X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Finifcns_nstdsums.cpp;h=3d1f8bcf45b37b721f090bfc962c4634ebe262dd;hp=fbc4a3faba81e4f24cb2a53ab14a1dc6d03d8bda;hb=da64e515abf7243bc4c84ca3631470931c4e6691;hpb=0962930400269497db414baba12740496157a63b diff --git a/ginac/inifcns_nstdsums.cpp b/ginac/inifcns_nstdsums.cpp index fbc4a3fa..3d1f8bcf 100644 --- a/ginac/inifcns_nstdsums.cpp +++ b/ginac/inifcns_nstdsums.cpp @@ -1,48 +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(m_1,...,m_k),lst(x_1,...,x_k)) - * nielsen's generalized polylogarithm S(n,p,x) + * 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 publications: * [Kol] Nielsen's Generalized Polylogarithms, K.S.Kolbig, SIAM J.Math.Anal. 17 (1986), pp. 1232-1258. - * [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172. - * [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754 - * [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941 + * [Cra] Fast Evaluation of Multiple Zeta Sums, R.E.Crandall, Math.Comp. 67 (1998), pp. 1163-1172. + * [ReV] Harmonic Polylogarithms, E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. A15 (2000), pp. 725-754 + * [BBB] Special Values of Multiple Polylogarithms, J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941 + * [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 nummerically 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 order of parameters and arguments of H, Li and zeta is defined according to their order in the - * nested sums representation. - * - * - Except for the multiple polylogarithm all functions can be nummerically evaluated with arguments in - * the whole complex plane. Multiple polylogarithms evaluate only if each argument x_i is smaller than - * one. The parameters for every function (n, p or n_i) must be positive integers. - * - * - The calculation of classical polylogarithms is speed up by using Bernoulli numbers and + * - The calculation of classical polylogarithms is speeded up by using Bernoulli numbers and * look-up tables. S uses look-up tables as well. The zeta function applies the algorithms in - * [Cra] and [BBB] for speed up. - * - * - The functions have no series expansion as nested sums. To do it, you have to convert these functions - * into the appropriate objects from the nestedsums library, do the expansion and convert the - * result back. - * + * [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. 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. + * 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) 1999-2005 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 @@ -56,9 +61,10 @@ * * 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 @@ -98,6 +104,9 @@ namespace { // lookup table for factors built from Bernoulli numbers // see fill_Xn() std::vector > Xn; +// initial size of Xn that should suffice for 32bit machines (must be even) +const int xninitsizestep = 26; +int xninitsize = xninitsizestep; int xnsize = 0; @@ -113,17 +122,14 @@ int xnsize = 0; // The second index in Xn corresponds to the index from the actual sum. void fill_Xn(int n) { - // rule of thumb. needs to be improved. TODO - const int initsize = Digits * 3 / 2; - if (n>1) { // calculate X_2 and higher (corresponding to Li_4 and higher) - std::vector buf(initsize); + std::vector buf(xninitsize); std::vector::iterator it = buf.begin(); cln::cl_N result; *it = -(cln::expt(cln::cl_I(2),n+1) - 1) / cln::expt(cln::cl_I(2),n+1); // i == 1 it++; - for (int i=2; i<=initsize; i++) { + for (int i=2; i<=xninitsize; i++) { if (i&1) { result = 0; // k == 0 } else { @@ -143,14 +149,14 @@ void fill_Xn(int n) Xn.push_back(buf); } else if (n==1) { // special case to handle the X_0 correct - std::vector buf(initsize); + std::vector buf(xninitsize); std::vector::iterator it = buf.begin(); cln::cl_N result; *it = cln::cl_I(-3)/cln::cl_I(4); // i == 1 it++; *it = cln::cl_I(17)/cln::cl_I(36); // i == 2 it++; - for (int i=3; i<=initsize; i++) { + for (int i=3; i<=xninitsize; i++) { if (i & 1) { result = -Xn[0][(i-3)/2]/2; *it = (cln::binomial(i,1)/cln::cl_I(2) + cln::binomial(i,i-1)/cln::cl_I(i))*result; @@ -167,9 +173,9 @@ void fill_Xn(int n) Xn.push_back(buf); } else { // calculate X_0 - std::vector buf(initsize/2); + std::vector buf(xninitsize/2); std::vector::iterator it = buf.begin(); - for (int i=1; i<=initsize/2; i++) { + for (int i=1; i<=xninitsize/2; i++) { *it = bernoulli(i*2).to_cl_N(); it++; } @@ -179,13 +185,59 @@ void fill_Xn(int n) xnsize++; } +// doubles the number of entries in each Xn[] +void double_Xn() +{ + const int pos0 = xninitsize / 2; + // X_0 + for (int i=1; i<=xninitsizestep/2; ++i) { + Xn[0].push_back(bernoulli((i+pos0)*2).to_cl_N()); + } + if (Xn.size() > 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); + } + } + } + xninitsize += xninitsizestep; +} + // calculates Li(2,x) without Xn cln::cl_N Li2_do_sum(const cln::cl_N& x) { cln::cl_N res = x; cln::cl_N resbuf; - cln::cl_N num = x; + cln::cl_N num = x * cln::cl_float(1, cln::float_format(Digits)); cln::cl_I den = 1; // n^2 = 1 unsigned i = 3; do { @@ -203,17 +255,23 @@ cln::cl_N Li2_do_sum(const cln::cl_N& x) cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x) { std::vector::const_iterator it = Xn[0].begin(); + std::vector::const_iterator xend = Xn[0].end(); cln::cl_N u = -cln::log(1-x); - cln::cl_N factor = u; - cln::cl_N 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; } @@ -222,7 +280,7 @@ cln::cl_N Li2_do_sum_Xn(const cln::cl_N& x) // calculates Li(n,x), n>2 without Xn cln::cl_N Lin_do_sum(int n, const cln::cl_N& x) { - cln::cl_N factor = x; + cln::cl_N factor = x * cln::cl_float(1, cln::float_format(Digits)); cln::cl_N res = x; cln::cl_N resbuf; int i=2; @@ -240,8 +298,9 @@ cln::cl_N Lin_do_sum(int n, const cln::cl_N& x) cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x) { std::vector::const_iterator it = Xn[n-2].begin(); + std::vector::const_iterator xend = Xn[n-2].end(); cln::cl_N u = -cln::log(1-x); - cln::cl_N factor = u; + cln::cl_N factor = u * cln::cl_float(1, cln::float_format(Digits)); cln::cl_N res = u; cln::cl_N resbuf; unsigned i=2; @@ -249,8 +308,12 @@ cln::cl_N Lin_do_sum_Xn(int n, const cln::cl_N& x) resbuf = res; factor = factor * u / i; res = res + (*it) * factor; - it++; // should we check it? or rely on initsize? ... i++; + if (++it == xend) { + double_Xn(); + it = Xn[n-2].begin() + (i-2); + xend = Xn[n-2].end(); + } } while (res != resbuf); return res; } @@ -309,7 +372,7 @@ cln::cl_N Li_projection(int n, const cln::cl_N& x, const cln::float_format_t& pr cln::cl_N result = -cln::expt(cln::log(x), n-1) * cln::log(1-x) / cln::factorial(n-1); for (int j=0; j(x).to_cl_N(); + 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) { const int j = s.size(); @@ -407,281 +480,1273 @@ cln::cl_N multipleLi_do_sum(const std::vector& s, const std::vector=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]); } - } while ((t[0] != t0buf) || (q<10)); - + // ... and do it again (to avoid premature drop out due to special arguments) + 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]); + } + } while (t[0] != t0buf); + return t[0]; } -} // end of anonymous namespace +// converts parameter types and calls multipleLi_do_sum (convenience function for G_numeric) +cln::cl_N mLi_do_summation(const lst& m, const lst& x) +{ + std::vector m_int; + std::vector x_cln; + for (lst::const_iterator itm = m.begin(), itx = x.begin(); itm != m.end(); ++itm, ++itx) { + m_int.push_back(ex_to(*itm).to_int()); + x_cln.push_back(ex_to(*itx).to_cl_N()); + } + return multipleLi_do_sum(m_int, x_cln); +} -////////////////////////////////////////////////////////////////////// -// -// Classical polylogarithm and multiple polylogarithm Li(n,x) -// -// GiNaC function -// -////////////////////////////////////////////////////////////////////// +// forward declaration for Li_eval() +lst convert_parameter_Li_to_H(const lst& m, const lst& x, ex& pf); + + +// holding dummy-symbols for the G/Li transformations +std::vector gsyms; + +// type used by the transformation functions for G +typedef std::vector Gparameter; -static ex Li_eval(const ex& x1, const ex& x2) + +// G_eval1-function for G transformations +ex G_eval1(int a, int scale) { - if (x2.is_zero()) { - return _ex0; - } - else { - if (x2.info(info_flags::numeric) && (!x2.info(info_flags::crational))) - return Li_num(ex_to(x1).to_int(), ex_to(x2)); - if (is_a(x2)) { - for (int i=0; i(x2.op(i))) { - return Li(x1,x2).hold(); - } - } - return Li(x1,x2).evalf(); + 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); } - return Li(x1,x2).hold(); + } else { + return log(gsyms[std::abs(scale)]); } } -static ex Li_evalf(const ex& x1, const ex& x2) +// G_eval-function for G transformations +ex G_eval(const Gparameter& a, int scale) { - // 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)) { - ex conv = 1; - for (int i=0; i(x2.op(i))) { - return Li(x1,x2).hold(); - } - conv *= x2.op(i); - if ((conv > 1) || ((conv == 1) && (x1.op(0) == 1))) { - return Li(x1,x2).hold(); + if (all_ones) { + ++count_ones; } + } else { + all_ones = false; } + } - 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()); + // care about divergent G: shuffle to separate divergencies that will be canceled + // later on in the transformation + if (newa.nops() > 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); + } + ex result = G_eval1(a.front(), scale) * G_eval(short_a, scale); + 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), 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()); + } -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); + // 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); } -static ex Li_deriv(const ex& x1, const ex& x2, unsigned deriv_param) +// converts data for G: pending_integrals -> a +Gparameter convert_pending_integrals_G(const Gparameter& pending_integrals) { - GINAC_ASSERT(deriv_param < 2); - if (deriv_param == 0) { - return _ex0; - } - if (x1 > 0) { - return Li(x1-1, x2) / x2; + GINAC_ASSERT(pending_integrals.size() != 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 { - return 1/(1-x2); + // just return empty parameter list + Gparameter new_a; + return new_a; } } -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(n,p,x) -// -// helper functions -// -////////////////////////////////////////////////////////////////////// - +// 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; + } + } + return ++lastnonzero; +} -// anonymous namespace for helper functions -namespace { +// add scale to pending_integrals if pending_integrals is empty +Gparameter prepare_pending_integrals(const Gparameter& pending_integrals, int scale) +{ + GINAC_ASSERT(pending_integrals.size() != 1); -// 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] + if (pending_integrals.size() > 0) { + return pending_integrals; + } else { + Gparameter new_pending_integrals; + new_pending_integrals.push_back(scale); + return new_pending_integrals; + } +} -// 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) +// handles trailing zeroes for an otherwise convergent integral +ex trailing_zeros_G(const Gparameter& a, int scale) { - const int initsize = ynlength; - //const int initsize = initsize_Yn; - cln::cl_N one = cln::cl_float(1, prec); + bool convergent; + int depth, trailing_zeros; + Gparameter::const_iterator last, dummyit; + last = check_parameter_G(a, scale, convergent, depth, trailing_zeros, dummyit); - 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++; + GINAC_ASSERT(convergent); + + if ((trailing_zeros > 0) && (depth > 0)) { + ex result; + Gparameter new_a(a.begin(), a.end()-1); + result += G_eval1(0, scale) * trailing_zeros_G(new_a, scale); + 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); } - Yn.push_back(buf); + + return result / trailing_zeros; } 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); + return G_eval(a, scale); } - ynsize++; } -// make Yn longer ... -void make_Yn_longer(int newsize, 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) { + // pendint = ( y1, b1, ..., br ) + // a = ( 0, ..., 0, amin ) + // scale = y2 + // + // int_0^y1 ds1/(s1-b1) ... int dsr/(sr-br) G(0, ..., 0, sr; y2) + // where sr replaces amin + + GINAC_ASSERT(a.back() != 0); + GINAC_ASSERT(a.size() > 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()); + } + + // G(y2_{-+}; sr) + result += trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), new_pending_integrals.front()); + + // G(0; sr) + new_pending_integrals.back() = 0; + result -= trailing_zeros_G(convert_pending_integrals_G(new_pending_integrals), new_pending_integrals.front()); - 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++; + return result; } - 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++; - } + // 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()); } - ynlength = newsize; + // 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); + + // 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()) + * depth_one_trafo_G(new_pending_integrals_2, new_a, scale); + } else { + result += depth_one_trafo_G(new_pending_integrals_2, new_a, scale); + } + + return result; } -// helper function for S(n,p,x) -// [Kol] (7.2) -cln::cl_N C(int n, int p) -{ - cln::cl_N result; +// forward declaration +ex shuffle_G(const Gparameter & a0, const Gparameter & a1, const Gparameter & a2, + const Gparameter& pendint, const Gparameter& a_old, int scale); - for (int k=0; k 0) { + result *= trailing_zeros_G(convert_pending_integrals_G(pendint), pendint.front()); + } + return result; } - int np = n+p; - if ((np-1) & 1) { - if (((np)/2+n) & 1) { - result = -result - cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p)); + + // handle trailing zeros + if (trailing_zeros > 0) { + ex result; + Gparameter new_a(a.begin(), a.end()-1); + result += G_eval1(0, scale) * G_transform(pendint, new_a, scale); + 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); } - else { - result = -result + cln::expt(cln::pi(),np) / (np * cln::factorial(n-1) * cln::factorial(p)); + return result / trailing_zeros; + } + + // convergence case + if (convergent) { + if (pendint.size() > 0) { + return G_eval(convert_pending_integrals_G(pendint), pendint.front()) * G_eval(a, scale); + } else { + return G_eval(a, scale); } } - return result; -} + // call basic transformation for depth equal one + if (depth == 1) { + return depth_one_trafo_G(pendint, a, scale); + } + // 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) -// helper function for S(n,p,x) -// [Kol] remark to (9.1) -cln::cl_N a_k(int k) -{ - cln::cl_N result; + // 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()); - if (k == 0) { - return 1; + ex result = G_transform(pendint,a2,scale)*G_transform(empty,a1,scale); + + result -= shuffle_G(empty,a1,a2,pendint,a,scale); + return result; } - result = 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); + if (pendint.size() > 0) { + result *= trailing_zeros_G(convert_pending_integrals_G(pendint), pendint.front()); + } + + // 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()) + * G_transform(empty, new_a, scale); + int buffer = *changeit; + *changeit = *min_it; + result += G_transform(new_pendint, new_a, scale); + *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()) + * G_transform(empty, new_a, scale); + *changeit = *min_it; + result -= G_transform(new_pendint, new_a, scale); + } else { + // smallest at the front + new_pendint.push_back(scale); + result += trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front()) + * G_transform(empty, new_a, scale); + new_pendint.back() = *changeit; + result -= trailing_zeros_G(convert_pending_integrals_G(new_pendint), new_pendint.front()) + * G_transform(empty, new_a, scale); + *changeit = *min_it; + result += G_transform(new_pendint, new_a, scale); + } + 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) +{ + 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); + } + + 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); + } + + 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); + } + + 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) + + shuffle_G(a02,a1,a2_removed,pendint,a_old,scale); +} + + +// handles the transformations and the numerical evaluation of G +// the parameter x, s and y must only contain numerics +ex G_numeric(const lst& x, const lst& s, const ex& y) +{ + // check for convergence and necessary accelerations + bool need_trafo = false; + bool need_hoelder = false; + int depth = 0; + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) { + if (!(*it).is_zero()) { + ++depth; + if (abs(*it) - y < -pow(10,-Digits+2)) { + need_trafo = true; + break; + } + if (abs((abs(*it) - y)/y) < 0.01) { + need_hoelder = true; + } + } + } + if (x.op(x.nops()-1).is_zero()) { + need_trafo = true; + } + if (depth == 1 && !need_trafo) { + return -Li(x.nops(), y / x.op(x.nops()-1)).evalf(); + } + + // convergence transformation + if (need_trafo) { + + // sort (|x|<->position) to determine indices + std::multimap sortmap; + int size = 0; + for (int i=0; i(abs(x[i]), i)); + ++size; + } + } + // include upper limit (scale) + sortmap.insert(std::pair(abs(y), x.nops())); + + // generate missing dummy-symbols + int i = 1; + gsyms.clear(); + gsyms.push_back(symbol("GSYMS_ERROR")); + ex lastentry; + for (std::multimap::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) { + if (it != sortmap.begin()) { + if (it->second < x.nops()) { + 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.nops()) { + lastentry = x[it->second]; + } else { + lastentry = y; + } + } + + // fill position data according to sorted indices and prepare substitution list + Gparameter a(x.nops()); + lst subslst; + int pos = 1; + int scale; + for (std::multimap::const_iterator it = sortmap.begin(); it != sortmap.end(); ++it) { + if (it->second < x.nops()) { + if (s[it->second] > 0) { + a[it->second] = pos; + } else { + a[it->second] = -pos; + } + subslst.append(gsyms[pos] == x[it->second]); + } else { + scale = pos; + subslst.append(gsyms[pos] == y); + } + ++pos; + } + + // do transformation + Gparameter pendint; + ex result = G_transform(pendint, a, scale); + // replace dummy symbols with their values + result = result.eval().expand(); + result = result.subs(subslst).evalf(); + + return result; + } + + // do acceleration transformation (hoelder convolution [BBB]) + if (need_hoelder) { + + ex result; + const int size = x.nops(); + lst newx; + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) { + newx.append(*it / y); + } + + for (int r=0; r<=size; ++r) { + ex buffer = pow(-1, r); + ex p = 2; + bool adjustp; + do { + adjustp = false; + for (lst::const_iterator it = newx.begin(); it != newx.end(); ++it) { + if (*it == 1/p) { + p += (3-p)/2; + adjustp = true; + continue; + } + } + } while (adjustp); + ex q = p / (p-1); + lst qlstx; + lst qlsts; + for (int j=r; j>=1; --j) { + qlstx.append(1-newx.op(j-1)); + if (newx.op(j-1).info(info_flags::real) && newx.op(j-1) > 1 && newx.op(j-1) <= 2) { + qlsts.append( s.op(j-1)); + } else { + qlsts.append( -s.op(j-1)); + } + } + if (qlstx.nops() > 0) { + buffer *= G_numeric(qlstx, qlsts, 1/q); + } + lst plstx; + lst plsts; + for (int j=r+1; j<=size; ++j) { + plstx.append(newx.op(j-1)); + plsts.append(s.op(j-1)); + } + if (plstx.nops() > 0) { + buffer *= G_numeric(plstx, plsts, 1/p); + } + result += buffer; + } + return result; + } + + // do summation + lst newx; + lst m; + int mcount = 1; + ex sign = 1; + ex factor = y; + for (lst::const_iterator it = x.begin(); it != x.end(); ++it) { + if ((*it).is_zero()) { + ++mcount; + } else { + newx.append(factor / (*it)); + factor = *it; + m.append(mcount); + mcount = 1; + sign = -sign; + } + } + + return sign * numeric(mLi_do_summation(m, newx)); +} + + +ex mLi_numeric(const lst& m, const lst& x) +{ + // let G_numeric do the transformation + lst newx; + lst s; + ex 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.append(0); + s.append(1); + } + newx.append(factor / *itx); + factor /= *itx; + s.append(1); + } + return pow(-1, m.nops()) * G_numeric(newx, s, _ex1); +} + + +} // 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::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(); + } + lst s; + 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; + } + s.append(+1); + } + if (all_zero) { + return pow(log(y), x.nops()) / factorial(x.nops()); + } + return G_numeric(x, s, y); +} + + +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::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(); + } + lst s; + 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; + } + s.append(+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(); + } + return G_numeric(x, s, y); +} + + +unsigned G2_SERIAL::serial = function::register_new(function_options("G", 2). + evalf_func(G2_evalf). + eval_func(G2_eval). + do_not_evalf_params(). + 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::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(); + } + lst sn; + 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 (*its >= 0) { + sn.append(+1); + } else { + sn.append(-1); + } + } + if (all_zero) { + return pow(log(y), x.nops()) / factorial(x.nops()); + } + return G_numeric(x, sn, y); +} + + +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::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(); + } + lst sn; + 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 (*its >= 0) { + sn.append(+1); + } else { + sn.append(-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(); + } + return G_numeric(x, sn, y); +} + + +unsigned G3_SERIAL::serial = function::register_new(function_options("G", 3). + evalf_func(G3_evalf). + eval_func(G3_eval). + do_not_evalf_params(). + 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)) { + return Lin_numeric(ex_to(m_).to_int(), ex_to(x_)); + } else { + // try to numerically evaluate second argument + ex x_val = x_.evalf(); + if (x_val.info(info_flags::numeric)) { + return Lin_numeric(ex_to(m_).to_int(), ex_to(x_val)); + } + } + } + // 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) { + return zeta(m_,x_); + } + 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)) { + return Lin_numeric(ex_to(m_).to_int(), ex_to(x_)); + } + + return Li(m_, x_).hold(); +} + + +static ex Li_series(const ex& m, const ex& x, const relational& rel, int order, unsigned options) +{ + epvector seq; + seq.push_back(expair(Li(m, x), 0)); + return pseries(rel, seq); +} + + +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 << "\\mbox{Li}_{"; + lst::const_iterator itm = m.begin(); + (*itm).print(c); + itm++; + for (; itm != m.end(); itm++) { + c.s << ","; + (*itm).print(c); + } + c.s << "}("; + lst::const_iterator itx = x.begin(); + (*itx).print(c); + itx++; + for (; itx != x.end(); itx++) { + c.s << ","; + (*itx).print(c); + } + c.s << ")"; +} + + +REGISTER_FUNCTION(Li, + 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 cln::cl_F("0.5")) { cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(x),n) - * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p); + * cln::expt(cln::log(1-x),p) / cln::factorial(n) / cln::factorial(p); for (int s=0; s(cln::imagpart(value))); - // [Kol] (5.3) - if (cln::realpart(value) < -0.5) { + if ((cln::realpart(value) < -0.5) || (n == 0)) { cln::cl_N result = cln::expt(cln::cl_I(-1),p) * cln::expt(cln::log(value),n) - * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p); + * cln::expt(cln::log(1-value),p) / cln::factorial(n) / cln::factorial(p); for (int s=0; s(x1).to_int(), ex_to(x2).to_int(), ex_to(x3)); + if (n.info(info_flags::posint) && p.info(info_flags::posint)) { + if (is_a(x)) { + return S_num(ex_to(n).to_int(), ex_to(p).to_int(), ex_to(x)); + } else { + ex x_val = x.evalf(); + if (is_a(x_val)) { + return S_num(ex_to(n).to_int(), ex_to(p).to_int(), ex_to(x_val)); + } + } } - return S(x1,x2,x3).hold(); + return S(n, p, x).hold(); } -static ex S_evalf(const ex& x1, const ex& x2, const ex& x3) +static ex S_eval(const ex& n, const ex& p, const ex& x) { - if (is_a(x1) && is_a(x2) && is_a(x3)) { - return S_num(ex_to(x1).to_int(), ex_to(x2).to_int(), ex_to(x3)); + if (n.info(info_flags::posint) && p.info(info_flags::posint)) { + if (x == 0) { + return _ex0; + } + if (x == 1) { + lst m(n+1); + for (int i=ex_to(p).to_int()-1; i>0; i--) { + m.append(1); + } + return zeta(m); + } + if (p == 1) { + return Li(n+1, x); + } + if (x.info(info_flags::numeric) && (!x.info(info_flags::crational))) { + return S_num(ex_to(n).to_int(), ex_to(p).to_int(), ex_to(x)); + } + } + if (n.is_zero()) { + // [Kol] (5.3) + return pow(-log(1-x), p) / factorial(p); } - return S(x1,x2,x3).hold(); + return S(n, p, x).hold(); } -static ex S_series(const ex& x1, const ex& x2, const ex& x3, const relational& rel, int order, unsigned options) +static ex S_series(const ex& n, const ex& p, const ex& x, const relational& rel, int order, unsigned options) { epvector seq; - seq.push_back(expair(S(x1,x2,x3), 0)); - return pseries(rel,seq); + seq.push_back(expair(S(n, p, x), 0)); + return pseries(rel, seq); } -static ex S_deriv(const ex& x1, const ex& x2, const ex& x3, unsigned deriv_param) +static ex S_deriv(const ex& n, const ex& p, const ex& x, unsigned deriv_param) { GINAC_ASSERT(deriv_param < 3); if (deriv_param < 2) { return _ex0; } - if (x1 > 0) { - return S(x1-1, x2, x3) / x3; + if (n > 0) { + return S(n-1, p, x) / x; } else { - return S(x1, x2-1, x3) / (1-x3); + return S(n, p-1, x) / (1-x); } } +static void S_print_latex(const ex& n, const ex& p, const ex& x, const print_context& c) +{ + c.s << "\\mbox{S}_{"; + n.print(c); + c.s << ","; + p.print(c); + c.s << "}("; + x.print(c); + c.s << ")"; +} + + REGISTER_FUNCTION(S, - eval_func(S_eval). - evalf_func(S_evalf). - do_not_evalf_params(). - series_func(S_series). - derivative_func(S_deriv)); + evalf_func(S_evalf). + eval_func(S_eval). + series_func(S_series). + derivative_func(S_deriv). + print_func(S_print_latex). + do_not_evalf_params()); ////////////////////////////////////////////////////////////////////// // // Harmonic polylogarithm H(m,x) // -// helper function +// helper functions // ////////////////////////////////////////////////////////////////////// @@ -940,9 +2041,245 @@ REGISTER_FUNCTION(S, // anonymous namespace for helper functions namespace { + +// regulates the pole (used by 1/x-transformation) +symbol H_polesign("IMSIGN"); -// forward declaration -ex convert_from_RV(const lst& parameterlst, const ex& arg); + +// convert parameters from H to Li representation +// parameters are expected to be in expanded form, i.e. only 0, 1 and -1 +// returns true if some parameters are negative +bool convert_parameter_H_to_Li(const lst& l, lst& m, lst& s, ex& pf) +{ + // expand parameter list + lst mexp; + for (lst::const_iterator it = l.begin(); it != l.end(); it++) { + if (*it > 1) { + for (ex count=*it-1; count > 0; count--) { + mexp.append(0); + } + mexp.append(1); + } else if (*it < -1) { + for (ex count=*it+1; count < 0; count++) { + mexp.append(0); + } + mexp.append(-1); + } else { + mexp.append(*it); + } + } + + ex signum = 1; + pf = 1; + bool has_negative_parameters = false; + ex acc = 1; + for (lst::const_iterator it = mexp.begin(); it != mexp.end(); it++) { + if (*it == 0) { + acc++; + continue; + } + if (*it > 0) { + m.append((*it+acc-1) * signum); + } else { + m.append((*it-acc+1) * signum); + } + acc = 1; + signum = *it; + pf *= *it; + if (pf < 0) { + has_negative_parameters = true; + } + } + if (has_negative_parameters) { + for (int i=0; i(e) || is_a(e)) { + return e.map(*this); + } + if (is_a(e)) { + std::string name = ex_to(e).get_name(); + if (name == "H") { + lst parameter; + if (is_a(e.op(0))) { + parameter = ex_to(e.op(0)); + } else { + parameter = lst(e.op(0)); + } + ex arg = e.op(1); + + lst m; + lst s; + ex pf; + if (convert_parameter_H_to_Li(parameter, m, s, pf)) { + s.let_op(0) = s.op(0) * arg; + return pf * Li(m, s).hold(); + } else { + for (int i=0; i(e) || is_a(e)) { + return e.map(*this); + } + if (is_a(e)) { + std::string name = ex_to(e).get_name(); + if (name == "H") { + lst parameter; + if (is_a(e.op(0))) { + parameter = ex_to(e.op(0)); + } else { + parameter = lst(e.op(0)); + } + + lst m; + lst s; + ex pf; + if (convert_parameter_H_to_Li(parameter, m, s, pf)) { + return pf * zeta(m, s); + } else { + return zeta(m); + } + } + } + return e; + } +}; + + +// remove trailing zeros from H-parameters +struct map_trafo_H_reduce_trailing_zeros : public map_function +{ + ex operator()(const ex& e) + { + if (is_a(e) || is_a(e)) { + return e.map(*this); + } + if (is_a(e)) { + std::string name = ex_to(e).get_name(); + if (name == "H") { + lst parameter; + if (is_a(e.op(0))) { + parameter = ex_to(e.op(0)); + } else { + parameter = lst(e.op(0)); + } + ex arg = e.op(1); + if (parameter.op(parameter.nops()-1) == 0) { + + // + if (parameter.nops() == 1) { + return log(arg); + } + + // + lst::const_iterator it = parameter.begin(); + while ((it != parameter.end()) && (*it == 0)) { + it++; + } + if (it == parameter.end()) { + return pow(log(arg),parameter.nops()) / factorial(parameter.nops()); + } + + // + parameter.remove_last(); + int lastentry = parameter.nops(); + while ((lastentry > 0) && (parameter[lastentry-1] == 0)) { + lastentry--; + } + + // + ex result = log(arg) * H(parameter,arg).hold(); + ex acc = 0; + for (ex i=0; i 0) { + parameter[i]++; + result -= (acc + parameter[i]-1) * H(parameter, arg).hold(); + parameter[i]--; + acc = 0; + } else if (parameter[i] < 0) { + parameter[i]--; + result -= (acc + abs(parameter[i]+1)) * H(parameter, arg).hold(); + parameter[i]++; + acc = 0; + } else { + acc++; + } + } + + if (lastentry < parameter.nops()) { + result = result / (parameter.nops()-lastentry+1); + return result.map(*this); + } else { + return result; + } + } + } + } + return e; + } +}; + + +// returns an expression with zeta functions corresponding to the parameter list for H +ex convert_H_to_zeta(const lst& 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()) { + signum *= (*itx > 0) ? 1 : -1; + pf *= signum; + res.append((*itm) * signum); + itm++; + itx++; + } + return res; +} // multiplies an one-dimensional H with another H @@ -1045,9 +2382,9 @@ struct map_trafo_H_mult : public map_function }; -// do integration [ReV] (49) -// put parameter 1 in front of existing parameters -ex trafo_H_prepend_one(const ex& e, const ex& arg) +// do integration [ReV] (55) +// put parameter 0 in front of existing parameters +ex trafo_H_1tx_prepend_zero(const ex& e, const ex& arg) { ex h; std::string name; @@ -1068,17 +2405,18 @@ ex trafo_H_prepend_one(const ex& e, const ex& arg) } if (h != 0) { lst newparameter = ex_to(h.op(0)); - newparameter.prepend(1); - return e.subs(h == H(newparameter, h.op(1)).hold()); + newparameter.prepend(0); + ex addzeta = convert_H_to_zeta(newparameter); + return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand(); } else { - return e * H(lst(1),1-arg).hold(); + return e * (-H(lst(0),1/arg).hold()); } } // do integration [ReV] (55) -// put parameter 0 in front of existing parameters -ex trafo_H_prepend_zero(const ex& e, const ex& arg) +// put parameter -1 in front of existing parameters +ex trafo_H_1tx_prepend_minusone(const ex& e, const ex& arg) { ex h; std::string name; @@ -1099,24 +2437,87 @@ ex trafo_H_prepend_zero(const ex& e, const ex& arg) } if (h != 0) { lst newparameter = ex_to(h.op(0)); - newparameter.prepend(0); - ex addzeta = convert_from_RV(newparameter, 1).subs(H(wild(1),wild(2))==zeta(wild(1))); + newparameter.prepend(-1); + ex addzeta = convert_H_to_zeta(newparameter); return e.subs(h == (addzeta-H(newparameter, h.op(1)).hold())).expand(); } else { - return e * (-H(lst(0),1/arg).hold()); + ex addzeta = convert_H_to_zeta(lst(-1)); + return (e * (addzeta - H(lst(-1),1/arg).hold())).expand(); + } +} + + +// do integration [ReV] (55) +// put parameter -1 in front of existing parameters +ex trafo_H_1mxt1px_prepend_minusone(const ex& e, const ex& arg) +{ + ex h; + std::string name; + if (is_a(e)) { + name = ex_to(e).get_name(); + } + if (name == "H") { + h = e; + } else { + for (int i=0; i(e.op(i))) { + std::string name = ex_to(e.op(i)).get_name(); + if (name == "H") { + h = e.op(i); + } + } + } + } + if (h != 0) { + lst newparameter = ex_to(h.op(0)); + newparameter.prepend(-1); + return e.subs(h == H(newparameter, h.op(1)).hold()).expand(); + } else { + return (e * H(lst(-1),(1-arg)/(1+arg)).hold()).expand(); + } +} + + +// do integration [ReV] (55) +// put parameter 1 in front of existing parameters +ex trafo_H_1mxt1px_prepend_one(const ex& e, const ex& arg) +{ + ex h; + std::string name; + if (is_a(e)) { + name = ex_to(e).get_name(); + } + if (name == "H") { + h = e; + } else { + for (int i=0; i(e.op(i))) { + std::string name = ex_to(e.op(i)).get_name(); + if (name == "H") { + h = e.op(i); + } + } + } + } + if (h != 0) { + lst newparameter = ex_to(h.op(0)); + newparameter.prepend(1); + return e.subs(h == H(newparameter, h.op(1)).hold()).expand(); + } else { + return (e * H(lst(1),(1-arg)/(1+arg)).hold()).expand(); } } -// do x -> 1-x transformation -struct map_trafo_H_1mx : public map_function +// do x -> 1/x transformation +struct map_trafo_H_1overx : public map_function { ex operator()(const ex& e) { if (is_a(e) || is_a(e)) { return e.map(*this); } - + if (is_a(e)) { std::string name = ex_to(e).get_name(); if (name == "H") { @@ -1124,45 +2525,83 @@ struct map_trafo_H_1mx : public map_function lst parameter = ex_to(e.op(0)); ex arg = e.op(1); - // if all parameters are either zero or one return the transformed function - if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) { - lst newparameter; - for (int i=parameter.nops(); i>0; i--) { - newparameter.append(0); + // special cases if all parameters are either 0, 1 or -1 + bool allthesame = true; + if (parameter.op(0) == 0) { + for (int i=1; i0; i--) { - newparameter.append(1); + if (allthesame) { + return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold(); + } + } else if (parameter.op(0) == -1) { + for (int i=1; i(buffer)) { + for (int i=0; i(buffer)) { for (int i=0; i 1/x transformation -struct map_trafo_H_1overx : public map_function +// do x -> (1-x)/(1+x) transformation +struct map_trafo_H_1mxt1px : public map_function { ex operator()(const ex& e) { @@ -1207,37 +2646,83 @@ struct map_trafo_H_1overx : public map_function lst parameter = ex_to(e.op(0)); ex arg = e.op(1); - // if all parameters are either zero or one return the transformed function - if (find(parameter.begin(), parameter.end(), 0) == parameter.end()) { - map_trafo_H_mult unify; - return unify((pow(H(lst(1),1/arg).hold() + H(lst(0),1/arg).hold() - I*Pi, parameter.nops()) / - factorial(parameter.nops())).expand()); - } else if (find(parameter.begin(), parameter.end(), 1) == parameter.end()) { - return pow(-1, parameter.nops()) * H(parameter, 1/arg).hold(); + // special cases if all parameters are either 0, 1 or -1 + bool allthesame = true; + if (parameter.op(0) == 0) { + for (int i=1; i(buffer)) { + for (int i=0; i(buffer)) { for (int i=0; i(e) || is_a(e)) { - return e.map(*this); - } - if (is_a(e)) { - std::string name = ex_to(e).get_name(); - if (name == "H") { - lst parameter; - if (is_a(e.op(0))) { - parameter = ex_to(e.op(0)); - } else { - parameter = lst(e.op(0)); - } - ex arg = e.op(1); - if (parameter.op(parameter.nops()-1) == 0) { - - // - if (parameter.nops() == 1) { - return log(arg); - } - - // - lst::const_iterator it = parameter.begin(); - while ((it != parameter.end()) && (*it == 0)) { - it++; - } - if (it == parameter.end()) { - return pow(log(arg),parameter.nops()) / factorial(parameter.nops()); - } - - // - parameter.remove_last(); - int lastentry = parameter.nops(); - while ((lastentry > 0) && (parameter[lastentry-1] == 0)) { - lastentry--; - } - - // - ex result = log(arg) * H(parameter,arg).hold(); - for (ex i=0; i(e) || is_a(e) || is_a(e)) { - return e.map(*this); - } - if (is_a(e)) { - std::string name = ex_to(e).get_name(); - if (name == "H") { - lst parameter = ex_to(e.op(0)); - ex arg = e.op(1); - return convert_from_RV(parameter, arg); - } - } - return e; - } -}; - - -// translate notation from nested sums to Remiddi/Vermaseren -lst convert_to_RV(const lst& o) -{ - lst res; - for (lst::const_iterator it = o.begin(); it != o.end(); it++) { - for (ex i=0; i<(*it)-1; i++) { - res.append(0); - } - res.append(1); - } - return res; -} - - -// translate notation from Remiddi/Vermaseren to nested sums -ex convert_from_RV(const lst& parameterlst, const ex& arg) -{ - lst newparameterlst; - - lst::const_iterator it = parameterlst.begin(); - int count = 1; - while (it != parameterlst.end()) { - if (*it == 0) { - count++; - } else { - newparameterlst.append((*it>0) ? count : -count); - count = 1; - } - it++; - } - for (int i=1; i& s, const cln::cl_N& x) +cln::cl_N H_do_sum(const std::vector& m, const cln::cl_N& x) { - const int j = s.size(); + const int j = m.size(); std::vector t(j); @@ -1405,14 +2767,14 @@ cln::cl_N H_do_sum(const std::vector& s, const cln::cl_N& x) do { t0buf = t[0]; q++; - t[j-1] = t[j-1] + 1 / cln::expt(cln::cl_I(q),s[j-1]); + 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), s[k]); + t[k] = t[k] + t[k+1] / cln::expt(cln::cl_I(q+j-1-k), m[k]); } - t[0] = t[0] + t[1] * factor / cln::expt(cln::cl_I(q+j-1), s[0]); + 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]; } @@ -1429,117 +2791,330 @@ cln::cl_N H_do_sum(const std::vector& s, const cln::cl_N& x) ////////////////////////////////////////////////////////////////////// -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)) { + 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 (int i=0; i(x2).to_cl_N(); - if (x == 1) { - return zeta(x1).evalf(); + return H(x1, x2).hold(); } - // choose trafo - if (cln::abs(x) > 1) { + const lst& morg = ex_to(x1); + // remove trailing zeros ... + if (*(--morg.end()) == 0) { symbol xtemp("xtemp"); - map_trafo_H_1overx trafo; - ex res = trafo(H(convert_to_RV(ex_to(x1)), xtemp)); - map_trafo_H_convert converter; - res = converter(res); - return res.subs(xtemp==x2).evalf(); + map_trafo_H_reduce_trailing_zeros filter; + return filter(H(x1, xtemp).hold()).subs(xtemp==x2).evalf(); + } + // ... and expand parameter notation + lst m; + for (lst::const_iterator it = morg.begin(); it != morg.end(); it++) { + if (*it > 1) { + for (ex count=*it-1; count > 0; count--) { + m.append(0); + } + m.append(1); + } else if (*it < -1) { + for (ex count=*it+1; count < 0; count++) { + m.append(0); + } + m.append(-1); + } else { + m.append(*it); + } } - // since the x->1-x transformation produces a lot of terms, it is only - // efficient for argument near one. - if (cln::realpart(x) > 0.95) { - symbol xtemp("xtemp"); - map_trafo_H_1mx trafo; - ex res = trafo(H(convert_to_RV(ex_to(x1)), xtemp)); - map_trafo_H_convert converter; - res = converter(res); - return res.subs(xtemp==x2).evalf(); + // since the transformations produce a lot of terms, they are only efficient for + // argument near one. + // no transformation needed -> do summation + if (cln::abs(x) < 0.95) { + lst m_lst; + lst s_lst; + ex pf; + if (convert_parameter_H_to_Li(m, m_lst, s_lst, pf)) { + // negative parameters -> s_lst is filled + std::vector m_int; + std::vector x_cln; + for (lst::const_iterator it_int = m_lst.begin(), it_cln = s_lst.begin(); + it_int != m_lst.end(); it_int++, it_cln++) { + m_int.push_back(ex_to(*it_int).to_int()); + x_cln.push_back(ex_to(*it_cln).to_cl_N()); + } + x_cln.front() = x_cln.front() * x; + return pf * numeric(multipleLi_do_sum(m_int, x_cln)); + } else { + // only positive parameters + //TODO + if (m_lst.nops() == 1) { + return Li(m_lst.op(0), x2).evalf(); + } + std::vector m_int; + for (lst::const_iterator it = m_lst.begin(); it != m_lst.end(); it++) { + m_int.push_back(ex_to(*it).to_int()); + } + return numeric(H_do_sum(m_int, x)); + } } - // no trafo -> do summation - int count = x1.nops(); - std::vector r(count); - for (int i=0; i(x1.op(i)).to_int(); + ex res = 1; + + // ensure that the realpart of the argument is positive + if (cln::realpart(x) < 0) { + x = -x; + for (int i=0; i (1-x)/(1+x) + map_trafo_H_1mxt1px trafo; + res *= trafo(H(m, xtemp)); + } else { + // x -> 1/x + map_trafo_H_1overx trafo; + res *= trafo(H(m, xtemp)); + if (cln::imagpart(x) <= 0) { + res = res.subs(H_polesign == -I*Pi); + } else { + res = res.subs(H_polesign == I*Pi); + } } - return numeric(H_do_sum(r,x)); + // simplify result +// TODO +// map_trafo_H_convert converter; +// res = converter(res).expand(); +// lst ll; +// res.find(H(wild(1),wild(2)), ll); +// res.find(zeta(wild(1)), ll); +// res.find(zeta(wild(1),wild(2)), ll); +// res = res.collect(ll); + + return res.subs(xtemp == numeric(x)).evalf(); } return H(x1,x2).hold(); } -static ex H_series(const ex& x1, const ex& x2, const relational& rel, int order, unsigned options) +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; - seq.push_back(expair(H(x1,x2), 0)); - return pseries(rel,seq); + seq.push_back(expair(H(m, x), 0)); + return pseries(rel, seq); } -static ex H_deriv(const ex& x1, const ex& x2, unsigned deriv_param) +static ex H_deriv(const ex& m_, const ex& x, unsigned deriv_param) { GINAC_ASSERT(deriv_param < 2); if (deriv_param == 0) { return _ex0; } - if (is_a(x1)) { - lst newparameter = ex_to(x1); - if (x1.op(0) == 1) { - newparameter.remove_first(); - return 1/(1-x2) * H(newparameter, x2); - } else { - newparameter[0]--; - return H(newparameter, x2).hold() / x2; - } + lst m; + if (is_a(m_)) { + m = ex_to(m_); } else { - if (x1 == 1) { - return 1/(1-x2); - } else { - return H(x1-1, x2).hold() / x2; - } + m = lst(m_); + } + ex mb = *m.begin(); + if (mb > _ex1) { + m[0]--; + return H(m, x) / x; + } + if (mb < _ex_1) { + m[0]++; + return H(m, x) / x; + } + m.remove_first(); + if (mb == _ex1) { + return 1/(1-x) * H(m, x); + } else if (mb == _ex_1) { + return 1/(1+x) * H(m, x); + } else { + return H(m, x) / x; } } +static void H_print_latex(const ex& m_, const ex& x, const print_context& c) +{ + lst m; + if (is_a(m_)) { + m = ex_to(m_); + } else { + m = lst(m_); + } + c.s << "\\mbox{H}_{"; + lst::const_iterator itm = m.begin(); + (*itm).print(c); + itm++; + for (; itm != m.end(); itm++) { + c.s << ","; + (*itm).print(c); + } + c.s << "}("; + x.print(c); + c.s << ")"; +} + + REGISTER_FUNCTION(H, - eval_func(H_eval). - evalf_func(H_evalf). - do_not_evalf_params(). - series_func(H_series). - derivative_func(H_deriv)); + evalf_func(H_evalf). + eval_func(H_eval). + series_func(H_series). + derivative_func(H_deriv). + print_func(H_print_latex). + do_not_evalf_params()); + + +// takes a parameter list for H and returns an expression with corresponding multiple polylogarithms +ex convert_H_to_Li(const ex& 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())); + } +} ////////////////////////////////////////////////////////////////////// @@ -1851,16 +3426,12 @@ cln::cl_N zeta_do_Hoelder_convolution(const std::vector& m_, const std::vec } 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() * cln::cl_N("1/2"); + s_p.front() = s_p.front() * spbuf; } s.erase(s.begin()); - for (int i=0; i 0) break; - - } m_q.insert(m_q.begin(), 1); if (s_q.size() > 0) { s_q.front() = s_q.front() * 4; @@ -1881,12 +3452,12 @@ cln::cl_N zeta_do_Hoelder_convolution(const std::vector& m_, const std::vec if (m_p.size() == 0) break; res = res + signum * multipleLi_do_sum(m_p, s_p) * multipleLi_do_sum(m_q, s_q); - + } while (true); // last term res = res + signum * multipleLi_do_sum(m_q, s_q); - + return res; } @@ -1938,7 +3509,7 @@ static ex zeta1_evalf(const ex& x) return numeric(zeta_do_sum_simple(r)); } } - + // single zeta value if (is_exactly_a(x) && (x != 1)) { try { @@ -1950,28 +3521,28 @@ static ex zeta1_evalf(const ex& x) } -static ex zeta1_eval(const ex& x) +static ex zeta1_eval(const ex& m) { - if (is_exactly_a(x)) { - if (x.nops() == 1) { - return zeta(x.op(0)); + if (is_exactly_a(m)) { + if (m.nops() == 1) { + return zeta(m.op(0)); } - return zeta(x).hold(); + return zeta(m).hold(); } - if (x.info(info_flags::numeric)) { - const numeric& y = ex_to(x); + if (m.info(info_flags::numeric)) { + const numeric& y = ex_to(m); // trap integer arguments: if (y.is_integer()) { if (y.is_zero()) { return _ex_1_2; } if (y.is_equal(_num1)) { - return zeta(x).hold(); + return zeta(m).hold(); } if (y.info(info_flags::posint)) { if (y.info(info_flags::odd)) { - return zeta(x).hold(); + return zeta(m).hold(); } else { return abs(bernoulli(y)) * pow(Pi, y) * pow(_num2, y-_num1) / factorial(y); } @@ -1984,33 +3555,52 @@ static ex zeta1_eval(const ex& x) } } // zeta(float) - if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) - return zeta1_evalf(x); + if (y.info(info_flags::numeric) && !y.info(info_flags::crational)) { + return zeta1_evalf(m); + } } - return zeta(x).hold(); + return zeta(m).hold(); } -static ex zeta1_deriv(const ex& x, unsigned deriv_param) +static ex zeta1_deriv(const ex& m, unsigned deriv_param) { GINAC_ASSERT(deriv_param==0); - if (is_exactly_a(x)) { + if (is_exactly_a(m)) { return _ex0; } else { - return zeta(_ex1, x); + return zetaderiv(_ex1, m); + } +} + + +static void zeta1_print_latex(const ex& m_, const print_context& c) +{ + c.s << "\\zeta("; + if (is_a(m_)) { + const lst& m = ex_to(m_); + lst::const_iterator it = m.begin(); + (*it).print(c); + it++; + for (; it != m.end(); it++) { + c.s << ","; + (*it).print(c); + } + } else { + m_.print(c); } + c.s << ")"; } -unsigned zeta1_SERIAL::serial = - function::register_new(function_options("zeta"). - eval_func(zeta1_eval). - evalf_func(zeta1_evalf). - do_not_evalf_params(). - derivative_func(zeta1_deriv). - latex_name("\\zeta"). - overloaded(2)); +unsigned zeta1_SERIAL::serial = function::register_new(function_options("zeta", 1). + evalf_func(zeta1_evalf). + eval_func(zeta1_eval). + derivative_func(zeta1_deriv). + print_func(zeta1_print_latex). + do_not_evalf_params(). + overloaded(2)); ////////////////////////////////////////////////////////////////////// @@ -2062,56 +3652,92 @@ static ex zeta2_evalf(const ex& x, const ex& s) // use Hoelder convolution return numeric(zeta_do_Hoelder_convolution(xi, si)); } - + return zeta(x, s).hold(); } -static ex zeta2_eval(const ex& x, const ex& s) +static ex zeta2_eval(const ex& m, const ex& s_) { - if (is_exactly_a(s)) { - const lst& l = ex_to(s); - lst::const_iterator it = l.begin(); - while (it != l.end()) { - if ((*it).info(info_flags::negative)) { - return zeta(x, s).hold(); + if (is_exactly_a(s_)) { + const lst& s = ex_to(s_); + for (lst::const_iterator it = s.begin(); it != s.end(); it++) { + if ((*it).info(info_flags::positive)) { + continue; } - it++; - } - return zeta(x); - } else { - if (s.info(info_flags::positive)) { - return zeta(x); + return zeta(m, s_).hold(); } + return zeta(m); + } else if (s_.info(info_flags::positive)) { + return zeta(m); } - return zeta(x, s).hold(); + return zeta(m, s_).hold(); } -static ex zeta2_deriv(const ex& x, const ex& s, unsigned deriv_param) +static ex zeta2_deriv(const ex& m, const ex& s, unsigned deriv_param) { GINAC_ASSERT(deriv_param==0); - if (is_exactly_a(x)) { + if (is_exactly_a(m)) { return _ex0; } else { - if ((is_exactly_a(s) && (s.op(0) > 0)) || (s > 0)) { - return zeta(_ex1, x); + if ((is_exactly_a(s) && s.op(0).info(info_flags::positive)) || s.info(info_flags::positive)) { + return zetaderiv(_ex1, m); } return _ex0; } } -unsigned zeta2_SERIAL::serial = - function::register_new(function_options("zeta"). - eval_func(zeta2_eval). - evalf_func(zeta2_evalf). - do_not_evalf_params(). - derivative_func(zeta2_deriv). - latex_name("\\zeta"). - overloaded(2)); +static void zeta2_print_latex(const ex& m_, const ex& s_, const print_context& c) +{ + lst m; + if (is_a(m_)) { + m = ex_to(m_); + } else { + m = lst(m_); + } + lst s; + if (is_a(s_)) { + s = ex_to(s_); + } else { + s = lst(s_); + } + c.s << "\\zeta("; + lst::const_iterator itm = m.begin(); + lst::const_iterator its = s.begin(); + if (*its < 0) { + c.s << "\\overline{"; + (*itm).print(c); + c.s << "}"; + } else { + (*itm).print(c); + } + its++; + itm++; + for (; itm != m.end(); itm++, its++) { + c.s << ","; + if (*its < 0) { + c.s << "\\overline{"; + (*itm).print(c); + c.s << "}"; + } else { + (*itm).print(c); + } + } + c.s << ")"; +} + + +unsigned zeta2_SERIAL::serial = function::register_new(function_options("zeta", 2). + evalf_func(zeta2_evalf). + eval_func(zeta2_eval). + derivative_func(zeta2_deriv). + print_func(zeta2_print_latex). + do_not_evalf_params(). + overloaded(2)); } // namespace GiNaC