}
-
- cln::float_format_t prec = cln::default_float_format;
- cln::cl_N res = cln::complex(cln::cl_float(0, prec), 0);
- cln::cl_N resbuf;
- for (int i=nops; true; i++) {
- // to infinity and beyond ... timewise
- resbuf = res;
- res = res + cln::recip(cln::expt(i,m_1)) * numeric_harmonic(i, m);
- if (cln::zerop(res-resbuf))
- break;
+ if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
+ return Li(m_,x_).hold();
}
- return numeric(res);
+ for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
+ if (!(*itm).info(info_flags::posint)) {
+ return Li(m_, x_).hold();
+ }
+ if (!(*itx).info(info_flags::numeric)) {
+ return Li(m_, x_).hold();
+ }
+ if (*itx == _ex0) {
+ return _ex0;
+ }
+ }
+ return mLi_numeric(m, x);
}
- return mZeta(x1).hold();
+ return Li(m_,x_).hold();
}
-static ex mZeta_series(const ex& x1, const relational& rel, int order, unsigned options)
+
+static ex Li_eval(const ex& m_, const ex& x_)
{
- epvector seq;
- seq.push_back(expair(mZeta(x1), 0));
- return pseries(rel,seq);
-}
+ if (is_a<lst>(m_)) {
+ if (is_a<lst>(x_)) {
+ // multiple polylogs
+ const lst& m = ex_to<lst>(m_);
+ const lst& x = ex_to<lst>(x_);
+ if (m.nops() != x.nops()) {
+ return Li(m_,x_).hold();
+ }
+ if (x.nops() == 0) {
+ return _ex1;
+ }
+ bool is_H = true;
+ bool is_zeta = true;
+ bool do_evalf = true;
+ bool crational = true;
+ for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
+ if (!(*itm).info(info_flags::posint)) {
+ return Li(m_,x_).hold();
+ }
+ if ((*itx != _ex1) && (*itx != _ex_1)) {
+ if (itx != x.begin()) {
+ is_H = false;
+ }
+ is_zeta = false;
+ }
+ if (*itx == _ex0) {
+ return _ex0;
+ }
+ if (!(*itx).info(info_flags::numeric)) {
+ do_evalf = false;
+ }
+ if (!(*itx).info(info_flags::crational)) {
+ crational = false;
+ }
+ }
+ if (is_zeta) {
+ lst newx;
+ for (lst::const_iterator itx = x.begin(); itx != x.end(); ++itx) {
+ GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
+ // XXX: 1 + 0.0*I is considered equal to 1. However
+ // the former is a not automatically converted
+ // to a real number. Do the conversion explicitly
+ // to avoid the "numeric::operator>(): complex inequality"
+ // exception (and similar problems).
+ newx.append(*itx != _ex_1 ? _ex1 : _ex_1);
+ }
+ return zeta(m_, newx);
+ }
+ if (is_H) {
+ ex prefactor;
+ lst newm = convert_parameter_Li_to_H(m, x, prefactor);
+ return prefactor * H(newm, x[0]);
+ }
+ if (do_evalf && !crational) {
+ return mLi_numeric(m,x);
+ }
+ }
+ return Li(m_, x_).hold();
+ } else if (is_a<lst>(x_)) {
+ return Li(m_, x_).hold();
+ }
+
+ // classical polylogs
+ if (x_ == _ex0) {
+ return _ex0;
+ }
+ if (x_ == _ex1) {
+ return zeta(m_);
+ }
+ if (x_ == _ex_1) {
+ return (pow(2,1-m_)-1) * zeta(m_);
+ }
+ if (m_ == _ex1) {
+ return -log(1-x_);
+ }
+ if (m_ == _ex2) {
+ if (x_.is_equal(I)) {
+ return power(Pi,_ex2)/_ex_48 + Catalan*I;
+ }
+ if (x_.is_equal(-I)) {
+ return power(Pi,_ex2)/_ex_48 - Catalan*I;
+ }
+ }
+ if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
+ int m__ = ex_to<numeric>(m_).to_int();
+ const cln::cl_N x__ = ex_to<numeric>(x_).to_cl_N();
+ const cln::cl_N result = Lin_numeric(m__, x__);
+ return numeric(result);
+ }
+
+ return Li(m_, x_).hold();
+}
+
+
+static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
+{
+ if (is_a<lst>(m) || is_a<lst>(x)) {
+ // multiple polylog
+ epvector seq;
+ seq.push_back(expair(Li(m, x), 0));
+ return pseries(rel, seq);
+ }
+
+ // classical polylog
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
+ if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) {
+ // First special case: x==0 (derivatives have poles)
+ if (x_pt.is_zero()) {
+ const symbol s;
+ ex ser;
+ // manually construct the primitive expansion
+ for (int i=1; i<order; ++i)
+ ser += pow(s,i) / pow(numeric(i), m);
+ // substitute the argument's series expansion
+ ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
+ // maybe that was terminating, so add a proper order term
+ epvector nseq;
+ nseq.push_back(expair(Order(_ex1), order));
+ ser += pseries(rel, nseq);
+ // reexpanding it will collapse the series again
+ return ser.series(rel, order);
+ }
+ // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
+ throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!");
+ }
+ // all other cases should be safe, by now:
+ throw do_taylor(); // caught by function::series()
+}
+
+
+static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param < 2);
+ if (deriv_param == 0) {
+ return _ex0;
+ }
+ if (m_.nops() > 1) {
+ throw std::runtime_error("don't know how to derivate multiple polylogarithm!");
+ }
+ ex m;
+ if (is_a<lst>(m_)) {
+ m = m_.op(0);
+ } else {
+ m = m_;
+ }
+ ex x;
+ if (is_a<lst>(x_)) {
+ x = x_.op(0);
+ } else {
+ x = x_;
+ }
+ if (m > 0) {
+ return Li(m-1, x) / x;
+ } else {
+ return 1/(1-x);
+ }
+}
+
+
+static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c)
+{
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
+ }
+ lst x;
+ if (is_a<lst>(x_)) {
+ x = ex_to<lst>(x_);
+ } else {
+ x = lst(x_);
+ }
+ c.s << "\\mathrm{Li}_{";
+ lst::const_iterator itm = m.begin();
+ (*itm).print(c);
+ itm++;
+ for (; itm != m.end(); itm++) {
+ c.s << ",";
+ (*itm).print(c);
+ }
+ c.s << "}(";
+ lst::const_iterator itx = x.begin();
+ (*itx).print(c);
+ itx++;
+ for (; itx != x.end(); itx++) {
+ c.s << ",";
+ (*itx).print(c);
+ }
+ c.s << ")";
+}
+
+
+REGISTER_FUNCTION(Li,
+ evalf_func(Li_evalf).
+ eval_func(Li_eval).
+ series_func(Li_series).
+ derivative_func(Li_deriv).
+ print_func<print_latex>(Li_print_latex).
+ do_not_evalf_params());
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Nielsen's generalized polylogarithm S(n,p,x)
+//
+// helper functions
+//
+//////////////////////////////////////////////////////////////////////
+
+
+// anonymous namespace for helper functions
+namespace {
+
+
+// lookup table for special Euler-Zagier-Sums (used for S_n,p(x))
+// see fill_Yn()
+std::vector<std::vector<cln::cl_N> > Yn;
+int ynsize = 0; // number of Yn[]
+int ynlength = 100; // initial length of all Yn[i]
+
+
+// This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x).
+// The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum
+// representing S_{n,p}(x).
+// The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the
+// equivalent Z-sum.
+// The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum
+// representing S_{n,p}(x).
+// The calculation of Y_n uses the values from Y_{n-1}.
+void fill_Yn(int n, const cln::float_format_t& prec)
+{
+ const int initsize = ynlength;
+ //const int initsize = initsize_Yn;
+ cln::cl_N one = cln::cl_float(1, prec);
+
+ if (n) {
+ std::vector<cln::cl_N> buf(initsize);
+ std::vector<cln::cl_N>::iterator it = buf.begin();
+ std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
+ *it = (*itprev) / cln::cl_N(n+1) * one;
+ it++;
+ itprev++;
+ // sums with an index smaller than the depth are zero and need not to be calculated.
+ // calculation starts with depth, which is n+2)
+ for (int i=n+2; i<=initsize+n; i++) {
+ *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
+ it++;
+ itprev++;
+ }
+ Yn.push_back(buf);
+ } else {
+ std::vector<cln::cl_N> buf(initsize);
+ std::vector<cln::cl_N>::iterator it = buf.begin();
+ *it = 1 * one;
+ it++;
+ for (int i=2; i<=initsize; i++) {
+ *it = *(it-1) + 1 / cln::cl_N(i) * one;
+ it++;
+ }
+ Yn.push_back(buf);
+ }
+ ynsize++;
+}
+
+
+// make Yn longer ...
+void make_Yn_longer(int newsize, const cln::float_format_t& prec)
+{
+
+ cln::cl_N one = cln::cl_float(1, prec);
+
+ Yn[0].resize(newsize);
+ std::vector<cln::cl_N>::iterator it = Yn[0].begin();
+ it += ynlength;
+ for (int i=ynlength+1; i<=newsize; i++) {
+ *it = *(it-1) + 1 / cln::cl_N(i) * one;
+ it++;
+ }
+
+ for (int n=1; n<ynsize; n++) {
+ Yn[n].resize(newsize);
+ std::vector<cln::cl_N>::iterator it = Yn[n].begin();
+ std::vector<cln::cl_N>::iterator itprev = Yn[n-1].begin();
+ it += ynlength;
+ itprev += ynlength;
+ for (int i=ynlength+n+1; i<=newsize+n; i++) {
+ *it = *(it-1) + (*itprev) / cln::cl_N(i) * one;
+ it++;
+ itprev++;
+ }
+ }
+
+ ynlength = newsize;
+}
+
+
+// helper function for S(n,p,x)
+// [Kol] (7.2)
+cln::cl_N C(int n, int p)
+{
+ cln::cl_N result;
+
+ for (int k=0; k<p; k++) {
+ for (int j=0; j<=(n+k-1)/2; j++) {
+ if (k == 0) {
+ if (n & 1) {
+ if (j & 1) {
+ result = result - 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
+ }
+ else {
+ result = result + 2 * cln::expt(cln::pi(),2*j) * S_num(n-2*j,p,1) / cln::factorial(2*j);
+ }
+ }
+ }
+ else {
+ if (k & 1) {
+ if (j & 1) {
+ result = result + cln::factorial(n+k-1)
+ * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
+ / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
+ }
+ else {
+ result = result - cln::factorial(n+k-1)
+ * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
+ / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
+ }
+ }
+ else {
+ if (j & 1) {
+ result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
+ / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
+ }
+ else {
+ result = result + cln::factorial(n+k-1)
+ * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1)
+ / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j));
+ }
+ }
+ }
+ }
+ }
+ int np = n+p;
+ if ((np-1) & 1) {
+ if (((np)/2+n) & 1) {
+ result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
+ }
+ else {
+ result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p));
+ }
+ }
+
+ return result;
+}
+
+
+// helper function for S(n,p,x)
+// [Kol] remark to (9.1)
+cln::cl_N a_k(int k)
+{
+ cln::cl_N result;
+
+ if (k == 0) {
+ return 1;
+ }
+
+ result = result;
+ for (int m=2; m<=k; m++) {
+ result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m);
+ }
+
+ return -result / k;
+}
+
+
+// helper function for S(n,p,x)
+// [Kol] remark to (9.1)
+cln::cl_N b_k(int k)
+{
+ cln::cl_N result;
+
+ if (k == 0) {
+ return 1;
+ }
+
+ result = result;
+ for (int m=2; m<=k; m++) {
+ result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m);
+ }
+
+ return result / k;
+}
+
+
+// helper function for S(n,p,x)
+cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
+{
+ static cln::float_format_t oldprec = cln::default_float_format;
+
+ if (p==1) {
+ return Li_projection(n+1, x, prec);
+ }
+
+ // precision has changed, we need to clear lookup table Yn
+ if ( oldprec != prec ) {
+ Yn.clear();
+ ynsize = 0;
+ ynlength = 100;
+ oldprec = prec;
+ }
+
+ // check if precalculated values are sufficient
+ if (p > ynsize+1) {
+ for (int i=ynsize; i<p-1; i++) {
+ fill_Yn(i, prec);
+ }
+ }
+
+ // should be done otherwise
+ cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
+ cln::cl_N xf = x * one;
+ //cln::cl_N xf = x * cln::cl_float(1, prec);
+
+ cln::cl_N res;
+ cln::cl_N resbuf;
+ cln::cl_N factor = cln::expt(xf, p);
+ int i = p;
+ do {
+ resbuf = res;
+ if (i-p >= ynlength) {
+ // make Yn longer
+ make_Yn_longer(ynlength*2, prec);
+ }
+ 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? ...
+ //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ...
+ factor = factor * xf;
+ i++;
+ } while (res != resbuf);
+
+ return res;
+}
+
+
+// helper function for S(n,p,x)
+cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
+{
+ // [Kol] (5.3)
+ if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) {
+
+ cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n)
+ * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p);
+
+ for (int s=0; s<n; s++) {
+ cln::cl_N res2;
+ for (int r=0; r<p; r++) {
+ res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-x),r)
+ * S_do_sum(p-r,n-s,1-x,prec) / cln::factorial(r);
+ }
+ result = result + cln::expt(cln::log(x),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
+ }
+
+ return result;
+ }
+
+ return S_do_sum(n, p, x, prec);
+}
+
+
+// helper function for S(n,p,x)
+const cln::cl_N S_num(int n, int p, const cln::cl_N& x)
+{
+ if (x == 1) {
+ if (n == 1) {
+ // [Kol] (2.22) with (2.21)
+ return cln::zeta(p+1);
+ }
+
+ if (p == 1) {
+ // [Kol] (2.22)
+ return cln::zeta(n+1);
+ }
+
+ // [Kol] (9.1)
+ cln::cl_N result;
+ for (int nu=0; nu<n; nu++) {
+ for (int rho=0; rho<=p; rho++) {
+ result = result + b_k(n-nu-1) * b_k(p-rho) * a_k(nu+rho+1)
+ * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
+ }
+ }
+ result = result * cln::expt(cln::cl_I(-1),n+p-1);
+
+ return result;
+ }
+ else if (x == -1) {
+ // [Kol] (2.22)
+ if (p == 1) {
+ return -(1-cln::expt(cln::cl_I(2),-n)) * cln::zeta(n+1);
+ }
+// throw std::runtime_error("don't know how to evaluate this function!");
+ }
+
+ // what is the desired float format?
+ // first guess: default format
+ cln::float_format_t prec = cln::default_float_format;
+ const cln::cl_N value = x;
+ // second guess: the argument's format
+ if (!instanceof(realpart(value), cln::cl_RA_ring))
+ prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
+ else if (!instanceof(imagpart(value), cln::cl_RA_ring))
+ prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
+
+ // [Kol] (5.3)
+ // the condition abs(1-value)>1 avoids an infinite recursion in the region abs(value)<=1 && abs(value)>0.95 && abs(1-value)<=1 && abs(1-value)>0.95
+ // we don't care here about abs(value)<1 && real(value)>0.5, this will be taken care of in S_projection
+ if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95) && (cln::abs(1-value) > 1) )) {
+
+ cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n)
+ * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p);
+
+ for (int s=0; s<n; s++) {
+ cln::cl_N res2;
+ for (int r=0; r<p; r++) {
+ res2 = res2 + cln::expt(cln::cl_I(-1),r) * cln::expt(cln::log(1-value),r)
+ * S_num(p-r,n-s,1-value) / cln::factorial(r);
+ }
+ result = result + cln::expt(cln::log(value),s) * (S_num(n-s,p,1) - res2) / cln::factorial(s);
+ }
+
+ return result;
+
+ }
+ // [Kol] (5.12)
+ if (cln::abs(value) > 1) {
+
+ cln::cl_N result;
+
+ for (int s=0; s<p; s++) {
+ for (int r=0; r<=s; r++) {
+ result = result + cln::expt(cln::cl_I(-1),s) * cln::expt(cln::log(-value),r) * cln::factorial(n+s-r-1)
+ / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
+ * S_num(n+s-r,p-s,cln::recip(value));
+ }
+ }
+ result = result * cln::expt(cln::cl_I(-1),n);
+
+ cln::cl_N res2;
+ for (int r=0; r<n; r++) {
+ res2 = res2 + cln::expt(cln::log(-value),r) * C(n-r,p) / cln::factorial(r);
+ }
+ res2 = res2 + cln::expt(cln::log(-value),n+p) / cln::factorial(n+p);
+
+ result = result + cln::expt(cln::cl_I(-1),p) * res2;
+
+ return result;
+ }
+
+ if ((cln::abs(value) > 0.95) && (cln::abs(value-9.53) < 9.47)) {
+ lst m;
+ m.append(n+1);
+ for (int s=0; s<p-1; s++)
+ m.append(1);
+
+ ex res = H(m,numeric(value)).evalf();
+ return ex_to<numeric>(res).to_cl_N();
+ }
+ else {
+ return S_projection(n, p, value, prec);
+ }
+}
+
+
+} // end of anonymous namespace
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Nielsen's generalized polylogarithm S(n,p,x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+static ex S_evalf(const ex& n, const ex& p, const ex& x)
+{
+ if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
+ const int n_ = ex_to<numeric>(n).to_int();
+ const int p_ = ex_to<numeric>(p).to_int();
+ if (is_a<numeric>(x)) {
+ const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
+ const cln::cl_N result = S_num(n_, p_, x_);
+ return numeric(result);
+ } else {
+ ex x_val = x.evalf();
+ if (is_a<numeric>(x_val)) {
+ const cln::cl_N x_val_ = ex_to<numeric>(x_val).to_cl_N();
+ const cln::cl_N result = S_num(n_, p_, x_val_);
+ return numeric(result);
+ }
+ }
+ }
+ return S(n, p, x).hold();
+}
+
+
+static ex S_eval(const ex& n, const ex& p, const ex& x)
+{
+ if (n.info(info_flags::posint) && p.info(info_flags::posint)) {
+ if (x == 0) {
+ return _ex0;
+ }
+ if (x == 1) {
+ lst m(n+1);
+ for (int i=ex_to<numeric>(p).to_int()-1; i>0; i--) {
+ m.append(1);
+ }
+ return zeta(m);
+ }
+ if (p == 1) {
+ return Li(n+1, x);
+ }
+ if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
+ int n_ = ex_to<numeric>(n).to_int();
+ int p_ = ex_to<numeric>(p).to_int();
+ const cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
+ const cln::cl_N result = S_num(n_, p_, x_);
+ return numeric(result);
+ }
+ }
+ if (n.is_zero()) {
+ // [Kol] (5.3)
+ return pow(-log(1-x), p) / factorial(p);
+ }
+ return S(n, p, x).hold();
+}
+
+
+static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
+{
+ if (p == _ex1) {
+ return Li(n+1, x).series(rel, order, options);
+ }
+
+ const ex x_pt = x.subs(rel, subs_options::no_pattern);
+ if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) {
+ // First special case: x==0 (derivatives have poles)
+ if (x_pt.is_zero()) {
+ const symbol s;
+ ex ser;
+ // manually construct the primitive expansion
+ // subsum = Euler-Zagier-Sum is needed
+ // dirty hack (slow ...) calculation of subsum:
+ std::vector<ex> presubsum, subsum;
+ subsum.push_back(0);
+ for (int i=1; i<order-1; ++i) {
+ subsum.push_back(subsum[i-1] + numeric(1, i));
+ }
+ for (int depth=2; depth<p; ++depth) {
+ presubsum = subsum;
+ for (int i=1; i<order-1; ++i) {
+ subsum[i] = subsum[i-1] + numeric(1, i) * presubsum[i-1];
+ }
+ }
+
+ for (int i=1; i<order; ++i) {
+ ser += pow(s,i) / pow(numeric(i), n+1) * subsum[i-1];
+ }
+ // substitute the argument's series expansion
+ ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
+ // maybe that was terminating, so add a proper order term
+ epvector nseq;
+ nseq.push_back(expair(Order(_ex1), order));
+ ser += pseries(rel, nseq);
+ // reexpanding it will collapse the series again
+ return ser.series(rel, order);
+ }
+ // TODO special cases: x==1 (branch point) and x real, >=1 (branch cut)
+ throw std::runtime_error("S_series: don't know how to do the series expansion at this point!");
+ }
+ // all other cases should be safe, by now:
+ throw do_taylor(); // caught by function::series()
+}
+
+
+static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param < 3);
+ if (deriv_param < 2) {
+ return _ex0;
+ }
+ if (n > 0) {
+ return S(n-1, p, x) / x;
+ } else {
+ return S(n, p-1, x) / (1-x);
+ }
+}
+
+
+static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
+{
+ c.s << "\\mathrm{S}_{";
+ n.print(c);
+ c.s << ",";
+ p.print(c);
+ c.s << "}(";
+ x.print(c);
+ c.s << ")";
+}
+
+
+REGISTER_FUNCTION(S,
+ evalf_func(S_evalf).
+ eval_func(S_eval).
+ series_func(S_series).
+ derivative_func(S_deriv).
+ print_func<print_latex>(S_print_latex).
+ do_not_evalf_params());
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Harmonic polylogarithm H(m,x)
+//
+// helper functions
+//
+//////////////////////////////////////////////////////////////////////
+
+
+// anonymous namespace for helper functions
+namespace {
+
+
+// regulates the pole (used by 1/x-transformation)
+symbol H_polesign("IMSIGN");
+
+
+// convert parameters from H to Li representation
+// parameters are expected to be in expanded form, i.e. only 0, 1 and -1
+// returns true if some parameters are negative
+bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf)
+{
+ // expand parameter list
+ lst mexp;
+ for (lst::const_iterator it = l.begin(); it != l.end(); it++) {
+ if (*it > 1) {
+ for (ex count=*it-1; count > 0; count--) {
+ mexp.append(0);
+ }
+ mexp.append(1);
+ } else if (*it < -1) {
+ for (ex count=*it+1; count < 0; count++) {
+ mexp.append(0);
+ }
+ mexp.append(-1);
+ } else {
+ mexp.append(*it);
+ }
+ }
+
+ ex signum = 1;
+ pf = 1;
+ bool has_negative_parameters = false;
+ ex acc = 1;
+ for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) {
+ if (*it == 0) {
+ acc++;
+ continue;
+ }
+ if (*it > 0) {
+ m.append((*it+acc-1) * signum);
+ } else {
+ m.append((*it-acc+1) * signum);
+ }
+ acc = 1;
+ signum = *it;
+ pf *= *it;
+ if (pf < 0) {
+ has_negative_parameters = true;
+ }
+ }
+ if (has_negative_parameters) {
+ for (std::size_t i=0; i<m.nops(); i++) {
+ if (m.op(i) < 0) {
+ m.let_op(i) = -m.op(i);
+ s.append(-1);
+ } else {
+ s.append(1);
+ }
+ }
+ }
+
+ return has_negative_parameters;
+}
+
+
+// recursivly transforms H to corresponding multiple polylogarithms
+struct map_trafo_H_convert_to_Li : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+ lst parameter;
+ if (is_a<lst>(e.op(0))) {
+ parameter = ex_to<lst>(e.op(0));
+ } else {
+ parameter = lst(e.op(0));
+ }
+ ex arg = e.op(1);
+
+ lst m;
+ lst s;
+ ex pf;
+ if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
+ s.let_op(0) = s.op(0) * arg;
+ return pf * Li(m, s).hold();
+ } else {
+ for (std::size_t i=0; i<m.nops(); i++) {
+ s.append(1);
+ }
+ s.let_op(0) = s.op(0) * arg;
+ return Li(m, s).hold();
+ }
+ }
+ }
+ return e;
+ }
+};
+
+
+// recursivly transforms H to corresponding zetas
+struct map_trafo_H_convert_to_zeta : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+ lst parameter;
+ if (is_a<lst>(e.op(0))) {
+ parameter = ex_to<lst>(e.op(0));
+ } else {
+ parameter = lst(e.op(0));
+ }
+
+ lst m;
+ lst s;
+ ex pf;
+ if (convert_parameter_H_to_Li(parameter, m, s, pf)) {
+ return pf * zeta(m, s);
+ } else {
+ return zeta(m);
+ }
+ }
+ }
+ return e;
+ }
+};
+
+
+// remove trailing zeros from H-parameters
+struct map_trafo_H_reduce_trailing_zeros : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+ lst parameter;
+ if (is_a<lst>(e.op(0))) {
+ parameter = ex_to<lst>(e.op(0));
+ } else {
+ parameter = lst(e.op(0));
+ }
+ ex arg = e.op(1);
+ if (parameter.op(parameter.nops()-1) == 0) {
+
+ //
+ if (parameter.nops() == 1) {
+ return log(arg);
+ }
+
+ //
+ lst::const_iterator it = parameter.begin();
+ while ((it != parameter.end()) && (*it == 0)) {
+ it++;
+ }
+ if (it == parameter.end()) {
+ return pow(log(arg),parameter.nops()) / factorial(parameter.nops());
+ }
+
+ //
+ parameter.remove_last();
+ std::size_t lastentry = parameter.nops();
+ while ((lastentry > 0) && (parameter[lastentry-1] == 0)) {
+ lastentry--;
+ }
+
+ //
+ ex result = log(arg) * H(parameter,arg).hold();
+ ex acc = 0;
+ for (ex i=0; i<lastentry; i++) {
+ if (parameter[i] > 0) {
+ parameter[i]++;
+ result -= (acc + parameter[i]-1) * H(parameter, arg).hold();
+ parameter[i]--;
+ acc = 0;
+ } else if (parameter[i] < 0) {
+ parameter[i]--;
+ result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold();
+ parameter[i]++;
+ acc = 0;
+ } else {
+ acc++;
+ }
+ }
+
+ if (lastentry < parameter.nops()) {
+ result = result / (parameter.nops()-lastentry+1);
+ return result.map(*this);
+ } else {
+ return result;
+ }
+ }
+ }
+ }
+ return e;
+ }
+};
+
+
+// returns an expression with zeta functions corresponding to the parameter list for H
+ex convert_H_to_zeta(const lst& m)
+{
+ symbol xtemp("xtemp");
+ map_trafo_H_reduce_trailing_zeros filter;
+ map_trafo_H_convert_to_zeta filter2;
+ return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1);
+}
+
+
+// convert signs form Li to H representation
+lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
+{
+ lst res;
+ lst::const_iterator itm = m.begin();
+ lst::const_iterator itx = ++x.begin();
+ int signum = 1;
+ pf = _ex1;
+ res.append(*itm);
+ itm++;
+ while (itx != x.end()) {
+ GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1));
+ // XXX: 1 + 0.0*I is considered equal to 1. However the former
+ // is not automatically converted to a real number.
+ // Do the conversion explicitly to avoid the
+ // "numeric::operator>(): complex inequality" exception.
+ signum *= (*itx != _ex_1) ? 1 : -1;
+ pf *= signum;
+ res.append((*itm) * signum);
+ itm++;
+ itx++;
+ }
+ return res;
+}
+
+
+// multiplies an one-dimensional H with another H
+// [ReV] (18)
+ex trafo_H_mult(const ex& h1, const ex& h2)
+{
+ ex res;
+ ex hshort;
+ lst hlong;
+ ex h1nops = h1.op(0).nops();
+ ex h2nops = h2.op(0).nops();
+ if (h1nops > 1) {
+ hshort = h2.op(0).op(0);
+ hlong = ex_to<lst>(h1.op(0));
+ } else {
+ hshort = h1.op(0).op(0);
+ if (h2nops > 1) {
+ hlong = ex_to<lst>(h2.op(0));
+ } else {
+ hlong = h2.op(0).op(0);
+ }
+ }
+ for (std::size_t i=0; i<=hlong.nops(); i++) {
+ lst newparameter;
+ std::size_t j=0;
+ for (; j<i; j++) {
+ newparameter.append(hlong[j]);
+ }
+ newparameter.append(hshort);
+ for (; j<hlong.nops(); j++) {
+ newparameter.append(hlong[j]);
+ }
+ res += H(newparameter, h1.op(1)).hold();
+ }
+ return res;
+}
+
+
+// applies trafo_H_mult recursively on expressions
+struct map_trafo_H_mult : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e)) {
+ return e.map(*this);
+ }
+
+ if (is_a<mul>(e)) {
+
+ ex result = 1;
+ ex firstH;
+ lst Hlst;
+ for (std::size_t pos=0; pos<e.nops(); pos++) {
+ if (is_a<power>(e.op(pos)) && is_a<function>(e.op(pos).op(0))) {
+ std::string name = ex_to<function>(e.op(pos).op(0)).get_name();
+ if (name == "H") {
+ for (ex i=0; i<e.op(pos).op(1); i++) {
+ Hlst.append(e.op(pos).op(0));
+ }
+ continue;
+ }
+ } else if (is_a<function>(e.op(pos))) {
+ std::string name = ex_to<function>(e.op(pos)).get_name();
+ if (name == "H") {
+ if (e.op(pos).op(0).nops() > 1) {
+ firstH = e.op(pos);
+ } else {
+ Hlst.append(e.op(pos));
+ }
+ continue;
+ }
+ }
+ result *= e.op(pos);
+ }
+ if (firstH == 0) {
+ if (Hlst.nops() > 0) {
+ firstH = Hlst[Hlst.nops()-1];
+ Hlst.remove_last();
+ } else {
+ return e;
+ }
+ }
+
+ if (Hlst.nops() > 0) {
+ ex buffer = trafo_H_mult(firstH, Hlst.op(0));
+ result *= buffer;
+ for (std::size_t i=1; i<Hlst.nops(); i++) {
+ result *= Hlst.op(i);
+ }
+ result = result.expand();
+ map_trafo_H_mult recursion;
+ return recursion(result);
+ } else {
+ return e;
+ }
+
+ }
+ return e;
+ }
+};
+
+
+// do integration [ReV] (55)
+// put parameter 0 in front of existing parameters
+ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg)
+{
+ ex h;
+ std::string name;
+ if (is_a<function>(e)) {
+ name = ex_to<function>(e).get_name();
+ }
+ if (name == "H") {
+ h = e;
+ } else {
+ for (std::size_t i=0; i<e.nops(); i++) {
+ if (is_a<function>(e.op(i))) {
+ std::string name = ex_to<function>(e.op(i)).get_name();
+ if (name == "H") {
+ h = e.op(i);
+ }
+ }
+ }
+ }
+ if (h != 0) {
+ lst newparameter = ex_to<lst>(h.op(0));
+ newparameter.prepend(0);
+ ex addzeta = convert_H_to_zeta(newparameter);
+ return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
+ } else {
+ return e * (-H(lst(ex(0)),1/arg).hold());
+ }
+}
+
+
+// do integration [ReV] (49)
+// put parameter 1 in front of existing parameters
+ex trafo_H_prepend_one(const ex& e, const ex& arg)
+{
+ ex h;
+ std::string name;
+ if (is_a<function>(e)) {
+ name = ex_to<function>(e).get_name();
+ }
+ if (name == "H") {
+ h = e;
+ } else {
+ for (std::size_t i=0; i<e.nops(); i++) {
+ if (is_a<function>(e.op(i))) {
+ std::string name = ex_to<function>(e.op(i)).get_name();
+ if (name == "H") {
+ h = e.op(i);
+ }
+ }
+ }
+ }
+ if (h != 0) {
+ lst newparameter = ex_to<lst>(h.op(0));
+ newparameter.prepend(1);
+ return e.subs(h == H(newparameter, h.op(1)).hold());
+ } else {
+ return e * H(lst(ex(1)),1-arg).hold();
+ }
+}
+
+
+// do integration [ReV] (55)
+// put parameter -1 in front of existing parameters
+ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg)
+{
+ ex h;
+ std::string name;
+ if (is_a<function>(e)) {
+ name = ex_to<function>(e).get_name();
+ }
+ if (name == "H") {
+ h = e;
+ } else {
+ for (std::size_t i=0; i<e.nops(); i++) {
+ if (is_a<function>(e.op(i))) {
+ std::string name = ex_to<function>(e.op(i)).get_name();
+ if (name == "H") {
+ h = e.op(i);
+ }
+ }
+ }
+ }
+ if (h != 0) {
+ lst newparameter = ex_to<lst>(h.op(0));
+ newparameter.prepend(-1);
+ ex addzeta = convert_H_to_zeta(newparameter);
+ return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
+ } else {
+ ex addzeta = convert_H_to_zeta(lst(ex(-1)));
+ return (e * (addzeta - H(lst(ex(-1)),1/arg).hold())).expand();
+ }
+}
+
+
+// do integration [ReV] (55)
+// put parameter -1 in front of existing parameters
+ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg)
+{
+ ex h;
+ std::string name;
+ if (is_a<function>(e)) {
+ name = ex_to<function>(e).get_name();
+ }
+ if (name == "H") {
+ h = e;
+ } else {
+ for (std::size_t i = 0; i < e.nops(); i++) {
+ if (is_a<function>(e.op(i))) {
+ std::string name = ex_to<function>(e.op(i)).get_name();
+ if (name == "H") {
+ h = e.op(i);
+ }
+ }
+ }
+ }
+ if (h != 0) {
+ lst newparameter = ex_to<lst>(h.op(0));
+ newparameter.prepend(-1);
+ return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
+ } else {
+ return (e * H(lst(ex(-1)),(1-arg)/(1+arg)).hold()).expand();
+ }
+}
+
+
+// do integration [ReV] (55)
+// put parameter 1 in front of existing parameters
+ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg)
+{
+ ex h;
+ std::string name;
+ if (is_a<function>(e)) {
+ name = ex_to<function>(e).get_name();
+ }
+ if (name == "H") {
+ h = e;
+ } else {
+ for (std::size_t i = 0; i < e.nops(); i++) {
+ if (is_a<function>(e.op(i))) {
+ std::string name = ex_to<function>(e.op(i)).get_name();
+ if (name == "H") {
+ h = e.op(i);
+ }
+ }
+ }
+ }
+ if (h != 0) {
+ lst newparameter = ex_to<lst>(h.op(0));
+ newparameter.prepend(1);
+ return e.subs(h == H(newparameter, h.op(1)).hold()).expand();
+ } else {
+ return (e * H(lst(ex(1)),(1-arg)/(1+arg)).hold()).expand();
+ }
+}
+
+
+// do x -> 1-x transformation
+struct map_trafo_H_1mx : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+
+ lst parameter = ex_to<lst>(e.op(0));
+ ex arg = e.op(1);
+
+ // special cases if all parameters are either 0, 1 or -1
+ bool allthesame = true;
+ if (parameter.op(0) == 0) {
+ for (std::size_t i = 1; i < parameter.nops(); i++) {
+ if (parameter.op(i) != 0) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ lst newparameter;
+ for (int i=parameter.nops(); i>0; i--) {
+ newparameter.append(1);
+ }
+ return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
+ }
+ } else if (parameter.op(0) == -1) {
+ throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!");
+ } else {
+ for (std::size_t i = 1; i < parameter.nops(); i++) {
+ if (parameter.op(i) != 1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ lst newparameter;
+ for (int i=parameter.nops(); i>0; i--) {
+ newparameter.append(0);
+ }
+ return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
+ }
+ }
+
+ lst newparameter = parameter;
+ newparameter.remove_first();
+
+ if (parameter.op(0) == 0) {
+
+ // leading zero
+ ex res = convert_H_to_zeta(parameter);
+ //ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
+ map_trafo_H_1mx recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (std::size_t i = 0; i < buffer.nops(); i++) {
+ res -= trafo_H_prepend_one(buffer.op(i), arg);
+ }
+ } else {
+ res -= trafo_H_prepend_one(buffer, arg);
+ }
+ return res;
+
+ } else {
+
+ // leading one
+ map_trafo_H_1mx recursion;
+ map_trafo_H_mult unify;
+ ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
+ std::size_t firstzero = 0;
+ while (parameter.op(firstzero) == 1) {
+ firstzero++;
+ }
+ for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) {
+ lst newparameter;
+ std::size_t j=0;
+ for (; j<=i; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ newparameter.append(1);
+ for (; j<parameter.nops()-1; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ res -= H(newparameter, arg).hold();
+ }
+ res = recursion(res).expand() / firstzero;
+ return unify(res);
+ }
+ }
+ }
+ return e;
+ }
+};
+
+
+// do x -> 1/x transformation
+struct map_trafo_H_1overx : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+
+ lst parameter = ex_to<lst>(e.op(0));
+ ex arg = e.op(1);
+
+ // special cases if all parameters are either 0, 1 or -1
+ bool allthesame = true;
+ if (parameter.op(0) == 0) {
+ for (std::size_t i = 1; i < parameter.nops(); i++) {
+ if (parameter.op(i) != 0) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
+ }
+ } else if (parameter.op(0) == -1) {
+ for (std::size_t i = 1; i < parameter.nops(); i++) {
+ if (parameter.op(i) != -1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ map_trafo_H_mult unify;
+ return unify((pow(H(lst(ex(-1)),1/arg).hold() - H(lst(ex(0)),1/arg).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ } else {
+ for (std::size_t i = 1; i < parameter.nops(); i++) {
+ if (parameter.op(i) != 1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ map_trafo_H_mult unify;
+ return unify((pow(H(lst(ex(1)),1/arg).hold() + H(lst(ex(0)),1/arg).hold() + H_polesign, parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ }
+
+ lst newparameter = parameter;
+ newparameter.remove_first();
+
+ if (parameter.op(0) == 0) {
+
+ // leading zero
+ ex res = convert_H_to_zeta(parameter);
+ map_trafo_H_1overx recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (std::size_t i = 0; i < buffer.nops(); i++) {
+ res += trafo_H_1tx_prepend_zero(buffer.op(i), arg);
+ }
+ } else {
+ res += trafo_H_1tx_prepend_zero(buffer, arg);
+ }
+ return res;
+
+ } else if (parameter.op(0) == -1) {
+
+ // leading negative one
+ ex res = convert_H_to_zeta(parameter);
+ map_trafo_H_1overx recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (std::size_t i = 0; i < buffer.nops(); i++) {
+ res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg);
+ }
+ } else {
+ res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg);
+ }
+ return res;
+
+ } else {
+
+ // leading one
+ map_trafo_H_1overx recursion;
+ map_trafo_H_mult unify;
+ ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
+ std::size_t firstzero = 0;
+ while (parameter.op(firstzero) == 1) {
+ firstzero++;
+ }
+ for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) {
+ lst newparameter;
+ std::size_t j = 0;
+ for (; j<=i; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ newparameter.append(1);
+ for (; j<parameter.nops()-1; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ res -= H(newparameter, arg).hold();
+ }
+ res = recursion(res).expand() / firstzero;
+ return unify(res);
+
+ }
+
+ }
+ }
+ return e;
+ }
+};
+
+
+// do x -> (1-x)/(1+x) transformation
+struct map_trafo_H_1mxt1px : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
+
+ if (is_a<function>(e)) {
+ std::string name = ex_to<function>(e).get_name();
+ if (name == "H") {
+
+ lst parameter = ex_to<lst>(e.op(0));
+ ex arg = e.op(1);
+
+ // special cases if all parameters are either 0, 1 or -1
+ bool allthesame = true;
+ if (parameter.op(0) == 0) {
+ for (std::size_t i = 1; i < parameter.nops(); i++) {
+ if (parameter.op(i) != 0) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ map_trafo_H_mult unify;
+ return unify((pow(-H(lst(ex(1)),(1-arg)/(1+arg)).hold() - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ } else if (parameter.op(0) == -1) {
+ for (std::size_t i = 1; i < parameter.nops(); i++) {
+ if (parameter.op(i) != -1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ map_trafo_H_mult unify;
+ return unify((pow(log(2) - H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ } else {
+ for (std::size_t i = 1; i < parameter.nops(); i++) {
+ if (parameter.op(i) != 1) {
+ allthesame = false;
+ break;
+ }
+ }
+ if (allthesame) {
+ map_trafo_H_mult unify;
+ return unify((pow(-log(2) - H(lst(ex(0)),(1-arg)/(1+arg)).hold() + H(lst(ex(-1)),(1-arg)/(1+arg)).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ }
+
+ lst newparameter = parameter;
+ newparameter.remove_first();
+
+ if (parameter.op(0) == 0) {
+
+ // leading zero
+ ex res = convert_H_to_zeta(parameter);
+ map_trafo_H_1mxt1px recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (std::size_t i = 0; i < buffer.nops(); i++) {
+ res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
+ }
+ } else {
+ res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg);
+ }
+ return res;
+
+ } else if (parameter.op(0) == -1) {
+
+ // leading negative one
+ ex res = convert_H_to_zeta(parameter);
+ map_trafo_H_1mxt1px recursion;
+ ex buffer = recursion(H(newparameter, arg).hold());
+ if (is_a<add>(buffer)) {
+ for (std::size_t i = 0; i < buffer.nops(); i++) {
+ res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg);
+ }
+ } else {
+ res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg);
+ }
+ return res;
+
+ } else {
+
+ // leading one
+ map_trafo_H_1mxt1px recursion;
+ map_trafo_H_mult unify;
+ ex res = H(lst(ex(1)), arg).hold() * H(newparameter, arg).hold();
+ std::size_t firstzero = 0;
+ while (parameter.op(firstzero) == 1) {
+ firstzero++;
+ }
+ for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) {
+ lst newparameter;
+ std::size_t j=0;
+ for (; j<=i; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ newparameter.append(1);
+ for (; j<parameter.nops()-1; j++) {
+ newparameter.append(parameter[j+1]);
+ }
+ res -= H(newparameter, arg).hold();
+ }
+ res = recursion(res).expand() / firstzero;
+ return unify(res);
+
+ }
+
+ }
+ }
+ return e;
+ }
+};
+
+
+// do the actual summation.
+cln::cl_N H_do_sum(const std::vector<int>& m, const cln::cl_N& x)
+{
+ const int j = m.size();
+
+ std::vector<cln::cl_N> t(j);
+
+ cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
+ cln::cl_N factor = cln::expt(x, j) * one;
+ cln::cl_N t0buf;
+ int q = 0;
+ do {
+ t0buf = t[0];
+ q++;
+ t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]);
+ for (int k=j-2; k>=1; k--) {
+ t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]);
+ }
+ t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]);
+ factor = factor * x;
+ } while (t[0] != t0buf);
+
+ return t[0];
+}
+
+
+} // end of anonymous namespace
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Harmonic polylogarithm H(m,x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+static ex H_evalf(const ex& x1, const ex& x2)
+{
+ if (is_a<lst>(x1)) {
+
+ cln::cl_N x;
+ if (is_a<numeric>(x2)) {
+ x = ex_to<numeric>(x2).to_cl_N();
+ } else {
+ ex x2_val = x2.evalf();
+ if (is_a<numeric>(x2_val)) {
+ x = ex_to<numeric>(x2_val).to_cl_N();
+ }
+ }
+
+ for (std::size_t i = 0; i < x1.nops(); i++) {
+ if (!x1.op(i).info(info_flags::integer)) {
+ return H(x1, x2).hold();
+ }
+ }
+ if (x1.nops() < 1) {
+ return H(x1, x2).hold();
+ }
+
+ const lst& morg = ex_to<lst>(x1);
+ // remove trailing zeros ...
+ if (*(--morg.end()) == 0) {
+ symbol xtemp("xtemp");
+ map_trafo_H_reduce_trailing_zeros filter;
+ return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf();
+ }
+ // ... and expand parameter notation
+ bool has_minus_one = false;
+ lst m;
+ for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) {
+ if (*it > 1) {
+ for (ex count=*it-1; count > 0; count--) {
+ m.append(0);
+ }
+ m.append(1);
+ } else if (*it <= -1) {
+ for (ex count=*it+1; count < 0; count++) {
+ m.append(0);
+ }
+ m.append(-1);
+ has_minus_one = true;
+ } else {
+ m.append(*it);
+ }
+ }
+
+ // do summation
+ if (cln::abs(x) < 0.95) {
+ lst m_lst;
+ lst s_lst;
+ ex pf;
+ if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) {
+ // negative parameters -> s_lst is filled
+ std::vector<int> m_int;
+ std::vector<cln::cl_N> x_cln;
+ for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin();
+ it_int != m_lst.end(); it_int++, it_cln++) {
+ m_int.push_back(ex_to<numeric>(*it_int).to_int());
+ x_cln.push_back(ex_to<numeric>(*it_cln).to_cl_N());
+ }
+ x_cln.front() = x_cln.front() * x;
+ return pf * numeric(multipleLi_do_sum(m_int, x_cln));
+ } else {
+ // only positive parameters
+ //TODO
+ if (m_lst.nops() == 1) {
+ return Li(m_lst.op(0), x2).evalf();
+ }
+ std::vector<int> m_int;
+ for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) {
+ m_int.push_back(ex_to<numeric>(*it).to_int());
+ }
+ return numeric(H_do_sum(m_int, x));
+ }
+ }
+
+ symbol xtemp("xtemp");
+ ex res = 1;
+
+ // ensure that the realpart of the argument is positive
+ if (cln::realpart(x) < 0) {
+ x = -x;
+ for (std::size_t i = 0; i < m.nops(); i++) {
+ if (m.op(i) != 0) {
+ m.let_op(i) = -m.op(i);
+ res *= -1;
+ }
+ }
+ }
+
+ // x -> 1/x
+ if (cln::abs(x) >= 2.0) {
+ map_trafo_H_1overx trafo;
+ res *= trafo(H(m, xtemp).hold());
+ if (cln::imagpart(x) <= 0) {
+ res = res.subs(H_polesign == -I*Pi);
+ } else {
+ res = res.subs(H_polesign == I*Pi);
+ }
+ return res.subs(xtemp == numeric(x)).evalf();
+ }
+
+ // check transformations for 0.95 <= |x| < 2.0
+
+ // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47
+ if (cln::abs(x-9.53) <= 9.47) {
+ // x -> (1-x)/(1+x)
+ map_trafo_H_1mxt1px trafo;
+ res *= trafo(H(m, xtemp).hold());
+ } else {
+ // x -> 1-x
+ if (has_minus_one) {
+ map_trafo_H_convert_to_Li filter;
+ return filter(H(m, numeric(x)).hold()).evalf();
+ }
+ map_trafo_H_1mx trafo;
+ res *= trafo(H(m, xtemp).hold());
+ }
+
+ return res.subs(xtemp == numeric(x)).evalf();
+ }
+
+ return H(x1,x2).hold();
+}
+
+
+static ex H_eval(const ex& m_, const ex& x)
+{
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
+ }
+ if (m.nops() == 0) {
+ return _ex1;
+ }
+ ex pos1;
+ ex pos2;
+ ex n;
+ ex p;
+ int step = 0;
+ if (*m.begin() > _ex1) {
+ step++;
+ pos1 = _ex0;
+ pos2 = _ex1;
+ n = *m.begin()-1;
+ p = _ex1;
+ } else if (*m.begin() < _ex_1) {
+ step++;
+ pos1 = _ex0;
+ pos2 = _ex_1;
+ n = -*m.begin()-1;
+ p = _ex1;
+ } else if (*m.begin() == _ex0) {
+ pos1 = _ex0;
+ n = _ex1;
+ } else {
+ pos1 = *m.begin();
+ p = _ex1;
+ }
+ for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) {
+ if ((*it).info(info_flags::integer)) {
+ if (step == 0) {
+ if (*it > _ex1) {
+ if (pos1 == _ex0) {
+ step = 1;
+ pos2 = _ex1;
+ n += *it-1;
+ p = _ex1;
+ } else {
+ step = 2;
+ }
+ } else if (*it < _ex_1) {
+ if (pos1 == _ex0) {
+ step = 1;
+ pos2 = _ex_1;
+ n += -*it-1;
+ p = _ex1;
+ } else {
+ step = 2;
+ }
+ } else {
+ if (*it != pos1) {
+ step = 1;
+ pos2 = *it;
+ }
+ if (*it == _ex0) {
+ n++;
+ } else {
+ p++;
+ }
+ }
+ } else if (step == 1) {
+ if (*it != pos2) {
+ step = 2;
+ } else {
+ if (*it == _ex0) {
+ n++;
+ } else {
+ p++;
+ }
+ }
+ }
+ } else {
+ // if some m_i is not an integer
+ return H(m_, x).hold();
+ }
+ }
+ if ((x == _ex1) && (*(--m.end()) != _ex0)) {
+ return convert_H_to_zeta(m);
+ }
+ if (step == 0) {
+ if (pos1 == _ex0) {
+ // all zero
+ if (x == _ex0) {
+ return H(m_, x).hold();
+ }
+ return pow(log(x), m.nops()) / factorial(m.nops());
+ } else {
+ // all (minus) one
+ return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops());
+ }
+ } else if ((step == 1) && (pos1 == _ex0)){
+ // convertible to S
+ if (pos2 == _ex1) {
+ return S(n, p, x);
+ } else {
+ return pow(-1, p) * S(n, p, -x);
+ }
+ }
+ if (x == _ex0) {
+ return _ex0;
+ }
+ if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
+ return H(m_, x).evalf();
+ }
+ return H(m_, x).hold();
+}
+
+
+static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
+{
+ epvector seq;
+ seq.push_back(expair(H(m, x), 0));
+ return pseries(rel, seq);
+}
+
+
+static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param < 2);
+ if (deriv_param == 0) {
+ return _ex0;
+ }
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
+ }
+ ex mb = *m.begin();
+ if (mb > _ex1) {
+ m[0]--;
+ return H(m, x) / x;
+ }
+ if (mb < _ex_1) {
+ m[0]++;
+ return H(m, x) / x;
+ }
+ m.remove_first();
+ if (mb == _ex1) {
+ return 1/(1-x) * H(m, x);
+ } else if (mb == _ex_1) {
+ return 1/(1+x) * H(m, x);
+ } else {
+ return H(m, x) / x;
+ }
+}
+
+
+static void H_print_latex(const ex& m_, const ex& x, const print_context& c)
+{
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
+ }
+ c.s << "\\mathrm{H}_{";
+ lst::const_iterator itm = m.begin();
+ (*itm).print(c);
+ itm++;
+ for (; itm != m.end(); itm++) {
+ c.s << ",";
+ (*itm).print(c);
+ }
+ c.s << "}(";
+ x.print(c);
+ c.s << ")";
+}
+
+
+REGISTER_FUNCTION(H,
+ evalf_func(H_evalf).
+ eval_func(H_eval).
+ series_func(H_series).
+ derivative_func(H_deriv).
+ print_func<print_latex>(H_print_latex).
+ do_not_evalf_params());
+
+
+// takes a parameter list for H and returns an expression with corresponding multiple polylogarithms
+ex convert_H_to_Li(const ex& m, const ex& x)
+{
+ map_trafo_H_reduce_trailing_zeros filter;
+ map_trafo_H_convert_to_Li filter2;
+ if (is_a<lst>(m)) {
+ return filter2(filter(H(m, x).hold()));
+ } else {
+ return filter2(filter(H(lst(m), x).hold()));
+ }
+}
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Multiple zeta values zeta(x) and zeta(x,s)
+//
+// helper functions
+//
+//////////////////////////////////////////////////////////////////////
+
+
+// anonymous namespace for helper functions
+namespace {
+
+
+// parameters and data for [Cra] algorithm
+const cln::cl_N lambda = cln::cl_N("319/320");
+
+void halfcyclic_convolute(const std::vector<cln::cl_N>& a, const std::vector<cln::cl_N>& b, std::vector<cln::cl_N>& c)
+{
+ const int size = a.size();
+ for (int n=0; n<size; n++) {
+ c[n] = 0;
+ for (int m=0; m<=n; m++) {
+ c[n] = c[n] + a[m]*b[n-m];
+ }
+ }
+}
+
+
+// [Cra] section 4
+static void initcX(std::vector<cln::cl_N>& crX,
+ const std::vector<int>& s,
+ const int L2)
+{
+ std::vector<cln::cl_N> crB(L2 + 1);
+ for (int i=0; i<=L2; i++)
+ crB[i] = bernoulli(i).to_cl_N() / cln::factorial(i);
+
+ int Sm = 0;
+ int Smp1 = 0;
+ std::vector<std::vector<cln::cl_N> > crG(s.size() - 1, std::vector<cln::cl_N>(L2 + 1));
+ for (int m=0; m < (int)s.size() - 1; m++) {
+ Sm += s[m];
+ Smp1 = Sm + s[m+1];
+ for (int i = 0; i <= L2; i++)
+ crG[m][i] = cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2);
+ }
+
+ crX = crB;
+
+ for (std::size_t m = 0; m < s.size() - 1; m++) {
+ std::vector<cln::cl_N> Xbuf(L2 + 1);
+ for (int i = 0; i <= L2; i++)
+ Xbuf[i] = crX[i] * crG[m][i];
+
+ halfcyclic_convolute(Xbuf, crB, crX);
+ }
+}
+
+
+// [Cra] section 4
+static cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk,
+ const std::vector<cln::cl_N>& crX)
+{
+ cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
+ cln::cl_N factor = cln::expt(lambda, Sqk);
+ cln::cl_N res = factor / Sqk * crX[0] * one;
+ cln::cl_N resbuf;
+ int N = 0;
+ do {
+ resbuf = res;
+ factor = factor * lambda;
+ N++;
+ res = res + crX[N] * factor / (N+Sqk);
+ } while ((res != resbuf) || cln::zerop(crX[N]));
+ return res;
+}
+
+
+// [Cra] section 4
+static void calc_f(std::vector<std::vector<cln::cl_N> >& f_kj,
+ const int maxr, const int L1)
+{
+ cln::cl_N t0, t1, t2, t3, t4;
+ int i, j, k;
+ std::vector<std::vector<cln::cl_N> >::iterator it = f_kj.begin();
+ cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
+
+ t0 = cln::exp(-lambda);
+ t2 = 1;
+ for (k=1; k<=L1; k++) {
+ t1 = k * lambda;
+ t2 = t0 * t2;
+ for (j=1; j<=maxr; j++) {
+ t3 = 1;
+ t4 = 1;
+ for (i=2; i<=j; i++) {
+ t4 = t4 * (j-i+1);
+ t3 = t1 * t3 + t4;
+ }
+ (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one);
+ }
+ it++;
+ }
+}
+
+
+// [Cra] (3.1)
+static cln::cl_N crandall_Z(const std::vector<int>& s,
+ const std::vector<std::vector<cln::cl_N> >& f_kj)
+{
+ const int j = s.size();
+
+ if (j == 1) {
+ cln::cl_N t0;
+ cln::cl_N t0buf;
+ int q = 0;
+ do {
+ t0buf = t0;
+ q++;
+ t0 = t0 + f_kj[q+j-2][s[0]-1];
+ } while (t0 != t0buf);
+
+ return t0 / cln::factorial(s[0]-1);
+ }
+
+ std::vector<cln::cl_N> t(j);
+
+ cln::cl_N t0buf;
+ int q = 0;
+ do {
+ t0buf = t[0];
+ q++;
+ t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]);
+ for (int k=j-2; k>=1; k--) {
+ t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]);
+ }
+ t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1];
+ } while (t[0] != t0buf);
+
+ return t[0] / cln::factorial(s[0]-1);
+}
+
+
+// [Cra] (2.4)
+cln::cl_N zeta_do_sum_Crandall(const std::vector<int>& s)
+{
+ std::vector<int> r = s;
+ const int j = r.size();
+
+ std::size_t L1;
+
+ // decide on maximal size of f_kj for crandall_Z
+ if (Digits < 50) {
+ L1 = 150;
+ } else {
+ L1 = Digits * 3 + j*2;
+ }
+
+ std::size_t L2;
+ // decide on maximal size of crX for crandall_Y
+ if (Digits < 38) {
+ L2 = 63;
+ } else if (Digits < 86) {
+ L2 = 127;
+ } else if (Digits < 192) {
+ L2 = 255;
+ } else if (Digits < 394) {
+ L2 = 511;
+ } else if (Digits < 808) {
+ L2 = 1023;
+ } else {
+ L2 = 2047;
+ }
+
+ cln::cl_N res;
+
+ int maxr = 0;
+ int S = 0;
+ for (int i=0; i<j; i++) {
+ S += r[i];
+ if (r[i] > maxr) {
+ maxr = r[i];
+ }
+ }
+
+ std::vector<std::vector<cln::cl_N> > f_kj(L1);
+ calc_f(f_kj, maxr, L1);
+
+ const cln::cl_N r0factorial = cln::factorial(r[0]-1);
+
+ std::vector<int> rz;
+ int skp1buf;
+ int Srun = S;
+ for (int k=r.size()-1; k>0; k--) {
+
+ rz.insert(rz.begin(), r.back());
+ skp1buf = rz.front();
+ Srun -= skp1buf;
+ r.pop_back();
+
+ std::vector<cln::cl_N> crX;
+ initcX(crX, r, L2);
+
+ for (int q=0; q<skp1buf; q++) {
+
+ cln::cl_N pp1 = crandall_Y_loop(Srun+q-k, crX);
+ cln::cl_N pp2 = crandall_Z(rz, f_kj);
+
+ rz.front()--;
+
+ if (q & 1) {
+ res = res - pp1 * pp2 / cln::factorial(q);
+ } else {
+ res = res + pp1 * pp2 / cln::factorial(q);
+ }
+ }
+ rz.front() = skp1buf;
+ }
+ rz.insert(rz.begin(), r.back());
+
+ std::vector<cln::cl_N> crX;
+ initcX(crX, rz, L2);
+
+ res = (res + crandall_Y_loop(S-j, crX)) / r0factorial
+ + crandall_Z(rz, f_kj);
+
+ return res;
+}
+
+
+cln::cl_N zeta_do_sum_simple(const std::vector<int>& r)
+{
+ const int j = r.size();
+
+ // buffer for subsums
+ std::vector<cln::cl_N> t(j);
+ cln::cl_F one = cln::cl_float(1, cln::float_format(Digits));
+
+ cln::cl_N t0buf;
+ int q = 0;
+ do {
+ t0buf = t[0];
+ q++;
+ t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]);
+ for (int k=j-2; k>=0; k--) {
+ t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]);
+ }
+ } while (t[0] != t0buf);
+
+ return t[0];
+}
+
+
+// does Hoelder convolution. see [BBB] (7.0)
+cln::cl_N zeta_do_Hoelder_convolution(const std::vector<int>& m_, const std::vector<int>& s_)
+{
+ // prepare parameters
+ // holds Li arguments in [BBB] notation
+ std::vector<int> s = s_;
+ std::vector<int> m_p = m_;
+ std::vector<int> m_q;
+ // holds Li arguments in nested sums notation
+ std::vector<cln::cl_N> s_p(s.size(), cln::cl_N(1));
+ s_p[0] = s_p[0] * cln::cl_N("1/2");
+ // convert notations
+ int sig = 1;
+ for (std::size_t i = 0; i < s_.size(); i++) {
+ if (s_[i] < 0) {
+ sig = -sig;
+ s_p[i] = -s_p[i];
+ }
+ s[i] = sig * std::abs(s[i]);
+ }
+ std::vector<cln::cl_N> s_q;
+ cln::cl_N signum = 1;
+
+ // first term
+ cln::cl_N res = multipleLi_do_sum(m_p, s_p);
+
+ // middle terms
+ do {
+
+ // change parameters
+ if (s.front() > 0) {
+ if (m_p.front() == 1) {
+ m_p.erase(m_p.begin());
+ s_p.erase(s_p.begin());
+ if (s_p.size() > 0) {
+ s_p.front() = s_p.front() * cln::cl_N("1/2");
+ }
+ s.erase(s.begin());
+ m_q.front()++;
+ } else {
+ m_p.front()--;
+ m_q.insert(m_q.begin(), 1);
+ if (s_q.size() > 0) {
+ s_q.front() = s_q.front() * 2;
+ }
+ s_q.insert(s_q.begin(), cln::cl_N("1/2"));
+ }
+ } else {
+ if (m_p.front() == 1) {
+ m_p.erase(m_p.begin());
+ cln::cl_N spbuf = s_p.front();
+ s_p.erase(s_p.begin());
+ if (s_p.size() > 0) {
+ s_p.front() = s_p.front() * spbuf;
+ }
+ s.erase(s.begin());
+ m_q.insert(m_q.begin(), 1);
+ if (s_q.size() > 0) {
+ s_q.front() = s_q.front() * 4;
+ }
+ s_q.insert(s_q.begin(), cln::cl_N("1/4"));
+ signum = -signum;
+ } else {
+ m_p.front()--;
+ m_q.insert(m_q.begin(), 1);
+ if (s_q.size() > 0) {
+ s_q.front() = s_q.front() * 2;
+ }
+ s_q.insert(s_q.begin(), cln::cl_N("1/2"));
+ }
+ }
+
+ // exiting the loop
+ if (m_p.size() == 0) break;
+
+ res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q);
+
+ } while (true);
+
+ // last term
+ res = res + signum * multipleLi_do_sum(m_q, s_q);
+
+ return res;
+}
+
+
+} // end of anonymous namespace
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Multiple zeta values zeta(x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+static ex zeta1_evalf(const ex& x)
+{
+ if (is_exactly_a<lst>(x) && (x.nops()>1)) {
+
+ // multiple zeta value
+ const int count = x.nops();
+ const lst& xlst = ex_to<lst>(x);
+ std::vector<int> r(count);
+
+ // check parameters and convert them
+ lst::const_iterator it1 = xlst.begin();
+ std::vector<int>::iterator it2 = r.begin();
+ do {
+ if (!(*it1).info(info_flags::posint)) {
+ return zeta(x).hold();
+ }
+ *it2 = ex_to<numeric>(*it1).to_int();
+ it1++;
+ it2++;
+ } while (it2 != r.end());
+
+ // check for divergence
+ if (r[0] == 1) {
+ return zeta(x).hold();
+ }
+
+ // decide on summation algorithm
+ // this is still a bit clumsy
+ int limit = (Digits>17) ? 10 : 6;
+ if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) {
+ return numeric(zeta_do_sum_Crandall(r));
+ } else {
+ return numeric(zeta_do_sum_simple(r));
+ }
+ }
+
+ // single zeta value
+ if (is_exactly_a<numeric>(x) && (x != 1)) {
+ try {
+ return zeta(ex_to<numeric>(x));
+ } catch (const dunno &e) { }
+ }
+
+ return zeta(x).hold();
+}
+
+
+static ex zeta1_eval(const ex& m)
+{
+ if (is_exactly_a<lst>(m)) {
+ if (m.nops() == 1) {
+ return zeta(m.op(0));
+ }
+ return zeta(m).hold();
+ }
+
+ if (m.info(info_flags::numeric)) {
+ const numeric& y = ex_to<numeric>(m);
+ // trap integer arguments:
+ if (y.is_integer()) {
+ if (y.is_zero()) {
+ return _ex_1_2;
+ }
+ if (y.is_equal(*_num1_p)) {
+ return zeta(m).hold();
+ }
+ if (y.info(info_flags::posint)) {
+ if (y.info(info_flags::odd)) {
+ return zeta(m).hold();
+ } else {
+ return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y);
+ }
+ } else {
+ if (y.info(info_flags::odd)) {
+ return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y);
+ } else {
+ return _ex0;
+ }
+ }
+ }
+ // zeta(float)
+ if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) {
+ return zeta1_evalf(m);
+ }
+ }
+ return zeta(m).hold();
+}
+
+
+static ex zeta1_deriv(const ex& m, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param==0);
+
+ if (is_exactly_a<lst>(m)) {
+ return _ex0;
+ } else {
+ return zetaderiv(_ex1, m);
+ }
+}
+
+
+static void zeta1_print_latex(const ex& m_, const print_context& c)
+{
+ c.s << "\\zeta(";
+ if (is_a<lst>(m_)) {
+ const lst& m = ex_to<lst>(m_);
+ lst::const_iterator it = m.begin();
+ (*it).print(c);
+ it++;
+ for (; it != m.end(); it++) {
+ c.s << ",";
+ (*it).print(c);
+ }
+ } else {
+ m_.print(c);
+ }
+ c.s << ")";
+}
+
+
+unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1).
+ evalf_func(zeta1_evalf).
+ eval_func(zeta1_eval).
+ derivative_func(zeta1_deriv).
+ print_func<print_latex>(zeta1_print_latex).
+ do_not_evalf_params().
+ overloaded(2));
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Alternating Euler sum zeta(x,s)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+static ex zeta2_evalf(const ex& x, const ex& s)
+{
+ if (is_exactly_a<lst>(x)) {
+
+ // alternating Euler sum
+ const int count = x.nops();
+ const lst& xlst = ex_to<lst>(x);
+ const lst& slst = ex_to<lst>(s);
+ std::vector<int> xi(count);
+ std::vector<int> si(count);
+
+ // check parameters and convert them
+ lst::const_iterator it_xread = xlst.begin();
+ lst::const_iterator it_sread = slst.begin();
+ std::vector<int>::iterator it_xwrite = xi.begin();
+ std::vector<int>::iterator it_swrite = si.begin();
+ do {
+ if (!(*it_xread).info(info_flags::posint)) {
+ return zeta(x, s).hold();
+ }
+ *it_xwrite = ex_to<numeric>(*it_xread).to_int();
+ if (*it_sread > 0) {
+ *it_swrite = 1;
+ } else {
+ *it_swrite = -1;
+ }
+ it_xread++;
+ it_sread++;
+ it_xwrite++;
+ it_swrite++;
+ } while (it_xwrite != xi.end());
+
+ // check for divergence
+ if ((xi[0] == 1) && (si[0] == 1)) {
+ return zeta(x, s).hold();
+ }
+
+ // use Hoelder convolution
+ return numeric(zeta_do_Hoelder_convolution(xi, si));
+ }
+
+ return zeta(x, s).hold();
+}
+
+
+static ex zeta2_eval(const ex& m, const ex& s_)
+{
+ if (is_exactly_a<lst>(s_)) {
+ const lst& s = ex_to<lst>(s_);
+ for (lst::const_iterator it = s.begin(); it != s.end(); it++) {
+ if ((*it).info(info_flags::positive)) {
+ continue;
+ }
+ return zeta(m, s_).hold();
+ }
+ return zeta(m);
+ } else if (s_.info(info_flags::positive)) {
+ return zeta(m);
+ }
+
+ return zeta(m, s_).hold();
+}
+
+
+static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param==0);
+
+ if (is_exactly_a<lst>(m)) {
+ return _ex0;
+ } else {
+ if ((is_exactly_a<lst>(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) {
+ return zetaderiv(_ex1, m);
+ }
+ return _ex0;
+ }
+}
+
+
+static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c)
+{
+ lst m;
+ if (is_a<lst>(m_)) {
+ m = ex_to<lst>(m_);
+ } else {
+ m = lst(m_);
+ }
+ lst s;
+ if (is_a<lst>(s_)) {
+ s = ex_to<lst>(s_);
+ } else {
+ s = lst(s_);
+ }
+ c.s << "\\zeta(";
+ lst::const_iterator itm = m.begin();
+ lst::const_iterator its = s.begin();
+ if (*its < 0) {
+ c.s << "\\overline{";
+ (*itm).print(c);
+ c.s << "}";
+ } else {
+ (*itm).print(c);
+ }
+ its++;
+ itm++;
+ for (; itm != m.end(); itm++, its++) {
+ c.s << ",";
+ if (*its < 0) {
+ c.s << "\\overline{";
+ (*itm).print(c);
+ c.s << "}";
+ } else {
+ (*itm).print(c);
+ }
+ }
+ c.s << ")";
+}
+
-REGISTER_FUNCTION(mZeta, eval_func(mZeta_eval).evalf_func(mZeta_evalf).do_not_evalf_params().series_func(mZeta_series));
+unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2).
+ evalf_func(zeta2_evalf).
+ eval_func(zeta2_eval).
+ derivative_func(zeta2_deriv).
+ print_func<print_latex>(zeta2_print_latex).
+ do_not_evalf_params().
+ overloaded(2));
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff