X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_nstdsums.cpp;h=82a0bbd87903b831d2f04ad3a8ccd5d8f6674261;hp=c89554553455cd4ccd8f5d7892138b40eb94a193;hb=81a315ac1de10724ad963e2a167b7f618b81ac0f;hpb=d6372945e85c0f53e2f49f26ded912935a491ea3 diff --git a/ginac/inifcns_nstdsums.cpp b/ginac/inifcns_nstdsums.cpp index c8955455..82a0bbd8 100644 --- a/ginac/inifcns_nstdsums.cpp +++ b/ginac/inifcns_nstdsums.cpp @@ -1,33 +1,53 @@ /** @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(n_1,...,n_k),lst(x_1,...,x_k) - * nielsen's generalized polylogarithm S(n,p,x) - * harmonic polylogarithm H(lst(m_1,...,m_k),x) - * multiple zeta value mZeta(lst(m_1,...,m_k)) + * multiple polylogarithm Li(lst{m_1,...,m_k},lst{x_1,...,x_k}) + * G(lst{a_1,...,a_k},y) or G(lst{a_1,...,a_k},lst{s_1,...,s_k},y) + * Nielsen's generalized polylogarithm S(n,p,x) + * harmonic polylogarithm H(m,x) or H(lst{m_1,...,m_k},x) + * multiple zeta value zeta(m) or zeta(lst{m_1,...,m_k}) + * alternating Euler sum zeta(m,s) or zeta(lst{m_1,...,m_k},lst{s_1,...,s_k}) * * Some remarks: - * - All formulae used can be looked up in the following publication: - * Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258. - * This document will be referenced as [Kol] throughout this source code. - * - Classical polylogarithms (Li) and nielsen's generalized polylogarithms (S) can be numerically - * evaluated in the whole complex plane. And of course, there is still room for speed optimizations ;-). - * - The calculation of classical polylogarithms is speed up by using Euler-Maclaurin summation (EuMac). - * - The remaining functions can only be numerically evaluated with arguments lying in the unit sphere - * at the moment. Sorry. The evaluation especially for mZeta is very slow ... better not use it - * right now. - * - The functions have no series expansion. To do it, you have to convert these functions - * into the appropriate objects from the nestedsums library, do the expansion and convert the - * result back. + * + * - 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 + * [VSW] Numerical evaluation of multiple polylogarithms, J.Vollinga, S.Weinzierl, hep-ph/0410259 + * + * - 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. + * + * - All functions can be numerically evaluated with arguments in the whole complex plane. The parameters + * for Li, zeta and S must be positive integers. If you want to have an alternating Euler sum, you have + * to give the signs of the parameters as a second argument s to zeta(m,s) containing 1 and -1. + * + * - The calculation of classical polylogarithms is speeded up by using Bernoulli numbers and + * look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in + * [Cra] and [BBB] for speed up. Multiple polylogarithms use Hoelder convolution [BBB]. + * + * - The functions have no means to do a series expansion into nested sums. To do this, you have to convert + * these functions into the appropriate objects from the nestedsums library, do the expansion and convert + * the result back. + * * - Numerical testing of this implementation has been performed by doing a comparison of results - * between this software and the commercial M.......... 4.1. + * between this software and the commercial M.......... 4.1. Multiple zeta values have been checked + * by means of evaluations into simple zeta values. Harmonic polylogarithms have been checked by + * comparison to S(n,p,x) for corresponding parameter combinations and by continuity checks + * around |x|=1 along with comparisons to corresponding zeta functions. Multiple polylogarithms were + * checked against H and zeta and by means of shuffle and quasi-shuffle relations. * */ /* - * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2015 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 @@ -41,45 +61,56 @@ * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software - * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ -#include -#include -#include - #include "inifcns.h" + +#include "add.h" +#include "constant.h" #include "lst.h" +#include "mul.h" #include "numeric.h" #include "operators.h" -#include "relational.h" +#include "power.h" #include "pseries.h" +#include "relational.h" +#include "symbol.h" +#include "utils.h" +#include "wildcard.h" +#include +#include +#include +#include namespace GiNaC { - -// lookup table for Euler-Maclaurin optimization -// see fill_Xn() -std::vector > Xn; -int xnsize = 0; +////////////////////////////////////////////////////////////////////// +// +// Classical polylogarithm Li(n,x) +// +// helper functions +// +////////////////////////////////////////////////////////////////////// -// lookup table for special Euler-Zagier-Sums (used for S_n,p(x)) -// see fill_Yn() -std::vector > Yn; -int ynsize = 0; // number of Yn[] -int ynlength = 100; // initial length of all Yn[i] +// anonymous namespace for helper functions +namespace { -////////////////////// -// helper functions // -////////////////////// +// lookup table for factors built from Bernoulli numbers +// see fill_Xn() +std::vector> Xn; +// initial size of Xn that should suffice for 32bit machines (must be even) +const int xninitsizestep = 26; +int xninitsize = xninitsizestep; +int xnsize = 0; -// This function calculates the X_n. The X_n are needed for the Euler-Maclaurin summation (EMS) of -// classical polylogarithms. -// With EMS the polylogs can be calculated as follows: + +// This function calculates the X_n. The X_n are needed for speed up of classical polylogarithms. +// With these numbers the polylogs can be calculated as follows: // Li_p (x) = \sum_{n=0}^\infty X_{p-2}(n) u^{n+1}/(n+1)! with u = -log(1-x) // X_0(n) = B_n (Bernoulli numbers) // X_p(n) = \sum_{k=0}^n binomial(n,k) B_{n-k} / (k+1) * X_{p-1}(k) @@ -87,20 +118,17 @@ int ynlength = 100; // initial length of all Yn[i] // X_0 is special, it holds only the non-zero Bernoulli numbers with index 2 or greater. // This results in a slightly more complicated algorithm for the X_n. // The first index in Xn corresponds to the index of the polylog minus 2. -// The second index in Xn corresponds to the index from the EMS. -static void fill_Xn(int n) +// The second index in Xn corresponds to the index from the actual sum. +void fill_Xn(int n) { - // rule of thumb. needs to be improved. TODO - const int initsize = Digits * 3 / 2; - 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 { @@ -120,14 +148,14 @@ static 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; @@ -144,9 +172,9 @@ static 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++; } @@ -156,88 +184,59 @@ static void fill_Xn(int n) xnsize++; } - -// This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x). -// The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum -// representing S_{n,p}(x). -// The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the -// equivalent Z-sum. -// The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum -// representing S_{n,p}(x). -// The calculation of Y_n uses the values from Y_{n-1}. -static void fill_Yn(int n, const cln::float_format_t& prec) -{ - const int initsize = ynlength; - //const int initsize = initsize_Yn; - cln::cl_N one = cln::cl_float(1, prec); - - if (n) { - std::vector buf(initsize); - std::vector::iterator it = buf.begin(); - std::vector::iterator itprev = Yn[n-1].begin(); - *it = (*itprev) / cln::cl_N(n+1) * one; - it++; - itprev++; - // sums with an index smaller than the depth are zero and need not to be calculated. - // calculation starts with depth, which is n+2) - for (int i=n+2; i<=initsize+n; i++) { - *it = *(it-1) + (*itprev) / cln::cl_N(i) * one; - it++; - itprev++; - } - Yn.push_back(buf); - } else { - std::vector buf(initsize); - std::vector::iterator it = buf.begin(); - *it = 1 * one; - it++; - for (int i=2; i<=initsize; i++) { - *it = *(it-1) + 1 / cln::cl_N(i) * one; - it++; - } - Yn.push_back(buf); - } - ynsize++; -} - - -// make Yn longer ... -static void make_Yn_longer(int newsize, const cln::float_format_t& prec) +// doubles the number of entries in each Xn[] +void double_Xn() { - - cln::cl_N one = cln::cl_float(1, prec); - - Yn[0].resize(newsize); - std::vector::iterator it = Yn[0].begin(); - it += ynlength; - for (int i=ynlength+1; i<=newsize; i++) { - *it = *(it-1) + 1 / cln::cl_N(i) * one; - it++; + 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()); } - - for (int n=1; n::iterator it = Yn[n].begin(); - std::vector::iterator itprev = Yn[n-1].begin(); - it += ynlength; - itprev += ynlength; - for (int i=ynlength+n+1; i<=newsize+n; i++) { - *it = *(it-1) + (*itprev) / cln::cl_N(i) * one; - it++; - itprev++; + if (Xn.size() > 1) { + 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); + } } } - - ynlength = newsize; + xninitsize += xninitsizestep; } -// calculates Li(2,x) without EuMac -static cln::cl_N Li2_series(const cln::cl_N& x) +// calculates Li(2,x) without Xn +cln::cl_N Li2_do_sum(const cln::cl_N& x) { cln::cl_N res = x; cln::cl_N resbuf; - 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 { @@ -251,30 +250,36 @@ static cln::cl_N Li2_series(const cln::cl_N& x) } -// calculates Li(2,x) with EuMac -static cln::cl_N Li2_series_EuMac(const cln::cl_N& x) +// calculates Li(2,x) with Xn +cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x) { std::vector::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 res = u - u*u/4; + cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits)); + cln::cl_N uu = cln::square(u); + cln::cl_N res = u - uu/4; cln::cl_N resbuf; unsigned i = 1; do { resbuf = res; - factor = factor * u*u / (2*i * (2*i+1)); + factor = factor * uu / (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; } -// calculates Li(n,x), n>2 without EuMac -static cln::cl_N Lin_series(int n, const cln::cl_N& x) +// calculates Li(n,x), n>2 without Xn +cln::cl_N Lin_do_sum(int n, const cln::cl_N& x) { - cln::cl_N factor = x; + cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits)); cln::cl_N res = x; cln::cl_N resbuf; int i=2; @@ -288,12 +293,13 @@ static cln::cl_N Lin_series(int n, const cln::cl_N& x) } -// calculates Li(n,x), n>2 with EuMac -static cln::cl_N Lin_series_EuMac(int n, const cln::cl_N& x) +// calculates Li(n,x), n>2 with Xn +cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x) { std::vector::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; @@ -301,19 +307,23 @@ static cln::cl_N Lin_series_EuMac(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; } // forward declaration needed by function Li_projection and C below -static numeric S_num(int n, int p, const numeric& x); +const cln::cl_N S_num(int n, int p, const cln::cl_N& x); // helper function for classical polylog Li -static cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec) +cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& prec) { // treat n=2 as special case if (n == 2) { @@ -329,16 +339,20 @@ static cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_forma // it solves also the problem with precision due to the u=-log(1-x) transformation if (cln::abs(cln::realpart(x)) < 0.25) { - return Li2_series(x); + return Li2_do_sum(x); } else { - return Li2_series_EuMac(x); + return Li2_do_sum_Xn(x); } } else { // choose the faster algorithm if (cln::abs(cln::realpart(x)) > 0.75) { - return -Li2_series(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2); + if ( x == 1 ) { + return cln::zeta(2); + } else { + return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2); + } } else { - return -Li2_series_EuMac(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2); + return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2); } } } else { @@ -351,32 +365,32 @@ static cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_forma if (cln::realpart(x) < 0.5) { // choose the faster algorithm - // with n>=12 the "normal" summation always wins against EuMac + // with n>=12 the "normal" summation always wins against the method with Xn if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) { - return Lin_series(n, x); + return Lin_do_sum(n, x); } else { - return Lin_series_EuMac(n, x); + return Lin_do_sum_Xn(n, x); } } else { - cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1); + cln::cl_N result = 0; + if ( x != 1 ) result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1); for (int j=0; j(cln::realpart(value))); - else if (!x.imag().is_rational()) + else if (!instanceof(imagpart(x), cln::cl_RA_ring)) prec = cln::float_format(cln::the(cln::imagpart(value))); // [Kol] (5.15) @@ -421,7 +443,7 @@ static numeric Li_num(int n, const numeric& x) cln::cl_N add; for (int j=0; j& s, const std::vector& x) { - if (step) { - cln::cl_N res; - for (int i=1; i 0. + for (std::vector::const_iterator it = x.begin(); it != x.end(); ++it) { + if ( *it == 0 ) return cln::cl_float(0, cln::float_format(Digits)); } + + const int j = s.size(); + bool flag_accidental_zero = false; + + std::vector t(j); + cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); + + cln::cl_N t0buf; + int q = 0; + do { + t0buf = t[0]; + q++; + t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one; + for (int k=j-2; k>=0; k--) { + t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]); + } + q++; + t[j-1] = t[j-1] + cln::expt(x[j-1], q) / cln::expt(cln::cl_I(q),s[j-1]) * one; + for (int k=j-2; k>=0; k--) { + flag_accidental_zero = cln::zerop(t[k+1]); + t[k] = t[k] + t[k+1] * cln::expt(x[k], q+j-1-k) / cln::expt(cln::cl_I(q+j-1-k), s[k]); + } + } while ( (t[0] != t0buf) || cln::zerop(t[0]) || flag_accidental_zero ); + + return t[0]; } -// helper function for S(n,p,x) -// [Kol] (7.2) -static cln::cl_N C(int n, int p) +// forward declaration for Li_eval() +lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf); + + +// type used by the transformation functions for G +typedef std::vector Gparameter; + + +// G_eval1-function for G transformations +ex G_eval1(int a, int scale, const exvector& gsyms) { - cln::cl_N result; + if (a != 0) { + const ex& scs = gsyms[std::abs(scale)]; + const ex& as = gsyms[std::abs(a)]; + if (as != scs) { + return -log(1 - scs/as); + } else { + return -zeta(1); + } + } else { + return log(gsyms[std::abs(scale)]); + } +} - for (int k=0; k 1 && newa.op(0) == sc && !all_ones && a.front()!=0) { + // do shuffle + Gparameter short_a; + Gparameter::const_iterator it = a.begin(); + ++it; + for (; it != a.end(); ++it) { + short_a.push_back(*it); } - else { - result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p)); + ex result = G_eval1(a.front(), scale, gsyms) * G_eval(short_a, scale, gsyms); + it = short_a.begin(); + for (int i=1; i G({1};y)^k / k! + if (all_ones && a.size() > 1) { + return pow(G_eval1(a.front(),scale, gsyms), count_ones) / factorial(count_ones); + } + + // G({0,...,0};y) -> log(y)^k / k! + if (all_zero) { + return pow(log(gsyms[std::abs(scale)]), a.size()) / factorial(a.size()); + } + + // no special cases anymore -> convert it into Li + lst m; + lst x; + ex argbuf = gsyms[std::abs(scale)]; + ex mval = _ex1; + for (Gparameter::const_iterator it=a.begin(); it!=a.end(); ++it) { + if (*it != 0) { + const ex& sym = gsyms[std::abs(*it)]; + x.append(argbuf / sym); + m.append(mval); + mval = _ex1; + argbuf = sym; + } else { + ++mval; + } + } + return pow(-1, x.nops()) * Li(m, x); } -// helper function for S(n,p,x) -// [Kol] remark to (9.1) -static cln::cl_N a_k(int k) +// converts data for G: pending_integrals -> a +Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals) { - cln::cl_N result; + GINAC_ASSERT(pending_integrals.size() != 1); - if (k == 0) { - return 1; + if (pending_integrals.size() > 0) { + // get rid of the first element, which would stand for the new upper limit + Gparameter new_a(pending_integrals.begin()+1, pending_integrals.end()); + return new_a; + } else { + // just return empty parameter list + Gparameter new_a; + return new_a; } +} - result = result; - for (int m=2; m<=k; m++) { - result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * a_k(k-m); - } - return -result / k; +// check the parameters a and scale for G and return information about convergence, depth, etc. +// convergent : true if G(a,scale) is convergent +// depth : depth of G(a,scale) +// trailing_zeros : number of trailing zeros of a +// min_it : iterator of a pointing on the smallest element in a +Gparameter::const_iterator check_parameter_G(const Gparameter& a, int scale, + bool& convergent, int& depth, int& trailing_zeros, Gparameter::const_iterator& min_it) +{ + convergent = true; + depth = 0; + trailing_zeros = 0; + min_it = a.end(); + Gparameter::const_iterator lastnonzero = a.end(); + for (Gparameter::const_iterator it = a.begin(); it != a.end(); ++it) { + if (std::abs(*it) > 0) { + ++depth; + trailing_zeros = 0; + lastnonzero = it; + if (std::abs(*it) < scale) { + convergent = false; + if ((min_it == a.end()) || (std::abs(*it) < std::abs(*min_it))) { + min_it = it; + } + } + } else { + ++trailing_zeros; + } + } + if (lastnonzero == a.end()) + return a.end(); + return ++lastnonzero; } -// helper function for S(n,p,x) -// [Kol] remark to (9.1) -static cln::cl_N b_k(int k) +// add scale to pending_integrals if pending_integrals is empty +Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale) { - cln::cl_N result; + GINAC_ASSERT(pending_integrals.size() != 1); - if (k == 0) { - return 1; + if (pending_integrals.size() > 0) { + return pending_integrals; + } else { + Gparameter new_pending_integrals; + new_pending_integrals.push_back(scale); + return new_pending_integrals; } +} - result = result; - for (int m=2; m<=k; m++) { - result = result + cln::expt(cln::cl_N(-1),m) * cln::zeta(m) * b_k(k-m); - } - return result / k; +// handles trailing zeroes for an otherwise convergent integral +ex trailing_zeros_G(const Gparameter& a, int scale, const exvector& gsyms) +{ + bool convergent; + int depth, trailing_zeros; + Gparameter::const_iterator last, dummyit; + last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit); + + GINAC_ASSERT(convergent); + + if ((trailing_zeros > 0) && (depth > 0)) { + ex result; + Gparameter new_a(a.begin(), a.end()-1); + result += G_eval1(0, scale, gsyms) * trailing_zeros_G(new_a, scale, gsyms); + for (Gparameter::const_iterator it = a.begin(); it != last; ++it) { + Gparameter new_a(a.begin(), it); + new_a.push_back(0); + new_a.insert(new_a.end(), it, a.end()-1); + result -= trailing_zeros_G(new_a, scale, gsyms); + } + + return result / trailing_zeros; + } else { + return G_eval(a, scale, gsyms); + } } -// helper function for S(n,p,x) -static cln::cl_N S_series(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec) +// G transformation [VSW] (57),(58) +ex depth_one_trafo_G(const Gparameter& pending_integrals, const Gparameter& a, int scale, const exvector& gsyms) { - if (p==1) { - return Li_projection(n+1, x, prec); - } - - // check if precalculated values are sufficient - if (p > ynsize+1) { - for (int i=ynsize; i 0); + + ex result; + Gparameter new_pending_integrals = prepare_pending_integrals(pending_integrals, std::abs(a.back())); + const int psize = pending_integrals.size(); + + // length == 1 + // G(sr_{+-}; y2 ) = G(y2_{-+}; sr) - G(0; sr) + ln(-y2_{-+}) + + if (a.size() == 1) { + + // ln(-y2_{-+}) + result += log(gsyms[ex_to(scale).to_int()]); + if (a.back() > 0) { + new_pending_integrals.push_back(-scale); + result += I*Pi; + } else { + new_pending_integrals.push_back(scale); + result -= I*Pi; + } + if (psize) { + result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), + pending_integrals.front(), + gsyms); + } + + // G(y2_{-+}; sr) + result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), + new_pending_integrals.front(), + gsyms); + + // G(0; sr) + new_pending_integrals.back() = 0; + result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), + new_pending_integrals.front(), + gsyms); + + return result; + } + + // length > 1 + // G_m(sr_{+-}; y2) = -zeta_m + int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t ) + // - int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t ) + + //term zeta_m + result -= zeta(a.size()); + if (psize) { + result *= trailing_zeros_G(convert_pending_integrals_G(pending_integrals), + pending_integrals.front(), + gsyms); + } + + // term int_0^sr dt/t G_{m-1}( (1/y2)_{+-}; 1/t ) + // = int_0^sr dt/t G_{m-1}( t_{+-}; y2 ) + Gparameter new_a(a.begin()+1, a.end()); + new_pending_integrals.push_back(0); + result -= depth_one_trafo_G(new_pending_integrals, new_a, scale, gsyms); + + // term int_0^y2 dt/t G_{m-1}( (1/y2)_{+-}; 1/t ) + // = int_0^y2 dt/t G_{m-1}( t_{+-}; y2 ) + Gparameter new_pending_integrals_2; + new_pending_integrals_2.push_back(scale); + new_pending_integrals_2.push_back(0); + if (psize) { + result += trailing_zeros_G(convert_pending_integrals_G(pending_integrals), + pending_integrals.front(), + gsyms) + * depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms); + } else { + result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale, gsyms); + } + + return result; +} + + +// forward declaration +ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2, + const Gparameter& pendint, const Gparameter& a_old, int scale, + const exvector& gsyms, bool flag_trailing_zeros_only); + + +// G transformation [VSW] +ex G_transform(const Gparameter& pendint, const Gparameter& a, int scale, + const exvector& gsyms, bool flag_trailing_zeros_only) +{ + // main recursion routine + // + // pendint = ( y1, b1, ..., br ) + // a = ( a1, ..., amin, ..., aw ) + // scale = y2 + // + // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2) + // where sr replaces amin + + // find smallest alpha, determine depth and trailing zeros, and check for convergence + bool convergent; + int depth, trailing_zeros; + Gparameter::const_iterator min_it; + Gparameter::const_iterator firstzero = + check_parameter_G(a, scale, convergent, depth, trailing_zeros, min_it); + int min_it_pos = min_it - a.begin(); + + // special case: all a's are zero + if (depth == 0) { + ex result; + + if (a.size() == 0) { + result = 1; + } else { + result = G_eval(a, scale, gsyms); + } + if (pendint.size() > 0) { + result *= trailing_zeros_G(convert_pending_integrals_G(pendint), + pendint.front(), + gsyms); + } + return result; + } + + // handle trailing zeros + if (trailing_zeros > 0) { + ex result; + Gparameter new_a(a.begin(), a.end()-1); + result += G_eval1(0, scale, gsyms) * G_transform(pendint, new_a, scale, gsyms, flag_trailing_zeros_only); + for (Gparameter::const_iterator it = a.begin(); it != firstzero; ++it) { + Gparameter new_a(a.begin(), it); + new_a.push_back(0); + new_a.insert(new_a.end(), it, a.end()-1); + result -= G_transform(pendint, new_a, scale, gsyms, flag_trailing_zeros_only); + } + return result / trailing_zeros; + } + + // convergence case or flag_trailing_zeros_only + if (convergent || flag_trailing_zeros_only) { + if (pendint.size() > 0) { + return G_eval(convert_pending_integrals_G(pendint), + pendint.front(), gsyms)* + G_eval(a, scale, gsyms); + } else { + return G_eval(a, scale, gsyms); + } + } + + // call basic transformation for depth equal one + if (depth == 1) { + return depth_one_trafo_G(pendint, a, scale, gsyms); + } + + // do recursion + // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,sr,...,aw,y2) + // = int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(a1,...,0,...,aw,y2) + // + int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) int_0^{sr} ds_{r+1} d/ds_{r+1} G(a1,...,s_{r+1},...,aw,y2) + + // smallest element in last place + if (min_it + 1 == a.end()) { + do { --min_it; } while (*min_it == 0); + Gparameter empty; + Gparameter a1(a.begin(),min_it+1); + Gparameter a2(min_it+1,a.end()); + + ex result = G_transform(pendint, a2, scale, gsyms, flag_trailing_zeros_only)* + G_transform(empty, a1, scale, gsyms, flag_trailing_zeros_only); + + result -= shuffle_G(empty, a1, a2, pendint, a, scale, gsyms, flag_trailing_zeros_only); + return result; + } + + Gparameter empty; + Gparameter::iterator changeit; + + // first term G(a_1,..,0,...,a_w;a_0) + Gparameter new_pendint = prepare_pending_integrals(pendint, a[min_it_pos]); + Gparameter new_a = a; + new_a[min_it_pos] = 0; + ex result = G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only); + if (pendint.size() > 0) { + result *= trailing_zeros_G(convert_pending_integrals_G(pendint), + pendint.front(), gsyms); + } + + // other terms + changeit = new_a.begin() + min_it_pos; + changeit = new_a.erase(changeit); + if (changeit != new_a.begin()) { + // smallest in the middle + new_pendint.push_back(*changeit); + result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), + new_pendint.front(), gsyms)* + G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only); + int buffer = *changeit; + *changeit = *min_it; + result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only); + *changeit = buffer; + new_pendint.pop_back(); + --changeit; + new_pendint.push_back(*changeit); + result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), + new_pendint.front(), gsyms)* + G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only); + *changeit = *min_it; + result -= G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only); + } else { + // smallest at the front + new_pendint.push_back(scale); + result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), + new_pendint.front(), gsyms)* + G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only); + new_pendint.back() = *changeit; + result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), + new_pendint.front(), gsyms)* + G_transform(empty, new_a, scale, gsyms, flag_trailing_zeros_only); + *changeit = *min_it; + result += G_transform(new_pendint, new_a, scale, gsyms, flag_trailing_zeros_only); + } + return result; +} + + +// shuffles the two parameter list a1 and a2 and calls G_transform for every term except +// for the one that is equal to a_old +ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2, + const Gparameter& pendint, const Gparameter& a_old, int scale, + const exvector& gsyms, bool flag_trailing_zeros_only) +{ + if (a1.size()==0 && a2.size()==0) { + // veto the one configuration we don't want + if ( a0 == a_old ) return 0; + + return G_transform(pendint, a0, scale, gsyms, flag_trailing_zeros_only); + } + + if (a2.size()==0) { + Gparameter empty; + Gparameter aa0 = a0; + aa0.insert(aa0.end(),a1.begin(),a1.end()); + return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms, flag_trailing_zeros_only); + } + + if (a1.size()==0) { + Gparameter empty; + Gparameter aa0 = a0; + aa0.insert(aa0.end(),a2.begin(),a2.end()); + return shuffle_G(aa0, empty, empty, pendint, a_old, scale, gsyms, flag_trailing_zeros_only); + } + + Gparameter a1_removed(a1.begin()+1,a1.end()); + Gparameter a2_removed(a2.begin()+1,a2.end()); + + Gparameter a01 = a0; + Gparameter a02 = a0; + + a01.push_back( a1[0] ); + a02.push_back( a2[0] ); + + return shuffle_G(a01, a1_removed, a2, pendint, a_old, scale, gsyms, flag_trailing_zeros_only) + + shuffle_G(a02, a1, a2_removed, pendint, a_old, scale, gsyms, flag_trailing_zeros_only); +} + +// handles the transformations and the numerical evaluation of G +// the parameter x, s and y must only contain numerics +static cln::cl_N +G_numeric(const std::vector& x, const std::vector& s, + const cln::cl_N& y); + +// do acceleration transformation (hoelder convolution [BBB]) +// the parameter x, s and y must only contain numerics +static cln::cl_N +G_do_hoelder(std::vector x, /* yes, it's passed by value */ + const std::vector& s, const cln::cl_N& y) +{ + cln::cl_N result; + const std::size_t size = x.size(); + for (std::size_t i = 0; i < size; ++i) + x[i] = x[i]/y; + + for (std::size_t r = 0; r <= size; ++r) { + cln::cl_N buffer(1 & r ? -1 : 1); + cln::cl_RA p(2); + bool adjustp; + do { + adjustp = false; + for (std::size_t i = 0; i < size; ++i) { + if (x[i] == cln::cl_RA(1)/p) { + p = p/2 + cln::cl_RA(3)/2; + adjustp = true; + continue; + } + } + } while (adjustp); + cln::cl_RA q = p/(p-1); + std::vector qlstx; + std::vector qlsts; + for (std::size_t j = r; j >= 1; --j) { + qlstx.push_back(cln::cl_N(1) - x[j-1]); + if (instanceof(x[j-1], cln::cl_R_ring) && realpart(x[j-1]) > 1) { + qlsts.push_back(1); + } else { + qlsts.push_back(-s[j-1]); + } + } + if (qlstx.size() > 0) { + buffer = buffer*G_numeric(qlstx, qlsts, 1/q); + } + std::vector plstx; + std::vector plsts; + for (std::size_t j = r+1; j <= size; ++j) { + plstx.push_back(x[j-1]); + plsts.push_back(s[j-1]); + } + if (plstx.size() > 0) { + buffer = buffer*G_numeric(plstx, plsts, 1/p); + } + result = result + buffer; + } + return result; +} + +class less_object_for_cl_N +{ +public: + bool operator() (const cln::cl_N & a, const cln::cl_N & b) const + { + // absolute value? + if (abs(a) != abs(b)) + return (abs(a) < abs(b)) ? true : false; + + // complex phase? + if (phase(a) != phase(b)) + return (phase(a) < phase(b)) ? true : false; + + // equal, therefore "less" is not true + return false; + } +}; + + +// convergence transformation, used for numerical evaluation of G function. +// the parameter x, s and y must only contain numerics +static cln::cl_N +G_do_trafo(const std::vector& x, const std::vector& s, + const cln::cl_N& y, bool flag_trailing_zeros_only) +{ + // sort (|x|<->position) to determine indices + typedef std::multimap sortmap_t; + sortmap_t sortmap; + std::size_t size = 0; + for (std::size_t i = 0; i < x.size(); ++i) { + if (!zerop(x[i])) { + sortmap.insert(std::make_pair(x[i], i)); + ++size; + } + } + // include upper limit (scale) + sortmap.insert(std::make_pair(y, x.size())); + + // generate missing dummy-symbols + int i = 1; + // holding dummy-symbols for the G/Li transformations + exvector gsyms; + gsyms.push_back(symbol("GSYMS_ERROR")); + cln::cl_N lastentry(0); + for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) { + if (it != sortmap.begin()) { + if (it->second < x.size()) { + if (x[it->second] == lastentry) { + gsyms.push_back(gsyms.back()); + continue; + } + } else { + if (y == lastentry) { + gsyms.push_back(gsyms.back()); + continue; + } + } + } + std::ostringstream os; + os << "a" << i; + gsyms.push_back(symbol(os.str())); + ++i; + if (it->second < x.size()) { + lastentry = x[it->second]; + } else { + lastentry = y; + } + } + + // fill position data according to sorted indices and prepare substitution list + Gparameter a(x.size()); + exmap subslst; + std::size_t pos = 1; + int scale = pos; + for (sortmap_t::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) { + if (it->second < x.size()) { + if (s[it->second] > 0) { + a[it->second] = pos; + } else { + a[it->second] = -int(pos); + } + subslst[gsyms[pos]] = numeric(x[it->second]); + } else { + scale = pos; + subslst[gsyms[pos]] = numeric(y); + } + ++pos; + } + + // do transformation + Gparameter pendint; + ex result = G_transform(pendint, a, scale, gsyms, flag_trailing_zeros_only); + // replace dummy symbols with their values + result = result.expand(); + result = result.subs(subslst).evalf(); + if (!is_a(result)) + throw std::logic_error("G_do_trafo: G_transform returned non-numeric result"); + + cln::cl_N ret = ex_to(result).to_cl_N(); + return ret; +} + +// handles the transformations and the numerical evaluation of G +// the parameter x, s and y must only contain numerics +static cln::cl_N +G_numeric(const std::vector& x, const std::vector& s, + const cln::cl_N& y) +{ + // check for convergence and necessary accelerations + bool need_trafo = false; + bool need_hoelder = false; + bool have_trailing_zero = false; + std::size_t depth = 0; + for (std::size_t i = 0; i < x.size(); ++i) { + if (!zerop(x[i])) { + ++depth; + const cln::cl_N x_y = abs(x[i]) - y; + if (instanceof(x_y, cln::cl_R_ring) && + realpart(x_y) < cln::least_negative_float(cln::float_format(Digits - 2))) + need_trafo = true; + + if (abs(abs(x[i]/y) - 1) < 0.01) + need_hoelder = true; + } + } + if (zerop(x.back())) { + have_trailing_zero = true; + need_trafo = true; + } + + if (depth == 1 && x.size() == 2 && !need_trafo) + return - Li_projection(2, y/x[1], cln::float_format(Digits)); + + // do acceleration transformation (hoelder convolution [BBB]) + if (need_hoelder && !have_trailing_zero) + return G_do_hoelder(x, s, y); + + // convergence transformation + if (need_trafo) + return G_do_trafo(x, s, y, have_trailing_zero); + + // do summation + std::vector newx; + newx.reserve(x.size()); + std::vector m; + m.reserve(x.size()); + int mcount = 1; + int sign = 1; + cln::cl_N factor = y; + for (std::size_t i = 0; i < x.size(); ++i) { + if (zerop(x[i])) { + ++mcount; + } else { + newx.push_back(factor/x[i]); + factor = x[i]; + m.push_back(mcount); + mcount = 1; + sign = -sign; + } + } + + return sign*multipleLi_do_sum(m, newx); +} + + +ex mLi_numeric(const lst& m, const lst& x) +{ + // let G_numeric do the transformation + std::vector newx; + newx.reserve(x.nops()); + std::vector s; + s.reserve(x.nops()); + cln::cl_N factor(1); + for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) { + for (int i = 1; i < *itm; ++i) { + newx.push_back(cln::cl_N(0)); + s.push_back(1); + } + const cln::cl_N xi = ex_to(*itx).to_cl_N(); + factor = factor/xi; + newx.push_back(factor); + if ( !instanceof(factor, cln::cl_R_ring) && imagpart(factor) < 0 ) { + s.push_back(-1); + } + else { + s.push_back(1); + } + } + return numeric(cln::cl_N(1 & m.nops() ? - 1 : 1)*G_numeric(newx, s, cln::cl_N(1))); +} + + +} // end of anonymous namespace + + +////////////////////////////////////////////////////////////////////// +// +// Generalized multiple polylogarithm G(x, y) and G(x, s, y) +// +// GiNaC function +// +////////////////////////////////////////////////////////////////////// + + +static ex G2_evalf(const ex& x_, const ex& y) +{ + if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) { + return G(x_, y).hold(); + } + lst x = is_a(x_) ? ex_to(x_) : lst{x_}; + if (x.nops() == 0) { + return _ex1; + } + if (x.op(0) == y) { + return G(x_, y).hold(); + } + std::vector s; + s.reserve(x.nops()); + bool all_zero = true; + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) { + if (!(*it).info(info_flags::numeric)) { + return G(x_, y).hold(); + } + if (*it != _ex0) { + all_zero = false; + } + if ( !ex_to(*it).is_real() && ex_to(*it).imag() < 0 ) { + s.push_back(-1); + } + else { + s.push_back(1); + } + } + if (all_zero) { + return pow(log(y), x.nops()) / factorial(x.nops()); + } + std::vector xv; + xv.reserve(x.nops()); + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) + xv.push_back(ex_to(*it).to_cl_N()); + cln::cl_N result = G_numeric(xv, s, ex_to(y).to_cl_N()); + return numeric(result); +} + + +static ex G2_eval(const ex& x_, const ex& y) +{ + //TODO eval to MZV or H or S or Lin + + if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) { + return G(x_, y).hold(); + } + lst x = is_a(x_) ? ex_to(x_) : lst{x_}; + if (x.nops() == 0) { + return _ex1; + } + if (x.op(0) == y) { + return G(x_, y).hold(); + } + std::vector s; + s.reserve(x.nops()); + bool all_zero = true; + bool crational = true; + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) { + if (!(*it).info(info_flags::numeric)) { + return G(x_, y).hold(); + } + if (!(*it).info(info_flags::crational)) { + crational = false; + } + if (*it != _ex0) { + all_zero = false; + } + if ( !ex_to(*it).is_real() && ex_to(*it).imag() < 0 ) { + s.push_back(-1); + } + else { + s.push_back(+1); + } + } + if (all_zero) { + return pow(log(y), x.nops()) / factorial(x.nops()); + } + if (!y.info(info_flags::crational)) { + crational = false; + } + if (crational) { + return G(x_, y).hold(); + } + std::vector xv; + xv.reserve(x.nops()); + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) + xv.push_back(ex_to(*it).to_cl_N()); + cln::cl_N result = G_numeric(xv, s, ex_to(y).to_cl_N()); + return numeric(result); +} + + +// option do_not_evalf_params() removed. +unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2). + evalf_func(G2_evalf). + eval_func(G2_eval). + overloaded(2)); +//TODO +// derivative_func(G2_deriv). +// print_func(G2_print_latex). + + +static ex G3_evalf(const ex& x_, const ex& s_, const ex& y) +{ + if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) { + return G(x_, s_, y).hold(); + } + lst x = is_a(x_) ? ex_to(x_) : lst{x_}; + lst s = is_a(s_) ? ex_to(s_) : lst{s_}; + if (x.nops() != s.nops()) { + return G(x_, s_, y).hold(); + } + if (x.nops() == 0) { + return _ex1; + } + if (x.op(0) == y) { + return G(x_, s_, y).hold(); + } + std::vector sn; + sn.reserve(s.nops()); + bool all_zero = true; + for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) { + if (!(*itx).info(info_flags::numeric)) { + return G(x_, y).hold(); + } + if (!(*its).info(info_flags::real)) { + return G(x_, y).hold(); + } + if (*itx != _ex0) { + all_zero = false; + } + if ( ex_to(*itx).is_real() ) { + if ( ex_to(*itx).is_positive() ) { + if ( *its >= 0 ) { + sn.push_back(1); + } + else { + sn.push_back(-1); + } + } else { + sn.push_back(1); + } + } + else { + if ( ex_to(*itx).imag() > 0 ) { + sn.push_back(1); + } + else { + sn.push_back(-1); + } + } + } + if (all_zero) { + return pow(log(y), x.nops()) / factorial(x.nops()); + } + std::vector xn; + xn.reserve(x.nops()); + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) + xn.push_back(ex_to(*it).to_cl_N()); + cln::cl_N result = G_numeric(xn, sn, ex_to(y).to_cl_N()); + return numeric(result); +} + + +static ex G3_eval(const ex& x_, const ex& s_, const ex& y) +{ + //TODO eval to MZV or H or S or Lin + + if ((!y.info(info_flags::numeric)) || (!y.info(info_flags::positive))) { + return G(x_, s_, y).hold(); + } + lst x = is_a(x_) ? ex_to(x_) : lst{x_}; + lst s = is_a(s_) ? ex_to(s_) : lst{s_}; + if (x.nops() != s.nops()) { + return G(x_, s_, y).hold(); + } + if (x.nops() == 0) { + return _ex1; + } + if (x.op(0) == y) { + return G(x_, s_, y).hold(); + } + std::vector sn; + sn.reserve(s.nops()); + bool all_zero = true; + bool crational = true; + for (lst::const_iterator itx = x.begin(), its = s.begin(); itx != x.end(); ++itx, ++its) { + if (!(*itx).info(info_flags::numeric)) { + return G(x_, s_, y).hold(); + } + if (!(*its).info(info_flags::real)) { + return G(x_, s_, y).hold(); + } + if (!(*itx).info(info_flags::crational)) { + crational = false; + } + if (*itx != _ex0) { + all_zero = false; + } + if ( ex_to(*itx).is_real() ) { + if ( ex_to(*itx).is_positive() ) { + if ( *its >= 0 ) { + sn.push_back(1); + } + else { + sn.push_back(-1); + } + } else { + sn.push_back(1); + } + } + else { + if ( ex_to(*itx).imag() > 0 ) { + sn.push_back(1); + } + else { + sn.push_back(-1); + } + } + } + if (all_zero) { + return pow(log(y), x.nops()) / factorial(x.nops()); + } + if (!y.info(info_flags::crational)) { + crational = false; + } + if (crational) { + return G(x_, s_, y).hold(); + } + std::vector xn; + xn.reserve(x.nops()); + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) + xn.push_back(ex_to(*it).to_cl_N()); + cln::cl_N result = G_numeric(xn, sn, ex_to(y).to_cl_N()); + return numeric(result); +} + + +// option do_not_evalf_params() removed. +// This is safe: in the code above it only matters if s_ > 0 or s_ < 0, +// s_ is allowed to be of floating type. +unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3). + evalf_func(G3_evalf). + eval_func(G3_eval). + overloaded(2)); +//TODO +// derivative_func(G3_deriv). +// print_func(G3_print_latex). + + +////////////////////////////////////////////////////////////////////// +// +// Classical polylogarithm and multiple polylogarithm Li(m,x) +// +// GiNaC function +// +////////////////////////////////////////////////////////////////////// + + +static ex Li_evalf(const ex& m_, const ex& x_) +{ + // classical polylogs + if (m_.info(info_flags::posint)) { + if (x_.info(info_flags::numeric)) { + int m__ = ex_to(m_).to_int(); + const cln::cl_N x__ = ex_to(x_).to_cl_N(); + const cln::cl_N result = Lin_numeric(m__, x__); + return numeric(result); + } else { + // try to numerically evaluate second argument + ex x_val = x_.evalf(); + if (x_val.info(info_flags::numeric)) { + int m__ = ex_to(m_).to_int(); + const cln::cl_N x__ = ex_to(x_val).to_cl_N(); + const cln::cl_N result = Lin_numeric(m__, x__); + return numeric(result); + } + } + } + // multiple polylogs + if (is_a(m_) && is_a(x_)) { + + const lst& m = ex_to(m_); + const lst& x = ex_to(x_); + if (m.nops() != x.nops()) { + return Li(m_,x_).hold(); + } + if (x.nops() == 0) { + return _ex1; + } + if ((m.op(0) == _ex1) && (x.op(0) == _ex1)) { + return Li(m_,x_).hold(); + } + + for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) { + if (!(*itm).info(info_flags::posint)) { + return Li(m_, x_).hold(); + } + if (!(*itx).info(info_flags::numeric)) { + return Li(m_, x_).hold(); + } + if (*itx == _ex0) { + return _ex0; + } + } + + return mLi_numeric(m, x); + } + + return Li(m_,x_).hold(); +} + + +static ex Li_eval(const ex& m_, const ex& x_) +{ + if (is_a(m_)) { + if (is_a(x_)) { + // multiple polylogs + const lst& m = ex_to(m_); + const lst& x = ex_to(x_); + if (m.nops() != x.nops()) { + return Li(m_,x_).hold(); + } + if (x.nops() == 0) { + return _ex1; + } + bool is_H = true; + bool is_zeta = true; + bool do_evalf = true; + bool crational = true; + for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) { + if (!(*itm).info(info_flags::posint)) { + return Li(m_,x_).hold(); + } + if ((*itx != _ex1) && (*itx != _ex_1)) { + if (itx != x.begin()) { + is_H = false; + } + is_zeta = false; + } + if (*itx == _ex0) { + return _ex0; + } + if (!(*itx).info(info_flags::numeric)) { + do_evalf = false; + } + if (!(*itx).info(info_flags::crational)) { + crational = false; + } + } + if (is_zeta) { + lst newx; + for (lst::const_iterator itx = x.begin(); itx != x.end(); ++itx) { + GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1)); + // XXX: 1 + 0.0*I is considered equal to 1. However + // the former is a not automatically converted + // to a real number. Do the conversion explicitly + // to avoid the "numeric::operator>(): complex inequality" + // exception (and similar problems). + newx.append(*itx != _ex_1 ? _ex1 : _ex_1); + } + return zeta(m_, newx); + } + if (is_H) { + ex prefactor; + lst newm = convert_parameter_Li_to_H(m, x, prefactor); + return prefactor * H(newm, x[0]); + } + if (do_evalf && !crational) { + return mLi_numeric(m,x); + } + } + return Li(m_, x_).hold(); + } else if (is_a(x_)) { + return Li(m_, x_).hold(); + } + + // classical polylogs + if (x_ == _ex0) { + return _ex0; + } + if (x_ == _ex1) { + return zeta(m_); + } + if (x_ == _ex_1) { + return (pow(2,1-m_)-1) * zeta(m_); + } + if (m_ == _ex1) { + return -log(1-x_); + } + if (m_ == _ex2) { + if (x_.is_equal(I)) { + return power(Pi,_ex2)/_ex_48 + Catalan*I; + } + if (x_.is_equal(-I)) { + return power(Pi,_ex2)/_ex_48 - Catalan*I; + } + } + if (m_.info(info_flags::posint) && x_.info(info_flags::numeric) && !x_.info(info_flags::crational)) { + int m__ = ex_to(m_).to_int(); + const cln::cl_N x__ = ex_to(x_).to_cl_N(); + const cln::cl_N result = Lin_numeric(m__, x__); + return numeric(result); + } + + return Li(m_, x_).hold(); +} + + +static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options) +{ + if (is_a(m) || is_a(x)) { + // multiple polylog + epvector seq { expair(Li(m, x), 0) }; + return pseries(rel, std::move(seq)); + } + + // classical polylog + const ex x_pt = x.subs(rel, subs_options::no_pattern); + if (m.info(info_flags::numeric) && x_pt.info(info_flags::numeric)) { + // First special case: x==0 (derivatives have poles) + if (x_pt.is_zero()) { + const symbol s; + ex ser; + // manually construct the primitive expansion + for (int i=1; i=1 (branch cut) + throw std::runtime_error("Li_series: don't know how to do the series expansion at this point!"); + } + // all other cases should be safe, by now: + throw do_taylor(); // caught by function::series() +} + + +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 (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 { + 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 << "\\mathrm{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, + 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()); + + +////////////////////////////////////////////////////////////////////// +// +// Nielsen's generalized polylogarithm S(n,p,x) +// +// helper functions +// +////////////////////////////////////////////////////////////////////// + + +// anonymous namespace for helper functions +namespace { + + +// lookup table for special Euler-Zagier-Sums (used for S_n,p(x)) +// see fill_Yn() +std::vector> Yn; +int ynsize = 0; // number of Yn[] +int ynlength = 100; // initial length of all Yn[i] + + +// This function calculates the Y_n. The Y_n are needed for the evaluation of S_{n,p}(x). +// The Y_n are basically Euler-Zagier sums with all m_i=1. They are subsums in the Z-sum +// representing S_{n,p}(x). +// The first index in Y_n corresponds to the parameter p minus one, i.e. the depth of the +// equivalent Z-sum. +// The second index in Y_n corresponds to the running index of the outermost sum in the full Z-sum +// representing S_{n,p}(x). +// The calculation of Y_n uses the values from Y_{n-1}. +void fill_Yn(int n, const cln::float_format_t& prec) +{ + const int initsize = ynlength; + //const int initsize = initsize_Yn; + cln::cl_N one = cln::cl_float(1, prec); + + if (n) { + std::vector buf(initsize); + std::vector::iterator it = buf.begin(); + std::vector::iterator itprev = Yn[n-1].begin(); + *it = (*itprev) / cln::cl_N(n+1) * one; + it++; + itprev++; + // sums with an index smaller than the depth are zero and need not to be calculated. + // calculation starts with depth, which is n+2) + for (int i=n+2; i<=initsize+n; i++) { + *it = *(it-1) + (*itprev) / cln::cl_N(i) * one; + it++; + itprev++; + } + Yn.push_back(buf); + } else { + std::vector buf(initsize); + std::vector::iterator it = buf.begin(); + *it = 1 * one; + it++; + for (int i=2; i<=initsize; i++) { + *it = *(it-1) + 1 / cln::cl_N(i) * one; + it++; + } + Yn.push_back(buf); + } + ynsize++; +} + + +// make Yn longer ... +void make_Yn_longer(int newsize, const cln::float_format_t& prec) +{ + + cln::cl_N one = cln::cl_float(1, prec); + + Yn[0].resize(newsize); + std::vector::iterator it = Yn[0].begin(); + it += ynlength; + for (int i=ynlength+1; i<=newsize; i++) { + *it = *(it-1) + 1 / cln::cl_N(i) * one; + it++; + } + + for (int n=1; n::iterator it = Yn[n].begin(); + std::vector::iterator itprev = Yn[n-1].begin(); + it += ynlength; + itprev += ynlength; + for (int i=ynlength+n+1; i<=newsize+n; i++) { + *it = *(it-1) + (*itprev) / cln::cl_N(i) * one; + it++; + itprev++; + } + } + + ynlength = newsize; +} + + +// helper function for S(n,p,x) +// [Kol] (7.2) +cln::cl_N C(int n, int p) +{ + cln::cl_N result; + + for (int k=0; k ynsize+1) { + for (int i=ynsize; i= ynlength) { + // make Yn longer + make_Yn_longer(ynlength*2, prec); + } + res = res + factor / cln::expt(cln::cl_I(i),n+1) * Yn[p-2][i-p]; // should we check it? or rely on magic number? ... + //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ... + factor = factor * xf; + i++; + } while (res != resbuf); + + return res; +} + + +// helper function for S(n,p,x) +cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec) +{ + // [Kol] (5.3) + if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) { + + 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); + + for (int s=0; s(cln::realpart(value))); + else if (!instanceof(imagpart(value), cln::cl_RA_ring)) + prec = cln::float_format(cln::the(cln::imagpart(value))); + + // [Kol] (5.3) + // the condition abs(1-value)>1 avoids an infinite recursion in the region abs(value)<=1 && abs(value)>0.95 && abs(1-value)<=1 && abs(1-value)>0.95 + // we don't care here about abs(value)<1 && real(value)>0.5, this will be taken care of in S_projection + if ((cln::realpart(value) < -0.5) || (n == 0) || ((cln::abs(value) <= 1) && (cln::abs(value) > 0.95) && (cln::abs(1-value) > 1) )) { + + 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); + + for (int s=0; s 1) { + + cln::cl_N result; + + for (int s=0; s 0.95) && (cln::abs(value-9.53) < 9.47)) { + lst m; + m.append(n+1); + for (int s=0; s(res).to_cl_N(); + } + else { + return S_projection(n, p, value, prec); + } +} + + +} // end of anonymous namespace + + +////////////////////////////////////////////////////////////////////// +// +// Nielsen's generalized polylogarithm S(n,p,x) +// +// GiNaC function +// +////////////////////////////////////////////////////////////////////// + + +static ex S_evalf(const ex& n, const ex& p, const ex& x) +{ + if (n.info(info_flags::posint) && p.info(info_flags::posint)) { + const int n_ = ex_to(n).to_int(); + const int p_ = ex_to(p).to_int(); + if (is_a(x)) { + const cln::cl_N x_ = ex_to(x).to_cl_N(); + const cln::cl_N result = S_num(n_, p_, x_); + return numeric(result); + } else { + ex x_val = x.evalf(); + if (is_a(x_val)) { + const cln::cl_N x_val_ = ex_to(x_val).to_cl_N(); + const cln::cl_N result = S_num(n_, p_, x_val_); + return numeric(result); + } + } + } + return S(n, p, x).hold(); +} + + +static ex S_eval(const ex& n, const ex& p, const ex& x) +{ + 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))) { + int n_ = ex_to(n).to_int(); + int p_ = ex_to(p).to_int(); + const cln::cl_N x_ = ex_to(x).to_cl_N(); + const cln::cl_N result = S_num(n_, p_, x_); + return numeric(result); + } + } + if (n.is_zero()) { + // [Kol] (5.3) + return pow(-log(1-x), p) / factorial(p); + } + return S(n, p, x).hold(); +} + + +static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options) +{ + if (p == _ex1) { + return Li(n+1, x).series(rel, order, options); + } + + const ex x_pt = x.subs(rel, subs_options::no_pattern); + if (n.info(info_flags::posint) && p.info(info_flags::posint) && x_pt.info(info_flags::numeric)) { + // First special case: x==0 (derivatives have poles) + if (x_pt.is_zero()) { + const symbol s; + ex ser; + // manually construct the primitive expansion + // subsum = Euler-Zagier-Sum is needed + // dirty hack (slow ...) calculation of subsum: + std::vector presubsum, subsum; + subsum.push_back(0); + for (int i=1; i=1 (branch cut) + throw std::runtime_error("S_series: don't know how to do the series expansion at this point!"); + } + // all other cases should be safe, by now: + throw do_taylor(); // caught by function::series() +} + + +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 (n > 0) { + return S(n-1, p, x) / x; + } else { + 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 << "\\mathrm{S}_{"; + n.print(c); + c.s << ","; + p.print(c); + c.s << "}("; + x.print(c); + c.s << ")"; +} + + +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()); + + +////////////////////////////////////////////////////////////////////// +// +// Harmonic polylogarithm H(m,x) +// +// helper functions +// +////////////////////////////////////////////////////////////////////// + + +// 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 +// 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 (std::size_t 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); + + 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 (std::size_t 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) override + { + 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(); + std::size_t 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(); + int signum = 1; + pf = _ex1; + res.append(*itm); + itm++; + while (itx != x.end()) { + GINAC_ASSERT((*itx == _ex1) || (*itx == _ex_1)); + // XXX: 1 + 0.0*I is considered equal to 1. However the former + // is not automatically converted to a real number. + // Do the conversion explicitly to avoid the + // "numeric::operator>(): complex inequality" exception. + signum *= (*itx != _ex_1) ? 1 : -1; + 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 = lst{h2.op(0).op(0)}; + } + } + for (std::size_t i=0; i<=hlong.nops(); i++) { + lst newparameter; + std::size_t j=0; + for (; j(e)) { + return e.map(*this); + } + + if (is_a(e)) { + + ex result = 1; + ex firstH; + lst Hlst; + for (std::size_t 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 (std::size_t i=1; i(e)) { + name = ex_to(e).get_name(); + } + if (name == "H") { + h = e; + } else { + for (std::size_t 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(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{ex(0)},1/arg).hold()); + } +} - cln::cl_N res; - cln::cl_N resbuf; - cln::cl_N factor = cln::expt(xf, p); - int i = p; + +// do integration [ReV] (49) +// put parameter 1 in front of existing parameters +ex trafo_H_prepend_one(const ex& e, const ex& arg) +{ + ex h; + std::string name; + if (is_a(e)) { + name = ex_to(e).get_name(); + } + if (name == "H") { + h = e; + } else { + for (std::size_t 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()); + } else { + return e * H(lst{ex(1)},1-arg).hold(); + } +} + + +// 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 h; + std::string name; + if (is_a(e)) { + name = ex_to(e).get_name(); + } + if (name == "H") { + h = e; + } else { + for (std::size_t 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); + 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{ex(-1)}); + return (e * (addzeta - H(lst{ex(-1)},1/arg).hold())).expand(); + } +} + + +// 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 (std::size_t i = 0; i < e.nops(); i++) { + if (is_a(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{ex(-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 (std::size_t i = 0; i < e.nops(); i++) { + if (is_a(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{ex(1)},(1-arg)/(1+arg)).hold()).expand(); + } +} + + +// do x -> 1-x transformation +struct map_trafo_H_1mx : public map_function +{ + ex operator()(const ex& e) override + { + 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); + + // special cases if all parameters are either 0, 1 or -1 + bool allthesame = true; + if (parameter.op(0) == 0) { + for (std::size_t i = 1; i < parameter.nops(); i++) { + if (parameter.op(i) != 0) { + allthesame = false; + break; + } + } + if (allthesame) { + lst newparameter; + for (int i=parameter.nops(); i>0; i--) { + newparameter.append(1); + } + return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold(); + } + } else if (parameter.op(0) == -1) { + throw std::runtime_error("map_trafo_H_1mx: cannot handle weights equal -1!"); + } else { + for (std::size_t i = 1; i < parameter.nops(); i++) { + if (parameter.op(i) != 1) { + allthesame = false; + break; + } + } + if (allthesame) { + lst newparameter; + for (int i=parameter.nops(); i>0; i--) { + newparameter.append(0); + } + return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold(); + } + } + + lst newparameter = parameter; + newparameter.remove_first(); + + if (parameter.op(0) == 0) { + + // leading zero + ex res = convert_H_to_zeta(parameter); + //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 (std::size_t i = 0; i < buffer.nops(); i++) { + res -= trafo_H_prepend_one(buffer.op(i), arg); + } + } else { + res -= trafo_H_prepend_one(buffer, arg); + } + return res; + + } else { + + // leading one + map_trafo_H_1mx recursion; + map_trafo_H_mult unify; + ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold(); + std::size_t firstzero = 0; + while (parameter.op(firstzero) == 1) { + firstzero++; + } + for (std::size_t i = firstzero-1; i < parameter.nops()-1; i++) { + lst newparameter; + std::size_t j=0; + for (; j<=i; j++) { + newparameter.append(parameter[j+1]); + } + newparameter.append(1); + for (; j 1/x transformation +struct map_trafo_H_1overx : public map_function +{ + ex operator()(const ex& e) override + { + 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); + + // special cases if all parameters are either 0, 1 or -1 + bool allthesame = true; + if (parameter.op(0) == 0) { + for (std::size_t i = 1; i < parameter.nops(); i++) { + if (parameter.op(i) != 0) { + allthesame = false; + break; + } + } + if (allthesame) { + return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold(); + } + } else if (parameter.op(0) == -1) { + for (std::size_t i = 1; i < parameter.nops(); i++) { + if (parameter.op(i) != -1) { + allthesame = false; + break; + } + } + if (allthesame) { + map_trafo_H_mult unify; + return unify((pow(H(lst{ex(-1)},1/arg).hold() - H(lst{ex(0)},1/arg).hold(), parameter.nops()) + / factorial(parameter.nops())).expand()); + } + } else { + for (std::size_t i = 1; i < parameter.nops(); i++) { + if (parameter.op(i) != 1) { + allthesame = false; + break; + } + } + if (allthesame) { + map_trafo_H_mult unify; + return unify((pow(H(lst{ex(1)},1/arg).hold() + H(lst{ex(0)},1/arg).hold() + H_polesign, parameter.nops()) + / factorial(parameter.nops())).expand()); + } + } + + lst newparameter = parameter; + newparameter.remove_first(); + + if (parameter.op(0) == 0) { + + // leading zero + ex res = convert_H_to_zeta(parameter); + map_trafo_H_1overx recursion; + ex buffer = recursion(H(newparameter, arg).hold()); + if (is_a(buffer)) { + for (std::size_t i = 0; i < buffer.nops(); i++) { + res += trafo_H_1tx_prepend_zero(buffer.op(i), arg); + } + } else { + res += trafo_H_1tx_prepend_zero(buffer, arg); + } + return res; + + } else if (parameter.op(0) == -1) { + + // leading negative one + ex res = convert_H_to_zeta(parameter); + map_trafo_H_1overx recursion; + ex buffer = recursion(H(newparameter, arg).hold()); + if (is_a(buffer)) { + for (std::size_t i = 0; i < buffer.nops(); i++) { + res += trafo_H_1tx_prepend_zero(buffer.op(i), arg) - trafo_H_1tx_prepend_minusone(buffer.op(i), arg); + } + } else { + res += trafo_H_1tx_prepend_zero(buffer, arg) - trafo_H_1tx_prepend_minusone(buffer, arg); + } + return res; + + } else { + + // leading one + map_trafo_H_1overx recursion; + map_trafo_H_mult unify; + ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold(); + std::size_t firstzero = 0; + while (parameter.op(firstzero) == 1) { + firstzero++; + } + for (std::size_t i = firstzero-1; i < parameter.nops() - 1; i++) { + lst newparameter; + std::size_t j = 0; + for (; j<=i; j++) { + newparameter.append(parameter[j+1]); + } + newparameter.append(1); + for (; j (1-x)/(1+x) transformation +struct map_trafo_H_1mxt1px : public map_function +{ + ex operator()(const ex& e) override + { + 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); + + // special cases if all parameters are either 0, 1 or -1 + bool allthesame = true; + if (parameter.op(0) == 0) { + for (std::size_t i = 1; i < parameter.nops(); i++) { + if (parameter.op(i) != 0) { + allthesame = false; + break; + } + } + if (allthesame) { + map_trafo_H_mult unify; + return unify((pow(-H(lst{ex(1)},(1-arg)/(1+arg)).hold() - H(lst{ex(-1)},(1-arg)/(1+arg)).hold(), parameter.nops()) + / factorial(parameter.nops())).expand()); + } + } else if (parameter.op(0) == -1) { + for (std::size_t i = 1; i < parameter.nops(); i++) { + if (parameter.op(i) != -1) { + allthesame = false; + break; + } + } + if (allthesame) { + map_trafo_H_mult unify; + return unify((pow(log(2) - H(lst{ex(-1)},(1-arg)/(1+arg)).hold(), parameter.nops()) + / factorial(parameter.nops())).expand()); + } + } else { + for (std::size_t i = 1; i < parameter.nops(); i++) { + if (parameter.op(i) != 1) { + allthesame = false; + break; + } + } + if (allthesame) { + map_trafo_H_mult unify; + return unify((pow(-log(2) - H(lst{ex(0)},(1-arg)/(1+arg)).hold() + H(lst{ex(-1)},(1-arg)/(1+arg)).hold(), parameter.nops()) + / factorial(parameter.nops())).expand()); + } + } + + lst newparameter = parameter; + newparameter.remove_first(); + + 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 (std::size_t i = 0; i < buffer.nops(); i++) { + res -= trafo_H_1mxt1px_prepend_one(buffer.op(i), arg) + trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg); + } + } else { + res -= trafo_H_1mxt1px_prepend_one(buffer, arg) + trafo_H_1mxt1px_prepend_minusone(buffer, arg); + } + return res; + + } else if (parameter.op(0) == -1) { + + // 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 (std::size_t i = 0; i < buffer.nops(); i++) { + res -= trafo_H_1mxt1px_prepend_minusone(buffer.op(i), arg); + } + } else { + res -= trafo_H_1mxt1px_prepend_minusone(buffer, arg); + } + return res; + + } else { + + // leading one + map_trafo_H_1mxt1px recursion; + map_trafo_H_mult unify; + ex res = H(lst{ex(1)}, arg).hold() * H(newparameter, arg).hold(); + std::size_t firstzero = 0; + while (parameter.op(firstzero) == 1) { + firstzero++; + } + for (std::size_t i = firstzero - 1; i < parameter.nops() - 1; i++) { + lst newparameter; + std::size_t j=0; + for (; j<=i; j++) { + newparameter.append(parameter[j+1]); + } + newparameter.append(1); + for (; j& m, const cln::cl_N& x) +{ + const int j = m.size(); + + std::vector t(j); + + cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); + cln::cl_N factor = cln::expt(x, j) * one; + cln::cl_N t0buf; + int q = 0; do { - resbuf = res; - if (i-p >= ynlength) { - // make Yn longer - make_Yn_longer(ynlength*2, prec); + t0buf = t[0]; + q++; + t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),m[j-1]); + for (int k=j-2; k>=1; k--) { + t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]); } - res = res + factor / cln::expt(cln::cl_I(i),n+1) * Yn[p-2][i-p]; // should we check it? or rely on magic number? ... - //res = res + factor / cln::expt(cln::cl_I(i),n+1) * (*it); // should we check it? or rely on magic number? ... - factor = factor * xf; - i++; - } while (res != resbuf); - - return res; + 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]; +} + + +} // end of anonymous namespace + + +////////////////////////////////////////////////////////////////////// +// +// Harmonic polylogarithm H(m,x) +// +// GiNaC function +// +////////////////////////////////////////////////////////////////////// + + +static ex H_evalf(const ex& x1, const ex& x2) +{ + if (is_a(x1)) { + + cln::cl_N x; + if (is_a(x2)) { + x = ex_to(x2).to_cl_N(); + } else { + ex x2_val = x2.evalf(); + if (is_a(x2_val)) { + x = ex_to(x2_val).to_cl_N(); + } + } + + for (std::size_t i = 0; i < x1.nops(); i++) { + if (!x1.op(i).info(info_flags::integer)) { + return H(x1, x2).hold(); + } + } + if (x1.nops() < 1) { + return H(x1, x2).hold(); + } + + 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 + bool has_minus_one = false; + 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); + has_minus_one = true; + } else { + m.append(*it); + } + } + + // 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)); + } + } + + symbol xtemp("xtemp"); + ex res = 1; + + // ensure that the realpart of the argument is positive + if (cln::realpart(x) < 0) { + x = -x; + for (std::size_t i = 0; i < m.nops(); i++) { + if (m.op(i) != 0) { + m.let_op(i) = -m.op(i); + res *= -1; + } + } + } + + // x -> 1/x + if (cln::abs(x) >= 2.0) { + map_trafo_H_1overx trafo; + res *= trafo(H(m, xtemp).hold()); + if (cln::imagpart(x) <= 0) { + res = res.subs(H_polesign == -I*Pi); + } else { + res = res.subs(H_polesign == I*Pi); + } + return res.subs(xtemp == numeric(x)).evalf(); + } + + // check transformations for 0.95 <= |x| < 2.0 + + // |(1-x)/(1+x)| < 0.9 -> circular area with center=9.53+0i and radius=9.47 + if (cln::abs(x-9.53) <= 9.47) { + // x -> (1-x)/(1+x) + map_trafo_H_1mxt1px trafo; + res *= trafo(H(m, xtemp).hold()); + } else { + // x -> 1-x + if (has_minus_one) { + map_trafo_H_convert_to_Li filter; + return filter(H(m, numeric(x)).hold()).evalf(); + } + map_trafo_H_1mx trafo; + res *= trafo(H(m, xtemp).hold()); + } + + return res.subs(xtemp == numeric(x)).evalf(); + } + + return H(x1,x2).hold(); +} + + +static ex H_eval(const ex& m_, const ex& x) +{ + 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 { + 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(); + } + } + 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& m, const ex& x, const relational& rel, int order, unsigned options) +{ + epvector seq { expair(H(m, x), 0) }; + return pseries(rel, std::move(seq)); +} + + +static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param < 2); + if (deriv_param == 0) { + return _ex0; + } + lst m; + if (is_a(m_)) { + m = ex_to(m_); + } else { + 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; + } } -// helper function for S(n,p,x) -static cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec) +static void H_print_latex(const ex& m_, const ex& x, const print_context& c) { - // [Kol] (5.3) - if (cln::abs(cln::realpart(x)) > cln::cl_F("0.5")) { + lst m; + if (is_a(m_)) { + m = ex_to(m_); + } else { + m = lst{m_}; + } + c.s << "\\mathrm{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 << ")"; +} - 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); - for (int s=0; s(H_print_latex). + do_not_evalf_params()); - return result; + +// 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) +{ + 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())); } - - return S_series(n, p, x, prec); } -// helper function for S(n,p,x) -static numeric S_num(int n, int p, const numeric& x) -{ - if (x == 1) { - if (n == 1) { - // [Kol] (2.22) with (2.21) - return cln::zeta(p+1); - } +////////////////////////////////////////////////////////////////////// +// +// Multiple zeta values zeta(x) and zeta(x,s) +// +// helper functions +// +////////////////////////////////////////////////////////////////////// - if (p == 1) { - // [Kol] (2.22) - return cln::zeta(n+1); - } - // [Kol] (9.1) - cln::cl_N result; - for (int nu=0; nu& a, const std::vector& b, std::vector& c) +{ + const int size = a.size(); + for (int n=0; n(cln::realpart(value))); - else if (!x.imag().is_rational()) - prec = cln::float_format(cln::the(cln::imagpart(value))); +// [Cra] section 4 +static void initcX(std::vector& crX, + const std::vector& s, + const int L2) +{ + std::vector crB(L2 + 1); + for (int i=0; i<=L2; i++) + crB[i] = bernoulli(i).to_cl_N() / cln::factorial(i); - // [Kol] (5.3) - if (cln::realpart(value) < -0.5) { + int Sm = 0; + int Smp1 = 0; + std::vector> crG(s.size() - 1, std::vector(L2 + 1)); + for (int m=0; m < (int)s.size() - 1; m++) { + Sm += s[m]; + Smp1 = Sm + s[m+1]; + for (int i = 0; i <= L2; i++) + crG[m][i] = cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2); + } - 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); + crX = crB; - for (int s=0; s Xbuf(L2 + 1); + for (int i = 0; i <= L2; i++) + Xbuf[i] = crX[i] * crG[m][i]; - return result; - + halfcyclic_convolute(Xbuf, crB, crX); } - // [Kol] (5.12) - if (cln::abs(value) > 1) { - - cln::cl_N result; +} - for (int s=0; s& crX) +{ + cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); + cln::cl_N factor = cln::expt(lambda, Sqk); + cln::cl_N res = factor / Sqk * crX[0] * one; + cln::cl_N resbuf; + int N = 0; + do { + resbuf = res; + factor = factor * lambda; + N++; + res = res + crX[N] * factor / (N+Sqk); + } while ((res != resbuf) || cln::zerop(crX[N])); + return res; +} - result = result + cln::expt(cln::cl_I(-1),p) * res2; - return result; - } - else { - return S_projection(n, p, value, prec); +// [Cra] section 4 +static void calc_f(std::vector>& f_kj, + const int maxr, const int L1) +{ + cln::cl_N t0, t1, t2, t3, t4; + int i, j, k; + std::vector>::iterator it = f_kj.begin(); + cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); + + t0 = cln::exp(-lambda); + t2 = 1; + for (k=1; k<=L1; k++) { + t1 = k * lambda; + t2 = t0 * t2; + for (j=1; j<=maxr; j++) { + t3 = 1; + t4 = 1; + for (i=2; i<=j; i++) { + t4 = t4 * (j-i+1); + t3 = t1 * t3 + t4; + } + (*it).push_back(t2 * t3 * cln::expt(cln::cl_I(k),-j) * one); + } + it++; } } -// helper function for multiple polylogarithm -static cln::cl_N numeric_zsum(int n, std::vector& x, std::vector& m) +// [Cra] (3.1) +static cln::cl_N crandall_Z(const std::vector& s, + const std::vector>& f_kj) { - cln::cl_N res; - if (x.empty()) { - return 1; - } - for (int i=1; i::iterator be; - std::vector::iterator en; - be = x.begin(); - be++; - en = x.end(); - std::vector xbuf(be, en); - be = m.begin(); - be++; - en = m.end(); - std::vector mbuf(be, en); - res = res + cln::expt(x[0],i) / cln::expt(i,m[0]) * numeric_zsum(i, xbuf, mbuf); + const int j = s.size(); + + if (j == 1) { + cln::cl_N t0; + cln::cl_N t0buf; + int q = 0; + do { + t0buf = t0; + q++; + t0 = t0 + f_kj[q+j-2][s[0]-1]; + } while (t0 != t0buf); + + return t0 / cln::factorial(s[0]-1); } - return res; + + std::vector t(j); + + cln::cl_N t0buf; + int q = 0; + do { + t0buf = t[0]; + q++; + t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]); + for (int k=j-2; k>=1; k--) { + t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), s[k]); + } + t[0] = t[0] + t[1] * f_kj[q+j-2][s[0]-1]; + } while (t[0] != t0buf); + + return t[0] / cln::factorial(s[0]-1); } -// helper function for harmonic polylogarithm -static cln::cl_N numeric_harmonic(int n, std::vector& m) +// [Cra] (2.4) +cln::cl_N zeta_do_sum_Crandall(const std::vector& s) { + std::vector r = s; + const int j = r.size(); + + std::size_t L1; + + // decide on maximal size of f_kj for crandall_Z + if (Digits < 50) { + L1 = 150; + } else { + L1 = Digits * 3 + j*2; + } + + std::size_t L2; + // decide on maximal size of crX for crandall_Y + if (Digits < 38) { + L2 = 63; + } else if (Digits < 86) { + L2 = 127; + } else if (Digits < 192) { + L2 = 255; + } else if (Digits < 394) { + L2 = 511; + } else if (Digits < 808) { + L2 = 1023; + } else { + L2 = 2047; + } + cln::cl_N res; - if (m.empty()) { - return 1; + + int maxr = 0; + int S = 0; + for (int i=0; i maxr) { + maxr = r[i]; + } } - for (int i=1; i::iterator be; - std::vector::iterator en; - be = m.begin(); - be++; - en = m.end(); - std::vector mbuf(be, en); - res = res + cln::recip(cln::expt(i,m[0])) * numeric_harmonic(i, mbuf); + + std::vector> f_kj(L1); + calc_f(f_kj, maxr, L1); + + const cln::cl_N r0factorial = cln::factorial(r[0]-1); + + std::vector rz; + int skp1buf; + int Srun = S; + for (int k=r.size()-1; k>0; k--) { + + rz.insert(rz.begin(), r.back()); + skp1buf = rz.front(); + Srun -= skp1buf; + r.pop_back(); + + std::vector crX; + initcX(crX, r, L2); + + for (int q=0; q crX; + initcX(crX, rz, L2); + + res = (res + crandall_Y_loop(S-j, crX)) / r0factorial + + crandall_Z(rz, f_kj); + return res; } -///////////////////////////// -// end of helper functions // -///////////////////////////// +cln::cl_N zeta_do_sum_simple(const std::vector& r) +{ + const int j = r.size(); + // buffer for subsums + std::vector t(j); + cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); -// Polylogarithm and multiple polylogarithm + cln::cl_N t0buf; + int q = 0; + do { + t0buf = t[0]; + q++; + t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]); + for (int k=j-2; k>=0; k--) { + t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]); + } + } while (t[0] != t0buf); -static ex Li_eval(const ex& x1, const ex& x2) -{ - if (x2.is_zero()) { - return 0; - } - else { - if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) - return Li_num(ex_to(x1).to_int(), ex_to(x2)); - return Li(x1,x2).hold(); - } + return t[0]; } -static ex Li_evalf(const ex& x1, const ex& x2) + +// does Hoelder convolution. see [BBB] (7.0) +cln::cl_N zeta_do_Hoelder_convolution(const std::vector& m_, const std::vector& s_) { - // classical polylogs - if (is_a(x1) && is_a(x2)) { - return Li_num(ex_to(x1).to_int(), ex_to(x2)); - } - // multiple polylogs - else if (is_a(x1) && is_a(x2)) { - for (int i=0; i(x1.op(i))) - return Li(x1,x2).hold(); - if (!is_a(x2.op(i))) - return Li(x1,x2).hold(); - if (x2.op(i) >= 1) - return Li(x1,x2).hold(); + // prepare parameters + // holds Li arguments in [BBB] notation + std::vector s = s_; + std::vector m_p = m_; + std::vector m_q; + // holds Li arguments in nested sums notation + std::vector s_p(s.size(), cln::cl_N(1)); + s_p[0] = s_p[0] * cln::cl_N("1/2"); + // convert notations + int sig = 1; + for (std::size_t i = 0; i < s_.size(); i++) { + if (s_[i] < 0) { + sig = -sig; + s_p[i] = -s_p[i]; } + s[i] = sig * std::abs(s[i]); + } + std::vector s_q; + cln::cl_N signum = 1; - cln::cl_N m_1 = ex_to(x1.op(x1.nops()-1)).to_cl_N(); - cln::cl_N x_1 = ex_to(x2.op(x2.nops()-1)).to_cl_N(); - std::vector x; - std::vector m; - const int nops = ex_to(x1.nops()).to_int(); - for (int i=nops-2; i>=0; i--) { - m.push_back(ex_to(x1.op(i)).to_cl_N()); - x.push_back(ex_to(x2.op(i)).to_cl_N()); - } + // first term + cln::cl_N res = multipleLi_do_sum(m_p, s_p); + + // middle terms + do { - cln::cl_N res; - cln::cl_N resbuf; - for (int i=nops; true; i++) { - resbuf = res; - res = res + cln::expt(x_1,i) / cln::expt(i,m_1) * numeric_zsum(i, x, m); - if (cln::zerop(res-resbuf)) - break; + // change parameters + if (s.front() > 0) { + if (m_p.front() == 1) { + m_p.erase(m_p.begin()); + s_p.erase(s_p.begin()); + if (s_p.size() > 0) { + s_p.front() = s_p.front() * cln::cl_N("1/2"); + } + s.erase(s.begin()); + m_q.front()++; + } else { + m_p.front()--; + m_q.insert(m_q.begin(), 1); + if (s_q.size() > 0) { + s_q.front() = s_q.front() * 2; + } + s_q.insert(s_q.begin(), cln::cl_N("1/2")); + } + } else { + if (m_p.front() == 1) { + m_p.erase(m_p.begin()); + cln::cl_N spbuf = s_p.front(); + s_p.erase(s_p.begin()); + if (s_p.size() > 0) { + s_p.front() = s_p.front() * spbuf; + } + s.erase(s.begin()); + m_q.insert(m_q.begin(), 1); + if (s_q.size() > 0) { + s_q.front() = s_q.front() * 4; + } + s_q.insert(s_q.begin(), cln::cl_N("1/4")); + signum = -signum; + } else { + m_p.front()--; + m_q.insert(m_q.begin(), 1); + if (s_q.size() > 0) { + s_q.front() = s_q.front() * 2; + } + s_q.insert(s_q.begin(), cln::cl_N("1/2")); + } } - return numeric(res); + // exiting the loop + if (m_p.size() == 0) break; - } + res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q); - return Li(x1,x2).hold(); -} + } while (true); -static ex Li_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options) -{ - epvector seq; - seq.push_back(expair(Li(x1,x2), 0)); - return pseries(rel,seq); + // last term + res = res + signum * multipleLi_do_sum(m_q, s_q); + + return res; } -REGISTER_FUNCTION(Li, eval_func(Li_eval).evalf_func(Li_evalf).do_not_evalf_params().series_func(Li_series)); + +} // end of anonymous namespace -// Nielsen's generalized polylogarithm +////////////////////////////////////////////////////////////////////// +// +// Multiple zeta values zeta(x) +// +// GiNaC function +// +////////////////////////////////////////////////////////////////////// -static ex S_eval(const ex& x1, const ex& x2, const ex& x3) + +static ex zeta1_evalf(const ex& x) { - if (x2 == 1) { - return Li(x1+1,x3); + if (is_exactly_a(x) && (x.nops()>1)) { + + // multiple zeta value + const int count = x.nops(); + const lst& xlst = ex_to(x); + std::vector r(count); + + // check parameters and convert them + lst::const_iterator it1 = xlst.begin(); + std::vector::iterator it2 = r.begin(); + do { + if (!(*it1).info(info_flags::posint)) { + return zeta(x).hold(); + } + *it2 = ex_to(*it1).to_int(); + it1++; + it2++; + } while (it2 != r.end()); + + // check for divergence + if (r[0] == 1) { + return zeta(x).hold(); + } + + // decide on summation algorithm + // this is still a bit clumsy + int limit = (Digits>17) ? 10 : 6; + if ((r[0] < limit) || ((count > 3) && (r[1] < limit/2))) { + return numeric(zeta_do_sum_Crandall(r)); + } else { + return numeric(zeta_do_sum_simple(r)); + } } - 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)); + + // single zeta value + if (is_exactly_a(x) && (x != 1)) { + try { + return zeta(ex_to(x)); + } catch (const dunno &e) { } } - return S(x1,x2,x3).hold(); + + return zeta(x).hold(); } -static ex S_evalf(const ex& x1, const ex& x2, const ex& x3) + +static ex zeta1_eval(const ex& m) { - if (is_a(x1) && is_a(x2) && is_a(x3)) { - if ((x3 == -1) && (x2 != 1)) { - // no formula to evaluate this ... sorry -// return S(x1,x2,x3).hold(); + if (is_exactly_a(m)) { + if (m.nops() == 1) { + return zeta(m.op(0)); } - return S_num(ex_to(x1).to_int(), ex_to(x2).to_int(), ex_to(x3)); + return zeta(m).hold(); } - return S(x1,x2,x3).hold(); -} -static ex S_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options) -{ - epvector seq; - seq.push_back(expair(S(x1,x2,x3), 0)); - return pseries(rel,seq); + 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_p)) { + return zeta(m).hold(); + } + if (y.info(info_flags::posint)) { + if (y.info(info_flags::odd)) { + return zeta(m).hold(); + } else { + return abs(bernoulli(y)) * pow(Pi, y) * pow(*_num2_p, y-(*_num1_p)) / factorial(y); + } + } else { + if (y.info(info_flags::odd)) { + return -bernoulli((*_num1_p)-y) / ((*_num1_p)-y); + } else { + return _ex0; + } + } + } + // zeta(float) + if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) { + return zeta1_evalf(m); + } + } + return zeta(m).hold(); } -REGISTER_FUNCTION(S, eval_func(S_eval).evalf_func(S_evalf).do_not_evalf_params().series_func(S_series)); - - -// Harmonic polylogarithm -static ex H_eval(const ex& x1, const ex& x2) +static ex zeta1_deriv(const ex& m, unsigned deriv_param) { - if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) { - return H(x1,x2).evalf(); + GINAC_ASSERT(deriv_param==0); + + if (is_exactly_a(m)) { + return _ex0; + } else { + return zetaderiv(_ex1, m); } - return H(x1,x2).hold(); } -static ex H_evalf(const ex& x1, const ex& x2) + +static void zeta1_print_latex(const ex& m_, const print_context& c) { - if (is_a(x1) && is_a(x2)) { - for (int i=0; i(x1.op(i))) - return H(x1,x2).hold(); - } - if (x2 >= 1) { - return H(x1,x2).hold(); + 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 << ")"; +} - cln::cl_N m_1 = ex_to(x1.op(x1.nops()-1)).to_cl_N(); - cln::cl_N x_1 = ex_to(x2).to_cl_N(); - std::vector m; - const int nops = ex_to(x1.nops()).to_int(); - for (int i=nops-2; i>=0; i--) { - m.push_back(ex_to(x1.op(i)).to_cl_N()); - } - cln::cl_N res; - cln::cl_N resbuf; - for (int i=nops; true; i++) { - resbuf = res; - res = res + cln::expt(x_1,i) / cln::expt(i,m_1) * numeric_harmonic(i, m); - if (cln::zerop(res-resbuf)) - break; - } +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)); - return numeric(res); - } +////////////////////////////////////////////////////////////////////// +// +// Alternating Euler sum zeta(x,s) +// +// GiNaC function +// +////////////////////////////////////////////////////////////////////// - return H(x1,x2).hold(); -} -static ex H_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options) +static ex zeta2_evalf(const ex& x, const ex& s) { - epvector seq; - seq.push_back(expair(H(x1,x2), 0)); - return pseries(rel,seq); -} + if (is_exactly_a(x)) { -REGISTER_FUNCTION(H, eval_func(H_eval).evalf_func(H_evalf).do_not_evalf_params().series_func(H_series)); + // alternating Euler sum + const int count = x.nops(); + const lst& xlst = ex_to(x); + const lst& slst = ex_to(s); + std::vector xi(count); + std::vector si(count); + // check parameters and convert them + lst::const_iterator it_xread = xlst.begin(); + lst::const_iterator it_sread = slst.begin(); + std::vector::iterator it_xwrite = xi.begin(); + std::vector::iterator it_swrite = si.begin(); + do { + if (!(*it_xread).info(info_flags::posint)) { + return zeta(x, s).hold(); + } + *it_xwrite = ex_to(*it_xread).to_int(); + if (*it_sread > 0) { + *it_swrite = 1; + } else { + *it_swrite = -1; + } + it_xread++; + it_sread++; + it_xwrite++; + it_swrite++; + } while (it_xwrite != xi.end()); -// Multiple zeta value + // check for divergence + if ((xi[0] == 1) && (si[0] == 1)) { + return zeta(x, s).hold(); + } -static ex mZeta_eval(const ex& x1) -{ - return mZeta(x1).hold(); + // use Hoelder convolution + return numeric(zeta_do_Hoelder_convolution(xi, si)); + } + + return zeta(x, s).hold(); } -static ex mZeta_evalf(const ex& x1) + +static ex zeta2_eval(const ex& m, const ex& s_) { - if (is_a(x1)) { - for (int i=0; i(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; + } + return zeta(m, s_).hold(); } + return zeta(m); + } else if (s_.info(info_flags::positive)) { + return zeta(m); + } - const int j = x1.nops(); - - std::vector r(j); - for (int i=0; i(x1.op(i)).to_int(); - } + return zeta(m, s_).hold(); +} - // check for divergence - if (r[0] == 1) { - return mZeta(x1).hold(); - } - - // buffer for subsums - std::vector t(j); - cln::cl_F one = cln::cl_float(1, cln::float_format(Digits)); - cln::cl_N t0buf; - int q = 0; - do { - t0buf = t[0]; - q++; - t[j-1] = t[j-1] + one / cln::expt(cln::cl_I(q),r[j-1]); - for (int k=j-2; k>=0; k--) { - t[k] = t[k] + one * t[k+1] / cln::expt(cln::cl_I(q+j-1-k), r[k]); - } - } while (t[0] != t0buf); +static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param==0); - return numeric(t[0]); + if (is_exactly_a(m)) { + return _ex0; + } else { + if ((is_exactly_a(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) { + return zetaderiv(_ex1, m); + } + return _ex0; } - - return mZeta(x1).hold(); } -static ex mZeta_series(const ex& x1, const relational& rel, int order, unsigned options) + +static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c) { - epvector seq; - seq.push_back(expair(mZeta(x1), 0)); - return pseries(rel,seq); + 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 << ")"; } -REGISTER_FUNCTION(mZeta, eval_func(mZeta_eval).evalf_func(mZeta_evalf).do_not_evalf_params().series_func(mZeta_series)); + +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