/** @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)
+ * multiple polylogarithm Li(lst(m_1,...,m_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))
+ * 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 except for S(n,p,-1) when p is not unit (no formula yet
- * to tackle these points). And of course, there is still room for speed optimizations ;-).
- * - 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
+ *
+ * - 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
+ *
+ * - 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.
+ *
+ * - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in
+ * the whole complex plane. Multiple polylogarithms evaluate only if for each argument x_i the product
+ * x_1 * x_2 * ... * x_i is smaller than one. 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 speed 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.
+ *
+ * - The functions have no 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.
+ * 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.
*
*/
#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 "power.h"
+#include "pseries.h"
#include "relational.h"
+#include "symbol.h"
+#include "utils.h"
+#include "wildcard.h"
namespace GiNaC {
-
-//////////////////////
-// helper functions //
-//////////////////////
+//////////////////////////////////////////////////////////////////////
+//
+// Classical polylogarithm Li(n,x)
+//
+// helper functions
+//
+//////////////////////////////////////////////////////////////////////
-// helper function for classical polylog Li
-static cln::cl_N Li_series(int n, const cln::cl_N& x, const cln::float_format_t& prec)
+
+// anonymous namespace for helper functions
+namespace {
+
+
+// lookup table for factors built from Bernoulli numbers
+// see fill_Xn()
+std::vector<std::vector<cln::cl_N> > Xn;
+int xnsize = 0;
+
+
+// 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)
+// The calculation of Xn depends on X0 and X{n-1}.
+// 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 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>::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++) {
+ 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
+
+ *it = result;
+ it++;
+ }
+ 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>::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++) {
+ 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;
+ it++;
+ } 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);
+ }
+ *it = result;
+ it++;
+ }
+ }
+ Xn.push_back(buf);
+ } else {
+ // calculate X_0
+ std::vector<cln::cl_N> buf(initsize/2);
+ std::vector<cln::cl_N>::iterator it = buf.begin();
+ for (int i=1; i<=initsize/2; i++) {
+ *it = bernoulli(i*2).to_cl_N();
+ it++;
+ }
+ Xn.push_back(buf);
+ }
+
+ xnsize++;
+}
+
+
+// calculates Li(2,x) without Xn
+cln::cl_N Li2_do_sum(const cln::cl_N& x)
{
- // Note: argument must be in the unit circle
- cln::cl_N aug, acc;
- cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
- cln::cl_N den = 0;
- int i = 1;
+ cln::cl_N res = x;
+ cln::cl_N resbuf;
+ 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 {
+ resbuf = res;
num = num * x;
- cln::cl_R ii = i;
- den = cln::expt(ii, n);
+ den = den + i; // n^2 = 4, 9, 16, ...
+ i += 2;
+ res = res + num / den;
+ } while (res != resbuf);
+ return res;
+}
+
+
+// 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();
+ cln::cl_N u = -cln::log(1-x);
+ cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits));
+ cln::cl_N res = u - u*u/4;
+ cln::cl_N resbuf;
+ unsigned i = 1;
+ do {
+ resbuf = res;
+ factor = factor * u*u / (2*i * (2*i+1));
+ res = res + (*it) * factor;
+ it++; // should we check it? or rely on initsize? ...
+ i++;
+ } while (res != resbuf);
+ return res;
+}
+
+
+// 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_float(1, cln::float_format(Digits));
+ cln::cl_N res = x;
+ cln::cl_N resbuf;
+ int i=2;
+ do {
+ resbuf = res;
+ factor = factor * x;
+ res = res + factor / cln::expt(cln::cl_I(i),n);
+ i++;
+ } while (res != resbuf);
+ return res;
+}
+
+
+// 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();
+ cln::cl_N u = -cln::log(1-x);
+ 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;
+ do {
+ resbuf = res;
+ factor = factor * u / i;
+ res = res + (*it) * factor;
+ it++; // should we check it? or rely on initsize? ...
i++;
- aug = num / den;
- acc = acc + aug;
- } while (acc != acc+aug);
- return acc;
+ } while (res != resbuf);
+ return res;
}
+// forward declaration needed by function Li_projection and C below
+numeric S_num(int n, int p, const numeric& 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)
{
- return Li_series(n, x, prec);
+ // treat n=2 as special case
+ if (n == 2) {
+ // check if precalculated X0 exists
+ if (xnsize == 0) {
+ fill_Xn(0);
+ }
+
+ if (cln::realpart(x) < 0.5) {
+ // choose the faster algorithm
+ // the switching point was empirically determined. the optimal point
+ // depends on hardware, Digits, ... so an approx value is okay.
+ // 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_do_sum(x);
+ } else {
+ return Li2_do_sum_Xn(x);
+ }
+ } else {
+ // choose the faster algorithm
+ if (cln::abs(cln::realpart(x)) > 0.75) {
+ return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
+ } else {
+ return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2);
+ }
+ }
+ } else {
+ // check if precalculated Xn exist
+ if (n > xnsize+1) {
+ for (int i=xnsize; i<n-1; i++) {
+ fill_Xn(i);
+ }
+ }
+
+ if (cln::realpart(x) < 0.5) {
+ // choose the faster algorithm
+ // 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_do_sum(n, x);
+ } else {
+ 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);
+ 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);
+ }
+ return result;
+ }
+ }
}
// helper function for classical polylog Li
-static numeric Li_num(int n, const numeric& x)
+numeric Li_num(int n, const numeric& x)
{
if (n == 1) {
// just a log
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);
+ * Li_num(n-j,1).to_cl_N() * 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 {
+
+
+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);
+ const int j = s.size();
+
+ 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];
+ // do it once ...
+ 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]);
+ }
+ // ... and do it again (to avoid premature drop out due to special arguments)
+ 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]);
}
- return res;
+ } while (t[0] != t0buf);
+
+ return t[0];
+}
+
+// forward declaration for Li_eval()
+lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
+
+
+} // end of anonymous namespace
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Classical polylogarithm and multiple polylogarithm Li(n,x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+static ex Li_evalf(const ex& x1, const ex& x2)
+{
+ // classical polylogs
+ if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
+ return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
}
- else {
- return 1;
+ // multiple polylogs
+ else if (is_a<lst>(x1) && is_a<lst>(x2)) {
+ ex conv = 1;
+ for (int i=0; i<x1.nops(); i++) {
+ if (!x1.op(i).info(info_flags::posint)) {
+ return Li(x1, x2).hold();
+ }
+ if (!is_a<numeric>(x2.op(i))) {
+ return Li(x1, x2).hold();
+ }
+ conv *= x2.op(i);
+ if (abs(conv) >= 1) {
+ return Li(x1, x2).hold();
+ }
+ }
+
+ std::vector<int> m;
+ std::vector<cln::cl_N> x;
+ for (int i=0; i<ex_to<numeric>(x1.nops()).to_int(); i++) {
+ m.push_back(ex_to<numeric>(x1.op(i)).to_int());
+ x.push_back(ex_to<numeric>(x2.op(i)).to_cl_N());
+ }
+
+ return numeric(multipleLi_do_sum(m, x));
+ }
+
+ return Li(x1,x2).hold();
+}
+
+
+static ex Li_eval(const ex& m_, const ex& x_)
+{
+ if (m_.nops() < 2) {
+ 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 (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.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
+ return Li_num(ex_to<numeric>(m).to_int(), ex_to<numeric>(x));
+ }
+ } else {
+ bool ish = true;
+ bool iszeta = true;
+ bool iszero = false;
+ bool doevalf = false;
+ bool doevalfveto = true;
+ const lst& m = ex_to<lst>(m_);
+ const lst& x = ex_to<lst>(x_);
+ lst::const_iterator itm = m.begin();
+ lst::const_iterator itx = x.begin();
+ for (; 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()) {
+ ish = false;
+ }
+ iszeta = false;
+ }
+ if (*itx == _ex0) {
+ iszero = true;
+ }
+ if (!(*itx).info(info_flags::numeric)) {
+ doevalfveto = false;
+ }
+ if (!(*itx).info(info_flags::crational)) {
+ doevalf = true;
+ }
+ }
+ if (iszeta) {
+ return zeta(m_, x_);
+ }
+ if (iszero) {
+ return _ex0;
+ }
+ if (ish) {
+ ex pf;
+ lst newm = convert_parameter_Li_to_H(m, x, pf);
+ return pf * H(newm, x[0]);
+ }
+ if (doevalfveto && doevalf) {
+ return Li(m_, x_).evalf();
+ }
+ }
+ return Li(m_, x_).hold();
+}
+
+
+static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
+{
+ epvector seq;
+ seq.push_back(expair(Li(m, x), 0));
+ return pseries(rel, seq);
+}
+
+
+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 << "\\mbox{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++;
}
-// forward declaration needed by function C below
-static numeric S_num(int n, int p, const numeric& x);
+// 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)
-static cln::cl_N C(int n, int p)
+cln::cl_N C(int n, int p)
{
cln::cl_N result;
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));
+ * 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));
+ * 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));
+ / (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));
+ * 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));
}
}
}
// helper function for S(n,p,x)
// [Kol] remark to (9.1)
-static cln::cl_N a_k(int k)
+cln::cl_N a_k(int k)
{
cln::cl_N result;
// helper function for S(n,p,x)
// [Kol] remark to (9.1)
-static cln::cl_N b_k(int k)
+cln::cl_N b_k(int k)
{
cln::cl_N result;
// 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)
+cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
{
- n++;
+ if (p==1) {
+ return Li_projection(n+1, x, 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;
- p--;
- cln::cl_N aug, acc;
- cln::cl_N num = cln::expt(x,p);
- cln::cl_N converter = cln::complex(cln::cl_float(1, prec), 0);
- cln::cl_N den = 0;
do {
- num = num * x;
- den = cln::expt(cln::cl_I(i), n);
- aug = num / den * numeric_nielsen(i, p);
+ 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++;
- acc = acc + aug;
- } while (acc != acc+aug);
-
- return acc;
+ } while (res != resbuf);
+
+ return res;
}
// 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)
+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);
+ * 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_series(p-r,n-s,1-x,prec) / cln::factorial(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).to_cl_N() - res2) / cln::factorial(s);
}
return result;
}
- return S_series(n, p, x, prec);
+ return S_do_sum(n, p, x, prec);
}
// helper function for S(n,p,x)
-static numeric S_num(int n, int p, const numeric& x)
+numeric S_num(int n, int p, const numeric& x)
{
if (x == 1) {
if (n == 1) {
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);
+ * cln::factorial(nu+rho+1) / cln::factorial(rho) / cln::factorial(nu+1);
}
}
result = result * cln::expt(cln::cl_I(-1),n+p-1);
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!");
+// throw std::runtime_error("don't know how to evaluate this function!");
}
// what is the desired float format?
if (cln::realpart(value) < -0.5) {
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);
+ * 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);
+ * 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);
}
}
// [Kol] (5.12)
- else if (cln::abs(value) > 1) {
+ 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();
+ / cln::factorial(r) / cln::factorial(s-r) / cln::factorial(n-1)
+ * S_num(n+s-r,p-s,cln::recip(value)).to_cl_N();
}
}
result = result * cln::expt(cln::cl_I(-1),n);
}
-// 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)
+} // 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)
{
- cln::cl_N res;
- if (x.empty()) {
- return 1;
- }
- 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);
+ if (n.info(info_flags::posint) && p.info(info_flags::posint) && is_a<numeric>(x)) {
+ return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
}
- return res;
+ return S(n, p, x).hold();
}
-// helper function for harmonic polylogarithm
-static cln::cl_N numeric_harmonic(int n, std::vector<cln::cl_N>& m)
+static ex S_eval(const ex& n, const ex& p, const ex& x)
{
- cln::cl_N res;
- if (m.empty()) {
- return 1;
- }
- 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 (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))) {
+ return S_num(ex_to<numeric>(n).to_int(), ex_to<numeric>(p).to_int(), ex_to<numeric>(x));
+ }
}
- return res;
+ return S(n, p, x).hold();
}
-/////////////////////////////
-// end of helper functions //
-/////////////////////////////
-
+static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options)
+{
+ epvector seq;
+ seq.push_back(expair(S(n, p, x), 0));
+ return pseries(rel, seq);
+}
-// Polylogarithm and multiple polylogarithm
-static ex Li_eval(const ex& x1, const ex& x2)
+static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param)
{
- if (x2.is_zero()) {
- return 0;
+ GINAC_ASSERT(deriv_param < 3);
+ if (deriv_param < 2) {
+ return _ex0;
}
- else {
- return Li(x1,x2).hold();
+ if (n > 0) {
+ return S(n-1, p, x) / x;
+ } else {
+ return S(n, p-1, x) / (1-x);
}
}
-static ex Li_evalf(const ex& x1, const ex& x2)
-{
- // classical polylogs
- if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
- return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
- }
- // 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 >= 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());
- }
+static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c)
+{
+ c.s << "\\mbox{S}_{";
+ n.print(c);
+ c.s << ",";
+ p.print(c);
+ c.s << "}(";
+ x.print(c);
+ c.s << ")";
+}
- 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;
- }
- return numeric(res);
+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());
- }
- return Li(x1,x2).hold();
-}
+//////////////////////////////////////////////////////////////////////
+//
+// Harmonic polylogarithm H(m,x)
+//
+// helper functions
+//
+//////////////////////////////////////////////////////////////////////
-REGISTER_FUNCTION(Li, eval_func(Li_eval).evalf_func(Li_evalf).do_not_evalf_params());
+// anonymous namespace for helper functions
+namespace {
-// Nielsen's generalized polylogarithm
-static ex S_eval(const ex& x1, const ex& x2, const ex& x3)
+// 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)
{
- if (x2 == 1) {
- return Li(x1+1,x3);
+ // 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);
+ }
}
- 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();
+
+ 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;
}
- return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
}
- return S(x1,x2,x3).hold();
+ if (has_negative_parameters) {
+ for (int 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;
}
-REGISTER_FUNCTION(S, eval_func(S_eval).evalf_func(S_evalf).do_not_evalf_params());
+// 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 (int i=0; i<m.nops(); i++) {
+ s.append(1);
+ }
+ s.let_op(0) = s.op(0) * arg;
+ return Li(m, s).hold();
+ }
+ }
+ }
+ return e;
+ }
+};
-// Harmonic polylogarithm
-static ex H_eval(const ex& x1, const ex& x2)
+// recursivly transforms H to corresponding zetas
+struct map_trafo_H_convert_to_zeta : public map_function
{
- return H(x1,x2).hold();
-}
+ 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();
+ int 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();
+ ex signum = _ex1;
+ pf = _ex1;
+ res.append(*itm);
+ itm++;
+ while (itx != x.end()) {
+ signum *= *itx;
+ 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 (int i=0; i<=hlong.nops(); i++) {
+ lst newparameter;
+ int 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 (int 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 (int 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 (int 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(0),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 (int 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(-1));
+ return (e * (addzeta - H(lst(-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 (int 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(-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 (int 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(1),(1-arg)/(1+arg)).hold()).expand();
+ }
+}
+
+
+// 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 (int 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 (int 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(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ } else {
+ for (int 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(1),1/arg).hold() + H(lst(0),1/arg).hold() - I*Pi, 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 (int 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 (int 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(1), arg).hold() * H(newparameter, arg).hold();
+ int firstzero = 0;
+ while (parameter.op(firstzero) == 1) {
+ firstzero++;
+ }
+ for (int i=firstzero-1; i<parameter.nops()-1; i++) {
+ lst newparameter;
+ int 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 (int 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(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ } else if (parameter.op(0) == -1) {
+ for (int 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(-1),(1-arg)/(1+arg)).hold(), parameter.nops())
+ / factorial(parameter.nops())).expand());
+ }
+ } else {
+ for (int 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(0),(1-arg)/(1+arg)).hold() + H(lst(-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 (int 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 (int 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(1), arg).hold() * H(newparameter, arg).hold();
+ int firstzero = 0;
+ while (parameter.op(firstzero) == 1) {
+ firstzero++;
+ }
+ for (int i=firstzero-1; i<parameter.nops()-1; i++) {
+ lst newparameter;
+ int 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) && is_a<numeric>(x2)) {
for (int i=0; i<x1.nops(); i++) {
- if (!is_a<numeric>(x1.op(i)))
+ if (!x1.op(i).info(info_flags::integer)) {
return H(x1,x2).hold();
+ }
+ }
+ if (x1.nops() < 1) {
+ return H(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).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());
+ cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
+
+ 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
+ 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);
+ } else {
+ m.append(*it);
+ }
+ }
+
+ // since the transformations produce a lot of terms, they are only efficient for
+ // argument near one.
+ // no transformation needed -> 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));
+ }
}
- 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;
+ ex res = 1;
+
+ // ensure that the realpart of the argument is positive
+ if (cln::realpart(x) < 0) {
+ x = -x;
+ for (int i=0; i<m.nops(); i++) {
+ if (m.op(i) != 0) {
+ m.let_op(i) = -m.op(i);
+ res *= -1;
+ }
+ }
}
- return numeric(res);
+ // choose transformations
+ symbol xtemp("xtemp");
+ if (cln::abs(x-1) < 1.4142) {
+ // x -> (1-x)/(1+x)
+ map_trafo_H_1mxt1px trafo;
+ res *= trafo(H(m, xtemp));
+ } else {
+ // x -> 1/x
+ map_trafo_H_1overx trafo;
+ res *= trafo(H(m, xtemp));
+ }
+ // simplify result
+// TODO
+// map_trafo_H_convert converter;
+// res = converter(res).expand();
+// lst ll;
+// res.find(H(wild(1),wild(2)), ll);
+// res.find(zeta(wild(1)), ll);
+// res.find(zeta(wild(1),wild(2)), ll);
+// res = res.collect(ll);
+
+ return res.subs(xtemp == numeric(x)).evalf();
}
return H(x1,x2).hold();
}
-REGISTER_FUNCTION(H, eval_func(H_eval).evalf_func(H_evalf).do_not_evalf_params());
+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();
+}
-// Multiple zeta value
-static ex mZeta_eval(const ex& x1)
+static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options)
{
- return mZeta(x1).hold();
+ epvector seq;
+ seq.push_back(expair(H(m, x), 0));
+ return pseries(rel, seq);
}
-static ex mZeta_evalf(const ex& x1)
+
+static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param)
{
- if (is_a<lst>(x1)) {
- for (int i=0; i<x1.nops(); i++) {
- if (!is_a<numeric>(x1.op(i)))
- return mZeta(x1).hold();
+ 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 << "\\mbox{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");
+int L1;
+int L2;
+std::vector<std::vector<cln::cl_N> > f_kj;
+std::vector<cln::cl_N> crB;
+std::vector<std::vector<cln::cl_N> > crG;
+std::vector<cln::cl_N> crX;
+
+
+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
+void initcX(const std::vector<int>& s)
+{
+ const int k = s.size();
- cln::cl_N m_1 = ex_to<numeric>(x1.op(x1.nops()-1)).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());
+ crX.clear();
+ crG.clear();
+ crB.clear();
+
+ for (int i=0; i<=L2; i++) {
+ crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i));
+ }
+
+ int Sm = 0;
+ int Smp1 = 0;
+ for (int m=0; m<k-1; m++) {
+ std::vector<cln::cl_N> crGbuf;
+ Sm = Sm + s[m];
+ Smp1 = Sm + s[m+1];
+ for (int i=0; i<=L2; i++) {
+ crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2));
}
+ crG.push_back(crGbuf);
+ }
+
+ crX = crB;
- 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;
+ for (int m=0; m<k-1; m++) {
+ std::vector<cln::cl_N> Xbuf;
+ for (int i=0; i<=L2; i++) {
+ Xbuf.push_back(crX[i] * crG[m][i]);
}
+ halfcyclic_convolute(Xbuf, crB, crX);
+ }
+}
+
+
+// [Cra] section 4
+cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk)
+{
+ 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;
+}
- return numeric(res);
+// [Cra] section 4
+void calc_f(int maxr)
+{
+ f_kj.clear();
+ f_kj.resize(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)
+cln::cl_N crandall_Z(const std::vector<int>& s)
+{
+ 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();
+
+ // decide on maximal size of f_kj for crandall_Z
+ if (Digits < 50) {
+ L1 = 150;
+ } else {
+ L1 = Digits * 3 + j*2;
+ }
+
+ // 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];
+ }
+ }
+
+ calc_f(maxr);
+
+ const cln::cl_N r0factorial = cln::factorial(r[0]-1);
- return mZeta(x1).hold();
+ 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();
+
+ initcX(r);
+
+ for (int q=0; q<skp1buf; q++) {
+
+ cln::cl_N pp1 = crandall_Y_loop(Srun+q-k);
+ cln::cl_N pp2 = crandall_Z(rz);
+
+ 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());
+
+ initcX(rz);
+
+ res = (res + crandall_Y_loop(S-j)) / r0factorial + crandall_Z(rz);
+
+ return res;
}
-REGISTER_FUNCTION(mZeta, eval_func(mZeta_eval).evalf_func(mZeta_evalf).do_not_evalf_params());
+
+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 (int 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)) {
+ 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, y-_num1) / factorial(y);
+ }
+ } else {
+ if (y.info(info_flags::odd)) {
+ return -bernoulli(_num1-y) / (_num1-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").
+ 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 << ")";
+}
+
+
+unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta").
+ 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