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=020ff333b01399329df89dd03e5fc34490087992;hb=d7d0bcda91b647db9588f3aa1a465f1570d088c4;hpb=168dd38dd35fcaaf61e177afb8dbe0288c0eb521 diff --git a/ginac/inifcns_nstdsums.cpp b/ginac/inifcns_nstdsums.cpp index 020ff333..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) @@ -11,32 +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 + * 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. - * + * number --- notation. + * * - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in * the whole complex plane. Multiple polylogarithms evaluate only if for each argument x_i the product * x_1 * x_2 * ... * x_i is smaller than one. The parameters for Li, zeta and S must be positive integers. * If you want to have an alternating Euler sum, you have to give the signs of the parameters as a * second argument s to zeta(m,s) containing 1 and -1. - * - * - The calculation of classical polylogarithms is speed up by using Bernoulli numbers and + * + * - 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 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 @@ -46,7 +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 @@ -102,6 +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; @@ -117,17 +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 { @@ -147,14 +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; @@ -171,9 +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++; } @@ -184,12 +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 { @@ -207,8 +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; @@ -216,8 +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; } @@ -226,7 +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; @@ -244,8 +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; @@ -253,8 +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; } @@ -313,7 +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(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(); } } @@ -477,45 +537,118 @@ static ex Li_evalf(const ex& x1, const ex& x2) } -static ex Li_eval(const ex& x1, const ex& x2) +static ex Li_eval(const ex& m_, const ex& x_) { - if (x2.is_zero()) { - return _ex0; - } - else { - if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) - return Li_num(ex_to(x1).to_int(), ex_to(x2)); - if (is_a(x2)) { - for (int i=0; i(x2.op(i))) { - return Li(x1,x2).hold(); + 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; } - return Li(x1,x2).evalf(); + 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(x1,x2).hold(); } + return Li(m_, x_).hold(); } -static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options) +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); } } @@ -555,12 +688,12 @@ static void Li_print_latex(const ex& m_, const ex& x_, const print_context& c) REGISTER_FUNCTION(Li, - 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()); + 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()); ////////////////////////////////////////////////////////////////////// @@ -680,24 +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)); } } } @@ -770,7 +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; @@ -799,13 +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) && 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) && 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_eval(const ex& x1, const ex& x2, const ex& x3) +static ex S_eval(const ex& n, const ex& p, const ex& x) { - if (x2 == 1) { - return Li(x1+1,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 (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational)) && - x1.info(info_flags::posint) && x2.info(info_flags::posint)) { - return S_num(ex_to(x1).to_int(), ex_to(x2).to_int(), ex_to(x3)); + 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); } } @@ -979,12 +1128,12 @@ static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_con REGISTER_FUNCTION(S, - 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()); + 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()); ////////////////////////////////////////////////////////////////////// @@ -999,6 +1148,10 @@ REGISTER_FUNCTION(S, // anonymous namespace for helper functions namespace { + +// regulates the pole (used by 1/x-transformation) +symbol H_polesign("IMSIGN"); + // convert parameters from H to Li representation // parameters are expected to be in expanded form, i.e. only 0, 1 and -1 @@ -1054,10 +1207,6 @@ bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf) } } } - for (; acc > 1; acc--) { - throw std::runtime_error("ERROR!"); - m.append(0); - } return has_negative_parameters; } @@ -1148,7 +1297,7 @@ struct map_trafo_H_reduce_trailing_zeros : public map_function if (name == "H") { lst parameter; if (is_a(e.op(0))) { - parameter = ex_to(e.op(0)); + parameter = ex_to(e.op(0)); } else { parameter = lst(e.op(0)); } @@ -1210,32 +1359,32 @@ struct map_trafo_H_reduce_trailing_zeros : public map_function // returns an expression with zeta functions corresponding to the parameter list for H -ex convert_H_to_zeta(const lst& l) +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(l, xtemp).hold())).subs(xtemp == 1); + return filter2(filter(H(m, xtemp).hold())).subs(xtemp == 1); } // convert signs form Li to H representation -// not used yet! -lst convert_parameter_Li_to_H(const lst& l, ex& pf) +lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf) { lst res; - lst::const_iterator it = l.begin(); - ex signum = *it; - pf = *it; - res.append(*it); - it++; - while (it != l.end()) { - signum = *it * signum; - res.append(signum); + 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; - it++; + res.append((*itm) * signum); + itm++; + itx++; } - return res; } @@ -1504,8 +1653,8 @@ struct map_trafo_H_1overx : public map_function } if (allthesame) { map_trafo_H_mult unify; - return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops()) / - factorial(parameter.nops())).expand()); + return unify((pow(H(lst(-1),1/arg).hold() - H(lst(0),1/arg).hold(), parameter.nops()) + / factorial(parameter.nops())).expand()); } } else { for (int i=1; i 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); + } } // simplify result @@ -1862,54 +2016,157 @@ static ex H_evalf(const ex& x1, const ex& x2) } -static ex H_eval(const ex& x1, const ex& x2) +static ex H_eval(const ex& m_, const ex& x) { - if (x2 == 0) { - return 0; + lst m; + if (is_a(m_)) { + m = ex_to(m_); + } else { + m = lst(m_); } -//TODO -// 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(); + if (m.nops() == 0) { + return _ex1; + } + ex pos1; + ex pos2; + ex n; + ex p; + int step = 0; + if (*m.begin() > _ex1) { + step++; + pos1 = _ex0; + pos2 = _ex1; + n = *m.begin()-1; + p = _ex1; + } else if (*m.begin() < _ex_1) { + step++; + pos1 = _ex0; + pos2 = _ex_1; + n = -*m.begin()-1; + p = _ex1; + } else if (*m.begin() == _ex0) { + pos1 = _ex0; + n = _ex1; + } else { + pos1 = *m.begin(); + p = _ex1; + } + for (lst::const_iterator it = ++m.begin(); it != m.end(); it++) { + if ((*it).info(info_flags::integer)) { + if (step == 0) { + if (*it > _ex1) { + if (pos1 == _ex0) { + step = 1; + pos2 = _ex1; + n += *it-1; + p = _ex1; + } else { + step = 2; + } + } else if (*it < _ex_1) { + if (pos1 == _ex0) { + step = 1; + pos2 = _ex_1; + n += -*it-1; + p = _ex1; + } else { + step = 2; + } + } else { + if (*it != pos1) { + step = 1; + pos2 = *it; + } + if (*it == _ex0) { + n++; + } else { + p++; + } + } + } else if (step == 1) { + if (*it != pos2) { + step = 2; + } else { + if (*it == _ex0) { + n++; + } else { + p++; + } + } + } + } else { + // if some m_i is not an integer + return H(m_, x).hold(); + } } - return H(x1,x2).hold(); + if ((x == _ex1) && (*(--m.end()) != _ex0)) { + return convert_H_to_zeta(m); + } + if (step == 0) { + if (pos1 == _ex0) { + // all zero + if (x == _ex0) { + return H(m_, x).hold(); + } + return pow(log(x), m.nops()) / factorial(m.nops()); + } else { + // all (minus) one + return pow(-pos1*log(1-pos1*x), m.nops()) / factorial(m.nops()); + } + } else if ((step == 1) && (pos1 == _ex0)){ + // convertible to S + if (pos2 == _ex1) { + return S(n, p, x); + } else { + return pow(-1, p) * S(n, p, -x); + } + } + if (x == _ex0) { + return _ex0; + } + if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) { + return H(m_, x).evalf(); + } + return H(m_, x).hold(); } -static ex H_series(const ex& x1, const ex& x2, 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), 0)); - return pseries(rel,seq); + seq.push_back(expair(H(m, x), 0)); + return pseries(rel, seq); } -static ex H_deriv(const ex& x1, const ex& x2, unsigned deriv_param) +static ex H_deriv(const ex& m_, const ex& x, 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 { - 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; } } @@ -1937,23 +2194,23 @@ static void H_print_latex(const ex& m_, const ex& x, const print_context& c) 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()); + 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()); // takes a parameter list for H and returns an expression with corresponding multiple polylogarithms -ex convert_H_to_Li(const ex& parameterlst, const ex& arg) +ex convert_H_to_Li(const ex& m, const ex& x) { map_trafo_H_reduce_trailing_zeros filter; map_trafo_H_convert_to_Li filter2; - if (is_a(parameterlst)) { - return filter2(filter(H(parameterlst, arg).hold())).eval(); + if (is_a(m)) { + return filter2(filter(H(m, x).hold())); } else { - return filter2(filter(H(lst(parameterlst), arg).hold())).eval(); + return filter2(filter(H(lst(m), x).hold())); } } @@ -2293,12 +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; } @@ -2350,7 +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 { @@ -2362,28 +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); } @@ -2396,53 +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& x, const print_context& c) +static void zeta1_print_latex(const ex& m_, const print_context& c) { c.s << "\\zeta("; - if (is_a(x)) { - lst arg; - arg = ex_to(x); - lst::const_iterator it = arg.begin(); + if (is_a(m_)) { + const lst& m = ex_to(m_); + lst::const_iterator it = m.begin(); (*it).print(c); it++; - for (; it != arg.end(); it++) { + for (; it != m.end(); it++) { c.s << ","; (*it).print(c); } } else { - x.print(c); + m_.print(c); } c.s << ")"; } -unsigned zeta1_SERIAL::serial = - function::register_new(function_options("zeta"). - evalf_func(zeta1_evalf). - eval_func(zeta1_eval). - derivative_func(zeta1_deriv). - print_func(zeta1_print_latex). - do_not_evalf_params(). - 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)); ////////////////////////////////////////////////////////////////////// @@ -2494,96 +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; } } -static void zeta2_print_latex(const ex& x, const ex& s, const print_context& c) +static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c) { - lst arg; - if (is_a(x)) { - arg = ex_to(x); + lst m; + if (is_a(m_)) { + m = ex_to(m_); } else { - arg = lst(x); + m = lst(m_); } - lst sig; - if (is_a(s)) { - sig = ex_to(s); + lst s; + if (is_a(s_)) { + s = ex_to(s_); } else { - sig = lst(s); + s = lst(s_); } c.s << "\\zeta("; - lst::const_iterator itarg = arg.begin(); - lst::const_iterator itsig = sig.begin(); - if (*itsig < 0) { + lst::const_iterator itm = m.begin(); + lst::const_iterator its = s.begin(); + if (*its < 0) { c.s << "\\overline{"; - (*itarg).print(c); + (*itm).print(c); c.s << "}"; } else { - (*itarg).print(c); + (*itm).print(c); } - itsig++; - itarg++; - for (; itarg != arg.end(); itarg++, itsig++) { + its++; + itm++; + for (; itm != m.end(); itm++, its++) { c.s << ","; - if (*itsig < 0) { + if (*its < 0) { c.s << "\\overline{"; - (*itarg).print(c); + (*itm).print(c); c.s << "}"; } else { - (*itarg).print(c); + (*itm).print(c); } } c.s << ")"; } -unsigned zeta2_SERIAL::serial = - function::register_new(function_options("zeta"). - evalf_func(zeta2_evalf). - eval_func(zeta2_eval). - derivative_func(zeta2_deriv). - print_func(zeta2_print_latex). - do_not_evalf_params(). - overloaded(2)); +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