Synced changes to HEAD.
authorJens Vollinga <vollinga@thep.physik.uni-mainz.de>
Mon, 3 Nov 2003 21:04:29 +0000 (21:04 +0000)
committerJens Vollinga <vollinga@thep.physik.uni-mainz.de>
Mon, 3 Nov 2003 21:04:29 +0000 (21:04 +0000)
ginac/inifcns.h
ginac/inifcns_nstdsums.cpp
ginac/inifcns_zeta.cpp

index 258ee2e2ef681f725ea05768a53127778055cd9e..3531c8a07dd776a1dd97dbd7831ebf4167c9d2a2 100644 (file)
@@ -89,7 +89,7 @@ DECLARE_FUNCTION_1P(Li2)
 DECLARE_FUNCTION_1P(Li3)
 
 // overloading at work: we cannot use the macros here
-/** Riemann's Zeta-function. */
+/** Multiple zeta value including Riemann's zeta-function. */
 class zeta1_SERIAL { public: static unsigned serial; };
 template<typename T1>
 inline function zeta(const T1 & p1) {
index c89554553455cd4ccd8f5d7892138b40eb94a193..22deea9e5baf0c854765ac8b5b4fd010b8f6cb92 100644 (file)
@@ -1,28 +1,41 @@
 /** @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(n_1,...,n_k),lst(x_1,...,x_k))
  *    nielsen's generalized polylogarithm  S(n,p,x)
- *    harmonic polylogarithm               H(lst(m_1,...,m_k),x)
- *    multiple zeta value                  mZeta(lst(m_1,...,m_k))
+ *    harmonic polylogarithm               H(n,x) or H(lst(n_1,...,n_k),x)
+ *    multiple zeta value                  zeta(n) or zeta(lst(n_1,...,n_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
+ *
+ *    - 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 algorithm in
+ *      [Cra] 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
+//
+// helper functions
+//
+//////////////////////////////////////////////////////////////////////
 
-// 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]
+
+// 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)
@@ -87,8 +108,8 @@ int ynlength = 100; // initial length of all Yn[i]
 // 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;
@@ -157,83 +178,8 @@ static void fill_Xn(int n)
 }
 
 
-// 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)
-{
-       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;
@@ -251,8 +197,8 @@ static cln::cl_N Li2_series(const cln::cl_N& x)
 }
 
 
-// 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);
@@ -271,8 +217,8 @@ static cln::cl_N Li2_series_EuMac(const cln::cl_N& 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;
@@ -288,8 +234,8 @@ static cln::cl_N Lin_series(int n, const cln::cl_N& 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);
@@ -309,11 +255,11 @@ static cln::cl_N Lin_series_EuMac(int n, const cln::cl_N& 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) {
@@ -329,16 +275,16 @@ static cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_forma
                        // 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 {
@@ -351,11 +297,11 @@ static cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_forma
 
                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);
@@ -370,7 +316,7 @@ static cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_forma
 
 
 // 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
@@ -432,25 +378,239 @@ static numeric Li_num(int n, const numeric& x)
 }
 
 
-// helper function for S(n,p,x)
-static cln::cl_N numeric_nielsen(int n, int step)
+} // end of anonymous namespace
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Multiple polylogarithm  Li
+//
+// 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];
+               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
+//
+// 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)) {
+               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();
+                       }
+                       if (x2.op(i) >= 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
+//
+// 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;
 
@@ -509,7 +669,7 @@ static cln::cl_N C(int n, int p)
 
 // 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;
 
@@ -528,7 +688,7 @@ static cln::cl_N a_k(int k)
 
 // 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;
 
@@ -546,7 +706,7 @@ static cln::cl_N b_k(int k)
 
 
 // helper function for S(n,p,x)
-static cln::cl_N S_series(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec)
+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);
@@ -583,7 +743,7 @@ static cln::cl_N S_series(int n, int p, const cln::cl_N& x, const cln::float_for
 
 
 // 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")) {
@@ -595,7 +755,7 @@ static cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float
                        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);
                }
@@ -603,12 +763,12 @@ static cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float
                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) {
@@ -700,203 +860,643 @@ static numeric S_num(int n, int p, const numeric& x)
 }
 
 
-// helper function for multiple polylogarithm
-static cln::cl_N numeric_zsum(int n, std::vector<cln::cl_N>& x, std::vector<cln::cl_N>& m)
-{
-       cln::cl_N res;
-       if (x.empty()) {
-               return 1;
-       }
-       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);
-       }
-       return res;
-}
+} // end of anonymous namespace
+
 
+//////////////////////////////////////////////////////////////////////
+//
+// Nielsen's generalized polylogarithm  S
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
 
-// 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& x1, const ex& x2, const ex& x3)
 {
-       cln::cl_N res;
-       if (m.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 = 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 (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();
 }
 
 
-/////////////////////////////
-// end of helper functions //
-/////////////////////////////
+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)) {
+               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);
+}
 
-// Polylogarithm and multiple polylogarithm
 
-static ex Li_eval(const ex& x1, const ex& x2)
+static ex S_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param)
 {
-       if (x2.is_zero()) {
-               return 0;
+       GINAC_ASSERT(deriv_param < 3);
+       if (deriv_param < 2) {
+               return _ex0;
        }
-       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();
+       if (x1 > 0) {
+               return S(x1-1, x2, x3) / x3;
+       } else {
+               return S(x1, x2-1, x3) / (1-x3);
        }
 }
 
-static ex Li_evalf(const ex& x1, const ex& x2)
+
+REGISTER_FUNCTION(S,
+               eval_func(S_eval).
+               evalf_func(S_evalf).
+               do_not_evalf_params().
+               series_func(S_series).
+               derivative_func(S_deriv));
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Harmonic polylogarithm  H
+//
+// helper function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+// 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)
 {
-       // 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 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);
+               }
        }
-       // 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();
+       for (int i=0; i<=hlong.nops(); i++) {
+               lst newparameter;
+               int j=0;
+               for (; j<i; j++) {
+                       newparameter.append(hlong[j]);
                }
-
-               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());
+               newparameter.append(hshort);
+               for (; j<hlong.nops(); j++) {
+                       newparameter.append(hlong[j]);
                }
+               res += H(newparameter, h1.op(1)).hold();
+       }
+       return res;
+}
 
-               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;
+
+// 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);
                }
 
-               return numeric(res);
+               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;
        }
+};
 
-       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();
+       }
 }
 
-static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options)
+
+// do integration [ReV] (55)
+// put parameter 0 in front of existing parameters
+ex trafo_H_prepend_zero(const ex& e, const ex& arg)
 {
-       epvector seq;
-       seq.push_back(expair(Li(x1,x2), 0));
-       return pseries(rel,seq);
+       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());
+       }
 }
 
-REGISTER_FUNCTION(Li, eval_func(Li_eval).evalf_func(Li_evalf).do_not_evalf_params().series_func(Li_series));
 
+// 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;
 
-// Nielsen's generalized polylogarithm
+                               } else {
 
-static ex S_eval(const ex& x1, const ex& x2, const ex& x3)
+                                       // 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;
+       }
+};
+
+
+// do x -> 1/x transformation
+struct map_trafo_H_1overx : 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 = 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()) {
+                                       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();
+                               }
+
+                               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;
        }
-       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));
+};
+
+
+// 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();
+                                       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;
        }
-       return S(x1,x2,x3).hold();
+};
+
+
+// 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;
+       }
+};
+
+
+// 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
+//////////////////////////////////////////////////////////////////////
+//
+// Harmonic polylogarithm  H
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
 
 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;
@@ -904,16 +1504,439 @@ static ex H_series(const ex& x1, const ex& x2, const relational& rel, int order,
        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
+//
+// 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();
+
+       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;
+
+       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;
+}
+
+
+// [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);
+
+       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];
+}
+
+
+} // end of anonymous namespace
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Multiple zeta values  zeta
+//
+// 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& x)
+{
+       if (is_exactly_a<lst>(x)) {
+               if (x.nops() == 1) {
+                       return zeta(x.op(0));
+               }
+               return zeta(x).hold();
+       }
+
+       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();
+}
 
 
-// Multiple zeta value
+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));
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Multiple zeta values  mZeta
+//
+// The use of mZeta is deprecated! This function will be removed
+// from GiNaC source soon. Use zeta instead!!
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
 
 static ex mZeta_eval(const ex& x1)
 {
        return mZeta(x1).hold();
 }
 
+
 static ex mZeta_evalf(const ex& x1)
 {
        if (is_a<lst>(x1)) {
@@ -933,36 +1956,39 @@ static ex mZeta_evalf(const ex& x1)
                if (r[0] == 1) {
                        return mZeta(x1).hold();
                }
-               
-               // 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 numeric(t[0]);
+               // if only one argument, use cln::zeta
+               if (j == 1) {
+                       return numeric(cln::zeta(r[0]));
+               }
+               
+               // decide on summation algorithm
+               // this is still a bit clumsy
+               int limit = (Digits>17) ? 10 : 6;
+               if (r[0]<limit || (j>3 && r[1]<limit/2)) {
+                       return numeric(zeta_do_sum_Crandall(r));
+               } else {
+                       return numeric(zeta_do_sum_simple(r));
+               }
+       } else if (x1.info(info_flags::posint) && (x1 != 1)) {
+               return numeric(cln::zeta(ex_to<numeric>(x1).to_int()));
        }
 
        return mZeta(x1).hold();
 }
 
-static ex mZeta_series(const ex& x1, const relational& rel, int order, unsigned options)
+
+static ex mZeta_deriv(const ex& x, unsigned deriv_param)
 {
-       epvector seq;
-       seq.push_back(expair(mZeta(x1), 0));
-       return pseries(rel,seq);
+       return 0;
 }
 
-REGISTER_FUNCTION(mZeta, eval_func(mZeta_eval).evalf_func(mZeta_evalf).do_not_evalf_params().series_func(mZeta_series));
+
+REGISTER_FUNCTION(mZeta, 
+               eval_func(mZeta_eval).
+               evalf_func(mZeta_evalf).
+               do_not_evalf_params().
+               derivative_func(mZeta_deriv));
 
 
 } // namespace GiNaC
index 365f35e22aeb9501541b8fb559a9d44168d171c0..2dba976bf4c2faf365aca062ab5de1a453b13f52 100644 (file)
 
 namespace GiNaC {
 
-//////////
-// Riemann's Zeta-function
-//////////
-
-static ex zeta1_evalf(const ex & x)
-{
-       if (is_exactly_a<numeric>(x)) {
-               try {
-                       return zeta(ex_to<numeric>(x));
-               } catch (const dunno &e) { }
-       }
-       
-       return zeta(x).hold();
-}
-
-static ex zeta1_eval(const ex & x)
-{
-       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))
-                               throw(std::domain_error("zeta(1): infinity"));
-                       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();
-}
-
-static ex zeta1_deriv(const ex & x, unsigned deriv_param)
-{
-       GINAC_ASSERT(deriv_param==0);
-       
-       return zeta(_ex1, x);
-}
-
-unsigned zeta1_SERIAL::serial =
-       function::register_new(function_options("zeta").
-                              eval_func(zeta1_eval).
-                              evalf_func(zeta1_evalf).
-                              derivative_func(zeta1_deriv).
-                              latex_name("\\zeta").
-                              overloaded(2));
-
 //////////
 // Derivatives of Riemann's Zeta-function  zeta(0,x)==zeta(x)
 //////////