X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_nstdsums.cpp;h=17db2c885dfd63a8ea880d61ff85ec4be5e8ea0f;hp=9bff997694da654394944c2b6dbcc9cf2c625203;hb=547db2f0380c03c7d013aa8bda36aa33ff5559e1;hpb=a450af1f438d53e924a074c936c648991eddfc71 diff --git a/ginac/inifcns_nstdsums.cpp b/ginac/inifcns_nstdsums.cpp index 9bff9976..17db2c88 100644 --- a/ginac/inifcns_nstdsums.cpp +++ b/ginac/inifcns_nstdsums.cpp @@ -1,8 +1,8 @@ /** @file inifcns_nstdsums.cpp * * Implementation of some special functions that have a representation as nested sums. - * - * The functions are: + * + * The functions are: * classical polylogarithm Li(n,x) * multiple polylogarithm Li(lst(m_1,...,m_k),lst(x_1,...,x_k)) * nielsen's generalized polylogarithm S(n,p,x) @@ -11,28 +11,32 @@ * alternating Euler sum zeta(m,s) or zeta(lst(m_1,...,m_k),lst(s_1,...,s_k)) * * Some remarks: - * + * * - All formulae used can be looked up in the following publications: * [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258. - * [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172. - * [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754 - * [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941 + * [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172. + * [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754 + * [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941 + * + * - The order of parameters and arguments of Li and zeta is defined according to the nested sums + * representation. The parameters for H are understood as in [ReV]. They can be in expanded --- only + * 0, 1 and -1 --- or in compactified --- a string with zeros in front of 1 or -1 is written as a single + * number --- notation. * - * - 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 + * the whole complex plane. Multiple polylogarithms evaluate only if for each argument x_i the product + * x_1 * x_2 * ... * x_i is smaller than one. The parameters for Li, zeta and S must be positive integers. + * If you want to have an alternating Euler sum, you have to give the signs of the parameters as a + * second argument s to zeta(m,s) containing 1 and -1. + * + * - The calculation of classical polylogarithms is 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. - * - * - The functions have no series expansion as nested sums. To do it, you have to convert these functions + * + * - The functions have no series expansion into nested sums. To do this, you have to convert these functions * into the appropriate objects from the nestedsums library, do the expansion and convert the - * result back. - * + * result back. + * * - Numerical testing of this implementation has been performed by doing a comparison of results * between this software and the commercial M.......... 4.1. Multiple zeta values have been checked * by means of evaluations into simple zeta values. Harmonic polylogarithms have been checked by @@ -42,7 +46,7 @@ */ /* - * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by @@ -79,11 +83,6 @@ #include "wildcard.h" -//DEBUG -#include -using namespace std; - - namespace GiNaC { @@ -103,6 +102,9 @@ namespace { // lookup table for factors built from Bernoulli numbers // see fill_Xn() std::vector > Xn; +// initial size of Xn that should suffice for 32bit machines (must be even) +const int xninitsizestep = 26; +int xninitsize = xninitsizestep; int xnsize = 0; @@ -118,17 +120,14 @@ int xnsize = 0; // The second index in Xn corresponds to the index from the actual sum. void fill_Xn(int n) { - // rule of thumb. needs to be improved. TODO - const int initsize = Digits * 3 / 2; - if (n>1) { // calculate X_2 and higher (corresponding to Li_4 and higher) - std::vector buf(initsize); + std::vector buf(xninitsize); std::vector::iterator it = buf.begin(); cln::cl_N result; *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1 it++; - for (int i=2; i<=initsize; i++) { + for (int i=2; i<=xninitsize; i++) { if (i&1) { result = 0; // k == 0 } else { @@ -148,14 +147,14 @@ void fill_Xn(int n) Xn.push_back(buf); } else if (n==1) { // special case to handle the X_0 correct - std::vector buf(initsize); + std::vector buf(xninitsize); std::vector::iterator it = buf.begin(); cln::cl_N result; *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1 it++; *it = cln::cl_I(17)/cln::cl_I(36); // i == 2 it++; - for (int i=3; i<=initsize; i++) { + for (int i=3; i<=xninitsize; i++) { if (i & 1) { result = -Xn[0][(i-3)/2]/2; *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result; @@ -172,9 +171,9 @@ void fill_Xn(int n) Xn.push_back(buf); } else { // calculate X_0 - std::vector buf(initsize/2); + std::vector buf(xninitsize/2); std::vector::iterator it = buf.begin(); - for (int i=1; i<=initsize/2; i++) { + for (int i=1; i<=xninitsize/2; i++) { *it = bernoulli(i*2).to_cl_N(); it++; } @@ -185,12 +184,59 @@ void fill_Xn(int n) } +// doubles the number of entries in each Xn[] +void double_Xn() +{ + const int pos0 = xninitsize / 2; + // X_0 + for (int i=1; i<=xninitsizestep/2; ++i) { + Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N()); + } + if (Xn.size() > 0) { + int xend = xninitsize + xninitsizestep; + cln::cl_N result; + // X_1 + for (int i=xninitsize+1; i<=xend; ++i) { + if (i & 1) { + result = -Xn[0][(i-3)/2]/2; + Xn[1].push_back((cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result); + } else { + result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1); + for (int k=1; k 1)) ) { + result = result + cln::binomial(i,k) * Xn[0][(i-k)/2-1] * Xn[n-1][k-1] / (k+1); + } + } + result = result - cln::binomial(i,i-1) * Xn[n-1][i-2] / 2 / i; // k == i-1 + result = result + Xn[n-1][i-1] / (i+1); // k == i + Xn[n].push_back(result); + } + } + } + xninitsize += xninitsizestep; +} + + // calculates Li(2,x) without Xn cln::cl_N Li2_do_sum(const cln::cl_N& x) { cln::cl_N res = x; cln::cl_N resbuf; - cln::cl_N num = x; + cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits)); cln::cl_I den = 1; // n^2 = 1 unsigned i = 3; do { @@ -208,8 +254,9 @@ cln::cl_N Li2_do_sum(const cln::cl_N& x) cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x) { std::vector::const_iterator it = Xn[0].begin(); + std::vector::const_iterator xend = Xn[0].end(); cln::cl_N u = -cln::log(1-x); - cln::cl_N factor = u; + cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits)); cln::cl_N res = u - u*u/4; cln::cl_N resbuf; unsigned i = 1; @@ -217,8 +264,12 @@ cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x) resbuf = res; factor = factor * u*u / (2*i * (2*i+1)); res = res + (*it) * factor; - it++; // should we check it? or rely on initsize? ... i++; + if (++it == xend) { + double_Xn(); + it = Xn[0].begin() + (i-1); + xend = Xn[0].end(); + } } while (res != resbuf); return res; } @@ -227,7 +278,7 @@ cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x) // calculates Li(n,x), n>2 without Xn cln::cl_N Lin_do_sum(int n, const cln::cl_N& x) { - cln::cl_N factor = x; + cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits)); cln::cl_N res = x; cln::cl_N resbuf; int i=2; @@ -245,8 +296,9 @@ cln::cl_N Lin_do_sum(int n, const cln::cl_N& x) cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x) { std::vector::const_iterator it = Xn[n-2].begin(); + std::vector::const_iterator xend = Xn[n-2].end(); cln::cl_N u = -cln::log(1-x); - cln::cl_N factor = u; + cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits)); cln::cl_N res = u; cln::cl_N resbuf; unsigned i=2; @@ -254,8 +306,12 @@ cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x) resbuf = res; factor = factor * u / i; res = res + (*it) * factor; - it++; // should we check it? or rely on initsize? ... i++; + if (++it == xend) { + double_Xn(); + it = Xn[n-2].begin() + (i-2); + xend = Xn[n-2].end(); + } } while (res != resbuf); return res; } @@ -314,7 +370,7 @@ cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& pr cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1); for (int j=0; j& s, const std::vector t(j); cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); @@ -438,12 +482,12 @@ cln::cl_N multipleLi_do_sum(const std::vector& s, const std::vector& s, const std::vector(x1).to_int(), ex_to(x2)); - if (is_a(x2)) { - for (int i=0; i(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 @@ -489,14 +512,14 @@ static ex Li_evalf(const ex& x1, const ex& x2) ex conv = 1; for (int i=0; i(x2.op(i))) { - return Li(x1,x2).hold(); + return Li(x1, x2).hold(); } conv *= x2.op(i); - if ((conv > 1) || ((conv == 1) && (x1.op(0) == 1))) { - return Li(x1,x2).hold(); + if (abs(conv) >= 1) { + return Li(x1, x2).hold(); } } @@ -514,34 +537,163 @@ static ex Li_evalf(const ex& x1, const ex& x2) } -static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options) +static ex Li_eval(const ex& m_, const ex& x_) +{ + if (m_.nops() < 2) { + ex m; + if (is_a(m_)) { + m = m_.op(0); + } else { + m = m_; + } + ex x; + if (is_a(x_)) { + x = x_.op(0); + } else { + x = x_; + } + if (x == _ex0) { + return _ex0; + } + if (x == _ex1) { + return zeta(m); + } + if (x == _ex_1) { + return (pow(2,1-m)-1) * zeta(m); + } + if (m == _ex1) { + return -log(1-x); + } + if (m.info(info_flags::posint) && x.info(info_flags::numeric) && (!x.info(info_flags::crational))) { + return Li_num(ex_to(m).to_int(), ex_to(x)); + } + } else { + bool ish = true; + bool iszeta = true; + bool iszero = false; + bool doevalf = false; + bool doevalfveto = true; + const lst& m = ex_to(m_); + const lst& x = ex_to(x_); + lst::const_iterator itm = m.begin(); + lst::const_iterator itx = x.begin(); + for (; itm != m.end(); itm++, itx++) { + if (!(*itm).info(info_flags::posint)) { + return Li(m_, x_).hold(); + } + if ((*itx != _ex1) && (*itx != _ex_1)) { + if (itx != x.begin()) { + ish = false; + } + iszeta = false; + } + if (*itx == _ex0) { + iszero = true; + } + if (!(*itx).info(info_flags::numeric)) { + doevalfveto = false; + } + if (!(*itx).info(info_flags::crational)) { + doevalf = true; + } + } + if (iszeta) { + return zeta(m_, x_); + } + if (iszero) { + return _ex0; + } + if (ish) { + ex pf; + lst newm = convert_parameter_Li_to_H(m, x, pf); + return pf * H(newm, x[0]); + } + if (doevalfveto && doevalf) { + return Li(m_, x_).evalf(); + } + } + return Li(m_, x_).hold(); +} + + +static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options) { epvector seq; - seq.push_back(expair(Li(x1,x2), 0)); - return pseries(rel,seq); + seq.push_back(expair(Li(m, x), 0)); + return pseries(rel, seq); } -static ex Li_deriv(const ex& x1, const ex& x2, unsigned deriv_param) +static ex Li_deriv(const ex& m_, const ex& x_, unsigned deriv_param) { GINAC_ASSERT(deriv_param < 2); if (deriv_param == 0) { return _ex0; } - if (x1 > 0) { - return Li(x1-1, x2) / x2; + if (m_.nops() > 1) { + throw std::runtime_error("don't know how to derivate multiple polylogarithm!"); + } + ex m; + if (is_a(m_)) { + m = m_.op(0); } else { - return 1/(1-x2); + m = m_; } + ex x; + if (is_a(x_)) { + x = x_.op(0); + } else { + x = x_; + } + if (m > 0) { + return Li(m-1, x) / x; + } else { + return 1/(1-x); + } +} + + +static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c) +{ + lst m; + if (is_a(m_)) { + m = ex_to(m_); + } else { + m = lst(m_); + } + lst x; + if (is_a(x_)) { + x = ex_to(x_); + } else { + x = lst(x_); + } + c.s << "\\mbox{Li}_{"; + lst::const_iterator itm = m.begin(); + (*itm).print(c); + itm++; + for (; itm != m.end(); itm++) { + c.s << ","; + (*itm).print(c); + } + c.s << "}("; + lst::const_iterator itx = x.begin(); + (*itx).print(c); + itx++; + for (; itx != x.end(); itx++) { + c.s << ","; + (*itx).print(c); + } + c.s << ")"; } REGISTER_FUNCTION(Li, - eval_func(Li_eval). - evalf_func(Li_evalf). - do_not_evalf_params(). - series_func(Li_series). - derivative_func(Li_deriv)); + evalf_func(Li_evalf). + eval_func(Li_eval). + series_func(Li_series). + derivative_func(Li_deriv). + print_func(Li_print_latex). + do_not_evalf_params()); ////////////////////////////////////////////////////////////////////// @@ -661,24 +813,24 @@ cln::cl_N C(int n, int p) if (k & 1) { if (j & 1) { result = result + cln::factorial(n+k-1) - * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N() - / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); + * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N() + / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); } else { result = result - cln::factorial(n+k-1) - * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N() - / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); + * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N() + / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); } } else { if (j & 1) { result = result - cln::factorial(n+k-1) * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N() - / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); + / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); } else { result = result + cln::factorial(n+k-1) - * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N() - / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); + * cln::expt(cln::pi(),2*j) * S_num(n+k-2*j,p-k,1).to_cl_N() + / (cln::factorial(k) * cln::factorial(n-1) * cln::factorial(2*j)); } } } @@ -751,7 +903,9 @@ cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& } // should be done otherwise - cln::cl_N xf = x * cln::cl_float(1, prec); + cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); + cln::cl_N xf = x * one; + //cln::cl_N xf = x * cln::cl_float(1, prec); cln::cl_N res; cln::cl_N resbuf; @@ -780,13 +934,13 @@ cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) { cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n) - * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p); + * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p); for (int s=0; s(cln::imagpart(value))); - // [Kol] (5.3) - if (cln::realpart(value) < -0.5) { + if ((cln::realpart(value) < -0.5) || (n == 0)) { cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n) - * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p); + * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p); for (int s=0; s(x1).to_int(), ex_to(x2).to_int(), ex_to(x3)); + if (n.info(info_flags::posint) && p.info(info_flags::posint) && is_a(x)) { + return S_num(ex_to(n).to_int(), ex_to(p).to_int(), ex_to(x)); } - return S(x1,x2,x3).hold(); + return S(n, p, x).hold(); } -static ex S_evalf(const ex& x1, const ex& x2, const ex& x3) +static ex S_eval(const ex& n, const ex& p, const ex& x) { - if (is_a(x1) && is_a(x2) && is_a(x3)) { - return S_num(ex_to(x1).to_int(), ex_to(x2).to_int(), ex_to(x3)); + if (n.info(info_flags::posint) && p.info(info_flags::posint)) { + if (x == 0) { + return _ex0; + } + if (x == 1) { + lst m(n+1); + for (int i=ex_to(p).to_int()-1; i>0; i--) { + m.append(1); + } + return zeta(m); + } + if (p == 1) { + return Li(n+1, x); + } + if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) { + return S_num(ex_to(n).to_int(), ex_to(p).to_int(), ex_to(x)); + } + } + if (n.is_zero()) { + // [Kol] (5.3) + return pow(-log(1-x), p) / factorial(p); } - return S(x1,x2,x3).hold(); + return S(n, p, x).hold(); } -static ex S_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options) +static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options) { epvector seq; - seq.push_back(expair(S(x1,x2,x3), 0)); - return pseries(rel,seq); + seq.push_back(expair(S(n, p, x), 0)); + return pseries(rel, seq); } -static ex S_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param) +static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param) { GINAC_ASSERT(deriv_param < 3); if (deriv_param < 2) { return _ex0; } - if (x1 > 0) { - return S(x1-1, x2, x3) / x3; + if (n > 0) { + return S(n-1, p, x) / x; } else { - return S(x1, x2-1, x3) / (1-x3); + return S(n, p-1, x) / (1-x); } } +static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c) +{ + c.s << "\\mbox{S}_{"; + n.print(c); + c.s << ","; + p.print(c); + c.s << "}("; + x.print(c); + c.s << ")"; +} + + REGISTER_FUNCTION(S, - eval_func(S_eval). - evalf_func(S_evalf). - do_not_evalf_params(). - series_func(S_series). - derivative_func(S_deriv)); + evalf_func(S_evalf). + eval_func(S_eval). + series_func(S_series). + derivative_func(S_deriv). + print_func(S_print_latex). + do_not_evalf_params()); ////////////////////////////////////////////////////////////////////// // -// Harmonic polylogarithm H(m,x) and H(m,s,x) +// Harmonic polylogarithm H(m,x) // -// helper function +// helper functions // ////////////////////////////////////////////////////////////////////// @@ -967,151 +1148,356 @@ REGISTER_FUNCTION(S, // anonymous namespace for helper functions namespace { - -// forward declaration -ex convert_from_RV(const lst& parameterlst, const ex& arg); + +// regulates the pole (used by 1/x-transformation) +symbol H_polesign("IMSIGN"); -// multiplies an one-dimensional H with another H -// [ReV] (18) -ex trafo_H_mult(const ex& h1, const ex& h2) +// convert parameters from H to Li representation +// parameters are expected to be in expanded form, i.e. only 0, 1 and -1 +// returns true if some parameters are negative +bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf) { - 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(h1.op(0)); - } else { - hshort = h1.op(0).op(0); - if (h2nops > 1) { - hlong = ex_to(h2.op(0)); + // expand parameter list + lst mexp; + for (lst::const_iterator it = l.begin(); it != l.end(); it++) { + if (*it > 1) { + for (ex count=*it-1; count > 0; count--) { + mexp.append(0); + } + mexp.append(1); + } else if (*it < -1) { + for (ex count=*it+1; count < 0; count++) { + mexp.append(0); + } + mexp.append(-1); } else { - hlong = h2.op(0).op(0); + mexp.append(*it); } } - for (int i=0; i<=hlong.nops(); i++) { - lst newparameter; - int j=0; - for (; j 0) { + m.append((*it+acc-1) * signum); + } else { + m.append((*it-acc+1) * signum); + } + acc = 1; + signum = *it; + pf *= *it; + if (pf < 0) { + has_negative_parameters = true; } - res += H(newparameter, h1.op(1)).hold(); } - return res; + if (has_negative_parameters) { + for (int i=0; i(e)) { + if (is_a(e) || is_a(e)) { return e.map(*this); } - - if (is_a(e)) { - - ex result = 1; - ex firstH; - lst Hlst; - for (int pos=0; pos(e.op(pos)) && is_a(e.op(pos).op(0))) { - std::string name = ex_to(e.op(pos).op(0)).get_name(); - if (name == "H") { - for (ex i=0; i(e.op(pos))) { - std::string name = ex_to(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(); + if (is_a(e)) { + std::string name = ex_to(e).get_name(); + if (name == "H") { + lst parameter; + if (is_a(e.op(0))) { + parameter = ex_to(e.op(0)); } else { - return e; + parameter = lst(e.op(0)); } - } + ex arg = e.op(1); - if (Hlst.nops() > 0) { - ex buffer = trafo_H_mult(firstH, Hlst.op(0)); - result *= buffer; - for (int i=1; i(e)) { - name = ex_to(e).get_name(); - } - if (name == "H") { - h = e; - } else { - for (int i=0; i(e.op(i))) { - std::string name = ex_to(e.op(i)).get_name(); - if (name == "H") { - h = e.op(i); + ex operator()(const ex& e) + { + if (is_a(e) || is_a(e)) { + return e.map(*this); + } + if (is_a(e)) { + std::string name = ex_to(e).get_name(); + if (name == "H") { + lst parameter; + if (is_a(e.op(0))) { + parameter = ex_to(e.op(0)); + } else { + parameter = lst(e.op(0)); + } + + lst m; + lst s; + ex pf; + if (convert_parameter_H_to_Li(parameter, m, s, pf)) { + return pf * zeta(m, s); + } else { + return zeta(m); } } } + return e; } - if (h != 0) { - lst newparameter = ex_to(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(); - } -} +}; -// do integration [ReV] (55) -// put parameter 0 in front of existing parameters -ex trafo_H_prepend_zero(const ex& e, const ex& arg) +// remove trailing zeros from H-parameters +struct map_trafo_H_reduce_trailing_zeros : public map_function { - ex h; - std::string name; - if (is_a(e)) { - name = ex_to(e).get_name(); - } + ex operator()(const ex& e) + { + if (is_a(e) || is_a(e)) { + return e.map(*this); + } + if (is_a(e)) { + std::string name = ex_to(e).get_name(); + if (name == "H") { + lst parameter; + if (is_a(e.op(0))) { + parameter = ex_to(e.op(0)); + } else { + parameter = lst(e.op(0)); + } + ex arg = e.op(1); + if (parameter.op(parameter.nops()-1) == 0) { + + // + if (parameter.nops() == 1) { + return log(arg); + } + + // + lst::const_iterator it = parameter.begin(); + while ((it != parameter.end()) && (*it == 0)) { + it++; + } + if (it == parameter.end()) { + return pow(log(arg),parameter.nops()) / factorial(parameter.nops()); + } + + // + parameter.remove_last(); + int lastentry = parameter.nops(); + while ((lastentry > 0) && (parameter[lastentry-1] == 0)) { + lastentry--; + } + + // + ex result = log(arg) * H(parameter,arg).hold(); + ex acc = 0; + for (ex i=0; i 0) { + parameter[i]++; + result -= (acc + parameter[i]-1) * H(parameter, arg).hold(); + parameter[i]--; + acc = 0; + } else if (parameter[i] < 0) { + parameter[i]--; + result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold(); + parameter[i]++; + acc = 0; + } else { + acc++; + } + } + + if (lastentry < parameter.nops()) { + result = result / (parameter.nops()-lastentry+1); + return result.map(*this); + } else { + return result; + } + } + } + } + return e; + } +}; + + +// returns an expression with zeta functions corresponding to the parameter list for H +ex convert_H_to_zeta(const lst& m) +{ + symbol xtemp("xtemp"); + map_trafo_H_reduce_trailing_zeros filter; + map_trafo_H_convert_to_zeta filter2; + return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1); +} + + +// convert signs form Li to H representation +lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf) +{ + lst res; + lst::const_iterator itm = m.begin(); + lst::const_iterator itx = ++x.begin(); + ex signum = _ex1; + pf = _ex1; + res.append(*itm); + itm++; + while (itx != x.end()) { + signum *= *itx; + pf *= signum; + res.append((*itm) * signum); + itm++; + itx++; + } + return res; +} + + +// multiplies an one-dimensional H with another H +// [ReV] (18) +ex trafo_H_mult(const ex& h1, const ex& h2) +{ + ex res; + ex hshort; + lst hlong; + ex h1nops = h1.op(0).nops(); + ex h2nops = h2.op(0).nops(); + if (h1nops > 1) { + hshort = h2.op(0).op(0); + hlong = ex_to(h1.op(0)); + } else { + hshort = h1.op(0).op(0); + if (h2nops > 1) { + hlong = ex_to(h2.op(0)); + } else { + hlong = h2.op(0).op(0); + } + } + for (int i=0; i<=hlong.nops(); i++) { + lst newparameter; + int j=0; + for (; j(e)) { + return e.map(*this); + } + + if (is_a(e)) { + + ex result = 1; + ex firstH; + lst Hlst; + for (int pos=0; pos(e.op(pos)) && is_a(e.op(pos).op(0))) { + std::string name = ex_to(e.op(pos).op(0)).get_name(); + if (name == "H") { + for (ex i=0; i(e.op(pos))) { + std::string name = ex_to(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(e)) { + name = ex_to(e).get_name(); + } if (name == "H") { h = e; } else { @@ -1127,7 +1513,7 @@ ex trafo_H_prepend_zero(const ex& e, const ex& arg) if (h != 0) { lst newparameter = ex_to(h.op(0)); newparameter.prepend(0); - ex addzeta = convert_from_RV(newparameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1))); + ex addzeta = convert_H_to_zeta(newparameter); return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand(); } else { return e * (-H(lst(0),1/arg).hold()); @@ -1135,87 +1521,99 @@ ex trafo_H_prepend_zero(const ex& e, const ex& arg) } -// do x -> 1-x transformation -struct map_trafo_H_1mx : public map_function +// do integration [ReV] (55) +// put parameter -1 in front of existing parameters +ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg) { - ex operator()(const ex& e) - { - if (is_a(e) || is_a(e)) { - return e.map(*this); + ex h; + std::string name; + if (is_a(e)) { + name = ex_to(e).get_name(); + } + if (name == "H") { + h = e; + } else { + for (int i=0; i(e.op(i))) { + std::string name = ex_to(e.op(i)).get_name(); + if (name == "H") { + h = e.op(i); + } + } } - - if (is_a(e)) { - std::string name = ex_to(e).get_name(); - if (name == "H") { + } + if (h != 0) { + lst newparameter = ex_to(h.op(0)); + newparameter.prepend(-1); + ex addzeta = convert_H_to_zeta(newparameter); + return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand(); + } else { + ex addzeta = convert_H_to_zeta(lst(-1)); + return (e * (addzeta - H(lst(-1),1/arg).hold())).expand(); + } +} - lst parameter = ex_to(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(); +// do integration [ReV] (55) +// put parameter -1 in front of existing parameters +ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg) +{ + ex h; + std::string name; + if (is_a(e)) { + name = ex_to(e).get_name(); + } + if (name == "H") { + h = e; + } else { + for (int i=0; i(e.op(i))) { + std::string name = ex_to(e.op(i)).get_name(); + if (name == "H") { + h = e.op(i); } + } + } + } + if (h != 0) { + lst newparameter = ex_to(h.op(0)); + newparameter.prepend(-1); + return e.subs(h == H(newparameter, h.op(1)).hold()).expand(); + } else { + return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand(); + } +} - 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(buffer)) { - for (int i=0; i(e)) { + name = ex_to(e).get_name(); + } + if (name == "H") { + h = e; + } else { + for (int i=0; i(e.op(i))) { + std::string name = ex_to(e.op(i)).get_name(); + if (name == "H") { + h = e.op(i); } - } } - return e; } -}; + if (h != 0) { + lst newparameter = ex_to(h.op(0)); + newparameter.prepend(1); + return e.subs(h == H(newparameter, h.op(1)).hold()).expand(); + } else { + return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand(); + } +} // do x -> 1/x transformation @@ -1234,13 +1632,42 @@ struct map_trafo_H_1overx : public map_function lst parameter = ex_to(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(); + // special cases if all parameters are either 0, 1 or -1 + bool allthesame = true; + if (parameter.op(0) == 0) { + for (int i=1; i(buffer)) { for (int i=0; i(buffer)) { + for (int i=0; i(e) || is_a(e)) { - return e.map(*this); - } - if (is_a(e)) { - std::string name = ex_to(e).get_name(); - if (name == "H") { - lst parameter; - if (is_a(e.op(0))) { - parameter = ex_to(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 0) { - parameter[i]++; - result -= (parameter[i]-1) * H(parameter, arg).hold(); - parameter[i]--; - } else { - parameter[i]--; - result -= abs(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; @@ -1364,116 +1737,127 @@ struct map_trafo_H_reduce_trailing_zeros : public map_function }; -// transform H(m,x) with signed m to H(m,s,x) -struct map_trafo_H_convert_signed_m : public map_function +// do x -> (1-x)/(1+x) transformation +struct map_trafo_H_1mxt1px : public map_function { ex operator()(const ex& e) { if (is_a(e) || is_a(e)) { return e.map(*this); } + if (is_a(e)) { std::string name = ex_to(e).get_name(); if (name == "H") { - lst parameter; - if (is_a(e.op(0))) { - parameter = ex_to(e.op(0)); - } else { - parameter = lst(e.op(0)); - } + + lst parameter = ex_to(e.op(0)); ex arg = e.op(1); - bool signedflag = false; - for (int i=0; i 0) { - signs.append(1); - } else { - signs.append(-1); - parameter.let_op(i) = -parameter.op(i); + if (allthesame) { + map_trafo_H_mult unify; + return unify((pow(-H(lst(1),(1-arg)/(1+arg)).hold() - H(lst(-1),(1-arg)/(1+arg)).hold(), parameter.nops()) + / factorial(parameter.nops())).expand()); + } + } else if (parameter.op(0) == -1) { + for (int i=1; i(e) || is_a(e) || is_a(e)) { - return e.map(*this); - } - if (is_a(e)) { - std::string name = ex_to(e).get_name(); - if (name == "H") { - lst parameter = ex_to(e.op(0)); - ex arg = e.op(1); - return convert_from_RV(parameter, arg); - } - } - return e; - } -}; + if (parameter.op(0) == 0) { + + // leading zero + ex res = convert_H_to_zeta(parameter); + map_trafo_H_1mxt1px recursion; + ex buffer = recursion(H(newparameter, arg).hold()); + if (is_a(buffer)) { + for (int i=0; i 0) { - for (ex i=0; i<(*it)-1; i++) { - res.append(0); - } - res.append(1); - } else { - for (ex i=0; i<(-*it)-1; i++) { - res.append(0); - } - res.append(-1); - } - } - return res; -} + // leading negative one + ex res = convert_H_to_zeta(parameter); + map_trafo_H_1mxt1px recursion; + ex buffer = recursion(H(newparameter, arg).hold()); + if (is_a(buffer)) { + for (int i=0; i0) ? count : -count); - count = 1; + } + + } } - it++; - } - for (int i=1; i& m, const cln::cl_N& x) t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), m[0]); factor = factor * x; } while (t[0] != t0buf); - + return t[0]; } @@ -1514,276 +1898,320 @@ cln::cl_N H_do_sum(const std::vector& m, const cln::cl_N& x) ////////////////////////////////////////////////////////////////////// -static ex H2_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 H2_evalf(const ex& x1, const ex& x2) +static ex H_evalf(const ex& x1, const ex& x2) { if (is_a(x1) && is_a(x2)) { for (int i=0; i(x2).to_cl_N(); - if (x == 1) { - return zeta(x1).evalf(); + + const lst& morg = ex_to(x1); + // remove trailing zeros ... + if (*(--morg.end()) == 0) { + symbol xtemp("xtemp"); + map_trafo_H_reduce_trailing_zeros filter; + return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf(); + } + // ... and expand parameter notation + lst m; + for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) { + if (*it > 1) { + for (ex count=*it-1; count > 0; count--) { + m.append(0); + } + m.append(1); + } else if (*it < -1) { + for (ex count=*it+1; count < 0; count++) { + m.append(0); + } + m.append(-1); + } else { + m.append(*it); + } } - // choose trafo - if (cln::abs(x) > 1) { - symbol xtemp("xtemp"); - map_trafo_H_1overx trafo; - ex res = trafo(H(convert_to_RV(ex_to(x1)), xtemp)); - map_trafo_H_convert converter; - res = converter(res); - return res.subs(xtemp==x2).evalf(); + // since the transformations produce a lot of terms, they are only efficient for + // argument near one. + // no transformation needed -> do summation + if (cln::abs(x) < 0.95) { + lst m_lst; + lst s_lst; + ex pf; + if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) { + // negative parameters -> s_lst is filled + std::vector m_int; + std::vector x_cln; + for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin(); + it_int != m_lst.end(); it_int++, it_cln++) { + m_int.push_back(ex_to(*it_int).to_int()); + x_cln.push_back(ex_to(*it_cln).to_cl_N()); + } + x_cln.front() = x_cln.front() * x; + return pf * numeric(multipleLi_do_sum(m_int, x_cln)); + } else { + // only positive parameters + //TODO + if (m_lst.nops() == 1) { + return Li(m_lst.op(0), x2).evalf(); + } + std::vector m_int; + for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) { + m_int.push_back(ex_to(*it).to_int()); + } + return numeric(H_do_sum(m_int, x)); + } } - // 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(x1)), xtemp)); - map_trafo_H_convert converter; - res = converter(res); - return res.subs(xtemp==x2).evalf(); + ex res = 1; + + // ensure that the realpart of the argument is positive + if (cln::realpart(x) < 0) { + x = -x; + for (int i=0; i do summation - int count = x1.nops(); - std::vector r(count); - for (int i=0; i(x1.op(i)).to_int(); + // choose transformations + symbol xtemp("xtemp"); + if (cln::abs(x-1) < 1.4142) { + // x -> (1-x)/(1+x) + map_trafo_H_1mxt1px trafo; + res *= trafo(H(m, xtemp)); + } else { + // x -> 1/x + map_trafo_H_1overx trafo; + res *= trafo(H(m, xtemp)); + if (cln::imagpart(x) <= 0) { + res = res.subs(H_polesign == -I*Pi); + } else { + res = res.subs(H_polesign == I*Pi); + } } - return numeric(H_do_sum(r,x)); + // simplify result +// TODO +// map_trafo_H_convert converter; +// res = converter(res).expand(); +// lst ll; +// res.find(H(wild(1),wild(2)), ll); +// res.find(zeta(wild(1)), ll); +// res.find(zeta(wild(1),wild(2)), ll); +// res = res.collect(ll); + + return res.subs(xtemp == numeric(x)).evalf(); } return H(x1,x2).hold(); } -static ex H2_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options) +static ex H_eval(const ex& m_, const ex& x) { - epvector seq; - seq.push_back(expair(H(x1,x2), 0)); - return pseries(rel,seq); -} - - -static ex H2_deriv(const ex& x1, const ex& x2, unsigned deriv_param) -{ - GINAC_ASSERT(deriv_param < 2); - if (deriv_param == 0) { - return _ex0; - } - if (is_a(x1)) { - lst newparameter = ex_to(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; - } + lst m; + if (is_a(m_)) { + m = ex_to(m_); + } else { + m = lst(m_); + } + if (m.nops() == 0) { + return _ex1; + } + ex pos1; + ex pos2; + ex n; + ex p; + int step = 0; + if (*m.begin() > _ex1) { + step++; + pos1 = _ex0; + pos2 = _ex1; + n = *m.begin()-1; + p = _ex1; + } else if (*m.begin() < _ex_1) { + step++; + pos1 = _ex0; + pos2 = _ex_1; + n = -*m.begin()-1; + p = _ex1; + } else if (*m.begin() == _ex0) { + pos1 = _ex0; + n = _ex1; } else { - if (x1 == 1) { - return 1/(1-x2); + pos1 = *m.begin(); + p = _ex1; + } + for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) { + if ((*it).info(info_flags::integer)) { + if (step == 0) { + if (*it > _ex1) { + if (pos1 == _ex0) { + step = 1; + pos2 = _ex1; + n += *it-1; + p = _ex1; + } else { + step = 2; + } + } else if (*it < _ex_1) { + if (pos1 == _ex0) { + step = 1; + pos2 = _ex_1; + n += -*it-1; + p = _ex1; + } else { + step = 2; + } + } else { + if (*it != pos1) { + step = 1; + pos2 = *it; + } + if (*it == _ex0) { + n++; + } else { + p++; + } + } + } else if (step == 1) { + if (*it != pos2) { + step = 2; + } else { + if (*it == _ex0) { + n++; + } else { + p++; + } + } + } } else { - return H(x1-1, x2).hold() / x2; + // if some m_i is not an integer + return H(m_, x).hold(); } } -} - - -unsigned H2_SERIAL::serial = - function::register_new(function_options("H"). - eval_func(H2_eval). - evalf_func(H2_evalf). - do_not_evalf_params(). - derivative_func(H2_deriv). - latex_name("\\mbox{H}"). - overloaded(2)); - - -////////////////////////////////////////////////////////////////////// -// -// Harmonic polylogarithm H(m,s,x) -// -// GiNaC function -// -////////////////////////////////////////////////////////////////////// - - -static ex H3_eval(const ex& x1, const ex& x2, const ex& x3) -{ - if (x3 == 0) { - return 0; - } - if (x3 == 1) { - return zeta(x1, x2); + if ((x == _ex1) && (*(--m.end()) != _ex0)) { + return convert_H_to_zeta(m); } - if (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational))) { - return H(x1, x2, x3).evalf(); - } - return H(x1, x2, x3).hold(); -} - - -static ex H3_evalf(const ex& x1, const ex& x2, const ex& x3) -{ - if (is_a(x1) && is_a(x3)) { - for (int i=0; i(x3).to_cl_N(); - if (x == 1) { - return zeta(x1, x2).evalf(); - } - - // choose trafo - if (cln::abs(x) > 1) { - //TODO - return H(x1, x2, x3).hold(); -// symbol xtemp("xtemp"); -// lst para = ex_to(x1); -// for (int i=0; i1-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(x1)), xtemp)); -// map_trafo_H_convert converter; -// res = converter(res); -// return res.subs(xtemp==x2).evalf(); -// } - - // no trafo -> do summation - int count = x1.nops(); - std::vector m(count); - std::vector s(count); - cln::cl_N signbuf = 1; - for (int i=0; i(x1.op(i)).to_int(); - signbuf = signbuf * ex_to(x2.op(i)).to_cl_N(); - s[i] = signbuf; - } - s[0] = s[0] * ex_to(x3).to_cl_N(); - - return numeric(signbuf * multipleLi_do_sum(m, s)); - } - - return H(x1, x2, x3).hold(); + } + if (x == _ex0) { + return _ex0; + } + if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) { + return H(m_, x).evalf(); + } + return H(m_, x).hold(); } -static ex H3_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options) +static ex H_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options) { epvector seq; - seq.push_back(expair(H(x1, x2, x3), 0)); + seq.push_back(expair(H(m, x), 0)); return pseries(rel, seq); } -static ex H3_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param) +static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param) { - //TODO - GINAC_ASSERT(deriv_param < 2); if (deriv_param == 0) { return _ex0; } - if (is_a(x1)) { - lst newparameter = ex_to(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; - } + lst m; + if (is_a(m_)) { + m = ex_to(m_); } else { - if (x1 == 1) { - return 1/(1-x2); - } else { - return H(x1-1, x2).hold() / x2; - } + m = lst(m_); + } + ex mb = *m.begin(); + if (mb > _ex1) { + m[0]--; + return H(m, x) / x; + } + if (mb < _ex_1) { + m[0]++; + return H(m, x) / x; + } + m.remove_first(); + if (mb == _ex1) { + return 1/(1-x) * H(m, x); + } else if (mb == _ex_1) { + return 1/(1+x) * H(m, x); + } else { + return H(m, x) / x; } } -unsigned H3_SERIAL::serial = - function::register_new(function_options("H"). - eval_func(H3_eval). - evalf_func(H3_evalf). - do_not_evalf_params(). - derivative_func(H3_deriv). - latex_name("\\mbox{H}"). - overloaded(2)); +static void H_print_latex(const ex& m_, const ex& x, const print_context& c) +{ + lst m; + if (is_a(m_)) { + m = ex_to(m_); + } else { + m = lst(m_); + } + c.s << "\\mbox{H}_{"; + lst::const_iterator itm = m.begin(); + (*itm).print(c); + itm++; + for (; itm != m.end(); itm++) { + c.s << ","; + (*itm).print(c); + } + c.s << "}("; + x.print(c); + c.s << ")"; +} + +REGISTER_FUNCTION(H, + evalf_func(H_evalf). + eval_func(H_eval). + series_func(H_series). + derivative_func(H_deriv). + print_func(H_print_latex). + do_not_evalf_params()); -ex convert_H_notation(const ex& parameterlst, const ex& arg) + +// takes a parameter list for H and returns an expression with corresponding multiple polylogarithms +ex convert_H_to_Li(const ex& m, const ex& x) { - if (is_a(parameterlst)) { - for (int i=0; i(parameterlst), arg).eval(); - } - if (parameterlst == 1) { - return -log(1-arg); - } - if (parameterlst == 0) { - return log(arg); - } - if (parameterlst == -1) { - return log(1+arg); + map_trafo_H_reduce_trailing_zeros filter; + map_trafo_H_convert_to_Li filter2; + if (is_a(m)) { + return filter2(filter(H(m, x).hold())); + } else { + return filter2(filter(H(lst(m), x).hold())); } - throw std::runtime_error("first parameter has to be a list containing only 0, 1 or -1!"); } @@ -2122,12 +2550,12 @@ cln::cl_N zeta_do_Hoelder_convolution(const std::vector& m_, const std::vec if (m_p.size() == 0) break; res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q); - + } while (true); // last term res = res + signum * multipleLi_do_sum(m_q, s_q); - + return res; } @@ -2179,7 +2607,7 @@ static ex zeta1_evalf(const ex& x) return numeric(zeta_do_sum_simple(r)); } } - + // single zeta value if (is_exactly_a(x) && (x != 1)) { try { @@ -2191,28 +2619,28 @@ static ex zeta1_evalf(const ex& x) } -static ex zeta1_eval(const ex& x) +static ex zeta1_eval(const ex& m) { - if (is_exactly_a(x)) { - if (x.nops() == 1) { - return zeta(x.op(0)); + if (is_exactly_a(m)) { + if (m.nops() == 1) { + return zeta(m.op(0)); } - return zeta(x).hold(); + return zeta(m).hold(); } - if (x.info(info_flags::numeric)) { - const numeric& y = ex_to(x); + if (m.info(info_flags::numeric)) { + const numeric& y = ex_to(m); // trap integer arguments: if (y.is_integer()) { if (y.is_zero()) { return _ex_1_2; } if (y.is_equal(_num1)) { - return zeta(x).hold(); + return zeta(m).hold(); } if (y.info(info_flags::posint)) { if (y.info(info_flags::odd)) { - return zeta(x).hold(); + return zeta(m).hold(); } else { return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y); } @@ -2225,33 +2653,52 @@ static ex zeta1_eval(const ex& x) } } // zeta(float) - if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) - return zeta1_evalf(x); + if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) { + return zeta1_evalf(m); + } } - return zeta(x).hold(); + return zeta(m).hold(); } -static ex zeta1_deriv(const ex& x, unsigned deriv_param) +static ex zeta1_deriv(const ex& m, unsigned deriv_param) { GINAC_ASSERT(deriv_param==0); - if (is_exactly_a(x)) { + if (is_exactly_a(m)) { return _ex0; } else { - return zeta(_ex1, x); + return zetaderiv(_ex1, m); + } +} + + +static void zeta1_print_latex(const ex& m_, const print_context& c) +{ + c.s << "\\zeta("; + if (is_a(m_)) { + const lst& m = ex_to(m_); + lst::const_iterator it = m.begin(); + (*it).print(c); + it++; + for (; it != m.end(); it++) { + c.s << ","; + (*it).print(c); + } + } else { + m_.print(c); } + c.s << ")"; } -unsigned zeta1_SERIAL::serial = - function::register_new(function_options("zeta"). - eval_func(zeta1_eval). - evalf_func(zeta1_evalf). - do_not_evalf_params(). - derivative_func(zeta1_deriv). - latex_name("\\zeta"). - overloaded(2)); +unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1). + evalf_func(zeta1_evalf). + eval_func(zeta1_eval). + derivative_func(zeta1_deriv). + print_func(zeta1_print_latex). + do_not_evalf_params(). + overloaded(2)); ////////////////////////////////////////////////////////////////////// @@ -2303,56 +2750,92 @@ static ex zeta2_evalf(const ex& x, const ex& s) // use Hoelder convolution return numeric(zeta_do_Hoelder_convolution(xi, si)); } - + return zeta(x, s).hold(); } -static ex zeta2_eval(const ex& x, const ex& s) +static ex zeta2_eval(const ex& m, const ex& s_) { - if (is_exactly_a(s)) { - const lst& l = ex_to(s); - lst::const_iterator it = l.begin(); - while (it != l.end()) { - if ((*it).info(info_flags::negative)) { - return zeta(x, s).hold(); + if (is_exactly_a(s_)) { + const lst& s = ex_to(s_); + for (lst::const_iterator it = s.begin(); it != s.end(); it++) { + if ((*it).info(info_flags::positive)) { + continue; } - it++; - } - return zeta(x); - } else { - if (s.info(info_flags::positive)) { - return zeta(x); + return zeta(m, s_).hold(); } + return zeta(m); + } else if (s_.info(info_flags::positive)) { + return zeta(m); } - return zeta(x, s).hold(); + return zeta(m, s_).hold(); } -static ex zeta2_deriv(const ex& x, const ex& s, unsigned deriv_param) +static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param) { GINAC_ASSERT(deriv_param==0); - if (is_exactly_a(x)) { + if (is_exactly_a(m)) { return _ex0; } else { - if ((is_exactly_a(s) && (s.op(0) > 0)) || (s > 0)) { - return zeta(_ex1, x); + if ((is_exactly_a(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) { + return zetaderiv(_ex1, m); } return _ex0; } } -unsigned zeta2_SERIAL::serial = - function::register_new(function_options("zeta"). - eval_func(zeta2_eval). - evalf_func(zeta2_evalf). - do_not_evalf_params(). - derivative_func(zeta2_deriv). - latex_name("\\zeta"). - overloaded(2)); +static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c) +{ + lst m; + if (is_a(m_)) { + m = ex_to(m_); + } else { + m = lst(m_); + } + lst s; + if (is_a(s_)) { + s = ex_to(s_); + } else { + s = lst(s_); + } + c.s << "\\zeta("; + lst::const_iterator itm = m.begin(); + lst::const_iterator its = s.begin(); + if (*its < 0) { + c.s << "\\overline{"; + (*itm).print(c); + c.s << "}"; + } else { + (*itm).print(c); + } + its++; + itm++; + for (; itm != m.end(); itm++, its++) { + c.s << ","; + if (*its < 0) { + c.s << "\\overline{"; + (*itm).print(c); + c.s << "}"; + } else { + (*itm).print(c); + } + } + c.s << ")"; +} + + +unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2). + evalf_func(zeta2_evalf). + eval_func(zeta2_eval). + derivative_func(zeta2_deriv). + print_func(zeta2_print_latex). + do_not_evalf_params(). + overloaded(2)); } // namespace GiNaC