/** @file inifcns_nstdsums.cpp
*
* Implementation of some special functions that have a representation as nested sums.
+ *
* 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. 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
+ *
+ * - 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 H, Li and zeta is defined according to their order in the
+ * nested sums representation.
+ *
+ * - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in
+ * the whole complex plane. Multiple polylogarithms evaluate only if each argument x_i is smaller than
+ * one. The parameters for every function (n, p or n_i) must be positive integers.
+ *
+ * - 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 as nested sums. 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.
+ *
* - 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 "relational.h"
+#include "power.h"
#include "pseries.h"
+#include "relational.h"
+#include "symbol.h"
+#include "utils.h"
+#include "wildcard.h"
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;
+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;
}
-// 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)
-{
-
- 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;
-}
-
-
-// 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;
}
-// 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();
cln::cl_N u = -cln::log(1-x);
}
-// 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 res = x;
}
-// 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();
cln::cl_N u = -cln::log(1-x);
// forward declaration needed by function Li_projection and C below
-static numeric S_num(int n, int p, const numeric& x);
+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)
{
// 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);
+ 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);
// 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
}
-// 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];
+}
+
+
+} // end of anonymous namespace
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Classical polylogarithm and multiple polylogarithm Li(n,x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+static ex Li_eval(const ex& x1, const ex& x2)
+{
+ if (x2.is_zero()) {
+ return _ex0;
}
else {
- return 1;
+ if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational)))
+ return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
+ if (is_a<lst>(x2)) {
+ for (int i=0; i<x2.nops(); i++) {
+ if (!is_a<numeric>(x2.op(i))) {
+ return Li(x1,x2).hold();
+ }
+ }
+ return Li(x1,x2).evalf();
+ }
+ return Li(x1,x2).hold();
+ }
+}
+
+
+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)) {
+ 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 ((conv > 1) || ((conv == 1) && (x1.op(0) == 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_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
+{
+ epvector seq;
+ seq.push_back(expair(Li(x1,x2), 0));
+ return pseries(rel,seq);
+}
+
+
+static ex Li_deriv(const ex& x1, const ex& x2, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param < 2);
+ if (deriv_param == 0) {
+ return _ex0;
+ }
+ if (x1 > 0) {
+ return Li(x1-1, x2) / x2;
+ } else {
+ return 1/(1-x2);
+ }
+}
+
+
+REGISTER_FUNCTION(Li,
+ eval_func(Li_eval).
+ evalf_func(Li_evalf).
+ do_not_evalf_params().
+ series_func(Li_series).
+ derivative_func(Li_deriv));
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// 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)
-static cln::cl_N C(int n, int p)
+cln::cl_N C(int n, int p)
{
cln::cl_N result;
// 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)
{
if (p==1) {
return Li_projection(n+1, x, prec);
}
- // 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++) {
// should be done otherwise
cln::cl_N xf = x * cln::cl_float(1, prec);
- cln::cl_N result;
- cln::cl_N resultbuffer;
- int i;
- for (i=p; true; i++) {
- resultbuffer = result;
+ 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);
}
- 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;
- }
- }
+ 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 result;
+ 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 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) {
}
-// 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_eval(const ex& x1, const ex& x2, const ex& x3)
{
- cln::cl_N res;
- if (x.empty()) {
- return 1;
+ if (x2 == 1) {
+ return Li(x1+1,x3);
}
- 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 (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));
}
- return res;
+ return S(x1,x2,x3).hold();
}
-// helper function for harmonic polylogarithm
-static cln::cl_N numeric_harmonic(int n, std::vector<cln::cl_N>& m)
+static ex S_evalf(const ex& x1, const ex& x2, const ex& x3)
{
- cln::cl_N res;
- if (m.empty()) {
- return 1;
+ if (is_a<numeric>(x1) && is_a<numeric>(x2) && is_a<numeric>(x3)) {
+ 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();
+}
+
+
+static ex S_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options)
+{
+ epvector seq;
+ seq.push_back(expair(S(x1,x2,x3), 0));
+ return pseries(rel,seq);
+}
+
+
+static ex S_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param < 3);
+ if (deriv_param < 2) {
+ return _ex0;
}
- 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 (x1 > 0) {
+ return S(x1-1, x2, x3) / x3;
+ } else {
+ return S(x1, x2-1, x3) / (1-x3);
}
- return res;
}
-/////////////////////////////
-// end of helper functions //
-/////////////////////////////
+REGISTER_FUNCTION(S,
+ eval_func(S_eval).
+ evalf_func(S_evalf).
+ do_not_evalf_params().
+ series_func(S_series).
+ derivative_func(S_deriv));
-// Polylogarithm and multiple polylogarithm
+//////////////////////////////////////////////////////////////////////
+//
+// Harmonic polylogarithm H(m,x)
+//
+// helper function
+//
+//////////////////////////////////////////////////////////////////////
-static ex Li_eval(const ex& x1, const ex& x2)
+
+// anonymous namespace for helper functions
+namespace {
+
+
+// forward declaration
+ex convert_from_RV(const lst& parameterlst, const ex& arg);
+
+
+// multiplies an one-dimensional H with another H
+// [ReV] (18)
+ex trafo_H_mult(const ex& h1, const ex& h2)
{
- if (x2.is_zero()) {
- return 0;
+ 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);
+ }
}
- 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();
+ 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;
}
-static ex Li_evalf(const ex& x1, const ex& x2)
+
+// applies trafo_H_mult recursively on expressions
+struct map_trafo_H_mult : public map_function
{
- // classical polylogs
- if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
- return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
+ 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;
}
- // 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();
+};
+
+
+// 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 (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());
+ } else {
+ return e * H(lst(1),1-arg).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());
+
+// do integration [ReV] (55)
+// put parameter 0 in front of existing parameters
+ex trafo_H_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_from_RV(newparameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
+ return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand();
+ } else {
+ return e * (-H(lst(0),1/arg).hold());
+ }
+}
- 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;
+
+// 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);
+
+ // if all parameters are either zero or one return the transformed function
+ if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) {
+ lst newparameter;
+ for (int i=parameter.nops(); i>0; i--) {
+ newparameter.append(0);
+ }
+ return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold();
+ } else if (find(parameter.begin(), parameter.end(), 1) == parameter.end()) {
+ lst newparameter;
+ for (int i=parameter.nops(); i>0; i--) {
+ newparameter.append(1);
+ }
+ 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_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 (int 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;
- return numeric(res);
+ } else {
+
+ // leading one
+ map_trafo_H_1mx recursion;
+ map_trafo_H_mult unify;
+ ex res;
+ 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();
+ }
+ return (unify((-H(lst(0), 1-arg).hold() * recursion(H(newparameter, arg).hold())).expand()) +
+ recursion(res)) / firstzero;
+ }
+
+ }
+ }
+ return e;
}
+};
- return Li(x1,x2).hold();
-}
-static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
+// do x -> 1/x transformation
+struct map_trafo_H_1overx : public map_function
{
- epvector seq;
- seq.push_back(expair(Li(x1,x2), 0));
- return pseries(rel,seq);
-}
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e)) {
+ return e.map(*this);
+ }
-REGISTER_FUNCTION(Li, eval_func(Li_eval).evalf_func(Li_evalf).do_not_evalf_params().series_func(Li_series));
+ 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);
-// Nielsen's generalized polylogarithm
+ // if all parameters are either zero or one return the transformed function
+ if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) {
+ 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());
+ } else if (find(parameter.begin(), parameter.end(), 1) == parameter.end()) {
+ return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold();
+ }
-static ex S_eval(const ex& x1, const ex& x2, const ex& x3)
+ lst newparameter = parameter;
+ newparameter.remove_first();
+
+ if (parameter.op(0) == 0) {
+
+ // leading zero
+ ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1)));
+ 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_prepend_zero(buffer.op(i), arg);
+ }
+ } else {
+ res += trafo_H_prepend_zero(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;
+ }
+};
+
+
+// remove trailing zeros from H-parameters
+struct map_trafo_H_reduce_trailing_zeros : public map_function
{
- if (x2 == 1) {
- return Li(x1+1,x3);
+ 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();
+ for (ex i=0; i<lastentry; i++) {
+ parameter[i]++;
+ result -= (parameter[i]-1) * H(parameter, arg).hold();
+ parameter[i]--;
+ }
+
+ if (lastentry < parameter.nops()) {
+ result = result / (parameter.nops()-lastentry+1);
+ return result.map(*this);
+ } else {
+ return result;
+ }
+ }
+ }
+ }
+ return e;
}
- 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));
+};
+
+
+// recursively call convert_from_RV on expression
+struct map_trafo_H_convert : public map_function
+{
+ ex operator()(const ex& e)
+ {
+ if (is_a<add>(e) || is_a<mul>(e) || is_a<power>(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);
+ return convert_from_RV(parameter, arg);
+ }
+ }
+ return e;
}
- return S(x1,x2,x3).hold();
+};
+
+
+// translate notation from nested sums to Remiddi/Vermaseren
+lst convert_to_RV(const lst& o)
+{
+ lst res;
+ for (lst::const_iterator it = o.begin(); it != o.end(); it++) {
+ for (ex i=0; i<(*it)-1; i++) {
+ res.append(0);
+ }
+ res.append(1);
+ }
+ return res;
}
-static ex S_evalf(const ex& x1, const ex& x2, const ex& x3)
+
+// translate notation from Remiddi/Vermaseren to nested sums
+ex convert_from_RV(const lst& parameterlst, const ex& arg)
{
- 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();
+ lst newparameterlst;
+
+ lst::const_iterator it = parameterlst.begin();
+ int count = 1;
+ while (it != parameterlst.end()) {
+ if (*it == 0) {
+ count++;
+ } else {
+ newparameterlst.append((*it>0) ? count : -count);
+ count = 1;
}
- return S_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2).to_int(), ex_to<numeric>(x3));
+ it++;
}
- return S(x1,x2,x3).hold();
+ for (int i=1; i<count; i++) {
+ newparameterlst.append(0);
+ }
+
+ map_trafo_H_reduce_trailing_zeros filter;
+ return filter(H(newparameterlst, arg).hold());
}
-static ex S_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options)
+
+// do the actual summation.
+cln::cl_N H_do_sum(const std::vector<int>& s, const cln::cl_N& x)
{
- epvector seq;
- seq.push_back(expair(S(x1,x2,x3), 0));
- return pseries(rel,seq);
+ 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 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),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] * factor / cln::expt(cln::cl_I(q+j-1), s[0]);
+ factor = factor * x;
+ } while (t[0] != t0buf);
+
+ return t[0];
}
-REGISTER_FUNCTION(S, eval_func(S_eval).evalf_func(S_evalf).do_not_evalf_params().series_func(S_series));
+} // end of anonymous namespace
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Harmonic polylogarithm H(m,x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
-// Harmonic polylogarithm
static ex H_eval(const ex& x1, const ex& x2)
{
+ if (x2 == 0) {
+ return 0;
+ }
+ if (x2 == 1) {
+ return zeta(x1);
+ }
+ if (x1.nops() == 1) {
+ return Li(x1.op(0), x2);
+ }
if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) {
return H(x1,x2).evalf();
}
return H(x1,x2).hold();
}
+
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::posint)) {
return H(x1,x2).hold();
+ }
}
- if (x2 >= 1) {
- return H(x1,x2).hold();
+ if (x1.nops() < 1) {
+ return _ex1;
+ }
+ if (x1.nops() == 1) {
+ return Li(x1.op(0), x2).evalf();
+ }
+ cln::cl_N x = ex_to<numeric>(x2).to_cl_N();
+ if (x == 1) {
+ return zeta(x1).evalf();
}
- 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());
+ // choose trafo
+ if (cln::abs(x) > 1) {
+ symbol xtemp("xtemp");
+ map_trafo_H_1overx trafo;
+ ex res = trafo(H(convert_to_RV(ex_to<lst>(x1)), xtemp));
+ map_trafo_H_convert converter;
+ res = converter(res);
+ return res.subs(xtemp==x2).evalf();
}
- 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;
+ // since the x->1-x transformation produces a lot of terms, it is only
+ // efficient for argument near one.
+ if (cln::realpart(x) > 0.95) {
+ symbol xtemp("xtemp");
+ map_trafo_H_1mx trafo;
+ ex res = trafo(H(convert_to_RV(ex_to<lst>(x1)), xtemp));
+ map_trafo_H_convert converter;
+ res = converter(res);
+ return res.subs(xtemp==x2).evalf();
}
- return numeric(res);
+ // no trafo -> do summation
+ int count = x1.nops();
+ std::vector<int> r(count);
+ for (int i=0; i<count; i++) {
+ r[i] = ex_to<numeric>(x1.op(i)).to_int();
+ }
+ return numeric(H_do_sum(r,x));
}
return H(x1,x2).hold();
}
+
static ex H_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
{
epvector seq;
return pseries(rel,seq);
}
-REGISTER_FUNCTION(H, eval_func(H_eval).evalf_func(H_evalf).do_not_evalf_params().series_func(H_series));
+static ex H_deriv(const ex& x1, const ex& x2, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param < 2);
+ if (deriv_param == 0) {
+ return _ex0;
+ }
+ if (is_a<lst>(x1)) {
+ lst newparameter = ex_to<lst>(x1);
+ if (x1.op(0) == 1) {
+ newparameter.remove_first();
+ return 1/(1-x2) * H(newparameter, x2);
+ } else {
+ newparameter[0]--;
+ return H(newparameter, x2).hold() / x2;
+ }
+ } else {
+ if (x1 == 1) {
+ return 1/(1-x2);
+ } else {
+ return H(x1-1, x2).hold() / x2;
+ }
+ }
+}
+
+
+REGISTER_FUNCTION(H,
+ eval_func(H_eval).
+ evalf_func(H_evalf).
+ do_not_evalf_params().
+ series_func(H_series).
+ derivative_func(H_deriv));
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// 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();
-// Multiple zeta value
+ crX.clear();
+ crG.clear();
+ crB.clear();
-static ex mZeta_eval(const ex& x1)
+ 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;
+
+ 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)
{
- return mZeta(x1).hold();
+ 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;
}
-static ex mZeta_evalf(const ex& x1)
+
+// [Cra] section 4
+void calc_f(int maxr)
{
- if (is_a<lst>(x1)) {
- for (int i=0; i<x1.nops(); i++) {
- if (!is_a<numeric>(x1.op(i)))
- return mZeta(x1).hold();
+ 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);
+
+ 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;
+}
+
+
+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"));
+ }
}
- cln::cl_N m_1 = ex_to<numeric>(x1.op(x1.nops()-1)).to_cl_N();
+ // 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 (m_1 == 1) {
- return mZeta(x1).hold();
+ 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));
}
+ }
- 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());
+ // 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& x)
+{
+ if (is_exactly_a<lst>(x)) {
+ if (x.nops() == 1) {
+ return zeta(x.op(0));
}
+ return zeta(x).hold();
+ }
- 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 (x.info(info_flags::numeric)) {
+ const numeric& y = ex_to<numeric>(x);
+ // trap integer arguments:
+ if (y.is_integer()) {
+ if (y.is_zero()) {
+ return _ex_1_2;
+ }
+ if (y.is_equal(_num1)) {
+ return zeta(x).hold();
+ }
+ if (y.info(info_flags::posint)) {
+ if (y.info(info_flags::odd)) {
+ return zeta(x).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(x);
+ }
+ return zeta(x).hold();
+}
+
- return numeric(res);
+static ex zeta1_deriv(const ex& x, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param==0);
+ if (is_exactly_a<lst>(x)) {
+ return _ex0;
+ } else {
+ return zeta(_ex1, x);
}
+}
+
+
+unsigned zeta1_SERIAL::serial =
+ function::register_new(function_options("zeta").
+ eval_func(zeta1_eval).
+ evalf_func(zeta1_evalf).
+ do_not_evalf_params().
+ derivative_func(zeta1_deriv).
+ latex_name("\\zeta").
+ 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());
- return mZeta(x1).hold();
+ // 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 mZeta_series(const ex& x1, const relational& rel, int order, unsigned options)
+
+static ex zeta2_eval(const ex& x, const ex& s)
{
- epvector seq;
- seq.push_back(expair(mZeta(x1), 0));
- return pseries(rel,seq);
+ if (is_exactly_a<lst>(s)) {
+ const lst& l = ex_to<lst>(s);
+ lst::const_iterator it = l.begin();
+ while (it != l.end()) {
+ if ((*it).info(info_flags::negative)) {
+ return zeta(x, s).hold();
+ }
+ it++;
+ }
+ return zeta(x);
+ } else {
+ if (s.info(info_flags::positive)) {
+ return zeta(x);
+ }
+ }
+
+ return zeta(x, s).hold();
+}
+
+
+static ex zeta2_deriv(const ex& x, const ex& s, unsigned deriv_param)
+{
+ GINAC_ASSERT(deriv_param==0);
+
+ if (is_exactly_a<lst>(x)) {
+ return _ex0;
+ } else {
+ if ((is_exactly_a<lst>(s) && (s.op(0) > 0)) || (s > 0)) {
+ return zeta(_ex1, x);
+ }
+ return _ex0;
+ }
}
-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").
+ eval_func(zeta2_eval).
+ evalf_func(zeta2_evalf).
+ do_not_evalf_params().
+ derivative_func(zeta2_deriv).
+ latex_name("\\zeta").
+ overloaded(2));
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff