From: Jens Vollinga Date: Sat, 29 Nov 2003 17:33:41 +0000 (+0000) Subject: * Removed H(m,s,x) X-Git-Tag: release_1-2-0~59 X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=commitdiff_plain;h=168dd38dd35fcaaf61e177afb8dbe0288c0eb521;hp=e6e621f1d80bcdf0abc69cef84dbb58168a14ff6 * Removed H(m,s,x) * H(m,s) now handles also negative parameters in RV notation * H uses now x -> (1-x)/(1+x) transformation * Removed convert_H_notation, added convert_H_to_Li * Using the new printing options every function now has a nice LaTeX output --- diff --git a/ginac/inifcns.h b/ginac/inifcns.h index 426a7e09..412eed53 100644 --- a/ginac/inifcns.h +++ b/ginac/inifcns.h @@ -116,24 +116,8 @@ DECLARE_FUNCTION_2P(Li) /** Nielsen's generalized polylogarithm. */ DECLARE_FUNCTION_3P(S) -// overloading at work: we cannot use the macros here -/** Harmonic polylogarithm with only positive parameters. */ -class H2_SERIAL { public: static unsigned serial; }; -template -inline function H(const T1& p1, const T2& p2) { - return function(H2_SERIAL::serial, ex(p1), ex(p2)); -} -/** Harmonic polylogarithm with signed parameters. */ -class H3_SERIAL { public: static unsigned serial; }; -template -inline function H(const T1& p1, const T2& p2, const T3& p3) { - return function(H3_SERIAL::serial, ex(p1), ex(p2), ex(p3)); -} -class H_SERIAL; -template<> inline bool is_the_function(const ex& x) -{ - return is_the_function(x) || is_the_function(x); -} +/** Harmonic polylogarithm. */ +DECLARE_FUNCTION_2P(H) /** Gamma-function. */ DECLARE_FUNCTION_1P(lgamma) @@ -181,7 +165,7 @@ inline bool is_order_function(const ex & e) /** Converts a given list containing parameters for H in Remiddi/Vermaseren notation into * the corresponding GiNaC functions. */ -ex convert_H_notation(const ex& parameterlst, const ex& arg); +ex convert_H_to_Li(const ex& parameterlst, const ex& arg); } // namespace GiNaC diff --git a/ginac/inifcns_nstdsums.cpp b/ginac/inifcns_nstdsums.cpp index 9bff9976..020ff333 100644 --- a/ginac/inifcns_nstdsums.cpp +++ b/ginac/inifcns_nstdsums.cpp @@ -18,18 +18,22 @@ * [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754 * [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941 * - * - The order of parameters and arguments of H, Li and zeta is defined according to their order in the - * nested sums representation. + * - 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 each argument x_i is smaller than - * one. The parameters for every function (n, p or n_i) must be positive integers. + * 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 * 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. * @@ -79,11 +83,6 @@ #include "wildcard.h" -//DEBUG -#include -using namespace std; - - namespace GiNaC { @@ -405,18 +404,6 @@ cln::cl_N multipleLi_do_sum(const std::vector& s, const std::vector t(j); cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); @@ -438,9 +425,6 @@ 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 @@ -514,6 +477,27 @@ static ex Li_evalf(const ex& x1, const ex& x2) } +static ex Li_eval(const ex& x1, const ex& x2) +{ + 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(); + } + } + return Li(x1,x2).evalf(); + } + return Li(x1,x2).hold(); + } +} + + static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options) { epvector seq; @@ -536,12 +520,47 @@ static ex Li_deriv(const ex& x1, const ex& x2, unsigned deriv_param) } +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(). + eval_func(Li_eval). series_func(Li_series). - derivative_func(Li_deriv)); + derivative_func(Li_deriv). + print_func(Li_print_latex). + do_not_evalf_params()); ////////////////////////////////////////////////////////////////////// @@ -903,22 +922,22 @@ numeric S_num(int n, int p, const numeric& x) ////////////////////////////////////////////////////////////////////// -static ex S_eval(const ex& x1, const ex& x2, const ex& x3) +static ex S_evalf(const ex& x1, const ex& x2, const ex& x3) { - if (x2 == 1) { - return Li(x1+1,x3); - } - if (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational)) && - x1.info(info_flags::posint) && x2.info(info_flags::posint)) { + 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)); } return S(x1,x2,x3).hold(); } -static ex S_evalf(const ex& x1, const ex& x2, const ex& x3) +static ex S_eval(const ex& x1, const ex& x2, const ex& x3) { - if (is_a(x1) && is_a(x2) && is_a(x3)) { + if (x2 == 1) { + return Li(x1+1,x3); + } + 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)); } return S(x1,x2,x3).hold(); @@ -947,19 +966,32 @@ static ex S_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param } +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(). + eval_func(S_eval). series_func(S_series). - derivative_func(S_deriv)); + 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 // ////////////////////////////////////////////////////////////////////// @@ -968,8 +1000,244 @@ REGISTER_FUNCTION(S, namespace { -// forward declaration -ex convert_from_RV(const lst& parameterlst, const ex& arg); +// 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) +{ + // 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 { + mexp.append(*it); + } + } + + ex signum = 1; + pf = 1; + bool has_negative_parameters = false; + ex acc = 1; + for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) { + if (*it == 0) { + acc++; + continue; + } + if (*it > 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; + } + } + if (has_negative_parameters) { + for (int i=0; i 1; acc--) { + throw std::runtime_error("ERROR!"); + m.append(0); + } + + return has_negative_parameters; +} + + +// recursivly transforms H to corresponding multiple polylogarithms +struct map_trafo_H_convert_to_Li : 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)); + } + ex arg = e.op(1); + + lst m; + lst s; + ex pf; + if (convert_parameter_H_to_Li(parameter, m, s, pf)) { + s.let_op(0) = s.op(0) * arg; + return pf * Li(m, s).hold(); + } else { + 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)); + } + + 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; + } +}; + + +// remove trailing zeros from H-parameters +struct map_trafo_H_reduce_trailing_zeros : 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)); + } + 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& l) +{ + 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); +} + + +// convert signs form Li to H representation +// not used yet! +lst convert_parameter_Li_to_H(const lst& l, 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); + pf *= signum; + it++; + } + + return res; +} // multiplies an one-dimensional H with another H @@ -1072,9 +1340,9 @@ struct map_trafo_H_mult : public map_function }; -// do integration [ReV] (49) -// put parameter 1 in front of existing parameters -ex trafo_H_prepend_one(const ex& e, const ex& arg) +// do integration [ReV] (55) +// put parameter 0 in front of existing parameters +ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg) { ex h; std::string name; @@ -1095,17 +1363,18 @@ ex trafo_H_prepend_one(const ex& e, const ex& arg) } if (h != 0) { lst newparameter = ex_to(h.op(0)); - newparameter.prepend(1); - return e.subs(h == H(newparameter, h.op(1)).hold()); + newparameter.prepend(0); + ex addzeta = convert_H_to_zeta(newparameter); + return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand(); } else { - return e * H(lst(1),1-arg).hold(); + return e * (-H(lst(0),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) +// put parameter -1 in front of existing parameters +ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg) { ex h; std::string name; @@ -1126,70 +1395,171 @@ 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))); + newparameter.prepend(-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()); + ex addzeta = convert_H_to_zeta(lst(-1)); + return (e * (addzeta - H(lst(-1),1/arg).hold())).expand(); } } -// 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_1mxt1px_prepend_minusone(const ex& e, const ex& arg) { - 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") { - + 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(); + } +} + + +// do integration [ReV] (55) +// put parameter 1 in front of existing parameters +ex trafo_H_1mxt1px_prepend_one(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(); + } +} + + +// do x -> 1/x transformation +struct map_trafo_H_1overx : 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 = 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); + // special cases if all parameters are either 0, 1 or -1 + bool allthesame = true; + if (parameter.op(0) == 0) { + for (int i=1; i0; i--) { - newparameter.append(1); + if (allthesame) { + return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold(); + } + } else if (parameter.op(0) == -1) { + for (int i=1; i(buffer)) { + for (int i=0; i(buffer)) { for (int i=0; i 1/x transformation -struct map_trafo_H_1overx : public map_function +// do x -> (1-x)/(1+x) transformation +struct map_trafo_H_1mxt1px : public map_function { ex operator()(const ex& e) { @@ -1234,37 +1604,83 @@ 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; - } -}; - - -// transform H(m,x) with signed m to H(m,s,x) -struct map_trafo_H_convert_signed_m : 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)); - } - 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); - } - } - return H(parameter, signs, arg).hold(); - } - } - } - return e; - } -}; - - -// recursively call convert_from_RV on expression -struct map_trafo_H_convert : public map_function -{ - ex operator()(const ex& e) - { - if (is_a(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; - } -}; - - -// translate notation from nested sums to Remiddi/Vermaseren -lst convert_to_RV(const lst& o) -{ - lst res; - for (lst::const_iterator it = o.begin(); it != o.end(); it++) { - if (*it > 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; -} - - -// translate notation from Remiddi/Vermaseren to nested sums -ex convert_from_RV(const lst& parameterlst, const ex& arg) -{ - lst newparameterlst; - - lst::const_iterator it = parameterlst.begin(); - int count = 1; - while (it != parameterlst.end()) { - if (*it == 0) { - count++; - } else { - newparameterlst.append((*it>0) ? count : -count); - count = 1; - } - it++; - } - for (int i=1; i& m, const cln::cl_N& x) { @@ -1497,7 +1732,7 @@ cln::cl_N H_do_sum(const std::vector& 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]; } @@ -1514,222 +1749,148 @@ 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(); + return H(x1,x2).hold(); } - // choose trafo - if (cln::abs(x) > 1) { + cln::cl_N x = ex_to(x2).to_cl_N(); + + const lst& morg = ex_to(x1); + // remove trailing zeros ... + if (*(--morg.end()) == 0) { 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(); + 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); + } } - // 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(); + // 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)); + } } - // no trafo -> do summation - int count = x1.nops(); - std::vector r(count); - for (int i=0; i(x1.op(i)).to_int(); + ex res = 1; + + // ensure that the realpart of the argument is positive + if (cln::realpart(x) < 0) { + x = -x; + for (int i=0; i(x1)) { - lst newparameter = ex_to(x1); - if (x1.op(0) == 1) { - newparameter.remove_first(); - return 1/(1-x2) * H(newparameter, x2); + // 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 { - newparameter[0]--; - return H(newparameter, x2).hold() / x2; - } - } else { - if (x1 == 1) { - return 1/(1-x2); - } else { - return H(x1-1, x2).hold() / x2; + // x -> 1/x + map_trafo_H_1overx trafo; + res *= trafo(H(m, xtemp)); } - } -} - -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)); + // 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(); + } -////////////////////////////////////////////////////////////////////// -// -// Harmonic polylogarithm H(m,s,x) -// -// GiNaC function -// -////////////////////////////////////////////////////////////////////// + return H(x1,x2).hold(); +} -static ex H3_eval(const ex& x1, const ex& x2, const ex& x3) +static ex H_eval(const ex& x1, const ex& x2) { - if (x3 == 0) { + if (x2 == 0) { return 0; } - if (x3 == 1) { - return zeta(x1, x2); - } - if (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational))) { - return H(x1, x2, x3).evalf(); +//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(); } - 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(); + return H(x1,x2).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& x1, const ex& x2, const relational& rel, int order, unsigned options) { epvector seq; - seq.push_back(expair(H(x1, x2, x3), 0)); - return pseries(rel, seq); + seq.push_back(expair(H(x1,x2), 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& x1, const ex& x2, unsigned deriv_param) { - //TODO - GINAC_ASSERT(deriv_param < 2); if (deriv_param == 0) { return _ex0; @@ -1753,37 +1914,47 @@ static ex H3_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_para } -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& parameterlst, const ex& arg) { + map_trafo_H_reduce_trailing_zeros filter; + map_trafo_H_convert_to_Li filter2; 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); + return filter2(filter(H(parameterlst, arg).hold())).eval(); + } else { + return filter2(filter(H(lst(parameterlst), arg).hold())).eval(); } - throw std::runtime_error("first parameter has to be a list containing only 0, 1 or -1!"); } @@ -2244,13 +2415,33 @@ static ex zeta1_deriv(const ex& x, unsigned deriv_param) } +static void zeta1_print_latex(const ex& x, const print_context& c) +{ + c.s << "\\zeta("; + if (is_a(x)) { + lst arg; + arg = ex_to(x); + lst::const_iterator it = arg.begin(); + (*it).print(c); + it++; + for (; it != arg.end(); it++) { + c.s << ","; + (*it).print(c); + } + } else { + x.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(). + eval_func(zeta1_eval). derivative_func(zeta1_deriv). - latex_name("\\zeta"). + print_func(zeta1_print_latex). + do_not_evalf_params(). overloaded(2)); @@ -2345,13 +2536,53 @@ static ex zeta2_deriv(const ex& x, const ex& s, unsigned deriv_param) } +static void zeta2_print_latex(const ex& x, const ex& s, const print_context& c) +{ + lst arg; + if (is_a(x)) { + arg = ex_to(x); + } else { + arg = lst(x); + } + lst sig; + if (is_a(s)) { + sig = ex_to(s); + } else { + sig = lst(s); + } + c.s << "\\zeta("; + lst::const_iterator itarg = arg.begin(); + lst::const_iterator itsig = sig.begin(); + if (*itsig < 0) { + c.s << "\\overline{"; + (*itarg).print(c); + c.s << "}"; + } else { + (*itarg).print(c); + } + itsig++; + itarg++; + for (; itarg != arg.end(); itarg++, itsig++) { + c.s << ","; + if (*itsig < 0) { + c.s << "\\overline{"; + (*itarg).print(c); + c.s << "}"; + } else { + (*itarg).print(c); + } + } + c.s << ")"; +} + + unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta"). - eval_func(zeta2_eval). evalf_func(zeta2_evalf). - do_not_evalf_params(). + eval_func(zeta2_eval). derivative_func(zeta2_deriv). - latex_name("\\zeta"). + print_func(zeta2_print_latex). + do_not_evalf_params(). overloaded(2));