* new G function (multiple polylogarithm).
authorJens Vollinga <vollinga@thep.physik.uni-mainz.de>
Tue, 19 Oct 2004 15:20:33 +0000 (15:20 +0000)
committerJens Vollinga <vollinga@thep.physik.uni-mainz.de>
Tue, 19 Oct 2004 15:20:33 +0000 (15:20 +0000)
* Li now evaluates for arbitrary arguments.

ginac/inifcns.h
ginac/inifcns_nstdsums.cpp

index a17c8d2dfb6761713db6dd6e4c0ce356d4972738..cb42c859a81e087398c31f6bf37f129b9d5a10f7 100644 (file)
@@ -113,6 +113,25 @@ template<> inline bool is_the_function<class zeta_SERIAL>(const ex& x)
        return is_the_function<zeta1_SERIAL>(x) || is_the_function<zeta2_SERIAL>(x);
 }
 
+// overloading at work: we cannot use the macros here
+/** Generalized multiple polylogarithm. */
+class G2_SERIAL { public: static unsigned serial; };
+template<typename T1, typename T2>
+inline function G(const T1& x, const T2& y) {
+       return function(G2_SERIAL::serial, ex(x), ex(y));
+}
+/** Generalized multiple polylogarithm with explicit imaginary parts. */
+class G3_SERIAL { public: static unsigned serial; };
+template<typename T1, typename T2, typename T3>
+inline function G(const T1& x, const T2& s, const T3& y) {
+       return function(G3_SERIAL::serial, ex(x), ex(s), ex(y));
+}
+class G_SERIAL;
+template<> inline bool is_the_function<class G_SERIAL>(const ex& x)
+{
+       return is_the_function<G2_SERIAL>(x) || is_the_function<G3_SERIAL>(x);
+}
+
 /** Polylogarithm and multiple polylogarithm. */
 DECLARE_FUNCTION_2P(Li)
 
index 94d0fb0c1d3af9442977375b4f90120c972a0b1f..b1f227d0c3be5179ff5beec7be805efb2c3005c1 100644 (file)
@@ -5,7 +5,8 @@
  *  The functions are:
  *    classical polylogarithm              Li(n,x)
  *    multiple polylogarithm               Li(lst(m_1,...,m_k),lst(x_1,...,x_k))
- *    nielsen's generalized polylogarithm  S(n,p,x)
+ *                                         G(lst(a_1,...,a_k),y) or G(lst(a_1,...,a_k),lst(s_1,...,s_k),y)
+ *    Nielsen's generalized polylogarithm  S(n,p,x)
  *    harmonic polylogarithm               H(m,x) or H(lst(m_1,...,m_k),x)
  *    multiple zeta value                  zeta(m) or zeta(lst(m_1,...,m_k))
  *    alternating Euler sum                zeta(m,s) or zeta(lst(m_1,...,m_k),lst(s_1,...,s_k))
  *      [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172.
  *      [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754
  *      [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
+ *      [VSW] Numerical evalutation of multiple polylogarithms, J.Vollinga, S.Weinzierl
  *
  *    - The order of parameters and arguments of Li and zeta is defined according to the nested sums
  *      representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only
  *      0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single
  *      number --- notation.
  *
- *    - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in
- *      the whole complex plane. Multiple polylogarithms evaluate only if for each argument x_i the product
- *      x_1 * x_2 * ... * x_i is smaller than one. The parameters for Li, zeta and S must be positive integers.
- *      If you want to have an alternating Euler sum, you have to give the signs of the parameters as a
- *      second argument s to zeta(m,s) containing 1 and -1.
+ *    - All functions can be nummerically evaluated with arguments in the whole complex plane. The parameters
+ *      for Li, zeta and S must be positive integers. If you want to have an alternating Euler sum, you have
+ *      to give the signs of the parameters as a second argument s to zeta(m,s) containing 1 and -1.
  *
  *    - The calculation of classical polylogarithms is speeded up by using Bernoulli numbers and 
  *      look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in
- *      [Cra] and [BBB] for speed up.
+ *      [Cra] and [BBB] for speed up. Multiple polylogarithms use Hoelder convolution [BBB].
  *
- *    - The functions have no series expansion into nested sums. To do this, you have to convert these functions
- *      into the appropriate objects from the nestedsums library, do the expansion and convert the
- *      result back.
+ *    - The functions have no means to do a series expansion into nested sums. To do this, you have to convert
+ *      these functions into the appropriate objects from the nestedsums library, do the expansion and convert
+ *      the result back.
  *
  *    - Numerical testing of this implementation has been performed by doing a comparison of results
  *      between this software and the commercial M.......... 4.1. 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.
+ *      around |x|=1 along with comparisons to corresponding zeta functions. Multiple polylogarithms were
+ *      checked against H and zeta and by means of shuffle and quasi-shuffle relations.
  *
  */
 
@@ -63,6 +64,7 @@
  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  */
 
+#include <sstream>
 #include <stdexcept>
 #include <vector>
 #include <cln/cln.h>
@@ -183,7 +185,6 @@ void fill_Xn(int n)
        xnsize++;
 }
 
-
 // doubles the number of entries in each Xn[]
 void double_Xn()
 {
@@ -192,7 +193,7 @@ void double_Xn()
        for (int i=1; i<=xninitsizestep/2; ++i) {
                Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N());
        }
-       if (Xn.size() > 0) {
+       if (Xn.size() > 1) {
                int xend = xninitsize + xninitsizestep;
                cln::cl_N result;
                // X_1
@@ -380,7 +381,7 @@ cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& pr
 
 
 // helper function for classical polylog Li
-numeric Li_num(int n, const numeric& x)
+numeric Lin_numeric(int n, const numeric& x)
 {
        if (n == 1) {
                // just a log
@@ -397,7 +398,16 @@ numeric Li_num(int n, const numeric& x)
                // [Kol] (2.22)
                return -(1-cln::expt(cln::cl_I(2),1-n)) * cln::zeta(n);
        }
-       
+       if (abs(x.real()) < 0.4 && abs(abs(x)-1) < 0.01) {
+               cln::cl_N x_ = ex_to<numeric>(x).to_cl_N();
+               cln::cl_N result = -cln::expt(cln::log(x_), n-1) * cln::log(1-x_) / cln::factorial(n-1);
+               for (int j=0; j<n-1; j++) {
+                       result = result + (S_num(n-j-1, 1, 1).to_cl_N() - S_num(1, n-j-1, 1-x_).to_cl_N())
+                               * cln::expt(cln::log(x_), j) / cln::factorial(j);
+               }
+               return result;
+       }
+
        // what is the desired float format?
        // first guess: default format
        cln::float_format_t prec = cln::default_float_format;
@@ -431,7 +441,7 @@ numeric Li_num(int n, const numeric& x)
                cln::cl_N add;
                for (int j=0; j<n-1; j++) {
                        add = add + (1+cln::expt(cln::cl_I(-1),n-j)) * (1-cln::expt(cln::cl_I(2),1-n+j))
-                                   * Li_num(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
+                                   * Lin_numeric(n-j,1).to_cl_N() * cln::expt(cln::log(-value),j) / cln::factorial(j);
                }
                result = result - add;
                return result;
@@ -458,6 +468,7 @@ numeric Li_num(int n, const numeric& x)
 namespace {
 
 
+// performs the actual series summation for multiple polylogarithms
 cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl_N>& x)
 {
        const int j = s.size();
@@ -486,142 +497,1007 @@ cln::cl_N multipleLi_do_sum(const std::vector<int>& s, const std::vector<cln::cl
        return t[0];
 }
 
+
+// converts parameter types and calls multipleLi_do_sum (convenience function for G_numeric)
+cln::cl_N mLi_do_summation(const lst& m, const lst& x)
+{
+       std::vector<int> m_int;
+       std::vector<cln::cl_N> x_cln;
+       for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
+               m_int.push_back(ex_to<numeric>(*itm).to_int());
+               x_cln.push_back(ex_to<numeric>(*itx).to_cl_N());
+       }
+       return multipleLi_do_sum(m_int, x_cln);
+}
+
+
 // forward declaration for Li_eval()
 lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf);
 
 
-} // end of anonymous namespace
+// holding dummy-symbols for the G/Li transformations
+std::vector<ex> gsyms;
 
 
-//////////////////////////////////////////////////////////////////////
-//
-// Classical polylogarithm and multiple polylogarithm  Li(n,x)
-//
-// GiNaC function
-//
-//////////////////////////////////////////////////////////////////////
+// type used by the transformation functions for G
+typedef std::vector<int> Gparameter;
 
 
-static ex Li_evalf(const ex& x1, const ex& x2)
+// G_eval1-function for G transformations
+ex G_eval1(int a, int scale)
 {
-       // classical polylogs
-       if (is_a<numeric>(x1) && is_a<numeric>(x2)) {
-               return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2));
-       }
-       if (is_a<numeric>(x1) && !is_a<lst>(x2)) {
-               // try to numerically evaluate second argument
-               ex x2_val = x2.evalf();
-               if (is_a<numeric>(x2_val)) {
-                       return Li_num(ex_to<numeric>(x1).to_int(), ex_to<numeric>(x2_val));
+       if (a != 0) {
+               const ex& scs = gsyms[std::abs(scale)];
+               const ex& as = gsyms[std::abs(a)];
+               if (as != scs) {
+                       return -log(1 - scs/as);
                } else {
-                       return Li(x1, x2).hold();
+                       return -zeta(1);
                }
+       } else {
+               return log(gsyms[std::abs(scale)]);
        }
-       // 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();
+}
+
+
+// G_eval-function for G transformations
+ex G_eval(const Gparameter& a, int scale)
+{
+       // check for properties of G
+       ex sc = gsyms[std::abs(scale)];
+       lst newa;
+       bool all_zero = true;
+       bool all_ones = true;
+       int count_ones = 0;
+       for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
+               if (*it != 0) {
+                       const ex sym = gsyms[std::abs(*it)];
+                       newa.append(sym);
+                       all_zero = false;
+                       if (sym != sc) {
+                               all_ones = false;
                        }
-                       if (!is_a<numeric>(x2.op(i))) {
-                               return Li(x1, x2).hold();
+                       if (all_ones) {
+                               ++count_ones;
                        }
-                       conv *= x2.op(i);
-                       if (abs(conv) >= 1) {
-                               return Li(x1, x2).hold();
+               } else {
+                       all_ones = false;
+               }
+       }
+
+       // care about divergent G: shuffle to separate divergencies that will be canceled
+       // later on in the transformation
+       if (newa.nops() > 1 && newa.op(0) == sc && !all_ones && a.front()!=0) {
+               // do shuffle
+               Gparameter short_a;
+               Gparameter::const_iterator it = a.begin();
+               ++it;
+               for (; it != a.end(); ++it) {
+                       short_a.push_back(*it);
+               }
+               ex result = G_eval1(a.front(), scale) * G_eval(short_a, scale);
+               it = short_a.begin();
+               for (int i=1; i<count_ones; ++i) {
+                       ++it;
+               }
+               for (; it != short_a.end(); ++it) {
+
+                       Gparameter newa;
+                       Gparameter::const_iterator it2 = short_a.begin();
+                       for (--it2; it2 != it;) {
+                               ++it2;
+                               newa.push_back(*it2);
+                       }
+                       newa.push_back(a[0]);
+                       ++it2;
+                       for (; it2 != short_a.end(); ++it2) {
+                               newa.push_back(*it2);   
                        }
+                       result -= G_eval(newa, scale);
+               }
+               return result / count_ones;
+       }
+
+       // G({1,...,1};y) -> G({1};y)^k / k!
+       if (all_ones && a.size() > 1) {
+               return pow(G_eval1(a.front(),scale), count_ones) / factorial(count_ones);
+       }
+
+       // G({0,...,0};y) -> log(y)^k / k!
+       if (all_zero) {
+               return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size());
+       }
+
+       // no special cases anymore -> convert it into Li
+       lst m;
+       lst x;
+       ex argbuf = gsyms[std::abs(scale)];
+       ex mval = _ex1;
+       for (Gparameter::const_iterator it=a.begin(); it!=a.end(); ++it) {
+               if (*it != 0) {
+                       const ex& sym = gsyms[std::abs(*it)];
+                       x.append(argbuf / sym);
+                       m.append(mval);
+                       mval = _ex1;
+                       argbuf = sym;
+               } else {
+                       ++mval;
                }
+       }
+       return pow(-1, x.nops()) * Li(m, x);
+}
+
+
+// converts data for G: pending_integrals -> a
+Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals)
+{
+       GINAC_ASSERT(pending_integrals.size() != 1);
+
+       if (pending_integrals.size() > 0) {
+               // get rid of the first element, which would stand for the new upper limit
+               Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end());
+               return new_a;
+       } else {
+               // just return empty parameter list
+               Gparameter new_a;
+               return new_a;
+       }
+}
+
 
-               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());
+// check the parameters a and scale for G and return information about convergence, depth, etc.
+// convergent     : true if G(a,scale) is convergent
+// depth          : depth of G(a,scale)
+// trailing_zeros : number of trailing zeros of a
+// min_it         : iterator of a pointing on the smallest element in a
+Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale,
+               bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it)
+{
+       convergent = true;
+       depth = 0;
+       trailing_zeros = 0;
+       min_it = a.end();
+       Gparameter::const_iterator lastnonzero = a.end();
+       for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) {
+               if (std::abs(*it) > 0) {
+                       ++depth;
+                       trailing_zeros = 0;
+                       lastnonzero = it;
+                       if (std::abs(*it) < scale) {
+                               convergent = false;
+                               if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) {
+                                       min_it = it;
+                               }
+                       }
+               } else {
+                       ++trailing_zeros;
                }
+       }
+       return ++lastnonzero;
+}
+
 
-               return numeric(multipleLi_do_sum(m, x));
+// add scale to pending_integrals if pending_integrals is empty
+Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale)
+{
+       GINAC_ASSERT(pending_integrals.size() != 1);
+
+       if (pending_integrals.size() > 0) {
+               return pending_integrals;
+       } else {
+               Gparameter new_pending_integrals;
+               new_pending_integrals.push_back(scale);
+               return new_pending_integrals;
        }
+}
+
+
+// handles trailing zeroes for an otherwise convergent integral
+ex trailing_zeros_G(const Gparameter& a, int scale)
+{
+       bool convergent;
+       int depth, trailing_zeros;
+       Gparameter::const_iterator last, dummyit;
+       last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit);
+
+       GINAC_ASSERT(convergent);
 
-       return Li(x1,x2).hold();
+       if ((trailing_zeros > 0) && (depth > 0)) {
+               ex result;
+               Gparameter new_a(a.begin(), a.end()-1);
+               result += G_eval1(0, scale) * trailing_zeros_G(new_a, scale);
+               for (Gparameter::const_iterator it = a.begin(); it != last; ++it) {
+                       Gparameter new_a(a.begin(), it);
+                       new_a.push_back(0);
+                       new_a.insert(new_a.end(), it, a.end()-1);
+                       result -= trailing_zeros_G(new_a, scale);
+               }
+
+               return result / trailing_zeros;
+       } else {
+               return G_eval(a, scale);
+       }
 }
 
 
-static ex Li_eval(const ex& m_, const ex& x_)
+// G transformation [VSW] (57),(58)
+ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale)
 {
-       if (m_.nops() < 2) {
-               ex m;
-               if (is_a<lst>(m_)) {
-                       m = m_.op(0);
+       // pendint = ( y1, b1, ..., br )
+       //       a = ( 0, ..., 0, amin )
+       //   scale = y2
+       //
+       // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2)
+       // where sr replaces amin
+
+       GINAC_ASSERT(a.back() != 0);
+       GINAC_ASSERT(a.size() > 0);
+
+       ex result;
+       Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back()));
+       const int psize = pending_integrals.size();
+
+       // length == 1
+       // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+})
+
+       if (a.size() == 1) {
+
+         // ln(-y2_{-+})
+         result += log(gsyms[ex_to<numeric>(scale).to_int()]);
+               if (a.back() > 0) {
+                       new_pending_integrals.push_back(-scale);
+                       result += I*Pi;
                } else {
-                       m = m_;
+                       new_pending_integrals.push_back(scale);
+                       result -= I*Pi;
                }
-               ex x;
-               if (is_a<lst>(x_)) {
-                       x = x_.op(0);
+               if (psize) {
+                       result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front());
+               }
+               
+               // G(y2_{-+}; sr)
+               result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), new_pending_integrals.front());
+               
+               // G(0; sr)
+               new_pending_integrals.back() = 0;
+               result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), new_pending_integrals.front());
+
+               return result;
+       }
+
+       // length > 1
+       // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
+       //                            - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
+
+       //term zeta_m
+       result -= zeta(a.size());
+       if (psize) {
+               result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front());
+       }
+       
+       // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
+       //    = int_0^sr dt/t G_{m-1}( t_{+-}; y2 )
+       Gparameter new_a(a.begin()+1, a.end());
+       new_pending_integrals.push_back(0);
+       result -= depth_one_trafo_G(new_pending_integrals, new_a, scale);
+       
+       // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t )
+       //    = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 )
+       Gparameter new_pending_integrals_2;
+       new_pending_integrals_2.push_back(scale);
+       new_pending_integrals_2.push_back(0);
+       if (psize) {
+               result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals), pending_integrals.front())
+                         * depth_one_trafo_G(new_pending_integrals_2, new_a, scale);
+       } else {
+               result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale);
+       }
+
+       return result;
+}
+
+
+// forward declaration
+ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
+            const Gparameter& pendint, const Gparameter& a_old, int scale);
+
+
+// G transformation [VSW]
+ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale)
+{
+       // main recursion routine
+       //
+       // pendint = ( y1, b1, ..., br )
+       //       a = ( a1, ..., amin, ..., aw )
+       //   scale = y2
+       //
+       // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
+       // where sr replaces amin
+
+       // find smallest alpha, determine depth and trailing zeros, and check for convergence
+       bool convergent;
+       int depth, trailing_zeros;
+       Gparameter::const_iterator min_it;
+       Gparameter::const_iterator firstzero = 
+               check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it);
+       int min_it_pos = min_it - a.begin();
+
+       // special case: all a's are zero
+       if (depth == 0) {
+               ex result;
+
+               if (a.size() == 0) {
+                 result = 1;
                } else {
-                       x = x_;
+                 result = G_eval(a, scale);
                }
-               if (x == _ex0) {
-                       return _ex0;
+               if (pendint.size() > 0) {
+                 result *= trailing_zeros_G(convert_pending_integrals_G(pendint), pendint.front());
+               } 
+               return result;
+       }
+
+       // handle trailing zeros
+       if (trailing_zeros > 0) {
+               ex result;
+               Gparameter new_a(a.begin(), a.end()-1);
+               result += G_eval1(0, scale) * G_transform(pendint, new_a, scale);
+               for (Gparameter::const_iterator it = a.begin(); it != firstzero; ++it) {
+                       Gparameter new_a(a.begin(), it);
+                       new_a.push_back(0);
+                       new_a.insert(new_a.end(), it, a.end()-1);
+                       result -= G_transform(pendint, new_a, scale);
                }
-               if (x == _ex1) {
-                       return zeta(m);
+               return result / trailing_zeros;
+       }
+
+       // convergence case
+       if (convergent) {
+               if (pendint.size() > 0) {
+                       return G_eval(convert_pending_integrals_G(pendint), pendint.front()) * G_eval(a, scale);
+               } else {
+                       return G_eval(a, scale);
+               }
+       }
+
+       // call basic transformation for depth equal one
+       if (depth == 1) {
+               return depth_one_trafo_G(pendint, a, scale);
+       }
+
+       // do recursion
+       // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2)
+       //  =  int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2)
+       //   + int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) int_0^{sr} ds_{r+1} d/ds_{r+1} G(a1,...,s_{r+1},...,aw,y2)
+
+       // smallest element in last place
+       if (min_it + 1 == a.end()) {
+               do { --min_it; } while (*min_it == 0);
+               Gparameter empty;
+               Gparameter a1(a.begin(),min_it+1);
+               Gparameter a2(min_it+1,a.end());
+
+               ex result = G_transform(pendint,a2,scale)*G_transform(empty,a1,scale);
+
+               result -= shuffle_G(empty,a1,a2,pendint,a,scale);
+               return result;
+       }
+
+       Gparameter empty;
+       Gparameter::iterator changeit;
+
+       // first term G(a_1,..,0,...,a_w;a_0)
+       Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]);
+       Gparameter new_a = a;
+       new_a[min_it_pos] = 0;
+       ex result = G_transform(empty, new_a, scale);
+       if (pendint.size() > 0) {
+               result *= trailing_zeros_G(convert_pending_integrals_G(pendint), pendint.front());
+       }
+
+       // other terms
+       changeit = new_a.begin() + min_it_pos;
+       changeit = new_a.erase(changeit);
+       if (changeit != new_a.begin()) {
+               // smallest in the middle
+               new_pendint.push_back(*changeit);
+               result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
+                       * G_transform(empty, new_a, scale);
+               int buffer = *changeit;
+               *changeit = *min_it;
+               result += G_transform(new_pendint, new_a, scale);
+               *changeit = buffer;
+               new_pendint.pop_back();
+               --changeit;
+               new_pendint.push_back(*changeit);
+               result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
+                       * G_transform(empty, new_a, scale);
+               *changeit = *min_it;
+               result -= G_transform(new_pendint, new_a, scale);
+       } else {
+               // smallest at the front
+               new_pendint.push_back(scale);
+               result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
+                       * G_transform(empty, new_a, scale);
+               new_pendint.back() =  *changeit;
+               result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front())
+                       * G_transform(empty, new_a, scale);
+               *changeit = *min_it;
+               result += G_transform(new_pendint, new_a, scale);
+       }
+       return result;
+}
+
+
+// shuffles the two parameter list a1 and a2 and calls G_transform for every term except
+// for the one that is equal to a_old
+ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2,
+            const Gparameter& pendint, const Gparameter& a_old, int scale) 
+{
+       if (a1.size()==0 && a2.size()==0) {
+               // veto the one configuration we don't want
+               if ( a0 == a_old ) return 0;
+
+               return G_transform(pendint,a0,scale);
+       }
+
+       if (a2.size()==0) {
+               Gparameter empty;
+               Gparameter aa0 = a0;
+               aa0.insert(aa0.end(),a1.begin(),a1.end());
+               return shuffle_G(aa0,empty,empty,pendint,a_old,scale);
+       }
+
+       if (a1.size()==0) {
+               Gparameter empty;
+               Gparameter aa0 = a0;
+               aa0.insert(aa0.end(),a2.begin(),a2.end());
+               return shuffle_G(aa0,empty,empty,pendint,a_old,scale);
+       }
+
+       Gparameter a1_removed(a1.begin()+1,a1.end());
+       Gparameter a2_removed(a2.begin()+1,a2.end());
+
+       Gparameter a01 = a0;
+       Gparameter a02 = a0;
+
+       a01.push_back( a1[0] );
+       a02.push_back( a2[0] );
+
+       return shuffle_G(a01,a1_removed,a2,pendint,a_old,scale)
+            + shuffle_G(a02,a1,a2_removed,pendint,a_old,scale);
+}
+
+
+// handles the transformations and the numerical evaluation of G
+// the parameter x, s and y must only contain numerics
+ex G_numeric(const lst& x, const lst& s, const ex& y)
+{
+       // check for convergence and necessary accelerations
+       bool need_trafo = false;
+       bool need_hoelder = false;
+       int depth = 0;
+       for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+               if (!(*it).is_zero()) {
+                       ++depth;
+                       if (abs(*it) - y < -pow(10,-Digits+2)) {
+                               need_trafo = true;
+                               break;
+                       }
+                       if (abs((abs(*it) - y)/y) < 0.01) {
+                               need_hoelder = true;
+                       }
                }
-               if (x == _ex_1) {
-                       return (pow(2,1-m)-1) * zeta(m);
+       }
+       if (x.op(x.nops()-1).is_zero()) {
+               need_trafo = true;
+       }
+       if (depth == 1 && !need_trafo) {
+               return -Li(x.nops(), y / x.op(x.nops()-1)).evalf();
+       }
+       
+       // convergence transformation
+       if (need_trafo) {
+
+               // sort (|x|<->position) to determine indices
+               std::multimap<ex,int> sortmap;
+               int size = 0;
+               for (int i=0; i<x.nops(); ++i) {
+                       if (!x[i].is_zero()) {
+                               sortmap.insert(std::pair<ex,int>(abs(x[i]), i));
+                               ++size;
+                       }
+               }
+               // include upper limit (scale)
+               sortmap.insert(std::pair<ex,int>(abs(y), x.nops()));
+
+               // generate missing dummy-symbols
+               int i = 1;
+               gsyms.clear();
+               gsyms.push_back(symbol("GSYMS_ERROR"));
+               ex lastentry;
+               for (std::multimap<ex,int>::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
+                       if (it != sortmap.begin()) {
+                               if (it->second < x.nops()) {
+                                       if (x[it->second] == lastentry) {
+                                               gsyms.push_back(gsyms.back());
+                                               continue;
+                                       }
+                               } else {
+                                       if (y == lastentry) {
+                                               gsyms.push_back(gsyms.back());
+                                               continue;
+                                       }
+                               }
+                       }
+                       std::ostringstream os;
+                       os << "a" << i;
+                       gsyms.push_back(symbol(os.str()));
+                       ++i;
+                       if (it->second < x.nops()) {
+                               lastentry = x[it->second];
+                       } else {
+                               lastentry = y;
+                       }
                }
-               if (m == _ex1) {
-                       return -log(1-x);
+
+               // fill position data according to sorted indices and prepare substitution list
+               Gparameter a(x.nops());
+               lst subslst;
+               int pos = 1;
+               int scale;
+               for (std::multimap<ex,int>::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) {
+                       if (it->second < x.nops()) {
+                               if (s[it->second] > 0) {
+                                       a[it->second] = pos;
+                               } else {
+                                       a[it->second] = -pos;
+                               }
+                               subslst.append(gsyms[pos] == x[it->second]);
+                       } else {
+                               scale = pos;
+                               subslst.append(gsyms[pos] == y);
+                       }
+                       ++pos;
                }
-               if (m.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) {
-                       return Li_num(ex_to<numeric>(m).to_int(), ex_to<numeric>(x));
+
+               // do transformation
+               Gparameter pendint;
+               ex result = G_transform(pendint, a, scale);
+               // replace dummy symbols with their values
+               result = result.eval().expand();
+               result = result.subs(subslst).evalf();
+               
+               return result;
+       }
+
+       // do acceleration transformation (hoelder convolution [BBB])
+       if (need_hoelder) {
+               
+               ex result;
+               const int size = x.nops();
+               lst newx;
+               for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+                       newx.append(*it / y);
                }
-       } else {
-               bool ish = true;
-               bool iszeta = true;
-               bool iszero = false;
-               bool doevalf = false;
-               bool doevalfveto = true;
+               
+               for (int r=0; r<=size; ++r) {
+                       ex buffer = pow(-1, r);
+                       ex p = 2;
+                       bool adjustp;
+                       do {
+                               adjustp = false;
+                               for (lst::const_iterator it = newx.begin(); it != newx.end(); ++it) {
+                                       if (*it == 1/p) {
+                                               p += (3-p)/2; 
+                                               adjustp = true;
+                                               continue;
+                                       }
+                               }
+                       } while (adjustp);
+                       ex q = p / (p-1);
+                       lst qlstx;
+                       lst qlsts;
+                       for (int j=r; j>=1; --j) {
+                               qlstx.append(1-newx.op(j-1));
+                               if (newx.op(j-1).info(info_flags::real) && newx.op(j-1) > 1 && newx.op(j-1) <= 2) {
+                                       qlsts.append( s.op(j-1));
+                               } else {
+                                       qlsts.append( -s.op(j-1));
+                               }
+                       }
+                       if (qlstx.nops() > 0) {
+                               buffer *= G_numeric(qlstx, qlsts, 1/q);
+                       }
+                       lst plstx;
+                       lst plsts;
+                       for (int j=r+1; j<=size; ++j) {
+                               plstx.append(newx.op(j-1));
+                               plsts.append(s.op(j-1));
+                       }
+                       if (plstx.nops() > 0) {
+                               buffer *= G_numeric(plstx, plsts, 1/p);
+                       }
+                       result += buffer;
+               }
+               return result;
+       }
+       
+       // do summation
+       lst newx;
+       lst m;
+       int mcount = 1;
+       ex sign = 1;
+       ex factor = y;
+       for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+               if ((*it).is_zero()) {
+                       ++mcount;
+               } else {
+                       newx.append(factor / (*it));
+                       factor = *it;
+                       m.append(mcount);
+                       mcount = 1;
+                       sign = -sign;
+               }
+       }
+
+       return sign * numeric(mLi_do_summation(m, newx));
+}
+
+
+ex mLi_numeric(const lst& m, const lst& x)
+{
+       // let G_numeric do the transformation
+       lst newx;
+       lst s;
+       ex factor = 1;
+       for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
+               for (int i = 1; i < *itm; ++i) {
+                       newx.append(0);
+                       s.append(1);
+               }
+               newx.append(factor / *itx);
+               factor /= *itx;
+               s.append(1);
+       }
+       return pow(-1, m.nops()) * G_numeric(newx, s, _ex1);
+}
+
+
+} // end of anonymous namespace
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Generalized multiple polylogarithm  G(x, y) and G(x, s, y)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+static ex G2_evalf(const ex& x_, const ex& y)
+{
+       if (!y.info(info_flags::positive)) {
+               return G(x_, y).hold();
+       }
+       lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+       if (x.nops() == 0) {
+               return _ex1;
+       }
+       if (x.op(0) == y) {
+               return G(x_, y).hold();
+       }
+       lst s;
+       bool all_zero = true;
+       for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+               if (!(*it).info(info_flags::numeric)) {
+                       return G(x_, y).hold();
+               }
+               if (*it != _ex0) {
+                       all_zero = false;
+               }
+               s.append(+1);
+       }
+       if (all_zero) {
+               return pow(log(y), x.nops()) / factorial(x.nops());
+       }
+       return G_numeric(x, s, y);
+}
+
+
+static ex G2_eval(const ex& x_, const ex& y)
+{
+       //TODO eval to MZV or H or S or Lin
+
+       if (!y.info(info_flags::positive)) {
+               return G(x_, y).hold();
+       }
+       lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+       if (x.nops() == 0) {
+               return _ex1;
+       }
+       if (x.op(0) == y) {
+               return G(x_, y).hold();
+       }
+       lst s;
+       bool all_zero = true;
+       bool crational = true;
+       for (lst::const_iterator it = x.begin(); it != x.end(); ++it) {
+               if (!(*it).info(info_flags::numeric)) {
+                       return G(x_, y).hold();
+               }
+               if (!(*it).info(info_flags::crational)) {
+                       crational = false;
+               }
+               if (*it != _ex0) {
+                       all_zero = false;
+               }
+               s.append(+1);
+       }
+       if (all_zero) {
+               return pow(log(y), x.nops()) / factorial(x.nops());
+       }
+       if (!y.info(info_flags::crational)) {
+               crational = false;
+       }
+       if (crational) {
+               return G(x_, y).hold();
+       }
+       return G_numeric(x, s, y);
+}
+
+
+unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2).
+                                evalf_func(G2_evalf).
+                                eval_func(G2_eval).
+                                do_not_evalf_params().
+                                overloaded(2));
+//TODO
+//                                derivative_func(G2_deriv).
+//                                print_func<print_latex>(G2_print_latex).
+
+
+static ex G3_evalf(const ex& x_, const ex& s_, const ex& y)
+{
+       if (!y.info(info_flags::positive)) {
+               return G(x_, s_, y).hold();
+       }
+       lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+       lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
+       if (x.nops() != s.nops()) {
+               return G(x_, s_, y).hold();
+       }
+       if (x.nops() == 0) {
+               return _ex1;
+       }
+       if (x.op(0) == y) {
+               return G(x_, s_, y).hold();
+       }
+       lst sn;
+       bool all_zero = true;
+       for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
+               if (!(*itx).info(info_flags::numeric)) {
+                       return G(x_, y).hold();
+               }
+               if (!(*its).info(info_flags::real)) {
+                       return G(x_, y).hold();
+               }
+               if (*itx != _ex0) {
+                       all_zero = false;
+               }
+               if (*its >= 0) {
+                       sn.append(+1);
+               } else {
+                       sn.append(-1);
+               }
+       }
+       if (all_zero) {
+               return pow(log(y), x.nops()) / factorial(x.nops());
+       }
+       return G_numeric(x, sn, y);
+}
+
+
+static ex G3_eval(const ex& x_, const ex& s_, const ex& y)
+{
+       //TODO eval to MZV or H or S or Lin
+
+       if (!y.info(info_flags::positive)) {
+               return G(x_, s_, y).hold();
+       }
+       lst x = is_a<lst>(x_) ? ex_to<lst>(x_) : lst(x_);
+       lst s = is_a<lst>(s_) ? ex_to<lst>(s_) : lst(s_);
+       if (x.nops() != s.nops()) {
+               return G(x_, s_, y).hold();
+       }
+       if (x.nops() == 0) {
+               return _ex1;
+       }
+       if (x.op(0) == y) {
+               return G(x_, s_, y).hold();
+       }
+       lst sn;
+       bool all_zero = true;
+       bool crational = true;
+       for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) {
+               if (!(*itx).info(info_flags::numeric)) {
+                       return G(x_, s_, y).hold();
+               }
+               if (!(*its).info(info_flags::real)) {
+                       return G(x_, s_, y).hold();
+               }
+               if (!(*itx).info(info_flags::crational)) {
+                       crational = false;
+               }
+               if (*itx != _ex0) {
+                       all_zero = false;
+               }
+               if (*its >= 0) {
+                       sn.append(+1);
+               } else {
+                       sn.append(-1);
+               }
+       }
+       if (all_zero) {
+               return pow(log(y), x.nops()) / factorial(x.nops());
+       }
+       if (!y.info(info_flags::crational)) {
+               crational = false;
+       }
+       if (crational) {
+               return G(x_, s_, y).hold();
+       }
+       return G_numeric(x, sn, y);
+}
+
+
+unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3).
+                                evalf_func(G3_evalf).
+                                eval_func(G3_eval).
+                                do_not_evalf_params().
+                                overloaded(2));
+//TODO
+//                                derivative_func(G3_deriv).
+//                                print_func<print_latex>(G3_print_latex).
+
+
+//////////////////////////////////////////////////////////////////////
+//
+// Classical polylogarithm and multiple polylogarithm  Li(m,x)
+//
+// GiNaC function
+//
+//////////////////////////////////////////////////////////////////////
+
+
+static ex Li_evalf(const ex& m_, const ex& x_)
+{
+       // classical polylogs
+       if (m_.info(info_flags::posint)) {
+               if (x_.info(info_flags::numeric)) {
+                       return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_));
+               } else {
+                       // try to numerically evaluate second argument
+                       ex x_val = x_.evalf();
+                       if (x_val.info(info_flags::numeric)) {
+                               return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_val));
+                       }
+               }
+       }
+       // multiple polylogs
+       if (is_a<lst>(m_) && is_a<lst>(x_)) {
+
                const lst& m = ex_to<lst>(m_);
                const lst& x = ex_to<lst>(x_);
-               lst::const_iterator itm = m.begin();
-               lst::const_iterator itx = x.begin();
-               for (; itm != m.end(); itm++, itx++) {
+               if (m.nops() != x.nops()) {
+                       return Li(m_,x_).hold();
+               }
+               if (x.nops() == 0) {
+                       return _ex1;
+               }
+               if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) {
+                       return Li(m_,x_).hold();
+               }
+
+               for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
                        if (!(*itm).info(info_flags::posint)) {
                                return Li(m_, x_).hold();
                        }
-                       if ((*itx != _ex1) && (*itx != _ex_1)) {
-                               if (itx != x.begin()) {
-                                       ish = false;
-                               }
-                               iszeta = false;
+                       if (!(*itx).info(info_flags::numeric)) {
+                               return Li(m_, x_).hold();
                        }
                        if (*itx == _ex0) {
-                               iszero = true;
+                               return _ex0;
                        }
-                       if (!(*itx).info(info_flags::numeric)) {
-                               doevalfveto = false;
+               }
+
+               return mLi_numeric(m, x);
+       }
+
+       return Li(m_,x_).hold();
+}
+
+
+static ex Li_eval(const ex& m_, const ex& x_)
+{
+       if (is_a<lst>(m_)) {
+               if (is_a<lst>(x_)) {
+                       // multiple polylogs
+                       const lst& m = ex_to<lst>(m_);
+                       const lst& x = ex_to<lst>(x_);
+                       if (m.nops() != x.nops()) {
+                               return Li(m_,x_).hold();
+                       }
+                       if (x.nops() == 0) {
+                               return _ex1;
+                       }
+                       bool is_H = true;
+                       bool is_zeta = true;
+                       bool do_evalf = true;
+                       bool crational = true;
+                       for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) {
+                               if (!(*itm).info(info_flags::posint)) {
+                                       return Li(m_,x_).hold();
+                               }
+                               if ((*itx != _ex1) && (*itx != _ex_1)) {
+                                       if (itx != x.begin()) {
+                                               is_H = false;
+                                       }
+                                       is_zeta = false;
+                               }
+                               if (*itx == _ex0) {
+                                       return _ex0;
+                               }
+                               if (!(*itx).info(info_flags::numeric)) {
+                                       do_evalf = false;
+                               }
+                               if (!(*itx).info(info_flags::crational)) {
+                                       crational = false;
+                               }
                        }
-                       if (!(*itx).info(info_flags::crational)) {
-                               doevalf = true;
+                       if (is_zeta) {
+                               return zeta(m_,x_);
+                       }
+                       if (is_H) {
+                               ex prefactor;
+                               lst newm = convert_parameter_Li_to_H(m, x, prefactor);
+                               return prefactor * H(newm, x[0]);
+                       }
+                       if (do_evalf && !crational) {
+                               return mLi_numeric(m,x);
                        }
                }
-               if (iszeta) {
-                       return zeta(m_, x_);
-               }
-               if (iszero) {
-                       return _ex0;
-               }
-               if (ish) {
-                       ex pf;
-                       lst newm = convert_parameter_Li_to_H(m, x, pf);
-                       return pf * H(newm, x[0]);
+               return Li(m_, x_).hold();
+       } else if (is_a<lst>(x_)) {
+               return Li(m_, x_).hold();
+       }
+
+       // classical polylogs
+       if (x_ == _ex0) {
+               return _ex0;
+       }
+       if (x_ == _ex1) {
+               return zeta(m_);
+       }
+       if (x_ == _ex_1) {
+               return (pow(2,1-m_)-1) * zeta(m_);
+       }
+       if (m_ == _ex1) {
+               return -log(1-x_);
+       }
+       if (m_ == _ex2) {
+               if (x_.is_equal(I)) {
+                       return power(Pi,_ex2)/_ex_48 + Catalan*I;
                }
-               if (doevalfveto && doevalf) {
-                       return Li(m_, x_).evalf();
+               if (x_.is_equal(-I)) {
+                       return power(Pi,_ex2)/_ex_48 - Catalan*I;
                }
        }
+       if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) {
+               return Lin_numeric(ex_to<numeric>(m_).to_int(), ex_to<numeric>(x_));
+       }
+
        return Li(m_, x_).hold();
 }
 
@@ -1391,12 +2267,12 @@ lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf)
        lst res;
        lst::const_iterator itm = m.begin();
        lst::const_iterator itx = ++x.begin();
-       ex signum = _ex1;
+       int signum = 1;
        pf = _ex1;
        res.append(*itm);
        itm++;
        while (itx != x.end()) {
-               signum *= *itx;
+               signum *= (*itx > 0) ? 1 : -1;
                pf *= signum;
                res.append((*itm) * signum);
                itm++;