X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_nstdsums.cpp;h=22deea9e5baf0c854765ac8b5b4fd010b8f6cb92;hp=879bb639902204eac3dd5cb3d530c488ce3d505a;hb=ffad02322624ab79fdad1a23a3aa83cd67376151;hpb=bab935d3cbaccc2f1a77d16d712271063d085d42 diff --git a/ginac/inifcns_nstdsums.cpp b/ginac/inifcns_nstdsums.cpp index 879bb639..22deea9e 100644 --- a/ginac/inifcns_nstdsums.cpp +++ b/ginac/inifcns_nstdsums.cpp @@ -1,28 +1,41 @@ /** @file inifcns_nstdsums.cpp * * Implementation of some special functions that have a representation as nested sums. + * * The functions are: * classical polylogarithm Li(n,x) - * multiple polylogarithm Li(lst(n_1,...,n_k),lst(x_1,...,x_k) + * 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)) + * harmonic polylogarithm H(n,x) or H(lst(n_1,...,n_k),x) + * multiple zeta value zeta(n) or zeta(lst(n_1,...,n_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 except for S(n,p,-1) when p is not unit (no formula yet - * to tackle these points). And of course, there is still room for speed optimizations ;-). - * - 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 + * + * - 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 + * + * - The order of parameters and arguments of H, Li and zeta is defined according to their order in the + * nested sums representation. + * + * - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in + * the whole complex plane. Multiple polylogarithms evaluate only if each argument x_i is smaller than + * one. The parameters for every function (n, p or n_i) must be positive integers. + * + * - The calculation of classical polylogarithms is speed up by using Bernoulli numbers and + * look-up tables. S uses look-up tables as well. The zeta function applies the algorithm in + * [Cra] for speed up. + * + * - The functions have no series expansion as nested sums. 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. + * * - 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. * */ @@ -49,49 +62,261 @@ #include #include "inifcns.h" + +#include "add.h" +#include "constant.h" #include "lst.h" +#include "mul.h" #include "numeric.h" #include "operators.h" +#include "power.h" +#include "pseries.h" #include "relational.h" +#include "symbol.h" +#include "utils.h" +#include "wildcard.h" namespace GiNaC { - -////////////////////// -// helper functions // -////////////////////// +////////////////////////////////////////////////////////////////////// +// +// Classical polylogarithm Li +// +// helper functions +// +////////////////////////////////////////////////////////////////////// -// helper function for classical polylog Li -static cln::cl_N Li_series(int n, const cln::cl_N& x, const cln::float_format_t& prec) + +// anonymous namespace for helper functions +namespace { + + +// lookup table for factors built from Bernoulli numbers +// see fill_Xn() +std::vector > Xn; +int xnsize = 0; + + +// 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) +// The calculation of Xn depends on X0 and X{n-1}. +// 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 actual sum. +void fill_Xn(int n) { - // Note: argument must be in the unit circle - cln::cl_N aug, acc; - cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0); - cln::cl_N den = 0; - int i = 1; + // 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::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++) { + if (i&1) { + result = 0; // k == 0 + } else { + result = Xn[0][i/2-1]; // k == 0 + } + 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 + + *it = result; + it++; + } + Xn.push_back(buf); + } else if (n==1) { + // special case to handle the X_0 correct + std::vector buf(initsize); + 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++) { + 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; + it++; + } else { + result = Xn[0][i/2-1] + Xn[0][i/2-1]/(i+1); + for (int k=1; k buf(initsize/2); + std::vector::iterator it = buf.begin(); + for (int i=1; i<=initsize/2; i++) { + *it = bernoulli(i*2).to_cl_N(); + it++; + } + Xn.push_back(buf); + } + + xnsize++; +} + + +// 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_I den = 1; // n^2 = 1 + unsigned i = 3; do { + resbuf = res; num = num * x; - cln::cl_R ii = i; - den = cln::expt(ii, n); + den = den + i; // n^2 = 4, 9, 16, ... + i += 2; + res = res + num / den; + } while (res != resbuf); + return res; +} + + +// 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(); + cln::cl_N u = -cln::log(1-x); + cln::cl_N factor = u; + cln::cl_N res = u - u*u/4; + cln::cl_N resbuf; + unsigned i = 1; + do { + resbuf = res; + factor = factor * u*u / (2*i * (2*i+1)); + res = res + (*it) * factor; + it++; // should we check it? or rely on initsize? ... + i++; + } while (res != resbuf); + return res; +} + + +// 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 res = x; + cln::cl_N resbuf; + int i=2; + do { + resbuf = res; + factor = factor * x; + res = res + factor / cln::expt(cln::cl_I(i),n); + i++; + } while (res != resbuf); + return res; +} + + +// 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(); + cln::cl_N u = -cln::log(1-x); + cln::cl_N factor = u; + cln::cl_N res = u; + cln::cl_N resbuf; + unsigned i=2; + do { + resbuf = res; + factor = factor * u / i; + res = res + (*it) * factor; + it++; // should we check it? or rely on initsize? ... i++; - aug = num / den; - acc = acc + aug; - } while (acc != acc+aug); - return acc; + } while (res != resbuf); + return res; } +// forward declaration needed by function Li_projection and C below +numeric S_num(int n, int p, const numeric& 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) { - return Li_series(n, x, prec); + // treat n=2 as special case + if (n == 2) { + // check if precalculated X0 exists + if (xnsize == 0) { + fill_Xn(0); + } + + if (cln::realpart(x) < 0.5) { + // choose the faster algorithm + // the switching point was empirically determined. the optimal point + // depends on hardware, Digits, ... so an approx value is okay. + // 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_do_sum(x); + } else { + return Li2_do_sum_Xn(x); + } + } else { + // choose the faster algorithm + if (cln::abs(cln::realpart(x)) > 0.75) { + return -Li2_do_sum(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2); + } else { + return -Li2_do_sum_Xn(1-x) - cln::log(x) * cln::log(1-x) + cln::zeta(2); + } + } + } else { + // check if precalculated Xn exist + if (n > xnsize+1) { + for (int i=xnsize; i=12 the "normal" summation always wins against the method with Xn + if ((cln::abs(cln::realpart(x)) < 0.3) || (n >= 12)) { + return Lin_do_sum(n, x); + } else { + 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); + for (int j=0; j& s, const std::vector& x) { - if (step) { - cln::cl_N res; - for (int i=1; i 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]); } - return res; + } while (t[0] != t0buf); + + return t[0]; +} + + +} // end of anonymous namespace + + +////////////////////////////////////////////////////////////////////// +// +// Classical polylogarithm and multiple polylogarithm Li +// +// GiNaC function +// +////////////////////////////////////////////////////////////////////// + + +static ex Li_eval(const ex& x1, const ex& x2) +{ + if (x2.is_zero()) { + return _ex0; } else { - return 1; + if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) + return Li_num(ex_to(x1).to_int(), ex_to(x2)); + if (is_a(x2)) { + for (int i=0; i(x2.op(i))) { + return Li(x1,x2).hold(); + } + } + return Li(x1,x2).evalf(); + } + return Li(x1,x2).hold(); } } -// forward declaration needed by function C below -static numeric S_num(int n, int p, const numeric& x); +static ex Li_evalf(const ex& x1, const ex& x2) +{ + // 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(x2.op(i))) { + return Li(x1,x2).hold(); + } + if (x2.op(i) >= 1) { + return Li(x1,x2).hold(); + } + } + + std::vector m; + std::vector x; + for (int i=0; i(x1.nops()).to_int(); i++) { + m.push_back(ex_to(x1.op(i)).to_int()); + x.push_back(ex_to(x2.op(i)).to_cl_N()); + } + + return numeric(multipleLi_do_sum(m, x)); + } + + return Li(x1,x2).hold(); +} + + +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); +} + + +static ex Li_deriv(const ex& x1, const ex& x2, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param < 2); + if (deriv_param == 0) { + return _ex0; + } + if (x1 > 0) { + return Li(x1-1, x2) / x2; + } else { + return 1/(1-x2); + } +} + +REGISTER_FUNCTION(Li, + eval_func(Li_eval). + evalf_func(Li_evalf). + do_not_evalf_params(). + series_func(Li_series). + derivative_func(Li_deriv)); + + +////////////////////////////////////////////////////////////////////// +// +// Nielsen's generalized polylogarithm S +// +// 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) -static cln::cl_N C(int n, int p) +cln::cl_N C(int n, int p) { cln::cl_N result; @@ -234,7 +669,7 @@ static cln::cl_N C(int n, int p) // helper function for S(n,p,x) // [Kol] remark to (9.1) -static cln::cl_N a_k(int k) +cln::cl_N a_k(int k) { cln::cl_N result; @@ -253,7 +688,7 @@ static cln::cl_N a_k(int k) // helper function for S(n,p,x) // [Kol] remark to (9.1) -static cln::cl_N b_k(int k) +cln::cl_N b_k(int k) { cln::cl_N result; @@ -271,29 +706,44 @@ static cln::cl_N b_k(int k) // 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) +cln::cl_N S_do_sum(int n, int p, const cln::cl_N& x, const cln::float_format_t& prec) { - n++; + 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= 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++; - acc = acc + aug; - } while (acc != acc+aug); - - return acc; + } while (res != resbuf); + + return res; } // 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) +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")) { @@ -305,7 +755,7 @@ static cln::cl_N S_projection(int n, int p, const cln::cl_N& x, const cln::float cln::cl_N res2; for (int r=0; r 1) { + if (cln::abs(value) > 1) { cln::cl_N result; @@ -410,221 +860,1135 @@ static numeric S_num(int n, int p, const numeric& x) } -// helper function for multiple polylogarithm -static cln::cl_N numeric_zsum(int n, std::vector& x, std::vector& m) +} // end of anonymous namespace + + +////////////////////////////////////////////////////////////////////// +// +// Nielsen's generalized polylogarithm S +// +// GiNaC function +// +////////////////////////////////////////////////////////////////////// + + +static ex S_eval(const ex& x1, const ex& x2, const ex& x3) { - cln::cl_N res; - if (x.empty()) { - return 1; + if (x2 == 1) { + return Li(x1+1,x3); } - 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); + if (x3.info(info_flags::numeric) && (!x3.info(info_flags::crational)) && + x1.info(info_flags::posint) && x2.info(info_flags::posint)) { + return S_num(ex_to(x1).to_int(), ex_to(x2).to_int(), ex_to(x3)); } - return res; + return S(x1,x2,x3).hold(); } -// helper function for harmonic polylogarithm -static cln::cl_N numeric_harmonic(int n, std::vector& m) +static ex S_evalf(const ex& x1, const ex& x2, const ex& x3) { - cln::cl_N res; - if (m.empty()) { - return 1; + if (is_a(x1) && is_a(x2) && is_a(x3)) { + return S_num(ex_to(x1).to_int(), ex_to(x2).to_int(), ex_to(x3)); } - 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); + 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); +} + + +static ex S_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param < 3); + if (deriv_param < 2) { + return _ex0; + } + if (x1 > 0) { + return S(x1-1, x2, x3) / x3; + } else { + return S(x1, x2-1, x3) / (1-x3); + } +} + + +REGISTER_FUNCTION(S, + eval_func(S_eval). + evalf_func(S_evalf). + do_not_evalf_params(). + series_func(S_series). + derivative_func(S_deriv)); + + +////////////////////////////////////////////////////////////////////// +// +// Harmonic polylogarithm H +// +// helper function +// +////////////////////////////////////////////////////////////////////// + + +// anonymous namespace for helper functions +namespace { + + +// forward declaration +ex convert_from_RV(const lst& parameterlst, const ex& arg); + + +// multiplies an one-dimensional H with another H +// [ReV] (18) +ex trafo_H_mult(const ex& h1, const ex& h2) +{ + ex res; + ex hshort; + lst hlong; + ex h1nops = h1.op(0).nops(); + ex h2nops = h2.op(0).nops(); + if (h1nops > 1) { + hshort = h2.op(0).op(0); + hlong = ex_to(h1.op(0)); + } else { + hshort = h1.op(0).op(0); + if (h2nops > 1) { + hlong = ex_to(h2.op(0)); + } else { + hlong = h2.op(0).op(0); + } + } + for (int i=0; i<=hlong.nops(); i++) { + lst newparameter; + int j=0; + for (; j(e)) { + return e.map(*this); + } + + if (is_a(e)) { + + ex result = 1; + ex firstH; + lst Hlst; + for (int pos=0; pos(e.op(pos)) && is_a(e.op(pos).op(0))) { + std::string name = ex_to(e.op(pos).op(0)).get_name(); + if (name == "H") { + for (ex i=0; i(e.op(pos))) { + std::string name = ex_to(e.op(pos)).get_name(); + if (name == "H") { + if (e.op(pos).op(0).nops() > 1) { + firstH = e.op(pos); + } else { + Hlst.append(e.op(pos)); + } + continue; + } + } + result *= e.op(pos); + } + if (firstH == 0) { + if (Hlst.nops() > 0) { + firstH = Hlst[Hlst.nops()-1]; + Hlst.remove_last(); + } else { + return e; + } + } + + if (Hlst.nops() > 0) { + ex buffer = trafo_H_mult(firstH, Hlst.op(0)); + result *= buffer; + for (int i=1; i(e)) { + name = ex_to(e).get_name(); } - else { - return Li(x1,x2).hold(); + if (name == "H") { + h = e; + } else { + for (int i=0; i(e.op(i))) { + std::string name = ex_to(e.op(i)).get_name(); + if (name == "H") { + h = e.op(i); + } + } + } + } + if (h != 0) { + lst newparameter = ex_to(h.op(0)); + newparameter.prepend(1); + return e.subs(h == H(newparameter, h.op(1)).hold()); + } else { + return e * H(lst(1),1-arg).hold(); } } -static ex Li_evalf(const ex& x1, const ex& x2) + +// do integration [ReV] (55) +// put parameter 0 in front of existing parameters +ex trafo_H_prepend_zero(const ex& e, const ex& arg) { - // classical polylogs - if (is_a(x1) && is_a(x2)) { - return Li_num(ex_to(x1).to_int(), ex_to(x2)); + ex h; + std::string name; + if (is_a(e)) { + name = ex_to(e).get_name(); } - // 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 >= 1) - return Li(x1,x2).hold(); + if (name == "H") { + h = e; + } else { + for (int i=0; i(e.op(i))) { + std::string name = ex_to(e.op(i)).get_name(); + if (name == "H") { + h = e.op(i); + } + } + } + } + if (h != 0) { + lst newparameter = ex_to(h.op(0)); + newparameter.prepend(0); + ex addzeta = convert_from_RV(newparameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1))); + return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand(); + } else { + return e * (-H(lst(0),1/arg).hold()); + } +} + + +// do x -> 1-x transformation +struct map_trafo_H_1mx : public map_function +{ + ex operator()(const ex& e) + { + if (is_a(e) || is_a(e)) { + return e.map(*this); } + + if (is_a(e)) { + std::string name = ex_to(e).get_name(); + if (name == "H") { + + lst parameter = ex_to(e.op(0)); + ex arg = e.op(1); + + // if all parameters are either zero or one return the transformed function + if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) { + lst newparameter; + for (int i=parameter.nops(); i>0; i--) { + newparameter.append(0); + } + return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold(); + } else if (find(parameter.begin(), parameter.end(), 1) == parameter.end()) { + lst newparameter; + for (int i=parameter.nops(); i>0; i--) { + newparameter.append(1); + } + return pow(-1, parameter.nops()) * H(newparameter, 1-arg).hold(); + } - 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()); + lst newparameter = parameter; + newparameter.remove_first(); + + if (parameter.op(0) == 0) { + + // leading zero + ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1))); + map_trafo_H_1mx recursion; + ex buffer = recursion(H(newparameter, arg).hold()); + if (is_a(buffer)) { + for (int i=0; i 1/x transformation +struct map_trafo_H_1overx : public map_function +{ + ex operator()(const ex& e) + { + if (is_a(e) || is_a(e)) { + return e.map(*this); } - return numeric(res); + if (is_a(e)) { + std::string name = ex_to(e).get_name(); + if (name == "H") { + + lst parameter = ex_to(e.op(0)); + ex arg = e.op(1); + // if all parameters are either zero or one return the transformed function + if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) { + map_trafo_H_mult unify; + return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() - I*Pi, parameter.nops()) / + factorial(parameter.nops())).expand()); + } else if (find(parameter.begin(), parameter.end(), 1) == parameter.end()) { + return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold(); + } + + lst newparameter = parameter; + newparameter.remove_first(); + + if (parameter.op(0) == 0) { + + // leading zero + ex res = convert_from_RV(parameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1))); + map_trafo_H_1overx recursion; + ex buffer = recursion(H(newparameter, arg).hold()); + if (is_a(buffer)) { + for (int i=0; i(e) || is_a(e)) { + return e.map(*this); + } + if (is_a(e)) { + std::string name = ex_to(e).get_name(); + if (name == "H") { + lst parameter; + if (is_a(e.op(0))) { + parameter = ex_to(e.op(0)); + } else { + parameter = lst(e.op(0)); + } + ex arg = e.op(1); + if (parameter.op(parameter.nops()-1) == 0) { + + // + if (parameter.nops() == 1) { + return log(arg); + } + + // + lst::const_iterator it = parameter.begin(); + while ((it != parameter.end()) && (*it == 0)) { + it++; + } + if (it == parameter.end()) { + return pow(log(arg),parameter.nops()) / factorial(parameter.nops()); + } + + // + parameter.remove_last(); + int lastentry = parameter.nops(); + while ((lastentry > 0) && (parameter[lastentry-1] == 0)) { + lastentry--; + } + + // + ex result = log(arg) * H(parameter,arg).hold(); + for (ex i=0; i(e) || is_a(e) || is_a(e)) { + return e.map(*this); + } + if (is_a(e)) { + std::string name = ex_to(e).get_name(); + if (name == "H") { + lst parameter = ex_to(e.op(0)); + ex arg = e.op(1); + return convert_from_RV(parameter, arg); + } + } + return e; + } +}; -// Nielsen's generalized polylogarithm -static ex S_eval(const ex& x1, const ex& x2, const ex& x3) +// translate notation from nested sums to Remiddi/Vermaseren +lst convert_to_RV(const lst& o) { - if (x2 == 1) { - return Li(x1+1,x3); + lst res; + for (lst::const_iterator it = o.begin(); it != o.end(); it++) { + for (ex i=0; i<(*it)-1; i++) { + res.append(0); + } + res.append(1); } - return S(x1,x2,x3).hold(); + return res; } -static ex S_evalf(const ex& x1, const ex& x2, const ex& x3) + +// translate notation from Remiddi/Vermaseren to nested sums +ex convert_from_RV(const lst& parameterlst, const ex& arg) { - 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(); + lst newparameterlst; + + lst::const_iterator it = parameterlst.begin(); + int count = 1; + while (it != parameterlst.end()) { + if (*it == 0) { + count++; + } else { + newparameterlst.append((*it>0) ? count : -count); + count = 1; } - return S_num(ex_to(x1).to_int(), ex_to(x2).to_int(), ex_to(x3)); + it++; } - return S(x1,x2,x3).hold(); + for (int i=1; i& s, const cln::cl_N& x) +{ + const int j = s.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 { + 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] * factor / cln::expt(cln::cl_I(q+j-1), s[0]); + factor = factor * x; + } while (t[0] != t0buf); + + return t[0]; +} + + +} // end of anonymous namespace + + +////////////////////////////////////////////////////////////////////// +// +// Harmonic polylogarithm H +// +// GiNaC function +// +////////////////////////////////////////////////////////////////////// -// Harmonic polylogarithm static ex H_eval(const ex& x1, const ex& x2) { + if (x2 == 0) { + return 0; + } + if (x2 == 1) { + return zeta(x1); + } + if (x1.nops() == 1) { + return Li(x1.op(0), x2); + } + if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) { + return H(x1,x2).evalf(); + } return H(x1,x2).hold(); } + static ex H_evalf(const ex& x1, const ex& x2) { if (is_a(x1) && is_a(x2)) { for (int i=0; i(x1.op(i))) + if (!x1.op(i).info(info_flags::posint)) { return H(x1,x2).hold(); + } + } + if (x1.nops() < 1) { + return _ex1; + } + if (x1.nops() == 1) { + return Li(x1.op(0), x2).evalf(); + } + cln::cl_N x = ex_to(x2).to_cl_N(); + if (x == 1) { + return zeta(x1).evalf(); } - 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()); + // choose trafo + if (cln::abs(x) > 1) { + symbol xtemp("xtemp"); + map_trafo_H_1overx trafo; + ex res = trafo(H(convert_to_RV(ex_to(x1)), xtemp)); + map_trafo_H_convert converter; + res = converter(res); + return res.subs(xtemp==x2).evalf(); } - 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; + // since the x->1-x transformation produces a lot of terms, it is only + // efficient for argument near one. + if (cln::realpart(x) > 0.95) { + symbol xtemp("xtemp"); + map_trafo_H_1mx trafo; + ex res = trafo(H(convert_to_RV(ex_to(x1)), xtemp)); + map_trafo_H_convert converter; + res = converter(res); + return res.subs(xtemp==x2).evalf(); } - return numeric(res); + // no trafo -> do summation + int count = x1.nops(); + std::vector r(count); + for (int i=0; i(x1.op(i)).to_int(); + } + return numeric(H_do_sum(r,x)); } return H(x1,x2).hold(); } -REGISTER_FUNCTION(H, eval_func(H_eval).evalf_func(H_evalf).do_not_evalf_params()); +static ex H_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options) +{ + epvector seq; + seq.push_back(expair(H(x1,x2), 0)); + return pseries(rel,seq); +} + + +static ex H_deriv(const ex& x1, const ex& x2, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param < 2); + if (deriv_param == 0) { + return _ex0; + } + if (is_a(x1)) { + lst newparameter = ex_to(x1); + if (x1.op(0) == 1) { + newparameter.remove_first(); + return 1/(1-x2) * H(newparameter, x2); + } else { + newparameter[0]--; + return H(newparameter, x2).hold() / x2; + } + } else { + if (x1 == 1) { + return 1/(1-x2); + } else { + return H(x1-1, x2).hold() / x2; + } + } +} + + +REGISTER_FUNCTION(H, + eval_func(H_eval). + evalf_func(H_evalf). + do_not_evalf_params(). + series_func(H_series). + derivative_func(H_deriv)); + + +////////////////////////////////////////////////////////////////////// +// +// Multiple zeta values zeta +// +// helper functions +// +////////////////////////////////////////////////////////////////////// + + +// anonymous namespace for helper functions +namespace { + + +// parameters and data for [Cra] algorithm +const cln::cl_N lambda = cln::cl_N("319/320"); +int L1; +int L2; +std::vector > f_kj; +std::vector crB; +std::vector > crG; +std::vector crX; + + +void halfcyclic_convolute(const std::vector& a, const std::vector& b, std::vector& c) +{ + const int size = a.size(); + for (int n=0; n& s) +{ + const int k = s.size(); + + crX.clear(); + crG.clear(); + crB.clear(); + + for (int i=0; i<=L2; i++) { + crB.push_back(bernoulli(i).to_cl_N() / cln::factorial(i)); + } + + int Sm = 0; + int Smp1 = 0; + for (int m=0; m crGbuf; + Sm = Sm + s[m]; + Smp1 = Sm + s[m+1]; + for (int i=0; i<=L2; i++) { + crGbuf.push_back(cln::factorial(i + Sm - m - 2) / cln::factorial(i + Smp1 - m - 2)); + } + crG.push_back(crGbuf); + } + + crX = crB; + + for (int m=0; m Xbuf; + for (int i=0; i<=L2; i++) { + Xbuf.push_back(crX[i] * crG[m][i]); + } + halfcyclic_convolute(Xbuf, crB, crX); + } +} + + +// [Cra] section 4 +cln::cl_N crandall_Y_loop(const cln::cl_N& Sqk) +{ + 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; +} + + +// [Cra] section 4 +void calc_f(int maxr) +{ + f_kj.clear(); + f_kj.resize(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++; + } +} + + +// [Cra] (3.1) +cln::cl_N crandall_Z(const std::vector& s) +{ + 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); + } + + 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); +} + + +// [Cra] (2.4) +cln::cl_N zeta_do_sum_Crandall(const std::vector& s) +{ + std::vector r = s; + const int j = r.size(); + + // decide on maximal size of f_kj for crandall_Z + if (Digits < 50) { + L1 = 150; + } else { + L1 = Digits * 3 + j*2; + } + + // 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; + + int maxr = 0; + int S = 0; + for (int i=0; i maxr) { + maxr = r[i]; + } + } + + calc_f(maxr); + + 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(); + + initcX(r); + + for (int q=0; q& 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)); + + 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); + + return t[0]; +} + + +} // end of anonymous namespace + + +////////////////////////////////////////////////////////////////////// +// +// Multiple zeta values zeta +// +// GiNaC function +// +////////////////////////////////////////////////////////////////////// + + +static ex zeta1_evalf(const ex& x) +{ + 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)); + } + } + + // single zeta value + if (is_exactly_a(x) && (x != 1)) { + try { + return zeta(ex_to(x)); + } catch (const dunno &e) { } + } + + return zeta(x).hold(); +} + + +static ex zeta1_eval(const ex& x) +{ + if (is_exactly_a(x)) { + if (x.nops() == 1) { + return zeta(x.op(0)); + } + return zeta(x).hold(); + } + + if (x.info(info_flags::numeric)) { + const numeric& y = ex_to(x); + // trap integer arguments: + if (y.is_integer()) { + if (y.is_zero()) { + return _ex_1_2; + } + if (y.is_equal(_num1)) { + return zeta(x).hold(); + } + if (y.info(info_flags::posint)) { + if (y.info(info_flags::odd)) { + return zeta(x).hold(); + } else { + return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y); + } + } else { + if (y.info(info_flags::odd)) { + return -bernoulli(_num1-y) / (_num1-y); + } else { + return _ex0; + } + } + } + // zeta(float) + if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) + return zeta1_evalf(x); + } + return zeta(x).hold(); +} + + +static ex zeta1_deriv(const ex& x, unsigned deriv_param) +{ + GINAC_ASSERT(deriv_param==0); + + if (is_exactly_a(x)) { + return _ex0; + } else { + return zeta(_ex1, x); + } +} + + +unsigned zeta1_SERIAL::serial = + function::register_new(function_options("zeta"). + eval_func(zeta1_eval). + evalf_func(zeta1_evalf). + do_not_evalf_params(). + derivative_func(zeta1_deriv). + latex_name("\\zeta"). + overloaded(2)); + + +////////////////////////////////////////////////////////////////////// +// +// Multiple zeta values mZeta +// +// The use of mZeta is deprecated! This function will be removed +// from GiNaC source soon. Use zeta instead!! +// +// GiNaC function +// +////////////////////////////////////////////////////////////////////// -// Multiple zeta value static ex mZeta_eval(const ex& x1) { return mZeta(x1).hold(); } + static ex mZeta_evalf(const ex& x1) { if (is_a(x1)) { for (int i=0; i(x1.op(i))) + if (!x1.op(i).info(info_flags::posint)) return mZeta(x1).hold(); } - cln::cl_N m_1 = ex_to(x1.op(x1.nops()-1)).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()); - } + const int j = x1.nops(); - cln::float_format_t prec = cln::default_float_format; - cln::cl_N res = cln::complex(cln::cl_float(0, prec), 0); - cln::cl_N resbuf; - for (int i=nops; true; i++) { - // to infinity and beyond ... timewise - resbuf = res; - res = res + cln::recip(cln::expt(i,m_1)) * numeric_harmonic(i, m); - if (cln::zerop(res-resbuf)) - break; + std::vector r(j); + for (int i=0; i(x1.op(i)).to_int(); } - return numeric(res); + // check for divergence + if (r[0] == 1) { + return mZeta(x1).hold(); + } + // if only one argument, use cln::zeta + if (j == 1) { + return numeric(cln::zeta(r[0])); + } + + // decide on summation algorithm + // this is still a bit clumsy + int limit = (Digits>17) ? 10 : 6; + if (r[0]3 && r[1](x1).to_int())); } return mZeta(x1).hold(); } -REGISTER_FUNCTION(mZeta, eval_func(mZeta_eval).evalf_func(mZeta_evalf).do_not_evalf_params()); + +static ex mZeta_deriv(const ex& x, unsigned deriv_param) +{ + return 0; +} + + +REGISTER_FUNCTION(mZeta, + eval_func(mZeta_eval). + evalf_func(mZeta_evalf). + do_not_evalf_params(). + derivative_func(mZeta_deriv)); } // namespace GiNaC